diff --git a/.gitattributes b/.gitattributes index a3f10b028db4faf89c88d58a23488761d245316f..c7ae9807232f4c458592457357fda2f60675dc80 100644 --- a/.gitattributes +++ b/.gitattributes @@ -138,3 +138,31 @@ preprints/A-PD4-group-without-an-aspherical-manifold-model-September-24-2026/pap preprints/A-Pivotal-Amplitude-for-Voronoi-Percolation-from-Cardys-Formula-October-5-2026/voronoi-pivotal-amplitude.pdf filter=lfs diff=lfs merge=lfs -text preprints/A-pointwise-2-converse-for-elliptic-curves-with-rational-two-torsion-September-24-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text preprints/A-Polynomial-Time-2-Approximation-for-Shortest-Common-Superstring-September-24-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text +preprints/A-polynomial-time-algorithm-for-three-machine-unit-job-scheduling-September-24-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text +preprints/A-polynomial-time-construction-of-strong-thin-trees-September-23-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text 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+preprints/A-stable-coordinate-that-is-not-a-coordinate-in-four-variables-October-5-2026/stable-coordinate-four-variables.pdf filter=lfs diff=lfs merge=lfs -text +preprints/A-strict-four-row-permanent-inequality-and-permutation-moments-September-26-2026/main.pdf filter=lfs diff=lfs merge=lfs -text +preprints/A-strict-inverse-first-power-bound-for-univalent-functions-September-24-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text +preprints/A-superquadratic-separation-between-sensitivity-and-block-sensitivity-September-25-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text +preprints/A-surface-counterexample-to-Shafarevich-holomorphic-convexity-September-23-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text diff --git a/preprints/A-Power-Saving-for-Polynomial-Differences-at-Prime-Arguments-October-5-2026/README.md b/preprints/A-Power-Saving-for-Polynomial-Differences-at-Prime-Arguments-October-5-2026/README.md new file mode 100644 index 0000000000000000000000000000000000000000..47b0a20c7965ed8104041ef0b74b4469e6a3c592 --- /dev/null +++ b/preprints/A-Power-Saving-for-Polynomial-Differences-at-Prime-Arguments-October-5-2026/README.md @@ -0,0 +1,18 @@ +# [A Power Saving for Polynomial Differences at Prime Arguments](prime-argument-polynomial-differences.pdf) + +**Author:** OpenAI + +**Date:** October 5, 2026 + +## Citation + +```bibtex +@misc{OAI:A-Power-Saving-for-Polynomial-Differences-at-Prime-Arguments-October-5-2026, + author = {{OpenAI}}, + title = {{A Power Saving for Polynomial Differences at Prime Arguments}}, + howpublished = {OpenAI Math Release preprint + \href{https://github.com/openai/math/blob/main/preprints/A-Power-Saving-for-Polynomial-Differences-at-Prime-Arguments-October-5-2026/prime-argument-polynomial-differences.pdf}{OAI:A-Power-Saving-for-Polynomial-Differences-at-Prime-Arguments-October-5-2026}}, + year = {2026} +} +``` + diff --git a/preprints/A-Power-Saving-for-Polynomial-Differences-at-Prime-Arguments-October-5-2026/build/figures/list-concatenation.tex b/preprints/A-Power-Saving-for-Polynomial-Differences-at-Prime-Arguments-October-5-2026/build/figures/list-concatenation.tex new file mode 100644 index 0000000000000000000000000000000000000000..dac5018ac5ea4786a89510022725228350870d74 --- /dev/null +++ b/preprints/A-Power-Saving-for-Polynomial-Differences-at-Prime-Arguments-October-5-2026/build/figures/list-concatenation.tex @@ -0,0 +1,63 @@ +\begin{figure}[!bp] + \centering + \begin{tikzpicture}[ + x=1cm,y=1cm,>=Stealth, + every node/.style={font=\small}, + oldlist/.style={draw=black!65,rounded corners=1pt,minimum height=8mm, + minimum width=32mm,inner sep=2pt}, + guide/.style={draw=black!65,->,line width=.5pt}, + chosen/.style={fill=blue!7,draw=blue!55!black}, + index/.style={text=blue!55!black} + ] + \node[anchor=south] at (5.4,.9) + {Old plus lists whose addresses forget to $u$}; + \node[anchor=south] at (1.6,.42) {$(u,\sigma_1)$}; + \node[anchor=south] at (5.4,.42) {$(u,\sigma_a)$}; + \node[anchor=south] at (9.2,.42) {$(u,\sigma_{r_*})$}; + \node[oldlist] (first) at (1.6,0) + {$(Y_{(u,\sigma_1),i})_{i\in[\ell_s]}$}; + \node[oldlist,chosen] (middle) at (5.4,0) + {$(Y_{(u,\sigma_a),i})_{i\in[\ell_s]}$}; + \node[oldlist] (last) at (9.2,0) + {$(Y_{(u,\sigma_{r_*}),i})_{i\in[\ell_s]}$}; + \node at (3.5,0) {$\cdots$}; + \node at (7.3,0) {$\cdots$}; + \draw[guide] (first.south) -- (1.6,-1.18); + \draw[guide] (middle.south) -- (5.4,-1.18); + \draw[guide] (last.south) -- (9.2,-1.18); + \node[fill=white,inner sep=2pt] at (5.4,-.8) {concatenate}; + + \fill[blue!7] (3.8,-1.2) rectangle (7,-2); + \draw[black!65] (0,-1.2) rectangle (10.8,-2); + \draw[black!65] (3.2,-1.2) -- (3.2,-2); + \draw[black!65] (3.8,-1.2) -- (3.8,-2); + \draw[black!65] (7,-1.2) -- (7,-2); + \draw[black!65] (7.6,-1.2) -- (7.6,-2); + \node at (1.6,-1.6) {sublist $1$}; + \node at (3.5,-1.6) {$\cdots$}; + \node[index] at (5.4,-1.6) {sublist $a$}; + \node at (7.3,-1.6) {$\cdots$}; + \node at (9.2,-1.6) {sublist $r_*$}; + \draw[decorate,decoration={brace,mirror,amplitude=4pt}] + (0,-2.12) -- (10.8,-2.12); + \node[anchor=north] at (5.4,-2.33) + {$\widetilde Y_u:\quad \ell_{s+1}=r_*\ell_s\ \text{independent uniform entries}$}; + + \node[align=center,text width=11.4cm,anchor=north] at (5.4,-3.02) + {For $v\in V_+$ with $u_S(v)=(u,\sigma_a)$, retain only indices\\[3pt] + $\displaystyle j'_v\in\{(a-1)\ell_s+1,\ldots,a\ell_s\}, + \qquad \mathbb P(\text{retained at every }v\in V_+)=r_*^{-d/2}.$}; + \end{tikzpicture} + \caption{Marking the cut block $b$: the comparison between $S$ and + $S'=S\cup\{b\}$. For each new address $u$, concatenate its $r_*$ old + plus lists; the displayed sublists are schematic. Independence and + probabilities refer to sampling before weights are applied. + Restricting the new index at each plus slot to its former sublist + recovers the old distribution of selected labels, with probability + $r_*^{-d/2}$. Every edge between new addresses + gives an old edge between every pair of their plus sublists. Hence every + factor of $I_{S'}$ occurs in $I_+$, and $0\le W\le1$ gives + $I_{S'}\ge I_+$. These are the two ingredients of the comparison for + $Q_{S'}(J)$.} + \label{fig:list-concatenation} +\end{figure} diff --git a/preprints/A-Power-Saving-for-Polynomial-Differences-at-Prime-Arguments-October-5-2026/build/main.tex b/preprints/A-Power-Saving-for-Polynomial-Differences-at-Prime-Arguments-October-5-2026/build/main.tex new file mode 100644 index 0000000000000000000000000000000000000000..3409837deed0cc5af8fa0b4349bc4c53483e7a37 --- /dev/null +++ b/preprints/A-Power-Saving-for-Polynomial-Differences-at-Prime-Arguments-October-5-2026/build/main.tex @@ -0,0 +1,78 @@ +\documentclass[11pt]{article} +\ifdefined\pdfinfoomitdate\pdfinfoomitdate=1\fi +\ifdefined\pdftrailerid\pdftrailerid{}\fi +\ifdefined\pdfsuppressptexinfo\pdfsuppressptexinfo=15\fi +\usepackage[T1]{fontenc} +\usepackage{lmodern} +\usepackage[margin=1in]{geometry} +\usepackage{amsmath,amssymb,amsthm,mathtools} +\usepackage{microtype} +\usepackage{enumitem,booktabs} +\usepackage{xcolor,tikz} +\usetikzlibrary{arrows.meta,positioning,calc,decorations.pathreplacing,matrix,fit} +\usepackage[numbers,sort&compress]{natbib} +\usepackage[hyphens]{url} +\usepackage{hyperref} +\hypersetup{colorlinks=true,linkcolor=blue!45!black,citecolor=blue!45!black, + urlcolor=blue!45!black,pdfauthor={OpenAI}, + pdftitle={A Power Saving for Polynomial Differences at Prime Arguments}} +\newtheorem{theorem}{Theorem}[section] +\newtheorem{proposition}[theorem]{Proposition} +\newtheorem{lemma}[theorem]{Lemma} +\newtheorem{corollary}[theorem]{Corollary} +\theoremstyle{definition} +\newtheorem{definition}[theorem]{Definition} +\theoremstyle{remark} +\newtheorem{remark}[theorem]{Remark} +\numberwithin{equation}{section} +\newcommand{\R}{\mathbb R} +\newcommand{\C}{\mathbb C} +\newcommand{\N}{\mathbb N} +\newcommand{\Z}{\mathbb Z} +\newcommand{\Q}{\mathbb Q} +\newcommand{\F}{\mathbb F} +\newcommand{\E}{\mathbb E} +\newcommand{\Prb}{\mathbb P} +\newcommand{\1}{\mathbf 1} +\newcommand{\eps}{\varepsilon} +\newcommand{\norm}[1]{\lVert#1\rVert} +\newcommand{\abs}[1]{\lvert#1\rvert} +\newcommand{\articleid}[1]{\begingroup\urlstyle{same}\nolinkurl{#1}\endgroup} +\DeclareMathOperator{\supp}{supp} +\DeclareMathOperator{\rad}{rad} +\DeclareMathOperator{\id}{id} +\DeclareMathOperator{\diag}{diag} +\setlist[enumerate]{leftmargin=*,itemsep=4pt,topsep=5pt} +\setlist[itemize]{leftmargin=*,itemsep=4pt,topsep=5pt} +\setlength{\emergencystretch}{2em} +\allowdisplaybreaks[1] +\title{A Power Saving for Polynomial Differences\protect\\ at Prime Arguments} +\author{OpenAI} +\date{October 5, 2026} +\begin{document} +\maketitle +\begin{abstract} +Let $h$ be a fixed integer polynomial of degree at least two with positive +leading coefficient, having a unit root modulo every positive integer. +We prove that any set $A\subseteq\{1,\ldots,N\}$ whose differences avoid +all nonzero values $h(p)$ at primes satisfies $|A|\le C_hN^{1-c_h}$, +where $c_h>0$ and $C_h\ge1$ depend only on $h$. Thus the local unit-root +condition gives a fixed power saving even when polynomial arguments are +restricted to primes. The proof uses the companion zero-free half-plane +theorem for Dirichlet $L$-functions to obtain the required prime-distribution +estimates. +\end{abstract} +\input{sections/introduction} +\input{sections/local} +\input{sections/tuples} +\input{sections/lifts} +\input{sections/primes} +\input{sections/kernel} +\input{sections/descent} +\begingroup +\small\raggedright +\setlength{\bibsep}{4pt} +\bibliographystyle{plainnat} +\bibliography{references} +\endgroup +\end{document} diff --git a/preprints/A-Power-Saving-for-Polynomial-Differences-at-Prime-Arguments-October-5-2026/build/references.bib b/preprints/A-Power-Saving-for-Polynomial-Differences-at-Prime-Arguments-October-5-2026/build/references.bib new file mode 100644 index 0000000000000000000000000000000000000000..5547a7e54ad77af5c8b28918ba2a63a1221f8706 --- /dev/null +++ b/preprints/A-Power-Saving-for-Polynomial-Differences-at-Prime-Arguments-October-5-2026/build/references.bib @@ -0,0 +1,316 @@ +@article{Furstenberg1977, + author = {Furstenberg, H.}, + title = {Ergodic behavior of diagonal measures and a theorem of {Szemer{\'e}di} on arithmetic progressions}, + journal = {Journal d'Analyse Math{\'e}matique}, + volume = {31}, + pages = {204--256}, + year = {1977}, + doi = {10.1007/BF02813304}, + url = {https://doi.org/10.1007/BF02813304} +} + +@article{Sarkozy1978, + author = {S{\'a}rk{\"o}zy, A.}, + title = {On difference sets of sequences of integers. {I}}, + journal = {Acta Mathematica Academiae Scientiarum Hungaricae}, + volume = {31}, + number = {1--2}, + pages = {125--149}, + year = {1978}, + doi = {10.1007/BF01896079}, + url = {https://doi.org/10.1007/BF01896079} +} + +@article{KamaeMendesFrance1978, + author = {Kamae, T. and Mend{\`e}s France, M.}, + title = {Van der {Corput}'s difference theorem}, + journal = {Israel Journal of Mathematics}, + volume = {31}, + number = {3--4}, + pages = {335--342}, + year = {1978}, + doi = {10.1007/BF02761498}, + url = {https://doi.org/10.1007/BF02761498} +} + +@article{PintzSteigerSzemeredi1988, + author = {Pintz, J. and Steiger, W. 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C.}, + title = {An elementary method in prime number theory}, + journal = {Acta Arithmetica}, + volume = {37}, + pages = {111--115}, + year = {1980}, + doi = {10.4064/aa-37-1-111-115} +} + +@book{Vaughan1997, + author = {Vaughan, R. C.}, + title = {The {Hardy--Littlewood} Method}, + edition = {Second}, + series = {Cambridge Tracts in Mathematics}, + volume = {125}, + publisher = {Cambridge University Press}, + address = {Cambridge}, + year = {1997}, + doi = {10.1017/CBO9780511470929} +} + +@article{Weyl1916, + author = {Weyl, Hermann}, + title = {{\"U}ber die {Gleichverteilung} von {Zahlen} mod. {Eins}}, + journal = {Mathematische Annalen}, + volume = {77}, + number = {3}, + pages = {313--352}, + year = {1916}, + doi = {10.1007/BF01475864} +} + +@misc{IntersectivePower, + author = {{OpenAI}}, + title = {{A power saving for intersective polynomial differences with an exponent depending only on the degree}}, + howpublished = {OpenAI Math Release preprint + \href{https://github.com/openai/math/blob/main/preprints/A-power-saving-for-intersective-polynomial-differences-with-an-exponent-depending-only-on-the-degree-October-5-2026/power-saving-intersective-polynomial-differences.pdf}{OAI:A-power-saving-for-intersective-polynomial-differences-with-an-exponent-depending-only-on-the-degree-October-5-2026}}, + year = {2026} +} + +@misc{SquarePower, + author = {{OpenAI}}, + title = {{A power saving for square-difference-free sets}}, + howpublished = {OpenAI Math Release preprint + \href{https://github.com/openai/math/blob/main/preprints/A-power-saving-for-square-difference-free-sets-September-24-2026/paper.pdf}{OAI:A-power-saving-for-square-difference-free-sets-September-24-2026}}, + year = {2026} +} + +@misc{ZeroFree, + author = {{OpenAI}}, + title = {{The Quasi-Riemann Hypothesis: A Zero-Free Half-Plane $\mathrm{Re}(s)>7/8$}}, + howpublished = {OpenAI Math Release preprint + \href{https://github.com/openai/math/blob/main/preprints/The-Quasi-Riemann-Hypothesis-September-30-2026/paper.pdf}{OAI:The-Quasi-Riemann-Hypothesis-September-30-2026}}, + year = {2026} +} diff --git a/preprints/A-Power-Saving-for-Polynomial-Differences-at-Prime-Arguments-October-5-2026/build/sections/descent.tex b/preprints/A-Power-Saving-for-Polynomial-Differences-at-Prime-Arguments-October-5-2026/build/sections/descent.tex new file mode 100644 index 0000000000000000000000000000000000000000..064f86a1888801f726c55ab93d53fb46303ad7e9 --- /dev/null +++ b/preprints/A-Power-Saving-for-Polynomial-Differences-at-Prime-Arguments-October-5-2026/build/sections/descent.tex @@ -0,0 +1,486 @@ +\section{Descent to shorter intervals}\label{sec:descent} + +We now combine the pair cancellation estimate with the seminorm furnished +by the tuple laws. The quantity to be decreased is a diagonal tuple +integral of a Fourier lift. Restriction to progressions compares this +quantity with the same quantity on shorter intervals; avoidance makes a +fixed proportion of the residue assignments cancel, apart from the small +exceptional measures. This is the recurrence method of +\cite{IntersectivePower}, adapted here to preserve the prime argument +condition throughout the auxiliary family. + +All structural parameters, including $M_0,d,\gamma$ and those in +\eqref{eq:parameters}, remain fixed. At every integer scale $N\ge1$ we use +$\mathcal P_N,Q,H$ from \eqref{eq:kernel-cutoffs}. For $l\ge1$ and +$A\subseteq[N]$, define +\begin{equation}\label{eq:descent-quantity} + \begin{split} + Y_l(N,A)&=Z_{l,\mathcal P_N} + \bigl(\mathcal L_{[N],\mathcal P_N,Q}\mathbf1_A\bigr),\\ + \mathcal Y_l(N)&=\max_{\substack{A\subseteq[N]\\A\text{ avoiding at }l}} + Y_l(N,A). + \end{split} +\end{equation} +Here $Y_l$ is unrelated to the prime-sum range $Y$ used in the preceding +section. Lemma~\ref{lem:seminorm} and the constant Fourier coefficient of +the lift give the first bound in +\begin{equation}\label{eq:descent-trivial} + \left(\frac{|A|}{N}\right)^d\le Y_l(N,A), + \qquad 0\le\mathcal Y_l(N)\ll N^{o(1)}. +\end{equation} +For the upper bound, ordinary H\"older with the uniform single-slot +marginals gives $Z_{l,\mathcal P_N}(g)\le\|g\|_d^d$. Apply +Proposition~\ref{prop:lift-moments}; its length hypothesis holds for large +$N$ since $6\beta<1$. For bounded $N$, bounding each term of the finite +Fourier sum gives the same uniform conclusion. In particular, the upper +bound in \eqref{eq:descent-trivial} holds for every $l\ge1$, with no +restriction of the form $l\le N^\zeta$. + +For each $s\bmod M_0$ put +\[ + f_s(x)=M_0\mathbf1_A(x)\mathbf1_{\{x\equiv s\pmod{M_0}\}}, + \qquad G_s=\mathcal L_{[N],\mathcal P_N,Q}f_s. +\] +Averaging $f_s$ over uniform $s$ gives $\mathbf1_A$. Thus, for independent +uniform residues $s_v\bmod M_0$ indexed by $v\in V$, multilinearity gives +\begin{equation}\label{eq:residue-average} + Y_l(N,A)=\mathbb E_{(s_v)_{v\in V}} + \mathcal Z_{l,\mathcal P_N}((G_{s_v})_{v\in V}). +\end{equation} +We shall bound these integrals in two ways. The comparison with shorter +intervals holds for every $l$; cancellation will require the analytic +range $l\le N^\zeta$. + +\subsection{Specialization and progression coordinates} + +Fixing prime coordinates of a lift restricts its underlying set to a +residue class, provided the remaining Fourier denominator is sufficiently +small. The following exact formula records the normalization. + +\begin{lemma}[Residue specialization]\label{lem:residue-specialization} +Let $N\ge1$, $A\subseteq[N]$, $s\bmod M_0$, and let $I\subseteq[N]$ be a +nonempty consecutive interval. Let $B\subseteq\mathcal P_N$, write +$q_B=\prod_{p\in B}p$, $D=M_0q_B$, and +$\mathcal S=\mathcal P_N\setminus B$. Fix the coordinates indexed by $B$ +of $\mathcal L_{I,\mathcal P_N,Q}f_s$ at $b=(b_p)_{p\in B}$, obtaining +$g_I:X_{\mathcal S}\to\R$. Let $a'\bmod D$ be the unique residue +satisfying $a'\equiv s\pmod{M_0}$ and $a'\equiv b_p\pmod p$ for $p\in B$. +Then every frequency $\xi$ on $X_{\mathcal S}$ with $q(\xi)q_B\le Q$ +satisfies +\begin{equation}\label{eq:residue-specialization} + \widehat{g_I}(\xi)=\frac{D}{|I|} + \sum_{\substack{x\in I\\x\equiv a'\pmod D}} + \mathbf1_A(x)e(-x\xi). +\end{equation} +\end{lemma} + +\begin{proof} +The coefficient on the left is the sum of +$\widehat{f_s}_I(\xi+\omega)e(b\omega)$ over the frequencies $\omega$ on +$X_B$ retained by the lift. The denominator condition retains every +$\omega$, because $q(\xi+\omega)=q(\xi)q(\omega)\le Q$. Orthogonality of +these $q_B$ characters imposes $x\equiv b_p\pmod p$ for every $p\in B$ +and contributes the factor $q_B$. Together with the factor $M_0$ in +$f_s$, this gives \eqref{eq:residue-specialization}. +\end{proof} + +\begin{proposition}[Comparison with shorter intervals]\label{prop:shorter-scale} +For all sufficiently large $N$, every $l\ge1$, and every set $A\subseteq[N]$ +avoiding at $l$, the following holds. Let $B\subseteq\mathcal P_N$ satisfy +$q_B\le N^\tau$, and put +\[ + D=M_0q_B,\qquad L_D=\lambda(D),\qquad + n_B=\left\lceil\frac{N}{L_D}\right\rceil,\qquad + \Lambda_B=\frac{L_Dn_B}{N},\qquad + \mathcal S=\mathcal P_N\setminus B. +\] +For each $v\in V$, choose $s_v\bmod M_0$ and fix the $B$-coordinates of +$G_{s_v}$ arbitrarily, obtaining $g_v:X_{\mathcal S}\to\R$. Then +\begin{equation}\label{eq:shorter-scale} + \bigl|\mathcal Z_{l,\mathcal S}((g_v)_{v\in V})\bigr| + \le\Lambda_B^d\mathcal Y_{lD}(n_B)+O(N^{-\sigma}), +\end{equation} +uniformly in all these choices. The new scale and normalization satisfy +\begin{equation}\label{eq:shorter-scale-size} + N^{1/2}\le M_0^{-k}N^{1-k\tau}\le n_B\sigma$. + +The auxiliary-family identity transports avoidance along progressions +of step $L_D=\lambda(D)$. Since $D\mid L_D$, we split the prescribed +class modulo $D$ into its $L_D/D$ classes modulo $L_D$. +Let $a'\bmod D$ be the class determined by this specialization. For +$j\in[L_D]$ with $j\equiv a'\pmod D$, set +\[ + c_j=j-L_D,\qquad + A'_j=\{x'\in[n_B]:L_Dx'+c_j\in A\},\qquad + \mathcal H_j=\mathcal L_{[n_B],\mathcal S,H}\mathbf1_{A'_j}. +\] +Every integer of $A$ in the class $a'\bmod D$ has a unique representation +$L_Dx'+c_j$ of this form: take $j$ to be its representative in $[L_D]$. +The definition by membership in $A$ excludes any surplus points beyond +$N$ in these progressions. + +Each $A'_j$ avoids at $lD$. More explicitly, a nonzero difference +$h_{lD}(m)$ with $m\ge1$ and $T_{lD}(m)$ prime would give the difference +$\lambda(D)h_{lD}(m)=h_l(Dm-i)$ in $A$, where $0\le i0$. + +Since $q_BH\le Q$, Lemma~\ref{lem:residue-specialization} now gives the +exact identity +\begin{equation}\label{eq:progression-fibers} + g_{\le H}(z)=\frac{\Lambda_B}{L_D/D} + \sum_{\substack{j\in[L_D]\\j\equiv a'\pmod D}} + \mathcal H_j\bigl(L_D^{-1}(z-c_j)\bigr). +\end{equation} +To check it, every prime divisor of $L_D$ divides $D$, so multiplication +by $L_D$ is invertible on $X_{\mathcal S}$ and preserves Fourier +denominators. For $q(\xi)\le H$, the coefficient of the $j$th summand at $\xi$ is +$\widehat{\mathcal H_j}(L_D\xi)e(-c_j\xi)$. Multiplication by the prefactor +$\Lambda_B/(L_D/D)=Dn_B/N$ and summation over $j$ therefore gives +$(D/N)\sum_{x\in A,\ x\equiv a'\ (D)}e(-x\xi)$, which is exactly the +coefficient in \eqref{eq:residue-specialization} for $I=[N]$. +Both sides of \eqref{eq:progression-fibers} have zero coefficients at +denominators greater than $H$. + +There are $L_D/D$ summands. The triangle inequality from +Lemma~\ref{lem:seminorm} and its transport identity +\eqref{eq:lift-transport} imply +\begin{equation}\label{eq:fiber-seminorm} + Z_{l,\mathcal S}(g_{\le H}) + \le\Lambda_B^d\max_j Z_{lD,\mathcal S}(\mathcal H_j). +\end{equation} +Increase the cutoff in $\mathcal H_j$ from $H$ to $Q'$. The same +large-denominator argument, now without specialization, changes its +diagonal integral by $O(H^{-\gamma}N^{o(1)})=O(N^{-\sigma})$. +The moment hypothesis is valid at the new scale: +$n_B\ge10(Q')^6$ for large $N$, by $6\beta<1$ and +\eqref{eq:shorter-scale-size}. + +The extended lift is obtained from +$\mathcal L_{[n_B],\mathcal P_{n_B},Q'}\mathbf1_{A'_j}$ by averaging out +its coordinates in $B\cap\mathcal P_{n_B}$ and adding unused coordinates +in $\mathcal S\setminus\mathcal P_{n_B}$. Averaging contracts the +seminorm, and adding unused coordinates does not change it, by +Lemma~\ref{lem:seminorm}. Consequently its diagonal integral is at most +$Y_{lD}(n_B,A'_j)\le\mathcal Y_{lD}(n_B)$. +Combining this observation with \eqref{eq:specialization-tail} and +\eqref{eq:fiber-seminorm} proves the diagonal bound, hence the stated +multilinear bound. None of these steps restricts the size of $l$. +\end{proof} + +\subsection{Cancellation for admissible residue assignments} + +The preceding proposition controls every assignment in +\eqref{eq:residue-average}. We now obtain a stronger bound for those +assignments whose cycle increments can occur as local polynomial values. +Call $(s_v)_{v\in V}$ \emph{good at $l$} if every +$s_{v_{j+1}}-s_{v_j}$, $0\le j0. +\end{equation} +Indeed combine such a sequence at each prime power dividing $M_0$ by the +Chinese remainder theorem. After prescribing these $L$ increments, the +initial cycle residue and every residue off the cycle are free. + +\begin{proposition}[Cancellation on the nonexceptional laws] +\label{prop:descent-cancellation} +For all sufficiently large $N$, all $1\le l\le N^\zeta$, every +$A\subseteq[N]$ avoiding at $l$, and every good assignment $(s_v)_{v\in V}$, +\begin{equation}\label{eq:descent-cancellation} + \left|\int\prod_{v\in V}G_{s_v}(z_v)\, + d\bigotimes_{p\in\mathcal P_N}(\mu_{l,p}-\varepsilon_{l,p})\right| + \ll N^{-\sigma}. +\end{equation} +The implied constant is uniform in $l$, $A$, and the assignment. +\end{proposition} + +\begin{proof} +The aim is to leave just the two endpoint lifts on an ordered cycle +edge. We first average over short intervals, then truncate and expand +the other inputs. Each expansion term uses prime coordinates with +small product; conditioning on those coordinates leaves the pair to +which Proposition~\ref{prop:pair-cancellation} applies. + +It suffices to use the probability laws $\mu_{l,p}^\circ$, since restoring +the omitted masses multiplies the integral by +$\prod_p(1-\eta_{l,p})\le1$. +Partition $[N]$ into $m=\lfloor N^\tau\rfloor$ consecutive intervals +of lengths as equal as possible. Each $G_s$ is the weighted average of +$\mathcal L_{I,\mathcal P_N,Q}f_s$ over these intervals, with weights +$|I|/N$. Thus the integral is an average over independently chosen +intervals for all slots. Their lengths are comparable to $N^{1-\tau}$, +so the lift moment bound applies because $6\beta+\tau<1$. +Ordinary H\"older and uniform marginals bound every interval assignment +by $N^{o(1)}$ in absolute value. + +The probability that all $L$ cycle slots use the same interval is +$\sum_I(|I|/N)^L=O(N^{-(L-1)\tau})$. These assignments contribute +$O(N^{-\sigma})$. For any remaining assignment, the cyclic sequence of +interval indices is nonconstant, and hence has a strict increase along +some directed cycle edge. Choose such an edge, with first endpoint in +$I_1$ and second endpoint in the later interval $I_2$. + +Keep the two endpoint lifts unchanged and truncate every other lift to +denominators at most $T=N^\tau$. Telescoping with +Lemma~\ref{lem:large-denominator} and the lift moment bound costs +$O(T^{-\gamma}N^{o(1)})=O(N^{-\sigma})$. Expand the truncated $d-2$ +inputs into characters. Their coefficients have absolute value at most +$M_0$, and each input has $O(T^2)$ frequencies, so the sum of absolute +coefficients in this product expansion is $O(T^{2(d-2)})$. + +Fix one term. Let $B$ be the union of the prime supports of its +characters and write +\[ + R=q_B\le T^{d-2},\qquad D=M_0R,\qquad + \mathcal S=\mathcal P_N\setminus B. +\] +Condition on the entire tuple of labels at every prime in $B$, taking +any conditioning with positive probability under the product of the +$\mu_{l,p}^\circ$. The expanded characters become constants of modulus +one. The specialized endpoint lifts give functions $g_1,g_2$ on +$X_{\mathcal S}$, and the laws at primes in $\mathcal S$ remain unchanged. +Translation invariance shows that their pair integral is +\begin{equation}\label{eq:conditioned-pair} + \sum_{\xi\text{ on }X_{\mathcal S}} + \widehat g_1(-\xi)\widehat g_2(\xi)\Gamma(\xi), +\end{equation} +where $\Gamma$ is the multiplier for this directed cycle edge. Indeed +only opposite characters survive common translation, and their product +is the character of the second label minus the first. +By \eqref{eq:edge-multiplier-bound}, Parseval, and the specialized moment +bound, the part with $q(\xi)>H$ is at most +\begin{equation}\label{eq:conditioned-pair-tail} + H^{-\gamma}\|g_1\|_2\|g_2\|_2 + \ll H^{-\gamma}R N^{o(1)}. +\end{equation} + +For the remaining frequencies, $HR\le Q$ and $D\le N^\beta$ for large +$N$, since $(d-2)\tau<\beta/2$. Apply +Lemma~\ref{lem:residue-specialization} to the endpoint lifts, obtaining +classes $a'_1,a'_2\bmod D$. Their directed difference is admissible at +$l$: goodness supplies admissibility modulo $M_0$, and the cycle +property of $\mu_{l,p}^\circ$ supplies it at the conditioned primes +$p\in B$. +Set +\[ + J_i=\mathbf1_A\mathbf1_{I_i}\mathbf1_{\{x\equiv a'_i\pmod D\}}, + \qquad i=1,2, +\] +as functions on $[N]$. Every contributing pair $x,y$ has $x(3d+2)\tau+\sigma$. +These estimates are uniform in the conditioning, in the selected edge, +and in the interval assignment. Averaging, and including the earlier +truncation and coincident-interval errors, proves the proposition. +\end{proof} + +\subsection{The contracting recurrence} + +We have obtained cancellation after removing the exceptional part of +every local law. To return to the full laws in +\eqref{eq:residue-average}, we expand that removal. The total mass of +the resulting exceptional terms is small enough to pay for their +possibly shorter progression scales. +For $B\subseteq\mathcal P_N$ write +\[ + \eta_{l,B}=\prod_{p\in B}\eta_{l,p},\qquad + D_B=M_0q_B,\qquad + n_B=\left\lceil\frac{N}{\lambda(D_B)}\right\rceil. +\] +Thus $D_\varnothing=M_0$. +The mass estimate in Proposition~\ref{prop:tuple-laws} and the fixed +choice of $P$ give, uniformly in $l$, +\begin{equation}\label{eq:exceptional-budget} + \sum_{\varnothing\ne B\subseteq\mathcal P_N}q_B\eta_{l,B} + \le\prod_{p\ge P}(1+p^{-3})-1\le\frac14. +\end{equation} + +\begin{proposition}[Recurrence]\label{prop:descent-recurrence} +There is a constant $C_1$ such that for every sufficiently large $N$ and +$1\le l\le N^\zeta$, +\begin{equation}\label{eq:descent-recurrence} + \begin{split} + \mathcal Y_l(N)\le {}&C_1N^{-\sigma}+(1+u_N)\biggl[ + (1-\rho_l)\mathcal Y_{lM_0}(n_\varnothing)\\ + &\hspace{12mm}+\rho_l + \sum_{\substack{\varnothing\ne B\subseteq\mathcal P_N\\q_B\le N^\tau}} + \eta_{l,B}\mathcal Y_{lD_B}(n_B)\biggr],\\ + u_N={}&(1+M_0^kN^{k\tau-1})^d-1=o(1). + \end{split} +\end{equation} +The constant and the sufficiently-large threshold are independent of $l$. +\end{proposition} + +\begin{proof} +Fix an avoiding set $A$. For assignments in \eqref{eq:residue-average} +that are not good, apply Proposition~\ref{prop:shorter-scale} with +$B=\varnothing$. Their proportion is $1-\rho_l$. +For a good assignment, Proposition~\ref{prop:descent-cancellation} +handles the product of the measures $\mu_{l,p}-\varepsilon_{l,p}$. +The difference from the full product is +\[ + \bigotimes_{p\in\mathcal P_N}\mu_{l,p} + -\bigotimes_{p\in\mathcal P_N}(\mu_{l,p}-\varepsilon_{l,p}) + =\sum_{\varnothing\ne B\subseteq\mathcal P_N}(-1)^{|B|+1} + \left(\bigotimes_{p\in B}\varepsilon_{l,p}\right) + \otimes\left(\bigotimes_{p\notin B}\mu_{l,p}\right). +\] +The positive product measure in the $B$th summand has mass +$\eta_{l,B}$. If this mass is zero the term vanishes. Otherwise, +after normalization, every single-slot marginal is uniform by +simultaneous translation invariance. H\"older and +Proposition~\ref{prop:lift-moments} thus bound its integral by +$N^{o(1)}$ in absolute value. The terms with $q_B>N^\tau$ have total +mass at most $N^{-\tau}/4$ by \eqref{eq:exceptional-budget}, and hence +contribute $O(N^{-\sigma})$. + +For $q_B\le N^\tau$, condition on all labels at the primes in $B$. +The remaining law is the product of the full measures $\mu_{l,p}$, so +Proposition~\ref{prop:shorter-scale} applies to the specialized inputs. +Its factor $\Lambda_B^d$ is at most $1+u_N$ by +\eqref{eq:shorter-scale-size}, and its errors sum to $O(N^{-\sigma})$ +by \eqref{eq:exceptional-budget}. Taking absolute upper bounds in the +signed expansion and averaging over the good assignments gives their +contribution in \eqref{eq:descent-recurrence}. Finally maximize over +$A$ in \eqref{eq:residue-average}. +\end{proof} + +\subsection{Uniform induction and the density bound} + +The recurrence uses prime estimates only when $l\le N^\zeta$, but its +successors $lD_B$ need not lie in the corresponding analytic range at +$n_B$. We therefore prove a bound simultaneously for every auxiliary +parameter $l$. The factor $l^{b_0}$ below covers the complementary +range by the elementary bound \eqref{eq:descent-trivial}. + +\begin{proposition}[Uniform power decay]\label{prop:uniform-descent} +There are constants $a_0,b_0>0$ and $C_0\ge1$, depending only on $h$, such +that for all integers $l,N\ge1$, +\begin{equation}\label{eq:uniform-descent} + \mathcal Y_l(N)\le C_0l^{b_0}N^{-a_0}. +\end{equation} +\end{proposition} + +\begin{proof} +After all parameters in \eqref{eq:parameters} have been fixed, choose +$a_0>0$ sufficiently small and put $b_0=2a_0/\zeta$, so that +\begin{equation}\label{eq:induction-parameters} + a_0<\sigma,\qquad s_0:=ka_0+b_0\le1,\qquad + \theta_0:=M_0^{s_0}(1-3\rho_*/4)<1. +\end{equation} +This is possible because $s_0\to0$ with $a_0$ while $\rho_*>0$ is fixed. +If $l>N^\zeta$, then +\[ + l^{b_0}N^{-a_0}\ge N^{a_0}. +\] +The estimate \eqref{eq:descent-trivial}, used with exponent $a_0$, proves +\eqref{eq:uniform-descent} in this range after a fixed enlargement of +$C_0$. This treats all such $l,N$ before the induction. + +For the remaining pairs, induct on the integer $N$, simultaneously for +all $l\ge1$. At a sufficiently large scale with $l\le N^\zeta$, every +successor in \eqref{eq:descent-recurrence} satisfies $n_BN^\zeta$ and every $N0$ and +$C_h\ge1$ such that, for every positive integer $N$ and every +$A\subseteq[N]$, +\[ + (A-A)\cap\{h(p):p\in\mathbb P\}\subseteq\{0\} + \quad\Longrightarrow\quad + |A|\le C_h N^{1-c_h}. +\] +\end{theorem} + +The convention at zero allows a prime root of $h$: a value $h(p)=0$ +places no restriction on $A$. All constants in the theorem depend only +on the fixed polynomial. Increasing $C_h$ incorporates the finitely +many small values of $N$. + +\subsection{History and main ideas} + +The starting point is the Furstenberg--S\'ark\"ozy theorem: every subset +of the integers with positive upper density contains two elements whose +difference is a nonzero square. Furstenberg's ergodic proof and +S\'ark\"ozy's quantitative Fourier argument established this independently +\cite{Furstenberg1977,Sarkozy1978}. For a nonconstant integer polynomial, +the unavoidable local condition is a root modulo every positive integer. +Kamae and Mend\`es France's difference theorem implies that this +condition also suffices for density decay +\cite{KamaeMendesFrance1978}. Lucier made the general polynomial theorem +quantitative by organizing the iteration around compatible $p$-adic +roots and a family of auxiliary polynomials \cite{Lucier2006}. + +Much subsequent work sharpened the density bounds. For square +differences, the double iteration of Pintz, Steiger, and Szemer\'edi +\cite{PintzSteigerSzemeredi1988} was improved by Bloom and Maynard +\cite{BloomMaynard2022}, and Green and Sawhney obtained +$N\exp(-c\sqrt{\log N})$ using an arithmetic level-$d$ inequality +\cite{GreenSawhney2025}. Rice \cite{Rice2019} and Arala +\cite{Arala2024} extended the earlier bounds to general intersective +polynomials. Adajar and collaborators subsequently obtained +$N\exp(-c_{h,\mu}(\log N)^\mu)$ for every $0<\mu<1/2$ +\cite{AdajarEtAl2026}. These results concern unrestricted integer +arguments. Restricting the argument to primes makes avoidance a weaker +condition and introduces the unit-root requirement in our definition. + +The prime problem already has a substantial history. +S\'ark\"ozy treated the shifts $p-1$ and $p+1$ +\cite{Sarkozy1978III}. After successive improvements by Lucier, +Ruzsa--Sanders, and Wang +\cite{Lucier2008,RuzsaSanders2008,Wang2020}, Green proved a power saving +for $p-1$, noting that his argument also applies to $p+1$ +\cite[Theorem~1.1 and the following remarks]{Green2024}. +Thorner and Zaman made an admissible exponent explicit +\cite{ThornerZaman2024}. +For nonlinear polynomials, Li and Pan proved a quantitative result when +$h(1)=0$ \cite{LiPan2009}. Rice's theorem covers every +prime-intersective polynomial of degree $k\ge2$, with +\[ + |A|\ll_{h,c}N(\log N)^{-c} + \qquad\text{for every }c<\frac{1}{2k-2}, +\] +and gives a stronger bound for quadratics +\cite[Theorems~1 and~2]{Rice2013}. Rice also records Wierdl's earlier +qualitative characterization by the unit-root condition +\cite[Section~1.2]{Rice2013}. The auxiliary family used below, including +its prime-argument restriction, already appears in +\cite[Section~2.1]{Rice2013}. + +Our direct methodological predecessor is the companion +\emph{A power saving for intersective polynomial differences with an exponent depending only on the degree} +\cite{IntersectivePower}. We adapt its reflection-positive tuple laws, +Fourier lifts, signed kernels, and contracting induction to prime +arguments. In particular, the Fourier-lift moment estimate is reproduced +with its proof below. The new requirements concern the compatibility +of the tuple laws with unit residue classes and the realization of +their pair correlations by a signed kernel supported on values at +prime arguments. The example $h(x)=(x-1)^2$ gives a power saving when +only the differences $(p-1)^2$ are forbidden; it therefore implies a +power saving for sets avoiding all nonzero square differences as well. + +Here is the mechanism. For each sufficiently large prime we construct +a probability law on a fixed finite tuple of residue labels. A +designated cycle of tuple positions has increments satisfying the +congruence conditions imposed by prime arguments, apart from a component +of very small mass. Reflection positivity means that the joint distribution across +each prescribed cut of the tuple gives a positive semidefinite matrix. +The cuts are chosen so that successive reflections can make all inputs +agree. Repeated Cauchy--Schwarz inequalities then give a multilinear +H\"older inequality and a seminorm. This is the classical chessboard +method of reflection positivity +\cite[Theorem~4.1]{FILS1978}. For related H\"older inequalities and +folding arguments in graph-norm theory, see +\cite{Hatami2010,ConlonLee2017}. We prove the form needed for our +tuple laws directly. + +We next lift an interval indicator to the product of these residue +spaces by retaining its rational Fourier coefficients with bounded +denominator. Averaging a product of copies of this lift over the tuple +laws gives a nonnegative quantity that dominates a fixed power of the +set's density. To exploit avoidance, a signed weight on prime polynomial +values is compared with a model kernel encoding the correlations of a +cycle edge; their Fourier transforms are uniformly close. Major arcs match the local +unit distributions; minor arcs are controlled by Vaughan's identity +and Weyl differencing \cite{Vaughan1980,Vaughan1997,Weyl1916}. +Avoidance annihilates the prime-supported kernel on ordered pairs from the set. +The resulting cancellation, together with the seminorm estimate for +restriction to progressions, produces a contraction of the lifted +quantity on shorter intervals. Induction turns that contraction into +the claimed power saving. + +The prime-distribution estimate used in the major arcs is derived from +\cite[Theorem~1.1]{ZeroFree}: every Dirichlet $L$-function, uniformly +over all characters and moduli, has no zero in $\Re s>7/8$, with the +principal pole at $s=1$ allowed. This fixed half-plane is a theorem input +from that companion, stronger than the classical zero-free regions +near $\Re s=1$. We prove here its required consequence for primes in +arithmetic progressions and all subsequent steps of the argument. + +\subsection{Notation and organization} + +Throughout the proof, $h$ is fixed, its degree is $k\ge2$, and its +leading coefficient is $a>0$. Put $e(x)=\exp(2\pi i x)$. +For $\xi\in\Q/\Z$, let $q(\xi)$ be its reduced positive denominator, +with $q(0)=1$. Fourier coefficients use the negative sign in the +exponential. Averages and norms on finite sets use uniform probability +measure unless another measure is specified. Empty products have +value one, and an empty product of spaces is a singleton. + +Implicit constants and lower thresholds may depend on $h$ and on the +structural parameters fixed in terms of $h$, but are uniform in the +auxiliary parameter and residue classes introduced below. The notation +$N^{o(1)}$ denotes a quantity bounded by $C_\epsilon N^\epsilon$ for +every $\epsilon>0$, with this same uniformity. + +Section~\ref{sec:local} constructs the auxiliary polynomials and their +admissible local values. Section~\ref{sec:tuples} builds the +reflection-positive tuple laws, and Section~\ref{sec:lifts} develops +their multilinear estimates and proves the Fourier-lift moment bound. +Section~\ref{sec:primes} obtains the prime estimates from the stated +zero-free input and elementary differencing. Section~\ref{sec:kernel} +matches the tuple multipliers with a kernel supported on prime +polynomial values. Section~\ref{sec:descent} uses that comparison to +contract a multilinear density quantity and proves +Theorem~\ref{thm:main}. diff --git a/preprints/A-Power-Saving-for-Polynomial-Differences-at-Prime-Arguments-October-5-2026/build/sections/kernel.tex b/preprints/A-Power-Saving-for-Polynomial-Differences-at-Prime-Arguments-October-5-2026/build/sections/kernel.tex new file mode 100644 index 0000000000000000000000000000000000000000..ca45fdc08f9defaf3ce49c094959b07ca1ccb20e --- /dev/null +++ b/preprints/A-Power-Saving-for-Polynomial-Differences-at-Prime-Arguments-October-5-2026/build/sections/kernel.tex @@ -0,0 +1,459 @@ +\section{A kernel on differences at prime arguments}\label{sec:kernel} + +The tuple laws prescribe a distribution of admissible increments along each +edge of the distinguished cycle. We now realize its low-denominator Fourier +multipliers by a signed kernel supported on polynomial values at prime +arguments. This will turn avoidance of those values into cancellation for a +pair of Fourier lifts. The kernel comparison adapts the integer-argument +method of \cite{IntersectivePower}; the unit conditions in its local weights +and their realization using prime arguments are the additional ingredients +here. + +\subsection{Parameters and the pair estimate} + +Fix the following parameters, all depending only on $h$ and the tuple +construction: +\begin{equation}\label{eq:parameters} +\begin{split} + &\eta=\frac{1}{10000k},\qquad + c_m=c_m(k,\eta/4)\quad\text{from Proposition~\ref{prop:prime-minor-arcs}}, + \qquad \nu=\frac{\min(\eta,c_m/k)}{10},\\ + &\beta=\frac{\nu}{100},\qquad \zeta=\frac{\nu}{100},\qquad + \tau=\frac{\beta\gamma}{1000k(d+1)},\qquad + \sigma=\frac{\gamma\tau}{4}. +\end{split} +\end{equation} +Here $\eta$ controls the major arcs, $\nu$ is a common saving from prime +sums, $\beta$ controls the Fourier cutoffs, and $\zeta$ bounds the +auxiliary parameter $l$. The smaller exponents $\tau$ and $\sigma$ are +reserved for the descent in Section~\ref{sec:descent}. +The parameter $\eta$ is distinct from the exceptional masses $\eta_{l,p}$. +For a scale $N$ put +\begin{equation}\label{eq:kernel-cutoffs} + Q=N^\beta,\qquad H=N^{\beta/2},\qquad + \mathcal P_N=\{p\text{ prime}:P\le p\le Q\}. +\end{equation} + +Fix $l\ge1$, write $F=h_l$, and choose a directed cycle edge +$v_j\to v_{j+1}$. For each $p\ge P$, let $\pi_p$ be the distribution on +$\F_p$ of $z_{v_{j+1}}-z_{v_j}$ under $\mu_{l,p}^\circ$. It is +supported on admissible residues by Proposition~\ref{prop:tuple-laws}. +For $v\in\Z$ define +\[ + \Gamma_p(v/p)=\sum_{y\in\F_p}\pi_p(y)e(vy/p). +\] +For a nonzero character, the conditional mixing estimate +\eqref{eq:tuple-mixing}, applied to the second endpoint and paired with the +conjugate character at the first, gives $|\Gamma_p(v/p)|\le p^{-\gamma}$. +If $\xi$ has squarefree denominator supported on primes at least $P$, write +$\xi=\sum_{p\mid q(\xi)}\xi_p$ for its prime-denominator decomposition and +put $\Gamma(\xi)=\prod_{p\mid q(\xi)}\Gamma_p(\xi_p)$. Thus +\begin{equation}\label{eq:edge-multiplier-bound} + \Gamma(0)=1,\qquad |\Gamma(\xi)|\le q(\xi)^{-\gamma}. +\end{equation} +The multiplier may depend on $l$ and the chosen edge; all estimates below +are uniform in both. + +\begin{proposition}[Cancellation for an ordered pair]\label{prop:pair-cancellation} +Suppose $N$ is sufficiently large, $l\le N^\zeta$, and +$D=M_0\prod_{p\in B}p\le N^\beta$, where $B$ is a finite set of primes at +least $P$. Let $J_1,J_2:[N]\to\C$ have modulus at most one and be +supported on residue classes $a'_1,a'_2\pmod D$. Assume that +$r=a'_2-a'_1\pmod D$ is admissible at $l$ and that +\[ + J_1(x)J_2(y)\ne0\quad\Longrightarrow\quad + x0$, one allowed regular argument $x_y\pmod p$ satisfying +$F(x_y)=y$. Define a function $\psi_p$ modulo $p$ by +\[ + \psi_p(x_y)=n_p\pi_p(y),\qquad + \psi_p(x)=0\quad\text{at all other classes}. +\] +It has unit mean one, and its weighted $F$-value distribution is precisely +$\pi_p$. Put +\[ + B_p(m)=\psi_p(m)-1,\qquad + B_u(m)=\prod_{p\mid u}B_p(m),\qquad B_1=1, +\] +where $u$ is squarefree and $(u,D)=1$. Each $B_p$ has unit mean zero, +$B_u$ has period $u$, and $|B_u|\le u$. In particular these definitions +specify the weights also on nonunit arguments; those arguments will have +zero main term in the prime-progression estimate. + +Expanding a finite product of $\psi_p=1+B_p$ produces a sum of the $B_u$ +over squarefree products of those primes. We retain $u\le H$ to bound +the periods of the weights. In a unit average against a rational phase +$e(\alpha F(m))$, any prime factor of $u$ absent from $q(\alpha)$ makes +that term vanish. Thus, when the part of $q(\alpha)$ supported away from +$D$ is at most $H$, the truncation retains every potentially nonzero term +of the corresponding product expansion. The comparison below will also +bound the contribution from larger denominators. + +By Lemma~\ref{lem:auxiliary}, all coefficients of $F$ are $O_h(a_l)$. +Consequently there is an integer $x_*\ge1$, independent of $l$, such that +for $x\ge x_*$, +\begin{equation}\label{eq:kernel-polynomial-scale} + F(x)\asymp a_lx^k,\qquad F'(x)\asymp a_lx^{k-1},\qquad + |F''(x)|\ll a_lx^{k-2}, +\end{equation} +with $F,F'$ positive. Put $U=(N/a_l)^{1/k}$. Since $a_l$ is a positive +integer and $a_l\le al^k$, +\[ + U\le N^{1/k},\qquad U\gg N^{1/k-\zeta}. +\] +For large $N$ there is therefore a unique $X\ge x_*$ with $F(X)=N$, and +$X\asymp U$. With $\Lambda'(n)=(\log n)\mathbf1_{\{n\text{ prime}\}}$, +define the signed kernel below. The factor $\varphi(l)/l$ compensates +for the prime density in the progression $T_l(m)$, and $F'(m)/N$ turns +the resulting $m$-integral into normalized measure on the $F$-values: +\begin{equation}\label{eq:prime-kernel} +\begin{split} + k_1(n)=\sum_{x_*H$, the model +coefficient is zero. Lemma~\ref{lem:local-sums} gives +$|a_{p,j}|\le p^{-8\delta j}$, while +\eqref{eq:edge-multiplier-bound} and +\eqref{eq:kernel-local-realization} give +$|b_{p,j}|\le2p^{-\gamma j}$ (for $j\ge2$ one has $b_{p,j}=-a_{p,j}$). +Bounding the truncated product expansion by the full sum of absolute +values, and using \eqref{eq:prime-thresholds}, we obtain +\[ + |c(\alpha_0)|\le + \prod_{p^j\Vert q_1}(|a_{p,j}|+|b_{p,j}|) + \le\prod_{p^j\Vert q_1}3p^{-\gamma j} + \le q_1^{-\gamma/2}0\), take a maximizing assignment. Each of its +folded integrals is nonnegative and at most \(M\), while their product +is at least \(M^2\) by~\eqref{eq:fold-cs}. Both therefore equal +\(M\). Hence either fold preserves maximality. Apply the pullbacks +in reverse order, first \(\phi_m\), then \(\phi_{m-1}\), and so on. +The final assignment is +\(a\circ\phi_m\circ\cdots\circ\phi_1\), which is constant. It +follows that \(M\) is a diagonal value. Conversely, constant +assignments are included in the maximum. We have proved that the +maximum absolute mixed integral equals the largest diagonal value in +the list; this is also true when \(M=0\). + +For \(\epsilon>0\), apply this conclusion to the list +\[ + \frac{g_v}{(Z_{l,\mathcal P}(g_v)+\epsilon)^{1/d}}, + \qquad v\in V. +\] +Each diagonal value in this list is at most one. Multilinearity gives +\( |\mathcal Z_{l,\mathcal P}((g_v)_v)| + \le\prod_v(Z_{l,\mathcal P}(g_v)+\epsilon)^{1/d}\). +Let \(\epsilon\downarrow0\). This proves the second assertion, +including cases where some diagonal values vanish. +\end{proof} + +Define +\[ + \|g\|_{l,\mathcal P}=Z_{l,\mathcal P}(g)^{1/d} + \quad\text{for real }g. +\] +For \(B\subseteq\mathcal P\), let \(\mathbb E_B g\) mean uniform averaging +over the coordinates in \(B\), with all other coordinates held fixed. +The resulting function may be viewed on either \(X_{\mathcal P}\) or +\(X_{\mathcal P\setminus B}\). + +\begin{lemma}[Seminorm and changes of coordinates] +\label{lem:seminorm} +The map \(g\mapsto\|g\|_{l,\mathcal P}\) is a seminorm on the real +functions on \(X_{\mathcal P}\). It satisfies +\[ + |\mathbb E g|\le\|g\|_{l,\mathcal P}, + \qquad + \|\mathbb E_Bg\|_{l,\mathcal P\setminus B} + =\|\mathbb E_Bg\|_{l,\mathcal P} + \le\|g\|_{l,\mathcal P}. +\] +Translations of the input are isometries. Moreover, if \(D\) is a +positive integer coprime to every prime of \(\mathcal P\), then for +any \(a'\in X_{\mathcal P}\), +\begin{equation} + \left\|g\bigl(\lambda(D)^{-1}(\,\cdot-a')\bigr) + \right\|_{l,\mathcal P} + =\|g\|_{lD,\mathcal P}. + \label{eq:lift-transport} +\end{equation} +Here the affine change is interpreted separately at each prime. +\end{lemma} + +\begin{proof} +Nonnegativity follows from Proposition~\ref{prop:signed-holder}. +Since \(d\) is even, homogeneity of \(Z\) gives absolute homogeneity +of its \(d\)-th root. Expanding \(Z(g+h)\) by its slots and applying +\eqref{eq:signed-holder} termwise gives +\[ + Z_{l,\mathcal P}(g+h) + \le\sum_{S\subseteq V}\|g\|_{l,\mathcal P}^{|S|} + \|h\|_{l,\mathcal P}^{d-|S|} + =\bigl(\|g\|_{l,\mathcal P}+\|h\|_{l,\mathcal P}\bigr)^d. +\] +Taking roots proves the triangle inequality. +Putting \(g\) in one slot and 1 in all others in~\eqref{eq:signed-holder} +proves mean domination, since the single-slot marginal is uniform and +\(Z(1)=1\). + +The simultaneous translation invariance of the tuple laws proves that +translations are isometries. Let \(G_B\subseteq X_{\mathcal P}\) +be the additive subgroup of vectors supported on \(B\). Then +\[ + (\mathbb E_Bg)(z)=\frac1{|G_B|}\sum_{a\in G_B}g(z+a). +\] +The triangle inequality gives the asserted contraction. Since +\(\mathbb E_Bg\) is independent of \(B\), the probability laws at those +primes integrate out, proving the equality of the two seminorms. +Finally, \(\lambda(D)\) is a unit at each remaining prime, and +the covariance identity \eqref{eq:tuple-covariance} pushes the product law at \(l\) exactly to that at +\(lD\). This proves~\eqref{eq:lift-transport}. +\end{proof} + +\subsection{Decay at large Fourier denominators} + +Conditional mixing gives a different estimate, valid also after +removing the exceptional part of each prime law. Its proof requires +orthogonality between different exact supports; applying the triangle +inequality to all Fourier coefficients would lose this decay. + +\begin{lemma}[Large-denominator estimate] +\label{lem:large-denominator} +Fix \(l\ge1\) and a finite prime set \(\mathcal P\) as above. At +each prime choose either \(\rho_p=\mu_{l,p}\) or +\(\rho_p=\mu_{l,p}^{\circ}\). Let \(g_v:X_{\mathcal P}\to\C\) +be arbitrary functions. If, in one distinguished slot \(v\), every +Fourier coefficient at a denominator at most \(R\) vanishes, where +\(R\ge1\), then +\begin{equation} + \left|\int\prod_{u\in V}g_u(z_u)\, + d\bigotimes_{p\in\mathcal P}\rho_p\right| + \le R^{-\gamma}\|g_v\|_2 + \prod_{u\ne v}\|g_u\|_{2(d-1)}. + \label{eq:large-denominator} +\end{equation} +\end{lemma} + +\begin{proof} +At one prime, let \(\mathfrak m_p\) denote the marginal law of the labels in all +slots other than \(v\), and let +\[ + T_p f=\mathbb E_{\rho_p}\bigl(f(z_v)\mid(z_u)_{u\ne v}\bigr). +\] +This is an operator from uniform \(L^2(\F_p)\) to \(L^2(\mathfrak m_p)\). +It preserves constants. It sends uniform-mean-zero inputs to +\(\mathfrak m_p\)-mean-zero outputs, since \(\mathbb E_{\mathfrak m_p}T_pf=\mathbb E f\). +By the conditional mixing estimate \eqref{eq:tuple-mixing}, its +norm on the mean-zero subspace is at most \(p^{-\gamma}\). + +The product law makes the conditional operator for all primes equal +to \(T=\bigotimes_pT_p\): this identity holds first for products of +one-prime functions, by independence across primes, and then for every +function by linearity. The exact-support \(B\) Fourier space has a +mean-zero factor at primes in \(B\) and a constant factor elsewhere. +On this space the operator norm is at most \(q_B^{-\gamma}\). +Furthermore, the images of two distinct exact-support spaces are +orthogonal in \(L^2(\bigotimes_p\mathfrak m_p)\). At a prime in the symmetric +difference of the supports, one image factor is constant and the other +has mean zero. Their inner product is zero, and tensoring preserves +this orthogonality. + +Decompose \(g_v=\sum_{q_B>R}g_{v,B}\) by exact supports. Both the +domain summands and their images are orthogonal, so +\[ + \|Tg_v\|_2^2 + =\sum_{q_B>R}\|Tg_{v,B}\|_2^2 + \le\sum_{q_B>R}q_B^{-2\gamma}\|g_{v,B}\|_2^2 + \le R^{-2\gamma}\|g_v\|_2^2. +\] +Condition the original integral on the other slots and apply +Cauchy--Schwarz. The remaining factor is bounded by +\[ + \left\|\prod_{u\ne v}g_u(z_u)\right\|_2 + \le\prod_{u\ne v} + \left(\mathbb E|g_u(z_u)|^{2(d-1)}\right)^{1/(2(d-1))} + =\prod_{u\ne v}\|g_u\|_{2(d-1)}, +\] +using ordinary H\"older and the uniform single-slot marginals. +This proves~\eqref{eq:large-denominator}. If \(\mathcal P\) is empty, the frequency +hypothesis forces \(g_v=0\), and the conclusion is immediate. +\end{proof} + +\subsection{Fourier lifts and uniform moments} + +A bounded function on an integer interval gives Fourier coefficients at +rational frequencies. We transfer a finite set of these coefficients +to \(X_{\mathcal P}\). Such a transfer need not preserve pointwise +bounds, so we prove bounds for every fixed moment. The following +proposition and proof reproduce the uniform lift argument +of~\cite[section ``Multilinear estimates and Fourier lifts'']{IntersectivePower}. +For the square-difference setting, see~\cite[Lemmas~5.2--5.3]{SquarePower}. +We must also allow +arbitrary subfamilies of complete exact supports, because fixing some +residue coordinates changes which denominators remain. + +Let \(I\) be a nonempty consecutive interval of integers and let +\(f:I\to\C\). Put +\[ + \widehat f_I(\xi)=\frac1{|I|}\sum_{x\in I}f(x)e(-x\xi). +\] +For \(B\subseteq\mathcal P\), define its exact-support contribution +\[ + f_B(z)=\sum_{q(\xi)=q_B}\widehat f_I(\xi)e(z\xi), +\] +where the frequencies are characters of \(X_{\mathcal P}\). For +\(Q\ge1\), the Fourier lift is +\[ + \mathcal L_{I,\mathcal P,Q}f(z) + =\sum_{\substack{B\subseteq\mathcal P\\q_B\le Q}}f_B(z) + =\sum_{\substack{\xi\text{ on }X_{\mathcal P}\\q(\xi)\le Q}} + \widehat f_I(\xi)e(z\xi). +\] +Each \(f_B\) depends only on the coordinates in \(B\), and has mean +zero in each of those coordinates. If \(f\) is real, all these +functions are real because each summation set is closed under negation. +Retaining a support \(B\) always means retaining all its frequencies. + +\begin{proposition}[Uniform lift moments] +\label{prop:lift-moments} +Let \(\mathcal P\) be any finite set of primes, \(Q\ge1\), and let +\(I\) be a consecutive integer interval with \(|I|\ge10Q^6\). +For every integer \(r\ge1\), every \(\epsilon>0\), every function +\(f:I\to\C\), and every subfamily +\(\mathcal F\subseteq\{B\subseteq\mathcal P:q_B\le Q\}\), +\begin{equation} + \left\|\sum_{B\in\mathcal F}f_B\right\|_{2r} + \le C_{r,\epsilon}\|f\|_\infty Q^\epsilon. + \label{eq:lift-moments} +\end{equation} +The constant is independent of the prime set, the interval and its +location, the function, and the subfamily. +\end{proposition} + +\begin{proof} +The zero function is immediate; otherwise scale so that +\(\|f\|_\infty\le1\). Write \(N_I=|I|\), and let +\[ + u=\max\{|B|:B\subseteq\mathcal P,\ q_B\le Q\}. +\] +The empty support is included, so this maximum is defined even when +\(\mathcal P\) is empty. +We first control conditional sums of squares of the \(f_B\), then their +square-function moments. Symmetrization will recover the moment of +their sum. + +\smallskip +\noindent\emph{Conditional energy.} +For \(J\subseteq\mathcal P\) with \(q_J\le Q\), and any fixed +coordinate vector \(z_J\), we claim that +\begin{equation} + \sum_{\substack{B\in\mathcal F\\B\supseteq J}} + \mathbb E\bigl(|f_B|^2\mid z_J\bigr)\ll4^{|J|}. + \label{eq:conditional-energy} +\end{equation} +The conditional expectation here averages uniformly over all coordinates +outside \(J\). If \(B\supseteq J\), the coefficient of an outside +character \(\eta\) of exact support \(B\setminus J\), after fixing +\(z_J\), is +\[ + c_J(\eta;z_J) + =\frac1{N_I}\sum_{x\in I}f(x)e(-x\eta) + \prod_{p\in J}\bigl(p\mathbf 1_{\{x\equiv z_p\pmod p\}}-1\bigr). +\] +Indeed, the sum over the nonzero frequencies at \(p\) is +\(\sum_{a=1}^{p-1}e(a(z_p-x)/p) +=p\mathbf 1_{\{x\equiv z_p\pmod p\}}-1\). +The outside characters associated with \(B\) have exact support +\(B\setminus J\). These supports distinguish the sets \(B\supseteq J\), +so their character sets are disjoint. Conditional Parseval therefore +identifies the left side of \eqref{eq:conditional-energy} with the squared +Euclidean norm of this coefficient vector on the union of those sets. + +Expand the product in \(c_J\) over subsets \(J'\subseteq J\). +Apart from a sign, the corresponding vector has coordinates +\[ + \frac{q_{J'}}{N_I} + \sum_{\substack{x\in I\\x\equiv z_{J'}\pmod{q_{J'}}}} + f(x)e(-x\eta). +\] +The congruence denotes the class determined by the fixed coordinates in +\(J'\); for the empty subset it imposes no restriction. Write its +progression as \(x=a+q_{J'}n\), with \(n\) in a consecutive interval +of length \(N'\). Then +\[ + N'\ge0.9N_I/q_{J'}, + \qquad N'q_{J'}/N_I\le1.1. +\] +In these coordinates the frequencies are \(q_{J'}\eta\), up to +constant phases. They remain distinct because their denominators are +coprime to \(q_J\). Their denominators are at most \(Q\), so they +number at most \(Q^2\) and have pairwise circle distance at least +\(Q^{-2}\). + +The Gram matrix of these characters in normalized \(L^2\) on the +progression has diagonal 1. By geometric summation, every off-diagonal +entry has modulus at most \(Q^2/(2N')\). Its operator norm is +therefore at most its maximum absolute row sum, which is bounded by +\[ + 1+\frac{Q^4}{2N'} + \le1+\frac{Q^5}{1.8N_I} + \le1+\frac1{18Q}. +\] +To see how this bounds the coefficients, let the synthesis map send +\((b_\eta)_\eta\) to \(\sum_\eta b_\eta e(nq_{J'}\eta)\) +on the progression interval with normalized counting measure. Its +adjoint is the normalized Fourier analysis map, and its Gram matrix +is the matrix just estimated. The two maps therefore have norm squared +equal to that Gram-matrix norm. Since \(\|f\|_\infty\le1\) and +\(N'q_{J'}/N_I\le1.1\), the displayed coefficient vector has +Euclidean norm bounded by an absolute constant. The triangle inequality +over the \(2^{|J|}\) subsets \(J'\), followed by squaring, proves +\eqref{eq:conditional-energy}. + +\smallskip +\noindent\emph{Square-function moments.} +Set +\[ + S(z)=\left(\sum_{B\in\mathcal F}|f_B(z)|^2\right)^{1/2}. +\] +Expand \(\mathbb E S^{2r}\) and fix the supports of its first \(r-1\) +factors. Their union \(W'\) has size at most \((r-1)u\), and their +product depends only on \(z_{W'}\). For the final support \(B\), +the function \(f_B\) ignores every coordinate outside \(B\). Thus +integration outside \(W'\) leaves exactly +\(\mathbb E(|f_B|^2\mid z_J)\), where \(J=B\cap W'\). +Every intersection that occurs satisfies \(q_J\le q_B\le Q\). +For each fixed intersection \(J\), the sum over supports with +\(B\cap W'=J\) is bounded by the larger sum over all \(B\supseteq J\) +in~\eqref{eq:conditional-energy}. There are at most \(2^{(r-1)u}\) possible +intersections and each has at most \(u\) elements. Thus +\[ + \mathbb E S^{2r}\le C\,2^{(r-1)u}4^u\mathbb E S^{2r-2}. +\] +The same argument with \(r=1\) uses \(J=W'=\varnothing\). +Iteration gives +\begin{equation} + \|S\|_{2r}\le C_r^{1+u}. + \label{eq:lift-square-function} +\end{equation} +The overlaps among supports have now been controlled. It remains to +bound their sum by this square function with a loss exponential only in +\(u\). + +\smallskip +\noindent\emph{Recovering the sum.} +We color the prime coordinates and first retain only terms that use one +coordinate of each color. This will let us apply a scalar random-sign +estimate one color at a time and use consistent coordinate copies across +the retained terms. Fix a support size \(j\ge1\) and color every prime coordinate with one +of \(j\) colors. Call a support \emph{transversal} when it has exactly +one prime of each color, and let \(\mathcal T\) be the transversal +supports in \(\mathcal F\) for this coloring. Take independent uniform +pairs \(X_p^0,X_p^1\) at every prime. The operator +\(\operatorname{swap}_p\) interchanges these two copies. Define +\[ + D_B=\prod_{p\in B}(1-\operatorname{swap}_p)f_B(X^0), + \qquad B\in\mathcal T. +\] +Averaging over all second copies leaves \(f_B(X^0)\), since every +other term in this expansion has mean zero in at least one copied +coordinate. Jensen's inequality gives +\[ + \left\|\sum_{B\in\mathcal T}f_B\right\|_{2r} + \le\left\|\sum_{B\in\mathcal T}D_B\right\|_{L^{2r}(X^0,X^1)}. +\] +Independent swaps preserve the joint law of the copies and multiply +\(D_B\) by the product of the corresponding signs. Thus for independent +uniform signs \(\epsilon_p\), indexed by primes, +\[ + \left\|\sum_{B\in\mathcal T}D_B\right\|_{L^{2r}(X^0,X^1)} + =\left\|\sum_{B\in\mathcal T}D_B\prod_{p\in B}\epsilon_p + \right\|_{L^{2r}(X^0,X^1,\epsilon)}. +\] + +For completeness, the scalar sign estimate +\[ + \left\|\sum_i a_i\epsilon_i\right\|_{2r} + \le C_r\left(\sum_i|a_i|^2\right)^{1/2} +\] +follows by expanding the even moment. Surviving terms have even +multiplicity at every index. For nonnegative coefficients their sum is +at most +\[ + \frac{(2r)!}{2^r r!}\left(\sum_i a_i^2\right)^r, +\] +because \((2m)!\ge2^m m!\). For complex coefficients, take absolute +values of the surviving terms in +\((\sum_i a_i\epsilon_i)^r(\sum_i\overline{a_i}\epsilon_i)^r\) +and use the nonnegative case. Iterating this estimate over the color +classes gives, for fixed copies, +\[ + \left\|\sum_{B\in\mathcal T}D_B\prod_{p\in B}\epsilon_p + \right\|_{L^{2r}(\epsilon)} + \le C_r^j\left(\sum_{B\in\mathcal T}|D_B|^2\right)^{1/2}. +\] +At each step Minkowski is used in the form +\(\| (\sum_i|B_i|^2)^{1/2}\|_{2r} +\le(\sum_i\|B_i\|_{2r}^2)^{1/2}\). +The signs are indexed by individual primes; transversality ensures that +each monomial uses exactly one sign from each color class. + +In expanding the swaps in \(D_B\), index its \(2^j\) terms by a choice +of copy for each color. For any fixed such choice, made consistently +across all supports, the selected prime coordinates have the original +product law. The triangle inequality in +\(L^{2r}(\ell^2(\mathcal T))\) therefore gives +\[ + \left\|\left(\sum_{B\in\mathcal T}|D_B|^2\right)^{1/2}\right\|_{2r} + \le2^j\left\|\left(\sum_{B\in\mathcal T}|f_B|^2\right)^{1/2} + \right\|_{2r} + \le2^j\|S\|_{2r}. +\] +Consequently the transversal sum has norm at most +\((2C_r)^j\|S\|_{2r}\). + +Now average over independent uniform colorings. A fixed size-\(j\) support is +transversal with probability \(j!/j^j\ge e^{-j}\), where here \(e\) +is Euler's number; the inequality follows from +\(\log(j!)\ge\int_1^j\log x\,dx\). +Thus the full size-\(j\) sum is exactly the coloring average of its +transversal sum divided by this probability. Minkowski yields +\[ + \left\|\sum_{\substack{B\in\mathcal F\\|B|=j}}f_B\right\|_{2r} + \le(2eC_r)^j\|S\|_{2r}. +\] +Include the constant term of size zero and sum over \(j\le u\). +Together with~\eqref{eq:lift-square-function}, this bounds the norm of +the full sum by \((C'_r)^{1+u}\). +Finally, the product of \(u\) distinct primes is at least \((u+1)!\), +so \((u+1)!\le Q\). Hence \(u=o(\log Q)\) as \(Q\to\infty\), +uniformly in the prime set. For every fixed \(r\) and \(\epsilon>0\), +\((C'_r)^{1+u}\le C_{r,\epsilon}Q^\epsilon\). +Restoring \(\|f\|_\infty\) proves~\eqref{eq:lift-moments} with all the stated +uniformities. +\end{proof} + +\begin{lemma}[Specialization of residue coordinates] +\label{lem:lift-specialization} +Let \(B\subseteq\mathcal P\), fix \(z_B\in X_B\), and let +\(G:X_{\mathcal P}\to\C\). For \(1\le s<\infty\), +\[ + \|G(z_B,\cdot)\|_{L^s(X_{\mathcal P\setminus B})} + \le q_B^{1/s}\|G\|_{L^s(X_{\mathcal P})}. +\] +In particular, a Fourier lift or a subfamily of its complete supports +satisfies~\eqref{eq:lift-moments} after specialization with the additional factor +\(q_B^{1/(2r)}\). The same bound holds after retaining either the +remaining denominators at most a given \(H\ge1\), or those greater +than \(H\). +\end{lemma} + +\begin{proof} +The specified fiber has uniform probability \(1/q_B\). Its contribution +to \(\mathbb E|G|^s\) is +\(q_B^{-1}\mathbb E|G(z_B,\cdot)|^s\). Dropping the other nonnegative +contributions proves the first assertion. For the last assertion, +a character with original exact support \(C\) retains exact support +\(C\setminus B\) after specialization. The two cuts therefore arise, +before specialization, from the complete-support subfamilies +\[ + \{C\in\mathcal F:q_{C\setminus B}\le H\}, + \qquad + \{C\in\mathcal F:q_{C\setminus B}>H\}. +\] +Apply Proposition~\ref{prop:lift-moments} to these subfamilies and then use +the fiber bound. Combining coefficients can cancel frequencies but +cannot create any outside the indicated ranges. +\end{proof} + +In particular, specialize the same set \(B\) of coordinates in every +slot, with possibly different values in different slots. Applying +\eqref{eq:large-denominator} to the product law on the remaining primes +incurs total specialization factor +\[ + q_B^{1/2}\left(q_B^{1/(2(d-1))}\right)^{d-1}=q_B. +\] +Two specialized \(L^2\) norms incur the same factor. All moment orders +needed here are fixed after \(h\) is fixed. Finite products of their +subpower bounds remain subpower; none of these constants depends on +\(l\), the specialized residues, or the retained supports. diff --git a/preprints/A-Power-Saving-for-Polynomial-Differences-at-Prime-Arguments-October-5-2026/build/sections/local.tex b/preprints/A-Power-Saving-for-Polynomial-Differences-at-Prime-Arguments-October-5-2026/build/sections/local.tex new file mode 100644 index 0000000000000000000000000000000000000000..96aead948cd4495744e4e966b528f5c95d5ce225 --- /dev/null +++ b/preprints/A-Power-Saving-for-Polynomial-Differences-at-Prime-Arguments-October-5-2026/build/sections/local.tex @@ -0,0 +1,347 @@ +\section{Auxiliary polynomials and local admissibility}\label{sec:local} + +Throughout this section, $h\in\Z[x]$ is prime-intersective, of +degree $k\ge2$ and leading coefficient $a>0$. +Restricting a set to an arithmetic progression changes the polynomial +describing its forbidden differences. We first choose progression steps +for which the transformed polynomials belong to a common auxiliary family. +We then select local polynomial values whose +arguments satisfy the unit condition required of primes. These values will +supply the increments of the residue-label cycles used later. + +\subsection{A family compatible with progression restriction} + +We use the auxiliary-polynomial construction of Lucier~\cite{Lucier2006} +in its prime-intersective form in Rice~\cite[Section~2.1]{Rice2013}. +For every prime $p$, prime-intersectivity gives a root of $h$ in +$\Z_p^\times$, the units of the $p$-adic integers. Indeed, the unit +roots modulo $p^j$ form a finitely branching tree under reduction, with +vertices at every depth. Successively choosing a vertex that has descendants +at arbitrarily large depths gives a compatible sequence of roots. Fix its +limit $\mathfrak z_p\in\Z_p^\times$, and write +\begin{equation}\label{eq:root-expansion} + h(\mathfrak z_p+X)=\sum_{j=m_p}^k b_{j,p}X^j, + \qquad b_{m_p,p}\ne0,\qquad 1\le m_p\le k. +\end{equation} +The coefficients lie in $\Z_p$. Define a completely multiplicative +function $\lambda:\N\to\N$ by $\lambda(p)=p^{m_p}$. +For $l\ge1$, the Chinese remainder theorem specifies a unique integer +$r_l\in(-l,0]$ satisfying +\[ + r_l\equiv\mathfrak z_p\pmod{p^{v_p(l)}}\qquad(p\mid l). +\] +Here $v_p$ denotes the $p$-adic valuation. In particular $(r_l,l)=1$; +for $l=1$ the convention is $r_1=0$. Put +\begin{equation}\label{eq:auxiliary-definition} + T_l(x)=r_l+lx,\qquad + h_l(x)=\frac{h(T_l(x))}{\lambda(l)},\qquad + a_l=\frac{a l^k}{\lambda(l)}. +\end{equation} + +\begin{lemma}[Auxiliary family]\label{lem:auxiliary} +For every $l\ge1$, the polynomial $h_l$ belongs to $\Z[x]$, has +degree $k$ and positive leading coefficient $a_l$, and all its coefficients +are $O_h(a_l)$. We have $h_1=h$ and $T_1(x)=x$. +For every $D\ge1$, there is an integer $i\in\{0,\ldots,D-1\}$ such that +\begin{equation}\label{eq:auxiliary-change} + \begin{split} + T_l(Dx-i)&=T_{lD}(x),\\ + h_l(Dx-i)&=\lambda(D)h_{lD}(x),\qquad + D\mid\lambda(D),\quad \lambda(D)\le D^k. + \end{split} +\end{equation} +Moreover, $b_{m_p,p}$ is a $p$-adic unit for all sufficiently large $p$, +with a threshold depending only on $h$. +\end{lemma} + +\begin{proof} +To check integrality at $p\mid l$, write $u=v_p(l)$ and make the affine +change of variable +\[ + Y=\frac{r_l-\mathfrak z_p}{p^u}+\frac{l}{p^u}x. +\] +Its coefficients lie in $\Z_p$, and its slope is a unit. Since +$v_p(\lambda(l))=um_p$, \eqref{eq:root-expansion} gives +\begin{equation}\label{eq:auxiliary-padic} + h_l(x)= + \left(\frac{\lambda(l)}{p^{um_p}}\right)^{-1} + \sum_{j=m_p}^k b_{j,p}p^{u(j-m_p)}Y^j. +\end{equation} +Thus $h_l$ is integral at $p$. If $p\nmid l$, its denominator +$\lambda(l)$ is a $p$-adic unit, so integrality is immediate. This proves +$h_l\in\Z[x]$. Its leading coefficient is $a_l$, and expanding +$h(r_l+lx)$ with $|r_l|k$, it follows that $F'\bmod p$ is not identically zero. +Call an argument $x\in\F_p$ \emph{allowed regular} if +\[ + p\nmid T_l(x)\qquad\text{and}\qquad F'(x)\ne0\pmod p. +\] +The unit condition excludes at most one residue: if $p\mid l$, it +excludes none because $(r_l,l)=1$. + +\begin{lemma}[Local exponential sums]\label{lem:local-sums} +There is an integer $P_1>\max(k,2)$, depending only on $h$, such that for +every $l\ge1$ and every prime $p\ge P_1$, at least $p/2$ arguments in +$\F_p$ are allowed regular. For every integer $v$ with $p\nmid v$, +\begin{equation}\label{eq:local-sums} + \begin{split} + \left|\mathbb E_{\substack{x\bmod p^j\\p\nmid T_l(x)}} + e\left(\frac{vF(x)}{p^j}\right)\right| + &\le p^{-8\delta j} \qquad(j\ge1),\\ + \left|\mathbb E_{\substack{x\bmod p\\x\text{ allowed regular}}} + e\left(\frac{vF(x)}p\right)\right| + &\le p^{-4\delta}. + \end{split} +\end{equation} +Each expectation is normalized by the number of arguments in its stated +averaging set. +\end{lemma} + +\begin{proof} +Take the threshold large enough for the preceding nonconstancy statements. +The derivative excludes at most $k-1$ residues, so at most $k$ arguments +fail to be allowed regular. This gives the asserted cardinality after +increasing the threshold. + +We first work modulo $p$. If $k'$ is the degree of $F\bmod p$, then +the complete mean $S=\mathbb E_{x\bmod p}e(vF(x)/p)$ vanishes when +$k'=1$. For $k'\ge2$, repeated differencing gives +\[ + |S|^{2^{k'-1}} + \le\mathbb E_{u_1,\ldots,u_{k'-1}\bmod p} + \left|\mathbb E_{x\bmod p} + e\left(\frac{v\Delta_{u_1}\cdots\Delta_{u_{k'-1}}F(x)}p\right)\right| + \le\frac{k'-1}{p}, +\] +where $\Delta_u F(x)=F(x+u)-F(x)$. The first inequality follows by +squaring, grouping the two arguments by their difference, and iterating +the triangle and Cauchy--Schwarz inequalities. For the second, the final +phase is linear, with nonzero coefficient unless one of the $u_i$ is +zero: its coefficient is $v k'!$ times the leading coefficient of +$F\bmod p$ times $u_1\cdots u_{k'-1}$. A union bound gives the displayed +probability. Removing at most $k$ arguments changes a normalized mean +by $O_k(p^{-1})$. Since $8\delta<2^{-(k-1)}$, this proves both assertions +of \eqref{eq:local-sums} at conductor $p$, for sufficiently large $p$. + +For $j\ge2$, set $n=\lfloor j/2\rfloor$ and write +$x=x_0+p^{j-n}y$, with $x_0$ modulo $p^{j-n}$ and $y$ modulo $p^n$. +Taylor expansion over the integers gives +\[ + F(x_0+p^{j-n}y)\equiv F(x_0)+p^{j-n}yF'(x_0)\pmod{p^j}. +\] +The condition $p\nmid T_l(x)$ depends only on $x_0$. Averaging over $y$ +therefore annihilates every term for which $p^n\nmid F'(x_0)$. +We bound the proportion of the remaining $x_0$ by interpolation. + +Put $b=\lceil n/(k-1)\rceil$. The solutions of +$F'(x_0)\equiv0\pmod{p^n}$ occupy at most $k-1$ residue classes modulo +$p^b$. Otherwise, choose $k$ integer solutions $x_1,\ldots,x_k$ in +distinct such classes. In the Lagrange interpolation formula for the +degree-at-most-$(k-1)$ polynomial $F'$, every denominator +$\prod_{s\ne r}(x_r-x_s)$ has valuation at most +\[ + (k-1)(b-1)E_p$, the uniform distribution on this argument class +modulo $p^j$ pushes forward under $h_l$ to the uniform distribution on +\[ + \{y\bmod p^j:y\equiv h_l(x)\pmod{p^{E_p}}\}. +\] +\end{lemma} + +\begin{proof} +Fix $p2$ such that +\begin{equation}\label{eq:cycle-length} + 4\delta L>2,\qquad p^{E_p}\mid L\quad(pP_1$, apply Lemma~\ref{lem:admissible-cycle} at each $p7/8$. This includes imprimitive characters and the trivial +character of modulus $1$, whose $L$-function is $\zeta(s)$. A pole at +$s=1$ for a principal character is allowed. +\end{theorem} + +\begin{proposition}[Uniform distribution in residue classes] +\label{prop:prime-progressions} +Uniformly for $Y\ge2$, $0\le x\le Y$, integers $b\ge1$, and reduced +residue classes $c$ modulo $b$, +\begin{equation}\label{eq:prime-progression} + \sum_{\substack{n\le x\\n\equiv c\pmod b}}\Lambda'(n) + =\frac{x}{\varphi(b)} + +O\bigl(Y^{31/32}\log^2(2bY)\bigr). +\end{equation} +For a nonreduced class modulo $b>1$, the sum is $O(\log b)$ and has +zero main term. +\end{proposition} + +\begin{proof} +Set $\sigma_0=15/16$. We first prove the logarithmic derivative estimate +\begin{equation}\label{eq:log-derivative} + \frac{L'}{L}(s,\chi)\ll\log\bigl(2b(2+|\Im s|)\bigr) +\end{equation} +on $\Re s=\sigma_0$, and throughout $\sigma_0\le\Re s\le2$ when +$|\Im s|\ge1$. The constants are absolute, including for imprimitive +characters. + +Write $\delta_\chi=1$ for a principal character and $0$ otherwise, and +let +\[ + G_\chi(s)=(s-1)^{\delta_\chi}L(s,\chi). +\] +Fix radii +$2-\sigma_00$ as +\[ + L(s,\chi)=\overline\chi\,\frac{s}{s-1} + +s\int_1^\infty E_\chi(u)u^{-s-1}\,du. +\] +Consequently, on the disk, +$|G_\chi(s)|\le C[2b(2+|t|)]^C$. At its center the absolutely +convergent Euler product gives $|L(2+it,\chi)|\ge\zeta(4)/\zeta(2)$; +hence $|G_\chi(s_t)|$ is bounded below by the same positive constant. +Choose a holomorphic logarithm and subtract its value at $s_t$. +Its real part is bounded above by +$C\log(2b(2+|t|))$. Borel--Carath\'eodory on an intermediate disk, +followed by Cauchy's estimate on the disk of radius $r$, now gives +\[ + \frac{G_\chi'}{G_\chi}(s) + \ll\log(2b(2+|t|))\qquad (|s-s_t|\le r). +\] +Subtracting $\delta_\chi/(s-1)$ proves +\eqref{eq:log-derivative} in the asserted ranges. + +We next pass through a smoothed sum so that the contour integral is +absolutely convergent. Mellin inversion for $(1-u)_+$ gives, for $x\ge1$, +\begin{equation}\label{eq:smoothed-character-primes} + \sum_{n\le x}(x-n)\Lambda(n)\chi(n) + =\frac{1}{2\pi i}\int_{(2)} + -\frac{L'}{L}(s,\chi)\frac{x^{s+1}}{s(s+1)}\,ds. +\end{equation} +Here the interchange with the Dirichlet series is absolute. Move the +line to $\Re s=\sigma_0$. Estimate~\eqref{eq:log-derivative} bounds +the horizontal integrals by a constant times +$x^3\log(2b(2+T))/T^2$, which tends to zero with the height $T$. +The sole crossed pole is at $s=1$ for the principal character; its +residue is $x^2/2$. On the new vertical line, integration of +$\log(2b(2+|t|))/(1+t^2)$ gives +\begin{equation}\label{eq:smoothed-character-result} + \sum_{n\le x}(x-n)\Lambda(n)\chi(n) + =\delta_\chi\frac{x^2}{2} + +O\bigl(x^{1+\sigma_0}\log(2b)\bigr). +\end{equation} + +For $(c,b)=1$, character orthogonality yields +\[ + A_c(x):=\sum_{\substack{n\le x\\n\equiv c\pmod b}} + (x-n)\Lambda(n) + =\frac{x^2}{2\varphi(b)} + +O\bigl(x^{1+\sigma_0}\log(2b)\bigr). +\] +There are $\varphi(b)$ character errors and a factor $1/\varphi(b)$, +so the error is uniform in $b$. Crucially, sharp differencing is now +performed on this nonnegative progression sum, rather than on a complex +character sum. For $00$ that function is $O_{k,\epsilon}(Y^\epsilon)$: +allocating a prime exponent among finitely many factors gives a +polynomial in the exponent, which is bounded by any fixed positive +power of the prime power after allowing a constant for finitely many +small primes. + +The following two differencing bounds isolate the treatment of +coefficients and summation domains. They are forms of classical Weyl +differencing; see \cite{Weyl1916,Vaughan1997} for the method. + +\begin{lemma}[One-variable and mixed differencing] +\label{lem:prime-differencing} +Let $k\ge2$, and let $f\in\R[x]$ have degree at most $k$, with +coefficient $b_k$ of $x^k$. + +If $J$ is an interval of integers of length at most an integer $M\ge1$, +and $m$ is a positive integer, then +\begin{equation}\label{eq:type-one-differencing} + \left|\frac1M\sum_{n\in J}e(f(mn))\right|^{2^{k-1}} + \ll_k\frac1{M^k} + \sum_{|u_1|,\ldots,|u_{k-1}|Z}=g-g_{\le Z}$. Let $\mu$ denote the M\"obius function, let +$1$ denote the constant arithmetic function, and use $*$ for Dirichlet +convolution. Vaughan's identity, in the form used here, is +\begin{equation}\label{eq:vaughan-identity} + \Lambda=\Lambda_{\le Z}+\mu_{\le Z}*\log + -\mu_{\le Z}*\Lambda_{\le Z}*1 + +\mu_{>Z}*1*\Lambda_{>Z}. +\end{equation} +This decomposition is classical; see \cite[Chapter~3]{Vaughan1997}. +It follows directly from $\log=1*\Lambda$ and +$\mu*1=\mathbf 1_{\{1\}}$: split $\mu$ at $Z$ in $\mu*\log$, and then +split $\Lambda$ at $Z$ in the term $\mu_{>Z}*1*\Lambda$. + +\emph{The two sums with a short variable.} +The middle two terms of~\eqref{eq:vaughan-identity} reduce to sums +with an outer variable $m\le Z^2$ and an inner variable $n\le Y/m$. +The outer coefficient is either $\mu(m)$ or +$\sum_{ab=m,\,a,b\le Z}\mu(a)\Lambda(b)$, and therefore has size +$Y^{o(1)}$. The inner weight is respectively $\log n$ or $1$. +Fix $m$, let $M=\lceil Y/m\rceil$, and consider any interval subsum +$S_m=\sum_{n\in J}e(f(mn))$ of the inner sum. + +Apply~\eqref{eq:type-one-differencing}. For every integer $r$, +\[ + \min(M,\|k!b_kr\|^{-1}) + \le k!\min(M,\|\theta r\|^{-1}), +\] +since $\|k!u\|\le k!\|u\|$. Tuples with a zero shift contribute +$O_k(M^{-1})$ after normalization. For the others group by +$r=m^k|u_1\cdots u_{k-1}|\le m^kM^{k-1}$ and apply +Lemma~\ref{lem:rational-spacing}. The divisor bound gives +\begin{equation}\label{eq:type-one-saving} + \left|\frac{S_m}{M}\right|^{2^{k-1}} + \ll Y^{o(1)}\left( + \frac1M+\frac{m^k}{q}+\frac{m^k}{M} + +\frac1{M^{k-1}}+\frac q{M^k}\right). +\end{equation} +Here and below logarithmic factors are included in $Y^{o(1)}$. +For completeness, the four terms after $1/M$ result from expanding +\[ + M^{-k}\left(\frac{m^kM^{k-1}}q+1\right)(M+q\log(2q)). +\] +As $m\le Y^{2e_0}$ and $M\ge Y^{1-2e_0}$, +\[ + \frac{m^k}{q},\ \frac q{M^k}\le Y^{-\rho+2ke_0},\qquad + \frac{m^k}{M}\le Y^{-1+2(k+1)e_0}. +\] +The remaining terms also save a positive power of $Y$. Taking the +$2^{k-1}$-st root in~\eqref{eq:type-one-saving} therefore gives +$S_m\ll M Y^{-c}$ for some $c=c(k,\rho)>0$, uniformly in the subinterval. +Partial summation permits the weight $\log n$ with a further +$O(\log Y)$ factor. Since +\[ + \sum_{m\le Z^2}\lceil Y/m\rceil\ll Y\log(2Y), +\] +both short-variable terms save a fixed power of $Y$ after including +the outer coefficients. + +\emph{The bilinear sum.} +For the final term of~\eqref{eq:vaughan-identity}, group the factors as +$(\mu_{>Z}*1)(m)\Lambda_{>Z}(n)$. Each coefficient is $Y^{o(1)}$ +when $mn\le Y$, and a nonzero coefficient requires both $m>Z$ and +$n>Z$. Partition into dyadic boxes +\[ + A\le m<2A,\qquad B\le n<2B, + \qquad A,B\ge Z/2,\qquad AB\le Y. +\] +There are $O(\log^2(2Y))$ relevant boxes. Divide out the coefficient +bounds to reduce to $|a_m|,|d_n|\le1$. If $AB5$, and set +\[ + w=2t+1,\qquad K=L!,\qquad r_*=K/2. +\] +Let $\Pi$ be the set of permutation words of length $L$: a word is a +bijection from the positions $[L]$ to the symbols $[L]$. The slots of our +tuple are +\[ + V=\Pi^w,\qquad d=|V|=K^w. +\] +The entries of a slot are called its blocks. Fix an $L$-cycle $c$ on the +positions. Composition $\pi c$ therefore permutes the positions of a +word $\pi$. Identifying $c^j$ itself with a word, fix the distinct slots +\[ + v_j=(c^j,\ldots,c^j)\quad(0\le jP_1$, depending only on $h$, with the following +property. For every $l\ge1$ and prime $p\ge P$, there are a probability +measure $\mu_{l,p}$ on $\F_p^V$ and a submeasure +$0\le\varepsilon_{l,p}\le\mu_{l,p}$ such that: +\begin{enumerate} +\item $\mu_{l,p}$ is reflection positive. Both $\mu_{l,p}$ and +$\varepsilon_{l,p}$ are invariant under simultaneous translation of all +labels. In particular, any nonzero one of these measures, or their +difference, has uniform single-slot marginals after normalization. +\item Writing +\[ + \eta_{l,p}=\varepsilon_{l,p}(1),\qquad + \mu_{l,p}^{\circ}=\frac{\mu_{l,p}-\varepsilon_{l,p}}{1-\eta_{l,p}}, +\] +we have $\eta_{l,p}\le p^{-4}$. Under $\mu_{l,p}^{\circ}$ every +increment $z_{v_{j+1}}-z_{v_j}$, $0\le jw$ blocks. Consequently +the graph on individual list entries satisfies Lemma~\ref{lem:graph-estimates}. +For $s=t+1$ define $I_S=1$; these are precisely the components in which +we will dispense with the cycle constraints. + +Independently select indices $j_v\in[\ell_s]$ uniformly at all slots. +To obtain the mean-square lower bound when the cut block is marked, +we put a positive interaction on some pairs of these indices. Let +$\Delta=1/(10L^2)$. For distinct slots $x,y$, set $H^S_{xy}=\Delta^g$ +if they agree except in one marked block and the words in that block +differ by exchanging symbols at positions of distance $g$. In all other +cases set $H^S_{xy}=0$, and also set $H^S_{xx}=0$. Define +\begin{align} + R_S(j)&=\exp\left(\sum_{\{x,y\}\subseteq V} + H^S_{xy}\mathbf1_{\{j_x=j_y\}}\right), + \label{eq:tuple-index-weight}\\ + \nu_S(\Phi)&=\mathbb E_{Y,j} + I_S(Y)R_S(j)\, + \Phi\bigl((Y_{u_S(v),j_v})_{v\in V}\bigr). + \label{eq:tuple-component} +\end{align} +The sum in \eqref{eq:tuple-index-weight} is over unordered two-element +subsets. Each $\nu_S$ is a finite nonnegative measure, with mass bounded +above by a fixed constant: $I_S\le1$ and $1\le R_S\le C$ for a constant +depending only on the fixed slot set. + +Every component is invariant under each reflection $\theta$. To see +this, exchanging two symbols throughout one block preserves all +transposition distances used in $H^S$. For an unmarked block, the same +symbol exchange permutes addresses and preserves +\eqref{eq:tuple-address-edge}, since symbol permutations commute with +the right action of $c$ on positions. For a marked block, addresses are +unchanged. The uniform lists and indices are preserved in either case. +Thus the cut matrices of $\nu_S$ are symmetric. Their positivity is +the remaining issue. + +\subsection{Positive contributions and their comparison} + +Fix a cut in block $b$, and for a real function $J$ of the plus labels +write its quadratic form as +\[ + Q_S(J)=\int J((z_v)_{v\in V_+}) + J((z_{\theta v})_{v\in V_+})\,d\nu_S(z). +\] +We first prove that marking $b$ gives a strictly positive lower bound. +We then compare that lower bound with the possible negative contribution +when $b$ is unmarked. These two estimates explain both the list lengths +$\ell_s$ and the coefficients in the final mixture. + +Suppose first that $b\in S$. The matrix of interactions across the cut, +indexed by $x,y\in V_+$, is +\[ + \mathsf A_{xy}=H^S_{x,\theta y}. +\] +It is symmetric by reflection invariance. If the two cut symbols in +slot $x$ occupy positions of distance $g$, then +$\mathsf A_{xx}=\Delta^g$. Any other transposition taking $x$ across +the cut must move one cut symbol past the other, leaving the latter +fixed. It therefore exchanges positions at distance at least $g+1$. +Only a transposition in block $b$ can do this. There are fewer than +$L^2$ choices, whence +\[ + \sum_{y\ne x}|\mathsf A_{xy}|\le L^2\Delta^{g+1} + =\tfrac1{10}\Delta^g. +\] +By diagonal dominance, +$\mathsf A\succeq\kappa\operatorname{Id}$ with the fixed choice +$\kappa=(9/10)\Delta^{L-1}>0$. + +Here is why this positivity for interactions gives strict positivity for +the whole index kernel. Identify minus indices with plus indices using +$\theta$, and let $j,j'$ be two half-index assignments. Put +\[ + u(j)=(\mathbf1_{\{j_x=i\}})_{x\in V_+,\ i\in[\ell_s]}. +\] +The cross factor of $R_S$ is the matrix with entries +\[ + C(j,j')=\exp\bigl(u(j)^T(\mathsf A\otimes\operatorname{Id})u(j')\bigr). +\] +Split $\mathsf A$ as $\kappa\operatorname{Id}$ plus a positive +semidefinite matrix. The entrywise exponential of a positive +semidefinite Gram matrix is positive semidefinite: expand its power +series and apply the Schur product theorem to each power. The factor +from $\mathsf A-\kappa\operatorname{Id}$ therefore is positive +semidefinite and has diagonal entries at least one. The factor from +$\kappa\operatorname{Id}$ is the tensor product, over $x\in V_+$, of +\[ + \mathbf1\mathbf1^T+(e^\kappa-1)\operatorname{Id}. +\] +It dominates $(e^\kappa-1)^{d/2}\operatorname{Id}$. Taking its Schur +product with the other factor preserves this lower bound, since the +Schur product with the identity is the diagonal of that factor. +The internal interactions in the two halves multiply this matrix on +the left and right by the same diagonal matrix with entries at least one, +so the lower bound is still valid. + +Reflected slots have the same address when $b\in S$. For fixed lists +$Y$, the two factors involving $J$ consequently evaluate the same vector +indexed by half-index assignments. There are +$n_s=\ell_s^{d/2}$ such assignments. The uniform average over both +halves is $n_s^{-2}$ times the associated quadratic form; its diagonal +lower bound is thus $(e^\kappa-1)^{d/2}n_s^{-1}$ times the uniform +average of the squares. Since there are only finitely many possible +$s$, we obtain a fixed $c_*>0$ such that +\begin{equation}\label{eq:tuple-marked-positive} + Q_S(J)\ge c_*\, + \mathbb E_{Y,j_+}I_S(Y) + J((Y_{u_S(v),j_v})_{v\in V_+})^2\qquad(b\in S). +\end{equation} + +Now suppose $b\notin S$. The lists split into plus and minus addresses +according to the order of the cut symbols in block $b$. Write $Y_+$ +for the plus lists, and $I_+(Y_+)$ for the factors of $I_S$ whose edges +have both ends on that side. Set +\[ + \mathcal A_S(J)=\mathbb E_{Y_+,j_+} I_+(Y_+) + J((Y_{u_S(v),j_v})_{v\in V_+})^2. +\] +No index interaction crosses this cut, because changing a marked block +does not change the word in block $b$. Let $R_+$ be the internal index +weight on the plus side. After averaging the indices, $Q_S(J)$ is the +quadratic form of the kernel of cross-list edge weights applied to the +half-function +\[ + B(Y_+)=I_+(Y_+)\, + \mathbb E_{j_+}R_+(j_+)J((Y_{u_S(v),j_v})_{v\in V_+}). +\] +By Cauchy--Schwarz, $R_+\le C$, and $I_+^2\le I_+$, +$\|B\|_2^2\le C\mathcal A_S(J)$. +Lemma~\ref{lem:graph-estimates} replaces the cross kernel by its +nonnegative constant value with operator error $O(p^{-\delta})$. +The constant kernel contributes a nonnegative multiple of +$(\mathbb E B)^2$. Hence, uniformly over $S$ and the cut, +\begin{equation}\label{eq:tuple-unmarked-error} + Q_S(J)\ge-Cp^{-\delta}\mathcal A_S(J)\qquad(b\notin S). +\end{equation} +If $s=t+1$, the cross kernel is exactly $1$, and $Q_S(J)\ge0$. + +For $b\notin S$ and $s\le t$, let $S'=S\cup\{b\}$. We claim that +\begin{equation}\label{eq:tuple-concatenation} + Q_{S'}(J)\ge c_*r_*^{-d/2}\mathcal A_S(J). +\end{equation} +The new address forgets block $b$. Exactly $r_*=K/2$ old plus +addresses become each new address, one for each word in which $\alpha$ +precedes $\beta$. Enumerate those words as $\sigma_1,\ldots,\sigma_{r_*}$, +and write the old addresses above a new address $u$ as $(u,\sigma_a)$. +Concatenate their old lists by setting +\[ + \widetilde Y_{u,(a-1)\ell_s+i}=Y_{(u,\sigma_a),i} + \qquad(a\in[r_*],\ i\in[\ell_s]). +\] +These are independent uniform lists of length +$r_*\ell_s=\ell_{s+1}$, precisely as required by the definition of +$\nu_{S'}$. + +Under this identification of list variables, $I_{S'}\ge I_+$ pointwise. +Indeed, any edge between new addresses is witnessed by at least $t+1$ +blocks other than $b$, so every pair of their constituent old plus +addresses already had that directed edge. Thus every factor of +$I_{S'}$ occurs in $I_+$; the latter may have additional factors in +$[0,1]$. If $|S'|=t+1$, the same inequality follows from $I_{S'}=1$. +In the nonnegative right side of \eqref{eq:tuple-marked-positive} for +$S'$, denote the independently chosen uniform new indices by $j'_v$. +For a plus slot of old address $(u,\sigma_a)$, restrict $j'_v$ to +$\{(a-1)\ell_s+1,\ldots,a\ell_s\}$. Each restriction has probability +$1/r_*$, independently over the $d/2$ slots. Conditional on these +restrictions, the selected labels have exactly their old distribution. +Using $I_{S'}\ge I_+$ gives \eqref{eq:tuple-concatenation}. +Figure~\ref{fig:list-concatenation} illustrates this identification of +the lists and the index restrictions. + +\input{figures/list-concatenation} + +\subsection{The mixture and its cycle property} + +Define a finite measure and its normalization by +\begin{equation}\label{eq:tuple-mixture} + M=\sum_{\substack{S\subseteq[w]\\|S|\le t+1}} + p^{-|S|\delta/2}\nu_S, + \qquad \mu_{l,p}=\frac{M}{M(1)}. +\end{equation} +For a fixed cut in block $b$, pair each $S$ missing $b$, $|S|\le t$, +with $S'=S\cup\{b\}$. By +\eqref{eq:tuple-unmarked-error} and \eqref{eq:tuple-concatenation}, their +combined quadratic form is at least +\[ + p^{-s\delta/2} + \bigl(c_*r_*^{-d/2}p^{-\delta/2}-Cp^{-\delta}\bigr) + \mathcal A_S(J)\ge0 +\] +for all sufficiently large $p$. Every component containing $b$ is +used in exactly one such pair; the remaining components have +$|S|=t+1$, $b\notin S$, and nonnegative quadratic forms. This proves +reflection positivity of $M$ and hence of $\mu_{l,p}$. + +For $S=\varnothing$, every list has length one, all index weights are +one, and the component mass is a graph average. Therefore +\eqref{eq:tuple-graph-mean} and $c_p\ge1/(2k)$ show that +$\nu_\varnothing(1)\ge c_0>0$ for sufficiently large $p$, uniformly in +$l$. The uniform upper bounds on all component masses give +\[ + c_0\le M(1)\le C_0. +\] +Define the exceptional submeasure by +\begin{equation}\label{eq:tuple-exceptional} + \varepsilon_{l,p} + =\frac{p^{-(t+1)\delta/2}}{M(1)} + \sum_{|S|=t+1}\nu_S. +\end{equation} +It has mass $O(p^{-(t+1)\delta/2})\le p^{-4}$ after increasing the +prime threshold, by the choice of $t$. + +For $|S|\le t$ at least $t+1$ blocks remain unmarked, and the addresses +of $v_j,v_{j+1}$ satisfy \eqref{eq:tuple-address-edge} in every one of +them. The product $I_S$ therefore includes the weight on every pair of +their list entries. On its support, in particular, the entries selected +at those two slots have increment in the support of $W$, which consists +of admissible residues. This proves the cycle property for +$\mu_{l,p}^{\circ}$. Translating every list entry by the same element +of $\F_p$ leaves all the weights unchanged. Each component, +and hence both measures in the proposition, is translation invariant. +The assertion about one-slot marginals follows because translation acts +transitively on $\F_p$. + +\subsection{Covariance and conditional mixing} + +It remains to show that the construction follows the auxiliary family +under affine changes and that conditioning on the other slots reveals +little about a mean-zero function at one slot. The first property will +permit passage to progressions. The second will control large +denominators in the Fourier lifts. + +If $p\nmid D$, the auxiliary-family identity \eqref{eq:auxiliary-change} gives +\[ + W_{lD,p}(y)=W_{l,p}(\lambda(D)y). +\] +Indeed the substitution $x\mapsto Dx-i$ is bijective modulo $p$, +preserves $T_l(Dx-i)=T_{lD}(x)$, and preserves nonvanishing of the +derivative because both $D$ and $\lambda(D)$ are units modulo $p$. +Consequently changing every list entry to +$\lambda(D)^{-1}(Y_{u,i}-a')$ transforms each component for $l$ into +the component for $lD$, preserving its mass and all index weights. +The mixtures and their exceptional parts therefore satisfy +\eqref{eq:tuple-covariance}. + +For \eqref{eq:tuple-mixing}, first consider the normalized component +$\nu_\varnothing/\nu_\varnothing(1)$. At a fixed slot $v$ write +$x=z_v$ and $y=(z_u)_{u\ne v}$. Its unnormalized density, relative to +independent uniform labels, factors as +\[ + J_1(x,y)J_2(y), +\] +where $J_1$ is the product of the $m$ edge weights incident with $v$, +and $J_2$ contains the remaining edges. Put +\[ + A(y)=\mathbb E_xJ_1(x,y),\qquad + B(y)=\mathbb E_xf(x)J_1(x,y). +\] +Apply \eqref{eq:tuple-graph-operator} to the graph of incident edges, +whose two sides are $\{v\}$ and the other vertices. Since +$\mathbb E f=0$, it gives +\[ + \|A-c_p^m\|_2\le Cp^{-\delta},\qquad + \|B\|_2\le Cp^{-\delta}\|f\|_2. +\] +The conditional expectation at $y$ is $B(y)/A(y)$ when $A(y)>0$. +The marginal density of $y$ is +$J_2(y)A(y)/\nu_\varnothing(1)$, so the squared conditional norm is +\begin{equation}\label{eq:tuple-leading-conditional} + \frac1{\nu_\varnothing(1)} + \mathbb E_yJ_2(y)\frac{|B(y)|^2}{A(y)}, +\end{equation} +with the ratio defined as zero when $A=0$. +On $A\ge c_p^m/2$, this is bounded using the estimate for $B$. +The complementary set has uniform probability $O(p^{-2\delta})$ +by the estimate for $A$, since $c_p^m$ is bounded below. On that set, +weighted Cauchy--Schwarz and $J_1\le1$ give +\[ + \frac{|B(y)|^2}{A(y)} + \le\mathbb E_x|f(x)|^2J_1(x,y)\le\|f\|_2^2. +\] +Together with $J_2\le1$ and the lower bound on +$\nu_\varnothing(1)$, this proves that +\eqref{eq:tuple-leading-conditional} is +$O(p^{-2\delta})\|f\|_2^2$. + +Both $\mu_{l,p}$ and $\mu_{l,p}^{\circ}$ are mixtures of the normalized +components, and in either mixture the total probability of +$S\ne\varnothing$ is $O(p^{-\delta/2})$. To justify transfer of the +estimate, retain the component label as an additional random variable. +Conditional Jensen shows that conditioning on this label as well as on +all other slots can only increase the squared conditional norm. In each +normalized component the trivial bound is $\|f\|_2^2$, since that +component has a uniform marginal at slot $v$. Averaging the leading +estimate and these trivial bounds therefore yields +\[ + \left\|\mathbb E_{\rho_p} + [f(z_v)\mid(z_u)_{u\ne v}]\right\|_2^2 + \le C p^{-\delta/2}\|f\|_2^2. +\] +As $2\gamma=\delta/5<\delta/2$, increasing the prime threshold makes +this at most $p^{-2\gamma}\|f\|_2^2$. This proves +\eqref{eq:tuple-mixing} and completes the proof of +Proposition~\ref{prop:tuple-laws}. + +Fix henceforth a threshold $P>P_1$ for which the proposition holds, large +enough also that +\begin{equation}\label{eq:prime-thresholds} + p^{\gamma/2}\ge3\quad(p\ge P),\qquad + \prod_{p\ge P}(1+p^{-3})-1\le\tfrac14, +\end{equation} +where the product is over primes. Its tail tends to one because +$\sum_p p^{-3}<\infty$. Define +\begin{equation}\label{eq:small-prime-modulus} + M_0=\prod_{p0$. Hence +$\sigma\log(4/\sigma)\le\tfrac12\log8<2$, and +\[ + \rho^{-2k} + =m^{\sigma\log(4/\sigma)/32}\le m^{1/16}. +\] +If $I(h)\le m^{1/4}$, the hypothesis implies +\[ + \sum_i I_i(h)^{3/2} + \le\sqrt{\frac{I(h)}m}\,I(h) + \le m^{-3/8}I(h). +\] +The preceding estimates bound the desired sum by +$\sigma^2m^{-5/16}I(h)\le m^{-1/4}I(h)$, including when $I(h)=0$. +If $I(h)>m^{1/4}$, Parseval and Booleanity instead give +\[ + \sum_{1\le |S|\le k}\widehat h(S)^2 + \le\E h^2\le10$, +\begin{align} + \E_y\sum_{\substack{A\subseteq F\\1\le |A|\le k}} + \widehat{f_y}(A)^2 + &=\sum_{\substack{S\subseteq E_n\\1\le|S\cap F|\le k}} + \widehat f(S)^2, + \label{eq:restriction-mass}\\ + \E_y I(f_y)&=\sum_{e\in F}I_e(f). + \label{eq:restriction-influence} +\end{align} +\end{lemma} + +\begin{proof} +Expand $f$ in the product Fourier basis and integrate over $x$. +For each $A\subseteq F$ this gives the exact finite identity +\[ + \widehat{f_y}(A) + =\sum_{D\subseteq H}\widehat f(A\cup D)\chi_D(y). +\] +Orthonormality on the exterior coordinates therefore gives +$\E_y\widehat{f_y}(A)^2 +=\sum_{D\subseteq H}\widehat f(A\cup D)^2$. +Summing over $A$ proves \eqref{eq:restriction-mass}. In particular, +there is no constraint on the exterior Fourier degree $|D|$. + +For $e\in F$, the pivotal event for $f_y$ at $e$ is the pivotal event +for $f$ at $e$ with the exterior coordinates fixed to $y$. Averaging +over $y$ and the remaining free coordinates gives +$\E_y I_e(f_y)=I_e(f)$; summation proves +\eqref{eq:restriction-influence}. +\end{proof} + +\begin{proposition}[The block estimate] +\label{prop:block} +Let $3\le m\le n$ be an integer, let $B$ be uniform among the +$m$-element subsets of $[n]$, and set +\[ + F_B=\{e\in E_n:e\cap B\ne\varnothing\}, + \qquad k=\frac{\sigma\log m}{64}. +\] +Then +\begin{equation} + \sum_{S\subseteq E_n} + \Pr_B\{1\le |S\cap F_B|\le k\}\widehat f(S)^2 + \le \frac{2m}{n}\,m^{-1/4}I(f). + \label{eq:block-average} +\end{equation} +\end{proposition} + +\begin{proof} +Fix $B$, and fix an arbitrary assignment $y$ to +$H_B=E_n\setminus F_B$. The free set $F_B$ contains \emph{all} edges +with at least one endpoint in $B$: both edges within $B$ and edges +from $B$ to its complement. Every permutation supported on $B$ +fixes each edge of $H_B$ individually, and hence fixes $y$. +The restricted function $f_y$ is consequently invariant under this +permutation group even if the graph encoded by $y$ has no symmetry. + +The free-edge orbits are precisely the $\binom m2$ edges within $B$ +and, for each \emph{fixed} $w\in[n]\setminus B$, the $m$ edges +$\{\{u,w\}:u\in B\}$. Every orbit has size at least $m$, since +$m\ge3$. The product law and the pivotal events are preserved by +these permutations, so influences agree within each orbit. +It follows that +\[ + I_e(f_y)\le\frac{I(f_y)}m\qquad(e\in F_B). +\] +Lemma~\ref{lem:low-degree} applies separately to every $y$. Its +right-hand side is linear in $I(f_y)$, so averaging and using +Lemma~\ref{lem:restriction} gives +\[ + \sum_{\substack{S\subseteq E_n\\1\le |S\cap F_B|\le k}} + \widehat f(S)^2 + \le m^{-1/4}\sum_{e\in F_B}I_e(f). +\] +Finally average over $B$. Each endpoint of a fixed edge belongs to +$B$ with probability $m/n$, whence +$\Pr_B(e\in F_B)\le2m/n$. This proves +\eqref{eq:block-average}. +\end{proof} + +\begin{lemma}[Capturing low Fourier levels] +\label{lem:block-capture} +Suppose $t=\log n\ge16$, and put +\[ + m=\lfloor\sqrt n\rfloor, + \qquad k=\frac{\sigma\log m}{64}. +\] +Then +\begin{equation} + \sum_{\substack{S\subseteq E_n\\1\le |S|\le k^2/2}} + \widehat f(S)^2 + \le 4m^{-1/4}I(f). + \label{eq:captured-mass} +\end{equation} +\end{lemma} + +\begin{proof} +Fix a nonempty $S\subseteq E_n$ with $s=|S|\le k^2/2$, viewed as +the edge set of a simple graph. Let $V(S)$ be its set of nonisolated +vertices and write $v=|V(S)|$. Since $s\le\binom v20$. +For the remaining Fourier degrees, the Dirichlet identity +\eqref{eq:dirichlet} gives +\begin{equation} + \sum_{|S|>k^2/2}\widehat f(S)^2 + \le\frac2{k^2}\sum_S|S|\widehat f(S)^2 + =\frac{2\sigma^2}{k^2}I(f). + \label{eq:graph-tail} +\end{equation} +This remains valid for $k^2/2<1$, when the left side includes all +nonconstant coefficients. Combining \eqref{eq:parseval}, +\eqref{eq:captured-mass}, and \eqref{eq:graph-tail} yields +\begin{equation} + \Var(f)\le + \left(4m^{-1/4}+\frac{2\sigma^2}{k^2}\right)I(f). + \label{eq:graph-variance-preconstant} +\end{equation} + +For completeness, all constants can be bounded independently of +$p$ and $n$. Since $m\ge\sqrt n/2$ and $t\ge16$, +$\log m\ge t/2-\log2\ge t/3$. Thus +\[ + \frac{2\sigma^2}{k^2} + =\frac{8192}{(\log m)^2}\le\frac{73728}{t^2}. +\] +The cancellation of $\sigma^2$ is exact, even when $p$ is arbitrarily +close to $0$ or $1$. Also, $t^2e^{-t/8}$ is nonincreasing for +$t\ge16$, so +\[ + 4m^{-1/4}t^2 + \le4\,2^{1/4}t^2e^{-t/8} + \le4\,2^{1/4}\,256e^{-2}<256. +\] +Thus the right-hand side of \eqref{eq:graph-variance-preconstant} is at most +$(73728+256)I(f)/t^2$, which is at most +$2^{17}I(f)/t^2$. + +If instead $2\le n0$ and $0<\varepsilon<1/2$, their graph-property +bound is +\[ + p_{1-\varepsilon}(\mathcal P)-p_\varepsilon(\mathcal P) + \le C_{\eta,\varepsilon}(\log n)^{-2+\eta}. +\] +Thus their result approached the conjectured exponent arbitrarily +closely. Further progress through Fourier estimates is described below; +Friedgut gives a broader account of the influence method +in~\cite{Friedgut22}. + +Two companion papers address complementary threshold-location +questions: comparison of integral and fractional covers of increasing +families~\cite[Theorem~1.1]{ExpectationEquivalence2026}, and comparison +of graph-containment thresholds with expected subgraph +counts~\cite[Theorem~1.1]{SecondKahnKalai2026}. +The present theorem controls the width of a transition for arbitrary +increasing graph properties. Its proof does not use either comparison. + +Here is the stronger estimate that drives the proof. For a Boolean function $f:\{0,1\}^{E_n}\to\{0,1\}$, define its \emph{pivotal influence} at edge $e$ and its \emph{total influence} as follows. Here $x_{e\leftarrow b}$ denotes $x$ with coordinate $e$ set to $b\in\{0,1\}$, and the probability is over all coordinates other than $e$. +\[ + I_{e,p}(f)=\Pr_p\bigl(f(x_{e\leftarrow1})\ne f(x_{e\leftarrow0})\bigr), + \qquad I_p(f)=\sum_{e\in E_n}I_{e,p}(f). +\] + +\begin{theorem}[Variance and influence] +\label{thm:variance} +For every integer $n\ge2$, every Boolean function $f$ on $E_n$ invariant under vertex permutations, and every $00$ +\cite[arXiv version~2, Section~7.4]{KKLMS20}. +This is an influence estimate at $p=1/2$; a threshold-width bound +requires control throughout the transition interval. +Theorem~\ref{thm:variance} removes the iterated-logarithm loss and +holds uniformly for every $00$ throughout $(0,1)$. Thus, by continuity, every quantile $p_a$ is the unique interior point satisfying $\mu(p_a)=a$. + +On the interval $[p_\varepsilon,p_{1-\varepsilon}]$, division in \eqref{eq:differential} is legitimate and gives +\[ + \frac{d}{dp}\log\frac{\mu(p)}{1-\mu(p)} + =\frac{\mu'(p)}{\mu(p)(1-\mu(p))} + \ge\frac{(\log n)^2}{C_0}. +\] +Integration yields +\[ + p_{1-\varepsilon}-p_\varepsilon + \le\frac{2C_0}{(\log n)^2} + \log\frac{1-\varepsilon}{\varepsilon}. +\] +Finally, $4\varepsilon(1-\varepsilon)\le1$ implies +\[ + \log\frac{1-\varepsilon}{\varepsilon} + \le 2\log\frac1{2\varepsilon}. +\] +This proves \eqref{eq:threshold} with $4C_0=2^{19}$ for the entire range $0<\varepsilon<1/2$. +\end{proof} + +To recover the additive formulation of \cite[Conjecture~1.2]{FK96}, set +\[ + \delta=\frac{2^{19}}{(\log n)^2}\log\frac1{2\varepsilon}, +\] +and suppose $\mu(p)>\varepsilon$. Then $p>p_\varepsilon$, so +Theorem~\ref{thm:threshold} gives $p+\delta>p_{1-\varepsilon}$. +Hence $\mu(p+\delta)>1-\varepsilon$ whenever $p+\delta\le1$; +for larger increments, the endpoint $1$ already has probability one. +Conversely, when $p_\varepsilon+\delta<1$, apply this strict formulation +at $p>p_\varepsilon$ and let $p\downarrow p_\varepsilon$. +Continuity gives $\mu(p_\varepsilon+\delta)\ge1-\varepsilon$, which +is the quantile bound. When $p_\varepsilon+\delta\ge1$, that bound +is automatic. + +The same log-odds estimate describes the transition around its median +$p_*=p_{1/2}$. The log odds vanish at $p_*$, so integration from the +median gives, with $C_0=2^{17}$, +\[ +\begin{aligned} + \mu(p_*-s) + &\le \frac{1}{1+\exp\!\bigl(s(\log n)^2/C_0\bigr)} + &&(0\le s\le p_*),\\ + 1-\mu(p_*+s) + &\le \frac{1}{1+\exp\!\bigl(s(\log n)^2/C_0\bigr)} + &&(0\le s\le1-p_*). +\end{aligned} +\] +The endpoint cases follow by continuity. diff --git a/preprints/A-Sharp-Threshold-Bound-for-Monotone-Graph-Properties-September-25-2026/paper.pdf b/preprints/A-Sharp-Threshold-Bound-for-Monotone-Graph-Properties-September-25-2026/paper.pdf new file mode 100644 index 0000000000000000000000000000000000000000..486d54d42ccb826805542609e869b0e10a556864 --- /dev/null +++ b/preprints/A-Sharp-Threshold-Bound-for-Monotone-Graph-Properties-September-25-2026/paper.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:0e6a660e755ef2b6a30755fa40c18256a4ad7f54fe1040f79e13458c1ecc290a +size 394307 diff --git a/preprints/A-Smooth-Metric-with-No-Local-Isometric-Immersion-into-Three-Space-September-24-2026/README.md b/preprints/A-Smooth-Metric-with-No-Local-Isometric-Immersion-into-Three-Space-September-24-2026/README.md new file mode 100644 index 0000000000000000000000000000000000000000..cf7f6b7fbbea8046f286c2556c15f014829daeab --- /dev/null +++ b/preprints/A-Smooth-Metric-with-No-Local-Isometric-Immersion-into-Three-Space-September-24-2026/README.md @@ -0,0 +1,17 @@ +# [A Smooth Metric with No Local Isometric Immersion into Three-Space](paper.pdf) + +OpenAI +September 24, 2026 + +## Citation + +```bibtex +@misc{OAI:A-Smooth-Metric-with-No-Local-Isometric-Immersion-into-Three-Space-September-24-2026, + author = {{OpenAI}}, + title = {{A Smooth Metric with No Local Isometric Immersion into Three-Space}}, + howpublished = {OpenAI Math Release preprint + \href{https://github.com/openai/math/blob/main/preprints/A-Smooth-Metric-with-No-Local-Isometric-Immersion-into-Three-Space-September-24-2026/paper.pdf}{OAI:A-Smooth-Metric-with-No-Local-Isometric-Immersion-into-Three-Space-September-24-2026}}, + year = {2026} +} +``` + diff --git a/preprints/A-Smooth-Metric-with-No-Local-Isometric-Immersion-into-Three-Space-September-24-2026/build/figures/accumulation.tex b/preprints/A-Smooth-Metric-with-No-Local-Isometric-Immersion-into-Three-Space-September-24-2026/build/figures/accumulation.tex new file mode 100644 index 0000000000000000000000000000000000000000..fcf9b8c0c7ead67a1fd6adc61e8b86925520a602 --- /dev/null +++ b/preprints/A-Smooth-Metric-with-No-Local-Isometric-Immersion-into-Three-Space-September-24-2026/build/figures/accumulation.tex @@ -0,0 +1,41 @@ +\begin{figure}[H] +\centering +\begin{tikzpicture}[font=\small,>=Latex] + \begin{scope} + \draw[->,gray] (-.2,0)--(4.1,0) node[right,black] {\(X\)}; + \fill (0,0) circle (1.5pt) node[below=4pt] {\(0\)}; + \foreach \position/\radius/\disklabel in {3.2/.20/1,1.6/.10/2,.8/.05/3} { + \draw[fill=orange!18] (\position,0) circle (\radius); + \node[above=10pt] at (\position,0) {\(\mathcal D_{\disklabel}\)}; + } + \fill (.4,0) circle (.025); + \fill (.2,0) circle (.0125); + \draw[->,blue!65!black] (2.6,-.50)--(.6,-.50); + \node at (2,-1.05) {(a) Disks approaching the origin}; + \end{scope} + \begin{scope}[shift={(7.1,.18)}] + \draw[fill=orange!18] (0,0) circle (.67); + \draw[gray,dashed] (0,0) circle (.90); + \draw[gray,dashed] (0,0) circle (1.23); + \foreach \angle/\innerrotation/\outerrotation in { + 10/0/60,50/30/0,90/60/30,130/15/75,170/45/15, + 210/75/45,250/0/30,290/60/0,330/30/60} { + \begin{scope}[shift={(\angle:.923)},rotate=\innerrotation] + \draw[blue!65!black,fill=blue!8] (-.033,-.033) rectangle (.033,.033); + \end{scope} + \begin{scope}[shift={(\angle:1.25)},rotate=\outerrotation] + \draw[blue!65!black,fill=blue!8] (-.07,-.07) rectangle (.07,.07); + \end{scope} + } + \node at (0,0) {\(\mathcal D_j\)}; + \node at (0,-1.55) {(b) Patches approaching a disk}; + \end{scope} +\end{tikzpicture} +\caption{The two accumulation levels, schematically and not to scale. +The disks in (a) are disjoint and avoid the origin. In (b), disjoint +full patches occupy successive annuli outside one disk. Only a few +patches and annuli are shown. Patch orientations are independent of +their positions around the disk; the actual construction approaches +every boundary point in every direction of a fixed finite rotation net.} +\label{fig:accumulation} +\end{figure} diff --git a/preprints/A-Smooth-Metric-with-No-Local-Isometric-Immersion-into-Three-Space-September-24-2026/build/figures/cap.tex b/preprints/A-Smooth-Metric-with-No-Local-Isometric-Immersion-into-Three-Space-September-24-2026/build/figures/cap.tex new file mode 100644 index 0000000000000000000000000000000000000000..d3f827079eb23c353585544267e0331f3e2dc87a --- /dev/null +++ b/preprints/A-Smooth-Metric-with-No-Local-Isometric-Immersion-into-Three-Space-September-24-2026/build/figures/cap.tex @@ -0,0 +1,35 @@ +\begin{figure}[H] +\centering +\begin{tikzpicture}[font=\small,>=Latex,xscale=1.35,yscale=1.2] + \fill[blue!6] (-2,-2) rectangle (2,0); + \fill[orange!22] + (-1.12,0)--(-1.12,-.35) + .. controls (-1.12,-.83) and (-.57,-1.28) .. (0,-1.28) + .. controls (.57,-1.28) and (1.12,-.83) .. (1.12,-.35) + --(1.12,0)--cycle; + \draw[thick] + (-1.12,0)--(-1.12,-.35) + .. controls (-1.12,-.83) and (-.57,-1.28) .. (0,-1.28) + .. controls (.57,-1.28) and (1.12,-.83) .. (1.12,-.35) + --(1.12,0); + \draw[very thick,blue!65!black] (-2,0)--(-2,-2)--(2,-2)--(2,0); + \draw[thick,dashed] (-2,0)--(2,0); + \node[above=7pt] at (0,0) {artificial edge \(s=b\)}; + \node at (-.2,-.48) {\(K<0\)}; + \node at (0,-1.65) {\(K>0\)}; + \node[anchor=east] at (-1.15,-.85) {\(K=0\)}; + \draw[<->] (.66,-.04)--(.66,-.82); + \node[anchor=west] at (.72,-.58) {\(d\)}; + \draw[->,gray] (2.45,-1.8)--(2.45,-.65) node[above,black] {\(s\)}; + \draw[->,gray] (2.2,-1.55)--(3.05,-1.55) node[right,black] {\(t\)}; + \node[text=blue!65!black] at (0,-2.5) {three edges in positive curvature}; +\end{tikzpicture} +\caption{A lower-cap integration region in flow coordinates, schematically +and not to scale. The lateral and bottom edges are chosen in positive +curvature. The artificial edge \(s=b\) may cross the negative region; +no high-order data are prescribed there. The distance \(d=b-s\) +enters the vanishing weight. The actual curvature zero set depends on +the solution through the flow coordinates; only its position relative +to the three elliptic edges and the artificial edge is illustrated.} +\label{fig:cap} +\end{figure} diff --git a/preprints/A-Smooth-Metric-with-No-Local-Isometric-Immersion-into-Three-Space-September-24-2026/build/figures/geometry.tex b/preprints/A-Smooth-Metric-with-No-Local-Isometric-Immersion-into-Three-Space-September-24-2026/build/figures/geometry.tex new file mode 100644 index 0000000000000000000000000000000000000000..32aad834c2dee0689c15cee57f7797e9d027983f --- /dev/null +++ b/preprints/A-Smooth-Metric-with-No-Local-Isometric-Immersion-into-Three-Space-September-24-2026/build/figures/geometry.tex @@ -0,0 +1,37 @@ +\begin{figure}[H] +\centering +\begin{tikzpicture}[font=\small,>=Latex] + \begin{scope}[scale=1.12] + \fill[blue!5] (-1.7,-1.7) rectangle (1.7,1.7); + \fill[orange!18] + (0,1.35) .. controls (.80,1.35) and (1.02,.95) .. (1.02,.60) + -- (1.02,-.60) + .. controls (1.02,-.95) and (.80,-1.35) .. (0,-1.35) + .. controls (-.80,-1.35) and (-1.02,-.95) .. (-1.02,-.60) + -- (-1.02,.60) + .. controls (-1.02,.95) and (-.80,1.35) .. (0,1.35); + \draw[thick] + (0,1.35) .. controls (.80,1.35) and (1.02,.95) .. (1.02,.60) + -- (1.02,-.60) + .. controls (1.02,-.95) and (.80,-1.35) .. (0,-1.35) + .. controls (-.80,-1.35) and (-1.02,-.95) .. (-1.02,-.60) + -- (-1.02,.60) + .. controls (-1.02,.95) and (-.80,1.35) .. (0,1.35); + \draw[gray] (-1.7,-1.7) rectangle (1.7,1.7); + \draw[->,gray] (-1.87,0)--(1.91,0) node[right,black] {\(x\)}; + \draw[->,gray] (0,-1.87)--(0,1.91) node[above,black] {\(y\)}; + \fill[white] (-.25,-.25) rectangle (.25,.25); + \draw[thick] (-.25,-.25) rectangle (.25,.25); + \node at (.02,.01) {\(\mathcal P\)}; + \node at (.0,.76) {\(K<0\)}; + \node at (1.28,1.42) {\(K>0\)}; + \node[fill=white,inner sep=1pt] at (.90,-1.15) {\(K=0\)}; + \end{scope} +\end{tikzpicture} +\caption{The model patch, schematically and not to scale. +The curvature zero set lies outside the perturbation box +\(\mathcal P\). The positive exterior permits estimates on lower and +upper caps whose free artificial edges may cut through the negative +region. The lower-cap integration region is shown in \Cref{fig:cap}.} +\label{fig:geometry} +\end{figure} diff --git a/preprints/A-Smooth-Metric-with-No-Local-Isometric-Immersion-into-Three-Space-September-24-2026/build/figures/pulse.tex b/preprints/A-Smooth-Metric-with-No-Local-Isometric-Immersion-into-Three-Space-September-24-2026/build/figures/pulse.tex new file mode 100644 index 0000000000000000000000000000000000000000..0950b789c65d851fa8258833643ee372e0422e2a --- /dev/null +++ b/preprints/A-Smooth-Metric-with-No-Local-Isometric-Immersion-into-Three-Space-September-24-2026/build/figures/pulse.tex @@ -0,0 +1,30 @@ +\begin{figure}[H] +\centering +\begin{tikzpicture}[font=\small,>=Latex,xscale=1.25,yscale=1.12] + \fill[orange!8] (-1.6,-1.35) rectangle (1.6,1.35); + \draw[gray] (-1.6,-1.35) rectangle (1.6,1.35); + \draw[thick,blue!65!black] (-1.6,-1.35)--(1.6,-1.35); + \draw[thick,blue!65!black] (-1.6,1.35)--(1.6,1.35); + \draw[blue!65!black,dashed] (-1.6,-1.35)--(-.92,-.13); + \draw[blue!65!black,dashed] (1.6,-1.35)--(.92,-.13); + \draw[blue!65!black,dashed] (-1.6,1.35)--(-.92,.13); + \draw[blue!65!black,dashed] (1.6,1.35)--(.92,.13); + \fill[orange!45] (-.38,-.13) rectangle (.38,.13); + \draw[thick] (-.38,-.13) rectangle (.38,.13); + \draw[->,gray] (-1.8,0)--(1.88,0) node[right,black] {\(\xi\)}; + \draw[->,gray] (0,-1.70)--(0,1.75) node[above,black] {\(\theta\)}; + \node[anchor=east] at (-1.67,-1.35) {\(-r\)}; + \node[anchor=east] at (-1.67,1.35) {\(r\)}; + \node[fill=white,inner sep=1pt] at (0,1.05) {upper data}; + \node[fill=white,inner sep=1pt] at (0,-1.05) {lower data}; + \draw[->] (.9,.28)--(.34,.08); + \node[anchor=west] at (.72,.35) {pulse}; +\end{tikzpicture} +\caption{The hyperbolic slab, schematically and not to scale. +Cap regularity gives smooth data on the two cuts \(\theta=\pm r\). +The dashed lines indicate the shrinking spatial intervals used to +propagate those data toward both pulse edges. The pulse is supported +in \(\abs\theta<\delta/\tau\) and oscillates in \(\xi\). +The entire slab lies in negative curvature.} +\label{fig:pulse} +\end{figure} diff --git a/preprints/A-Smooth-Metric-with-No-Local-Isometric-Immersion-into-Three-Space-September-24-2026/build/main.tex b/preprints/A-Smooth-Metric-with-No-Local-Isometric-Immersion-into-Three-Space-September-24-2026/build/main.tex new file mode 100644 index 0000000000000000000000000000000000000000..25eaea9aa10176cddcb1959f31c823d0b76872e9 --- /dev/null +++ b/preprints/A-Smooth-Metric-with-No-Local-Isometric-Immersion-into-Three-Space-September-24-2026/build/main.tex @@ -0,0 +1,228 @@ +\documentclass[11pt]{amsart} +\usepackage[T1]{fontenc} +\usepackage{lmodern} +\input{glyphtounicode-cmex} +\pdfgentounicode=1 +\usepackage[margin=1in]{geometry} +\usepackage{amsmath,amssymb,amsthm,mathtools} +\usepackage{microtype} +\AtBeginDocument{\begingroup\fontencoding{OT1}\footnotesize\selectfont\endgroup} +\usepackage{enumitem} +\usepackage{float} +\usepackage{tikz} +\usetikzlibrary{arrows.meta,patterns,calc} +\usepackage[colorlinks=true,linkcolor=blue!45!black,citecolor=blue!45!black,urlcolor=blue!45!black]{hyperref} +\usepackage[capitalise,noabbrev]{cleveref} +\newtheorem{theorem}{Theorem}[section] +\newtheorem{proposition}[theorem]{Proposition} +\newtheorem{lemma}[theorem]{Lemma} +\newtheorem{corollary}[theorem]{Corollary} +\theoremstyle{definition} +\newtheorem{definition}[theorem]{Definition} +\theoremstyle{remark} +\newtheorem{remark}[theorem]{Remark} +\newtheorem*{maintheorem}{Theorem A} +\numberwithin{equation}{section} +\newcommand{\R}{\mathbb R} +\newcommand{\N}{\mathbb N} +\newcommand{\II}{\mathrm{II}} +\newcommand{\dd}{\,\mathrm d} +\newcommand{\pd}{\partial} +\newcommand{\norm}[1]{\left\lVert#1\right\rVert} +\newcommand{\abs}[1]{\left\lvert#1\right\rvert} +\DeclareMathOperator{\supp}{supp} +\DeclareMathOperator{\diag}{diag} +\title[A smooth metric with no local isometric immersion]{A Smooth Metric with No Local Isometric Immersion into Three-Space} +\author{OpenAI} +\date{September 24, 2026} +\hypersetup{ + pdftitle={A Smooth Metric with No Local Isometric Immersion into Three-Space}, + pdfauthor={OpenAI}, + pdfsubject={Smooth local isometric realization of Riemannian surfaces} +} +\begin{document} +\begin{abstract} +We construct a smooth positive-definite Riemannian metric on +\((-1,1)^2\) that agrees with the Euclidean metric to every order at +the origin, yet no neighborhood of the origin admits a smooth +isometric immersion into Euclidean three-space. This gives a negative +answer to the unrestricted smooth local isometric realization problem +for surfaces. +\end{abstract} +\maketitle + +\section{Introduction} + +The local isometric realization problem asks whether a Riemannian +surface \((\Sigma,g)\) can, near each point, be realized as a smooth +surface in Euclidean three-space. In local coordinates this means +finding a smooth map \(F\) with +\[ +\langle \partial_iF,\partial_jF\rangle=g_{ij}, +\qquad i,j\in\{1,2\}. +\] +Positive definiteness of \(g\) then forces \(\operatorname{rank}dF=2\). +The unrestricted smooth question is part of Yau's discussion of +isometric embedding~\cite[p.~236]{Yau2000} and is formulated explicitly +in Ghomi's survey~\cite[Section~1.5, Problem~1.9]{Ghomi2019}. +We give a negative answer to this local three-dimensional problem. + +\begin{maintheorem} +There is a \(C^\infty\) positive definite Riemannian metric \(g\) on +\((-1,1)^2\) such that no open neighborhood \(U\) of the origin admits a +\(C^\infty\) isometric immersion +\[ +F:(U,g)\longrightarrow\R^3. +\] +In particular, no such neighborhood admits a smooth isometric embedding. +\end{maintheorem} + +A smooth isometric immersion is an embedding after restriction to a +sufficiently small neighborhood of any point. Thus the two local +formulations have the same existence content. No completeness or +topological assumption is involved in Theorem~A. +The metric can moreover be chosen to have the same full Taylor jet +at the origin as the Euclidean metric; see \Cref{cor:flat-jet}. +Thus smooth local realizability is not determined by that jet. + +\subsection*{Curvature and regularity} +The analytic local problem is settled by the classical Janet--Cartan +theorem~\cite{Janet1926,Cartan1927}. For smooth metrics, nonzero +Gaussian curvature leads to elliptic or hyperbolic equations and +smooth local existence; zeros of curvature introduce degeneracy +and changes of type. Lin's foundational work treated nonnegative +curvature and a clean change of sign with finite differentiability +estimates~\cite{Lin1985,Lin1986}. Here a clean change of sign means +\(K(p)=0\) and \(dK(p)\ne0\). Smooth local existence under +this condition was established by Nakamura and Maeda~\cite{NakamuraMaeda1989}; +see also the regularity comparison in \cite[p.~320]{NY2008}. +More complicated zero sets have been treated under finite-order +and geometric restrictions, including transverse curves and +cusps~\cite[Theorem~1.1]{HanKhuri2010}\cite[Theorem~1.1]{Lin2015}. + +The regularity distinction is essential. Existence at every finite +differentiability order, on neighborhoods allowed to shrink with +that order, does not by itself give one smooth map on a fixed +neighborhood. At the other end of the scale, Nash--Kuiper +flexibility gives \(C^1\) local isometric realizations in +three-space~\cite{Nash1954,Kuiper1955}. That regularity does not +impose the second-order compatibility used for smooth immersions. +Pogorelov constructed a \(C^{2,1}\) metric with no local \(C^2\) +realization~\cite[paragraph~7]{Pogorelov1971}. +The published refinement by Nadirashvili and Yuan achieves +\(K\ge0\) with the same input and output +regularities~\cite[Theorem~1.1]{NY2008}. +In Theorem~A, positive definiteness refers to the metric, not its +curvature: the construction uses both signs of curvature. +It does not settle the smooth local-realizability problem under +either sign restriction \(K\ge0\) or \(K\le0\). + +An earlier preprint of Nadirashvili and Yuan~\cite[Theorem~1.1]{NY2002} +states a stronger +counterexample: a smooth metric with \(K\le0\) and no \(C^3\) +isometric embedding on any neighborhood of the distinguished +point. Their later published paper, however, explicitly leaves +the arbitrary smooth-metric question open~\cite[p.~320]{NY2008}, +as does the subsequent discussion in +\cite[p.~650]{HanKhuri2010}. The earlier preprint and these +later accounts therefore give different statuses for the unrestricted +smooth question. We leave this discrepancy unresolved. +We do not use the preprint's nonsolvability assertion; +the construction below supplies its own analytic obstruction. + +\subsection*{Ideas and proof structure} +The proof has two tasks: obstruct a scalar height on an entire fixed +square, and construct a metric for which any local immersion would +produce such a height on one of those squares. The scalar equation +is the Darboux equation. It follows from the Gauss equation because +the component of an immersion in a fixed ambient direction has +covariant Hessian equal to a scalar multiple of the second fundamental +form. We use only this necessary condition for an immersion. +The patch argument excludes heights with gradient norm below one, +a nonzero diagonal entry of the covariant Hessian, and a mixed entry +small relative to that diagonal entry. + +Two earlier approaches supply important parts of this strategy. +The geometric construction uses the boundary-saddle argument of +Nadirashvili and Yuan~\cite[Section~2, Step~3]{NY2002}: +a negatively curved disk cannot be a graph whose second fundamental +form vanishes along its entire boundary. On the analytic side, +Khuri's counterexamples for general Monge--Amp\`ere equations use +high-frequency integral testing~\cite{Khuri2007Counterexamples}. +His work on the Darboux equation develops solution-dependent flow +coordinates and a prescribed-curvature construction, while leaving +open the construction of the required nonsolvable curvature +profile~\cite[Theorem~1 and the following discussion]{Khuri2009}. +The distinction matters here: the Darboux coefficients come from +one metric, so a nonsolvable general Monge--Amp\`ere equation does +not by itself give an isometric-immersion obstruction. We perturb +the metric itself and estimate the resulting Darboux forcing. + +For the model curvature used here, the main analytic ingredient is +a uniform regularity estimate for Darboux heights on a \emph{cap}: +a region with three edges in positive +curvature and a fourth, artificial edge that may cut through negative +curvature. Starting from fixed low-order bounds, the estimate gives +all-orders bounds on compact subsets whenever the metric has such +bounds there. No high-order data are imposed on the artificial edge. +Differentiating the equation produces a transverse drift proportional +to the curvature derivative. A directed multiplier uses this drift +together with a small tangential term, needed where the transverse +curvature derivative vanishes. A weighted elliptic estimate controls +the remaining positive-curvature region. This is \Cref{prop:cap}. + +Two caps supply uniformly smooth data on opposite sides of a small +negative-curvature slab. Hyperbolic propagation carries these data +to the two edges of a thin, oscillating metric perturbation. +A finite Taylor comparison for the unperturbed metric isolates the +change in the height's transverse derivative caused by the perturbation. +Choose the strip width to be a small parameter times the inverse +oscillation frequency. The forced oscillatory moment is linear in +that parameter, while the leading competing spatial terms are quadratic +and the smooth endpoint data contribute a negligible high-frequency +moment. Fixing that parameter sufficiently small and then increasing +the frequency excludes density, in any nonempty open set, of metrics +admitting a height in a fixed bounded class. +Baire category then removes those bounds and gives \Cref{prop:patch}, +the obstruction on a whole square. + +The geometric step explains why that squarewise obstruction suffices. +For each orientation in a fixed finite net, place shrinking obstructed +patches approaching every boundary point of a disk with negative +curvature inside and zero curvature on its boundary. +The boundary-saddle +argument finds a point where any hypothetical graph immersion has a +nonzero rank-one second fundamental form. A height in the fixed ambient +normal direction at that point has zero differential there. +In a frame aligned with the +Hessian's kernel, its mixed entry vanishes and its transverse diagonal +entry is nonzero. A nearby orientation from the net preserves the +required strict inequalities on a sufficiently small full patch. +Finally, let such disks and their surrounding +patches accumulate at the origin, with amplitudes small enough to +preserve smoothness and the Euclidean Taylor jet there. + +\Cref{sec:model} formulates the scalar height equation and the +whole-patch obstruction. \Cref{sec:caps} proves the cap estimate, +and \Cref{sec:pulse} combines it with the pulse and category +argument. \Cref{sec:assembly} constructs one smooth metric and +completes the geometric contradiction. + +\subsection*{Conventions} +All metrics and functions are real and smooth unless another regularity +is specified. Coordinate derivatives are ordinary partial derivatives; +\(\nabla_g^2z\) is the covariant Hessian. For a family of functions, +a uniform bound on all derivatives means a separate finite bound for +each derivative order, independent of the member of the family. +Constants may depend on a fixed model patch and on the derivative +order. No estimate below is asserted uniformly as the curvature +amplitude of a patch tends to zero. + +\input{sections/model} +\input{sections/caps} +\input{sections/pulse} +\input{sections/assembly} + +\bibliographystyle{plain} +\bibliography{references} +\end{document} diff --git a/preprints/A-Smooth-Metric-with-No-Local-Isometric-Immersion-into-Three-Space-September-24-2026/build/references.bib b/preprints/A-Smooth-Metric-with-No-Local-Isometric-Immersion-into-Three-Space-September-24-2026/build/references.bib new file mode 100644 index 0000000000000000000000000000000000000000..484755d0f358646859ee79fa304adaf9d9dd58cf --- /dev/null +++ b/preprints/A-Smooth-Metric-with-No-Local-Isometric-Immersion-into-Three-Space-September-24-2026/build/references.bib @@ -0,0 +1,195 @@ +@article{Lin1985, + author = {Lin, Chang-Shou}, + title = {The local isometric embedding in {$\mathbb{R}^3$} of 2-dimensional {Riemannian} manifolds with nonnegative curvature}, + journal = {Journal of Differential Geometry}, + volume = {21}, + number = {2}, + year = {1985}, + pages = {213--230}, + doi = {10.4310/jdg/1214439563} +} + +@article{Lin1986, + author = {Lin, Chang-Shou}, + title = {The local isometric embedding in {$\mathbb{R}^3$} of two-dimensional {Riemannian} manifolds with {Gaussian} curvature changing sign cleanly}, + journal = {Communications on Pure and Applied Mathematics}, + volume = {39}, + number = {6}, + year = {1986}, + pages = {867--887}, + doi = {10.1002/cpa.3160390607} +} + +@article{NakamuraMaeda1989, + author = {Nakamura, Gen and Maeda, Yoshiaki}, + title = {Local smooth isometric embeddings of low-dimensional {Riemannian} manifolds into {Euclidean} spaces}, + journal = {Transactions of the American Mathematical Society}, + volume = {313}, + number = {1}, + year = {1989}, + pages = {1--51}, + doi = {10.1090/S0002-9947-1989-0992597-8} +} + +@article{Nash1954, + author = {Nash, John}, + title = {{$C^1$} isometric imbeddings}, + journal = {Annals of Mathematics}, + series = {2}, + volume = {60}, + number = {3}, + year = {1954}, + pages = {383--396}, + doi = {10.2307/1969840} +} + +@article{Kuiper1955, + author = {Kuiper, Nicolaas H.}, + title = {On {$C^1$}-isometric imbeddings. {I}}, + journal = {Nederl. Akad. Wetensch. Proc. Ser. A}, + volume = {58}, + year = {1955}, + pages = {545--556} +} + +@article{Khuri2007Counterexamples, + author = {Khuri, Marcus A.}, + title = {Counterexamples to the Local Solvability of {Monge--Amp\`ere} Equations in the Plane}, + journal = {Communications in Partial Differential Equations}, + volume = {32}, + number = {4}, + year = {2007}, + pages = {665--674}, + doi = {10.1080/03605300600635061} +} + +@book{Yau, + author = {Schoen, Richard and Yau, Shing-Tung}, + title = {Lectures on Differential Geometry}, + series = {Conference Proceedings and Lecture Notes in Geometry and Topology}, + volume = {1}, + publisher = {International Press}, + address = {Cambridge, MA}, + year = {1994}, + isbn = {1-57146-012-8}, + note = {Chapter IX, Problem 22, p.~375}, + url = {https://www.intlpress.com/site/pub/files/preview/bookpubs/00000308.pdf} +} + +@article{Pogorelov1971, + author = {Pogorelov, A. V.}, + title = {An example of a two-dimensional {Riemannian} metric admitting no local realization in {$E^3$}}, + journal = {Doklady Akademii Nauk SSSR}, + volume = {198}, + number = {1}, + year = {1971}, + pages = {42--43}, + note = {In Russian; English translation in Soviet Mathematics Doklady 12 (1971), 729--730}, + url = {https://www.mathnet.ru/eng/dan36135} +} + +@article{NY2008, + author = {Nadirashvili, Nikolai and Yuan, Yu}, + title = {Improving {Pogorelov}'s isometric embedding counterexample}, + journal = {Calculus of Variations and Partial Differential Equations}, + volume = {32}, + year = {2008}, + pages = {319--323}, + doi = {10.1007/s00526-007-0140-7}, + url = {https://sites.math.washington.edu/~yuan/papers/2008CVPDE.pdf} +} + +@misc{NY2002, + author = {Nadirashvili, Nikolai and Yuan, Yu}, + title = {Counterexamples for Local Isometric Embedding}, + year = {2002}, + eprint = {math/0208127}, + archivePrefix = {arXiv}, + primaryClass = {math.DG}, + note = {Preprint, arXiv:math/0208127v1, 16 August 2002}, + url = {https://arxiv.org/abs/math/0208127v1} +} + +@article{Khuri2009, + author = {Khuri, Marcus A.}, + title = {On the local solvability of {Darboux}'s equation}, + journal = {Discrete and Continuous Dynamical Systems}, + year = {2009}, + note = {Supplement}, + pages = {451--456}, + doi = {10.3934/proc.2009.2009.451}, + url = {https://www.math.stonybrook.edu/~khuri/Khuri_AIMS_Darboux.pdf} +} + +@article{HanKhuri2010, + author = {Han, Qing and Khuri, Marcus}, + title = {On the local isometric embedding in {$\mathbb{R}^3$} + of surfaces with {Gaussian} curvature of mixed sign}, + journal = {Communications in Analysis and Geometry}, + volume = {18}, + number = {4}, + year = {2010}, + pages = {649--704}, + doi = {10.4310/CAG.2010.v18.n4.a2}, + eprint = {1009.6214}, + archivePrefix = {arXiv}, + url = {https://arxiv.org/abs/1009.6214} +} + +@article{Lin2015, + author = {Lin, Tsung-Yin}, + title = {On the local isometric embedding in {$\mathbb{R}^3$} + of surfaces with zero sets of {Gaussian} curvature + forming cusp domains}, + journal = {Communications in Partial Differential Equations}, + volume = {41}, + number = {10}, + year = {2016}, + pages = {1553--1592}, + doi = {10.1080/03605302.2016.1227336}, + eprint = {1510.07505}, + archivePrefix = {arXiv}, + url = {https://arxiv.org/abs/1510.07505} +} + +@misc{Ghomi2019, + author = {Ghomi, Mohammad}, + title = {Open problems in geometry of curves and surfaces}, + year = {2019}, + note = {Author-hosted survey, revised 2 September 2019; Section 1.5}, + url = {https://ghomi.math.gatech.edu/Papers/op.pdf} +} + +@article{Janet1926, + author = {Janet, Maurice}, + title = {Sur la possibilit{\'e} de plonger un espace riemannien donn{\'e} + dans un espace euclidien}, + journal = {Annales de la Soci{\'e}t{\'e} Polonaise de Math{\'e}matique}, + volume = {5}, + year = {1926}, + pages = {38--43}, + url = {https://rcin.org.pl/dlibra/publication/31746/edition/15034} +} + +@article{Cartan1927, + author = {Cartan, {\'E}lie}, + title = {Sur la possibilit{\'e} de plonger un espace riemannien donn{\'e} + dans un espace euclidien}, + journal = {Annales de la Soci{\'e}t{\'e} Polonaise de Math{\'e}matique}, + volume = {6}, + year = {1927}, + pages = {1--7}, + url = {https://rcin.org.pl/dlibra/publication/32955/edition/18106} +} + +@article{Yau2000, + author = {Yau, Shing-Tung}, + title = {Review of geometry and analysis}, + journal = {Asian Journal of Mathematics}, + volume = {4}, + number = {1}, + year = {2000}, + pages = {235--278}, + doi = {10.4310/AJM.2000.v4.n1.a16}, + url = {https://archive.intlpress.com/site/pub/files/_fulltext/journals/ajm/2000/0004/0001/AJM-2000-0004-0001-a016.pdf} +} diff --git a/preprints/A-Smooth-Metric-with-No-Local-Isometric-Immersion-into-Three-Space-September-24-2026/build/sections/assembly.tex b/preprints/A-Smooth-Metric-with-No-Local-Isometric-Immersion-into-Three-Space-September-24-2026/build/sections/assembly.tex new file mode 100644 index 0000000000000000000000000000000000000000..1e618144e9745e071dd90e1bf21ca3c4b05393c4 --- /dev/null +++ b/preprints/A-Smooth-Metric-with-No-Local-Isometric-Immersion-into-Three-Space-September-24-2026/build/sections/assembly.tex @@ -0,0 +1,289 @@ +\section{Assembly of a smooth metric germ}\label{sec:assembly} + +We now turn the whole-patch obstruction into one metric germ. +There are two levels of accumulation: obstructed patches approach +every boundary point of each negative disk, and the disks approach +the origin. Summable bounds in each derivative order will preserve +smoothness at both levels. +We first recall the elementary prescribed-curvature construction +used in \cite[p.~453]{Khuri2009}. + +\begin{lemma}[Prescribing small curvature]\label{lem:prescribe} +Let \(K_0\in C^\infty((-1,1)^2)\) with +\(\norm{K_0}_\infty\le10^{-3}\). +There is a smooth positive metric +\[ +\bar g=\dd X^2+f(X,Y)^2\dd Y^2 +\] +whose curvature is \(K_0\). Its eigenvalues have positive uniform +lower and upper bounds. +\end{lemma} + +\begin{proof} +Solve the ordinary differential equation with parameter \(Y\), +\begin{equation}\label{eq:prescribe-ode} +f_{XX}=-K_0f,\qquad f(0,Y)=1,\qquad f_X(0,Y)=0. +\end{equation} +Smooth parameter dependence gives smoothness. On any segment +joining \(0\) to \(X\), let \(F\) be the supremum of \(\abs f\). +The integral equation gives +\[ +\sup\abs{f-1}\le10^{-3}F,\qquad F\le1+10^{-3}F. +\] +Thus \(f\) stays uniformly close to one and, in particular, +strictly positive. Computing the Christoffel symbols of +\(\dd X^2+f^2\dd Y^2\) gives \(K_{\bar g}=-f_{XX}/f\). +This formula holds also when \(f\) depends on \(Y\). +Equation \eqref{eq:prescribe-ode} proves the assertion. +\end{proof} + +\subsection{Placement and smooth summation} + +Use the fixed coordinates \(X,Y\) on \((-1,1)^2\). +Let \(\mathcal D_j\), \(j\ge1\), be the round disks with centers +\((2^{-j},0)\) and radii \(2^{-j-4}\). +They have disjoint closures and disjoint slightly larger +neighborhoods whose closures avoid the origin, and converge to the +origin without containing it. + +Choose a finite set of rotations whose first-axis lines form an +angular net of mesh less than \(1/1000\). +Place countably many affine patch maps +\[ +\Phi_n(x,y)=c_n+\lambda_nR_n(x,y),\qquad \lambda_n>0, +\] +where \(R_n\) belongs to this finite set, with the following properties: +\begin{enumerate}[label=(\roman*),leftmargin=2.3em] +\item the images of \([-4,4]^2\) have disjoint closures and lie +outside all the closed disks; +\item for every \(j\), every \(b\in\partial\mathcal D_j\), and every +rotation in the net, a sequence of full patches with that rotation +converges to \(b\), with diameters tending to zero. +\end{enumerate} +To obtain this placement, use disjoint thin annuli outside each +\(\mathcal D_j\), within its larger neighborhood, converging to +its boundary. In each annulus choose finitely many distinct centers +so that those assigned each rotation form a successively finer +angular net on the circle. Their number is finite, so all square +sizes can be chosen small enough to give disjoint closed patches +inside that annulus. This proves both properties. \Cref{fig:accumulation} separates the +two levels of the construction. + +\input{figures/accumulation} + +Now prescribe \(K_0\). In each \(\mathcal D_j\) choose a smooth +negative bump, strictly negative throughout the disk and flat on +its boundary. In each patch choose a smooth bump supported in +\(\Phi_n((-4,4)^2)\), equal on a neighborhood of +\(\Phi_n([-3,3]^2)\) to +\begin{equation}\label{eq:patch-K0} +K_0(\Phi_n(x,y))=\kappa_n(x^2-h(y)),\qquad \kappa_n>0. +\end{equation} +The amplitudes of all disk and patch bumps can be chosen so that +their sum is smooth and \(\norm{K_0}_\infty\le10^{-3}\). +Here is a precise choice. Enumerate the unscaled, globally smooth +bump profiles, extended by zero, as \(k_n\). Choose positive +numbers \(\epsilon_n\) such that +\[ +\norm{\epsilon_nk_n}_{C^m}\le10^{-3}2^{-n} +\qquad (0\le m\le n). +\] +There are only finitely many restrictions at stage \(n\). +For each fixed \(m\), the resulting series of derivatives has +a uniformly summable tail. Its sum is \(C^\infty\), including +on all accumulation sets. Disjointness of the supports preserves +the strict negativity inside each disk and the exact profiles +\eqref{eq:patch-K0}. Every summand is flat at each disk-boundary +point at which its support can accumulate, so +\[ +K_0=0\qquad\text{on }\partial\mathcal D_j +\] +for every \(j\). + +Apply \Cref{lem:prescribe} to obtain \(\bar g\). +For each patch, the actual pullback \(g_{0,n}=\Phi_n^*\bar g\) +has curvature +\[ +K_{g_{0,n}}=K_{\bar g}\circ\Phi_n +=\kappa_n(x^2-h(y)). +\] +Indeed, in coefficient matrices, +\(g_{0,n}=\lambda_n^2R_n^T(\bar g\circ\Phi_n)R_n\); +curvature is a scalar under pullback. +This is a background metric of the kind required in +\Cref{sec:model}. + +By \Cref{prop:patch}, choose a smooth tensor perturbation +\(\eta_n\) supported in the model box \(\mathcal P\) such that +\(g_{0,n}+\eta_n\) admits no admissible height. +Push \(\eta_n\) to the original coordinates and extend by zero, +obtaining \(\widehat\eta_n\). +The extension is smooth because its support is strictly inside +the full patch. In matrix notation the pushforward is +\[ +\widehat\eta_n +=\lambda_n^{-2}R_n(\eta_n\circ\Phi_n^{-1})R_n^T. +\] +Consequently, for each ordinary derivative order \(m\), +\[ +\norm{D^m\widehat\eta_n}_\infty +\le C_m\lambda_n^{-m-2}\norm{D^m\eta_n}_\infty, +\] +where \(C_m\) accounts only for the fixed-dimensional component +norms and rotation. The perturbation may be chosen arbitrarily +small in every prescribed finite collection of seminorms, so +these fixed scaling costs can be paid separately on each patch. +Require +\begin{equation}\label{eq:tensor-sum} +\norm{\widehat\eta_n}_{C^m}\le\epsilon\,2^{-n} +\qquad(0\le m\le n), +\end{equation} +where \(\epsilon>0\) is smaller than half a uniform lower +eigenvalue bound for \(\bar g\). +The metric +\begin{equation}\label{eq:final-metric} +g=\bar g+\sum_{n=1}^\infty\widehat\eta_n +\end{equation} +is therefore smooth and positive. +It retains the chosen patch obstruction on every patch: +all the other tensor perturbations vanish there. +Every tensor perturbation and all its derivatives vanish on +each closed disk. Uniform convergence of the derivative series +in \eqref{eq:tensor-sum} consequently preserves the background +metric jets on those disks and their boundaries. Thus +\begin{equation}\label{eq:disk-curvature} +K_g<0\text{ in }\mathcal D_j,\qquad +K_g=0\text{ on }\partial\mathcal D_j. +\end{equation} +This construction imposes no lower bound on the individual +\(\kappa_n\), \(\lambda_n\), or admissible Hessian separations. + +\subsection{The boundary obstruction} + +The following graph observation is also used in the geometric +construction of \cite[Section~2, Step~3]{NY2002}. We include its short proof. + +\begin{lemma}[A negatively curved disk]\label{lem:disk} +Suppose a smooth metric on a neighborhood of a closed disk +has negative curvature in the disk and zero curvature on its +boundary. If an isometric immersion of this neighborhood is +a graph over a plane, its second fundamental form is nonzero +at some boundary point. At that point it has rank one. +\end{lemma} + +\begin{proof} +Let \(u\) be the graph function and let \(\Omega\) be the planar +projection of the disk. This is a bounded domain with connected +smooth boundary, and \(u\) is smooth on a neighborhood of +\(\overline\Omega\). Suppose the second fundamental form vanishes +at every boundary point. The graph formula +\[ +\II_{ij}=\frac{u_{ij}}{\sqrt{1+\abs{Du}^2}} +\] +then gives \(D^2u=0\) along \(\partial\Omega\). Differentiation +along this connected curve shows that \(Du\) is a fixed vector +\(a\) there. Hence \(u=a\cdot x+c\) on the boundary for a constant +\(c\). + +The function \(v=u-a\cdot x-c\) is zero on \(\partial\Omega\) +and has +\[ +\det D^2v=\det D^2u<0\qquad\text{in }\Omega. +\] +It cannot vanish identically. If it is nonzero, compactness +gives either a positive interior maximum or a negative interior +minimum. Its Hessian is semidefinite there, with nonnegative +determinant, a contradiction. +Thus the second fundamental form is nonzero at some boundary +point. The Gauss equation and zero boundary curvature make +its determinant zero, so its rank is one. +\end{proof} + +\begin{proof}[Proof of Theorem~A] +Use the metric \eqref{eq:final-metric}. +Suppose that a neighborhood of the origin admits a smooth +isometric immersion \(F\) into \(\R^3\). +Projection onto the tangent plane at \(F(0)\) is a local +diffeomorphism. After restricting the domain, the immersion +is a graph and the projection is a diffeomorphism onto a +planar open set. Choose \(j\) so large that +\(\overline{\mathcal D_j}\) lies in this restricted domain. +By \eqref{eq:disk-curvature} and \Cref{lem:disk}, there is +\(b\in\partial\mathcal D_j\) where \(\II\) is nonzero and +has rank one. + +Let \(e\) be the fixed unit normal at \(F(b)\), and set +\(z=\langle F,e\rangle\). By \Cref{lem:height}, the covariant +Hessian \(H\) is nonzero and rank one at \(b\), while \(dz(b)=0\). +In particular \(1-\abs{dz}_g^2>0\) nearby. + +Regard \(H(b)\) as a symmetric form in the fixed \(X,Y\) +parameter plane. Choose a rotation in our finite net whose +first-axis line makes angle less than \(1/1000\) with its +kernel. In that rotated parameter frame, a rank-one form +has +\[ +H_{yy}(b)\ne0,\qquad +\abs{H_{xy}(b)/H_{yy}(b)} +=\abs{\tan\alpha}<1/100, +\] +where \(\alpha\) is the angular error. +Continuity gives the same strict inequalities in a +neighborhood of \(b\). + +There are full patches of this orientation converging to \(b\). +One such patch lies wholly in that neighborhood and in the +domain of \(F\). Its common dilation of the two axes multiplies +all Hessian entries by the same positive factor, leaving their +ratio unchanged. In model coordinates the pulled-back height +is therefore smooth on a neighborhood of \(S\), has \(E>0\), +\(H_{yy}\ne0\), and \(\abs q\le1/100\) throughout \(S\). +It satisfies the Darboux equation by \Cref{lem:height}. +It is an admissible height for a metric chosen to have none, +contradicting \Cref{prop:patch}. + +Thus no neighborhood of the origin admits the asserted smooth +isometric immersion. The open square with the smooth positive +metric \(g\) is the required Riemannian surface. +\end{proof} + +\begin{corollary}[Euclidean Taylor jet]\label{cor:flat-jet} +The metric in Theorem~A can be chosen so that, in the coordinates +\(X,Y\), +\[ +\partial^\alpha(g_{ij}-\delta_{ij})(0)=0 +\qquad +\text{for every multi-index }\alpha\in\mathbb N_0^2 +\text{ and }i,j\in\{1,2\}. +\] +Thus smooth local isometric realizability is not determined by +the full Taylor jet of the metric at the base point. +\end{corollary} + +\begin{proof} +Every individual curvature bump and every tensor +\(\widehat\eta_n\) vanishes on a neighborhood of the origin. +The uniform convergence of each derivative series therefore gives +\[ +\partial^\alpha K_0(0)=0, +\qquad +\partial^\alpha\Bigl(\sum_n\widehat\eta_n\Bigr)(0)=0 +\] +for every multi-index \(\alpha\). +It remains to check the background metric +\(\bar g=\dd X^2+f^2\dd Y^2\). +The initial conditions in \eqref{eq:prescribe-ode} give +\(\partial_X^a\partial_Y^b(f-1)(0)=0\) for \(a=0,1\) +and every \(b\ge0\). For \(a\ge2\), the same equation yields +\[ +\partial_X^a\partial_Y^b(f-1)(0) +=-\partial_X^{a-2}\partial_Y^b(K_0f)(0)=0, +\] +since every term in the Leibniz expansion contains a derivative +of \(K_0\) at the origin. Hence \(f-1\), and therefore +\(\bar g_{ij}-\delta_{ij}\), has zero derivatives of every order +there. Adding the tensor series preserves this property. +The Euclidean metric itself admits an isometric immersion into +\(\R^3\), whereas the constructed metric does not on any +neighborhood of the origin. +\end{proof} diff --git a/preprints/A-Smooth-Metric-with-No-Local-Isometric-Immersion-into-Three-Space-September-24-2026/build/sections/caps.tex b/preprints/A-Smooth-Metric-with-No-Local-Isometric-Immersion-into-Three-Space-September-24-2026/build/sections/caps.tex new file mode 100644 index 0000000000000000000000000000000000000000..faf0980ff6ddf741e5f1c27f5670cf3045c0b939 --- /dev/null +++ b/preprints/A-Smooth-Metric-with-No-Local-Isometric-Immersion-into-Three-Space-September-24-2026/build/sections/caps.tex @@ -0,0 +1,484 @@ +\section{Uniform regularity on caps}\label{sec:caps} + +The purpose of this section is to turn low-order control of an +admissible height into all-orders control wherever the metric has +all-orders control in a lower or upper cap. The edge facing the +center is artificial: no high-order boundary data will be imposed +there. + +Throughout this section we consider families of metrics \(g\in\mathcal U\) +and admissible heights \(z\). Assume fixed uniform bounds +\begin{equation}\label{eq:low-bounds} +\norm g_{C^8(S)}+\norm z_{C^8(S)}\le M,\qquad +g\ge cI,\qquad \abs{H_{yy}}\ge c,\qquad E\ge c, +\end{equation} +where \(M<\infty\) and \(c>0\). Constants in this section may depend +on these bounds, on \(\kappa,h\), and on the derivative order. + +Set +\begin{equation}\label{eq:DGa} +D=\pd_x-q\pd_y,\qquad G=\frac E{H_{yy}^2},\qquad a=KG. +\end{equation} +Use flow coordinates \(t,s\) defined by +\begin{equation}\label{eq:flow} +x=t,\qquad y_t(t,s)=-q(t,y(t,s)),\qquad y(0,s)=s. +\end{equation} +They exist on \([-2,2]^2\), have image in the interior of \(S\), +and satisfy +\begin{equation}\label{eq:flow-bounds} +\abs{y(t,s)-s}\le1/50,\qquad D=\pd_t,\qquad +00. +\] +The low bounds give uniform bounds for the derivatives of the chart +and its inverse through the orders used below, in particular through +order three. Denote this chart by \(\Phi_z(t,s)=(t,y(t,s))\). + +\begin{proposition}[Cap regularity]\label{prop:cap} +Fix \(-1/4l\) for \(l\ge8\). Thus all but at most one factor +use the supremum bounds through order \(l-2\); the remaining factor +uses its \(L^2\) bound. Metric derivatives are bounded by hypothesis, +and inverse powers of \(H_{yy}\) are bounded by +\eqref{eq:low-bounds}. The remainder is consequently bounded in +\(L^2\). + +Using +\[ +D^2=\pd_x^2-2q\pd_{xy}+q^2\pd_y^2-(Dq)\pd_y +\] +and \(\pd_x=D+q\pd_y\), the differentiated equation becomes +\begin{equation}\label{eq:pre-flow} +\left[ +D^2+a\pd_y^2+BD+ +\bigl(Dq-P_{z_x}q-P_{z_y}+la_y\bigr)\pd_y +\right]u=R_l. +\end{equation} +Now \(D=\pd_t\), \(\pd_y=s_y\pd_s\), and +\(\pd_y^2=s_y^2\pd_s^2+s_{yy}\pd_s\). +By \Cref{lem:drift}, the transverse coefficient is +\[ +C=a s_{yy}+s_y(K_yG+la_y+O(K)) +=(l+1)Gs_yK_y+O_l(K). +\] +Since +\[ +A_s=Gs_yK_y+K(Gs_y^2)_s, +\] +this is \eqref{eq:linear-coefficients}. The factor \(l+1\) has two +sources: the commutation identity contributes one copy of +\(Gs_yK_y\), and the term \(la_y\) contributes \(l\) copies. +Every other transverse term contains \(K\) times a bounded factor. +All asserted low coefficient +bounds follow from \eqref{eq:low-bounds} and the flow bounds. +\end{proof} + +\subsection{A weighted gradient estimate} + +The differentiated equation reduces the cap induction to a gain +of one derivative. Its principal part changes type, so an elliptic +estimate alone is insufficient. We combine a directed multiplier +near and inside the negative region with a weighted elliptic +estimate on its complement. + +\begin{lemma}[The free cap edge]\label{lem:gradient} +Retain the model geometry and flow coordinates fixed above. +Let \(l\ge8\), and let the coefficients satisfy +\eqref{eq:linear-coefficients} with the uniform bounds in +\Cref{lem:high-equation}. For every +\(R_0\Subset(-2,2)\times(-2,b_*)\), there is a larger compact +rectangle \(R_1\) in that cap and a constant \(C_{R_0,l}\) such that +every smooth solution of +\[ +u_{tt}+Au_{ss}+Bu_t+Cu_s=f +\] +on a neighborhood of \(R_1\) satisfies +\begin{equation}\label{eq:gradient-bound} +\norm{(u_t,u_s)}_{L^2(R_0)} +\le C_{R_0,l}\bigl(\norm u_{L^2(R_1)}+\norm f_{L^2(R_1)}\bigr). +\end{equation} +The constants are uniform over the families under consideration. +\end{lemma} + +\begin{proof} +Choose \(b0\). At the lower edge, +\eqref{eq:flow-bounds} gives \(y<-1\), so \(h(y)<0\). +There is room for nested cutoffs there and for a compact rectangle +\(R_1\) containing every integration set, including its upper edge. + +Set \(d=b-s\) and \(p=8\). We choose the directed weight to vanish +like \(d^p\) at the artificial edge; it remains positive on \(R_0\). +Differentiating such a weight costs a factor \(d^{-1}\), so the +two estimates will meet at \(K=c_1d\), for a small constant +\(c_1>0\) chosen below. The directed multiplier controls +\(K\le c_1d\), where its positive-curvature error is +\(O(K/d)\). On \(K\ge c_1d\), an elliptic estimate controls +the stronger weight \(d^{p-1}\), which pays for the weight and +cutoff derivatives. \Cref{fig:cap} shows the integration geometry. + +\input{figures/cap} + +\smallskip +\noindent\emph{The directed multiplier.} +The transverse multiplier uses the favorable curvature drift. +We add a small \(t\)-directed part to obtain positivity at those +zeros of curvature where \(K_y=0\): there the \(x^2\) term in +the model curvature must supply the estimate. +Put +\begin{equation}\label{eq:weights} +\begin{gathered} +I(t,s)=\int_0^tB(r,s)\dd r,\\ +W=d^p e^{-\lambda s+I(t,s)},\qquad +m=-W,\qquad n=\epsilon tW,\qquad H_0=\lambda+p/d . +\end{gathered} +\end{equation} +Here \(\epsilon>0\) will be small and \(\lambda>0\) large. +The factor \(e^I\) gives \(m_t=Bm\), canceling the corresponding +mixed term in the identity below. Also +\(m_s/W=H_0-I_s\), with \(I_s\) uniformly bounded. + +Testing against \(mu_s+nu_t\) and integrating by parts gives +the interior quadratic form +\begin{equation}\label{eq:quadratic} +T u_t^2+S_0u_s^2+Ju_tu_s, +\qquad +\begin{cases} +T=(m_s-n_t)/2+Bn,\\ +S_0=-(Am)_s/2+(An)_t/2+Cm,\\ +J=-m_t+Bm-(An)_s+Cn. +\end{cases} +\end{equation} +The symbol \(S_0\) denotes this coefficient, not the model square. +Substituting \eqref{eq:weights} gives, for large \(\lambda\), +\begin{equation}\label{eq:quadratic-bounds} +\begin{aligned} +T/W&\ge H_0/4,\\ +S_0/W&=-(l+1/2)A_s+\epsilon tA_t/2-H_0A/2+O_l(A),\\ +\abs J/W&\le C_l\epsilon(1+H_0\abs A). +\end{aligned} +\end{equation} + +\smallskip +\noindent\emph{Positivity near and inside the negative region.} +There are choices of these constants and \(c_1,c_2>0\) such that +\begin{equation}\label{eq:coercive} +T u_t^2+S_0u_s^2+Ju_tu_s +\ge c_2W(u_t^2+u_s^2) +\qquad\text{where }K\le c_1d. +\end{equation} +We verify the choices in detail. + +For \(\abs K\) sufficiently small, \eqref{eq:negative-box} +excludes \(\mathcal P\), so the prescribed curvature formula applies. +In the lower cap \(y\le1/50\). The plateau and monotonicity in +\eqref{eq:h} imply \(K_y\le0\) there. Also +\begin{equation}\label{eq:A-derivatives} +A_s=G_1K_y/s_y+O(A),\qquad +A_t=G_1(2\kappa t-qK_y)+O(A). +\end{equation} +Thus the first two terms in the second line of +\eqref{eq:quadratic-bounds}, up to \(O_l(A)\), equal +\[ +G_1\left[ +-\left(\frac{l+1/2}{s_y}+\frac{\epsilon tq}{2}\right)K_y ++\epsilon\kappa t^2 +\right]. +\] +For small fixed \(\epsilon>0\) this is bounded below by a positive +constant times \(-K_y+\epsilon t^2\). To make its uniform +positivity independent of the flow chart, consider the fixed +physical zero set +\[ +\mathcal Z=\{(x,y)\in[-2,2]\times[-101/50,1/50]:x^2=h(y)\}. +\] +It is compact and contains every curvature zero in the lower cap. +If \(x=0\) on \(\mathcal Z\), then \(y=-1\), where \(K_y<0\); +at every other point \(x^2>0\). Thus +\(-K_y+\epsilon x^2\) has a positive minimum on \(\mathcal Z\). +The uniform bounds for \(G_1,s_y,q\) transfer this minimum to +the geometric contribution on a common neighborhood of \(K=0\). + +The cross-term loss after absorbing part of \(T u_t^2\) is bounded by +\begin{equation}\label{eq:cross-loss} +\frac{J^2}{TW} +\le C_l'\epsilon^2\left(H_0^{-1}+H_0A^2\right). +\end{equation} +On \(K\le0\), the term \(-H_0A/2=H_0\abs A/2\) is favorable. +Since \(\abs A\) is bounded, choose \(\epsilon\) small enough +to absorb the second term of \eqref{eq:cross-loss} into a small +fraction of \(H_0\abs A\). Increase \(\lambda\) to absorb +the \(O_l(A)\) terms and the \(H_0^{-1}\) loss. +Away from a neighborhood of the zero set on the negative side, +the same increase of \(\lambda\) makes \(H_0\abs A\) dominate +all bounded adverse terms. + +Fix these \(\epsilon,\lambda\). On \(00\), \(A\ge c'd\). In the transition region of \(\phi\), +\(K=O(d)\), and hence derivatives of \(K/d\) are \(O(d^{-1})\). +It follows that +\[ +\abs{v_t}+\abs{v_s} +\le C\sqrt v\,d^{(p-2)/2}(1+d^{-1}). +\] +The squared cutoffs make this inequality valid also at their zeros. + +Integration by parts produces the positive term +\(\iint v(u_t^2+Au_s^2)\) and the error +\[ +\iint u\bigl[(v_t-vB)u_t+((Av)_s-vC)u_s\bigr]. +\] +On \(v>0\), the coefficient bounds and \(A\ge c'd\) give +\[ +\frac{(v_t-vB)^2}{v} ++\frac{((Av)_s-vC)^2}{Av} +\le C(1+d^{p-5}). +\] +Young's inequality absorbs half of the positive term; the remaining +cost is at most \(C\iint(1+d^{p-5})u^2\). +Since \(p=8\), this is bounded by \(C\norm u_2^2\). +The source term is at most \(C\norm f_2\norm u_2\). +We conclude that +\[ +\iint v(u_t^2+Au_s^2) +\le C(\norm u_2^2+\norm f_2^2). +\] +Where \(\chi=\phi=1\), \(v=d^{p-2}\gtrsim d^{p-1}\) +and \(vA\gtrsim d^{p-1}\), proving \eqref{eq:aux-elliptic}. +This estimate permits ellipticity to degenerate linearly as +\(s\) approaches \(b\). + +\smallskip +\noindent\emph{Combining the estimates.} +Multiply \(m,n\) by one nonnegative cutoff toward the other three +edges, equal to one on \(R_0\) and on the nonelliptic portion of +the integration region. Its derivative terms lie in fixed positive +curvature; decreasing \(c_1\), if needed, puts those supports +in the region controlled by \eqref{eq:aux-elliptic}. +The coefficients in \eqref{eq:quadratic} and the cutoff errors +are \(O(d^{p-1})\), since \(W\asymp d^p\) and \(H_0=O(1+d^{-1})\). +Thus \eqref{eq:aux-elliptic} controls all adverse terms outside +\eqref{eq:coercive}. + +On the coercive region the source \(f(mu_s+nu_t)\) is absorbed by +Young's inequality, at a cost \(C\norm f_2^2\). On the complementary +region use \eqref{eq:aux-elliptic} and +\(W^2/d^{p-1}=O(d^{p+1})\) to obtain the same bound together +with \(C\norm u_2^2\). Combining the coercive part with the +elliptic estimate on its complement, we obtain +\[ +\iint_{R_0}W(u_t^2+u_s^2) +\le C(\norm u_{L^2(R_1)}^2+\norm f_{L^2(R_1)}^2). +\] +The weight is bounded below on \(R_0\), so this proves +\eqref{eq:gradient-bound}. + +All integrations at the artificial edge can first be made below +\(s=b-\eta\) and then continued to \(\eta=0\). +For each individual smooth solution, the boundary terms vanish +because \(W\) and \(v\) vanish to positive order. The edge \(b\) +is strictly inside the smooth domain. No uniform bound on its +high derivatives is needed to obtain these identities; the uniform +estimates involve only the interior \(L^2\) norms just displayed. +\end{proof} + +\subsection{Completion of the cap induction} + +\begin{proof}[Proof of \Cref{prop:cap}] +We prove uniform \(L^2\) bounds for ordinary derivatives of \(z\) +through order \(l\), on images of all compact rectangles in the +cap. The case \(l=8\) follows from \eqref{eq:low-bounds}. +Assume the assertion at some \(l\ge8\). Apply +\Cref{lem:high-equation,lem:gradient} with \(f=R_l\). +The required bounds for \(u,R_l\) on the larger compact rectangle +are supplied by the induction hypothesis. Measures in the two +coordinate systems are uniformly comparable. + +The resulting gradient bound controls +\[ +\pd_y^{l+1}z=s_yu_s,\qquad +\pd_x\pd_y^lz=u_t+q s_yu_s. +\] +To recover \(\pd_x^k\pd_y^{l+1-k}z\) for \(k\ge2\), apply +\(\pd_x^{k-2}\pd_y^{l+1-k}\) to \eqref{eq:P}. +The two top terms on the right are +\[ +P_{z_{xy}}\pd_x^{k-1}\pd_y^{l+2-k}z +\quad\text{and}\quad +P_{z_{yy}}\pd_x^{k-2}\pd_y^{l+3-k}z. +\] +They have fewer \(x\) indices. All remaining solution factors +have order at most \(l\) and obey the same Sobolev product bounds +as in \Cref{lem:high-equation}. Increasing \(k\) therefore gives +every derivative of order \(l+1\). + +This closes the induction. For each fixed derivative order, local +Sobolev estimates on slightly larger compact rectangles convert +the \(L^2\) bounds into the asserted supremum bounds. The final +rectangle is arbitrary and fixed; only the constants and intermediate +margins depend on the derivative order. +Reflection \(y\mapsto-y\), using the evenness of \(h\), proves +the upper-cap assertion. +\end{proof} diff --git a/preprints/A-Smooth-Metric-with-No-Local-Isometric-Immersion-into-Three-Space-September-24-2026/build/sections/model.tex b/preprints/A-Smooth-Metric-with-No-Local-Isometric-Immersion-into-Three-Space-September-24-2026/build/sections/model.tex new file mode 100644 index 0000000000000000000000000000000000000000..64693882b445dc65bdd235f5780b2c175d3bd5fd --- /dev/null +++ b/preprints/A-Smooth-Metric-with-No-Local-Isometric-Immersion-into-Three-Space-September-24-2026/build/sections/model.tex @@ -0,0 +1,105 @@ +\section{The model patch and the height equation}\label{sec:model} + +The first task is to obstruct scalar height functions on a whole +closed square. We fix a curvature profile with a negative center +and a positive exterior, then specify the equation and inequalities +that the final geometric argument will force on one such square. + +Fix a smooth even function \(h:\R\to\R\) such that +\begin{equation}\label{eq:h} +\begin{gathered} +h=1 \quad\text{on }[-1/2,1/2],\qquad h'\le0\quad\text{on }[0,\infty),\\ +h>0\quad\text{on }(-1,1),\qquad h<0\quad\text{for }\abs y>1,\\ +h(1)=0,\qquad h'(1)<0. +\end{gathered} +\end{equation} +For example, let \(\rho\) be a nonnegative smooth function supported +in \([1/2,2]\), positive on \((1/2,3/2)\), and put +\[ +h(y)=1-\frac{\int_0^{\abs y}\rho(v)\dd v} + {\int_0^1\rho(v)\dd v}. +\] +This function has all the required properties. + +Write \(S=[-3,3]^2\), with coordinates \(x,y\), and fix a positive +constant \(\kappa\). Let \(g_0\) be a smooth positive metric on a +neighborhood of \(S\) with curvature +\begin{equation}\label{eq:model-curvature} +K_{g_0}(x,y)=\kappa\bigl(x^2-h(y)\bigr). +\end{equation} +We will construct such background metrics in \Cref{sec:assembly}. +Perturbations are smooth symmetric tensors supported in +\[ +\mathcal P=[-1/10,1/10]^2. +\] +Let \(\mathcal U\) be the open set of these perturbations for which +the resulting metric \(g\) is positive and +\begin{equation}\label{eq:negative-box} +K_g<-\kappa/2\qquad\text{on }[-1/5,1/5]^2. +\end{equation} +We regard \(\mathcal U\) as a set of metrics with its \(C^\infty\) +topology. It contains \(g_0\): on this smaller square, +\(h=1\) and \(x^2\le1/25\). Outside \(\mathcal P\), every metric +in \(\mathcal U\) still has curvature \eqref{eq:model-curvature}. + +For a function \(z\), write +\begin{equation}\label{eq:height-notation} +H_{ij}=z_{ij}-\Gamma^r_{ij}z_r,\qquad +E=\det(g)(1-\abs{dz}_g^2). +\end{equation} +Here \(K=K_g\), \(\Gamma^r_{ij}\) are the Christoffel symbols of +\(g\), and repeated indices are summed. Wherever \(H_{yy}\ne0\), +put \(q=H_{xy}/H_{yy}\). An \emph{admissible height} +for \(g\) is a function \(z\), smooth on a neighborhood of the closed square \(S\), satisfying throughout~\(S\) +\begin{equation}\label{eq:admissible} +H_{xx}H_{yy}-H_{xy}^2=KE,\qquad +E>0,\qquad H_{yy}\ne0,\qquad \abs q\le1/100. +\end{equation} +The scalar equation here is the Darboux equation; compare +\cite[Equation~(1) and Appendix]{Khuri2009}. We recall the necessary geometric +identity directly. + +\begin{lemma}[Height equation]\label{lem:height} +Let \(F\) be a smooth isometric immersion of a surface into \(\R^3\), +let \(e\) be a fixed ambient unit vector, and set \(z=\langle F,e\rangle\). +Then +\[ +\det(\nabla_g^2z)=K\det(g)(1-\abs{dz}_g^2). +\] +If \(e\) is normal to the immersed surface at a point \(b\), then +\(dz(b)=0\) and \(\nabla_g^2z(b)=\II(b)\), with the choice of unit +normal equal to \(e\) at \(b\). +\end{lemma} + +\begin{proof} +With \(n\) the chosen unit normal and \(\nu=\langle n,e\rangle\), +the Gauss formula gives \(\nabla_g^2z=\nu\II\). The tangential and +normal decomposition of \(e\) gives +\(\abs{dz}_g^2+\nu^2=1\). +The Gauss equation \(\det\II=K\det g\) now proves the identity, +and the final statement follows from \(\nu(b)=1\). +\end{proof} + +Only this necessary equation for a height is used. We do not need +the converse reconstruction of an immersion from a Darboux solution. + +\begin{proposition}[Patch obstruction]\label{prop:patch} +The metrics in \(\mathcal U\) admitting no admissible height form +a residual subset of \(\mathcal U\). In particular, for every +\(C^\infty\) neighborhood of \(g_0\) in \(\mathcal U\), there is +a metric in that neighborhood with no admissible height. +\end{proposition} + +The whole-square requirement is essential. Every curvature zero +in \(S\) of a metric in \(\mathcal U\) is a clean change of sign. Indeed, +\(K<0\) on \(\mathcal P\); outside it, +\(K_x=2\kappa x\), and a zero with \(x=0\) has +\(y=\pm1\), where \(K_y\ne0\). Thus the pointwise local-existence +results recalled in the introduction are compatible with this +proposition: they do not supply one height satisfying +\eqref{eq:admissible} throughout \(S\). + +We prove \Cref{prop:patch} in \Cref{sec:pulse}, after establishing estimates on +the two caps. \Cref{fig:geometry} shows the model geometry. + +\input{figures/geometry} diff --git a/preprints/A-Smooth-Metric-with-No-Local-Isometric-Immersion-into-Three-Space-September-24-2026/build/sections/pulse.tex b/preprints/A-Smooth-Metric-with-No-Local-Isometric-Immersion-into-Three-Space-September-24-2026/build/sections/pulse.tex new file mode 100644 index 0000000000000000000000000000000000000000..c81fe45a0c4e48441605a3c6feaf204967b9f220 --- /dev/null +++ b/preprints/A-Smooth-Metric-with-No-Local-Isometric-Immersion-into-Three-Space-September-24-2026/build/sections/pulse.tex @@ -0,0 +1,406 @@ +\section{The oscillatory pulse and the patch obstruction}\label{sec:pulse} + +We now prove \Cref{prop:patch} by a category argument. Assuming +density of a class of metrics admitting uniformly bounded heights, +we perturb a fixed metric inside a thin strip. Cap regularity and hyperbolic propagation give +smooth tangential data at both strip edges for every corresponding +height. A Taylor comparison for the fixed metric isolates the change +in the transverse derivative caused by the perturbation. Its +oscillatory moment is incompatible with smooth data at both edges. + +\subsection{A bounded solution class} + +The vector space of smooth symmetric tensors supported in +\(\mathcal P\) is a closed subspace of a smooth-function Fréchet +space. Its open subset \(\mathcal U\) is therefore a Baire space. +For \(M\in\N\), consider metrics admitting an admissible height with +\begin{equation}\label{eq:class-bounds} +\norm z_{C^8(S)}\le M,\qquad +M^{-1}\le\abs{H_{yy}}\le M,\qquad E\ge M^{-1}. +\end{equation} +Put +\begin{equation}\label{eq:speed-constants} +v_*=\sqrt{\kappa/2}\,M^{-3/2},\qquad +L=100(1+v_*^{-1}). +\end{equation} +For each rational \(q_0\), let \(\mathcal A_{M,q_0}\) be the +subclass in which such a height also satisfies +\begin{equation}\label{eq:q-class} +\abs{q(0,0)-q_0}<\frac1{100L}. +\end{equation} +These countably many classes cover every metric with an admissible +height. Indeed smoothness and the strict inequalities on the compact +square supply some \(M\), after which rational approximation supplies +\(q_0\). We will show that each class is nowhere dense. + +Suppose, to the contrary, that the closure of one class contains a +nonempty open set \(\mathcal O\subset\mathcal U\). Fix +\(g_*\in\mathcal O\). All test metrics below lie in \(\mathcal O\) +and converge to \(g_*\) in \(C^8\). Metrics from the class can +be chosen arbitrarily close to each test metric in the smooth +topology. We may consequently use uniform low bounds for the metric, +its inverse, and the first derivatives of \(q\). +Since \(\abs q\le1/100\), a nonempty class satisfies +\(\abs{q_0}<1/20\). + +Introduce linear coordinates +\begin{equation}\label{eq:shear} +\xi=x,\qquad \theta=y+q_0x. +\end{equation} +On a sufficiently small fixed neighborhood of the center, the +preceding low bounds imply +\begin{equation}\label{eq:central-bounds} +\abs{q-q_0}\le\frac1{10L}, +\qquad \sqrt{-a}\ge v_*. +\end{equation} +The second inequality follows from \eqref{eq:negative-box} +and \eqref{eq:class-bounds}. In these coordinates, +\begin{equation}\label{eq:sheared-hessian} +\begin{gathered} +H_{\theta\theta}=H_{yy},\qquad +H_{\xi\theta}=(q-q_0)H_{yy},\\ +H_{\xi\xi}=d_0H_{yy},\qquad d_0=(q-q_0)^2+a. +\end{gathered} +\end{equation} +Because \((10L)^{-1}\le v_*/1000\), \(d_0<0\) is separated +from zero. Solving the height equation in the other direction gives +\begin{equation}\label{eq:Q} +z_{\theta\theta} +=Q_g(\xi,\theta,z_i,z_{\xi\xi},z_{\theta\xi}) +=\Gamma^i_{\theta\theta}z_i+ +\frac{H_{\theta\xi}^2+(K\det g)(1-\abs{dz}_g^2)} + {H_{\xi\xi}}. +\end{equation} +Its second-jet partial derivatives, at a solution, are +\begin{equation}\label{eq:hyperbolic-coefficients} +P_1=Q_{z_{\theta\xi}}=\frac{2(q-q_0)}{d_0}, +\qquad S_1=Q_{z_{\xi\xi}}=-\frac1{d_0}>0. +\end{equation} +They have bounded first derivatives; \(S_1\) has a positive lower +bound. The following convenient speed bound follows from +\eqref{eq:central-bounds}: +\begin{equation}\label{eq:speed-bound} +\abs{P_1}+\sqrt{S_1}\le4/v_*0\) small enough that the slab +\begin{equation}\label{eq:slab} +-r\le\theta\le r,\qquad \abs\xi\le Lr +\end{equation} +and fixed margins lie in the preceding neighborhood and inside +\(\mathcal P\). We also require \(r<1/2\). +Along a flow line from \eqref{eq:flow}, +\(D\theta=q_0-q\), whereas \(\theta=s\) when \(t=0\). +The global bound \(\abs q\le1/100\) keeps these flow segments +in that neighborhood after a further shrinking of \(r\). Thus throughout +the slab +\begin{equation}\label{eq:s-theta} +\abs{s-\theta}\le r/4. +\end{equation} +In fact the bound \(r/10\) follows from its \(t\)-length at most \(Lr\). + +\subsection{The test metrics and smooth data at the pulse edges} + +Choose a nonzero real \(\chi_0\in C_c^\infty((-Lr/8,Lr/8))\) and +a nonnegative \(\phi_0\in C_c^\infty((-1,1))\) with +\(\int\phi_0>0\). For small fixed \(\delta>0\) and integers +\(\tau\to\infty\), the test metrics are +\begin{equation}\label{eq:pulse} +\widetilde g_\tau +=g_*+\tau^{-N}\chi_0(\xi)\phi_0(\tau\theta/\delta) + \cos(\tau\xi)\dd\theta^2. +\end{equation} +Choose the fixed integer \(N>10\) larger than every derivative order +in one basic neighborhood of \(g_*\) contained in \(\mathcal O\). +For each fixed \(\delta\), the test metrics then belong to +\(\mathcal O\) for all sufficiently large \(\tau\). Only finitely +many seminorms are involved in this assertion. + +By the closure assumption choose \(g_\tau\in\mathcal A_{M,q_0}\) +such that +\begin{equation}\label{eq:approximation} +\norm{g_\tau-\widetilde g_\tau}_{C^\tau(S)} +\le\tau^{-\tau}, +\end{equation} +and choose a corresponding height \(z_\tau\) with +\eqref{eq:class-bounds} and \eqref{eq:q-class}. +We suppress its subscript when deriving estimates. +All comparisons using finitely many low derivatives hold uniformly +for large \(\tau\). Their limiting constants can be chosen independently +of the fixed small \(\delta\), since \(N>10\). + +Away from the pulse rectangle +\[ +\abs\theta\le\delta/\tau,\qquad \abs\xi\le Lr/8, +\] +the metrics have uniform bounds at every fixed derivative order. +By \eqref{eq:s-theta}, every point of that rectangle has +\(\abs sj\). +Bound that factor in \(L^2\), if it occurs. All other factors +have order at most \(j-2\), so the one-dimensional Sobolev +inequality and the available \(H^{j-1}\) bound for \(V\) +control them in supremum norm. This gives the \(L^2\) bound +for the remainder in \eqref{eq:energy}. +The same moving interval is used at every derivative order; +its length has a fixed positive lower bound, so the one-dimensional +Sobolev constants are uniform. + +The initial first-jet bounds through spatial order seven follow +from the \(C^8\) class bound. For each larger order, the initial +energy on the cut \(\theta=-r\) is bounded by the cap estimate. +The metric derivatives are bounded on +\(-r\le\theta\le-\delta/\tau\), so \eqref{eq:energy} +closes the spatial induction there. The intervals start as +\([-Lr,Lr]\) and lose at most \(Lr/4\) at each end, leaving +room for Sobolev estimates on \(\abs\xi\le Lr/2\). +Reverse time and start at \(\theta=r\) to obtain the same bounds +at the upper pulse edge. +\end{proof} + +\subsection{A comparison across the thin strip} + +On \(\abs\xi\le Lr/2\), construct a smooth comparison \(z^0\) +on \(-\delta/\tau\le\theta\le\delta/\tau\) with the same data +\((z,z_\theta)\) at \(\theta=-\delta/\tau\), satisfying +\eqref{eq:Q} for the fixed metric \(g_*\) up to \(O(\tau^{-N})\). +For this purpose take the time Taylor polynomial of degree \(N+1\) +at the initial edge. Recursively determine its higher coefficients +from \eqref{eq:Q}, using \(g_*\). At step \(k\), for +\(0\le k\le N-1\), differentiating the right side \(k\) times +determines the time derivative of order \(k+2\) from those +through order \(k+1\) and their spatial derivatives. This is +possible because \(Q_g\) contains no second time derivative. +The denominator at the initial +data stays separated from zero by metric closeness and +\eqref{eq:sheared-hessian}. The coefficients and each fixed spatial +derivative are uniformly bounded by \Cref{lem:propagation}. +The denominator remains nonzero on the short strip. The recurrence +makes the residual and its first \(N-1\) time derivatives vanish +at the initial edge. Taylor's formula on a strip of length +\(2\delta/\tau\) gives the claimed \(O(\tau^{-N})\) bound +for fixed \(\delta\). In particular, \(z^0_\theta\) at the final +edge has bounds for all fixed spatial derivatives. + +Put \(w=z-z^0\). Move \(Q_{g_*}(z)-Q_{g_*}(z^0)\) to the +linearized side, using mean partial derivatives along the segments +between their argument vectors +\((z_i,z_{\xi\xi},z_{\theta\xi})\). This yields +\begin{equation}\label{eq:w-equation} +w_{\theta\theta}-P_2w_{\theta\xi}-S_2w_{\xi\xi} ++\beta^i\pd_iw +=Q_{g_\tau}(z)-Q_{g_*}(z)+O(\tau^{-N}), +\end{equation} +with zero Cauchy data at the lower edge. +Here \(S_2>0\) has a uniform lower bound; the coefficients +are bounded and \(P_2,S_2\) have bounded first derivatives. +To see the positivity, the \(C^8\) bound keeps the actual argument +vector close throughout the strip to its initial value. The same is +true of the polynomial, by its coefficient bounds. The argument +vectors agree at the initial edge, because the Cauchy data agree. +At this common value, \(Q_{g_*,z_{\xi\xi}}>0\) by metric closeness and +\eqref{eq:hyperbolic-coefficients}. Continuity keeps it positive +on all the intervening segments, on which the denominator also +remains separated from zero. It is not necessary that those +segments consist of solutions. + +A fixed speed bound larger than the one in \eqref{eq:speed-bound} +therefore applies to \eqref{eq:w-equation}. The limiting bounds +for the zeroth-order coefficient values and the positive lower +bound for \(S_2\) are independent of the fixed small \(\delta\). + +\subsection{The curvature forcing and its moment} + +In the linear coordinates \(\xi,\theta\), the curvature tensor formula +gives +\begin{equation}\label{eq:curvature-density} +K\det g +=\pd_{\xi\theta}g_{\xi\theta} +-\frac{\pd_\xi^2g_{\theta\theta}+\pd_\theta^2g_{\xi\xi}}2 ++\mathcal L(g,\pd g), +\end{equation} +where \(\mathcal L\) is smooth for positive metric matrices. +Since \eqref{eq:pulse} changes only \(g_{\theta\theta}\), +its principal contribution is \(-\tfrac12\pd_\xi^2\) +of the scalar pulse. Consequently the right side of +\eqref{eq:w-equation} equals +\begin{equation}\label{eq:forcing} +\tau^{2-N}D_\tau\chi_0(\xi)\phi_0(\tau\theta/\delta) + \cos(\tau\xi)+o(\tau^{2-N}), +\qquad +D_\tau=\frac{1-\abs{dz}_{g_*}^2}{2H_{\xi\xi}(g_*,z)}. +\end{equation} +The error is uniform in space and time as \(\tau\to\infty\) +for each fixed \(\delta\). Metric changes involving at most one +derivative are \(O_\delta(\tau^{1-N})\); the approximation +\eqref{eq:approximation} and the Taylor residual are smaller. + +There are constants \(c_D,C_D>0\), independent of fixed small +\(\delta\), such that for all large \(\tau\) +\[ +c_D\le\abs{D_\tau}\le C_D, +\] +and \(D_\tau\) has one sign throughout the strip used. +Indeed the numerator is bounded below by the \(E\) bound, +uniform metric bounds, and metric closeness. The denominator +has the sign opposite to \(H_{yy}\), by \(d_0<0\), and stays +nonzero when \(g_\tau\) is replaced by \(g_*\). +The sign of \(H_{yy}\) is constant on the connected square. +Upper and lower bounds follow from the same low estimates. + +Apply the zero-data estimate \eqref{eq:energy} to +\eqref{eq:w-equation}, shrinking intervals from +\([-Lr/2,Lr/2]\) at a fixed sufficiently large speed. +They contain \(\supp\chi_0\) throughout the strip for large +\(\tau\). Its source has \(L^2\) size \(O(\tau^{2-N})\), +so +\begin{equation}\label{eq:short-energy} +\norm{\pd w}_{L^2}\le(C_2+o(1))\delta\tau^{1-N} +\end{equation} +uniformly in strip time. +Here \(C_2\) can be chosen independently of small fixed +\(\delta\): the strip length is \(2\delta/\tau\), and every +Gronwall factor tends to one for fixed \(\delta\). +Finite coefficient derivative bounds depending on that \(\delta\) +are therefore harmless. + +Integrate \eqref{eq:w-equation} against +\(\psi_\tau(\xi)=\chi_0(\xi)\cos(\tau\xi)\) over space and strip time. +The \(w_{\theta\theta}\) term is +\[ +\int\psi_\tau(\xi)w_\theta(\xi,\delta/\tau)\dd\xi, +\] +because the initial data vanish. Both \(z_\theta\) and +\(z^0_\theta\) at the final edge have uniform spatial bounds of +every fixed order. Repeated integration by parts therefore makes +this term \(o(\tau^{1-N})\). + +For each other second-order term, integrate one spatial derivative +by parts. The derivative of \(\psi_\tau\) costs \(O(\tau)\). +By \eqref{eq:short-energy} and the strip length, the total +absolute cost is at most +\begin{equation}\label{eq:moment-error} +(C_3+o(1))\delta^2\tau^{1-N}. +\end{equation} +The constant \(C_3\) is independent of small fixed \(\delta\); +terms with a derivative on a coefficient and the first-order terms +have an additional factor \(\tau^{-1}\) for fixed \(\delta\). +The spatial test is compactly supported, so there is no spatial +boundary contribution. + +In contrast, \eqref{eq:forcing} paired with \(\psi_\tau\) +has absolute integral at least +\begin{equation}\label{eq:moment-lower} +(c_3+o(1))\delta\tau^{1-N}, +\qquad c_3>0, +\end{equation} +with \(c_3\) independent of small fixed \(\delta\). +This follows from the constant sign and lower bound of \(D_\tau\), +the nonnegativity and positive integral of \(\phi_0\), and +\[ +\int\chi_0(\xi)^2\cos^2(\tau\xi)\dd\xi +\longrightarrow \frac12\int\chi_0(\xi)^2\dd\xi>0. +\] +No convergence of \(D_\tau\) is needed. + +Choose \(\delta>0\) so small that \(C_3\delta0\), with \(C_3\delta=stealth,font=\small] + \draw[->] (0,0.15) -- (4.55,0.15) node[right] {$s$}; + \draw[->] (0.15,0) -- (0.15,3.05) node[above] {$\lambda$}; + \draw[gray,densely dotted] (2.2,0.15) -- (2.2,2.75); + \draw[very thick] (0.45,0.6) -- (3.95,2.6) node[right] {$A$}; + \draw[very thick,dashed] (0.45,2.6) -- (3.95,0.6) node[right] {$B$}; + \node[font=\scriptsize,below] at (2.2,0.15) {double}; + + \node at (7.45,2.75) {$[A,B,C,D]$}; + \node at (7.45,2.05) {$[B,A,C,D]$}; + \node at (7.45,1.35) {$[B,C,A,D]$}; + \node at (7.45,0.65) {$[B,C,D,A]$}; + \draw[->] (7.45,2.52) -- (7.45,2.28); + \draw[->] (7.45,1.82) -- (7.45,1.58); + \draw[->] (7.45,1.12) -- (7.45,0.88); +\end{tikzpicture} +\caption{Schematic of a continued pair at an isolated double (left) +and the increasing adjacent-swap schedule on four illustrative ranks +(right). The solid line $A$ changes from lower to upper ordered rank; +the four-rank schedule sends the first label to the last rank. No +numerical eigenvalues or phase durations are represented.} +\label{fig:ang-crossing-cycle} +\end{figure} diff --git a/preprints/A-Three-Dimensional-Counterexample-to-Integer-Degree-Harmonic-Dimension-Comparison-September-26-2026/build/paper.tex b/preprints/A-Three-Dimensional-Counterexample-to-Integer-Degree-Harmonic-Dimension-Comparison-September-26-2026/build/paper.tex new file mode 100644 index 0000000000000000000000000000000000000000..a1981fd331fea2f03425c49f953a99b98b10ed48 --- /dev/null +++ b/preprints/A-Three-Dimensional-Counterexample-to-Integer-Degree-Harmonic-Dimension-Comparison-September-26-2026/build/paper.tex @@ -0,0 +1,42 @@ +\documentclass[11pt]{article} +\pdftrailerid{} +\usepackage[T1]{fontenc} +\usepackage{lmodern} +\usepackage[margin=1.05in]{geometry} +\usepackage{amsmath,amssymb,amsthm,mathtools} +\usepackage{microtype} +\usepackage{tikz} +\usepackage[hidelinks]{hyperref} +\hypersetup{pdftitle={A Three-Dimensional Counterexample to Integer-Degree Harmonic Dimension Comparison},pdfauthor={OpenAI}} +\numberwithin{equation}{section} +\newtheorem{theorem}{Theorem}[section] +\newtheorem{proposition}[theorem]{Proposition} +\newtheorem{lemma}[theorem]{Lemma} +\newtheorem{corollary}[theorem]{Corollary} +\theoremstyle{remark} +\newtheorem{remark}[theorem]{Remark} +\title{A Three-Dimensional Counterexample to Integer-Degree Harmonic Dimension Comparison} +\author{OpenAI} +\date{September 26, 2026} +\begin{document} +\maketitle +\begin{abstract} +For every \(4/9=stealth, + job/.style={circle,draw,fill=white,minimum size=6.8mm,inner sep=0pt}] + \draw[black!55,rounded corners=2pt] (-.15,-.25) rectangle (10.75,2.85); + \node[anchor=west] at (-.15,3.13) {Parent $W$}; + \node at (4,3.13) {Child $W'$}; + \node at (9.1,3.13) {$b'\subset W$, $|b'|=3$}; + + \fill[black!5] (8.55,-.12) rectangle (9.65,2.72); + \foreach \yy/\ii/\xx/\cc in {2.2/1/3.55/blue!65!black,1.3/2/4/teal!65!black,.4/3/4.45/violet!70!black} { + \draw[draw=\cc,fill=\cc!5,rounded corners=3pt] + (.03,\yy-.34) rectangle (10.57,\yy+.34); + \node[text=\cc] at (1.05,\yy) {$I_{\ii}\cap W$}; + \node[text=\cc] at (2.5,\yy) {$\cdots$}; + \node[job,draw=\cc] (x\ii) at (\xx,\yy) {$x_{\ii}$}; + \node[job,draw=\cc] (z\ii) at (9.1,\yy) {$z_{\ii}$}; + \draw[->,draw=\cc] (x\ii) -- node[above,font=\footnotesize] {$\prec$} (z\ii); + \node[text=\cc] at (10.05,\yy) {$\cdots$}; + } + \draw[dashed,black!60,rounded corners=2pt] (3.03,-.12) rectangle (4.98,2.72); + \draw[black!60,rounded corners=2pt] (8.55,-.12) rectangle (9.65,2.72); + \node[anchor=west,font=\footnotesize] at (-.15,-.64) + {$x_i\in I_i,\ x_i\prec z_i\ \Longrightarrow\ z_i\in I_i$ + }; + \node[anchor=east,font=\footnotesize] at (10.75,-.64) + {$I_i\cap I_j\cap W=\varnothing\ (i\ne j)\ \Longrightarrow\ z_i\ne z_j$}; +\end{tikzpicture} diff --git a/preprints/A-polynomial-time-algorithm-for-three-machine-unit-job-scheduling-September-24-2026/build/figures/global-gap-filters.tex b/preprints/A-polynomial-time-algorithm-for-three-machine-unit-job-scheduling-September-24-2026/build/figures/global-gap-filters.tex new file mode 100644 index 0000000000000000000000000000000000000000..24c26942d613f6312e8a127c4a0b9983e51dd422 --- /dev/null +++ b/preprints/A-polynomial-time-algorithm-for-three-machine-unit-job-scheduling-September-24-2026/build/figures/global-gap-filters.tex @@ -0,0 +1,38 @@ +\begin{tikzpicture}[x=1cm,y=1cm,font=\small, + filter/.style={draw,rounded corners=2pt,minimum height=.75cm, + minimum width=1.8cm,inner sep=4pt,align=center}] + \node[anchor=west,font=\bfseries] at (-.92,2.18) {High jobs}; + \node[filter,draw=blue!60!black,fill=blue!5] (h1) at (0,1.35) + {$\displaystyle\bigcup_j K'_j$}; + \node[filter,draw=blue!60!black,fill=blue!5] (h2) at (2.35,1.35) + {$P_{b'}$}; + \node[filter,draw=blue!60!black,fill=blue!5,minimum width=2.15cm] (h3) at (4.9,1.35) + {$J\setminus\widehat P_{a'}$}; + \node at (1.175,1.35) {$\cap$}; + \node at (3.54,1.35) {$\cap$}; + \node[anchor=west] at (6.32,1.35) {$=\ W'\setminus S$}; + \node[font=\footnotesize,align=center] at (4.9,.49) + {boundary invariant\\excludes early qualifiers}; + + \node[anchor=west,font=\bfseries] at (-.92,-.36) {Low jobs}; + \node[filter,draw=teal!65!black,fill=teal!5,minimum width=2.15cm] (l1) at (.15,-1.18) + {$D_{a'}\cap P_b$}; + \node[filter,draw=teal!65!black,fill=teal!5,minimum width=1.4cm] (l2) at (2.5,-1.18) + {$\widetilde S$}; + \node[filter,draw=teal!65!black,fill=teal!5,minimum width=2.15cm] (l3) at (4.85,-1.18) + {$J\setminus\widehat D_v$}; + \node[filter,draw=teal!65!black,fill=teal!5,minimum width=2.15cm] (l4) at (7.55,-1.18) + {$J\setminus\widehat D_{b'}$}; + \node at (1.51,-1.18) {$\cap$}; + \node at (3.49,-1.18) {$\cap$}; + \node at (6.20,-1.18) {$\cap$}; + \node[anchor=west] at (8.82,-1.18) {$=\ W'\cap S$}; + \node[font=\footnotesize,align=center] at (.15,-2.08) + {localize\\inside $W$}; + \node[font=\footnotesize,align=center] at (2.5,-2.08) + {cutoff test\\valid on $W$}; + \node[font=\footnotesize,align=center] at (4.85,-2.08) + {center separator:\\before $v$}; + \node[font=\footnotesize,align=center] at (7.55,-2.08) + {edge separator:\\before $b'$}; +\end{tikzpicture} diff --git a/preprints/A-polynomial-time-algorithm-for-three-machine-unit-job-scheduling-September-24-2026/build/figures/shared-cutoff.tex b/preprints/A-polynomial-time-algorithm-for-three-machine-unit-job-scheduling-September-24-2026/build/figures/shared-cutoff.tex new file mode 100644 index 0000000000000000000000000000000000000000..c9e2e4040238db8345a79c2efc12853ade9d89fc --- /dev/null +++ b/preprints/A-polynomial-time-algorithm-for-three-machine-unit-job-scheduling-September-24-2026/build/figures/shared-cutoff.tex @@ -0,0 +1,21 @@ +\begin{tikzpicture}[x=1cm,y=1cm,font=\small,>=stealth] + \fill[blue!7] (3.15,-.25) rectangle (6.35,.62); + \draw[->,gray!75] (-.25,0) -- (10.6,0); + \node[anchor=east,font=\footnotesize] at (10.6,-.75) + {time in the working frame}; + \foreach \x in {0,3,6.5,9.6} + {\draw[thick] (\x,-.18) -- (\x,.62);} + \node[above] at (0,.62) {$a=s_0$}; + \node[above] at (3,.62) {$a'=s_i$}; + \node[above] at (6.5,.62) {$b'=s_{i+1}$}; + \node[above] at (9.6,.62) {$v=s_{q+1}$}; + \node at (4.75,.22) {$W'=(a',b')$}; + \node at (1.5,.22) {$\cdots$}; + \node at (8.05,.22) {$\cdots$}; + \draw[blue!65!black] (3,1.18) -- (3,1.4) -- (6.5,1.4) -- (6.5,1.18); + \node[above,text=blue!65!black] at (4.75,1.4) + {both endpoints separate at cutoff $S_i$}; + \node[anchor=north,align=center] at (4.75,-1.05) + {$W'\cap S_i\subseteq D_{a'}$\qquad + $W'\setminus S_i\subseteq P_{b'}$}; +\end{tikzpicture} diff --git a/preprints/A-polynomial-time-algorithm-for-three-machine-unit-job-scheduling-September-24-2026/build/figures/switch-localization.tex b/preprints/A-polynomial-time-algorithm-for-three-machine-unit-job-scheduling-September-24-2026/build/figures/switch-localization.tex new file mode 100644 index 0000000000000000000000000000000000000000..ac9ba18c18282b4b47ba142e43e730c0bc7fa48a --- /dev/null +++ b/preprints/A-polynomial-time-algorithm-for-three-machine-unit-job-scheduling-September-24-2026/build/figures/switch-localization.tex @@ -0,0 +1,24 @@ +\begin{tikzpicture}[x=1cm,y=1cm,font=\small,>=stealth] + \fill[teal!8] (4.35,-.25) rectangle (7.25,.65); + \draw[->,gray!75] (-.25,0) -- (10.6,0); + \foreach \x in {0,2.2,4.2,7.4,9.6} + {\draw[thick] (\x,-.18) -- (\x,.65);} + \node[above] at (0,.65) {$a_-$}; + \node[above] at (2.2,.65) {$v_-$}; + \node[above] at (4.2,.65) {$a$}; + \node[above] at (7.4,.65) {$b$}; + \node[above] at (9.6,.65) {$b_-$}; + \node at (5.8,.25) {$W=(a,b)$}; + \draw[gray] (0,1.24) -- (0,1.43) -- (9.6,1.43) -- (9.6,1.24); + \node[above] at (4.8,1.43) {$U=(a_-,b_-)$}; + \node[anchor=east,font=\footnotesize] at (10.6,-.7) + {time in the new frame}; + \node[anchor=north,align=center] at (4.8,-1.05) + {$G=D_{a_-}\cap (J\setminus\widehat P_a) + \cap (J\setminus\widehat P_{v_-})\cap H_-^*$}; + \node[draw=teal!60!black,fill=teal!3,rounded corners=2pt, + inner sep=8pt,align=center,font=\footnotesize] at (4.8,-2.33) + {If $z\in G$ were at or before $a$, then $z\in U$ and hence $z\in H_-$.\\[3pt] + At or before $v_-$: $z\in\widehat P_{v_-}$. + \qquad After $v_-$ through $a$: $z\in\widehat P_a$.}; +\end{tikzpicture} diff --git a/preprints/A-polynomial-time-algorithm-for-three-machine-unit-job-scheduling-September-24-2026/build/lists.tex b/preprints/A-polynomial-time-algorithm-for-three-machine-unit-job-scheduling-September-24-2026/build/lists.tex new file mode 100644 index 0000000000000000000000000000000000000000..5d34fd860e687f006df5463b31f8ee80c5d19613 --- /dev/null +++ b/preprints/A-polynomial-time-algorithm-for-three-machine-unit-job-scheduling-September-24-2026/build/lists.tex @@ -0,0 +1,372 @@ +\section{Short global lists and changes of direction}\label{sec:lists} + +Fix a nonempty child \(W'=(a',b')\) of \(W=(a,b)\), with center +\(v\), edge cutoff \(S\), and working frame as in +\eqref{eq:edge-data}. The child's tag is this working frame. +At the end of an update, \emph{prune} every entry whose intersection +with \(W'\) is empty, preserving the order of the retained entries. +We distinguish this pruning from the suffix operation used to form a +carry. + +\subsection{Carry and pruning} + +\begin{lemma}[Suffix carry]\label{lem:carry} +For a global upset list \(\K=(K_1,\ldots,K_\ell)\), fix +\(1\le d\le\ell\) and an integer \(0\le r\le N+1\), and form +\begin{equation} + \operatorname{carry}_{d,r}(\K) + =\bigl(K_d\cap\{\rk\ge r\},K_{d+1},\ldots,K_\ell\bigr). + \label{eq:carry} +\end{equation} +Its qualifier set consists exactly of the old global qualifiers with +old key at least \((d,r)\). Among those jobs the new list preserves +the old key order. Its entries are global upsets. +\end{lemma} +\begin{proof} +Let \(j\) be a job's largest old membership index. It is selected +exactly when \(j>d\), or \(j=d\) and its rank is at least \(r\). +Whenever selected, its largest old membership is retained. +Reindexing applies the same strictly increasing shift to these retained +largest indices, and leaves the rank tiebreak unchanged. Rank suffixes +are upsets because rank is a linear extension. +\end{proof} + +\Needspace{3\baselineskip} +\begin{lemma}[The comparison preserved by pruning]\label{lem:prune} +Let a list be pruned on \(W'\). Suppose \(x\in W'\) qualifies +and \(y\) is any global qualifier of the pruned list. If the pruned +key of \(x\) is smaller than the pruned key of \(y\), the same +comparison held before pruning. +\end{lemma} +\begin{proof} +Use original indices for the retained entries; their eventual +reindexing preserves order. Let \(M(z)\) be the largest membership +index before pruning and \(R(z)\) the largest retained one. +Every entry containing \(x\) survives, so \(R(x)=M(x)\), whereas +\(R(y)\le M(y)\). If \(R(x)M(x)\) or the rank comparison remains the +tiebreak. Both cases give the asserted old key comparison. +\end{proof} + +\subsection{The two updates} + +For a same-direction edge, the old list is in the working frame. +If \(O\setminus S=\varnothing\), its carry is empty. +Otherwise let \((d,r)\) be the smallest key of a job of +\(O\setminus S\), and use \eqref{eq:carry}. +Append the \emph{injection} +\begin{equation} + I=D_a\cap(J\setminus P_b). + \label{eq:injection} +\end{equation} +Then prune on \(W'\). + +For a switched edge, we shall construct a global upset \(G\) in the +new working frame with the following properties: +\begin{equation} + W\setminus D_a\subseteq G, + \qquad + G\cap\{x:\text{\(x\) is scheduled at or before \(a\)}\} + =\varnothing . + \label{eq:G-properties} +\end{equation} +Choose \(h\) so that the still-high part of +\(X=W\setminus D_a\) is exactly \(X\cap\{\rk\ge h\}\). +Take \(h=N+1\) if this part is empty. +Discard the old list and begin a new list with the ordered pair +\begin{equation} + G\cap\{\rk\ge h\},\qquad D_a. + \label{eq:switch-pair} +\end{equation} +Then prune on \(W'\). Notice that \eqref{eq:G-properties} covers +\emph{all} of the parent's \(X\), not only the jobs in this child. + +\begin{lemma}\label{lem:update-labels} +For either update, assuming \eqref{eq:G-properties} in the switch +case, the pre-pruning qualifier union on \(W\) is exactly +\(W\setminus S\). On these high jobs the pre-pruning key order +agrees with the side's priority order. After pruning, the qualifiers +on \(W'\) are exactly \(W'\setminus S\), and +Invariant~\ref{inv:cones} holds for the child. +\end{lemma} +\begin{proof} +In the same case, \Cref{lem:carry} gives exactly \(O\setminus S\) +on \(W\), in its old key order. The injection gives exactly \(Y\): +use \(Y\subseteq D_a\) in \eqref{eq:cover}. It is disjoint from +the old qualifier union on \(W\), which is contained in \(P_b\). +Since the injection is last, the pre-pruning order is the unpromoted +part of \(O\), followed by \(Y\) in rank order, as required. + +In the switch case, coverage of \(X\) makes the first entry select +exactly the unpromoted part of \(X\) there. The second entry selects +all of \(W\cap D_a\). Even if the two entries overlap, the larger +index of the second puts this whole group last. The ordering +therefore agrees with the switched priority. + +Pruning changes no membership on \(W'\). Its high jobs lie in +\(P_{b'}\), and its other jobs lie in \(D_{a'}\), by +\Cref{lem:edge-labels}. These are precisely the two assertions of +Invariant~\ref{inv:cones} for the child. +\end{proof} + +\subsection{Why only five entries survive} + +A \emph{run} is a chain of nodes with the same tag, starting at the +root or just after a switch. Its initial entries are the root's +single entry, or the two entries in \eqref{eq:switch-pair}. +Same-direction updates only take suffixes, restrict entries by rank +suffixes, and append injections. Multiple restrictions of an entry +by rank suffixes in the same frame combine into one tightest threshold. + +Figure~\ref{fig:capacity-three} isolates the use of machine capacity +in the following bound. + +\Needspace{3\baselineskip} +\begin{lemma}\label{lem:list-length} +Every pruned list produced by this construction has at most five entries. +At most three of them are injections from its current run. +\end{lemma} +\begin{proof} +The root and switch updates have at most two initial entries. +Consider a same update from \(W\) to \(W'\). +Injections retained into this child are pairwise disjoint on \(W\). +To see this, consider any pair and the time the later one was +introduced. It was disjoint on its then-current parent from all +previous carried entries, by \Cref{lem:update-labels}. +That parent contains the current \(W\), and subsequent carries +only restrict the older predicates. The disjointness persists on \(W\). + +Each surviving injection contains some \(x\in W'\). +It is a high job, so \(x\prec z\) for some \(z\in b'\). +Global upward closure puts \(z\) in that same injection. +The actual triple \(b'\) lies in \(W\). Pairwise disjointness on +\(W\) thus forces distinct surviving injections to contain distinct +members of \(b'\). There are only three such members. +Together with the at most two initial entries this proves the bound. +\end{proof} + +The freshly formed same-update list may have six entries before +pruning. We will use the five-entry bound only for pruned lists +or for a carry taken from such a list \emph{before} appending a +new injection. + +\begin{figure}[t] +\centering +\input{figures/capacity-three} +\caption{Why at most three injections survive a same-direction update. +For each surviving injection, choose a high job $x_i$ in the child +and a successor $z_i$ in its ending triple $b'$. +Upward closure puts $z_i$ in the same injection; disjointness on the +parent $W$ makes the chosen $z_i$ distinct. +The three-injection case is drawn; arrows denote precedence. +Only witness pairs are shown: +the injections may contain other jobs and may overlap outside $W$.} +\label{fig:capacity-three} +\end{figure} + + +\subsection{Flattening the history of a run} + +Only the first entry of a switch pair can carry a complicated predicate +from an earlier run. The following lemma removes it whenever we need +to reconstruct an old cutoff on a descendant parent. + +\begin{lemma}[Local flattening]\label{lem:flatten} +Suppose a run began with a switch out of a node \(Q_0\), using +left boundary \(a_0\) in the resulting run frame, and entries +\[ + G_0\cap\{\rk\ge h_0\},\qquad D_{a_0}, + \qquad Q_0\setminus D_{a_0}\subseteq G_0 . +\] +Let \(U\subseteq Q_0\) be a later parent in that run. If the entry derived from the first displayed entry +survives into an unpruned carry out of \(U\), write its +current form as \(G_0\cap\{\rk\ge k\}\). Then the carried portion +of this pair has the same union on \(U\) as the global upset +\begin{equation} + D_{a_0}\cup\{\rk\ge k\}. + \label{eq:flatten} +\end{equation} +\end{lemma} +\begin{proof} +Track the original entries through later reindexings. +As long as the first entry survives, a suffix operation cannot +trim or discard the second entry: a cutoff at that later index +would already discard the first, permanently. +Thus the second entry is either still carried untrimmed, or was +pruned because it was empty on some intermediate node containing +\(U\). In the latter case \(D_{a_0}\cap U=\varnothing\). + +On \(U\cap D_{a_0}\), the second entry, when this set is nonempty, +therefore selects every job. Every job of +\(U\setminus D_{a_0}\) lies in \(G_0\), because \(U\subseteq Q_0\). +On that remainder the first entry selects exactly the rank suffix +\(\rk\ge k\). This proves equality of the two unions on \(U\). +Both factors in \eqref{eq:flatten} are upsets, so their union is +globally upward closed, although it need not agree with the +carried pair outside \(U\). +\end{proof} + +A new switch out of a node \(W\) uses the edge \(U\to W\) +by which \(W\) was created. Let \(H_-\) be that incoming +edge's low cutoff, in its own working frame, and reserve \(S\) +for the cutoff of the new outgoing edge. We need an extension +of \(H_-\) that does not retain a complicated switch predicate +from an earlier run. Local flattening provides it using lists and +covering properties belonging to already constructed ancestors. + +\begin{lemma}[A short extension of an earlier low set]\label{lem:old-low} +Consider an already constructed edge from a parent \(U\) to a child +\(W\), in its old working frame. Its low cutoff \(H_-\subseteq U\) +has a global downset extension \(H_-^*\) agreeing with it on \(U\). +This extension can be chosen with at most \(21\) leaves. +\end{lemma} +\begin{proof} +Use the boundary notation of \(U\) in that old frame for this proof. +If the edge was a switch relative to the tag of \(U\), its low +cutoff is a rank prefix of \(U\setminus D_a\). Thus +\[ + (J\setminus D_a)\cap\{\rk\le r\} +\] +is the required global downset, using at most two leaves. + +If the edge was same, its low cutoff on \(U\) equals \(P_b\) +minus its unpruned carry union, without the fresh injection. +The old list on \(U\) has at most five entries by +\Cref{lem:list-length}. If its run began at the root, all carry +entries are simple: the root base or injections, possibly rank-trimmed. +If it began at a switch, the only possibly complicated carry +entry is the first of that switch pair. If it is present in the carry, +replace the carried pair by \eqref{eq:flatten}; if it is absent from +the carry, every carried entry is already simple. + +This gives a global upset agreeing with the carry union on \(U\). +It has at most five terms, each with at most four leaves. +Indeed an injection with a rank trim has three leaves; a root +base or a second switch base with a trim has at most two; and +the flattened pair has two. Taking \(P_b\) minus this union +gives a global downset with at most \(1+5\cdot4=21\) leaves. +An empty carry is represented by false. +\end{proof} + +\subsection{Constructing the new switch upset} + +Suppose a switch out of \(W\) produces a nonempty child. +By \Cref{lem:edge-labels}, \(W\) is not the root. +Let \(U\) be the parent that produced \(W\). +That preceding edge was taken in the tag frame of \(W\), which +is opposite to the current working frame. +In the current frame write \(a_-,b_-\) for the boundaries of \(U\) +and \(v_-\) for its center. The temporal arrangement is +\begin{equation} + a_-0\), Levey and Rothvoß +obtained a \((1+\varepsilon)\)-approximation using linear programming +(LP) hierarchies +\citep{LeveyRothvoss2016}; Garg improved the running time to +quasipolynomial \citep{Garg2018}. Li replaced LP conditioning by +combinatorial guesses and obtained the bound +\(n^{O((m^4/\varepsilon^3)(\log\log n)^3)}\) \citep{Li2021}. +Das and Wiese later gave a simpler combinatorial quasipolynomial-time +approximation scheme \citep{DasWiese2022}. +These works give a related perspective on the recursive structure of +nearly optimal schedules. The exact separator and global-description +statements used here are established separately. + +\paragraph{Why a finite search can suffice.} +After adding isolated jobs, we may assume that every slot contains three +jobs. If selected slots are removed from a feasible schedule, the remaining +open gaps can be scheduled independently: reordering the jobs of one gap +does not change their temporal relation to any exterior job. This suggests +a recursive search whose subproblems are the sets of jobs in those gaps. +The obstacle is their number. Naming a gap by its two boundary triples +does not by itself determine its job set, and repeatedly intersecting +descriptions inherited from its ancestors can require an unbounded formula. + +We prove that every feasible instance has a recursive decomposition in +which each gap is described \emph{globally}, as a subset of all jobs, by a +Boolean formula of bounded size. The permitted tests are precedence to or +from a triple of jobs, membership in that triple, and rank thresholds in +a fixed linear extension or its reverse. The bound is independent of the +depth of the decomposition. There are therefore only polynomially many +sets that the algorithm must consider. Compatible selected triples connect +smaller accepted gap sets, and their concatenation gives the schedule. + +The structural proof chooses lexicographically minimal schedules only as +existence witnesses; the algorithm enumerates the descriptions instead of +computing those witnesses. Fix a priority order extending precedence, +and call its current initial segment low and the remaining jobs high. +A selected slot separates this cutoff when every earlier high job is a +precedence predecessor of a job in that slot, and every later low job is +a precedence successor of one. In a normalized interval, we promote +jobs one at a time from high to low and move a separator forward. Each pair +of successive selected slots separates one common cutoff: each low job +in the intervening gap has a predecessor in its left slot, and each high +job has a successor in its right slot. + +The two sides of a center slot are treated by the same construction: +for one side we reverse both time and precedence. To describe a gap +without remembering all its ancestors, we maintain a short list of sets +closed under taking successors in the current direction. A boundary slot limits the number of new list entries that can +survive. A global boundary condition excludes distant jobs, while an +exchange condition keeps that control valid after later reorderings. +At a change of direction, a paired form of the inherited entries lets us +replace their history by a bounded formula. The distinction between +agreement inside a known interval and a description valid on all jobs is +what makes this last step necessary. + +\paragraph{Organization.} +\Cref{sec:preliminaries,sec:algorithm} specify a finite description family +and the uniform algorithm. \Cref{sec:separators} supplies common cutoffs, +and \Cref{sec:construction} uses them to form smaller child intervals. +\Cref{sec:lists} bounds inherited predicate information; +\Cref{sec:boundary} proves the global boundary invariants. +\Cref{sec:descriptions} describes whole intervals and marked +splits by bounded formulas, completing the hierarchy proof. +\Cref{sec:complexity} gives the explicit running-time bound. + +\section{Full schedules, frames, and descriptions}\label{sec:preliminaries} + +We write \(x\prec y\) if there is a directed path of positive length +from \(x\) to \(y\). Transitive reachability can be computed in polynomial +time. All restrictions of the precedence order below mean restrictions of +this transitive order, including when a path in the original graph uses +vertices outside the restricted set. + +\subsection{Padding and interval reorderings} + +If \(n>3T\), reject. Otherwise add \(3T-n\) isolated jobs and let \(J\) +be the resulting set, with \(N=|J|=3T\). +A \emph{full schedule} of a set \(W\subseteq J\) is an ordered sequence +of antichain triples partitioning \(W\), respecting \(\prec\). +Its length is necessarily \(|W|/3\). The empty set has the empty full +schedule. + +\begin{lemma}\label{lem:padding} +The original deadline is feasible if and only if \(J\) has a full schedule. +The padded instance has polynomial size in the stated input encoding. +\end{lemma} +\begin{proof} +A schedule of the original jobs leaves exactly \(3T-n\) vacant positions +among the three machines and \(T\) slots. Fill them with the isolated jobs. +Conversely, delete the isolated jobs from a full schedule. +Since \(T\le n\) and the vertices are explicitly listed, \(N=3T\le3n\) +is polynomially bounded. +\end{proof} + +A schedule's slots will also denote their job triples. An \emph{open +interval} \((a,b)\) consists of all slots strictly between two boundary +slots; we use the same symbol \(W\) for that interval's job set. +Virtual boundaries before the first and after the last slot are allowed. + +\begin{lemma}\label{lem:reorder} +Replacing the schedule inside an open interval by any full schedule of +its job set preserves feasibility of the complete schedule. +\end{lemma} +\begin{proof} +All exterior jobs are either before the entire interval or after it. +Their order relative to every interior job is unchanged. Interior +precedences hold by the assumed feasibility of the replacement. +\end{proof} + +\subsection{The two frames} + +Fix a linear extension of the order on \(J\), with rank +\(\rk:J\to\{1,\ldots,N\}\). A deterministic choice is obtained by +successively removing a minimal job, breaking ties by its identifier. +We also use the reversed order, reversed time, and reversed linear +extension. These are our two \emph{frames}. The two ranks of a job sum +to \(N+1\). Unless a frame is specified otherwise, all time comparisons, +precedence relations, and ranks in an argument belong to its working frame. + +For an actual slot \(a\), identified with its job triple, define global cones +\[ + D_a=\{x\in J:\exists y\in a,\ y\prec x\}, + \qquad + P_a=\{x\in J:\exists y\in a,\ x\prec y\}, + \qquad + \wD a=D_a\cup a,\quad \wP a=P_a\cup a . +\] +At the left sentinel take \(D_a=\wD a=J\) and +\(P_a=\wP a=\varnothing\). At the right sentinel use the opposite +convention. Reversal exchanges \(D\) and \(P\), and exchanges their +weak versions. In a fixed frame the \(D\)-cones and weak \(D\)-cones are +upward closed, whereas the \(P\)-cones and weak \(P\)-cones are downward +closed. + +Cone membership always concerns the whole poset \(J\), even when a formula +is evaluated on an interval. A job that is temporally earlier than a block +need not be a predecessor of that block. + +\subsection{A fixed finite family} + +Let \(\F\) be the family of subsets of \(J\) described by Boolean formulas +with at most \(K=10000\) leaves. Leaves are the following atomic tests or +their negations: +\begin{itemize} + \item true and false; + \item rank thresholds \(\rk\le k\) and \(\rk\ge k\), with + \(0\le k\le N+1\), in either frame; + \item membership in a triple of jobs, or in any of its four cones + \(P,D,\widehat P,\widehat D\), in either frame. +\end{itemize} +Internal nodes are binary conjunctions and disjunctions. Negating a formula +does not increase its number of leaves: apply De Morgan's laws and push +negations to the leaves. Sentinel tests are constants. +We distinguish a \emph{global description} \(W=F\), with \(F\in\F\), +from a \emph{relative description} \(L=W\cap F\). + +\begin{lemma}\label{lem:family} +The family \(\F\) can be generated deterministically in polynomial time +and represented by \(N\)-bit vectors. +\end{lemma} +\begin{proof} +There are \(O(N^3)\) atomic tests. The number of binary tree shapes and +connective assignments with at most \(K\) leaves is a constant depending +only on \(K\). Labelling the leaves therefore gives \(O(N^{3K})\) +formulas, with a constant implicit factor. Compute their truth sets after +precomputing reachability, then deduplicate by comparing the bit vectors +pairwise. This is a deterministic finite procedure with polynomial cost. +\end{proof} + +\section{The algorithm and its structural certificate}\label{sec:algorithm} + +The algorithm accepts or rejects each \(W\in\F\), in increasing order +of cardinality. Write \(\A(W)\) for its acceptance value, and regard +\(\A(U)\) as false when \(U\notin\F\). +Set \(\A(\varnothing)\) to true, and reject sets whose sizes are not +divisible by three. + +For each remaining nonempty \(W\), form states +\[ + (L,Z,R),\qquad W=L\dotcup Z\dotcup R, +\] +where \(Z\) is an antichain triple and \(L=W\cap F\) for some \(F\in\F\). +Keep a state only if the displayed order of its three parts is +\emph{compatible}: no job in a later part precedes a job in an earlier +part. This test uses the transitive order on \(J\). +Declare the state initial when \(\A(L)\) is true and final when +\(\A(R)\) is true. Add an edge +\[ + (L,Z,R)\longrightarrow(L',Z',R') +\] +exactly when +\begin{equation} + L\cup Z\subseteq L', + \qquad + \A\bigl(L'\setminus(L\cup Z)\bigr)=\mathrm{true}. + \label{eq:transition} +\end{equation} +All acceptance values consulted concern smaller sets. +Set \(\A(W)\) to true if an initial state reaches a final state, +allowing a path consisting of a single state. +Return \(\YES\) precisely when \(\A(J)\) is true. + +\begin{proposition}\label{prop:algorithm} +This is a uniform deterministic polynomial-time algorithm. Every set it +accepts has a full schedule, which the algorithm can explicitly return. +With \(K=10000\), its deadline decision and schedule construction take +\(O((L+2)^{150020})\) deterministic multitape steps. +\end{proposition} +\begin{proof} +For soundness, consider a path with triples \(Z_1,\ldots,Z_k\) and +left sets \(L_1,\ldots,L_k\). Put +\[ + A_0=L_1,\qquad + A_i=L_{i+1}\setminus(L_i\cup Z_i)\ (1\le is\) containing such a job, and choose one of them, \(x\). +Every domain predecessor of \(x\) lies strictly before \(s\). +Indeed, that predecessor is low because \(L\) is an ideal. +A predecessor at \(s\) would force \(x\in D_s\). A predecessor +strictly between \(s\) and \(t\) would descend from \(s\), by the +earliest choice of \(t\), and again force \(x\in D_s\). + +Let \(y\) be a high domain job at a slot \(r\) with \(s\le r0pt% + \pgf@arrows@inner@line@width@dep{#2}{#3}% + \else% + \advance\pgf@x by#2\pgflinewidth% + \fi% +} +\def\pgf@arrows@inner@line@width@dep#1#2{% + \pgf@xa\pgflinewidth% + \pgf@xb-.5\pgflinewidth% + \pgf@xb#2\pgf@xb% + \advance\pgf@xa by\pgf@xb% + \pgf@xb\pgfinnerlinewidth% + \pgf@xb-.5\pgf@xb + \advance\pgf@xa by#2\pgf@xb% + \advance\pgf@x by#1\pgf@xa% +} + + +\def\pgfarrowslengthdependent#1#2#3#4{% + #4#1% + \advance#4by#2\pgfarrowlength% + \advance#4by#3\pgflinewidth% +} + + + + +\def\pgf@arrow@fullname#1{pgf@@ar@#1@\the\pgflinewidth @\pgfinnerlinewidth @\ifpgfarrowreversed r\fi @\pgf@arrows@stroke@color @\pgf@arrows@fill@color @\csname pgf@ar@par@#1\endcsname} +\def\pgf@arrow@id@count{0}% + + +\def\pgfarrows@getid#1#2{% + \let\pgf@arrows@late@options\pgfutil@empty% + \csname pgf@ar@defaults@#1\endcsname% + \pgf@arrows@options@scope + #2\relax% + \pgf@arrows@nested@options% + \pgf@arrows@late@options% + \pgfarrows@scale@list% + \expandafter\global\expandafter\let\expandafter\pgf@arrow@id\csname\pgf@arrow@fullname{#1}\endcsname\relax% + \ifx\pgf@arrow@id\relax% + \pgf@arrows@instantiate{#1} + \fi% +} + +\def\pgf@arrows@instantiate#1{% + \pgfutil@tempcnta\pgf@arrow@id@count\relax% + \advance\pgfutil@tempcnta by1\relax% + \xdef\pgf@arrow@id@count{\the\pgfutil@tempcnta}% + \expandafter\global\expandafter\let\csname\pgf@arrow@fullname{#1}\endcsname\pgf@arrow@id@count% + \global\let\pgf@arrow@id\pgf@arrow@id@count% + {% + \pgfclearid% + \csname pgf@ar@setup@#1\endcsname% + \ifpgfarrowreversed% + \pgf@arrows@handle@reverse% + \fi% + \expandafter\global\expandafter\let\csname pgf@ar@saves@\pgf@arrow@id\endcsname\pgf@arrows@saves% + \expandafter\global\expandafter\let\csname pgf@ar@hull@\pgf@arrow@id\endcsname\pgf@arrows@convexhull% + \expandafter\xdef\csname pgf@ar@ends@\pgf@arrow@id\endcsname{{\pgf@arrows@the@tipend}{\pgf@arrows@the@backend}{\pgf@arrows@the@lineend}}% + \expandafter\xdef\csname pgf@ar@visual@\pgf@arrow@id\endcsname{{\pgf@arrows@the@visualtipend}{\pgf@arrows@the@visualbackend}{}}% + \edef\pgf@arrow@code{% + \ifpgfarrowreversed\noexpand\pgftransformxscale{-1}\fi% + \expandafter\noexpand\csname pgf@ar@code@#1\endcsname% + }% + \expandafter\global\expandafter\let\csname pgf@ar@inst@code@\pgf@arrow@id\endcsname\pgf@arrow@code% + \expandafter\ifx\csname pgf@ar@do@cache@#1\endcsname\pgf@truetext% + \pgfsysprotocol@getcurrentprotocol\pgf@arrow@temp% + {% + \pgfinterruptpath% + \let\pgfusepath=\pgf@nousepath@here% + \pgf@relevantforpicturesizefalse% + \pgftransformreset% + \pgfsysprotocol@setcurrentprotocol\pgfutil@empty% + \pgfsysprotocol@bufferedtrue% + \pgfscope% + \pgf@arrow@code% + \endpgfscope% + \pgfsysprotocol@getcurrentprotocol\pgf@@arrow@temp% + \expandafter\gdef\expandafter\pgf@arrow@code\expandafter{\expandafter\pgfsysprotocol@literal\expandafter{\pgf@@arrow@temp}}% + \endpgfinterruptpath% + }% + \pgfsysprotocol@setcurrentprotocol\pgf@arrow@temp% + \fi% + \expandafter\global\expandafter\let\csname pgf@ar@cache@\pgf@arrow@id\endcsname\pgf@arrow@code% + }% +} +\def\pgf@truetext{true} + +\def\pgf@arrows@copy@tipend{\pgf@arrows@the@tipend} +\def\pgf@arrows@copy@backend{\pgf@arrows@the@backend} +\def\pgf@arrows@copy@lineend{\pgf@arrows@the@lineend} +\def\pgf@arrows@zeropt{0pt}% + +\let\pgf@arrows@saves\pgfutil@empty% +\let\pgf@arrows@convexhull\pgfutil@empty% +\let\pgf@arrows@the@tipend\pgf@arrows@zeropt% +\let\pgf@arrows@the@backend\pgf@arrows@zeropt% +\let\pgf@arrows@the@lineend\pgf@arrows@zeropt% +\let\pgf@arrows@the@visualtipend\pgf@arrows@copy@tipend% +\let\pgf@arrows@the@visualbackend\pgf@arrows@copy@backend% + + +\def\pgf@arrows@handle@reverse{% + \pgf@x\pgf@arrows@the@visualtipend\pgf@x-\pgf@x% + \pgf@y\pgf@arrows@the@visualbackend\pgf@y-\pgf@y + \edef\pgf@arrows@the@visualtipend{\the\pgf@y}% + \edef\pgf@arrows@the@visualbackend{\the\pgf@x}% + \pgf@x\pgf@arrows@the@tipend\pgf@x-\pgf@x% + \pgf@y\pgf@arrows@the@backend\pgf@y-\pgf@y + \edef\pgf@arrows@the@tipend{\the\pgf@y}% + \edef\pgf@arrows@the@backend{\the\pgf@x}% + \pgf@x\pgf@arrows@the@lineend\pgf@x-\pgf@x\edef\pgf@arrows@the@lineend{\the\pgf@x}% + \let\pgf@temp\pgf@arrows@convexhull% + \let\pgf@arrows@convexhull\pgfutil@empty% + \let\pgf@arrow@hull@point\pgf@arrow@hull@point@reverse% + \pgf@temp% +} +\def\pgf@arrow@hull@point@reverse#1#2{% + \pgf@x#1% + \pgfarrowshullpoint{-\pgf@x}{#2}% +} + +\def\pgf@nousepath@here#1{\pgferror{The definition of an arrow may not use \string\pgfusepath}} + +\def\pgfarrowssavethe#1{% + \expandafter\expandafter\expandafter\def% + \expandafter\expandafter\expandafter\pgf@arrows@saves% + \expandafter\expandafter\expandafter{\expandafter\pgf@arrows@saves\expandafter#1\the#1\relax} +} + +\def\pgfarrowssave#1{% + \expandafter\expandafter\expandafter\def% + \expandafter\expandafter\expandafter\pgf@arrows@saves% + \expandafter\expandafter\expandafter{\expandafter\pgf@arrows@saves\expandafter\def\expandafter#1\expandafter{#1}} +} + +\def\pgfarrowshullpoint#1#2{% + \pgf@x#1\relax% + \pgf@y#2\relax% + \edef\pgf@arrows@temp{\noexpand\pgf@arrows@convexhull\noexpand\pgf@arrow@hull@point{\the\pgf@x}{\the\pgf@y}}% + \expandafter\expandafter\expandafter\def\expandafter\expandafter\expandafter\pgf@arrows@convexhull\expandafter\expandafter\expandafter{\pgf@arrows@temp}% +} + +\def\pgfarrowsupperhullpoint#1#2{% + \pgf@x#1\relax% + \pgf@y#2\relax% + \ifdim\pgf@y>0pt + \edef\pgf@arrows@temp{\noexpand\pgf@arrows@convexhull\noexpand\pgf@arrow@hull@point{\the\pgf@x}{\the\pgf@y}\ifpgfarrowharpoon\else\noexpand\pgf@arrow@hull@point{\the\pgf@x}{-\the\pgf@y}\fi}% + \else + \edef\pgf@arrows@temp{\noexpand\pgf@arrows@convexhull\noexpand\pgf@arrow@hull@point{\the\pgf@x}{\the\pgf@y}}% + \fi + \expandafter\expandafter\expandafter\def\expandafter\expandafter\expandafter\pgf@arrows@convexhull\expandafter\expandafter\expandafter{\pgf@arrows@temp}% +} + +\def\pgfarrowssettipend#1{\pgf@x#1\edef\pgf@arrows@the@tipend{\the\pgf@x}} +\def\pgfarrowssetbackend#1{\pgf@x#1\edef\pgf@arrows@the@backend{\the\pgf@x}} +\def\pgfarrowssetlineend#1{\pgf@x#1\edef\pgf@arrows@the@lineend{\the\pgf@x}} +\def\pgfarrowssetvisualtipend#1{\pgf@x#1\edef\pgf@arrows@the@visualtipend{\the\pgf@x}} +\def\pgfarrowssetvisualbackend#1{\pgf@x#1\edef\pgf@arrows@the@visualbackend{\the\pgf@x}} + + + +\def\pgf@arrow@compute@shortening#1{% + \let\pgf@arrow@handle\pgf@arrow@compute@shortening@ + \let\pgf@arrow@handle@dot\pgf@arrow@after@line@endtrue% + \pgf@arrow@after@line@endfalse% + \pgf@xa0pt% + \pgf@xb0pt% + #1% +} +\newif\ifpgf@arrow@after@line@end + +\def\pgf@arrow@compute@shortening@#1#2{% + {\pgfarrows@getid{#1}{#2}\ifnum\pgf@arrow@flex@mode>0\relax\global\pgf@precise@shorteningtrue\fi\expandafter}\expandafter\pgf@x\the\pgfarrowsep% + \expandafter\expandafter\expandafter\pgf@arrow@cont@short\csname pgf@ar@ends@\pgf@arrow@id\endcsname% +} +\def\pgf@dot@text{.} + +\def\pgf@arrow@cont@short#1#2#3{% + \advance\pgf@xb by#1% + \pgf@xc#2% + \advance\pgf@xb by-\pgf@xc% + \advance\pgf@xb by\pgf@x% + \ifpgf@arrow@after@line@end% + \advance\pgf@xa by#1% + \advance\pgf@xa by\pgf@x% + \pgf@xc#2% + \advance\pgf@xa by-\pgf@xc% + \else% + \pgf@xa#1% + \advance\pgf@xa by\pgf@x% + \pgf@xc#3% + \advance\pgf@xa by-\pgf@xc% + \fi% +} + + + +\def\pgf@arrow@rigid@arrow#1#2{% + {% + \pgf@xb=#2% + \pgftransformxshift{-\pgf@xb}% + \let\pgf@arrow@handle\pgf@arrow@rigid + \let\pgf@arrow@handle@dot\relax% + #1% + }% +} + + +\def\pgf@arrow@rigid#1#2{% + {% + \pgfarrows@getid{#1}{#2}% + \expandafter\expandafter\expandafter\pgf@arrow@drawer@shift\csname pgf@ar@ends@\pgf@arrow@id\endcsname% + \ifdim\pgfarrows@slant pt=0pt% + \else% + \pgftransformxslant{\pgfarrows@slant}% + \fi% + \ifpgfarrowswap% + \pgftransformyscale{-1}% + \fi% + {% + \csname pgf@ar@saves@\pgf@arrow@id\endcsname% + \pgf@arrows@rigid@hull% + }% + \expandafter}% + \expandafter\pgftransformxshift\expandafter{\the\pgf@xc}% +} + + + + + +\def\pgf@arrow@draw@arrow#1#2{% + {% + \pgf@xb=#2% + \pgftransformxshift{-\pgf@xb}% + \let\pgf@arrow@handle\pgf@arrow@drawer + \let\pgf@arrow@handle@dot\relax% + #1% + }% +} + + +\def\pgf@arrow@drawer#1#2{% + {% + \pgfarrows@getid{#1}{#2}% + \expandafter\expandafter\expandafter\pgf@arrow@drawer@shift\csname pgf@ar@ends@\pgf@arrow@id\endcsname% + \ifdim\pgfarrows@slant pt=0pt% + \else% + \pgftransformxslant{\pgfarrows@slant}% + \fi% + \ifpgfarrowswap% + \pgftransformyscale{-1}% + \fi% + {% + \csname pgf@ar@saves@\pgf@arrow@id\endcsname% + \pgfscope% + \pgf@arrows@color@setup% + \pgflowlevelsynccm\csname pgf@ar@cache@\pgf@arrow@id\endcsname% + \endpgfscope% + \pgf@arrows@rigid@hull% + }% + \expandafter}% + \expandafter\pgftransformxshift\expandafter{\the\pgf@xc}% +} + +\def\pgf@arrows@color@setup@restore@std@colors{% + \pgfutil@namelet{\string\color@pgfstrokecolor}{pgf@strokecolor@global}% + \pgfutil@namelet{\string\color@pgffillcolor}{pgf@fillcolor@global}% +}% + +\def\pgf@arrows@color@setup{% + \ifx\pgf@arrows@stroke@color\pgfutil@empty\else\pgfsetstrokecolor{\pgf@arrows@stroke@color}\fi% + \ifx\pgf@arrows@fill@color\pgfutil@empty% + \ifx\pgf@strokecolor@global\pgf@fillcolor@global + \else + \pgf@arrows@color@setup@restore@std@colors + \pgfsetfillcolor{pgfstrokecolor}% + \fi% + \else% + \pgf@arrows@color@setup@restore@std@colors + \pgfsetfillcolor{\pgf@arrows@fill@color}% + \fi% +} + +\def\pgf@arrow@drawer@shift#1#2#3{% + \pgf@xb#2\pgftransformxshift{-\pgf@xb}% + \pgf@xc#1% + \advance\pgf@xc by\pgfarrowsep% + \advance\pgf@xc by-\pgf@xb% +} + +\def\pgf@arrows@rigid@hull{% + \ifpgf@relevantforpicturesize% + \expandafter\let\expandafter\pgf@temp\csname pgf@ar@hull@\pgf@arrow@id\endcsname% + \ifx\pgf@temp\pgfutil@empty\else% + \let\pgf@arrow@hull@point\pgf@arrow@hull@point@first% + \pgf@temp% + \pgf@arrow@update@bb% + \fi% + \fi +} + + + +\def\pgf@arrow@hull@point@first#1#2{% + \pgf@x#1\pgf@y#2% + \pgf@xa\pgf@pt@aa\pgf@x% + \advance\pgf@xa by\pgf@pt@ba\pgf@y% + \pgf@ya\pgf@pt@ab\pgf@x% + \advance\pgf@ya by\pgf@pt@bb\pgf@y% + \pgf@xb\pgf@xa\pgf@yb\pgf@ya% + \let\pgf@arrow@hull@point\pgf@arrow@hull@point@other% +} +\def\pgf@arrow@hull@point@other#1#2{% + \pgfutil@tempdima#1\pgfutil@tempdimb#2% + \pgf@x\pgf@pt@aa\pgfutil@tempdima% + \advance\pgf@x by\pgf@pt@ba\pgfutil@tempdimb% + \pgf@y\pgf@pt@ab\pgfutil@tempdima% + \advance\pgf@y by\pgf@pt@bb\pgfutil@tempdimb% + \ifdim\pgf@x<\pgf@xa\pgf@xa\pgf@x\else\ifdim\pgf@x>\pgf@xb\pgf@xb\pgf@x\fi\fi% + \ifdim\pgf@y<\pgf@ya\pgf@ya\pgf@y\else\ifdim\pgf@y>\pgf@yb\pgf@yb\pgf@y\fi\fi% +} + +\def\pgf@arrow@update@bb{% + \advance\pgf@xa by\pgf@pt@x% + \advance\pgf@ya by\pgf@pt@y% + \advance\pgf@xb by\pgf@pt@x% + \advance\pgf@yb by\pgf@pt@y% + \ifdim\pgf@picmaxx=-16000pt\relax% + \global\pgf@picminx\pgf@xa% + \global\pgf@picmaxx\pgf@xb% + \global\pgf@picminy\pgf@ya% + \global\pgf@picmaxy\pgf@yb% + \else% + \ifdim\pgf@xa<\pgf@picminx\global\pgf@picminx\pgf@xa\fi% + \ifdim\pgf@ya<\pgf@picminy\global\pgf@picminy\pgf@ya\fi% + \ifdim\pgf@xb>\pgf@picmaxx\global\pgf@picmaxx\pgf@xb\fi% + \ifdim\pgf@yb>\pgf@picmaxy\global\pgf@picmaxy\pgf@yb\fi% + \fi% +} + + + + +\def\pgfsetarrows#1{\pgf@arrows#1\pgf@stop} +\def\pgf@arrows{\pgfutil@ifnextchar[\pgf@arrows@test@opt\pgf@arrows@}% +\def\pgf@arrows@#1-#2\pgf@stop{% + \pgfsetarrowsstart{#1}% + \pgfsetarrowsend{#2}% +} +\def\pgf@arrows@test@opt[{% + \pgfutil@ifnextchar-{\pgf@arrows@[}% + \pgf@arrows@test@opt@% +} +\def\pgf@arrows@test@opt@#1]{% + \pgfutil@ifnextchar\pgf@stop{% + \let\pgf@arrows@options\pgf@arrows@options@scope% + \pgfkeys{/pgf/arrow keys/.cd,#1}% + \let\pgf@arrows@options@scope\pgf@arrows@options% + \pgfutil@gobble% + }{% + \pgf@arrows@[#1]% + }% +} + +\let\pgf@arrows@options@scope\pgfutil@empty + + + + + + + +\pgfdeclarearrow{ + name = ., + drawing code = +} + + + +\pgfdeclarearrow{ + name = _, + defaults = {sep}, + drawing code = +} + + + + +\pgfdeclarearrow{ name = To, means = } % +\pgfdeclarearrow{ name = Bar, means = } % +\pgfkeys{ + <-> /.tip = {To}, + >-< /.tip = {>[reversed]}, + | /.tip = {Bar} +} + + + + +\pgfkeys {} + +\def\pgf@arrows@direct@def#1#2{\expandafter\pgf@arrows@direct@def@\expandafter{\csname pgf@ar@means@#2\endcsname}{#1}} +\def\pgf@arrows@direct@def@#1#2{\expandafter\def\csname pgf@arrows@direct@name@#2\endcsname{\pgf@arrow@handle@shorthand@empty{#1}}} +\pgf@arrows@direct@def{start@[}{]} +\pgf@arrows@direct@def{start@]}{[} +\pgf@arrows@direct@def{start@spaced [}{spaced ]} +\pgf@arrows@direct@def{start@spaced ]}{spaced [} +\pgf@arrows@direct@def{end@[}{[} +\pgf@arrows@direct@def{end@]}{]} +\pgf@arrows@direct@def{end@spaced [}{spaced [} +\pgf@arrows@direct@def{end@spaced ]}{spaced ]} + +\def\pgfsetarrowoptions#1#2{\expandafter\def\csname pgf@arrow@compat@opt@#1\endcsname{#2}} +\def\pgfgetarrowoptions#1{\csname pgf@arrow@compat@opt@#1\endcsname} +\def\pgfarrowsdeclare#1#2#3#4{% + \pgfdeclarearrow{ name ={#1}-{#2}, setup code ={#3}, drawing code={#4}, parameters/.expand once={\csname pgf@arrow@compat@opt@#2\endcsname}}% + \expandafter\def\csname pgf@arrow@compat@opt@#2\endcsname{0}} +\def\pgfarrowsleftextend#1{\pgfmathsetlength\pgf@xa{#1}\pgfarrowssetbackend{\pgf@xa}} +\def\pgfarrowsrightextend#1{\pgfmathsetlength\pgf@xb{#1}\pgfarrowssettipend{\pgf@xb}} +\def\pgfarrowsdeclarealias#1#2#3#4{\pgfdeclarearrow{ name ={#1}-{#2}, means ={#4} } } +\def\pgfarrowsdeclarereversed#1#2#3#4{\pgfdeclarearrow{ name ={#1}-{#2}, means={{#4}[reversed]}}} +\def\pgfarrowsdeclarecombine{\pgfutil@ifnextchar*{\pgfarrowsdeclarecombine@star}{\pgf@@arrowsdeclarecombine}} +\def\pgf@@arrowsdeclarecombine{\pgfutil@ifnextchar[{\pgf@arrowsdeclarecombine}{\pgf@arrowsdeclarecombine[0pt]}} +\def\pgf@arrowsdeclarecombine[#1]#2#3#4#5#6#7{\def\pgf@@temp{#1}\ifx\pgf@@temp\pgf@linewidthtext\def\pgf@@temp{+0pt 1}\fi% + \pgfdeclarearrow{name ={#2}-{#3}, means={#5[sep/.expand once=\pgf@@temp]#6}}} +\def\pgf@linewidth@text{\pgflinewidth} +\def\pgfarrowsdeclarecombine@star*{\pgfutil@ifnextchar[{\pgf@arrowsdeclarecombine@star}{\pgf@arrowsdeclarecombine@star[0pt]}} +\def\pgf@arrowsdeclarecombine@star[#1]#2#3#4#5#6#7{\def\pgf@@temp{#1}\ifx\pgf@@temp\pgf@linewidthtext\def\pgf@@temp{+0pt 1}\fi% + \pgfdeclarearrow{name ={#2}-{#3}, means={#5[sep/.expand once=\pgf@@temp].#6}}} +\def\pgfarrowsdeclaredouble{\pgfutil@ifnextchar[{\pgf@arrowsdeclaredouble}{\pgf@arrowsdeclaredouble[0pt]}} +\def\pgf@arrowsdeclaredouble[#1]#2#3#4#5{\pgfarrowsdeclarecombine[#1]{#2}{#3}{#4}{#5}{#4}{#5}} +\def\pgfarrowsdeclaretriple{\pgfutil@ifnextchar[{\pgf@arrowsdeclaretriple}{\pgf@arrowsdeclaretriple[0pt]}} +\def\pgf@arrowsdeclaretriple[#1]#2#3#4#5{ \pgfarrowsdeclarecombine[#1]{pgf@trip@#2}{pgf@trip@#3}{#4}{#5}{#4}{#5}% + \pgfarrowsdeclarecombine[#1]{#2}{#3}{pgf@trip@#2}{pgf@trip@#3}{#4}{#5}} +\def\pgf@linewidthtext{\pgflinewidth} + +\pgfarrowsdeclare{stealth}{stealth} +{ + \pgfutil@tempdima=0.28pt% + \pgfutil@tempdimb=\pgflinewidth% + \ifdim\pgfinnerlinewidth>0pt% + \pgfmathsetlength\pgfutil@tempdimb{.6\pgflinewidth-.4*\pgfinnerlinewidth}% + \fi% + \advance\pgfutil@tempdima by.3\pgfutil@tempdimb% + \pgfarrowsleftextend{+-3\pgfutil@tempdima} + \pgfarrowsrightextend{+5\pgfutil@tempdima} +} +{ + \pgfutil@tempdima=0.28pt% + \pgfutil@tempdimb=\pgflinewidth% + \ifdim\pgfinnerlinewidth>0pt% + \pgfmathsetlength\pgfutil@tempdimb{.6\pgflinewidth-.4*\pgfinnerlinewidth}% + \fi% + \advance\pgfutil@tempdima by.3\pgfutil@tempdimb% + \pgfpathmoveto{\pgfqpoint{5\pgfutil@tempdima}{0pt}} + \pgfpathlineto{\pgfqpoint{-3\pgfutil@tempdima}{4\pgfutil@tempdima}} + \pgfpathlineto{\pgfpointorigin} + \pgfpathlineto{\pgfqpoint{-3\pgfutil@tempdima}{-4\pgfutil@tempdima}} + \pgfusepathqfill +} +\pgfarrowsdeclarereversed{stealth reversed}{stealth reversed}{stealth}{stealth} +\pgfarrowsdeclare{to}{to} +{ + \pgfutil@tempdima=-0.84pt% + \advance\pgfutil@tempdima by-1.3\pgflinewidth% + \pgfutil@tempdimb=0.21pt% + \advance\pgfutil@tempdimb by.625\pgflinewidth% + \pgfarrowsleftextend{+\pgfutil@tempdima} + \pgfarrowsrightextend{+\pgfutil@tempdimb} +} +{ + \pgfutil@tempdima=0.28pt% + \advance\pgfutil@tempdima by.3\pgflinewidth% + \pgfsetlinewidth{0.8\pgflinewidth} + \pgfsetdash{}{+0pt} + \pgfsetroundcap + \pgfsetroundjoin + \pgfpathmoveto{\pgfqpoint{-3\pgfutil@tempdima}{4\pgfutil@tempdima}} + \pgfpathcurveto + {\pgfqpoint{-2.75\pgfutil@tempdima}{2.5\pgfutil@tempdima}} + {\pgfqpoint{0pt}{0.25\pgfutil@tempdima}} + {\pgfqpoint{0.75\pgfutil@tempdima}{0pt}} + \pgfpathcurveto + {\pgfqpoint{0pt}{-0.25\pgfutil@tempdima}} + {\pgfqpoint{-2.75\pgfutil@tempdima}{-2.5\pgfutil@tempdima}} + {\pgfqpoint{-3\pgfutil@tempdima}{-4\pgfutil@tempdima}} + \pgfusepathqstroke +} +\pgfkeys{To /.tip = to} +\pgfarrowsdeclare{to reversed}{to reversed} +{ + \pgfutil@tempdima=-0.21pt% + \advance\pgfutil@tempdima by-0.475\pgflinewidth% + \pgfutil@tempdimb=0.98pt% + \advance\pgfutil@tempdimb by1.45\pgflinewidth% + \pgfarrowsleftextend{+\pgfutil@tempdima} + \pgfarrowsrightextend{+\pgfutil@tempdimb} +} +{ + \pgfutil@tempdima=0.28pt% + \advance\pgfutil@tempdima by.3\pgflinewidth% + \pgfsetlinewidth{+0.8\pgflinewidth} + \pgfsetdash{}{+0pt} + \pgfsetroundcap + \pgfsetroundjoin + \pgfpathmoveto{\pgfqpoint{3.5\pgfutil@tempdima}{4\pgfutil@tempdima}} + \pgfpathcurveto + {\pgfqpoint{3.25\pgfutil@tempdima}{2.5\pgfutil@tempdima}} + {\pgfqpoint{0.5\pgfutil@tempdima}{0.25\pgfutil@tempdima}} + {\pgfqpoint{-0.25\pgfutil@tempdima}{0\pgfutil@tempdima}} + \pgfpathcurveto + {\pgfqpoint{0.5\pgfutil@tempdima}{-0.25\pgfutil@tempdima}} + {\pgfqpoint{3.25\pgfutil@tempdima}{-2.5\pgfutil@tempdima}} + {\pgfqpoint{3.5\pgfutil@tempdima}{-4\pgfutil@tempdima}} + \pgfusepathqstroke +} +\pgfarrowsdeclare{latex}{latex} +{ + \pgfutil@tempdima=0.28pt% + \pgfutil@tempdimb=\pgflinewidth% + \ifdim\pgfinnerlinewidth>0pt% + \pgfmathsetlength\pgfutil@tempdimb{.6\pgflinewidth-.4*\pgfinnerlinewidth}% + \fi% + \advance\pgfutil@tempdima by.3\pgfutil@tempdimb% + \pgfarrowsleftextend{+-1\pgfutil@tempdima} + \pgfarrowsrightextend{+9\pgfutil@tempdima} +} +{ + \pgfutil@tempdima=0.28pt% + \pgfutil@tempdimb=\pgflinewidth% + \ifdim\pgfinnerlinewidth>0pt% + \pgfmathsetlength\pgfutil@tempdimb{.6\pgflinewidth-.4*\pgfinnerlinewidth}% + \fi% + \advance\pgfutil@tempdima by.3\pgfutil@tempdimb% + \pgfpathmoveto{\pgfqpoint{9\pgfutil@tempdima}{0pt}} + \pgfpathcurveto + {\pgfqpoint{6.3333\pgfutil@tempdima}{.5\pgfutil@tempdima}} + {\pgfqpoint{2\pgfutil@tempdima}{2\pgfutil@tempdima}} + {\pgfqpoint{-1\pgfutil@tempdima}{3.75\pgfutil@tempdima}} + \pgfpathlineto{\pgfqpoint{-1\pgfutil@tempdima}{-3.75\pgfutil@tempdima}} + \pgfpathcurveto + {\pgfqpoint{2\pgfutil@tempdima}{-2\pgfutil@tempdima}} + {\pgfqpoint{6.3333\pgfutil@tempdima}{-.5\pgfutil@tempdima}} + {\pgfqpoint{9\pgfutil@tempdima}{0pt}} + \pgfusepathqfill +} +\pgfarrowsdeclarereversed{latex reversed}{latex reversed}{latex}{latex} +\pgfarrowsdeclare{@bar}{@bar} +{ + \pgfarrowsleftextend{+-0.25\pgflinewidth} + \pgfarrowsrightextend{+.75\pgflinewidth} +} +{ + \pgfutil@tempdima=2pt% + \advance\pgfutil@tempdima by1.5\pgflinewidth% + \pgfsetdash{}{+0pt} + \pgfsetrectcap + \pgfpathmoveto{\pgfqpoint{0.25\pgflinewidth}{-\pgfutil@tempdima}} + \pgfpathlineto{\pgfqpoint{0.25\pgflinewidth}{\pgfutil@tempdima}} + \pgfusepathqstroke +} +\pgfkeys{Bar /.tip = @bar} +\pgfarrowsdeclare{space}{space} +{ + \pgfutil@tempdima=0.88pt% + \advance\pgfutil@tempdima by.3\pgflinewidth% + \pgfarrowsleftextend{0pt} + \pgfarrowsrightextend{\pgfutil@tempdima} +} +{} + + + +\endinput diff --git a/preprints/A-polynomial-time-algorithm-for-three-machine-unit-job-scheduling-September-24-2026/build/tex/runtime/generic/pgf/basiclayer/pgfcoreexternal.code.tex b/preprints/A-polynomial-time-algorithm-for-three-machine-unit-job-scheduling-September-24-2026/build/tex/runtime/generic/pgf/basiclayer/pgfcoreexternal.code.tex new file mode 100644 index 0000000000000000000000000000000000000000..474d7aec96aa1e6bc405af84f51f9110adfdfb60 --- /dev/null +++ b/preprints/A-polynomial-time-algorithm-for-three-machine-unit-job-scheduling-September-24-2026/build/tex/runtime/generic/pgf/basiclayer/pgfcoreexternal.code.tex @@ -0,0 +1,357 @@ + +\ProvidesFileRCS{pgfcoreexternal.code.tex} + + + +\newif\ifpgfexternal@aux@in@dpth +\newif\ifpgfexternal@info + +\pgfkeys{ + /pgf/images/include external/.code={\pgfimage{#1}}, + /pgf/images/aux in dpth/.is if=pgfexternal@aux@in@dpth, + /pgf/images/aux in dpth/.default=true, + /pgf/images/trim external/.code n args={4}{% + \def\pgf@external@kern@left{#1}% + \def\pgf@external@kern@bottom{#2}% + \def\pgf@external@kern@right{#3}% + \def\pgf@external@kern@top{#4}% + \ifx\pgf@external@kern@left\pgfutil@empty\def\pgf@external@kern@left{0sp}\fi% + \ifx\pgf@external@kern@bottom\pgfutil@empty\def\pgf@external@kern@bottom{0sp}\fi% + \ifx\pgf@external@kern@right\pgfutil@empty\def\pgf@external@kern@right{0sp}\fi% + \ifx\pgf@external@kern@top\pgfutil@empty\def\pgf@external@kern@top{0sp}\fi% + }, + /pgf/images/trim external={1truein}{1truein}{1truein}{1truein}, + /pgf/images/external info/.is if=pgfexternal@info, + /pgf/images/external info/.default=true, +} + + + +\newif\ifpgfexternalreadmainaux +\pgfexternalreadmainauxtrue + + +\def\pgfrealjobname#1{% + \global\let\pgfactualjobname=\jobname + \edef\pgf@tempa{\expandafter\string\csname #1\endcsname}% + \edef\pgf@tempb{\expandafter\string\csname\jobname\endcsname}% + \ifx\pgf@tempa\pgf@tempb% + \else% + \pgf@external@grabshipouttrue% + \pgfexternal@nofiles% + \ifpgfexternalreadmainaux + \gdef\jobname{#1}% + \fi + \ifpgfexternal@aux@in@dpth + \csname newwrite\endcsname\w@pgfexternal@auxout + \fi + \pgfexternal@geometry@compatibility + \fi% + \gdef\pgfrealjobname##1{}% +} + +\def\pgfexternal@geometry@compatibility{% + \pgfutil@ifundefined{geometry}{% + }{% + \pgf@typeout{Package TikZ externalization: calling \string\geometry{driver=none} during externalization.^^J}% + \geometry{driver=none}% + }% +}% + +\def\pgfexternal@nofiles{% + \let\pgfexternal@nofiles@=\relax + \def\relax{\relax}% + \csname nofiles\endcsname + \let\relax=\pgfexternal@nofiles@ +}% + +\newif\ifpgf@external@grabshipout +\newbox\pgfexternal@startupbox + +\pgfutil@ifundefined{AtBeginDocument}{}{% + \AtBeginDocument{% + \ifpgf@external@grabshipout% + \global\let\pgfexternal@originalshipout=\shipout% + \global\def\shipout{\setbox\pgfpic=}% + \maxdeadcycles=10000% + {% + \output{\global\setbox\pgfexternal@startupbox=\vbox{\csname @begindvi\endcsname\unvbox255}}% + \hbox{}\eject% + }% + \fi% + } +}% + +\pgfutil@ifundefined{AtEndDocument}{}{% + \AtEndDocument{% + \ifpgf@external@grabshipout + \pgfutil@ifundefined{pgfexternal@did@a@shipout}{% + \pgfexternal@error@no@shipout + }{}% + \fi + }% +} + +\def\pgfexternal@error@no@shipout{% + \pgferror{Sorry, image externalization failed: the resulting image was EMPTY. I tried to externalize '\pgfactualjobname'. Perhaps there was a typo somewhere? Please check that your document contains '\string\beginpgfgraphicnamed{\pgfactualjobname} ... \string\endpgfgraphicnamed'}% +}% + + + +\def\pgf@externalbegin#1{% + \let\pgf@next=\pgf@external@normal% + \ifpgf@external@grabshipout% + \edef\pgf@tempa{\expandafter\string\csname #1\endcsname}% + \edef\pgf@tempb{\expandafter\string\csname\pgfactualjobname\endcsname}% + \ifx\pgf@tempa\pgf@tempb% + \let\pgf@next=\pgf@external@grab% + \fi% + \fi% + \pgf@next{#1}% +} + + +\AtBeginDocument{ + \let\beginpgfgraphicnamed=\pgf@externalbegin% + \let\endpgfgraphicnamed=\unskip +} + + +\def\pgf@external@normal#1{% + \let\pgf@filename=\pgfutil@empty% + \expandafter\pgf@findfile\pgfsys@imagesuffixlist:+{#1}% + \ifx\pgf@filename\pgfutil@empty% + \let\pgf@next=\relax% + \else% + \let\pgf@next=\pgf@replacepicturebygraphic% + \fi% + \pgf@next% +} + +\long\def\pgf@replacepicturebygraphic#1\endpgfgraphicnamed{% + \expandafter\pgfincludeexternalgraphics\expandafter{\pgf@filename}\unskip +} + +\def\pgfincludeexternalgraphics#1{% + \begingroup + \pgfexternalreaddpth{#1}% + \pgfkeysifdefined{/pgf/images/include external/#1/.@cmd}{% + \pgfkeysgetvalue{/pgf/images/include external/#1/.@cmd}\pgf@tempa + \pgfkeyslet{/pgf/images/include external/.@cmd}\pgf@tempa + }{}% + \setbox1=\hbox{\pgfkeysvalueof{/pgf/images/include external/.@cmd}{#1}\pgfeov}% + \ifdim\pgfexternaltrimleft=0sp \else \kern\pgfexternaltrimleft\relax \fi + \ifdim\pgfexternaldepth=0pt + \box1 + \else + \dimen0=\pgfexternaldepth\relax + \hbox{\lower\dimen0 \box1 }% + \fi + \ifdim\pgfexternaltrimright=0sp \else \kern\pgfexternaltrimright\relax \fi + \endgroup +} + +\def\pgfexternalreaddpth#1{% + \edef\pgfexternalreaddpth@restore{% + \noexpand\endlinechar=\the\endlinechar\space + \noexpand\catcode`\noexpand\@=\the\catcode`\@\space + }% + \def\pgfexternaldepth{0sp}% + \def\pgfexternaltrimleft{0sp}% + \def\pgfexternaltrimright{0sp}% + \endlinechar=-1 % + \catcode`\@=11 + \openin\r@pgf@reada=#1.dpth + \pgfincludeexternalgraphics@read@dpth + \pgfexternalreaddpth@restore +}% +\def\pgfincludeexternalgraphics@read@dpth{% + \ifeof\r@pgf@reada + \closein\r@pgf@reada + \else + \read\r@pgf@reada to\pgfincludeexternalgraphics@auxline + \ifx\pgfincludeexternalgraphics@auxline\pgfutil@empty + \else + \expandafter\pgfincludeexternalgraphics@read@dpth@line\pgfincludeexternalgraphics@auxline\pgfincludeexternalgraphics@read@dpth@line@EOI + \fi + \expandafter\pgfincludeexternalgraphics@read@dpth + \fi +}% + +\def\pgfexternal@restore#1{#1}% + +\long\def\pgfincludeexternalgraphics@read@dpth@line@getfirst#1#2\pgfincludeexternalgraphics@read@dpth@line@EOI{% + \def\pgf@first{#1}% +}% +\long\def\pgfincludeexternalgraphics@read@dpth@line#1\pgfincludeexternalgraphics@read@dpth@line@EOI{% + \pgfincludeexternalgraphics@read@dpth@line@getfirst#1\pgfincludeexternalgraphics@read@dpth@line@EOI + \expandafter\ifcat\expandafter\noexpand\pgf@first\relax + \expandafter\ifx\pgf@first\pgfexternal@restore + #1% + \else + \pgfutil@ifundefined{if@filesw}{% + }{% + \if@filesw + {% + \toks0={#1}% + \immediate\write\@auxout{% + \noexpand\def\noexpand\dpthimport{\the\toks0}\noexpand\dpthimport}% + }% + \fi + }% + \fi + \else% + \def\pgfexternaldepth{#1}% + \fi +}% + +\def\pgfexternalstorecommand#1{% +}% +\def\pgfexternalstorecommand@isexporting#1{% + \immediate\write\w@pgfexternal@auxout{\noexpand\pgfexternal@restore{#1}}% +}% + + + +\def\pgf@external@grab#1{% + \def\pgf@filename{#1}% + \ifpgfexternal@aux@in@dpth + \begingroup + \pgf@external@init@aux@in@dpth + \let\G@refundefinedtrue=\pgf@external@grab@refundefinedtrue + \fi + \gdef\pgf@trimleft@final{0sp}% + \gdef\pgf@trimright@final{0sp}% + \setbox\pgfpic=\hbox\bgroup\bgroup% + \let\endpgfgraphicnamed=\pgf@externalend% +} + +{\catcode`\^=12 \gdef\pgfexternal@hat{^}} + +\expandafter\let\expandafter\pgf@external@grab@refundefinedtrue@orig\csname G@refundefinedtrue\endcsname + +\def\pgf@external@grab@refundefinedtrue{% + \pgf@external@grab@refundefinedtrue@orig + \begingroup + \def\n{\pgfexternal@hat\pgfexternal@hat J}% + \pgfexternalstorecommand{% + \pgf@external@grab@refundefinedtrue@code + \noexpand\immediate\noexpand\write16{\pgf@external@grab@refundefinedtrue@warning}% + \noexpand\G@refundefinedtrue + }% + \endgroup +}% + +\def\pgf@external@grab@refundefinedtrue@code{} + +\def\pgf@external@grab@refundefinedtrue@warning{% + LaTeX Warning: External picture `\pgfactualjobname' contains undefined references\noexpand\on@line.\n +}% + +\let\pgf@external@@protected@write@orig=\protected@write +\long\def\pgf@external@@protected@write@immediate#1#2#3{% + \begingroup + \let\pgf@write@=\write + \def\write{\noexpand\immediate\pgf@write@}% + \pgf@external@@protected@write@orig{#1}{#2}{#3}% + \endgroup +}% + +\def\pgf@external@init@aux@in@dpth{% + \let\pgfexternalstorecommand=\pgfexternalstorecommand@isexporting + \csname @fileswtrue\endcsname + \immediate\openout\w@pgfexternal@auxout=\pgf@filename.dpth + \let\@auxout=\w@pgfexternal@auxout + \let\protected@write=\pgf@external@@protected@write@immediate +}% + +\def\pgf@externalend@storeshifts#1{% + \immediate\write#1{\the\dp\pgfpic}% + \ifx\pgf@trimleft@final\pgfutil@empty\else + \immediate\write#1{\noexpand\pgfexternal@restore{\noexpand\def\noexpand\pgfexternaltrimleft{\pgf@trimleft@final}}}% + \fi + \ifx\pgf@trimright@final\pgfutil@empty\else + \immediate\write#1{\noexpand\pgfexternal@restore{\noexpand\def\noexpand\pgfexternaltrimright{\pgf@trimright@final}}}% + \fi + \ifpgfexternal@info + \immediate\write#1{\noexpand\pgfexternal@restore{\noexpand\def\noexpand\pgfexternalwidth{\the\wd\pgfpic}}}% + \immediate\write#1{\noexpand\pgfexternal@restore{\noexpand\def\noexpand\pgfexternalheight{\the\dimen0}}}% + \fi +} +\def\pgf@externalend{% + \unskip\egroup\egroup% + {% + \def\pgf@external@trim{0}% + \def\pgf@external@store@dpth{0}% + \ifpgf@trim@lowlevel + \gdef\pgf@trimleft@final{}% + \gdef\pgf@trimright@final{}% + \else + \ifdim\pgf@trimleft@final=0sp \gdef\pgf@trimleft@final{} \else\def\pgf@external@trim{1}\fi + \ifdim\pgf@trimright@final=0sp \gdef\pgf@trimright@final{} \else\def\pgf@external@trim{1}\fi + \if1\pgf@external@trim + \setbox\pgfpic=\hbox{% + \ifx\pgf@trimleft@final\pgfutil@empty\else\kern-\pgf@trimleft@final\fi + \box\pgfpic + \ifx\pgf@trimright@final\pgfutil@empty\else\kern-\pgf@trimright@final\fi + }% + \def\pgf@external@store@dpth{1}% + \fi + \fi + \parindent0pt % + \hoffset=0pt % + \voffset=0pt + \dimen0\ht\pgfpic% + \advance\dimen0\dp\pgfpic% + \ifdim\dp\pgfpic=0pt\relax + \else% + \def\pgf@external@store@dpth{1}% + \fi + \ifpgfexternal@info + \def\pgf@external@store@dpth{1}% + \fi + \if1\pgf@external@store@dpth + \ifpgfexternal@aux@in@dpth + \pgf@externalend@storeshifts\@auxout + \immediate\closeout\@auxout + \else + \immediate\openout\pgf@plotwrite=\pgf@filename.dpth + \pgf@externalend@storeshifts\pgf@plotwrite + \immediate\closeout\pgf@plotwrite + \fi + \fi + \pgfsys@prepare@papersize{\the\wd\pgfpic}{\the\dimen0}% + \setbox0=\vbox{% + \kern -\pgf@external@kern@top\relax% + \hbox{% + \kern -\pgf@external@kern@left\relax% + \hbox to0pt{% + \wd\pgfexternal@startupbox=0pt % + \ht\pgfexternal@startupbox=0pt % + \dp\pgfexternal@startupbox=0pt % + \box\pgfexternal@startupbox% + \pgfsys@atbegindocument\hss}% + \box\pgfpic% + \kern\pgf@external@kern@right\relax% + }% + \kern\pgf@external@kern@bottom\relax% + }% + \let\pgfexternal@nointerlineskip@orig=\nointerlineskip + \let\nointerlineskip=\pgfexternal@nointerlineskip + \pgfexternal@originalshipout\box0 % + \gdef\pgfexternal@did@a@shipout{1}% + }% + \ifpgfexternal@aux@in@dpth + \endgroup + \fi +} + +\def\pgfexternal@nointerlineskip{% + \ifvmode + \pgfexternal@nointerlineskip@orig + \else + \relax% + \fi +}% + +\endinput diff --git a/preprints/A-polynomial-time-algorithm-for-three-machine-unit-job-scheduling-September-24-2026/build/tex/runtime/generic/pgf/basiclayer/pgfcoregraphicstate.code.tex b/preprints/A-polynomial-time-algorithm-for-three-machine-unit-job-scheduling-September-24-2026/build/tex/runtime/generic/pgf/basiclayer/pgfcoregraphicstate.code.tex new file mode 100644 index 0000000000000000000000000000000000000000..dd4e49d0ee8b3406c37de2b947d125448812b8d7 --- /dev/null +++ b/preprints/A-polynomial-time-algorithm-for-three-machine-unit-job-scheduling-September-24-2026/build/tex/runtime/generic/pgf/basiclayer/pgfcoregraphicstate.code.tex @@ -0,0 +1,162 @@ + +\ProvidesFileRCS{pgfcoregraphicstate.code.tex} + + +\newdimen\pgflinewidth + + + +\def\pgfsetlinewidth#1{% + \pgfmathsetlength\pgflinewidth{#1}% + \global\pgflinewidth=\pgflinewidth% + \pgfsys@setlinewidth{\the\pgflinewidth}% + \ignorespaces} + + + +\def\pgfsetinnerlinewidth#1{% + \pgfmathsetlength\pgf@x{#1}% + \edef\pgfinnerlinewidth{\the\pgf@x}% +} +\def\pgfinnerlinewidth{0pt} + + + +\def\pgfsetinnerstrokecolor#1{\def\pgfinnerstrokecolor{#1}} +\def\pgfinnerstrokecolor{white} + + + +\def\pgfsetmiterlimit#1{% + \ifdim#1pt<1pt\pgferror{miter limit cannot be less than 1}\fi% + \pgfsys@setmiterlimit{#1}% + \ignorespaces} + + + +\def\pgfsetdash#1#2{% + \def\pgf@temp{}% + \def\pgf@next{\pgf@strip}% + \pgf@strip#1{pgf@stop}% + \pgfmathsetlength\pgf@x{#2}% + \pgfsys@setdash{\pgf@temp}{\the\pgf@x}% + \ignorespaces} + +\def\pgf@strip#1{% + \def\pgf@@temp{#1}% + \ifx\pgf@@temp\pgf@stop% + \let\pgf@next=\relax% + \else% + \pgfmathsetlength\pgf@x{#1}% + \ifx\pgf@temp\pgfutil@empty% + \edef\pgf@temp{\the\pgf@x}% + \else% + \edef\pgf@temp{\pgf@temp,\the\pgf@x}% + \fi% + \fi\pgf@next} + +\def\pgf@stop{pgf@stop} + + + +\def\pgfsetstrokecolor#1{% + \pgfutil@colorlet{pgfstrokecolor}{#1}% + \pgfutil@ifundefined{applycolormixins}{}{\applycolormixins{pgfstrokecolor}}% + \expandafter\let\expandafter\pgf@temp\csname\string\color@pgfstrokecolor\endcsname + \global\let\pgf@strokecolor@global=\pgf@temp + \expandafter\pgf@setstrokecolor\pgf@temp% +} +\def\pgf@setstrokecolor#1#2#3#4#5{% + \expandafter\ifx\csname pgfsys@color@#4@stroke\endcsname\relax% + \pgferror{Unsupported color model `#4'. Sorry}% + \else% + \edef\pgf@colmarshal{\expandafter\noexpand\csname pgfsys@color@#4@stroke\endcsname}% + \pgf@uncomma#5,,% + \pgf@colmarshal% + \fi} + +\def\pgf@uncomma#1,{% + \def\pgf@coltest{#1}% + \ifx\pgf@coltest\pgfutil@empty% + \let\pgf@colnext=\relax% + \else% + \expandafter\def\expandafter\pgf@colmarshal\expandafter{\pgf@colmarshal{#1}}% + \let\pgf@colnext=\pgf@uncomma% + \fi% + \pgf@colnext% +} + + + +\def\pgfsetfillcolor#1{% + \pgfutil@colorlet{pgffillcolor}{#1}% + \pgfutil@ifundefined{applycolormixins}{}{\applycolormixins{pgffillcolor}}% + \expandafter\let\expandafter\pgf@temp\csname\string\color@pgffillcolor\endcsname + \global\let\pgf@fillcolor@global=\pgf@temp + \expandafter\pgf@setfillcolor\pgf@temp +} +\def\pgf@setfillcolor#1#2#3#4#5{% + \expandafter\ifx\csname pgfsys@color@#4@fill\endcsname\relax% + \pgferror{Unsupported color model `#4'. Sorry}% + \else% + \edef\pgf@colmarshal{\expandafter\noexpand\csname pgfsys@color@#4@fill\endcsname}% + \pgf@uncomma#5,,% + \pgf@colmarshal% + \fi} + + + + +\def\pgfsetcolor#1{% + \pgfutil@colorlet{pgfstrokecolor}{#1}% + \pgfutil@ifundefined{applycolormixins}{}{\applycolormixins{pgfstrokecolor}}% + \expandafter\let\expandafter\pgf@temp\csname\string\color@pgfstrokecolor\endcsname + \global\let\pgf@strokecolor@global=\pgf@temp + \global\let\pgf@fillcolor@global=\pgf@temp + \expandafter\pgf@setcolor\pgf@temp% +} +\def\pgf@setcolor#1#2#3#4#5{% + \expandafter\ifx\csname pgfsys@color@#4\endcsname\relax% + \pgferror{Unsupported color model `#4'. Sorry}% + \else% + \edef\pgf@colmarshal{\expandafter\noexpand\csname pgfsys@color@#4\endcsname}% + \pgf@uncomma#5,,% + \pgf@colmarshal% + \fi} + +\def\XC@bcolor{\pgfsys@color@reset} + + +\def\pgfsetbuttcap{\pgfsys@buttcap\ignorespaces} + + + +\def\pgfsetroundcap{\pgfsys@roundcap\ignorespaces} + + + +\def\pgfsetrectcap{\pgfsys@rectcap\ignorespaces} + + + +\def\pgfsetmiterjoin{\pgfsys@miterjoin\ignorespaces} + + + +\def\pgfsetbeveljoin{\pgfsys@beveljoin\ignorespaces} + + + +\def\pgfsetroundjoin{\pgfsys@roundjoin\ignorespaces} + + + +\def\pgfseteorule{\pgfsys@eoruletrue\ignorespaces} + + + +\def\pgfsetnonzerorule{\pgfsys@eorulefalse\ignorespaces} + + + +\endinput diff --git a/preprints/A-polynomial-time-algorithm-for-three-machine-unit-job-scheduling-September-24-2026/build/tex/runtime/generic/pgf/basiclayer/pgfcoreimage.code.tex b/preprints/A-polynomial-time-algorithm-for-three-machine-unit-job-scheduling-September-24-2026/build/tex/runtime/generic/pgf/basiclayer/pgfcoreimage.code.tex new file mode 100644 index 0000000000000000000000000000000000000000..28b82798aceeaf9e6843b1cbd26c0c04973e9e25 --- /dev/null +++ b/preprints/A-polynomial-time-algorithm-for-three-machine-unit-job-scheduling-September-24-2026/build/tex/runtime/generic/pgf/basiclayer/pgfcoreimage.code.tex @@ -0,0 +1,150 @@ + +\ProvidesFileRCS{pgfcoreimage.code.tex} + + + + +\def\pgfdeclareimage{\pgfutil@ifnextchar[{\pgf@declareimage}{\pgf@declareimage[]}} +\def\pgf@declareimage[#1]#2#3{% + \pgfqkeys{/pgf/images}{height=,width=,page=,interpolate=false,mask=,#1}% + \gdef\pgf@filename{}% + \ifx\pgf@imagepage\pgfutil@empty% + \else% + \expandafter\pgf@findfile\pgfsys@imagesuffixlist:+{#3.page\pgf@imagepage}% + \fi% + \ifx\pgf@filename\pgfutil@empty% + \expandafter\pgf@findfile\pgfsys@imagesuffixlist:+{#3}% + \else% + \pgfkeys{/pgf/images/page=}% + \fi% + \ifx\pgf@filename\pgfutil@empty% + \pgfwarning% + {File "#3" not found when defining image "#2". + Tried all extensions in "\pgfsys@imagesuffixlist"}% + \pgf@declaredraftimage{#2}{#3}% + \else% + \ifpgf@draftmode% + \pgf@declaredraftimage{#2}{#3}% + \else% + \pgfsys@defineimage% + \fi% + \fi% + \expandafter\global\expandafter\let\csname pgf@image@#2!\endcsname=\pgf@image% +} + + +\pgfkeys{ + /pgf/images/draft/.is if=pgf@draftmode, + /pgf/images/draft/.default=true} +\pgfkeys{/pgf/images/width/.estore in=\pgf@imagewidth} +\pgfkeys{/pgf/images/height/.estore in=\pgf@imageheight} +\pgfkeys{/pgf/images/page/.estore in=\pgf@imagepage} +\pgfkeys{/pgf/images/interpolate/.cd,.code=\edef\pgf@imageinterpolate{#1},.default=true} +\let\pgf@imageinterpolate\pgfutil@empty +\pgfkeys{/pgf/images/mask/.code=% + {% + \edef\pgf@imagemask{#1}% + \ifx\pgf@imagemask\pgfutil@empty% + \else% + \pgfutil@ifundefined{pgf@mask@#1}% + {\pgferror{Undefined mask "#1".}% + \edef\pgf@imagemask{}}% + {\edef\pgf@imagemask{\csname pgf@mask@#1\endcsname}% + }% + \fi% + }} + + +\def\pgf@findfile#1:#2+#3{% + \pgfutil@IfFileExists{#3#1}% + {\xdef\pgf@filename{#3#1}}% + {\def\pgf@mightbeempty{#2}% + \ifx\pgf@mightbeempty\pgfutil@empty\else% + \pgf@findfile#2+{#3}% + \fi}} + +\def\pgf@declaredraftimage#1#2{% + \ifx\pgf@imagewidth\pgfutil@empty% + \pgfwarning{Missing width for image "#1" ("#2") in draft mode. Using 1cm instead}% + \edef\pgf@imagewidth{1cm}% + \fi% + \ifx\pgf@imageheight\pgfutil@empty% + \pgfwarning{Missing height for image "#1" ("#2") in draft mode. Using 1cm instead}% + \edef\pgf@imageheight{1cm}% + \fi% + \ifx\pgf@imagepage\pgfutil@empty\else\edef\pgf@imagepagetext{ page \pgf@imagepage}\fi% + \edef\pgf@image{% + \hbox to \pgf@imagewidth{% + \vrule\kern-0.4pt% + \vbox to \pgf@imageheight{% + \hrule\vfil% + \hbox to \pgf@imagewidth{\hskip-10cm\hfil\noexpand\pgfutil@font@tiny#2\pgf@imagepage\hfil\hskip-10cm}% + \vfil\hrule}% + \kern-0.4pt\vrule}% + }% +} + + + + +\def\pgfdeclaremask{\pgfutil@ifnextchar[\pgf@declaremask{\pgf@declaremask[]}} +\def\pgf@declaremask[#1]#2#3{% + \pgfqkeys{/pgf/images/mask}{matte={1 1 1},#1}% + \def\pgf@mask{}% + \gdef\pgf@filename{}% + \expandafter\pgf@findfile\pgfsys@imagesuffixlist:+{#3}% + \ifx\pgf@filename\pgfutil@empty% + \pgfwarning% + {File "#3" not found when defining mask "#2". + Tried all extensions in "\pgfsys@imagesuffixlist"}% + \else% + \pgfsys@definemask% + \fi% + \expandafter\global\expandafter\let\csname pgf@mask@#2\endcsname=\pgf@mask% +} + +\pgfkeys{/pgf/images/mask/matte/.cd,.estore in=\pgf@maskmatte,.value required} + + + + +\def\pgfaliasimage#1#2{% + \expandafter\global\expandafter\let\expandafter\pgf@temp\expandafter=\csname pgf@image@#2!\endcsname% + \expandafter\global\expandafter\let\csname pgf@image@#1!\endcsname=\pgf@temp% + } + + + +\def\pgfuseimage#1{% + \def\pgf@imagename{pgf@image@#1}% + \pgf@tryextensions{\pgf@imagename}{\pgfalternateextension}% + \expandafter\pgfutil@ifundefined\expandafter{\pgf@imagename}% + {\pgferror{Undefined image "#1"}}% + {{\leavevmode\csname\pgf@imagename\endcsname}}} + +\def\pgf@tryextensions#1#2{% + \edef\pgf@args{[#2!]}\expandafter\pgf@@tryextensions\pgf@args{#1}} +\def\pgf@@tryextensions[#1!#2]#3{% + \expandafter\pgfutil@ifundefined\expandafter{#3#1!#2}% + {\def\pgf@temp{#2}% + \ifx\pgf@temp\pgfutil@empty% + \edef#3{#3!}% + \else% + \pgf@@tryextensions[#2]{#3}% + \fi}% + {\edef#3{#3#1!#2}}} + + + +\def\pgfalternateextension{} + + + + +\def\pgfimage{\pgfutil@ifnextchar[\pgf@imagecom{\pgf@imagecom[]}} +\def\pgf@imagecom[#1]#2{% + \pgfdeclareimage[#1]{pgflastimage}{#2}% + \pgfuseimage{pgflastimage}} + + +\endinput diff --git a/preprints/A-polynomial-time-algorithm-for-three-machine-unit-job-scheduling-September-24-2026/build/tex/runtime/generic/pgf/basiclayer/pgfcorelayers.code.tex b/preprints/A-polynomial-time-algorithm-for-three-machine-unit-job-scheduling-September-24-2026/build/tex/runtime/generic/pgf/basiclayer/pgfcorelayers.code.tex new file mode 100644 index 0000000000000000000000000000000000000000..5ab39e5f739afd3996261b8b63d8a09479a1aa96 --- /dev/null +++ b/preprints/A-polynomial-time-algorithm-for-three-machine-unit-job-scheduling-September-24-2026/build/tex/runtime/generic/pgf/basiclayer/pgfcorelayers.code.tex @@ -0,0 +1,161 @@ + +\ProvidesFileRCS{pgfcorelayers.code.tex} + + + +\def\pgfdeclarelayer#1{% + \pgfutil@ifundefined{pgf@layerbox@#1}{% + \expandafter\expandafter\csname pgf@newbox\endcsname\csname pgf@layerbox@#1\endcsname% + \expandafter\expandafter\csname pgf@newbox\endcsname\csname pgf@layerboxsaved@#1\endcsname% + }{}% +} +\let\pgf@newbox=\newbox % + + +\def\pgfsetlayers#1{% + \let\pgf@layerlist\pgfutil@empty% + \pgf@dosetlayer#1,,\relax% +} + +\def\pgf@dosetlayer#1,#2,\relax{% + \ifx\pgf@layerlist\pgfutil@empty% + \edef\pgf@layerlist{\pgfutil@trimspaces{#1}}% + \else + \edef\pgf@layerlist{\pgf@layerlist,\pgfutil@trimspaces{#1}}% + \fi + \def\pgf@test{#2}% + \ifx\pgf@test\pgfutil@empty% + \else% + \pgf@dosetlayer#2,\relax% + \fi% +} + +\pgfsetlayers{main} + + +\def\pgfonlayer@name{main} + +\def\pgfonlayer#1{% + \pgfutil@ifundefined{pgf@layerbox@#1}{% + \pgferror{Sorry, the requested layer '#1' could not be found. Maybe you misspelled it?}% + \bgroup + \begingroup + }{% + \begingroup + \edef\pgf@temp{#1}% + \ifx\pgf@temp\pgfonlayer@name + \def\pgf@temp{% + \bgroup + \begingroup + }% + \else + \let\pgfonlayer@name=\pgf@temp + \pgfonlayer@assert@is@active + \def\pgf@temp{% + \expandafter\global\expandafter% + \setbox\csname pgf@layerbox@#1\endcsname=\hbox to 0pt% + \bgroup% + \expandafter\box\csname pgf@layerbox@#1\endcsname% + \begingroup% + }% + \fi + \pgfsetlinewidth{0.4pt}% + \pgf@temp + }% +} +\def\endpgfonlayer{% + \endgroup% + \hss + \egroup% + \endgroup +} + +\let\startpgfonlayer=\pgfonlayer +\let\stoppgfonlayer=\endpgfonlayer + +\def\pgfdiscardlayername{discard} + +\def\pgfonlayer@assert@is@active{% + \ifx\pgfonlayer@name\pgfdiscardlayername + \else + \begingroup + \def\pgfonlayer@isactive{0}% + \expandafter\pgf@assert@layer@is@active@loop\pgf@layerlist,,\relax% + \if0\pgfonlayer@isactive + \pgfonlayer@assert@fail + \fi + \endgroup + \fi +}% +\def\pgfonlayer@assert@fail{% + \pgferror{Sorry, the requested layer '\pgfonlayer@name' is not + part of the layer list. Please verify that you provided + \string\pgfsetlayers\space and that '\pgfonlayer@name' is part of this list}% +}% +\def\pgf@assert@layer@is@active@loop#1,#2,\relax{% + \edef\pgf@test{#1}% + \ifx\pgf@test\pgfonlayer@name + \def\pgfonlayer@isactive{1}% + \else + \def\pgf@test{#2}% + \ifx\pgf@test\pgfutil@empty% + \else% + \pgf@assert@layer@is@active@loop#2,\relax% + \fi% + \fi +} + + +\def\pgf@insertlayers{% + \expandafter\pgf@dolayer\pgf@layerlist,,\relax% +} +\def\pgf@maintext{main}% +\def\pgf@dolayer#1,#2,\relax{% + \def\pgf@test{#1}% + \ifx\pgf@test\pgf@maintext% + \box\pgf@layerbox@main% + \else% + \pgfsys@beginscope% + \expandafter\box\csname pgf@layerbox@#1\endcsname% + \pgfsys@endscope% + \fi% + \def\pgf@test{#2}% + \ifx\pgf@test\pgfutil@empty% + \else% + \pgf@dolayer#2,\relax% + \fi% +} + +\def\pgf@savelayers{% + \expandafter\pgf@dosavelayer\pgf@layerlist,,\relax% +} +\def\pgf@dosavelayer#1,#2,\relax{% + \def\pgf@test{#1}% + \ifx\pgf@test\pgf@maintext% + \else% + \setbox\csname pgf@layerboxsaved@#1\endcsname=\box\csname pgf@layerbox@#1\endcsname% + \fi% + \def\pgf@test{#2}% + \ifx\pgf@test\pgfutil@empty% + \else% + \pgf@dosavelayer#2,\relax% + \fi% +} + +\def\pgf@restorelayers{% + \expandafter\pgf@dorestorelayer\pgf@layerlist,,\relax% +} +\def\pgf@dorestorelayer#1,#2,\relax{% + \def\pgf@test{#1}% + \ifx\pgf@test\pgf@maintext% + \else% + \global\setbox\csname pgf@layerbox@#1\endcsname=\box\csname pgf@layerboxsaved@#1\endcsname% + \fi% + \def\pgf@test{#2}% + \ifx\pgf@test\pgfutil@empty% + \else% + \pgf@dorestorelayer#2,\relax% + \fi% +} + +\endinput diff --git a/preprints/A-polynomial-time-algorithm-for-three-machine-unit-job-scheduling-September-24-2026/build/tex/runtime/generic/pgf/basiclayer/pgfcoreobjects.code.tex b/preprints/A-polynomial-time-algorithm-for-three-machine-unit-job-scheduling-September-24-2026/build/tex/runtime/generic/pgf/basiclayer/pgfcoreobjects.code.tex new file mode 100644 index 0000000000000000000000000000000000000000..6cde5bb0e6ffe703f51af07df9a15cf12bb60a8e --- /dev/null +++ b/preprints/A-polynomial-time-algorithm-for-three-machine-unit-job-scheduling-September-24-2026/build/tex/runtime/generic/pgf/basiclayer/pgfcoreobjects.code.tex @@ -0,0 +1,29 @@ + +\ProvidesFileRCS{pgfcoreobjects.code.tex} + + + + + +\def\pgfdefobject#1#2#3#4{% + \pgfinterruptpath% + \pgf@relevantforpicturesizefalse% + \pgfsys@defobject{#1}{#2}{#3}{#4}% + \endpgfinterruptpath% + \ignorespaces% +} + + + +\def\pgfuseobject#1{\pgfsys@beginscope\pgfsys@useobject{#1}{}\pgfsys@endscope} + + + +\def\pgfuseobjectmagnify#1{% + \pgfsys@beginscope% + \pgfsys@useobject{#1}{\pgfsys@transformxyscale{\pgf@sys@tonumber{\pgf@x}}{\pgf@sys@tonumber{\pgf@y}}}% + \pgfsys@endscope% +} + + +\endinput diff --git a/preprints/A-polynomial-time-algorithm-for-three-machine-unit-job-scheduling-September-24-2026/build/tex/runtime/generic/pgf/basiclayer/pgfcorepathconstruct.code.tex b/preprints/A-polynomial-time-algorithm-for-three-machine-unit-job-scheduling-September-24-2026/build/tex/runtime/generic/pgf/basiclayer/pgfcorepathconstruct.code.tex new file mode 100644 index 0000000000000000000000000000000000000000..19c95bbf5019002e72f89b3fac94e6396b60e6fd --- /dev/null +++ b/preprints/A-polynomial-time-algorithm-for-three-machine-unit-job-scheduling-September-24-2026/build/tex/runtime/generic/pgf/basiclayer/pgfcorepathconstruct.code.tex @@ -0,0 +1,997 @@ + +\ProvidesFileRCS{pgfcorepathconstruct.code.tex} + + +\newdimen\pgf@path@lastx +\newdimen\pgf@path@lasty + +\let\pgfgetpath=\pgfsyssoftpath@getcurrentpath +\let\pgfsetpath=\pgfsyssoftpath@setcurrentpath + + + + +\newif\ifpgf@arccorners + +\def\pgfsetcornersarced#1{% + \pgf@process{#1}% + \edef\pgf@corner@arc{{\the\pgf@x}{\the\pgf@y}}% + \pgf@arccornerstrue% + \ifdim\pgf@x=0pt% + \ifdim\pgf@y=0pt\relax% + \pgf@arccornersfalse% + \fi% + \fi% +} + +\def\pgf@roundcornerifneeded{% + \ifpgf@arccorners\expandafter\pgfsyssoftpath@specialround\pgf@corner@arc\fi% +} + + + + +\def\pgf@lt@moveto#1#2{% + \pgf@protocolsizes{#1}{#2}% + \pgfsyssoftpath@moveto{\the#1}{\the#2}% +} +\def\pgf@lt@lineto#1#2{% + \pgf@protocolsizes{#1}{#2}% + \pgfsyssoftpath@lineto{\the#1}{\the#2}% +} +\def\pgf@lt@curveto#1#2#3#4#5#6{% + \pgf@protocolsizes{#1}{#2}% + \pgf@protocolsizes{#3}{#4}% + \pgf@protocolsizes{#5}{#6}% + \pgfsyssoftpath@curveto{\the#1}{\the#2}{\the#3}{\the#4}{\the#5}{\the#6}% +} + +\let\pgf@lt@closepath\pgfsyssoftpath@closepath + +\let\pgf@nlt@moveto\pgf@lt@moveto +\let\pgf@nlt@lineto\pgf@lt@lineto +\let\pgf@nlt@curveto\pgf@lt@curveto +\let\pgf@nlt@closepath\pgf@lt@closepath + +\let\pgf@nlt@list\pgfutil@empty % + + + +\def\pgfpathmoveto#1{% + \pgfpointtransformed{#1}% + \pgf@nlt@moveto{\pgf@x}{\pgf@y}% + \global\pgf@path@lastx=\pgf@x% + \global\pgf@path@lasty=\pgf@y% +} + +\def\pgf@protocolsizes#1#2{% + \ifpgf@relevantforpicturesize% + \ifdim#1<\pgf@picminx\global\pgf@picminx#1\fi% + \ifdim#1>\pgf@picmaxx\global\pgf@picmaxx#1\fi% + \ifdim#2<\pgf@picminy\global\pgf@picminy#2\fi% + \ifdim#2>\pgf@picmaxy\global\pgf@picmaxy#2\fi% + \ifpgf@size@hooked% + \let\pgf@size@hook@x#1\let\pgf@size@hook@y#2\pgf@path@size@hook% + \fi% + \fi% + \ifdim#1<\pgf@pathminx\global\pgf@pathminx#1\fi% + \ifdim#1>\pgf@pathmaxx\global\pgf@pathmaxx#1\fi% + \ifdim#2<\pgf@pathminy\global\pgf@pathminy#2\fi% + \ifdim#2>\pgf@pathmaxy\global\pgf@pathmaxy#2\fi% +} +\newif\ifpgf@size@hooked +\let\pgf@path@size@hook=\pgfutil@empty% + +\def\pgf@resetpathsizes{% + \global\pgf@pathmaxx=-16000pt\relax% + \global\pgf@pathminx=16000pt\relax% + \global\pgf@pathmaxy=-16000pt\relax% + \global\pgf@pathminy=16000pt\relax% +} + +\def\pgf@getpathsizes#1{% + \edef#1{{\the\pgf@pathmaxx}{\the\pgf@pathminx}{\the\pgf@pathmaxy}{\the\pgf@pathminy}}% +} +\def\pgf@setpathsizes#1{% + \expandafter\pgf@@setpathsizes#1% +} +\def\pgf@@setpathsizes#1#2#3#4{% + \global\pgf@pathmaxx=#1\relax% + \global\pgf@pathminx=#2\relax% + \global\pgf@pathmaxy=#3\relax% + \global\pgf@pathminy=#4\relax% +} + + + + + + +\def\pgfpathlineto#1{% + \pgfpointtransformed{#1}% + \pgf@roundcornerifneeded% + \pgf@nlt@lineto{\pgf@x}{\pgf@y}% + \global\pgf@path@lastx=\pgf@x% + \global\pgf@path@lasty=\pgf@y% +} + + + + +\def\pgfpathclose{% + \pgf@roundcornerifneeded% + \pgf@nlt@closepath% +} + + + +\def\pgfpathcurveto#1#2#3{% + \pgfpointtransformed{#3}% + \pgf@xb=\pgf@x% + \pgf@yb=\pgf@y% + \pgfpointtransformed{#2}% + \pgf@xa=\pgf@x% + \pgf@ya=\pgf@y% + \pgfpointtransformed{#1}% + \pgf@roundcornerifneeded% + \pgf@nlt@curveto{\pgf@x}{\pgf@y}{\pgf@xa}{\pgf@ya}{\pgf@xb}{\pgf@yb}% + \global\pgf@path@lastx=\pgf@xb% + \global\pgf@path@lasty=\pgf@yb% +} + + + + +\def\pgfpathquadraticcurveto#1#2{% + \pgfpointtransformed{#2}% + \pgf@xb=\pgf@x% + \pgf@yb=\pgf@y% + \pgfpointtransformed{#1}% + \pgf@xc=.6666666\pgf@x% + \pgf@yc=.6666666\pgf@y% + \pgf@xa=.33333333\pgf@xb% + \pgf@ya=.33333333\pgf@yb% + \advance\pgf@xa by\pgf@xc% + \advance\pgf@ya by\pgf@yc% + \advance\pgf@xc by.3333333\pgf@path@lastx% + \advance\pgf@yc by.3333333\pgf@path@lasty% + \pgf@roundcornerifneeded% + \pgf@nlt@curveto{\pgf@xc}{\pgf@yc}{\pgf@xa}{\pgf@ya}{\pgf@xb}{\pgf@yb}% + \global\pgf@path@lastx=\pgf@xb% + \global\pgf@path@lasty=\pgf@yb% +} + + + + + +\def\pgfpatharc#1#2#3{% + {% + \pgfmathparse{#1}\let\pgf@temp@a=\pgfmathresult% + \pgfmathparse{#2}\let\pgf@temp@b=\pgfmathresult% + \pgfutil@in@{and }{#3}% + \ifpgfutil@in@% + \pgf@arc@get@radii#3\pgf@arc@stop% + \else + \pgf@arc@get@radii#3and #3\pgf@arc@stop% + \fi% + \pgf@arc@local@angle@a=\pgf@temp@a pt% + \pgf@arc@local@angle@b=\pgf@temp@b pt% + \loop% + \pgfutil@tempdima=\pgf@arc@local@angle@a% + \advance\pgfutil@tempdima by-\pgf@arc@local@angle@b\relax% + \ifdim\pgfutil@tempdima<0pt\relax% + \pgfutil@tempdima=-\pgfutil@tempdima\relax% + \fi% + \ifdim\pgfutil@tempdima>90pt\relax% + \ifdim\pgfutil@tempdima>115pt\relax% + \pgf@arc@temp=90pt% + \else% + \pgf@arc@temp=60pt% + \fi% + \ifdim\pgf@arc@local@angle@b>\pgf@arc@local@angle@a\relax% + {% + \pgf@arc@local@angle@b=\pgf@arc@local@angle@a\relax% + \advance\pgf@arc@local@angle@b by\pgf@arc@temp\relax% + \pgf@arc% + } + \advance\pgf@arc@local@angle@a by\pgf@arc@temp\relax% + \else + {% + \pgf@arc@local@angle@b=\pgf@arc@local@angle@a\relax% + \advance\pgf@arc@local@angle@b by-\pgf@arc@temp\relax% + \pgf@arc% + }% + \advance\pgf@arc@local@angle@a by-\pgf@arc@temp\relax% + \fi% + \repeat% + \pgf@roundcornerifneeded% + \pgf@arc% + }% +} +\dimendef\pgf@arc@local@angle@a=0 +\dimendef\pgf@arc@local@angle@b=1 +\dimendef\pgf@arc@temp=2 + +\def\pgf@arc@get@radii#1and #2\pgf@arc@stop{% + \pgfmathparse{#1}\let\pgf@arc@radius@a=\pgfmathresult% + \pgfmathparse{#2}\let\pgf@arc@radius@b=\pgfmathresult% +} + + +\def\pgf@arc{% + {% + \pgfutil@tempdima=\pgf@arc@radius@a pt% + \pgfutil@tempdimb=\pgf@arc@radius@b pt% + \pgf@xa=\pgf@arc@local@angle@a\relax% + \pgf@xb=\pgf@arc@local@angle@b\relax% + \advance\pgf@xb by-\pgf@xa\relax% + \ifdim\pgf@xb<0pt\relax% + \pgf@xb=-\pgf@xb\relax% + \fi% + \ifdim\pgf@xb=90.0pt% + \def\pgfmathresult{0.55228475}% + \else% + \pgfmathparse{1.333333333*tan(.25*\pgf@sys@tonumber{\pgf@xb})}% + \fi% + \pgfutil@tempdima=\pgfmathresult\pgfutil@tempdima% + \pgfutil@tempdimb=\pgfmathresult\pgfutil@tempdimb% + \pgf@xa=\pgf@arc@local@angle@a\relax% + \ifdim\pgf@arc@local@angle@b>\pgf@arc@local@angle@a\relax% + \advance\pgf@xa by 90pt\relax% + \else% + \advance\pgf@xa by -90pt\relax% + \fi% + \edef\pgf@arc@angle{\pgf@sys@tonumber{\pgf@xa}}% + \pgfpointtransformed{\pgfpointpolar{\pgf@arc@angle}{\pgfutil@tempdima and \pgfutil@tempdimb}}% + \advance\pgf@x by-\pgf@pt@x% + \advance\pgf@y by-\pgf@pt@y% + \pgf@xa=\pgf@path@lastx% + \pgf@ya=\pgf@path@lasty% + \advance\pgf@xa by \pgf@x% + \advance\pgf@ya by \pgf@y% + \pgfpointtransformed{\pgfpointpolar{\pgf@sys@tonumber{\pgf@arc@local@angle@a}}{\pgf@arc@radius@a pt and \pgf@arc@radius@b pt}}% + \pgf@xb=\pgf@path@lastx% + \pgf@yb=\pgf@path@lasty% + \advance\pgf@xb by -\pgf@x% + \advance\pgf@yb by -\pgf@y% + \pgfpointtransformed{\pgfpointpolar{\pgf@sys@tonumber{\pgf@arc@local@angle@b}}{\pgf@arc@radius@a pt and \pgf@arc@radius@b pt}}% + \advance\pgf@xb by \pgf@x% + \advance\pgf@yb by \pgf@y% + \ifdim\pgf@arc@local@angle@b>\pgf@arc@local@angle@a\relax% + \advance\pgf@arc@local@angle@b by -90pt\relax% + \else% + \advance\pgf@arc@local@angle@b by 90pt\relax% + \fi% + \pgfpointtransformed{\pgfpointpolar{\pgf@sys@tonumber{\pgf@arc@local@angle@b}}{\pgfutil@tempdima and \pgfutil@tempdimb}}% + \advance\pgf@x by-\pgf@pt@x% + \advance\pgf@y by-\pgf@pt@y% + \pgf@xc=\pgf@xb\relax% + \pgf@yc=\pgf@yb\relax% + \advance \pgf@xc by \pgf@x\relax% + \advance \pgf@yc by \pgf@y\relax% + \pgf@nlt@curveto{\pgf@xa}{\pgf@ya}{\pgf@xc}{\pgf@yc}{\pgf@xb}{\pgf@yb}% + \global\pgf@path@lastx=\pgf@xb% + \global\pgf@path@lasty=\pgf@yb% + }% +} + + + +\def\pgfpatharcaxes#1#2#3#4{% + {% + \pgftransformtriangle{\pgfpointorigin}{#3}{#4}% + \pgfpatharc{#1}{#2}{1pt}% + }% +} + + + + +\def\pgfpatharcto#1#2#3#4#5#6{% + {% + \pgfmathsetmacro\pgf@arcto@rx{abs(#1)}% + \pgfmathsetmacro\pgf@arcto@ry{abs(#2)}% + \ifdim\pgf@arcto@rx pt=0pt% + \gdef\pgf@marshal{\pgfpathlineto{#6}}% + \else + \ifdim\pgf@arcto@ry pt=0pt% + \gdef\pgf@marshal{\pgfpathlineto{#6}}% + \else + \pgfmathsetmacro\pgf@arcto@phi{#3}% + \pgfmathsetmacro\pgf@arcto@fA{#4}% + \ifdim\pgf@arcto@fA pt=0pt + \else + \pgfmathsetmacro\pgf@arcto@fA{1.0} % + \fi + \pgfmathsetmacro\pgf@arcto@fS{#5}% + \ifdim\pgf@arcto@fS pt=0pt + \else + \pgfmathsetmacro\pgf@arcto@fS{1.0} % + \fi + \pgf@process{#6} + \edef\pgf@arcto@xb{\the\pgf@x}% + \edef\pgf@arcto@yb{\the\pgf@y}% + \pgftransforminvert% + \pgf@process{\pgfpointtransformed{\pgfqpoint{\pgf@path@lastx}{\pgf@path@lasty}}} + \edef\pgf@arcto@xa{\the\pgf@x} + \edef\pgf@arcto@ya{\the\pgf@y} + \edef\pgf@temp@a{\pgf@arcto@xa,\pgf@arcto@ya} + \edef\pgf@temp@b{\pgf@arcto@xb,\pgf@arcto@yb} + \ifx\pgf@temp@a\pgf@temp@b% + \global\let\pgf@marshal\pgfutil@empty + \else + \pgftransformreset + \pgftransformrotate{-\pgf@arcto@phi} + \pgf@process{ + \pgfpointtransformed{\pgfpointscale{.5}{\pgfpointdiff + {\pgfqpoint{\pgf@arcto@xb}{\pgf@arcto@yb}} + {\pgfqpoint{\pgf@arcto@xa}{\pgf@arcto@ya}} + } + } + } + \edef\pgf@arcto@xaprime{\pgf@sys@tonumber\pgf@x} + \edef\pgf@arcto@yaprime{\pgf@sys@tonumber\pgf@y} + \pgfmathsetmacro\pgf@arcto@frac@x{\pgf@arcto@xaprime/\pgf@arcto@rx} + \pgfmathsetmacro\pgf@arcto@frac@y{\pgf@arcto@yaprime/\pgf@arcto@ry} + \pgfmathsetmacro\pgf@arcto@lambda{ + \pgf@arcto@frac@x*\pgf@arcto@frac@x+\pgf@arcto@frac@y*\pgf@arcto@frac@y + } + \ifdim\pgf@arcto@lambda pt>1pt% + \pgfmathsetmacro\pgf@arcto@sqrt@lambda{sqrt(\pgf@arcto@lambda)} + \pgfmathsetmacro\pgf@arcto@rx{\pgf@arcto@sqrt@lambda*\pgf@arcto@rx} + \pgfmathsetmacro\pgf@arcto@ry{\pgf@arcto@sqrt@lambda*\pgf@arcto@ry} + \fi + \pgfmathsetmacro\pgf@arcto@xaprime@abs{abs(\pgf@arcto@xaprime)} + \pgfmathsetmacro\pgf@arcto@yaprime@abs{abs(\pgf@arcto@yaprime)} + \pgfmathmax@{\pgf@arcto@rx,\pgf@arcto@ry,\pgf@arcto@xaprime@abs,\pgf@arcto@yaprime@abs} + \pgfmathsetmacro\pgf@arcto@scaling{20/\pgfmathresult} + \pgfmathsetmacro\pgf@arcto@rx@scaled{\pgf@arcto@scaling*\pgf@arcto@rx} + \pgfmathsetmacro\pgf@arcto@ry@scaled{\pgf@arcto@scaling*\pgf@arcto@ry} + \pgfmathsetmacro\pgf@arcto@xaprime@scaled{\pgf@arcto@scaling*\pgf@arcto@xaprime} + \pgfmathsetmacro\pgf@arcto@yaprime@scaled{\pgf@arcto@scaling*\pgf@arcto@yaprime} + \ifdim\pgf@arcto@rx pt>\pgf@arcto@ry pt% + \pgfmathsetmacro\pgf@arcto@rx@over@ry{\pgf@arcto@rx/\pgf@arcto@ry} + \pgfmathsetmacro\pgf@arcto@ry@over@rx{\pgf@arcto@ry/\pgf@arcto@rx} + \pgfmathsetmacro\pgf@arcto@temp{\pgf@arcto@ry@over@rx*\pgf@arcto@xaprime@scaled} + \pgfmathsetmacro\pgf@arcto@numerator{ + \pgf@arcto@ry@scaled*\pgf@arcto@ry@scaled- + \pgf@arcto@yaprime@scaled*\pgf@arcto@yaprime@scaled- + \pgf@arcto@temp*\pgf@arcto@temp + } + \pgfmathsetmacro\pgf@arcto@denominator{ + \pgf@arcto@yaprime@scaled*\pgf@arcto@yaprime@scaled+ + \pgf@arcto@temp*\pgf@arcto@temp + } + \else + \pgfmathsetmacro\pgf@arcto@rx@over@ry{\pgf@arcto@rx/\pgf@arcto@ry} + \pgfmathsetmacro\pgf@arcto@ry@over@rx{\pgf@arcto@ry/\pgf@arcto@rx} + \pgfmathsetmacro\pgf@arcto@temp{\pgf@arcto@rx@over@ry*\pgf@arcto@yaprime@scaled} + \pgfmathsetmacro\pgf@arcto@numerator{ + \pgf@arcto@rx@scaled*\pgf@arcto@rx@scaled- + \pgf@arcto@xaprime@scaled*\pgf@arcto@xaprime@scaled- + \pgf@arcto@temp*\pgf@arcto@temp + } + \pgfmathsetmacro\pgf@arcto@denominator{ + \pgf@arcto@xaprime@scaled*\pgf@arcto@xaprime@scaled+ + \pgf@arcto@temp*\pgf@arcto@temp + } + \fi + \pgfmathsetmacro\pgf@arcto@frac{ + \pgf@arcto@numerator/\pgf@arcto@denominator + } + \ifdim\pgf@arcto@frac pt<0pt + \pgfmathsetmacro\pgf@arcto@factor{0} + \else + \pgfmathsetmacro\pgf@arcto@factor{sqrt(\pgf@arcto@frac)} + \fi + \ifx\pgf@arcto@fA\pgf@arcto@fS + \pgfmathsetmacro\pgf@arcto@factor{-\pgf@arcto@factor} + \fi + \pgfmathsetmacro\pgf@arcto@cxprime{ + \pgf@arcto@factor*\pgf@arcto@rx@over@ry*\pgf@arcto@yaprime + } + \pgfmathsetmacro\pgf@arcto@cyprime{ + -\pgf@arcto@factor*\pgf@arcto@ry@over@rx*\pgf@arcto@xaprime + } + \pgftransformreset + \pgftransformrotate{\pgf@arcto@phi} + \pgf@process{ + \pgfpointtransformed{\pgfqpoint{\pgf@arcto@cxprime pt}{\pgf@arcto@cyprime pt}} + } + \edef\pgf@arcto@temp{\noexpand\pgfqpoint{\the\pgf@x}{\the\pgf@y}} + \pgf@process{\pgfpointadd{\pgf@arcto@temp}{ + \pgfpointscale{.5}{ + \pgfpointadd + {\pgfqpoint{\pgf@arcto@xa}{\pgf@arcto@ya}} + {\pgfqpoint{\pgf@arcto@xb}{\pgf@arcto@yb}} } + } + } + \edef\pgf@arcto@cx{\the\pgf@x} + \edef\pgf@arcto@cy{\the\pgf@y} + \pgfmathsetmacro\pgf@arcto@vec@x{(\pgf@arcto@xaprime-\pgf@arcto@cxprime)/\pgf@arcto@rx} + \pgfmathsetmacro\pgf@arcto@vec@y{(\pgf@arcto@yaprime-\pgf@arcto@cyprime)/\pgf@arcto@ry} + \pgfmathsetmacro\pgf@arcto@denominator{veclen(\pgf@arcto@vec@x,\pgf@arcto@vec@y)} + \pgfmathsetmacro\pgf@arcto@frac{\pgf@arcto@vec@x/\pgf@arcto@denominator} + \pgfmathsetmacro\pgf@arcto@theta@start{acos(\pgf@arcto@frac)} + \ifdim\pgf@arcto@vec@y pt<0pt + \pgfmathsetmacro\pgf@arcto@theta@start{-\pgf@arcto@theta@start} + \fi + \pgfmathsetmacro\pgf@arcto@vec@x{(-\pgf@arcto@xaprime-\pgf@arcto@cxprime)/\pgf@arcto@rx} + \pgfmathsetmacro\pgf@arcto@vec@y{(-\pgf@arcto@yaprime-\pgf@arcto@cyprime)/\pgf@arcto@ry} + \pgfmathsetmacro\pgf@arcto@denominator{veclen(\pgf@arcto@vec@x,\pgf@arcto@vec@y)} + \pgfmathsetmacro\pgf@arcto@frac{\pgf@arcto@vec@x/\pgf@arcto@denominator} + \pgfmathsetmacro\pgf@arcto@theta@end{acos(\pgf@arcto@frac)} + \ifdim\pgf@arcto@vec@y pt<0pt + \pgfmathsetmacro\pgf@arcto@theta@end{-\pgf@arcto@theta@end} + \fi + \pgfmathsetmacro\pgf@arcto@delta@theta{abs(\pgf@arcto@theta@start-\pgf@arcto@theta@end)} + \ifdim\pgf@arcto@fA pt=0pt% + \ifdim\pgf@arcto@delta@theta pt>180pt% + \ifdim\pgf@arcto@theta@end pt>\pgf@arcto@theta@start pt + \pgfmathsetmacro\pgf@arcto@theta@end{\pgf@arcto@theta@end-360} + \else + \pgfmathsetmacro\pgf@arcto@theta@end{\pgf@arcto@theta@end+360} + \fi + \fi + \else + \ifdim\pgf@arcto@delta@theta pt<180pt% + \ifdim\pgf@arcto@theta@end pt>\pgf@arcto@theta@start pt + \pgfmathsetmacro\pgf@arcto@theta@end{\pgf@arcto@theta@end-360} + \else + \pgfmathsetmacro\pgf@arcto@theta@end{\pgf@arcto@theta@end+360} + \fi + \fi + \fi + \xdef\pgf@marshal{\noexpand + \pgfpatharcaxes{\pgf@arcto@theta@start}{\pgf@arcto@theta@end} + {\noexpand\pgfpointpolar{\pgf@arcto@phi}{\pgf@arcto@rx}} + {\noexpand\pgfpointpolar{\pgf@arcto@phi+90}{\pgf@arcto@ry}} + } + \fi\fi\fi + } + \pgf@marshal +} + + + +\def\pgfpatharctomaxstepsize{45} + +\def\pgfpatharctoprecomputed#1#2#3#4#5#6#7#8{% + \begingroup + \pgf@process{#1}% + \edef\pgfpath@center@x{\the\pgf@x}% + \edef\pgfpath@center@y{\the\pgf@y}% + \def\pgfpath@completearcend{#4}% + \pgf@xa=#2pt + \pgf@xb=#3pt + \edef\pgfpath@startangle{#2pt}% + \edef\pgfpath@endangle{\pgf@sys@tonumber\pgf@xb}% + \pgf@ya=\pgf@xb + \advance\pgf@ya by-\pgf@xa + \ifx\pgfpatharctomaxstepsize\pgfutil@empty + \def\pgfpath@N{1}% + \pgf@xc=\pgf@ya + \else + \pgf@xc=\pgf@ya% + \ifdim\pgf@xc<0pt + \multiply\pgf@xc by-1 + \fi + \divide\pgf@xc by\pgfpatharctomaxstepsize\relax + \afterassignment\pgfutil@gobble@until@relax + \c@pgf@counta=\the\pgf@xc\relax + \advance\c@pgf@counta by1 + \edef\pgfpath@N{\the\c@pgf@counta}% + \pgf@xc=\pgf@ya + \divide\pgf@xc by\c@pgf@counta + \fi + \edef\pgfpath@h{\pgf@sys@tonumber\pgf@xc}% + \pgf@xc=0.116355283466289\pgf@xc % + \divide\pgf@xc by20 + \pgf@xa=#7\pgf@xc + \edef\pgfpath@scale@A{\pgf@sys@tonumber\pgf@xa}% + \pgf@xa=#8\pgf@xc + \edef\pgfpath@scale@B{\pgf@sys@tonumber\pgf@xa}% + \c@pgf@countd=1 + \pgfutil@loop + \ifnum\c@pgf@countd<\pgfpath@N\relax + \pgf@xa=\pgfpath@startangle % + \pgf@xb=\pgfpath@h pt + \multiply\pgf@xb by\c@pgf@countd + \advance\pgf@xa by\pgf@xb + \edef\pgfpath@angle@i{\pgf@sys@tonumber\pgf@xa}% + \pgfpatharcofellipse@{% + \pgfpoint + {\pgfpath@center@x + #5*cos(\pgfpath@angle@i)} + {\pgfpath@center@y + #6*sin(\pgfpath@angle@i)}% + }% + \advance\c@pgf@countd by1 + \pgfutil@repeat + \pgfpatharcofellipse@{\pgfpath@completearcend}% + \endgroup +}% +\def\pgfpatharcofellipse@#1{% + \begingroup + \pgf@process{#1}% + \edef\pgfpath@endpt{\global\pgf@x=\the\pgf@x\space\global\pgf@y=\the\pgf@y\space}% + \pgfpathcurveto{ + \begingroup + \global\pgf@x=\pgf@path@lastx + \global\pgf@y=\pgf@path@lasty + \pgf@xa=\pgf@x \advance\pgf@xa by-\pgfpath@center@x + \pgf@ya=\pgf@y \advance\pgf@ya by-\pgfpath@center@y + \global\advance\pgf@x by-\pgfpath@scale@A\pgf@ya + \global\advance\pgf@y by \pgfpath@scale@B\pgf@xa + \endgroup + }{% + \begingroup + \pgfpath@endpt + \pgf@xa=\pgf@x \advance\pgf@xa by-\pgfpath@center@x + \pgf@ya=\pgf@y \advance\pgf@ya by-\pgfpath@center@y + \global\advance\pgf@x by \pgfpath@scale@A\pgf@ya + \global\advance\pgf@y by-\pgfpath@scale@B\pgf@xa + \endgroup + }{% + \pgfpath@endpt + }% + \endgroup +} + + + + + + + + + + +\def\pgfpathellipse#1#2#3{% + \pgfpointtransformed{#1}% + \pgf@xc=\pgf@x% + \pgf@yc=\pgf@y% + \pgfpointtransformed{#2}% + \pgf@xa=\pgf@x% + \pgf@ya=\pgf@y% + \advance\pgf@xa by-\pgf@pt@x% + \advance\pgf@ya by-\pgf@pt@y% + \pgfpointtransformed{#3}% + \pgf@xb=\pgf@x% + \pgf@yb=\pgf@y% + \advance\pgf@xb by-\pgf@pt@x% + \advance\pgf@yb by-\pgf@pt@y% + {% + \advance\pgf@xa by\pgf@xc% + \advance\pgf@ya by\pgf@yc% + \pgf@nlt@moveto{\pgf@xa}{\pgf@ya}% + }% + \pgf@x=0.55228475\pgf@xb% + \pgf@y=0.55228475\pgf@yb% + \advance\pgf@x by\pgf@xa% + \advance\pgf@y by\pgf@ya% + \advance\pgf@x by\pgf@xc% + \advance\pgf@y by\pgf@yc% + \edef\pgf@temp{\pgf@xc\the\pgf@x\pgf@yc\the\pgf@y}% + \pgf@x=0.55228475\pgf@xa% + \pgf@y=0.55228475\pgf@ya% + \advance\pgf@x by\pgf@xb% + \advance\pgf@y by\pgf@yb% + {% + \advance\pgf@x by\pgf@xc% + \advance\pgf@y by\pgf@yc% + \advance\pgf@xb by\pgf@xc% + \advance\pgf@yb by\pgf@yc% + \pgf@temp% + \pgf@nlt@curveto{\pgf@xc}{\pgf@yc}{\pgf@x}{\pgf@y}{\pgf@xb}{\pgf@yb}% + }% + \pgf@xa=-\pgf@xa% + \pgf@ya=-\pgf@ya% + \pgf@x=0.55228475\pgf@xa% + \pgf@y=0.55228475\pgf@ya% + \advance\pgf@x by\pgf@xb% + \advance\pgf@y by\pgf@yb% + \advance\pgf@x by\pgf@xc% + \advance\pgf@y by\pgf@yc% + \edef\pgf@temp{\pgf@xc\the\pgf@x\pgf@yc\the\pgf@y}% + \pgf@x=0.55228475\pgf@xb% + \pgf@y=0.55228475\pgf@yb% + \advance\pgf@x by\pgf@xa% + \advance\pgf@y by\pgf@ya% + {% + \advance\pgf@x by\pgf@xc% + \advance\pgf@y by\pgf@yc% + \advance\pgf@xa by\pgf@xc% + \advance\pgf@ya by\pgf@yc% + \pgf@temp% + \pgf@nlt@curveto{\pgf@xc}{\pgf@yc}{\pgf@x}{\pgf@y}{\pgf@xa}{\pgf@ya}% + }% + \pgf@xb=-\pgf@xb% + \pgf@yb=-\pgf@yb% + \pgf@x=0.55228475\pgf@xb% + \pgf@y=0.55228475\pgf@yb% + \advance\pgf@x by\pgf@xa% + \advance\pgf@y by\pgf@ya% + \advance\pgf@x by\pgf@xc% + \advance\pgf@y by\pgf@yc% + \edef\pgf@temp{\pgf@xc\the\pgf@x\pgf@yc\the\pgf@y}% + \pgf@x=0.55228475\pgf@xa% + \pgf@y=0.55228475\pgf@ya% + \advance\pgf@x by\pgf@xb% + \advance\pgf@y by\pgf@yb% + {% + \advance\pgf@x by\pgf@xc% + \advance\pgf@y by\pgf@yc% + \advance\pgf@xb by\pgf@xc% + \advance\pgf@yb by\pgf@yc% + \pgf@temp% + \pgf@nlt@curveto{\pgf@xc}{\pgf@yc}{\pgf@x}{\pgf@y}{\pgf@xb}{\pgf@yb}% + }% + \pgf@xa=-\pgf@xa% + \pgf@ya=-\pgf@ya% + \pgf@x=0.55228475\pgf@xa% + \pgf@y=0.55228475\pgf@ya% + \advance\pgf@x by\pgf@xb% + \advance\pgf@y by\pgf@yb% + \advance\pgf@x by\pgf@xc% + \advance\pgf@y by\pgf@yc% + \edef\pgf@temp{\pgf@xc\the\pgf@x\pgf@yc\the\pgf@y}% + \pgf@x=0.55228475\pgf@xb% + \pgf@y=0.55228475\pgf@yb% + \advance\pgf@x by\pgf@xa% + \advance\pgf@y by\pgf@ya% + {% + \advance\pgf@x by\pgf@xc% + \advance\pgf@y by\pgf@yc% + \advance\pgf@xa by\pgf@xc% + \advance\pgf@ya by\pgf@yc% + \pgf@temp% + \pgf@nlt@curveto{\pgf@xc}{\pgf@yc}{\pgf@x}{\pgf@y}{\pgf@xa}{\pgf@ya}% + }% + \pgf@nlt@closepath% + \pgf@nlt@moveto{\pgf@xc}{\pgf@yc}% +} + + + + +\def\pgfpathcircle#1#2{\pgfpathellipse{#1}{\pgfpoint{#2}{0pt}}{\pgfpoint{0pt}{#2}}} + + + + + +\def\pgfpathrectangle{% + \let\pgfrect@next=\pgf@specialrect% + \ifpgf@pt@identity% + \ifpgf@arccorners% + \else% + \ifx\pgf@nlt@list\pgfutil@empty% + \let\pgfrect@next=\pgf@normalrect% + \fi% + \fi% + \fi% + \pgfrect@next% +} + +\def\pgf@normalrect#1#2{% + \pgf@process{#2}% + \pgf@xa=\pgf@x% + \pgf@ya=\pgf@y% + \pgfpointtransformed{#1}% + \pgfsyssoftpath@rect{\the\pgf@x}{\the\pgf@y}{\the\pgf@xa}{\the\pgf@ya}% + \pgf@protocolsizes{\pgf@x}{\pgf@y}% + \advance\pgf@x by\pgf@xa\relax% + \advance\pgf@y by\pgf@ya\relax% + \pgf@protocolsizes{\pgf@x}{\pgf@y}% +} + +\def\pgf@specialrect#1#2{% + \pgf@process{#2}% + \pgf@xa=\pgf@x% + \pgf@ya=\pgf@y% + \pgf@process{#1}% + \pgf@xb=\pgf@x% + \pgf@yb=\pgf@y% + \advance\pgf@xa by\pgf@xb% + \advance\pgf@ya by\pgf@yb% + \pgfpathmoveto{\pgfqpoint{\pgf@xa}{\pgf@ya}}% + \pgfpathlineto{\pgfqpoint{\pgf@xb}{\pgf@ya}}% + \pgfpathlineto{\pgfqpoint{\pgf@xb}{\pgf@yb}}% + \pgfpathlineto{\pgfqpoint{\pgf@xa}{\pgf@yb}}% + \pgfpathclose% + \pgfpathmoveto{\pgfqpoint{\pgf@xb}{\pgf@yb}}% +} + + +\def\pgfpathrectanglecorners#1#2{% + \pgf@process{#2}% + \pgf@xc=\pgf@x% + \pgf@yc=\pgf@y% + \pgf@process{#1}% + \advance\pgf@xc by-\pgf@x% + \advance\pgf@yc by-\pgf@y% + \pgfpathrectangle{#1}{\pgfqpoint{\pgf@xc}{\pgf@yc}}% +} + + + +\pgfkeys{ + /pgf/stepx/.initial=1cm, + /pgf/stepy/.initial=1cm, + /pgf/step/.code={\pgf@process{#1}\pgfkeysalso{/pgf/stepx/.expanded=\the\pgf@x,/pgf/stepy/.expanded=\the\pgf@y}}, + /pgf/step/.value required +} + +\def\pgfpathgrid{\pgfutil@ifnextchar[{\pgf@pathgrid}{\pgf@pathgrid[]}} +\def\pgf@pathgrid[#1]#2#3{% + \pgfset{#1}% + \pgfmathsetlength\pgf@xc{\pgfkeysvalueof{/pgf/stepx}}% + \pgfmathsetlength\pgf@yc{\pgfkeysvalueof{/pgf/stepy}}% + \pgf@process{#3}% + \pgf@xb=\pgf@x% + \pgf@yb=\pgf@y% + \pgf@process{#2}% + \pgf@xa=\pgf@x% + \pgf@ya=\pgf@y% + \ifdim\pgf@xa>\pgf@xb% + \pgf@x=\pgf@xb% + \pgf@xb=\pgf@xa% + \pgf@xa=\pgf@x% + \fi% + \ifdim\pgf@ya>\pgf@yb% + \pgf@y=\pgf@yb% + \pgf@yb=\pgf@ya% + \pgf@ya=\pgf@y% + \fi% + \ifdim \pgf@yc > .01pt\relax% + \c@pgf@counta=\pgf@ya\relax% + \c@pgf@countb=\pgf@yc\relax% + \divide\c@pgf@counta by\c@pgf@countb\relax% + \pgfutil@tempdima=\c@pgf@counta\pgf@yc\relax% + \ifdim\pgfutil@tempdima<\pgf@ya% + \advance\pgfutil@tempdima by\pgf@yc% + \fi% + \pgfutil@tempdimb\pgf@x + \pgfutil@loop% + {% + \pgf@xa=\pgfutil@tempdimb% + \pgf@ya=\pgfutil@tempdima% + \pgf@pos@transform{\pgf@xa}{\pgf@ya} + \pgf@nlt@moveto{\pgf@xa}{\pgf@ya}% + \pgf@xa=\pgf@xb% + \pgf@ya=\pgfutil@tempdima% + \pgf@pos@transform{\pgf@xa}{\pgf@ya} + \pgf@nlt@lineto{\pgf@xa}{\pgf@ya}% + }% + \advance\pgfutil@tempdima by\pgf@yc% + \ifdim\pgfutil@tempdima<\pgf@yb% + \pgfutil@repeat% + \advance\pgfutil@tempdima by-0.01pt\relax% + \ifdim\pgfutil@tempdima<\pgf@yb% + {% + \pgf@xa=\pgfutil@tempdimb% + \pgf@ya=\pgfutil@tempdima% + \pgf@pos@transform{\pgf@xa}{\pgf@ya} + \pgf@nlt@moveto{\pgf@xa}{\pgf@ya}% + \pgf@xa=\pgf@xb% + \pgf@ya=\pgfutil@tempdima% + \pgf@pos@transform{\pgf@xa}{\pgf@ya} + \pgf@nlt@lineto{\pgf@xa}{\pgf@ya}% + }% + \fi% + \fi% + \ifdim \pgf@xc > .01pt\relax% + \c@pgf@counta=\pgf@xa\relax% + \c@pgf@countb=\pgf@xc\relax% + \divide\c@pgf@counta by\c@pgf@countb\relax% + \pgfutil@tempdimb=\c@pgf@counta\pgf@xc\relax% + \ifdim\pgfutil@tempdimb<\pgf@xa% + \advance\pgfutil@tempdimb by\pgf@xc% + \fi% + \pgfutil@loop% + {% + \pgf@xc=\pgfutil@tempdimb% + \pgf@yc=\pgf@ya% + \pgf@pos@transform{\pgf@xc}{\pgf@yc} + \pgf@nlt@moveto{\pgf@xc}{\pgf@yc}% + \pgf@xc=\pgfutil@tempdimb% + \pgf@yc=\pgf@yb% + \pgf@pos@transform{\pgf@xc}{\pgf@yc} + \pgf@nlt@lineto{\pgf@xc}{\pgf@yc}% + }% + \advance\pgfutil@tempdimb by\pgf@xc% + \ifdim\pgfutil@tempdimb<\pgf@xb% + \pgfutil@repeat% + \advance\pgfutil@tempdimb by-0.01pt\relax% + \ifdim\pgfutil@tempdimb<\pgf@xb% + {% + \pgf@xc=\pgfutil@tempdimb% + \pgf@yc=\pgf@ya% + \pgf@pos@transform{\pgf@xc}{\pgf@yc} + \pgf@nlt@moveto{\pgf@xc}{\pgf@yc}% + \pgf@xc=\pgfutil@tempdimb% + \pgf@yc=\pgf@yb% + \pgf@pos@transform{\pgf@xc}{\pgf@yc} + \pgf@nlt@lineto{\pgf@xc}{\pgf@yc}% + }% + \fi% + \fi% + \pgf@process{#3}% + \pgf@pos@transform{\pgf@x}{\pgf@y}% + \pgf@nlt@moveto{\pgf@x}{\pgf@y}% +} + + + + +\def\pgfpathparabola#1#2{% + {% + \pgf@process{#2}% + \pgf@xb=\pgf@x% + \pgf@yb=\pgf@y% + \pgf@process{#1}% + \pgf@xc=\pgf@x% + \pgf@yc=\pgf@y% + \pgfutil@tempswatrue% + \ifdim\pgf@xb=0pt\relax% + \ifdim\pgf@yb=0pt\relax% + \pgfutil@tempswafalse% + \fi% + \fi% + {% + \ifpgfutil@tempswa% + \pgf@arccornersfalse + \else% + \fi% + \pgfutil@tempswatrue% + \ifdim\pgf@xc=0pt\relax% + \ifdim\pgf@yc=0pt\relax% + \pgfutil@tempswafalse% + \fi% + \fi% + \ifpgfutil@tempswa + {% + \pgf@pt@x=\pgf@path@lastx% + \pgf@pt@y=\pgf@path@lasty% + \pgfpathcurveto% + {\pgfqpoint{.1125\pgf@xc}{.225\pgf@yc}}% + {\pgfqpoint{.5\pgf@xc}{\pgf@yc}}% + {\pgfqpoint{\pgf@xc}{\pgf@yc}}% + }% + \fi% + }% + \ifpgfutil@tempswa% + \pgf@xc=\pgf@xb% + \pgf@yc=\pgf@yb% + {% + \pgf@pt@x=\pgf@path@lastx% + \pgf@pt@y=\pgf@path@lasty% + \pgfpathcurveto% + {\pgfqpoint{.5\pgf@xc}{0\pgf@yc}}% + {\pgfqpoint{.8875\pgf@xc}{.775\pgf@yc}}% + {\pgfqpoint{\pgf@xc}{\pgf@yc}}% + }% + \fi% + }% +} + + + + + +\def\pgfpathsine#1{% + {% + \pgf@process{#1}% + \pgf@xc=\pgf@x% + \pgf@yc=\pgf@y% + \pgf@pt@x=\pgf@path@lastx% + \pgf@pt@y=\pgf@path@lasty% + \pgfpathcurveto% + {\pgfqpoint{.3260\pgf@xc}{.5120\pgf@yc}}% + {\pgfqpoint{.6380\pgf@xc}{\pgf@yc}}% + {\pgfqpoint{\pgf@xc}{\pgf@yc}}% + }% +} + + +\def\pgfpathcosine#1{% + {% + \pgf@process{#1}% + \pgf@xc=\pgf@x% + \pgf@yc=\pgf@y% + \pgf@pt@x=\pgf@path@lastx% + \pgf@pt@y=\pgf@path@lasty% + \pgfpathcurveto% + {\pgfqpoint{.3620\pgf@xc}{0pt}}% + {\pgfqpoint{.6740\pgf@xc}{.4880\pgf@yc}}% + {\pgfqpoint{\pgf@xc}{\pgf@yc}}% + }% +} + + + +\def\pgfpathcurvebetweentime{\pgf@ignoremovetofalse\pgf@@pathcurvebetweentime} +\def\pgfpathcurvebetweentimecontinue{\pgf@ignoremovetotrue\pgf@@pathcurvebetweentime} +\newif\ifpgf@ignoremoveto + +\def\pgf@@pathcurvebetweentime#1#2#3#4#5#6{% + \pgfmathparse{#1}% + \let\pgf@time@s=\pgfmathresult% + \pgfmathparse{#2}% + \let\pgf@time@t=\pgfmathresult% + \ifdim\pgf@time@s pt>\pgf@time@t pt\relax% + \pgfmathsetmacro\pgf@time@s{1-#1}% + \pgfmathsetmacro\pgf@time@t{1-#2}% + \pgf@@@pathcurvebetweentime{\pgf@time@t}{#6}{#5}{#4}{#3}% + \else% + \pgf@@@pathcurvebetweentime{\pgf@time@t}{#3}{#4}{#5}{#6}% + \fi% +} + +\def\pgf@@@pathcurvebetweentime#1#2#3#4#5{% + \pgf@process{#2}% + \pgf@xc=\pgf@x% + \pgf@yc=\pgf@y% + \pgf@process{% + \pgf@process{#3}% + \pgf@xa=#1\pgf@x% + \pgf@ya=#1\pgf@y% + \pgf@process{#2}% + \pgf@xb=\pgf@x% + \pgf@yb=\pgf@y% + \advance\pgf@x by-#1\pgf@xb% + \advance\pgf@y by-#1\pgf@yb% + \advance\pgf@x by\pgf@xa% + \advance\pgf@y by\pgf@ya% + }% + \pgf@xb=\pgf@x% + \pgf@yb=\pgf@y% + \pgf@process{% + \pgf@process{#4}% + \pgf@xa=#1\pgf@x% + \pgf@ya=#1\pgf@y% + \pgf@process{#3}% + \pgf@xc=\pgf@x% + \pgf@yc=\pgf@y% + \advance\pgf@xc by-#1\pgf@x% + \advance\pgf@yc by-#1\pgf@y% + \pgf@x=\pgf@xb% + \pgf@y=\pgf@yb% + \advance\pgf@x by#1\pgf@xa% + \advance\pgf@y by#1\pgf@ya% + \advance\pgf@x by-#1\pgf@xb% + \advance\pgf@y by-#1\pgf@yb% + \advance\pgf@x by#1\pgf@xc% + \advance\pgf@y by#1\pgf@yc% + }% + \pgf@xa=\pgf@x% + \pgf@ya=\pgf@y% + \pgf@process{\pgfpointcurveattime{#1}{#2}{#3}{#4}{#5}}% + \ifx#1\pgf@time@t% + \pgfmathdivide@{\pgf@time@s}{\pgf@time@t}% + \pgfmathadd@{-\pgfmathresult}{1}% + \let\pgf@time@s=\pgfmathresult% + \edef\pgf@marshal{% + \noexpand\pgf@@@pathcurvebetweentime{\noexpand\pgf@time@s}% + {\noexpand\pgfqpoint{\the\pgf@x}{\the\pgf@y}}{\noexpand\pgfqpoint{\the\pgf@xa}{\the\pgf@ya}}% + {\noexpand\pgfqpoint{\the\pgf@xb}{\the\pgf@yb}}{\noexpand\pgfqpoint{\the\pgf@xc}{\the\pgf@yc}}% + }% + \else% + \ifpgf@ignoremoveto% + \edef\pgf@marshal{% + \noexpand\pgfpathcurveto{\noexpand\pgfqpoint{\the\pgf@xa}{\the\pgf@ya}}% + {\noexpand\pgfqpoint{\the\pgf@xb}{\the\pgf@yb}}{\noexpand\pgfqpoint{\the\pgf@xc}{\the\pgf@yc}}% + }% + \else% + \edef\pgf@marshal{% + \noexpand\pgfpathmoveto{\noexpand\pgfqpoint{\the\pgf@x}{\the\pgf@y}}% + \noexpand\pgfpathcurveto{\noexpand\pgfqpoint{\the\pgf@xa}{\the\pgf@ya}}% + {\noexpand\pgfqpoint{\the\pgf@xb}{\the\pgf@yb}}{\noexpand\pgfqpoint{\the\pgf@xc}{\the\pgf@yc}}% + }% + \fi% + \fi% + \pgf@marshal% +} + + +\endinput diff --git a/preprints/A-polynomial-time-algorithm-for-three-machine-unit-job-scheduling-September-24-2026/build/tex/runtime/generic/pgf/basiclayer/pgfcorepathprocessing.code.tex b/preprints/A-polynomial-time-algorithm-for-three-machine-unit-job-scheduling-September-24-2026/build/tex/runtime/generic/pgf/basiclayer/pgfcorepathprocessing.code.tex new file mode 100644 index 0000000000000000000000000000000000000000..53ed8df15f694da9691d9e94a3074f9f1f63f32a --- /dev/null +++ b/preprints/A-polynomial-time-algorithm-for-three-machine-unit-job-scheduling-September-24-2026/build/tex/runtime/generic/pgf/basiclayer/pgfcorepathprocessing.code.tex @@ -0,0 +1,417 @@ + +\ProvidesFileRCS{pgfcorepathprocessing.code.tex} + + + +\def\pgfprocesssplitpath#1{% + \let\pgfprocessresultpathprefix\pgfutil@empty% + \let\pgfprocessresultpathsuffix\pgfutil@empty% + \let\pgf@next\pgf@process@split% + \expandafter\pgf@process@split#1\pgfsyssoftpath@movetotoken{}{}\pgf@stop% +} + +\def\pgf@process@split#1\pgfsyssoftpath@movetotoken#2#3#4{% + \ifx#4\pgf@stop% + \expandafter\def\expandafter\pgfprocessresultpathsuffix\expandafter{\pgfprocessresultpathsuffix#1}% + \let\pgf@next\pgfutil@gobble% + \else% + \expandafter\expandafter\expandafter\def% + \expandafter\expandafter\expandafter\pgfprocessresultpathprefix% + \expandafter\expandafter\expandafter{\expandafter\pgfprocessresultpathprefix\pgfprocessresultpathsuffix#1}% + \def\pgfprocessresultpathsuffix{\pgfsyssoftpath@movetotoken{#2}{#3}}% + \fi% + \pgf@next#4% +} + + + + +\def\pgfprocesssplitsubpath#1{% + \let\pgf@tempa\pgfutil@emtpy% + \let\pgf@tempb\pgfutil@emtpy% + \let\pgf@tempc\pgfutil@emtpy% + \let\pgf@tempd\pgfutil@emtpy% + \let\pgfprocessresultsubpathprefix\pgfutil@empty% + \let\pgfprocessresultsubpathsuffix\pgfutil@empty% + \let\pgf@next\pgf@split@subpath% + \expandafter\pgf@split@subpath#1\pgf@stop% +} + +\newif\ifpgf@split@move + +\def\pgf@split@subpath#1#2#3#4{% + \ifx#4\pgf@stop% + \expandafter\def\expandafter\pgfprocessresultsubpathsuffix\expandafter{\pgfprocessresultsubpathsuffix#1{#2}{#3}}% + \let\pgf@next\pgfutil@gobble% + \else% + \pgf@split@movetrue% + \ifx#1\pgfsyssoftpath@curvetosupportatoken% + \pgf@split@movefalse% + \else + \ifx#1\pgfsyssoftpath@curvetosupportbtoken% + \pgf@split@movefalse% + \fi% + \fi% + \ifpgf@split@move% + \expandafter\expandafter\expandafter\def% + \expandafter\expandafter\expandafter\pgfprocessresultsubpathprefix% + \expandafter\expandafter\expandafter{% + \expandafter\pgfprocessresultsubpathprefix\pgfprocessresultsubpathsuffix}% + \def\pgfprocessresultsubpathsuffix{#1{#2}{#3}}% + \else% + \expandafter\def\expandafter\pgfprocessresultsubpathsuffix\expandafter{\pgfprocessresultsubpathsuffix#1{#2}{#3}}% + \fi% + \fi% + \pgf@next#4% +} + + + + + +\def\pgfprocesspathextractpoints#1{% + \ifx#1\pgfutil@empty% + \let\pgfpointfirstonpath=\pgfpointorigin% + \let\pgfpointsecondonpath=\pgfpointorigin% + \let\pgfpointsecondlastonpath=\pgfpointorigin% + \let\pgfpointlastonpath=\pgfpointorigin% + \else% + \expandafter\pgf@extractprocessorfirst#1\pgf@stop% + \fi% +} + +\def\pgf@extractprocessorfirst#1#2#3#4{% + \def\pgfpointfirstonpath{\pgfqpoint{#2}{#3}}% + \ifx#4\pgf@stop% + \let\pgfpointsecondonpath=\pgfpointfirstonpath% + \let\pgfpointsecondlastonpath=\pgfpointfirstonpath% + \let\pgfpointlastonpath=\pgfpointfirstonpath% + \let\next=\pgfutil@gobble% + \else + \let\next=\pgf@extractprocessorsecond% + \fi% + \next#4% +} + +\def\pgf@extractprocessorsecond#1{% + \ifx#1\pgfsyssoftpath@movetotoken% + \let\pgf@next=\pgf@extractprocessorfirst% + \else% + \let\pgf@next=\pgf@@extractprocessorsecond% + \fi% + \pgf@next#1% +} +\def\pgf@@extractprocessorsecond#1#2#3#4{% + \def\pgfpointsecondonpath{\pgfqpoint{#2}{#3}}% + \let\pgfpointsecondlastonpath=\pgfpointfirstonpath% + \let\pgfpointlastonpath=\pgfpointsecondonpath% + \ifx#4\pgf@stop% + \let\next=\pgfutil@gobble% + \else + \let\next=\pgf@extractprocessorother% + \fi% + \next#4% +} + +\def\pgf@extractprocessorother#1{% + \ifx#1\pgfsyssoftpath@movetotoken% + \let\pgf@next=\pgf@extractprocessorfirst% + \else% + \let\pgf@next=\pgf@@extractprocessorother% + \fi% + \pgf@next#1% +} +\def\pgf@@extractprocessorother#1#2#3#4{% + \let\pgfpointsecondlastonpath=\pgfpointlastonpath% + \def\pgfpointlastonpath{\pgfqpoint{#2}{#3}}% + \ifx#4\pgf@stop% + \let\next=\pgfutil@gobble% + \fi% + \next#4% +} + + + + +\def\pgfprocesscheckclosed#1#2{% + {% + \global\let\pgf@proc@todo=\relax% + \let\pgfsyssoftpath@movetotoken=\pgfutil@gobbletwo% + \let\pgfsyssoftpath@linetotoken=\pgfutil@gobbletwo% + \let\pgfsyssoftpath@curvetosupportatoken=\pgfutil@gobbletwo% + \let\pgfsyssoftpath@curvetosupportbtoken=\pgfutil@gobbletwo% + \let\pgfsyssoftpath@curvetotoken=\pgfutil@gobbletwo% + \def\pgfsyssoftpath@rectcornertoken{\gdef\pgf@proc@todo{#2}\pgfutil@gobbletwo}% + \def\pgfsyssoftpath@rectsizetoken{\gdef\pgf@proc@todo{#2}\pgfutil@gobbletwo}% + \def\pgfsyssoftpath@closepathtoken{\gdef\pgf@proc@todo{#2}\pgfutil@gobbletwo}% + #1% + }% + \pgf@proc@todo% +} + + + +\def\pgfprocessround#1#2{% + \expandafter\pgf@processroundcheck\expandafter#1\expandafter#2#1\pgfsyssoftpath@specialroundtoken\pgf@stop% +} + +\def\pgf@processroundcheck#1#2#3\pgfsyssoftpath@specialroundtoken#4\pgf@stop{% + \def\pgf@test{#4}% + \ifx\pgf@test\pgfutil@empty% + \let#2=#1% + \else% + \pgf@@processround#1#2% + \fi% +} + + +\def\pgf@@processround#1#2{% + \let\pgfprocess@newpath\pgfutil@empty% + \let\pgfprocess@moveto\pgfutil@empty% + \let\pgfprocess@firstto\pgfutil@empty% + \let\pgfprocess@segment\pgfutil@empty% + \pgf@xa=0pt% + \pgf@ya=0pt% + \expandafter\pgfprocess@round#1\pgf@stop{0pt}{0pt}% + \expandafter\pgfutil@g@addto@macro\expandafter\pgfprocess@newpath\expandafter{\pgfprocess@moveto}% + \expandafter\pgfutil@g@addto@macro\expandafter\pgfprocess@newpath\expandafter{\pgfprocess@segment}% + \let#2=\pgfprocess@newpath% +} + +\def\pgfprocess@round#1#2#3{% + \ifx#1\pgf@stop% + \let\next=\relax% + \else% + \ifx#1\pgfsyssoftpath@specialroundtoken% + \def\pgfprocess@savex{#2}% + \def\pgfprocess@savey{#3}% + \let\next=\pgfprocess@specialround% + \ifdim#2=0pt% + \ifdim#3=0pt% + \let\next=\pgfprocess@round% + \fi% + \fi% + \else% + \ifx\pgfprocess@firstto\pgfutil@empty% + \def\pgfprocess@firstto{{#2}{#3}}% + \fi% + \pgf@xa=#2% + \pgf@ya=#3% + \ifx#1\pgfsyssoftpath@movetotoken% + \expandafter\pgfutil@g@addto@macro\expandafter\pgfprocess@newpath\expandafter{\pgfprocess@moveto}% + \expandafter\pgfutil@g@addto@macro\expandafter\pgfprocess@newpath\expandafter{\pgfprocess@segment}% + \def\pgfprocess@moveto{#1{#2}{#3}}% + \let\pgfprocess@firstto=\pgfutil@empty% + \let\pgfprocess@segment=\pgfutil@empty% + \else% + \pgfutil@g@addto@macro\pgfprocess@segment{#1{#2}{#3}}% + \fi% + \let\next=\pgfprocess@round% + \fi% + \fi% + \next% +} + +\def\pgfprocess@specialround#1#2#3{% + \ifx\pgfprocess@firstto\pgfutil@empty% + \def\pgfprocess@firstto{{#2}{#3}}% + \fi% + \let\next=\pgfprocess@round% + \ifx#1\pgfsyssoftpath@linetotoken% + \let\next=\pgfprocess@specialroundcope% + \else% + \ifx#1\pgfsyssoftpath@curvetosupportatoken% + \let\next=\pgfprocess@specialroundcurveto% + \else% + \ifx#1\pgfsyssoftpath@closepathtoken% + \let\next=\pgfprocess@specialroundclose% + \fi% + \fi% + \fi% + \next#1{#2}{#3}% +}% + +\def\pgfprocess@specialroundcope#1#2#3#4#5#6{% + \let\next=\pgfprocess@round% + \let\pgfprocess@continueafterrounding=\pgfprocess@round% + \ifx#4\pgfsyssoftpath@linetotoken% + \pgf@xc=#5% + \pgf@yc=#6% + \let\next=\pgfprocess@@specialround% + \else% + \ifx#4\pgfsyssoftpath@curvetosupportatoken% + \pgf@xc=#5% + \pgf@yc=#6% + \let\next=\pgfprocess@@specialround% + \else% + \ifx#4\pgfsyssoftpath@closepathtoken% + \pgf@xc=#5% + \pgf@yc=#6% + \let\next=\pgfprocess@@specialround% + \else + \ifx#4\pgfsyssoftpath@specialroundtoken% + \let\next=\pgfprocess@@findrightcoordinates% + \fi% + \fi% + \fi% + \fi% + \next#1{#2}{#3}#4{#5}{#6}% +} + +\def\pgfprocess@@findrightcoordinates#1#2#3#4#5#6#7#8#9{% + \pgf@xc=#8% + \pgf@yc=#9% + \pgfprocess@@specialround#1{#2}{#3}#4{#5}{#6}#7{#8}{#9}% +} + +\def\pgfprocess@@specialround#1#2#3{% + \pgfutil@g@addto@macro\pgfprocess@segment{#1}% + \pgf@xb=#2% + \pgf@yb=#3% + \edef\pgf@marshal% + {\noexpand\pgfpointlineatdistance{\pgfprocess@savex}% + {\noexpand\pgfqpoint{\the\pgf@xb}{\the\pgf@yb}}% + {\noexpand\pgfqpoint{\the\pgf@xa}{\the\pgf@ya}}}% + \pgf@process{\pgf@marshal}% + \pgf@xa=\pgf@x% + \pgf@ya=\pgf@y% + \edef\pgfprocess@addition{{\the\pgf@x}{\the\pgf@y}}% + \expandafter\pgfutil@g@addto@macro\expandafter\pgfprocess@segment\expandafter{\pgfprocess@addition}% + \edef\pgf@marshal% + {\noexpand\pgfpointlineatdistance{\pgfprocess@savey}% + {\noexpand\pgfqpoint{\the\pgf@xb}{\the\pgf@yb}}% + {\noexpand\pgfqpoint{\the\pgf@xc}{\the\pgf@yc}}}% + \pgf@process{\pgf@marshal}% + \pgf@xc=\pgf@x% + \pgf@yc=\pgf@y% + {% + \pgf@x=\pgf@xb% + \pgf@y=\pgf@yb% + \advance\pgf@x by-\pgf@xa% + \advance\pgf@y by-\pgf@ya% + \advance\pgf@xa by 0.5522847\pgf@x% + \advance\pgf@ya by 0.5522847\pgf@y% + \edef\pgfprocess@addition{\noexpand\pgfsyssoftpath@curvetosupportatoken{\the\pgf@xa}{\the\pgf@ya}}% + \expandafter\pgfutil@g@addto@macro\expandafter\pgfprocess@segment\expandafter{\pgfprocess@addition}% + }% + {% + \pgf@x=\pgf@xb% + \pgf@y=\pgf@yb% + \advance\pgf@x by-\pgf@xc% + \advance\pgf@y by-\pgf@yc% + \advance\pgf@xc by 0.5522847\pgf@x% + \advance\pgf@yc by 0.5522847\pgf@y% + \edef\pgfprocess@addition{\noexpand\pgfsyssoftpath@curvetosupportbtoken{\the\pgf@xc}{\the\pgf@yc}}% + \expandafter\pgfutil@g@addto@macro\expandafter\pgfprocess@segment\expandafter{\pgfprocess@addition}% + }% + \edef\pgfprocess@addition{\noexpand\pgfsyssoftpath@curvetotoken{\the\pgf@xc}{\the\pgf@yc}}% + \expandafter\pgfutil@g@addto@macro\expandafter\pgfprocess@segment\expandafter{\pgfprocess@addition}% + \pgf@xa=#2% + \pgf@ya=#3% + \pgfprocess@continueafterrounding% +} + +\def\pgfprocess@specialroundcurveto#1#2#3#4#5#6{% + \pgfutil@g@addto@macro\pgfprocess@segment{#1{#2}{#3}#4{#5}{#6}}% + \pgf@xa=#5% + \pgf@ya=#6% + \pgfprocess@specialroundcope} + +\def\pgfprocess@specialroundclose{% + \let\next=\pgfprocess@round% + \ifx\pgfprocess@moveto\pgfutil@empty% + \else% + \ifx\pgfprocess@firstto\pgfutil@empty% + \else% + \expandafter\pgfprocess@@specialclosetarget\pgfprocess@firstto% + \let\pgfprocess@continueafterrounding=\pgfprocess@cleanupafterclose% + \edef\next{% + \noexpand\pgfprocess@@specialround% + \expandafter\expandafter\expandafter\noexpand\expandafter\pgfprocess@@specialclosemoveto\pgfprocess@moveto% + }% + \fi% + \fi% + \next% +} + +\def\pgfprocess@@specialclosetarget#1#2{% + \pgf@xc=#1% + \pgf@yc=#2% +} + +\def\pgfprocess@@specialclosemoveto#1#2#3{% + \pgfsyssoftpath@linetotoken{#2}{#3}% +} + + +\def\pgfprocess@cleanupafterclose#1#2#3{% + \edef\pgfprocess@moveto{\noexpand\pgfsyssoftpath@movetotoken{\the\pgf@xc}{\the\pgf@yc}}% + \edef\pgf@marshal{\noexpand\pgfprocess@round\noexpand\pgfsyssoftpath@closepathtoken{\the\pgf@xc}{\the\pgf@yc}}% + \pgf@marshal% +} + + + +\def\pgfprocesspathreplacestartandend#1#2#3{% + \ifx#1\pgfutil@empty% + \else% + {% + \pgfsyssoftpath@getcurrentpath\pgf@replace@origpath% + \let\pgf@prepath=\pgfutil@empty% + \pgf@process{#3}% + \pgf@xa=\pgf@x% + \pgf@ya=\pgf@y% + \pgf@process{#2}% + \expandafter\pgf@replaceprocessfirst#1\pgf@stop% + \pgfsyssoftpath@getcurrentpath\pgf@restpath% + \expandafter\expandafter\expandafter\expandafter\expandafter\expandafter\expandafter\gdef% + \expandafter\expandafter\expandafter\expandafter\expandafter\expandafter\expandafter#1% + \expandafter\expandafter\expandafter\expandafter\expandafter\expandafter\expandafter{% + \expandafter\expandafter\expandafter\pgf@prepath% + \expandafter\pgf@newstart\pgf@restpath}% + \pgfsyssoftpath@setcurrentpath\pgf@replace@origpath% + }% + \fi% +} + +\def\pgf@replaceprocessfirst#1#2#3#4{% + \def\pgf@origstart{#1{#2}{#3}}% + \edef\pgf@newstart{\noexpand#1{\the\pgf@x}{\the\pgf@y}}% + \pgfsyssoftpath@setcurrentpath\pgfutil@empty% + \ifx#4\pgf@stop% + \let\pgf@next=\pgfutil@gobble% + \else% + \let\pgf@next=\pgf@replaceprocessother% + \let\pgf@@next=\pgf@@replaceprocessother% + \fi% + \pgf@next#4% +} + +\def\pgf@replaceprocessother#1{% + \ifx#1\pgfsyssoftpath@movetotoken% + \pgf@replace@handle@move% + \fi% + \pgf@@next#1% +} +\def\pgf@@replaceprocessother#1#2#3#4{% + \ifx#4\pgf@stop% + \edef\pgf@restpathaddition{{\noexpand#1{\the\pgf@xa}{\the\pgf@ya}}}% + \expandafter\pgfsyssoftpath@addtocurrentpath\pgf@restpathaddition% + \let\pgf@next=\pgfutil@gobble% + \else% + \pgfsyssoftpath@addtocurrentpath{#1{#2}{#3}}% + \fi% + \pgf@next#4% +} + +\def\pgf@replace@handle@move{% + \pgfsyssoftpath@getcurrentpath\pgf@restpath% + \expandafter\expandafter\expandafter\expandafter\expandafter\expandafter\expandafter\gdef% + \expandafter\expandafter\expandafter\expandafter\expandafter\expandafter\expandafter\pgf@prepath% + \expandafter\expandafter\expandafter\expandafter\expandafter\expandafter\expandafter{% + \expandafter\expandafter\expandafter\pgf@prepath% + \expandafter\pgf@origstart\pgf@restpath}% + \let\pgf@@next=\pgf@replaceprocessfirst% +} + +\endinput diff --git a/preprints/A-polynomial-time-algorithm-for-three-machine-unit-job-scheduling-September-24-2026/build/tex/runtime/generic/pgf/basiclayer/pgfcorepathusage.code.tex b/preprints/A-polynomial-time-algorithm-for-three-machine-unit-job-scheduling-September-24-2026/build/tex/runtime/generic/pgf/basiclayer/pgfcorepathusage.code.tex new file mode 100644 index 0000000000000000000000000000000000000000..3258ae3a96fbde780ca05f56cfa67fdfacd829f3 --- /dev/null +++ b/preprints/A-polynomial-time-algorithm-for-three-machine-unit-job-scheduling-September-24-2026/build/tex/runtime/generic/pgf/basiclayer/pgfcorepathusage.code.tex @@ -0,0 +1,510 @@ + +\ProvidesFileRCS{pgfcorepathusage.code.tex} + + + +\pgfkeys{ + /pgf/stroke/.code=\let\pgf@up@stroke\pgf@up@stroke@text\pgf@up@path@neededtrue, + /pgf/draw/.code=\let\pgf@up@stroke\pgf@up@stroke@text\pgf@up@path@neededtrue, + /pgf/fill/.code=\let\pgf@up@fill\pgf@up@fill@text\pgf@up@path@neededtrue, + /pgf/clip/.code=\let\pgf@up@clip\pgf@up@clip@text\pgf@up@path@neededtrue, + /pgf/discard/.code=, + /pgf/use as bounding box/.code=\let\pgf@up@bb\pgf@do@up@bb, + /pgf/arrow keys/flex/.code=\pgferror{You need to say \string\usetikzlibrary{bending} for flexing and bending arrows}, + /pgf/arrow keys/flex'/.code=\pgferror{You need to say \string\usetikzlibrary{bending} for flexing and bending arrows}, + /pgf/arrow keys/bend/.code=\pgferror{You need to say \string\usetikzlibrary{bending} for flexing and bending arrows}, + /pgf/arrow keys/quick/.code=\pgfarrowsaddtooptions{\let\pgf@arrow@flex@mode\pgf@arrow@mode@is@quick}, +} + +\pgfkeys{ + /pgf/.cd, + tips/.is choice, + tips/.default=true, + tips/proper/.code=\let\pgf@tips@mode\pgf@tips@mode@onproper, + tips/on proper draw/.code=\let\pgf@tips@mode\pgf@tips@mode@onproperdraw, + tips/true/.code=\let\pgf@tips@mode\pgf@tips@mode@true, + tips/on draw/.code=\let\pgf@tips@mode\pgf@tips@mode@ondraw, + tips/never/.code=\let\pgf@tips@mode\pgf@tips@mode@false, + tips/false/.code=\let\pgf@tips@mode\pgf@tips@mode@false, +} +\def\pgf@tips@mode@onproper{4} +\def\pgf@tips@mode@onproperdraw{3} +\def\pgf@tips@mode@true{2} +\def\pgf@tips@mode@ondraw{1} +\def\pgf@tips@mode@false{0} +\let\pgf@tips@mode\pgf@tips@mode@ondraw + + +\newif\ifpgf@precise@shortening +\def\pgf@arrow@mode@is@quick{0} +\let\pgf@arrow@flex@mode\pgf@arrow@mode@is@quick + +\newif\ifpgf@up@path@needed +\newif\ifpgf@up@draw@arrows +\def\pgf@up@stroke@text{stroke} +\def\pgf@up@fill@text{fill} +\def\pgf@up@clip@text{clip} +\def\pgf@do@up@bb{\pgf@relevantforpicturesizefalse} + +\def\pgfusepath#1{% + \pgf@up@path@neededfalse% + \pgf@up@draw@arrowsfalse% + \let\pgf@up@stroke\pgfutil@empty% + \let\pgf@up@fill\pgfutil@empty% + \let\pgf@up@clip\pgfutil@empty% + \let\pgf@up@bb\pgfutil@empty% + \pgfset{#1}% + \expandafter\def\expandafter\pgf@up@action\expandafter{\csname pgfsys@\pgf@up@fill\pgf@up@stroke\endcsname}% + \ifnum\pgf@tips@mode=2\relax% + \pgf@up@path@neededtrue% + \fi% + \ifnum\pgf@tips@mode=4\relax% + \pgf@up@path@neededtrue% + \fi% + \ifpgf@up@path@needed% + \else% + \pgfsyssoftpath@setcurrentpath\pgfutil@empty% + \fi% + \ifx\pgf@up@stroke\pgfutil@empty% + \ifx\pgf@up@fill\pgfutil@empty% + \let\pgf@up@action=\pgfutil@empty% + \ifx\pgf@up@clip\pgfutil@empty% + \else% + \let\pgf@up@action=\pgfsys@discardpath% + \fi% + \fi% + \fi% + \pgfsyssoftpath@getcurrentpath\pgf@last@processed@path + \pgfprocessround{\pgf@last@processed@path}{\pgf@last@processed@path}% + \pgfsyssoftpath@setcurrentpath\pgf@last@processed@path% + \ifpgf@relevantforpicturesize% + \ifx\pgf@up@stroke\pgfutil@empty% + \else% + \ifdim\pgf@picmaxx=-16000pt\relax% + \else% + \pgf@x=\pgf@pathminx\advance\pgf@x by-.5\pgflinewidth% + \ifdim\pgf@x<\pgf@picminx\global\pgf@picminx\pgf@x\fi% + \pgf@y=\pgf@pathminy\advance\pgf@y by-.5\pgflinewidth% + \ifdim\pgf@y<\pgf@picminy\global\pgf@picminy\pgf@y\fi% + \pgf@x=\pgf@pathmaxx\advance\pgf@x by.5\pgflinewidth% + \ifdim\pgf@x>\pgf@picmaxx\global\pgf@picmaxx\pgf@x\fi% + \pgf@y=\pgf@pathmaxy\advance\pgf@y by.5\pgflinewidth% + \ifdim\pgf@y>\pgf@picmaxy\global\pgf@picmaxy\pgf@y\fi% + \fi% + \fi% + \fi% + \ifx\pgf@up@clip\pgfutil@empty% + \ifx\pgf@up@stroke\pgfutil@empty% + \ifx\pgf@up@action\pgfutil@empty% + \ifnum\pgf@tips@mode=2\relax% + \pgf@up@draw@arrows@only% + \fi% + \ifnum\pgf@tips@mode=4\relax% + \pgf@up@draw@arrows@only% + \fi% + \else% + \pgfsyssoftpath@invokecurrentpath% + \pgf@up@action% + \fi% + \else% + \pgfgetpath\pgf@arrowpath% + \pgf@path@setup@tempswa% + \pgfprocesscheckclosed{\pgf@arrowpath}{\pgfutil@tempswafalse}% + \pgf@path@check@proper% + \ifpgfutil@tempswa% + \pgf@check@for@arrow@and@animation% + \pgf@prepare@end@of@path% + \begingroup% + \pgf@prepare@start@of@path% + \fi% + \pgfsyssoftpath@invokecurrentpath% + \pgf@up@action% + \ifdim\pgfinnerlinewidth>0pt\relax% + \pgf@stroke@inner@line% + \fi% + \ifpgfutil@tempswa% + \pgf@add@arrow@at@start% + \endgroup% + \pgf@add@arrow@at@end% + \fi% + \fi% + \else% + \pgfsyssoftpath@invokecurrentpath% + \pgfsys@clipnext% + \pgf@up@action% + \pgf@relevantforpicturesizefalse% + \fi% + \pgf@up@bb% + \pgfsyssoftpath@setcurrentpath\pgfutil@empty% + \pgf@resetpathsizes% + \ignorespaces% +} +\def\pgf@path@setup@tempswa{% + \ifnum\pgf@tips@mode>0\relax + \pgfutil@tempswatrue% + \else + \pgfutil@tempswafalse% + \fi% +} +\def\pgf@up@draw@arrows@only{% + \pgfgetpath\pgf@arrowpath% + \pgfutil@tempswatrue% + \pgfprocesscheckclosed{\pgf@arrowpath}{\pgfutil@tempswafalse}% + \pgf@path@check@proper% + \ifpgfutil@tempswa% + \pgf@check@for@arrow@and@animation% + \pgf@prepare@end@of@path% + \begingroup% + \pgf@prepare@start@of@path% + \pgf@add@arrow@at@start% + \endgroup% + \pgf@add@arrow@at@end% + \fi% +} + +\def\pgf@check@for@arrow@and@animation{\pgfsys@if@fresh@currentid{\pgf@check@for@arrow@and@animation@}{}}% +\def\pgf@check@for@arrow@and@animation@{% + \expandafter\ifx\csname pgfsysanim@path@is@animated@\pgf@sys@id@current@id @\pgfsys@current@type\endcsname\pgfutil@empty% + \ifx\pgf@end@tip@sequence\pgfutil@empty% + \ifx\pgf@start@tip@sequence\pgfutil@empty% + \else% + \pgf@check@for@arrow@and@animation@error% + \fi% + \else% + \pgf@check@for@arrow@and@animation@error% + \fi% + \fi% +} +\def\pgf@check@for@arrow@and@animation@error{\pgferror{Animated path may not have normal arrow tips. Use a base path with arrow tips}} + +\def\pgf@path@check@proper{% + \ifpgfutil@tempswa% + \ifnum\pgf@tips@mode>2\relax% + \pgf@path@check@proper@% + \fi% + \fi% +} + +\def\pgf@path@check@proper@{% + {% + \pgf@x0pt\pgf@y\pgf@x% + \pgfutil@tempswatrue% + \let\pgfsyssoftpath@movetotoken\pgf@path@proper@init% + \let\pgfsyssoftpath@linetotoken\pgf@path@proper@test% + \let\pgfsyssoftpath@curvetosupportatoken\pgf@path@proper@test% + \let\pgfsyssoftpath@curvetosupportbtoken=\pgf@path@proper@test% + \let\pgfsyssoftpath@curvetotoken\pgf@path@proper@test% + \let\pgfsyssoftpath@rectcornertoken\pgf@path@proper@test% + \let\pgfsyssoftpath@rectsizetoken\pgf@path@proper@zero@test% + \let\pgfsyssoftpath@closepathtoken\pgf@path@proper@test% + \pgf@arrowpath% + \expandafter + }% + \ifpgfutil@tempswa\pgfutil@tempswafalse\else\pgfutil@tempswatrue\fi% +} + +\def\pgf@path@proper@init#1#2{\pgfutil@tempswatrue\pgf@x#1\pgf@y#2}% +\def\pgf@path@proper@test#1#2{% + \ifdim\pgf@x=#1\relax\else\pgfutil@tempswafalse\fi% + \ifdim\pgf@y=#2\relax\else\pgfutil@tempswafalse\fi} +\def\pgf@path@proper@zero@test#1#2{% + \ifdim\pgf@x=0pt\relax\else\pgfutil@tempswafalse\fi% + \ifdim\pgf@y=0pt\relax\else\pgfutil@tempswafalse\fi} + + + + +\def\pgf@stroke@inner@line{% + \let\pgf@temp@save=\pgf@strokecolor@global + \pgfsys@beginscope% + {% + \pgfsys@setlinewidth{\pgfinnerlinewidth}% + \pgfsetstrokecolor{\pgfinnerstrokecolor}% + \pgfsyssoftpath@invokecurrentpath% + \pgfsys@stroke% + }% + \pgfsys@endscope% + \global\let\pgf@strokecolor@global=\pgf@temp@save +} + +\let\pgf@prepare@start@of@path\relax% + + + + + +\def\pgfsetshortenstart#1{\pgfmathsetlength\pgf@shorten@start@additional{#1}} +\def\pgfsetshortenend#1{\pgfmathsetlength\pgf@shorten@end@additional{#1}} + +\newdimen\pgf@shorten@end@additional +\newdimen\pgf@shorten@start@additional + + + + +\newif\ifpgf@worry + + + +\def\pgf@prepare@end@of@path{% + \let\pgfprocessresultpathsuffix\relax% + \let\pgfprocessresultsubpathsuffix\relax% + \pgfsyssoftpath@getcurrentpath\pgf@arrowpath% + \ifx\pgf@arrowpath\pgfutil@empty\else% + \pgf@worryfalse% + \ifx\pgf@end@tip@sequence\pgfutil@empty% + \else% + \pgf@worrytrue% + \fi% + \pgf@precise@shorteningfalse% + \pgf@arrow@compute@shortening\pgf@end@tip@sequence% + \advance\pgf@xa by\pgf@shorten@end@additional% + \advance\pgf@xb by\pgf@shorten@end@additional% + \ifdim\pgf@xa=0pt\relax\else\pgf@worrytrue\fi% + \edef\pgf@path@shortening@distance{\the\pgf@xa}% + \edef\pgf@arrow@tip@total@length{\the\pgf@xb}% + \ifpgf@worry% + \pgfprocesssplitpath{\pgf@arrowpath}% + \pgfprocesssplitsubpath{\pgfprocessresultpathsuffix}% + \expandafter\pgf@parse@end\pgfprocessresultsubpathsuffix\pgf@stop\pgf@stop\pgf@stop% + \pgf@prep@end% + \ifdim\pgf@path@shortening@distance=0pt\else\pgf@do@shorten@end\fi% + \expandafter\expandafter\expandafter\def% + \expandafter\expandafter\expandafter\pgf@arrowpath% + \expandafter\expandafter\expandafter{\expandafter\pgfprocessresultpathprefix\pgfprocessresultpathsuffix}% + \pgfsyssoftpath@setcurrentpath\pgf@arrowpath% + \fi% + \fi% +} + +\def\pgf@parse@end#1#2#3#4#5#6{% + \ifx#4\pgf@stop% + \def\pgfpointlastonpath{\pgfqpoint{#2}{#3}}% + \let\pgf@next\relax% + \let\pgf@do@shorten@end\relax% + \let\pgf@do@draw@end\pgf@do@draw@straightend% + \let\pgf@prep@end\pgf@prep@movetoend% + \else\ifx#4\pgfsyssoftpath@curvetosupportatoken% + \def\pgfsubpathfourthlasttoken{#1}% + \def\pgfpointfourthlastonpath{\pgfqpoint{#2}{#3}}% + \def\pgfpointthirdlastonpath{\pgfqpoint{#5}{#6}}% + \ifpgf@precise@shortening% + \let\pgf@do@shorten@end\pgf@do@shorten@curvedend% + \let\pgf@do@draw@end\pgf@do@draw@curvedend% + \let\pgf@prep@end\pgf@prep@curveend% + \else% + \def\pgfsubpathstart{\noexpand#1{#2}{#3}\noexpand#4{#5}{#6}}% + \let\pgf@do@shorten@end\pgf@do@shorten@straightend% + \let\pgf@do@draw@end\pgf@do@draw@straightend% + \let\pgf@prep@end\pgf@prep@straightend% + \fi% + \let\pgf@next\pgf@parse@end@curve@cont% + \else% + \def\pgfsubpathstart{\noexpand#1{#2}{#3}\noexpand#4}% + \def\pgfpointsecondlastonpath{\pgfqpoint{#2}{#3}}% + \def\pgfpointlastonpath{\pgfqpoint{#5}{#6}}% + \let\pgfpointfourthlastonpath\relax% + \let\pgfpointthirdlastonpath\relax% + \let\pgf@next\pgf@parse@end@gobble@three% + \let\pgf@do@shorten@end\pgf@do@shorten@straightend% + \let\pgf@do@draw@end\pgf@do@draw@straightend% + \let\pgf@prep@end\pgf@prep@straightend% + \fi\fi% + \pgf@next% +} + +\def\pgf@parse@end@gobble@three#1#2#3{}% +\def\pgf@parse@end@curve@cont#1#2#3#4#5#6#7#8#9{% + \def\pgfpointsecondlastonpath{\pgfqpoint{#2}{#3}}% + \def\pgfpointlastonpath{\pgfqpoint{#5}{#6}}% + \ifpgf@precise@shortening% + \else% + \expandafter\def\expandafter\pgfsubpathstart\expandafter{\pgfsubpathstart\noexpand#1{#2}{#3}\noexpand#4}% + \fi% +} + +\def\pgf@prep@movetoend{% + \pgf@process{\pgfpointlastonpath}% + \pgf@xb\pgf@x + \pgf@yb\pgf@y + \pgf@xc\pgf@x + \pgf@yc\pgf@y +} +\def\pgf@prep@straightend{% + \let\pgf@ref\pgfpointsecondlastonpath + \ifx\pgfpointlastonpath\pgfpointsecondlastonpath% + \ifx\pgfpointthirdlastonpath\relax% + \else% + \ifx\pgfpointlastonpath\pgfpointthirdlastonpath% + \let\pgf@ref\pgfpointfourthlastonpath% + \else + \let\pgf@ref\pgfpointthirdlastonpath% + \fi% + \fi% + \fi% + \pgfpointlineatdistance{\pgf@path@shortening@distance}{\pgfpointlastonpath}{\pgf@ref}% + \pgf@xa\pgf@x% + \pgf@ya\pgf@y% +} + + +\def\pgf@do@shorten@straightend{% + \edef\pgfprocessresultsubpathsuffix{\pgfsubpathstart{\the\pgf@xa}{\the\pgf@ya}}% + \expandafter\expandafter\expandafter\def% + \expandafter\expandafter\expandafter\pgfprocessresultpathsuffix% + \expandafter\expandafter\expandafter{\expandafter\pgfprocessresultsubpathprefix\pgfprocessresultsubpathsuffix}% +} + + + +\def\pgf@add@arrow@at@end{% + \ifx\pgf@arrowpath\pgfutil@empty\else% + \ifx\pgf@end@tip@sequence\pgfutil@empty\else% + \pgf@do@draw@end% + \fi% + \fi% +} + +\def\pgf@do@draw@straightend{% + {% + \pgftransformreset% + \pgftransformarrow{\pgfqpoint{\pgf@xc}{\pgf@yc}}{\pgfqpoint{\pgf@xb}{\pgf@yb}}% + \pgf@arrow@draw@arrow\pgf@end@tip@sequence\pgf@arrow@tip@total@length% + }% +} + + + + + + + +\def\pgf@prepare@start@of@path{% + \ifx\pgf@arrowpath\pgfutil@empty\else% + \pgf@worryfalse% + \ifx\pgf@start@tip@sequence\pgfutil@empty\else\pgf@worrytrue\fi% + \pgf@precise@shorteningfalse% + \pgf@arrow@compute@shortening\pgf@start@tip@sequence% + \advance\pgf@xa by\pgf@shorten@start@additional% + \advance\pgf@xb by\pgf@shorten@start@additional% + \ifdim\pgf@xa=0pt\relax\else\pgf@worrytrue\fi% + \edef\pgf@path@shortening@distance{\the\pgf@xa}% + \edef\pgf@arrow@tip@total@length{\the\pgf@xb}% + \ifpgf@worry% + \ifx\pgfprocessresultpathsuffix\relax% + \pgfprocesssplitpath{\pgf@arrowpath}% + \fi% + \expandafter\pgf@parse@start\pgfprocessresultpathsuffix\pgf@stop\pgf@stop\pgf@stop% + \pgf@prep@start% + \ifdim\pgf@path@shortening@distance=0pt\else\pgf@do@shorten@start\fi% + \expandafter\expandafter\expandafter\def% + \expandafter\expandafter\expandafter\pgf@arrowpath% + \expandafter\expandafter\expandafter{\expandafter\pgfprocessresultpathprefix\pgfprocessresultpathsuffix}% + \pgfsyssoftpath@setcurrentpath\pgf@arrowpath% + \fi% + \fi% +} + +\def\pgf@parse@start#1#2#3#4#5#6{% + \def\pgfpointfirstonpath{\pgfqpoint{#2}{#3}}% + \def\pgfpointsecondonpath{\pgfqpoint{#5}{#6}}% + \let\pgfpointthirdonpath\relax% + \let\pgfpointfourthonpath\relax% + \def\pgfsubpathfirsttoken{\noexpand#1}% + \def\pgfsubpathsecondtoken{\noexpand#4}% + \ifx#4\pgf@stop% + \let\pgf@next\relax% + \let\pgf@do@shorten@start\relax% + \let\pgf@prep@start\pgf@prep@movetostart% + \let\pgf@do@draw@start\pgf@do@draw@straightstart% + \else\ifx#4\pgfsyssoftpath@curvetosupportatoken% + \ifpgf@precise@shortening% + \let\pgf@next\pgf@parse@start@curve@cont% + \let\pgf@do@shorten@start\pgf@do@shorten@curvedstart% + \let\pgf@do@draw@start\pgf@do@draw@curvedstart% + \let\pgf@prep@start\pgf@prep@curvedstart% + \else% + \let\pgf@next\pgf@parse@start@curve@cont@straight% + \let\pgf@do@shorten@start\pgf@do@shorten@straightstart% + \let\pgf@do@draw@start\pgf@do@draw@straightstart% + \let\pgf@prep@start\pgf@prep@straightstart% + \fi% + \else% + \let\pgf@next\pgf@parse@start@till@stop% + \let\pgf@do@shorten@start\pgf@do@shorten@straightstart% + \let\pgf@do@draw@start\pgf@do@draw@straightstart% + \let\pgf@prep@start\pgf@prep@straightstart% + \fi\fi% + \pgf@next% +} + +\def\pgf@parse@start@till@stop#1\pgf@stop\pgf@stop\pgf@stop{\def\pgfsubpathend{#1}}% + +\def\pgf@parse@start@curve@cont#1#2#3#4#5#6#7\pgf@stop\pgf@stop\pgf@stop{% + \def\pgfpointthirdonpath{\pgfqpoint{#2}{#3}}% + \def\pgfpointfourthonpath{\pgfqpoint{#5}{#6}}% + \def\pgfsubpathend{#7}% +} + +\def\pgf@parse@start@curve@cont@straight#1#2#3#4#5#6#7\pgf@stop\pgf@stop\pgf@stop{% + \def\pgfpointthirdonpath{\pgfqpoint{#2}{#3}}% + \def\pgfpointfourthonpath{\pgfqpoint{#5}{#6}}% + \def\pgfsubpathend{#1{#2}{#3}#4{#5}{#6}#7}% +} + +\def\pgf@prep@movetostart{% + \pgf@process{\pgfpointfirstonpath}% + \pgf@xb\pgf@x + \pgf@yb\pgf@y + \pgf@xc\pgf@x + \pgf@yc\pgf@y +} +\def\pgf@prep@straightstart{% + \let\pgf@ref\pgfpointsecondonpath + \ifx\pgfpointfirstonpath\pgfpointsecondonpath% + \ifx\pgfpointthirdonpath\relax% + \else% + \ifx\pgfpointfirstonpath\pgfpointthirdonpath% + \let\pgf@ref\pgfpointfourthonpath% + \else + \let\pgf@ref\pgfpointthirdonpath% + \fi% + \fi% + \fi% + \pgfpointlineatdistance{\pgf@path@shortening@distance}{\pgfpointfirstonpath}{\pgf@ref}% + \pgf@xa\pgf@x% + \pgf@ya\pgf@y% +} + +\def\pgf@do@shorten@straightstart{% + \edef\pgfprocessresultpathsuffix{\pgfsubpathfirsttoken{\the\pgf@xa}{\the\pgf@ya}\pgfsubpathsecondtoken{\the\pgf@xc}{\the\pgf@yc}}% + \expandafter\expandafter\expandafter\def% + \expandafter\expandafter\expandafter\pgfprocessresultpathsuffix% + \expandafter\expandafter\expandafter{\expandafter\pgfprocessresultpathsuffix\pgfsubpathend}% +} + + + +\def\pgf@add@arrow@at@start{% + \ifx\pgf@arrowpath\pgfutil@empty\else% + \ifx\pgf@start@tip@sequence\pgfutil@empty\else% + \pgf@do@draw@start% + \fi% + \fi% +} + +\def\pgf@do@draw@straightstart{% + {% + \pgftransformreset% + \pgftransformarrow{\pgfqpoint{\pgf@xc}{\pgf@yc}}{\pgfqpoint{\pgf@xb}{\pgf@yb}}% + \pgf@arrow@draw@arrow\pgf@start@tip@sequence\pgf@arrow@tip@total@length% + }% +} + + + +\let\pgf@shorten@end=\pgfutil@empty +\let\pgf@shorten@start=\pgfutil@empty + + +\endinput% diff --git a/preprints/A-polynomial-time-algorithm-for-three-machine-unit-job-scheduling-September-24-2026/build/tex/runtime/generic/pgf/basiclayer/pgfcorepatterns.code.tex b/preprints/A-polynomial-time-algorithm-for-three-machine-unit-job-scheduling-September-24-2026/build/tex/runtime/generic/pgf/basiclayer/pgfcorepatterns.code.tex new file mode 100644 index 0000000000000000000000000000000000000000..60152c5d064e1c313b839399353e33ed207494c2 --- /dev/null +++ b/preprints/A-polynomial-time-algorithm-for-three-machine-unit-job-scheduling-September-24-2026/build/tex/runtime/generic/pgf/basiclayer/pgfcorepatterns.code.tex @@ -0,0 +1,197 @@ + +\ProvidesFileRCS{pgfcorepatterns.code.tex} + +\def\pgfdeclarepatternformonly{% + \def\pgf@pattern@type{formonly}% + \pgfutil@ifnextchar[{\pgf@declarepattern@checkarg}{\pgf@declarepattern@checkarg[]}% +} + + +\def\pgfdeclarepatterninherentlycolored{% + \def\pgf@pattern@type{inherentlycolored}% + \pgfutil@ifnextchar[{\pgf@declarepattern@checkarg}{\pgf@declarepattern@checkarg[]}% +} + + + + + +\def\pgf@declarepattern@checkarg[#1]{% + \def\pgf@pattern@tempvars{#1}% + \ifx\pgf@pattern@tempvars\pgfutil@empty% + \expandafter\let\expandafter\pgf@next\csname pgf@declarepattern\pgf@pattern@type\endcsname% + \else% + \expandafter\let\expandafter\pgf@next\csname pgf@declarepattern\pgf@pattern@type @mutable\endcsname% + \fi% + \pgf@next} + +\long\def\pgf@declarepatternformonly#1#2#3#4#5{\pgf@declarepattern{#1}{#2}{#3}{#4}{#5}{0}} + +\long\def\pgf@declarepatterninherentlycolored#1#2#3#4#5{\pgf@declarepattern{#1}{#2}{#3}{#4}{#5}{1}} + +\long\def\pgf@declarepatternformonly@mutable#1#2#3#4#5{% + \def\pgf@marshal{\pgf@declarepatternmutable{#1}}% + \expandafter\pgf@marshal\expandafter{\pgf@pattern@tempvars}{#2}{#3}{#4}{#5}{2}% +} + +\long\def\pgf@declarepatterninherentlycolored@mutable#1#2#3#4#5#6{% + \def\pgf@marshal{\pgf@declarepatternmutable{#1}}% + \expandafter\pgf@marshal\expandafter{\pgf@pattern@tempvars}{#2}{#3}{#4}{#5}{3}% +} + +\def\pgfsys@patternmatrix{{1.0}{0.0}{0.0}{1.0}{0.0pt}{0.0pt}} +\def\pgf@declarepattern#1#2#3#4#5#6{% + \pgfutil@ifundefined{pgf@pattern@name@#1}{% + \pgfsysprotocol@getcurrentprotocol\pgf@pattern@temp% + {% + \pgfinterruptpath% + \pgfpicturetrue% + \pgf@relevantforpicturesizefalse% + \pgftransformreset% + \pgfsysprotocol@setcurrentprotocol\pgfutil@empty% + \pgfsysprotocol@bufferedtrue% + \pgfsys@beginscope% + \pgfsetarrows{-}% + #5% + \pgfsys@endscope% + \pgfsysprotocol@getcurrentprotocol\pgf@pattern@code% + \global\let\pgf@pattern@code=\pgf@pattern@code% + \endpgfinterruptpath% + \pgf@process{#2}% + \pgf@xa=\pgf@x% + \pgf@ya=\pgf@y% + \pgf@process{#3}% + \pgf@xb=\pgf@x% + \pgf@yb=\pgf@y% + \pgf@process{#4}% + \pgf@xc=\pgf@x% + \pgf@yc=\pgf@y% + \pgfutil@tempcnta=\pgf@pattern@number% + \advance\pgfutil@tempcnta by1\relax% + \xdef\pgf@pattern@number{\the\pgfutil@tempcnta}% + \expandafter\xdef\csname pgf@pattern@name@#1\endcsname{\the\pgfutil@tempcnta}% + \expandafter\gdef\csname pgf@pattern@type@#1\endcsname{#6}% + \xdef\pgf@marshal{\noexpand\pgfsys@declarepattern + {\csname pgf@pattern@name@#1\endcsname}% + {\the\pgf@xa}{\the\pgf@ya}{\the\pgf@xb}{\the\pgf@yb}% + {\the\pgf@xc}{\the\pgf@yc}% + \pgfsys@patternmatrix% + {\pgf@pattern@code}{#6}}% + }% + \pgfsysprotocol@setcurrentprotocol\pgf@pattern@temp% + \expandafter\global\expandafter\let\csname pgf@pattern@instantiate@#1\endcsname=\pgf@marshal% + } + {% + \pgferror{Pattern `#1' already defined}% + }% +} + +\def\pgf@pattern@number{0}% + +\def\pgf@declarepatternmutable#1#2#3#4#5#6#7{% + \pgfutil@ifundefined{pgf@pattern@name@#1}{% + \expandafter\gdef\csname pgf@pattern@name@#1\endcsname{#1}% + \expandafter\gdef\csname pgf@pattern@variables@#1\endcsname{#2}% + \expandafter\gdef\csname pgf@pattern@lowerleft@#1\endcsname{#3}% + \expandafter\gdef\csname pgf@pattern@upperright@#1\endcsname{#4}% + \expandafter\gdef\csname pgf@pattern@tilesize@#1\endcsname{#5}% + \expandafter\long\expandafter\gdef\csname pgf@pattern@code@#1\endcsname{#6}% + \expandafter\gdef\csname pgf@pattern@type@#1\endcsname{#7}% + }{\pgferror{The pattern `#1' is already defined}}% +} + +\def\pgf@ifpatternisinherentlycolored#1#2#3{% + \expandafter\ifodd\csname pgf@pattern@type@#1\endcsname\relax#2\else#3\fi% +} + +\def\pgf@ifpatternismutable#1#2#3{% + \expandafter\ifnum\csname pgf@pattern@type@#1\endcsname>1\relax#2\else#3\fi% +} + +\def\pgfpatternreleasename#1{\expandafter\let\csname pgf@pattern@name@#1\endcsname\relax} + + + + + + +\def\pgfsetfillpattern#1#2{% + \pgfutil@ifundefined{pgf@pattern@name@#1}% + {% + \pgferror{Undefined pattern `#1'}% + }% + {% + \pgf@ifpatternismutable{#1}% + {% + \let\pgf@pattern@tempvars\pgfutil@empty% + \expandafter\expandafter\expandafter\pgf@pattern@check@vars% + \csname pgf@pattern@variables@#1\endcsname,\pgf@stop,% + \edef\pgf@pattern@nametemp{pgf@pattern@#1@\pgf@pattern@tempvars}% + \expandafter\pgfutil@ifundefined\expandafter{\pgf@pattern@nametemp}% + {% + \pgf@ifpatternisinherentlycolored{#1}% + {\let\pgf@set@fillpattern@declare\pgfdeclarepatterninherentlycolored}% + {\let\pgf@set@fillpattern@declare\pgfdeclarepatternformonly}% + \pgf@set@fillpattern@declare% + {\pgf@pattern@nametemp}% + {\csname pgf@pattern@lowerleft@#1\endcsname}% + {\csname pgf@pattern@upperright@#1\endcsname}% + {\csname pgf@pattern@tilesize@#1\endcsname}% + {\csname pgf@pattern@code@#1\endcsname}% + }% + {% + }% + \expandafter\pgf@set@fillpattern\expandafter{\pgf@pattern@nametemp}{#2}% + \expandafter\gdef\csname\pgf@pattern@nametemp\endcsname{#1}% + }% + {% + \pgf@set@fillpattern{#1}{#2}% + }% + }% +} + + +\def\pgf@set@fillpattern#1#2{% + \pgfutil@ifundefined{pgf@pattern@name@#1}{% + \pgferror{Undefined pattern `#1'}} + {% + \csname pgf@pattern@instantiate@#1\endcsname% + \expandafter\global\expandafter\let\csname pgf@pattern@instantiate@#1\endcsname=\relax% + \pgf@ifpatternisinherentlycolored{#1}{% + \pgfsys@setpatterncolored{\csname pgf@pattern@name@#1\endcsname}% + }{% + \pgfutil@colorlet{pgf@tempcolor}{#2}% + \pgfutil@ifundefined{applycolormixins}{}{\applycolormixins{pgf@tempcolor}}% + \pgfutil@extractcolorspec{pgf@tempcolor}{\pgf@tempcolor}% + \expandafter\pgfutil@convertcolorspec\pgf@tempcolor{rgb}{\pgf@rgbcolor}% + \expandafter\pgf@set@fill@patternuncolored\pgf@rgbcolor\relax{#1}% + }% + }% +} + +\def\pgf@set@fill@patternuncolored#1,#2,#3\relax#4{% + \pgfsys@setpatternuncolored{\csname pgf@pattern@name@#4\endcsname}{#1}{#2}{#3}% +} + +\def\pgf@pattern@check@vars#1,{% + \ifx#1\pgf@stop% + \let\pgf@next\relax% + \else% + \afterassignment\pgfmath@gobbletilpgfmath@% + \expandafter\let\expandafter\pgf@pattern@char#1#1\pgfmath@% + \ifx\pgf@pattern@char/% + \edef\pgf@pattern@tempvar{(\pgfkeysvalueof{#1})}% + \else% + \expandafter\ifcat\pgf@pattern@char\relax% + \edef\pgf@pattern@tempvar{(\the#1)}% + \else% + \edef\pgf@pattern@tempvar{(#1)}% + \fi% + \fi% + \edef\pgf@pattern@tempvars{\pgf@pattern@tempvars\pgf@pattern@tempvar}% + \let\pgf@next\pgf@pattern@check@vars% + \fi% + \pgf@next% +} + +\endinput diff --git a/preprints/A-polynomial-time-algorithm-for-three-machine-unit-job-scheduling-September-24-2026/build/tex/runtime/generic/pgf/basiclayer/pgfcorepoints.code.tex b/preprints/A-polynomial-time-algorithm-for-three-machine-unit-job-scheduling-September-24-2026/build/tex/runtime/generic/pgf/basiclayer/pgfcorepoints.code.tex new file mode 100644 index 0000000000000000000000000000000000000000..ebc1f86d050c30cf277644f460057041551a022c --- /dev/null +++ b/preprints/A-polynomial-time-algorithm-for-three-machine-unit-job-scheduling-September-24-2026/build/tex/runtime/generic/pgf/basiclayer/pgfcorepoints.code.tex @@ -0,0 +1,724 @@ + +\ProvidesFileRCS{pgfcorepoints.code.tex} + +\newdimen\pgf@picminx +\newdimen\pgf@picmaxx +\newdimen\pgf@picminy +\newdimen\pgf@picmaxy + +\newdimen\pgf@pathminx +\newdimen\pgf@pathmaxx +\newdimen\pgf@pathminy +\newdimen\pgf@pathmaxy + +\newif\ifpgf@relevantforpicturesize + +\def\pgf@process#1{{#1\global\pgf@x=\pgf@x\global\pgf@y=\pgf@y}} + + +\def\pgfextract@process#1#2{% + \pgf@process{#2}% + \edef#1{\noexpand\global\pgf@x=\the\pgf@x\noexpand\relax\noexpand\global\pgf@y=\the\pgf@y\noexpand\relax}% +} +\let\pgfsavepgf@process\pgfextract@process% + + + +\def\pgfpoint#1#2{% + \pgfmathsetlength\pgf@x{#1}% + \pgfmathsetlength\pgf@y{#2}\ignorespaces} + + + +\def\pgfqpoint#1#2{\global\pgf@x=#1\relax\global\pgf@y=#2\relax} + + + + +\def\pgfpointorigin{\global\pgf@x=0pt \global\pgf@y=\pgf@x\ignorespaces} + + + + +\def\pgfpointtransformed#1{% + \pgf@process{% + #1% + \pgf@pos@transform@glob% + }% +} + + + +\def\pgfpointdiff#1#2{% + \pgf@process{#1}% + \pgf@xa=\pgf@x% + \pgf@ya=\pgf@y% + \pgf@process{#2}% + \global\advance\pgf@x by-\pgf@xa\relax% + \global\advance\pgf@y by-\pgf@ya\relax\ignorespaces} + + +\def\pgfpointadd#1#2{% + \pgf@process{#1}% + \pgf@xa=\pgf@x% + \pgf@ya=\pgf@y% + \pgf@process{#2}% + \global\advance\pgf@x by\pgf@xa% + \global\advance\pgf@y by\pgf@ya} + + + + +\def\pgfpointscale#1#2{% + \pgf@process{#2}% + \pgfmathparse{#1}% + \global\pgf@x=\pgfmathresult\pgf@x% + \global\pgf@y=\pgfmathresult\pgf@y% +} +\def\pgfqpointscale#1#2{% + \pgf@process{#2}% + \global\pgf@x=#1\pgf@x% + \global\pgf@y=#1\pgf@y% +} + + + +\def\pgfpointintersectionoflines#1#2#3#4{% + {% + \pgf@process{#2}% + \pgf@xa=\pgf@x% + \pgf@ya=\pgf@y% + \pgf@process{#1}% + \advance\pgf@xa by-\pgf@x% + \advance\pgf@ya by-\pgf@y% + \pgf@ya=-\pgf@ya% + \c@pgf@counta=\pgf@xa% + \ifnum\c@pgf@counta<0\relax% + \c@pgf@counta=-\c@pgf@counta\relax% + \fi% + \c@pgf@countb=\pgf@ya% + \ifnum\c@pgf@countb<0\relax% + \c@pgf@countb=-\c@pgf@countb\relax% + \fi% + \advance\c@pgf@counta by\c@pgf@countb\relax% + \divide\c@pgf@counta by 65536\relax% + \ifnum\c@pgf@counta>0\relax% + \divide\pgf@xa by\c@pgf@counta\relax% + \divide\pgf@ya by\c@pgf@counta\relax% + \fi% + \pgf@xc=\pgf@sys@tonumber{\pgf@ya}\pgf@x% + \advance\pgf@xc by\pgf@sys@tonumber{\pgf@xa}\pgf@y% + \pgf@process{#4}% + \pgf@xb=\pgf@x% + \pgf@yb=\pgf@y% + \pgf@process{#3}% + \advance\pgf@xb by-\pgf@x% + \advance\pgf@yb by-\pgf@y% + \pgf@yb=-\pgf@yb% + \c@pgf@counta=\pgf@xb% + \ifnum\c@pgf@counta<0\relax% + \c@pgf@counta=-\c@pgf@counta\relax% + \fi% + \c@pgf@countb=\pgf@yb% + \ifnum\c@pgf@countb<0\relax% + \c@pgf@countb=-\c@pgf@countb\relax% + \fi% + \advance\c@pgf@counta by\c@pgf@countb\relax% + \divide\c@pgf@counta by 65536\relax% + \ifnum\c@pgf@counta>0\relax% + \divide\pgf@xb by\c@pgf@counta\relax% + \divide\pgf@yb by\c@pgf@counta\relax% + \fi% + \pgf@yc=\pgf@sys@tonumber{\pgf@yb}\pgf@x% + \advance\pgf@yc by\pgf@sys@tonumber{\pgf@xb}\pgf@y% + \pgfsettransform{{\pgf@sys@tonumber\pgf@ya}{\pgf@sys@tonumber\pgf@yb}{\pgf@sys@tonumber\pgf@xa}{\pgf@sys@tonumber\pgf@xb}{0pt}{0pt}}% + \pgftransforminvert% + \pgf@process{\pgfpointtransformed{\pgfpoint{\pgf@xc}{\pgf@yc}}}% + }% +} + + + +\def\pgfpointintersectionofcircles#1#2#3#4#5{% + {% + \pgf@process{#1}% + \pgf@xa=\pgf@x% + \pgf@ya=\pgf@y% + \pgfmathsetlength{\pgf@xc}{#3}% + \pgf@process{#2}% + \pgf@xb=\pgf@x% + \pgf@yb=\pgf@y% + \pgfmathsetlength{\pgf@yc}{#4}% + \c@pgf@counta=1\relax% + \loop% + \pgf@scale@downfalse% + \ifdim\pgf@xc>50pt\relax% + \pgf@scale@downtrue% + \fi% + \ifdim\pgf@yc>50pt\relax% + \pgf@scale@downtrue% + \fi% + \ifpgf@scale@down% + \multiply\c@pgf@counta by2\relax% + \divide\pgf@xa by2\relax% + \divide\pgf@ya by2\relax% + \divide\pgf@xb by2\relax% + \divide\pgf@yb by2\relax% + \divide\pgf@xc by2\relax% + \divide\pgf@yc by2\relax% + \repeat% + \dimen0=\pgf@xb% + \advance\dimen0 by-\pgf@xa% + \dimen1=\pgf@yb% + \advance\dimen1 by-\pgf@ya% + \pgf@x=\dimen0\relax% + \pgf@x=\pgf@sys@tonumber{\pgf@x}\pgf@x% + \dimen3=\pgf@x% + \pgf@x=\dimen1\relax% + \advance\dimen3 by\pgf@sys@tonumber{\pgf@x}\pgf@x% + \pgfmathparse{sqrt(\the\dimen3)}% + \dimen2=\pgfmathresult pt% + \pgfmathreciprocal@{\pgfmathresult}% + \dimen5=\pgfmathresult pt% + \dimen4=\dimen3\relax% + \pgf@x=\pgf@xc% + \advance\dimen4 by\pgf@sys@tonumber{\pgf@x}\pgf@x\relax% + \pgf@x=\pgf@yc% + \advance\dimen4 by-\pgf@sys@tonumber{\pgf@x}\pgf@x\relax% + \dimen4=.5\dimen4% + \dimen4=\pgf@sys@tonumber{\dimen5}\dimen4% + \dimen7=\pgf@sys@tonumber{\dimen4}\dimen4\relax% + \pgfmathparse{sqrt(\pgf@sys@tonumber{\pgf@xc}\pgf@xc-\the\dimen7)}% + \dimen6=\pgfmathresult pt% + \dimen8=\pgf@sys@tonumber{\dimen4}\dimen5\relax% + \dimen9=\pgf@sys@tonumber{\dimen6}\dimen5\relax% + \ifnum#5=1\relax% + \pgf@x=\pgf@xa% + \advance\pgf@x by\pgf@sys@tonumber{\dimen0}\dimen8\relax% + \advance\pgf@x by\pgf@sys@tonumber{\dimen1}\dimen9\relax% + \pgf@y=\pgf@ya% + \advance\pgf@y by\pgf@sys@tonumber{\dimen1}\dimen8\relax% + \pgf@xb=\pgf@sys@tonumber{\dimen0}\dimen9% + \pgf@xb=-\pgf@xb% + \advance\pgf@y by\pgf@xb\relax% + \else% + \pgf@x=\pgf@xa% + \advance\pgf@x by\pgf@sys@tonumber{\dimen0}\dimen8\relax% + \pgf@xb=\pgf@sys@tonumber{\dimen1}\dimen9% + \pgf@xb=-\pgf@xb% + \advance\pgf@x by\pgf@xb\relax% + \pgf@y=\pgf@ya% + \advance\pgf@y by\pgf@sys@tonumber{\dimen1}\dimen8\relax% + \advance\pgf@y by\pgf@sys@tonumber{\dimen0}\dimen9\relax% + \fi% + \pgf@x=\c@pgf@counta\pgf@x% + \pgf@y=\c@pgf@counta\pgf@y% + \pgf@process{}% + }% +} +\newif\ifpgf@scale@down + + + +\def\pgfpointlineattime#1#2#3{% + \pgf@process{#3}% + \pgf@xa\pgf@x% + \pgf@ya\pgf@y% + \pgf@xc\pgf@x% + \pgf@yc\pgf@y% + \pgf@process{#2}% + \pgf@xb\pgf@x% + \pgf@yb\pgf@y% + \pgfmathsetmacro\pgf@temp{#1}% + \advance\pgf@xa by-\pgf@x% + \advance\pgf@ya by-\pgf@y% + \global\advance\pgf@x by\pgf@temp\pgf@xa% + \global\advance\pgf@y by\pgf@temp\pgf@ya% +} + + + +\def\pgfpointlineatdistance#1#2#3{% + \pgfmathsetlength\pgf@xa{#1}% + \pgf@process{#2}% + \pgf@xb\pgf@x% + \pgf@yb\pgf@y% + \pgf@process{#3}% + \pgf@xc\pgf@x% + \pgf@yc\pgf@y% + \global\advance\pgf@x by-\pgf@xb\relax% + \global\advance\pgf@y by-\pgf@yb\relax% + \pgf@process{\pgfpointnormalised{}}% + \pgf@ya=\pgf@xa\relax% + \pgf@xa=\pgf@sys@tonumber{\pgf@x}\pgf@xa% + \pgf@ya=\pgf@sys@tonumber{\pgf@y}\pgf@ya% + \global\pgf@x=\pgf@xb\relax% + \global\pgf@y=\pgf@yb\relax% + \global\advance\pgf@x by\pgf@xa\relax% + \global\advance\pgf@y by\pgf@ya\relax% +} + + + +\def\pgfpointcurveattime#1#2#3#4#5{% + \pgfmathparse{#1}% + \let\pgf@time@s=\pgfmathresult% + \global\pgf@x=\pgfmathresult pt% + \global\pgf@x=-\pgf@x% + \advance\pgf@x by 1pt% + \edef\pgf@time@t{\pgf@sys@tonumber{\pgf@x}}% + \pgf@process{#5}% + \pgf@xc=\pgf@x% + \pgf@yc=\pgf@y% + \pgf@process{#4}% + \pgf@xb=\pgf@x% + \pgf@yb=\pgf@y% + \pgf@process{#3}% + \pgf@xa=\pgf@x% + \pgf@ya=\pgf@y% + \pgf@process{#2}% + \global\pgf@x=\pgf@time@t\pgf@x\global\advance\pgf@x by\pgf@time@s\pgf@xa% + \global\pgf@y=\pgf@time@t\pgf@y\global\advance\pgf@y by\pgf@time@s\pgf@ya% + \pgf@xa=\pgf@time@t\pgf@xa\advance\pgf@xa by\pgf@time@s\pgf@xb% + \pgf@ya=\pgf@time@t\pgf@ya\advance\pgf@ya by\pgf@time@s\pgf@yb% + \pgf@xb=\pgf@time@t\pgf@xb\advance\pgf@xb by\pgf@time@s\pgf@xc% + \pgf@yb=\pgf@time@t\pgf@yb\advance\pgf@yb by\pgf@time@s\pgf@yc% + \global\pgf@x=\pgf@time@t\pgf@x\global\advance\pgf@x by\pgf@time@s\pgf@xa% + \global\pgf@y=\pgf@time@t\pgf@y\global\advance\pgf@y by\pgf@time@s\pgf@ya% + \pgf@xa=\pgf@time@t\pgf@xa\advance\pgf@xa by\pgf@time@s\pgf@xb% + \pgf@ya=\pgf@time@t\pgf@ya\advance\pgf@ya by\pgf@time@s\pgf@yb% + \pgf@xb=\pgf@x% + \pgf@yb=\pgf@y% + \global\pgf@x=\pgf@time@t\pgf@x\global\advance\pgf@x by\pgf@time@s\pgf@xa% + \global\pgf@y=\pgf@time@t\pgf@y\global\advance\pgf@y by\pgf@time@s\pgf@ya% +} + + + + +\def\pgfpointarcaxesattime#1#2#3#4#5#6{% + \pgfmathsetmacro\pgf@angle@start{#5}% + \pgfmathsetmacro\pgf@angle@end{#6}% + \pgfmathparse{#1}% + \global\pgf@x=\pgfmathresult pt% + \global\pgf@x=-\pgf@x% + \advance\pgf@x by 1pt% + \pgfmathsetmacro\pgf@angle@mid{\pgf@angle@end*\pgfmathresult+\pgf@angle@start*\pgf@sys@tonumber{\pgf@x}}% + \pgfmathcos@{\pgf@angle@mid}% + \let\pgf@angle@cos\pgfmathresult% + \pgfmathsin@{\pgf@angle@mid}% + \let\pgf@angle@sin\pgfmathresult% + \pgf@process{#3}% + \edef\pgf@angle@zero@axis{\noexpand\pgfqpoint{\the\pgf@x}{\the\pgf@y}} + \pgf@process{#4}% + \edef\pgf@angle@ninety@axis{\noexpand\pgfqpoint{\the\pgf@x}{\the\pgf@y}} + \pgf@process{\pgfpointadd{\pgfpointscale{\pgf@angle@sin}{\pgf@angle@zero@axis}}% + {\pgfpointscale{-\pgf@angle@cos}{\pgf@angle@ninety@axis}}}% + \ifdim\pgf@angle@start pt>\pgf@angle@end pt% + \pgf@xa=\pgf@x% + \pgf@ya=\pgf@y% + \else% + \pgf@xa=-\pgf@x% + \pgf@ya=-\pgf@y% + \fi% + \pgf@process{\pgfpointadd{#2}{% + \pgfpointadd{\pgfpointscale{\pgf@angle@cos}{\pgf@angle@zero@axis}}% + {\pgfpointscale{\pgf@angle@sin}{\pgf@angle@ninety@axis}}}}% +} + + + +\newdimen\pgf@xx +\newdimen\pgf@xy +\newdimen\pgf@yx +\newdimen\pgf@yy +\newdimen\pgf@zx +\newdimen\pgf@zy + + + + +\def\pgfpointpolar#1#2{% + \pgfutil@in@{and }{#2}% + \ifpgfutil@in@% + \pgf@polar@#2\@@% + \else% + \pgf@polar@#2 and #2\@@% + \fi% + \pgfmathparse{#1}% + \let\pgfpoint@angle=\pgfmathresult% + \pgfmathcos@{\pgfpoint@angle}% + \global\pgf@x=\pgfmathresult\pgf@x% + \pgfmathsin@{\pgfpoint@angle}% + \global\pgf@y=\pgfmathresult\pgf@y% +} + +\def\pgf@polar@#1and #2\@@{% + \pgfmathsetlength{\pgf@y}{#2}% + \pgfmathsetlength{\pgf@x}{#1}% +} + + +\def\pgfqpointpolar#1#2{% + \global\pgf@x=#2% + \global\pgf@y=\pgf@x% + \pgfmathcos@{#1}% + \global\pgf@x=\pgfmathresult\pgf@x% + \pgfmathsin@{#1}% + \global\pgf@y=\pgfmathresult\pgf@y\relax% +} + + + + + +\def\pgfpointpolarxy#1#2{% + \pgfutil@in@{and }{#2}% + \ifpgfutil@in@% + \pgf@polarxy@#2\@@% + \else% + \pgf@polarxy@#2and #2\@@% + \fi% + \pgfmathparse{#1}% + \let\pgfpoint@angle=\pgfmathresult% + \pgfmathcos@{\pgfpoint@angle}% + \pgf@xa=\pgfmathresult\pgf@xa% + \pgfmathsin@{\pgfpoint@angle}% + \pgf@ya=\pgfmathresult\pgf@ya% + \global\pgf@x=\pgf@sys@tonumber{\pgf@xa}\pgf@xx% + \global\advance\pgf@x by \pgf@sys@tonumber{\pgf@ya}\pgf@yx% + \global\pgf@y=\pgf@sys@tonumber{\pgf@xa}\pgf@xy% + \global\advance\pgf@y by \pgf@sys@tonumber{\pgf@ya}\pgf@yy} + +\def\pgf@polarxy@#1and #2\@@{% + \pgfmathsetlength{\pgf@xa}{#1}% + \pgfmathsetlength{\pgf@ya}{#2}% +} + + + + +\def\pgfpointcylindrical#1#2#3{% + \pgfpointpolarxy{#1}{#2}% + \pgfmathparse{#3}% + \global\advance\pgf@x by \pgfmathresult\pgf@zx% + \global\advance\pgf@y by \pgfmathresult\pgf@zy} + + + +\def\pgfpointspherical#1#2#3{% + \pgfmathparse{#1}% + \let\pgfpoint@angle=\pgfmathresult% + \pgfmathsin@{\pgfpoint@angle}% + \pgf@xb=\pgfmathresult\pgf@xx% + \pgf@yb=\pgfmathresult\pgf@xy% + \pgfmathcos@{\pgfpoint@angle}% + \advance\pgf@xb by \pgfmathresult\pgf@yx% + \advance\pgf@yb by \pgfmathresult\pgf@yy% + \pgfmathparse{#2}% + \let\pgfpoint@angle=\pgfmathresult% + \pgfmathcos@{\pgfpoint@angle}% + \pgf@xc=\pgfmathresult\pgf@xb% + \pgf@yc=\pgfmathresult\pgf@yb% + \pgfmathsin@{\pgfpoint@angle}% + \advance\pgf@xc by \pgfmathresult\pgf@zx% + \advance\pgf@yc by \pgfmathresult\pgf@zy% + \pgfmathparse{#3}% + \global\pgf@x=\pgfmathresult\pgf@xc% + \global\pgf@y=\pgfmathresult\pgf@yc\relax% +} + + + +\def\pgfpointxy#1#2{% + \pgfmathparse{#1}% + \let\pgftemp@x=\pgfmathresult% + \pgfmathparse{#2}% + \let\pgftemp@y=\pgfmathresult% + \global\pgf@x=\pgftemp@x\pgf@xx% + \global\advance\pgf@x by \pgftemp@y\pgf@yx% + \global\pgf@y=\pgftemp@x\pgf@xy% + \global\advance\pgf@y by \pgftemp@y\pgf@yy} + + +\def\pgfqpointxy#1#2{% + \global\pgf@x=#1\pgf@xx% + \global\advance\pgf@x by #2\pgf@yx% + \global\pgf@y=#1\pgf@xy% + \global\advance\pgf@y by #2\pgf@yy} + + +\def\pgfpointxyz#1#2#3{% + \pgfmathparse{#1}% + \let\pgftemp@x=\pgfmathresult% + \pgfmathparse{#2}% + \let\pgftemp@y=\pgfmathresult% + \pgfmathparse{#3}% + \let\pgftemp@z=\pgfmathresult% + \global\pgf@x=\pgftemp@x\pgf@xx% + \global\advance\pgf@x by \pgftemp@y\pgf@yx% + \global\advance\pgf@x by \pgftemp@z\pgf@zx% + \global\pgf@y=\pgftemp@x\pgf@xy% + \global\advance\pgf@y by \pgftemp@y\pgf@yy% + \global\advance\pgf@y by \pgftemp@z\pgf@zy} + +\def\pgfqpointxyz#1#2#3{% + \global\pgf@x=#1\pgf@xx% + \global\advance\pgf@x by #2\pgf@yx% + \global\advance\pgf@x by #3\pgf@zx% + \global\pgf@y=#1\pgf@xy% + \global\advance\pgf@y by #2\pgf@yy% + \global\advance\pgf@y by #3\pgf@zy} + + + + +\def\pgfsetxvec#1{% + \pgf@process{#1}% + \pgf@xx=\pgf@x% + \pgf@xy=\pgf@y% + \ignorespaces} + + + +\def\pgfsetyvec#1{% + \pgf@process{#1}% + \pgf@yx=\pgf@x% + \pgf@yy=\pgf@y% + \ignorespaces} + + + +\def\pgfsetzvec#1{% + \pgf@process{#1}% + \pgf@zx=\pgf@x% + \pgf@zy=\pgf@y% + \ignorespaces} + + + +\pgfsetxvec{\pgfpoint{1cm}{0cm}} +\pgfsetyvec{\pgfpoint{0cm}{1cm}} +\pgfsetzvec{\pgfpoint{-0.385cm}{-0.385cm}} + + + + + +\def\pgfpointnormalised#1{% + \pgf@process{#1}% + \pgfmathatantwo{\the\pgf@y}{\the\pgf@x}% + \let\pgf@tmp=\pgfmathresult% + \pgfmathcos@{\pgf@tmp}% + \pgf@x=\pgfmathresult pt\relax% + \pgfmathsin@{\pgf@tmp}% + \pgf@y=\pgfmathresult pt\relax% +} + + + + + + +\def\pgfpointborderrectangle#1#2{% + \pgf@process{#2}% + \pgf@xb=\pgf@x% + \pgf@yb=\pgf@y% + \pgf@process{#1}% + \pgf@xa=\pgf@x% + \pgf@ya=\pgf@y% + \ifnum\pgf@xa<0\relax% + \global\pgf@x=-\pgf@x% + \fi% + \ifnum\pgf@ya<0\relax% + \global\pgf@y=-\pgf@y% + \fi% + \pgf@xc=.125\pgf@x% + \pgf@yc=.125\pgf@y% + \c@pgf@counta=\pgf@xc% + \c@pgf@countb=\pgf@yc% + \ifnum\c@pgf@countb<\c@pgf@counta% + \ifnum\c@pgf@counta<255\relax% + \global\pgf@y=\pgf@yb\relax% + \global\pgf@x=0pt\relax% + \else% + \pgf@xc=8192pt% + \divide\pgf@xc by\c@pgf@counta% + \global\pgf@y=\pgf@sys@tonumber{\pgf@xc}\pgf@y% + \global\pgf@y=\pgf@sys@tonumber{\pgf@xb}\pgf@y% + \ifnum\pgf@y<\pgf@yb% + \global\pgf@x=\pgf@xb% + \else% + \ifnum\c@pgf@countb<255\relax% + \global\pgf@x=\pgf@xb\relax% + \global\pgf@y=0pt\relax% + \else% + \pgf@yc=8192pt% + \divide\pgf@yc by\c@pgf@countb% + \global\pgf@x=\pgf@sys@tonumber{\pgf@yc}\pgf@x% + \global\pgf@x=\pgf@sys@tonumber{\pgf@yb}\pgf@x% + \global\pgf@y=\pgf@yb% + \fi% + \fi% + \fi% + \else% + \ifnum\c@pgf@countb<255\relax% + \global\pgf@x=\pgf@xb\relax% + \global\pgf@y=0pt\relax% + \else% + \pgf@yc=8192pt% + \divide\pgf@yc by\c@pgf@countb% + \global\pgf@x=\pgf@sys@tonumber{\pgf@yc}\pgf@x% + \global\pgf@x=\pgf@sys@tonumber{\pgf@yb}\pgf@x% + \ifnum\pgf@x<\pgf@xb% + \global\pgf@y=\pgf@yb% + \else% + \ifnum\c@pgf@counta<255\relax% + \global\pgf@y=\pgf@yb\relax% + \global\pgf@x=0pt\relax% + \else% + \pgf@xc=8192pt% + \divide\pgf@xc by\c@pgf@counta% + \global\pgf@y=\pgf@sys@tonumber{\pgf@xc}\pgf@y% + \global\pgf@y=\pgf@sys@tonumber{\pgf@xb}\pgf@y% + \global\pgf@x=\pgf@xb% + \fi% + \fi% + \fi% + \fi% + \ifnum\pgf@xa<0\relax\global\pgf@x=-\pgf@x\fi% + \ifnum\pgf@ya<0\relax\global\pgf@y=-\pgf@y\fi% +} + + + + + +\def\pgfpointborderellipse#1#2{% + \pgf@process{#2}% + \pgf@xa=\pgf@x% + \pgf@ya=\pgf@y% + \ifdim\pgf@xa=\pgf@ya% + \pgf@process{\pgfpointnormalised{#1}}% + \global\pgf@x=\pgf@sys@tonumber{\pgf@xa}\pgf@x% + \global\pgf@y=\pgf@sys@tonumber{\pgf@xa}\pgf@y% + \else% + \ifdim\pgf@xa<\pgf@ya% + \c@pgf@countb=\pgf@ya% + \divide\c@pgf@countb by65536\relax% + \global\divide\pgf@x by\c@pgf@countb% + \global\divide\pgf@y by\c@pgf@countb% + \pgf@xc=\pgf@x% + \pgf@yc=8192pt% + \global\pgf@y=.125\pgf@y% + \c@pgf@countb=\pgf@y% + \divide\pgf@yc by\c@pgf@countb% + \pgf@process{#1}% + \global\pgf@y=\pgf@sys@tonumber{\pgf@yc}\pgf@y% + \global\pgf@y=\pgf@sys@tonumber{\pgf@xc}\pgf@y% + \pgf@process{\pgfpointnormalised{}}% + \global\pgf@x=\pgf@sys@tonumber{\pgf@xa}\pgf@x% + \global\pgf@y=\pgf@sys@tonumber{\pgf@ya}\pgf@y% + \else% + \c@pgf@countb=\pgf@xa% + \divide\c@pgf@countb by65536\relax% + \global\divide\pgf@x by\c@pgf@countb% + \global\divide\pgf@y by\c@pgf@countb% + \pgf@yc=\pgf@y% + \pgf@xc=8192pt% + \global\pgf@x=.125\pgf@x% + \c@pgf@countb=\pgf@x% + \divide\pgf@xc by\c@pgf@countb% + \pgf@process{#1}% + \global\pgf@x=\pgf@sys@tonumber{\pgf@yc}\pgf@x% + \global\pgf@x=\pgf@sys@tonumber{\pgf@xc}\pgf@x% + \pgf@process{\pgfpointnormalised{}}% + \global\pgf@x=\pgf@sys@tonumber{\pgf@xa}\pgf@x% + \global\pgf@y=\pgf@sys@tonumber{\pgf@ya}\pgf@y% + \fi% + \fi% +} + + + + + + +\def\pgfextractx#1#2{% + \pgf@process{#2}% + #1=\pgf@x\relax} + + + +\def\pgfextracty#1#2{% + \pgf@process{#2}% + #1=\pgf@y\relax} + + +\def\pgfgetlastxy#1#2{% + \edef#1{\the\pgf@x}% + \edef#2{\the\pgf@y}% +}% + + +\def\pgf@def#1#2#3{\expandafter\def\csname pgf@#1#2\endcsname{#3}} +\pgf@def{cosfrac}{0}{1} +\pgf@def{cosfrac}{1}{0.99995} \pgf@def{cosfrac}{2}{0.9998} +\pgf@def{cosfrac}{3}{0.99955} \pgf@def{cosfrac}{4}{0.999201} +\pgf@def{cosfrac}{5}{0.998752} \pgf@def{cosfrac}{6}{0.998205} +\pgf@def{cosfrac}{7}{0.997559} \pgf@def{cosfrac}{8}{0.996815} +\pgf@def{cosfrac}{9}{0.995974} \pgf@def{cosfrac}{10}{0.995037} +\pgf@def{cosfrac}{11}{0.994004} \pgf@def{cosfrac}{12}{0.992877} +\pgf@def{cosfrac}{13}{0.991656} \pgf@def{cosfrac}{14}{0.990342} +\pgf@def{cosfrac}{15}{0.988936} \pgf@def{cosfrac}{16}{0.987441} +\pgf@def{cosfrac}{17}{0.985856} \pgf@def{cosfrac}{18}{0.984183} +\pgf@def{cosfrac}{19}{0.982424} \pgf@def{cosfrac}{20}{0.980581} +\pgf@def{cosfrac}{21}{0.978653} \pgf@def{cosfrac}{22}{0.976644} +\pgf@def{cosfrac}{23}{0.974555} \pgf@def{cosfrac}{24}{0.972387} +\pgf@def{cosfrac}{25}{0.970143} \pgf@def{cosfrac}{26}{0.967823} +\pgf@def{cosfrac}{27}{0.965429} \pgf@def{cosfrac}{28}{0.962964} +\pgf@def{cosfrac}{29}{0.960429} \pgf@def{cosfrac}{30}{0.957826} +\pgf@def{cosfrac}{31}{0.955157} \pgf@def{cosfrac}{32}{0.952424} +\pgf@def{cosfrac}{33}{0.949629} \pgf@def{cosfrac}{34}{0.946773} +\pgf@def{cosfrac}{35}{0.943858} \pgf@def{cosfrac}{36}{0.940887} +\pgf@def{cosfrac}{37}{0.937862} \pgf@def{cosfrac}{38}{0.934784} +\pgf@def{cosfrac}{39}{0.931655} \pgf@def{cosfrac}{40}{0.928477} +\pgf@def{cosfrac}{41}{0.925252} \pgf@def{cosfrac}{42}{0.921982} +\pgf@def{cosfrac}{43}{0.918669} \pgf@def{cosfrac}{44}{0.915315} +\pgf@def{cosfrac}{45}{0.911922} \pgf@def{cosfrac}{46}{0.90849} +\pgf@def{cosfrac}{47}{0.905024} \pgf@def{cosfrac}{48}{0.901523} +\pgf@def{cosfrac}{49}{0.89799} \pgf@def{cosfrac}{50}{0.894427} +\pgf@def{cosfrac}{51}{0.890835} \pgf@def{cosfrac}{52}{0.887217} +\pgf@def{cosfrac}{53}{0.883573} \pgf@def{cosfrac}{54}{0.879905} +\pgf@def{cosfrac}{55}{0.876216} \pgf@def{cosfrac}{56}{0.872506} +\pgf@def{cosfrac}{57}{0.868777} \pgf@def{cosfrac}{58}{0.865031} +\pgf@def{cosfrac}{59}{0.861269} \pgf@def{cosfrac}{60}{0.857493} +\pgf@def{cosfrac}{61}{0.853704} \pgf@def{cosfrac}{62}{0.849903} +\pgf@def{cosfrac}{63}{0.846092} \pgf@def{cosfrac}{64}{0.842271} +\pgf@def{cosfrac}{65}{0.838444} \pgf@def{cosfrac}{66}{0.834609} +\pgf@def{cosfrac}{67}{0.83077} \pgf@def{cosfrac}{68}{0.826927} +\pgf@def{cosfrac}{69}{0.82308} \pgf@def{cosfrac}{70}{0.819232} +\pgf@def{cosfrac}{71}{0.815383} \pgf@def{cosfrac}{72}{0.811534} +\pgf@def{cosfrac}{73}{0.807687} \pgf@def{cosfrac}{74}{0.803842} +\pgf@def{cosfrac}{75}{0.8} \pgf@def{cosfrac}{76}{0.796162} +\pgf@def{cosfrac}{77}{0.792329} \pgf@def{cosfrac}{78}{0.788502} +\pgf@def{cosfrac}{79}{0.784682} \pgf@def{cosfrac}{80}{0.780869} +\pgf@def{cosfrac}{81}{0.777064} \pgf@def{cosfrac}{82}{0.773268} +\pgf@def{cosfrac}{83}{0.769481} \pgf@def{cosfrac}{84}{0.765705} +\pgf@def{cosfrac}{85}{0.761939} \pgf@def{cosfrac}{86}{0.758185} +\pgf@def{cosfrac}{87}{0.754443} \pgf@def{cosfrac}{88}{0.750714} +\pgf@def{cosfrac}{89}{0.746997} \pgf@def{cosfrac}{90}{0.743294} +\pgf@def{cosfrac}{91}{0.739605} \pgf@def{cosfrac}{92}{0.735931} +\pgf@def{cosfrac}{93}{0.732272} \pgf@def{cosfrac}{94}{0.728628} +\pgf@def{cosfrac}{95}{0.724999} \pgf@def{cosfrac}{96}{0.721387} +\pgf@def{cosfrac}{97}{0.717792} \pgf@def{cosfrac}{98}{0.714213} +\pgf@def{cosfrac}{99}{0.710651} \pgf@def{cosfrac}{100}{0.707107} + + + + + +\let\pgfpointtransformednonlinear\pgfpointtransformed + + +\endinput diff --git a/preprints/A-polynomial-time-algorithm-for-three-machine-unit-job-scheduling-September-24-2026/build/tex/runtime/generic/pgf/basiclayer/pgfcorequick.code.tex b/preprints/A-polynomial-time-algorithm-for-three-machine-unit-job-scheduling-September-24-2026/build/tex/runtime/generic/pgf/basiclayer/pgfcorequick.code.tex new file mode 100644 index 0000000000000000000000000000000000000000..49bd4f5b66c1e44feba662de724edfba868c03c1 --- /dev/null +++ b/preprints/A-polynomial-time-algorithm-for-three-machine-unit-job-scheduling-September-24-2026/build/tex/runtime/generic/pgf/basiclayer/pgfcorequick.code.tex @@ -0,0 +1,67 @@ + +\ProvidesFileRCS{pgfcorequick.code.tex} + + + +\def\pgfpathqmoveto#1#2{\pgfsyssoftpath@moveto{#1}{#2}} + + + +\def\pgfpathqlineto#1#2{\pgfsyssoftpath@lineto{#1}{#2}} + + + +\def\pgfpathqcurveto#1#2#3#4#5#6{\pgfsyssoftpath@curveto{#1}{#2}{#3}{#4}{#5}{#6}} + + + + + +\def\pgfpathqcircle#1{% + {% + \pgf@x=#1% + \pgf@y=0.5522847\pgf@x% + \pgfsyssoftpath@moveto{\the\pgf@x}{0pt}% + \pgfsyssoftpath@curveto{\the\pgf@x}{\the\pgf@y}{\the\pgf@y}{\the\pgf@x}{0pt}{\the\pgf@x}% + \pgfsyssoftpath@curveto{-\the\pgf@y}{\the\pgf@x}{-\the\pgf@x}{\the\pgf@y}{-\the\pgf@x}{0pt}% + \pgfsyssoftpath@curveto{-\the\pgf@x}{-\the\pgf@y}{-\the\pgf@y}{-\the\pgf@x}{0pt}{-\the\pgf@x}% + \pgfsyssoftpath@curveto{\the\pgf@y}{-\the\pgf@x}{\the\pgf@x}{-\the\pgf@y}{\the\pgf@x}{0pt}% + \pgfsyssoftpath@closepath% + }% +} + + + + +\def\pgfusepathqstroke{% + \pgfsyssoftpath@flushcurrentpath% + \pgfsys@stroke% + \pgf@resetpathsizes% +} + + + +\def\pgfusepathqfill{% + \pgfsyssoftpath@flushcurrentpath% + \pgfsys@fill% + \pgf@resetpathsizes% +} + + + +\def\pgfusepathqfillstroke{% + \pgfsyssoftpath@flushcurrentpath% + \pgfsys@fillstroke% + \pgf@resetpathsizes% +} + + +\def\pgfusepathqclip{% + \pgfsyssoftpath@flushcurrentpath% + \pgfsys@clipnext% + \pgfsys@discardpath% + \pgf@resetpathsizes% +} + + +\endinput diff --git a/preprints/A-polynomial-time-algorithm-for-three-machine-unit-job-scheduling-September-24-2026/build/tex/runtime/generic/pgf/basiclayer/pgfcorerdf.code.tex b/preprints/A-polynomial-time-algorithm-for-three-machine-unit-job-scheduling-September-24-2026/build/tex/runtime/generic/pgf/basiclayer/pgfcorerdf.code.tex new file mode 100644 index 0000000000000000000000000000000000000000..eef0d328b3a9d3938a85ef24d64a8d3d2c07792c --- /dev/null +++ b/preprints/A-polynomial-time-algorithm-for-three-machine-unit-job-scheduling-September-24-2026/build/tex/runtime/generic/pgf/basiclayer/pgfcorerdf.code.tex @@ -0,0 +1,83 @@ + +\ProvidesFileRCS{pgfcorerdf.code.tex} + + + + + + + + + +\def\pgfrdfabout#1{\pgfsys@rdf@about{#1}} + + + + +\def\pgfrdfcontent#1{\pgfsys@rdf@content{#1}} + + + + +\def\pgfrdfdatatype#1{\pgfsys@rdf@datatype{#1}} + + + + +\def\pgfrdfhref#1{\pgfsys@rdf@href{#1}} + + + + +\def\pgfrdfinlist{\pgfsys@rdf@inlist} + + + + + +\def\pgfrdfprefix#1{\pgfsys@rdf@prefix{#1}} + + + + + +\def\pgfrdfproperty#1{\pgfsys@rdf@property{#1}} + + + + +\def\pgfrdfrel#1{\pgfsys@rdf@rel{#1}} + + + +\def\pgfrdfresource#1{\pgfsys@rdf@resource{#1}} + + + + + +\def\pgfrdfrev#1{\pgfsys@rdf@rev{#1}} + + + + + +\def\pgfrdfsrc#1{\pgfsys@rdf@src{#1}} + + + + + +\def\pgfrdftypeof#1{\pgfsys@rdf@typeof{#1}} + + + + + +\def\pgfrdfvocab#1{\pgfsys@rdf@vocab{#1}} + + + + + +\endinput diff --git a/preprints/A-polynomial-time-algorithm-for-three-machine-unit-job-scheduling-September-24-2026/build/tex/runtime/generic/pgf/basiclayer/pgfcorescopes.code.tex b/preprints/A-polynomial-time-algorithm-for-three-machine-unit-job-scheduling-September-24-2026/build/tex/runtime/generic/pgf/basiclayer/pgfcorescopes.code.tex new file mode 100644 index 0000000000000000000000000000000000000000..4036c90ff3df07178ae730c610104701b81b5195 --- /dev/null +++ b/preprints/A-polynomial-time-algorithm-for-three-machine-unit-job-scheduling-September-24-2026/build/tex/runtime/generic/pgf/basiclayer/pgfcorescopes.code.tex @@ -0,0 +1,496 @@ + +\ProvidesFileRCS{pgfcorescopes.code.tex} + + +\newbox\pgfpic +\newbox\pgf@hbox + +\newbox\pgf@layerbox@main + +\newcount\pgf@picture@serial@count + + + +\newif\ifpgfrememberpicturepositiononpage + + + + + + + +\def\pgfscope{% + \pgfsyssoftpath@setcurrentpath\pgfutil@empty% + \pgfsys@beginscope% + \pgf@resetpathsizes% + \edef\pgfscope@linewidth{\the\pgflinewidth}% + \let\pgfscope@stroke@color=\pgf@strokecolor@global% + \let\pgfscope@fill@color=\pgf@fillcolor@global% + \begingroup} +\def\endpgfscope{% + \endgroup% + \global\pgflinewidth=\pgfscope@linewidth% + \global\let\pgf@strokecolor@global=\pgfscope@stroke@color% + \global\let\pgf@fillcolor@global=\pgfscope@fill@color% + \pgfsys@endscope} + + + + + +\def\pgfqbox#1{% + \pgfsys@hbox#1% +} + + + +\def\pgfqboxsynced#1{% + \pgfsys@hboxsynced#1% +} + + + +\def\pgftext{\pgfutil@ifnextchar[\pgf@text{\pgf@text[]}}% +\def\pgf@text[#1]{% + \def\pgf@text@options{#1}% + \pgf@maketext\pgf@after@text} +\def\pgf@after@text{% + {% + \def\pgf@text@hshift{center}% + \def\pgf@text@vshift{center}% + \expandafter\pgf@text@setkeys\expandafter{\pgf@text@options}% + \csname pgf@halign\pgf@text@hshift\endcsname% + \csname pgf@valign\pgf@text@vshift\endcsname% + \pgfapproximatenonlineartransformation% + \pgf@process{\pgfpointtransformed{\pgfqpoint{0pt}{\dp\pgf@hbox}}}% + \pgf@protocolsizes{\pgf@x}{\pgf@y}% + \pgf@process{\pgfpointtransformed{\pgfqpoint{\wd\pgf@hbox}{\dp\pgf@hbox}}}% + \pgf@protocolsizes{\pgf@x}{\pgf@y}% + \pgf@process{\pgfpointtransformed{\pgfqpoint{0pt}{\ht\pgf@hbox}}}% + \pgf@protocolsizes{\pgf@x}{\pgf@y}% + \pgf@process{\pgfpointtransformed{\pgfqpoint{\wd\pgf@hbox}{\ht\pgf@hbox}}}% + \pgf@protocolsizes{\pgf@x}{\pgf@y}% + \pgfqboxsynced{\pgf@hbox}% + }% +} +\def\pgf@text@setkeys{\pgfqkeys{/pgf/text}}% + +\pgfkeys{/pgf/text/left/.code=\def\pgf@text@hshift{left}} +\pgfkeys{/pgf/text/center/.code=} +\pgfkeys{/pgf/text/right/.code=\def\pgf@text@hshift{right}} +\pgfkeys{/pgf/text/top/.code=\def\pgf@text@vshift{top}} +\pgfkeys{/pgf/text/bottom/.code=\def\pgf@text@vshift{bottom}} +\pgfkeys{/pgf/text/base/.code=\def\pgf@text@vshift{base}} +\pgfkeys{/pgf/text/at/.cd,.code=\pgftransformshift{#1},.value required} +\pgfkeys{/pgf/text/x/.cd,.code=\pgftransformxshift{#1},.value required} +\pgfkeys{/pgf/text/y/.cd,.code=\pgftransformyshift{#1},.value required} +\pgfkeys{/pgf/text/rotate/.cd,.code=\pgftransformrotate{#1},.value required} + +\def\pgf@halignleft{}% +\def\pgf@haligncenter{\pgftransformxshift{+-.5\wd\pgf@hbox}} +\def\pgf@halignright{\pgftransformxshift{+-\wd\pgf@hbox}}% +\def\pgf@valignbase{}% +\def\pgf@valignbottom{\pgftransformyshift{+\dp\pgf@hbox}}% +\def\pgf@valigncenter{\pgftransformyshift{+.5\dp\pgf@hbox}\pgftransformyshift{+-.5\ht\pgf@hbox}}% +\def\pgf@valigntop{\pgftransformyshift{+-\ht\pgf@hbox}}% + + +\def\pgf@maketext#1{% + \def\pgf@@maketextafter{#1}% + \setbox\pgf@hbox=\hbox\bgroup% + \pgfinterruptpicture% + \bgroup% + \aftergroup\pgf@collectresetcolor% + \let\next=% +} +\def\pgf@collectresetcolor{% + \pgfutil@ifnextchar\reset@color% + {\reset@color\afterassignment\pgf@collectresetcolor\let\pgf@temp=}% + {\pgf@textdone}% +} +\def\pgf@textdone{% + \endpgfinterruptpicture% + \egroup% + \pgf@@maketextafter% +} + +\long\def\pgf@makehbox#1{% + \setbox\pgf@hbox=\hbox{{% + \pgfinterruptpicture% + #1% + \endpgfinterruptpicture% + }}} + + + + +\def\pgfresetboundingbox{% + \global\pgf@picmaxx=-16000pt\relax% + \global\pgf@picminx=16000pt\relax% + \global\pgf@picmaxy=-16000pt\relax% + \global\pgf@picminy=16000pt\relax% +}% + +\def\pgfpicture{% + \begingroup% + \pgfpicturetrue% + \global\advance\pgf@picture@serial@count by1\relax% + \edef\pgfpictureid{pgfid\the\pgf@picture@serial@count}% + \let\pgf@nodecallback=\pgfutil@gobble% + \pgf@picmaxx=-16000pt\relax% + \pgf@picminx=16000pt\relax% + \pgf@picmaxy=-16000pt\relax% + \pgf@picminy=16000pt\relax% + \pgf@relevantforpicturesizetrue% + \pgf@resetpathsizes% + \pgfutil@ifnextchar\bgroup\pgf@oldpicture\pgf@picture} +\def\pgf@oldpicture#1#2#3#4{% + \pgfmathsetlength\pgf@picminx{#1}% + \pgfmathsetlength\pgf@picminy{#2}% + \pgfmathsetlength\pgf@picmaxx{#3}% + \pgfmathsetlength\pgf@picmaxy{#4}% + \pgf@relevantforpicturesizefalse% + \pgf@picture} + +\def\pgf@picture{% + \setbox\pgfpic\hbox to0pt\bgroup% + \begingroup% + \pgfsys@beginpicture% + \pgfsys@beginscope% + \begingroup% + \pgfsetcolor{.}% + \pgfsetlinewidth{0.4pt}% + \pgftransformreset% + \pgfsyssoftpath@setcurrentpath\pgfutil@empty% + \begingroup% + \let\pgf@setlengthorig=\setlength% + \let\pgf@addtolengthorig=\addtolength% + \let\pgf@selectfontorig=\selectfont% + \let\setlength=\pgf@setlength% + \let\addtolength=\pgf@addtolength% + \let\selectfont=\pgf@selectfont% + \nullfont\spaceskip0pt\xspaceskip0pt% + \tracinglostchars=% + \ifnum\tracinglostchars>2 2\else\tracinglostchars\fi + \setbox\pgf@layerbox@main\hbox to0pt\bgroup% + \begingroup% + } +\def\endpgfpicture{% + \ifpgfrememberpicturepositiononpage% + \hbox to0pt{\pgfsys@markposition{\pgfpictureid}}% + \fi% + \ifdim\pgf@picmaxx=-16000pt\relax% + \global\pgf@picmaxx=0pt\relax% + \global\pgf@picminx=0pt\relax% + \global\pgf@picmaxy=0pt\relax% + \global\pgf@picminy=0pt\relax% + \fi% + \pgf@relevantforpicturesizefalse% + \pgf@process{\pgf@baseline}% + \xdef\pgf@shift@baseline{\the\pgf@y}% + \pgf@process{\pgf@trimleft}% + \global\advance\pgf@x by-\pgf@picminx + \xdef\pgf@trimleft@final{-\the\pgf@x}% + \pgf@process{\pgf@trimright}% + \global\advance\pgf@x by-\pgf@picmaxx + \xdef\pgf@trimright@final{\the\pgf@x}% + \pgf@remember@layerlist@globally + \endgroup% + \hss% + \egroup% + \pgf@restore@layerlist@from@global + \pgf@insertlayers% + \endgroup% + \pgfsys@discardpath% + \endgroup% + \pgfsys@endscope% + \pgfsys@endpicture% + \endgroup% + \hss + \egroup% + \pgfsys@typesetpicturebox\pgfpic% + \endgroup% +} + +\def\pgf@remember@layerlist@globally{% + \global\let\pgf@layerlist@=\pgf@layerlist +}% +\def\pgf@restore@layerlist@from@global{% + \let\pgf@layerlist=\pgf@layerlist@ +}% +\def\pgf@insertlayers{% + \box\pgf@layerbox@main% +} + +\def\pgf@selectfont{\pgf@selectfontorig\nullfont} + +\def\pgf@setlength#1#2{% + \begingroup% + \pgfutil@selectfont% + \pgf@setlengthorig#1{#2}% + \expandafter% + \endgroup% + \expandafter#1\expandafter=\the#1\relax} +\def\pgf@addtolength#1#2{% + \begingroup% + \pgfutil@selectfont% + \pgf@addtolengthorig#1{#2}% + \expandafter% + \endgroup% + \expandafter#1\expandafter=\the#1\relax} + + + +\def\pgfsetbaselinepointlater#1{\def\pgf@baseline{#1}} + + + +\def\pgfsetbaselinepointnow#1{% + \pgf@process{#1}% + \edef\pgf@setter@baseline{\noexpand\pgfpoint{\the\pgf@x}{\the\pgf@y}}% + \pgfsetbaselinepointlater{\pgf@setter@baseline}% +} + +\def\pgf@default@text{default}% + + +\def\pgfsetbaseline#1{% + \def\pgf@temp{#1}% + \ifx\pgf@temp\pgf@default@text + \pgfsetbaseline{\pgf@picminy}% + \else + \pgfsetbaselinepointlater{\pgfpoint{0pt}{#1}}% + \fi +} +\pgfsetbaseline{\pgf@picminy} + +\newif\ifpgf@trim@lowlevel +\pgfkeys{ + /pgf/trim lowlevel/.is if=pgf@trim@lowlevel, + /pgf/trim lowlevel/.default=true, +} + +\def\pgfsettrimleftpointlater#1{\def\pgf@trimleft{#1}} +\def\pgfsettrimleftpointnow#1{% + \pgf@process{#1}% + \edef\pgf@setter@baseline{\noexpand\pgfpoint{\the\pgf@x}{\the\pgf@y}}% + \pgfsettrimleftpointlater{\pgf@setter@baseline}% +} +\def\pgfsettrimleft#1{% + \def\pgf@temp{#1}% + \ifx\pgf@temp\pgf@default@text + \pgfsettrimleft{\pgf@picminx} + \else + \pgfsettrimleftpointlater{\pgfpoint{#1}{0pt}}% + \fi +} +\pgfsettrimleft{\pgf@picminx} + +\def\pgfsettrimrightpointlater#1{\def\pgf@trimright{#1}} +\def\pgfsettrimrightpointnow#1{% + \pgf@process{#1}% + \edef\pgf@setter@baseline{\noexpand\pgfpoint{\the\pgf@x}{\the\pgf@y}}% + \pgfsettrimrightpointlater{\pgf@setter@baseline}% +} +\def\pgfsettrimright#1{% + \def\pgf@temp{#1}% + \ifx\pgf@temp\pgf@default@text + \pgfsettrimright{\pgf@picmaxx}% + \else + \pgfsettrimrightpointlater{\pgfpoint{#1}{0pt}}% + \fi +} +\pgfsettrimright{\pgf@picmaxx} + + + +\def\pgfinterruptpath +{% + \begingroup% + \edef\pgf@interrupt@savex{\the\pgf@path@lastx}% + \edef\pgf@interrupt@savey{\the\pgf@path@lasty}% + \pgf@getpathsizes\pgf@interrupt@pathsizes% + \pgfsyssoftpath@getcurrentpath\pgf@interrupt@path% + \pgfsyssoftpath@setcurrentpath\pgfutil@empty% + \let\pgf@interrupt@lastmoveto=\pgfsyssoftpath@lastmoveto% + \begingroup% +} +\def\endpgfinterruptpath +{% + \endgroup% + \pgfsyssoftpath@setcurrentpath\pgf@interrupt@path% + \pgf@setpathsizes\pgf@interrupt@pathsizes% + \global\pgf@path@lastx\pgf@interrupt@savex% + \global\pgf@path@lasty\pgf@interrupt@savey% + \global\let\pgfsyssoftpath@lastmoveto\pgf@interrupt@lastmoveto% + \endgroup% +} + + + + +\def\pgfinterruptboundingbox +{% + \begingroup% + \edef\pgf@interrupt@savemaxx{\the\pgf@picmaxx}% + \edef\pgf@interrupt@saveminx{\the\pgf@picminx}% + \edef\pgf@interrupt@savemaxy{\the\pgf@picmaxy}% + \edef\pgf@interrupt@saveminy{\the\pgf@picminy}% + \pgf@picmaxx=-16000pt\relax% + \pgf@picminx=16000pt\relax% + \pgf@picmaxy=-16000pt\relax% + \pgf@picminy=16000pt\relax% + \pgf@size@hookedfalse% + \let\pgf@path@size@hook=\pgfutil@empty% +} +\def\endpgfinterruptboundingbox +{% + \global\pgf@picmaxx=\pgf@interrupt@savemaxx% + \global\pgf@picmaxy=\pgf@interrupt@savemaxy% + \global\pgf@picminx=\pgf@interrupt@saveminx% + \global\pgf@picminy=\pgf@interrupt@saveminy% + \endgroup% +} + + + + + +\def\pgfidscope{\pgfsys@begin@idscope} +\def\endpgfidscope{\pgfsys@end@idscope} + +\let\startpgfidscope\pgfidscope +\let\stoppgfidscope\pgfendidscope + + + + + +\def\pgfuseid#1{% + \edef\pgf@temp{#1}% + \ifx\pgf@temp\pgfutil@empty% + \else% + \pgf@lookup@id{#1}% + \pgfsys@use@id{\pgf@next@id}% + \expandafter\xdef\csname pgf@id@names@#1\endcsname{{\pgf@next@id}{}}% + \fi% +} + +\def\pgf@lookup@id#1{% + \expandafter\let\expandafter\pgf@id@name\csname pgf@id@names@#1\endcsname% + \ifx\pgf@id@name\relax% + \let\pgf@prev@id\pgfutil@empty% + \let\pgf@next@id\pgfutil@empty% + \else% + \expandafter\pgf@id@parse\pgf@id@name% + \fi% + \ifx\pgf@next@id\pgfutil@empty% + \pgfsys@new@id\pgf@next@id% + \expandafter\xdef\csname pgf@id@names@#1\endcsname{{\pgf@prev@id}{\pgf@next@id}}% + \fi% +} +\def\pgf@id@parse#1#2{% + \def\pgf@prev@id{#1}% + \def\pgf@next@id{#2}% +} + + + + +\def\pgfclearid{% + \pgfsys@clear@id% +} + + + +\def\pgfidrefnextuse#1#2{% + \edef\pgf@temp{#2}\ifx\pgf@temp\pgfutil@empty\pgferror{Missing name reference for ``\string#1''}\fi% + \pgf@lookup@id{#2}% + \let#1\pgf@next@id% +} + + + +\def\pgfidrefprevuse#1#2{% + \edef\pgf@temp{#2}\ifx\pgf@temp\pgfutil@empty\pgferror{Missing name reference for ``\string#1''}\fi% + \pgf@lookup@id{#2}% + \let#1\pgf@prev@id% +} + + + +\def\pgfusetype#1{% + \edef\pgf@temp{#1}% + \expandafter\pgfutil@ifnextchar\expandafter.\expandafter\pgf@id@use@type@add\expandafter\pgf@id@use@type@replace\pgf@temp\pgf@stop +} +\def\pgf@id@use@type@add.#1\pgf@stop{\pgfsys@append@type{#1}} +\def\pgf@id@use@type@replace#1\pgf@stop{\pgfsys@use@type{#1}} + + +\def\pgfpushtype{\pgfsys@push@type} + + + +\def\pgfpoptype{\pgfsys@pop@type} + + + + + +\def\pgfaliasid#1#2{% + \expandafter\let\expandafter\pgf@temp\csname pgf@id@names@#2\endcsname% + \expandafter\let\csname pgf@id@names@#1\endcsname\pgf@temp% +} +\def\pgfgaliasid#1#2{% + \expandafter\let\expandafter\pgf@temp\csname pgf@id@names@#2\endcsname% + \expandafter\global\expandafter\let\csname pgf@id@names@#1\endcsname\pgf@temp% +} + + + +\def\pgfifidreferenced#1#2#3{% + \expandafter\let\expandafter\pgf@id@name\csname pgf@id@names@#1\endcsname% + \ifx\pgf@id@name\relax% + #3% + \else + \expandafter\pgf@id@parse\pgf@id@name% + \ifx\pgf@next@id\pgfutil@empty% + #3% + \else% + #2% + \fi% + \fi% +} + + + +\def\pgfinterruptpicture +{% + \begingroup% + \csname tikz@inside@picturefalse\endcsname% + \pgfinterruptboundingbox% + \pgftransformreset% + \pgfinterruptpath% + \ifx\pgf@selectfontorig\@undefined% + \else% + \let\setlength\pgf@setlengthorig% + \let\addtolength\pgf@addtolengthorig% + \let\selectfont\pgf@selectfontorig% + \fi% + \pgfutil@selectfont% + \pgfpicturefalse% + \let\pgf@positionnodelater@macro\relax% + \pgf@savelayers% +} +\def\endpgfinterruptpicture +{% + \pgf@restorelayers% + \endpgfinterruptpath% + \endpgfinterruptboundingbox% + \endgroup% +} + +\let\pgf@savelayers=\relax +\let\pgf@restorelayers=\relax + +\endinput diff --git a/preprints/A-polynomial-time-algorithm-for-three-machine-unit-job-scheduling-September-24-2026/build/tex/runtime/generic/pgf/basiclayer/pgfcoreshade.code.tex b/preprints/A-polynomial-time-algorithm-for-three-machine-unit-job-scheduling-September-24-2026/build/tex/runtime/generic/pgf/basiclayer/pgfcoreshade.code.tex new file mode 100644 index 0000000000000000000000000000000000000000..a26221b163a67580b304050051e277cf0e87caff --- /dev/null +++ b/preprints/A-polynomial-time-algorithm-for-three-machine-unit-job-scheduling-September-24-2026/build/tex/runtime/generic/pgf/basiclayer/pgfcoreshade.code.tex @@ -0,0 +1,700 @@ + +\ProvidesFileRCS{pgfcoreshade.code.tex} + +\newdimen\pgf@max +\newcount\pgf@sys@shading@range@num +\def\pgf@parsefunc#1{% + \edef\temp{{#1}}% + \expandafter\pgf@convertstring\temp% + \edef\temp{{\pgf@conv}}% + \expandafter\pgf@@parsefunc\temp} +\def\pgf@@parsefunc#1{% + \let\pgf@bounds=\pgfutil@empty% + \let\pgf@funcs=\pgfutil@empty% + \let\pgf@psfuncs=\pgfutil@empty% + \let\pgf@encode=\pgfutil@empty% + \let\pgf@sys@shading@ranges=\pgfutil@empty% + \pgf@sys@shading@range@num=0\relax% + \csname pgf@parsefirst\pgf@shading@model\endcsname[#1; ]% + \csname pgf@parselastdom\pgf@shading@model\endcsname[#1; ]% + \csname pgf@parsemid\pgf@shading@model\endcsname[#1; ]% + \ifx\pgf@bounds\pgfutil@empty% + \edef\pgf@pdfparseddomain{0 1}% + \edef\pgf@pdfparsedfunction{\pgf@singlefunc\space}% + \else% + \edef\pgf@pdfparseddomain{\pgf@doma\space\pgf@domb}% + \edef\pgf@pdfparsedfunction{% + << /FunctionType 3 /Domain [\pgf@doma\space\pgf@domb] /Functions + [\pgf@funcs\space] /Bounds [\pgf@bounds] /Encode [0 1 \pgf@encode] + >> }% + \fi% + \xdef\pgf@psfuncs{\pgf@psfuncs}% + } +\def\pgf@parsefirstrgb[rgb(#1)=(#2,#3,#4)#5]{% + \pgfmathsetlength\pgf@x{#1}% + \edef\pgf@sys@shading@start@pos{\the\pgf@x}% + \pgf@sys@bp@correct\pgf@x% + \edef\pgf@doma{\pgf@sys@tonumber{\pgf@x}}% + \edef\pgf@prevx{\pgf@sys@tonumber{\pgf@x}}% + \pgf@getrgbtuplewithmixin{#2}{#3}{#4}% + \edef\pgf@sys@shading@start@rgb{\pgf@sys@rgb}% + \let\pgf@sys@prevcolor=\pgf@sys@shading@start@rgb% + \let\pgf@sys@prevpos=\pgf@sys@shading@start@pos% + \edef\pgf@prevcolor{\pgf@rgb}% + \edef\pgf@firstcolor{\pgf@rgb}} +\def\pgf@parselastdomrgb[rgb(#1)=(#2,#3,#4); {% + \pgfutil@ifnextchar]{% + \pgfmathsetlength\pgf@x{#1}% + \edef\pgf@sys@shading@end@pos{\the\pgf@x}% + \pgf@max=\pgf@x\relax% + \pgf@sys@bp@correct\pgf@x% + \edef\pgf@domb{\pgf@sys@tonumber{\pgf@x}}% + \pgf@getrgbtuplewithmixin{#2}{#3}{#4}% + \edef\pgf@sys@shading@end@rgb{\pgf@sys@rgb}% + \pgfutil@gobble}{\pgf@parselastdomrgb[}} +\def\pgf@parsemidrgb[rgb(#1)=(#2,#3,#4); {\pgf@parserestrgb[} +\def\pgf@parserestrgb[rgb(#1)=(#2,#3,#4); {% + \advance\pgf@sys@shading@range@num by1\relax% + \pgfutil@ifnextchar]{% + \pgf@getrgbtuplewithmixin{#2}{#3}{#4}% + \edef\pgf@singlefunc{\space% + << /FunctionType 2 /Domain [0 1] /C0 + [\pgf@prevcolor] /C1 [\pgf@rgb] /N 1 >> }% + \edef\pgf@funcs{\pgf@funcs\space% + << /FunctionType 2 /Domain [\pgf@doma\space\pgf@domb] /C0 + [\pgf@prevcolor] /C1 [\pgf@rgb] /N 1 >> }% + \edef\pgf@psfuncs{\pgf@prevx\space \pgf@rgb\space \pgf@prevcolor\space pgfshade \pgf@psfuncs}% + \pgfmathsetlength\pgf@x{#1}% + \edef\pgf@sys@shading@ranges{\pgf@sys@shading@ranges{{\pgf@sys@prevpos}{\the\pgf@x}{\pgf@sys@prevcolor}{\pgf@sys@rgb}}}% + \edef\pgf@sys@prevpos{\the\pgf@x}% + \let\pgf@sys@prevcolor=\pgf@sys@rgb% + \pgfutil@gobble}{% + \pgfmathsetlength\pgf@x{#1}% + \pgf@getrgbtuplewithmixin{#2}{#3}{#4}% + \edef\pgf@sys@shading@ranges{\pgf@sys@shading@ranges{{\pgf@sys@prevpos}{\the\pgf@x}{\pgf@sys@prevcolor}{\pgf@sys@rgb}}}% + \edef\pgf@sys@prevpos{\the\pgf@x}% + \let\pgf@sys@prevcolor=\pgf@sys@rgb% + \edef\pgf@psfuncs{\pgf@prevx\space \pgf@rgb\space \pgf@prevcolor\space pgfshade \pgf@psfuncs}% + \pgf@sys@bp@correct\pgf@x% + \edef\pgf@prevx{\pgf@sys@tonumber{\pgf@x}}% + \edef\pgf@bounds{\pgf@bounds\space\pgf@sys@tonumber{\pgf@x}}% + \edef\pgf@encode{\pgf@encode\space0 1}% + \edef\pgf@singlefunc{\space% + << /FunctionType 2 /Domain [0 1] /C0 + [\pgf@prevcolor] /C1 [\pgf@rgb] /N 1 >> }% + \edef\pgf@funcs{\pgf@funcs\space% + << /FunctionType 2 /Domain [\pgf@doma\space\pgf@domb] /C0 + [\pgf@prevcolor] /C1 [\pgf@rgb] /N 1 >> }% + \edef\pgf@prevcolor{\pgf@rgb}% + \pgf@parserestrgb[}} + +\def\pgf@getrgbtuplewithmixin#1#2#3{% + \pgfutil@definecolor{pgfshadetemp}{rgb}{#1,#2,#3}% + \pgfutil@ifundefined{applycolormixins}{}{\applycolormixins{pgfshadetemp}}% + \pgfutil@extractcolorspec{pgfshadetemp}{\pgf@tempcolor}% + \expandafter\pgfutil@convertcolorspec\pgf@tempcolor{rgb}{\pgf@rgbcolor}% + \expandafter\pgf@getrgb@@\pgf@rgbcolor!} +\def\pgf@getrgb@@#1,#2,#3!{% + \def\pgf@rgb{#1 #2 #3}% + \def\pgf@sys@rgb{{#1}{#2}{#3}}% +} + +\def\pgf@parsefirstcmyk[cmyk(#1)=(#2,#3,#4,#5)#6]{% + \pgfmathsetlength\pgf@x{#1}% + \edef\pgf@sys@shading@start@pos{\the\pgf@x}% + \pgf@sys@bp@correct\pgf@x% + \edef\pgf@doma{\pgf@sys@tonumber{\pgf@x}}% + \edef\pgf@prevx{\pgf@sys@tonumber{\pgf@x}}% + \pgf@getcmyktuplewithmixin{#2}{#3}{#4}{#5}% + \edef\pgf@sys@shading@start@cmyk{\pgf@sys@cmyk}% + \let\pgf@sys@prevcolor=\pgf@sys@shading@start@cmyk% + \let\pgf@sys@prevpos=\pgf@sys@shading@start@pos% + \edef\pgf@prevcolor{\pgf@cmyk}% + \edef\pgf@firstcolor{\pgf@cmyk}} +\def\pgf@parselastdomcmyk[cmyk(#1)=(#2,#3,#4,#5); {% + \pgfutil@ifnextchar]{% + \pgfmathsetlength\pgf@x{#1}% + \edef\pgf@sys@shading@end@pos{\the\pgf@x}% + \pgf@max=\pgf@x\relax% + \pgf@sys@bp@correct\pgf@x% + \edef\pgf@domb{\pgf@sys@tonumber{\pgf@x}}% + \pgf@getcmyktuplewithmixin{#2}{#3}{#4}{#5}% + \edef\pgf@sys@shading@end@cmyk{\pgf@sys@cmyk}% + \pgfutil@gobble}{\pgf@parselastdomcmyk[}} +\def\pgf@parsemidcmyk[cmyk(#1)=(#2,#3,#4,#5); {\pgf@parserestcmyk[} +\def\pgf@parserestcmyk[cmyk(#1)=(#2,#3,#4,#5); {% + \advance\pgf@sys@shading@range@num by1\relax% + \pgfutil@ifnextchar]{% + \pgf@getcmyktuplewithmixin{#2}{#3}{#4}{#5}% + \edef\pgf@singlefunc{\space% + << /FunctionType 2 /Domain [0 1] /C0 + [\pgf@prevcolor] /C1 [\pgf@cmyk] /N 1 >> }% + \edef\pgf@funcs{\pgf@funcs\space% + << /FunctionType 2 /Domain [\pgf@doma\space\pgf@domb] /C0 + [\pgf@prevcolor] /C1 [\pgf@cmyk] /N 1 >> }% + \edef\pgf@psfuncs{\pgf@prevx\space + \pgf@cmyk\space \pgf@prevcolor\space pgfshade \pgf@psfuncs}% + \pgfmathsetlength\pgf@x{#1}% + \edef\pgf@sys@shading@ranges{\pgf@sys@shading@ranges{% + {\pgf@sys@prevpos}{\the\pgf@x}{\pgf@sys@prevcolor}{\pgf@sys@cmyk}}}% + \edef\pgf@sys@prevpos{\the\pgf@x}% + \let\pgf@sys@prevcolor=\pgf@sys@cmyk% + \pgfutil@gobble}{% + \pgfmathsetlength\pgf@x{#1}% + \pgf@getcmyktuplewithmixin{#2}{#3}{#4}{#5}% + \edef\pgf@sys@shading@ranges{\pgf@sys@shading@ranges{% + {\pgf@sys@prevpos}{\the\pgf@x}{\pgf@sys@prevcolor}{\pgf@sys@cmyk}}}% + \edef\pgf@sys@prevpos{\the\pgf@x}% + \let\pgf@sys@prevcolor=\pgf@sys@cmyk% + \edef\pgf@psfuncs{\pgf@prevx\space \pgf@cmyk\space + \pgf@prevcolor\space pgfshade \pgf@psfuncs}% + \pgf@sys@bp@correct\pgf@x% + \edef\pgf@prevx{\pgf@sys@tonumber{\pgf@x}}% + \edef\pgf@bounds{\pgf@bounds\space\pgf@sys@tonumber{\pgf@x}}% + \edef\pgf@encode{\pgf@encode\space0 1}% + \edef\pgf@singlefunc{\space% + << /FunctionType 2 /Domain [0 1] /C0 + [\pgf@prevcolor] /C1 [\pgf@cmyk] /N 1 >> }% + \edef\pgf@funcs{\pgf@funcs\space% + << /FunctionType 2 /Domain [\pgf@doma\space\pgf@domb] /C0 + [\pgf@prevcolor] /C1 [\pgf@cmyk] /N 1 >> }% + \edef\pgf@prevcolor{\pgf@cmyk}% + \pgf@parserestcmyk[}} + +\def\pgf@getcmyktuplewithmixin#1#2#3#4{% + \pgfutil@definecolor{pgfshadetemp}{cmyk}{#1,#2,#3,#4}% + \pgfutil@ifundefined{applycolormixins}{}{\applycolormixins{pgfshadetemp}}% + \pgfutil@extractcolorspec{pgfshadetemp}{\pgf@tempcolor}% + \expandafter\pgfutil@convertcolorspec\pgf@tempcolor{cmyk}{\pgf@cmykcolor}% + \expandafter\pgf@getcmyk@@\pgf@cmykcolor!} +\def\pgf@getcmyk@@#1,#2,#3,#4!{% + \def\pgf@cmyk{#1 #2 #3 #4}% + \def\pgf@sys@cmyk{{#1}{#2}{#3}{#4}}% +} + +\def\pgf@parsefirstgray[gray(#1)=(#2)#3]{% + \pgfmathsetlength\pgf@x{#1}% + \edef\pgf@sys@shading@start@pos{\the\pgf@x}% + \pgf@sys@bp@correct\pgf@x% + \edef\pgf@doma{\pgf@sys@tonumber{\pgf@x}}% + \edef\pgf@prevx{\pgf@sys@tonumber{\pgf@x}}% + \pgf@getgraytuplewithmixin{#2}% + \edef\pgf@sys@shading@start@gray{\pgf@sys@gray}% + \let\pgf@sys@prevcolor=\pgf@sys@shading@start@gray% + \let\pgf@sys@prevpos=\pgf@sys@shading@start@pos% + \edef\pgf@prevcolor{\pgf@gray}% + \edef\pgf@firstcolor{\pgf@gray}} +\def\pgf@parselastdomgray[gray(#1)=(#2); {% + \pgfutil@ifnextchar]{% + \pgfmathsetlength\pgf@x{#1}% + \edef\pgf@sys@shading@end@pos{\the\pgf@x}% + \pgf@max=\pgf@x\relax% + \pgf@sys@bp@correct\pgf@x% + \edef\pgf@domb{\pgf@sys@tonumber{\pgf@x}}% + \pgf@getgraytuplewithmixin{#2}% + \edef\pgf@sys@shading@end@gray{\pgf@sys@gray}% + \pgfutil@gobble}{\pgf@parselastdomgray[}} +\def\pgf@parsemidgray[gray(#1)=(#2); {\pgf@parserestgray[} +\def\pgf@parserestgray[gray(#1)=(#2); {% + \advance\pgf@sys@shading@range@num by1\relax% + \pgfutil@ifnextchar]{% + \pgf@getgraytuplewithmixin{#2}% + \edef\pgf@singlefunc{\space% + << /FunctionType 2 /Domain [0 1] /C0 + [\pgf@prevcolor] /C1 [\pgf@gray] /N 1 >> }% + \edef\pgf@funcs{\pgf@funcs\space% + << /FunctionType 2 /Domain [\pgf@doma\space\pgf@domb] /C0 + [\pgf@prevcolor] /C1 [\pgf@gray] /N 1 >> }% + \edef\pgf@psfuncs{\pgf@prevx\space \pgf@gray\space + \pgf@prevcolor\space pgfshade \pgf@psfuncs}% + \pgfmathsetlength\pgf@x{#1}% + \edef\pgf@sys@shading@ranges{\pgf@sys@shading@ranges{% + {\pgf@sys@prevpos}{\the\pgf@x}{\pgf@sys@prevcolor}{\pgf@sys@gray}}}% + \edef\pgf@sys@prevpos{\the\pgf@x}% + \let\pgf@sys@prevcolor=\pgf@sys@gray% + \pgfutil@gobble}{% + \pgfmathsetlength\pgf@x{#1}% + \pgf@getgraytuplewithmixin{#2}% + \edef\pgf@sys@shading@ranges{\pgf@sys@shading@ranges{% + {\pgf@sys@prevpos}{\the\pgf@x}{\pgf@sys@prevcolor}{\pgf@sys@gray}}}% + \edef\pgf@sys@prevpos{\the\pgf@x}% + \let\pgf@sys@prevcolor=\pgf@sys@gray% + \edef\pgf@psfuncs{\pgf@prevx\space \pgf@gray\space + \pgf@prevcolor\space pgfshade \pgf@psfuncs}% + \pgf@sys@bp@correct\pgf@x% + \edef\pgf@prevx{\pgf@sys@tonumber{\pgf@x}}% + \edef\pgf@bounds{\pgf@bounds\space\pgf@sys@tonumber{\pgf@x}}% + \edef\pgf@encode{\pgf@encode\space0 1}% + \edef\pgf@singlefunc{\space% + << /FunctionType 2 /Domain [0 1] /C0 + [\pgf@prevcolor] /C1 [\pgf@gray] /N 1 >> }% + \edef\pgf@funcs{\pgf@funcs\space% + << /FunctionType 2 /Domain [\pgf@doma\space\pgf@domb] /C0 + [\pgf@prevcolor] /C1 [\pgf@gray] /N 1 >> }% + \edef\pgf@prevcolor{\pgf@gray}% + \pgf@parserestgray[}} + +\def\pgf@getgraytuplewithmixin#1{% + \pgfutil@definecolor{pgfshadetemp}{gray}{#1}% + \pgfutil@ifundefined{applycolormixins}{}{\applycolormixins{pgfshadetemp}}% + \pgfutil@extractcolorspec{pgfshadetemp}{\pgf@tempcolor}% + \expandafter\pgfutil@convertcolorspec\pgf@tempcolor{gray}{\pgf@graycolor}% + \expandafter\pgf@getgray@@\pgf@graycolor!} +\def\pgf@getgray@@#1!{% + \def\pgf@gray{#1}% + \def\pgf@sys@gray{{#1}}% +} + +\def\pgf@convertstring#1{% + \def\pgf@conv{}% + \pgf@convert#1]% + } +\def\pgf@convert{% + \pgfutil@ifnextchar]{\pgfutil@gobble}% + {% + \pgfutil@ifnextchar;{\pgf@grabsemicolor}% + {% + \pgfutil@ifnextchar c{\pgf@gobblec}% + {% + \pgfutil@ifnextchar g{\pgf@grabgray}% + {% + \pgfutil@ifnextchar o{\pgf@grabcolor}% + {% + \pgfutil@ifnextchar m{\pgf@grabcmyk}% + {% + \pgfutil@ifnextchar r{\pgf@grabrgb}% + {\pgferror{Illformed shading + specification}\pgf@convert}% + }% + }% + }% + }% + }% + }% +} +\def\pgf@savecolor#1{% + \pgfutil@extractcolorspec{pgf@tempcol}{\pgf@tempcolor}% + \expandafter\pgfutil@convertcolorspec\pgf@tempcolor + {\pgf@shading@model}{\pgf@color}% + \expandafter\pgf@convget@\expandafter{\pgf@color}{#1}% +} +\def\pgf@grabsemicolor;{% + \edef\pgf@conv{\pgf@conv; }\pgf@convert} +\def\pgf@gobblec c{\pgf@convert} +\def\pgf@grabrgb rgb(#1)=(#2,#3,#4){% + \pgfutil@definecolor{pgf@tempcol}{rgb}{#2,#3,#4}% + \pgf@savecolor{#1}% +} +\def\pgf@grabcmyk myk(#1)=(#2,#3,#4,#5){% + \pgfutil@definecolor{pgf@tempcol}{cmyk}{#2,#3,#4,#5}% + \pgf@savecolor{#1}% +} +\def\pgf@grabgray gray(#1)=(#2){% + \pgfutil@definecolor{pgf@tempcol}{gray}{#2}% + \pgf@savecolor{#1}% +} +\def\pgf@grabcolor olor(#1)=(#2){% + \pgfutil@colorlet{pgf@tempcol}{#2}% + \pgf@savecolor{#1}% +} +\def\pgf@convget@#1#2{% + \edef\pgf@conv{\pgf@conv \pgf@shading@model(#2)=(#1)}\pgf@convert} + + +\newcount\pgf@shadingcount +\pgf@shadingcount=0 +{ + \catcode`\/=0 + \catcode`\\=12 + /gdef/pgf@shadingnum{\/the/pgf@shadingcount\} +} + + +\newif\ifpgfshadingmodelrgb +\newif\ifpgfshadingmodelcmyk +\newif\ifpgfshadingmodelgray +\def\pgf@setup@shading@model{% + \pgfshadingmodelrgbtrue + \pgfshadingmodelcmykfalse + \pgfshadingmodelgrayfalse + \XC@sdef\pgf@mod@test{\XC@tgt@mod{natural}}% + \def\pgf@shading@device{/DeviceRGB}% + \def\pgf@shading@ps@device{setrgbcolor}% + \def\pgf@shading@functional@range{0 1 0 1 0 1}% + \def\pgf@shading@model{rgb}% + \ifx\pgf@mod@test\XC@mod@cmyk + \def\pgf@shading@device{/DeviceCMYK}% + \def\pgf@shading@ps@device{setcmykcolor}% + \def\pgf@shading@functional@range{0 1 0 1 0 1 0 1}% + \def\pgf@shading@model{cmyk}% + \pgfshadingmodelrgbfalse + \pgfshadingmodelcmyktrue + \fi + \ifx\pgf@mod@test\XC@mod@gray + \def\pgf@shading@device{/DeviceGray}% + \def\pgf@shading@ps@device{setgray}% + \def\pgf@shading@functional@range{0 1}% + \def\pgf@shading@model{gray}% + \pgfshadingmodelrgbfalse + \pgfshadingmodelgraytrue + \fi + \edef\pgf@sys@driver@dvisvgm{pgfsys-dvisvgm.def}% + \ifx\pgfsysdriver\pgf@sys@driver@dvisvgm + \def\pgf@shading@model{rgb}% + \fi + \edef\pgf@sys@driver@texforht{pgfsys-tex4ht.def}% + \ifx\pgfsysdriver\pgf@sys@driver@texforht + \def\pgf@shading@model{rgb}% + \fi +} + + + +\def\pgfdeclarehorizontalshading{% + \pgf@setup@shading@model + \pgfutil@ifnextchar[% + \pgf@declarehorizontalshading{\pgf@declarehorizontalshading[]}} +\def\pgf@declarehorizontalshading[#1]#2#3#4{% + \expandafter\def\csname pgf@deps@pgfshading#2!\endcsname{#1}% + \expandafter\ifx\csname pgf@deps@pgfshading#2!\endcsname\pgfutil@empty% + \global\expandafter\let\csname pgf@num@pgfshading#2!\endcsname\pgfutil@empty + \pgfsys@horishading{#2}{#3}{#4}% + \else% + \global\advance\pgf@shadingcount 1\relax + \global\expandafter\edef\csname pgf@num@pgfshading#2!\endcsname{\pgf@shadingnum}% + \expandafter\def\csname pgf@func@pgfshading#2!\endcsname{\pgfsys@horishading}% + \expandafter\def\csname pgf@args@pgfshading#2!\endcsname{{#3}{#4}}% + \expandafter\let\csname @pgfshading#2!\endcsname=\pgfutil@empty% + \fi} + + + +\def\pgfdeclareverticalshading{% + \pgf@setup@shading@model + \pgfutil@ifnextchar[% + \pgf@declareverticalshading{\pgf@declareverticalshading[]}} +\def\pgf@declareverticalshading[#1]#2#3#4{% + \expandafter\def\csname pgf@deps@pgfshading#2!\endcsname{#1}% + \expandafter\ifx\csname pgf@deps@pgfshading#2!\endcsname\pgfutil@empty% + \global\expandafter\let\csname pgf@num@pgfshading#2!\endcsname=\pgfutil@empty + \pgfsys@vertshading{#2}{#3}{#4}% + \else% + \global\advance\pgf@shadingcount 1\relax + \global\expandafter\edef\csname pgf@num@pgfshading#2!\endcsname{\pgf@shadingnum}% + \expandafter\def\csname pgf@func@pgfshading#2!\endcsname{\pgfsys@vertshading}% + \expandafter\def\csname pgf@args@pgfshading#2!\endcsname{{#3}{#4}}% + \expandafter\let\csname @pgfshading#2!\endcsname=\pgfutil@empty% + \fi} + + + +\def\pgfdeclareradialshading{% + \pgf@setup@shading@model + \pgfutil@ifnextchar[% + \pgf@declareradialshading{\pgf@declareradialshading[]}} +\def\pgf@declareradialshading[#1]#2#3#4{% + \expandafter\def\csname pgf@deps@pgfshading#2!\endcsname{#1}% + \expandafter\ifx\csname pgf@deps@pgfshading#2!\endcsname\pgfutil@empty% + \global\expandafter\let\csname pgf@num@pgfshading#2!\endcsname=\pgfutil@empty + \pgfsys@radialshading{#2}{#3}{#4}% + \else% + \global\advance\pgf@shadingcount 1\relax + \global\expandafter\edef\csname pgf@num@pgfshading#2!\endcsname{\pgf@shadingnum}% + \expandafter\def\csname pgf@func@pgfshading#2!\endcsname{\pgfsys@radialshading}% + \expandafter\def\csname pgf@args@pgfshading#2!\endcsname{{#3}{#4}}% + \expandafter\let\csname @pgfshading#2!\endcsname=\pgfutil@empty% + \fi} + + + + + +\def\pgfdeclarefunctionalshading{% + \pgf@setup@shading@model + \pgfutil@ifnextchar[% + \pgf@declarefunctionalshading{\pgf@declarefunctionalshading[]}} +\def\pgf@declarefunctionalshading[#1]#2#3#4#5#6{% + \expandafter\def\csname pgf@deps@pgfshading#2!\endcsname{#1}% + \expandafter\ifx\csname pgf@deps@pgfshading#2!\endcsname\pgfutil@empty% + \global\expandafter\let\csname pgf@num@pgfshading#2!\endcsname=\pgfutil@empty + \pgfshade@functionaldo{#2}{#3}{#4}{#5}{#6}% + \else% + \global\advance\pgf@shadingcount 1\relax + \global\expandafter\edef\csname pgf@num@pgfshading#2!\endcsname{\pgf@shadingnum}% + \expandafter\def\csname pgf@func@pgfshading#2!\endcsname{\pgfshade@functionaldo}% + \expandafter\def\csname pgf@args@pgfshading#2!\endcsname{{#3}{#4}{#5}{#6}}% + \expandafter\let\csname @pgfshading#2!\endcsname=\pgfutil@empty% + \fi} +\def\pgfshade@functionaldo#1#2#3#4#5{% + \begingroup + #4% + \pgfsys@functionalshading{#1}{#2}{#3}{#5}% + \expandafter\pgfmath@smuggleone\csname @pgfshading#1!\endcsname + \endgroup +} + +\def\pgfshadecolortorgb#1#2{% + \pgfutil@colorlet{pgf@tempcol}{#1}% + \pgfutil@extractcolorspec{pgf@tempcol}{\pgf@tempcolor}% + \expandafter\pgfutil@convertcolorspec\pgf@tempcolor{rgb}{\pgf@rgbcolor}% + \expandafter\pgfshading@rgb\pgf@rgbcolor\relax% + \edef#2{\pgf@sys@tonumber{\pgf@xa}\space\pgf@sys@tonumber{\pgf@xb}\space\pgf@sys@tonumber{\pgf@xc}\space}% + \c@pgf@counta\escapechar% + \escapechar-1\relax% + \expandafter\edef\csname\string#2red\endcsname{\pgf@sys@tonumber{\pgf@xa}\space}% + \expandafter\edef\csname\string#2green\endcsname{\pgf@sys@tonumber{\pgf@xb}\space}% + \expandafter\edef\csname\string#2blue\endcsname{\pgf@sys@tonumber{\pgf@xc}\space}% + \escapechar\c@pgf@counta% +} +\def\pgfshading@rgb#1,#2,#3\relax{% + \pgf@xa=#1pt% + \pgf@xb=#2pt% + \pgf@xc=#3pt% +} + +\def\pgfshadecolortocmyk#1#2{% + \pgfutil@colorlet{pgf@tempcol}{#1}% + \pgfutil@extractcolorspec{pgf@tempcol}{\pgf@tempcolor}% + \expandafter\pgfutil@convertcolorspec\pgf@tempcolor{cmyk}{\pgf@cmykcolor}% + \expandafter\pgfshading@cmyk\pgf@cmykcolor\relax% + \edef#2{\pgf@sys@tonumber{\pgf@xa}\space\pgf@sys@tonumber{\pgf@xb}\space + \pgf@sys@tonumber{\pgf@xc}\space\pgf@sys@tonumber{\pgf@xd}\space}% + \c@pgf@counta\escapechar% + \escapechar-1\relax% + \expandafter\edef\csname\string#2cyan\endcsname{% + \pgf@sys@tonumber{\pgf@xa}\space}% + \expandafter\edef\csname\string#2magenta\endcsname{% + \pgf@sys@tonumber{\pgf@xb}\space}% + \expandafter\edef\csname\string#2yellow\endcsname{% + \pgf@sys@tonumber{\pgf@xc}\space}% + \expandafter\edef\csname\string#2black\endcsname{% + \pgf@sys@tonumber{\pgf@xd}\space}% + \escapechar\c@pgf@counta +} +\def\pgfshading@cmyk#1,#2,#3,#4\relax{% + \pgf@xa=#1pt% + \pgf@xb=#2pt% + \pgf@xc=#3pt% + \pgf@xd=#4pt% +} + +\def\pgfshadecolortogray#1#2{% + \pgfutil@colorlet{pgf@tempcol}{#1}% + \pgfutil@extractcolorspec{pgf@tempcol}{\pgf@tempcolor}% + \expandafter\pgfutil@convertcolorspec\pgf@tempcolor{gray}{\pgf@graycolor}% + \expandafter\pgfshading@gray\pgf@graycolor\relax + \edef#2{\pgf@sys@tonumber{\pgf@xa}\space}% + \c@pgf@counta\escapechar + \escapechar-1\relax + \expandafter\edef\csname\string#2gray\endcsname{% + \pgf@sys@tonumber{\pgf@xa}\space}% + \escapechar\c@pgf@counta +} +\def\pgfshading@gray#1\relax{% + \pgf@xa=#1pt% +} + + +\def\pgffuncshadingrgbtocmyk{% + 1.0 exch sub 3 1 roll + 1.0 exch sub 3 1 roll + 1.0 exch sub 3 1 roll + 3 copy + 2 copy gt { exch } if pop + 2 copy gt { exch } if pop + dup 3 1 roll sub + 0.0 2 copy lt { exch } if pop + 1.0 2 copy gt { exch } if pop + 4 1 roll + dup 3 1 roll sub + 0.0 2 copy lt { exch } if pop + 1.0 2 copy gt { exch } if pop + 4 1 roll + dup 3 1 roll sub + 0.0 2 copy lt { exch } if pop + 1.0 2 copy gt { exch } if pop + 4 1 roll +} + +\def\pgffuncshadingrgbtogray{% + 0.11 mul exch 0.59 mul add exch 0.3 mul add +} + +\def\pgffuncshadingcmyktorgb{% + dup 3 1 roll add + 1.0 2 copy gt { exch } if pop + 4 1 roll + dup 3 1 roll add + 1.0 2 copy gt { exch } if pop + 4 1 roll + add + 1.0 2 copy gt { exch } if pop + 3 1 roll + 1.0 exch sub + 3 1 roll + 1.0 exch sub + 3 1 roll + 1.0 exch sub + 3 1 roll +} + +\def\pgffuncshadingcmyktogray{% + exch 0.11 mul add exch 0.59 mul add exch 0.3 mul add + 1.0 2 copy gt { exch } if pop + 1.0 exch sub +} + +\def\pgffuncshadinggraytorgb{% + dup dup +} + +\def\pgffuncshadinggraytocmyk{% + 0.0 0.0 0.0 + 4 3 roll +} + + +\def\pgfuseshading#1{% + \edef\pgf@shadingname{@pgfshading#1}% + \pgf@tryextensions{\pgf@shadingname}{\pgfalternateextension}% + \expandafter\pgfutil@ifundefined\expandafter{\pgf@shadingname}% + {\pgferror{Undefined shading "#1"}}% + {% + \edef\pgf@shade@adds{\csname pgf@num\pgf@shadingname\endcsname}% + \pgfutil@ifundefined{pgf@deps\pgf@shadingname}% + {}% + {% + \pgf@setup@shading@model + \begingroup + \pgfutil@globalcolorsfalse + \edef\@list{\csname pgf@deps\pgf@shadingname\endcsname}% + \pgfutil@for\@temp:=\@list\do{% + \pgfutil@ifundefined{applycolormixins}{}{\applycolormixins{\@temp}}% + \pgfutil@extractcolorspec{\@temp}{\pgf@tempcolor}% + \expandafter\pgfutil@convertcolorspec\pgf@tempcolor{\pgf@shading@model}{\pgf@color}% + \edef\pgf@shade@adds{\pgf@shade@adds,\pgf@color}% + }% + \pgfmath@smuggleone\pgf@shade@adds + \endgroup + }% + \expandafter\pgf@strip@shadename\pgf@shadingname!!% + \edef\pgf@shadingxname{@pgfshading\pgf@basename\pgf@shade@adds!}% + \pgfutil@ifundefined{\pgf@shadingxname}% + {% + \begingroup + \edef\@temp{\expandafter\noexpand\csname pgf@func\pgf@shadingname\endcsname}% + \edef\@args{{\pgf@basename\pgf@shade@adds}}% + \expandafter\expandafter\expandafter\def + \expandafter\expandafter\expandafter\@@args + \expandafter\expandafter\expandafter{\csname pgf@args\pgf@shadingname\endcsname}% + \expandafter\expandafter\expandafter\@temp\expandafter\@args\@@args + \expandafter\pgfmath@smuggleone\csname\pgf@shadingxname\endcsname + \endgroup + }% + {}% + \pgf@invokeshading{\csname\pgf@shadingxname\endcsname}% + }% +} + +\def\pgf@strip@shadename @pgfshading#1!!!{\def\pgf@basename{#1}} + +\def\pgf@invokeshading#1{% + \ifpgfpicture% + \pgfsys@shadinginsidepgfpicture{#1}% + \else% + \pgfsys@shadingoutsidepgfpicture{#1}% + \fi% +} + + + +\def\pgfaliasshading#1#2{% + \expandafter\let\expandafter\pgf@temp\expandafter=\csname @pgfshading#2!\endcsname% + \expandafter\let\csname @pgfshading#1!\endcsname=\pgf@temp% + \expandafter\let\expandafter\pgf@temp\expandafter=\csname pgf@num@pgfshading#2!\endcsname% + \expandafter\let\csname pgf@num@pgfshading#1!\endcsname=\pgf@temp% + \expandafter\let\expandafter\pgf@temp\expandafter=\csname pgf@deps@pgfshading#2!\endcsname% + \expandafter\let\csname pgf@deps@pgfshading#1!\endcsname=\pgf@temp% + \expandafter\let\expandafter\pgf@temp\expandafter=\csname pgf@func@pgfshading#2!\endcsname% + \expandafter\let\csname pgf@func@pgfshading#1!\endcsname=\pgf@temp% + \expandafter\let\expandafter\pgf@temp\expandafter=\csname pgf@args@pgfshading#2!\endcsname% + \expandafter\let\csname pgf@args@pgfshading#1!\endcsname=\pgf@temp% +} + + + + + +\def\pgfshadepath#1#2{% + \ifdim\pgf@pathminx=16000pt% + \pgfwarning{No path specified that can be filled}% + \else% + \begingroup% + \pgf@xb=.5\pgf@pathmaxx% + \advance\pgf@xb by.5\pgf@pathminx% + \pgf@yb=.5\pgf@pathmaxy% + \advance\pgf@yb by.5\pgf@pathminy% + \pgfmathparse{#2}% + \let\pgfshade@angle=\pgfmathresult + \pgfmathsin@\pgfshade@angle + \let\pgfshade@sin=\pgfmathresult + \pgfmathcos@\pgfshade@angle + \let\pgfshade@cos=\pgfmathresult + \pgf@xa=\pgf@pathmaxx + \advance\pgf@xa by-\pgf@pathminx + \pgf@ya=\pgf@pathmaxy + \advance\pgf@ya by-\pgf@pathminy + \pgfmathabs@\pgfshade@sin + \let\pgfshade@abssin=\pgfmathresult + \pgfmathabs@\pgfshade@cos + \let\pgfshade@abscos=\pgfmathresult + \pgf@xc=\pgfshade@abscos\pgf@xa + \advance\pgf@xc by\pgfshade@abssin\pgf@ya + \pgf@yc=\pgfshade@abssin\pgf@xa + \advance\pgf@yc by\pgfshade@abscos\pgf@ya + \pgfmathadd@\pgfshade@abscos\pgfshade@abssin + \pgfmathdivide@{.01992528}{\pgfmathresult}% + \pgf@xc=\pgfmathresult\pgf@xc + \pgf@yc=\pgfmathresult\pgf@yc + \ifdim\pgf@xc<0.0001pt + \ifdim\pgf@xc>-0.0001pt + \pgf@no@shadetrue + \fi + \fi + \ifdim\pgf@yc<0.0001pt + \ifdim\pgf@yc>-0.0001pt + \pgf@no@shadetrue + \fi + \fi + \ifpgf@no@shade\else + \pgfsys@beginscope + \pgfsyssoftpath@invokecurrentpath + \pgfsys@clipnext + \pgfsys@discardpath + \pgfsys@transformcm{1}{0}{0}{1}{\the\pgf@xb}{\the\pgf@yb}% + \pgf@x=\pgfshade@sin pt% + \pgf@xa=-\pgf@x% + \pgfsys@transformcm{\pgfshade@cos}{\pgfshade@sin}{\pgf@sys@tonumber{\pgf@xa}}{\pgfshade@cos}{0pt}{0pt}% + \pgfsys@transformcm{\pgf@sys@tonumber{\pgf@xc}}{0}{0}{\pgf@sys@tonumber{\pgf@yc}}{0pt}{0pt}% + \ifx\pgf@shade@extra@transform\pgfutil@empty% + \else% + \pgflowlevel{\pgf@shade@extra@transform}% + \fi% + \pgfuseshading{#1}% + \pgfsys@endscope% + \fi% + \endgroup% + \fi% +} + +\newif\ifpgf@no@shade + + +\def\pgfsetadditionalshadetransform#1{\def\pgf@shade@extra@transform{#1}} +\let\pgf@shade@extra@transform=\pgfutil@empty + + + + + +\endinput diff --git a/preprints/A-polynomial-time-algorithm-for-three-machine-unit-job-scheduling-September-24-2026/build/tex/runtime/generic/pgf/basiclayer/pgfcoretransformations.code.tex b/preprints/A-polynomial-time-algorithm-for-three-machine-unit-job-scheduling-September-24-2026/build/tex/runtime/generic/pgf/basiclayer/pgfcoretransformations.code.tex new file mode 100644 index 0000000000000000000000000000000000000000..35ace33ec43433a3066590603221058d82b4addc --- /dev/null +++ b/preprints/A-polynomial-time-algorithm-for-three-machine-unit-job-scheduling-September-24-2026/build/tex/runtime/generic/pgf/basiclayer/pgfcoretransformations.code.tex @@ -0,0 +1,503 @@ + +\ProvidesFileRCS{pgfcoretransformations.code.tex} + + + +\newdimen\pgf@pt@x +\newdimen\pgf@pt@y + +\newdimen\pgf@pt@temp + +\newif\ifpgf@pt@identity +\pgf@pt@identitytrue + +\def\pgf@pt@aa{1.0} \def\pgf@pt@ab{0.0} +\def\pgf@pt@ba{0.0} \def\pgf@pt@bb{1.0} + + + + +\def\pgfgettransform#1{% + \edef#1{{\pgf@pt@aa}{\pgf@pt@ab}{\pgf@pt@ba}{\pgf@pt@bb}{\the\pgf@pt@x}{\the\pgf@pt@y}}% +} + +\def\pgfgettransformentries#1#2#3#4#5#6{% + \edef#1{\pgf@pt@aa}% + \edef#2{\pgf@pt@ab}% + \edef#3{\pgf@pt@ba}% + \edef#4{\pgf@pt@bb}% + \edef#5{\the\pgf@pt@x}% + \edef#6{\the\pgf@pt@y}% +}% +\def\pgfsettransformentries#1#2#3#4#5#6{% + \pgfsettransform{{#1}{#2}{#3}{#4}{#5}{#6}}% +}% + + +\def\pgfsettransform#1{% + \edef\pgf@temp{#1}% + \expandafter\pgf@settransform\pgf@temp% +} +\def\pgf@settransform#1#2#3#4#5#6{% + \def\pgf@pt@aa{#1}% + \def\pgf@pt@ab{#2}% + \def\pgf@pt@ba{#3}% + \def\pgf@pt@bb{#4}% + \pgf@pt@x=#5% + \pgf@pt@y=#6% + \edef\pgf@idtest{#1,#2,#3,#4}% + \ifx\pgf@idtest\pgf@idmatrixtext% + \pgf@pt@identitytrue% + \else% + \pgf@pt@identityfalse% + \fi% +} + +\def\pgf@idmatrixtext{1.0,0.0,0.0,1.0} +\def\pgf@zerozerotext{0.0,0.0} +\def\pgf@one@text{1.0} +\def\pgf@zero@text{0.0} + + +\def\pgf@pos@transform#1#2{% + \ifpgf@pt@identity% + \else% + \pgf@pt@temp=#1% + #1=\pgf@pt@aa#1% + \advance#1 by\pgf@pt@ba#2% + #2=\pgf@pt@bb#2% + \advance#2 by\pgf@pt@ab\pgf@pt@temp% + \fi% + \advance#1 by\pgf@pt@x% + \advance#2 by\pgf@pt@y% +} +\def\pgf@pos@transform@glob{% + \ifpgf@pt@identity% + \else% + \pgf@pt@temp=\pgf@x% + \global\pgf@x=\pgf@pt@aa\pgf@x% + \global\advance\pgf@x by\pgf@pt@ba\pgf@y% + \global\pgf@y=\pgf@pt@bb\pgf@y% + \global\advance\pgf@y by\pgf@pt@ab\pgf@pt@temp% + \fi% + \global\advance\pgf@x by\pgf@pt@x% + \global\advance\pgf@y by\pgf@pt@y% +} + + + +\def\pgftransforminvert{% + \pgf@pt@x=-\pgf@pt@x + \pgf@pt@y=-\pgf@pt@y + \ifpgf@pt@identity% + \else% + \edef\pgf@zerotest{\pgf@pt@ab,\pgf@pt@ba}% + \ifx\pgf@zerotest\pgf@zerozerotext% + \pgfmathreciprocal@{\pgf@pt@aa}% + \let\pgf@pt@aa\pgfmathresult% + \pgfmathreciprocal@{\pgf@pt@bb}% + \let\pgf@pt@bb\pgfmathresult% + \else% + {% + \pgfutil@tempdima=\pgf@pt@aa pt% + \pgfutil@tempdima=\pgf@pt@bb\pgfutil@tempdima% + \pgfutil@tempdimb=\pgf@pt@ba pt% + \pgfutil@tempdimb=\pgf@pt@ab\pgfutil@tempdimb% + \advance\pgfutil@tempdima by -\pgfutil@tempdimb% + \pgfmathreciprocal@{\pgf@sys@tonumber{\pgfutil@tempdima}}% + \pgfutil@tempdima=\pgfmathresult pt% + \pgf@x=\pgf@pt@bb\pgfutil@tempdima% + \xdef\pgf@new@pt@aa{\pgf@sys@tonumber{\pgf@x}}% + \pgf@x=\pgf@pt@aa\pgfutil@tempdima% + \xdef\pgf@new@pt@bb{\pgf@sys@tonumber{\pgf@x}}% + \pgfutil@tempdima=-\pgfutil@tempdima% + \pgf@x=\pgf@pt@ba\pgfutil@tempdima% + \xdef\pgf@new@pt@ba{\pgf@sys@tonumber{\pgf@x}}% + \pgf@x=\pgf@pt@ab\pgfutil@tempdima% + \xdef\pgf@new@pt@ab{\pgf@sys@tonumber{\pgf@x}}% + }% + \let\pgf@pt@aa\pgf@new@pt@aa% + \let\pgf@pt@ab\pgf@new@pt@ab% + \let\pgf@pt@ba\pgf@new@pt@ba% + \let\pgf@pt@bb\pgf@new@pt@bb% + \fi% + \pgf@pt@temp=\pgf@pt@x% + \pgf@pt@x=\pgf@pt@aa\pgf@pt@x% + \advance\pgf@pt@x by\pgf@pt@ba\pgf@pt@y% + \pgf@pt@y=\pgf@pt@bb\pgf@pt@y% + \advance\pgf@pt@y by\pgf@pt@ab\pgf@pt@temp% + \fi% +} + + + + + +\def\pgftransformcm#1#2#3#4#5{% + \edef\pgf@marshal{\noexpand\pgf@transformcm{#1}{#2}{#3}{#4}}% + \pgf@marshal{#5}% +} +\def\pgf@transformcm#1#2#3#4#5{% + {% + \pgf@process{#5}% + \pgf@xc=\pgf@x% + \pgf@yc=\pgf@y% + \pgfmathsetlength\pgf@x{#1}% + \pgfmathsetlength\pgf@y{#2}% + \pgf@xa=\pgf@pt@aa\pgf@x% + \advance\pgf@xa by \pgf@pt@ba\pgf@y% + \pgf@ya=\pgf@pt@ab\pgf@x% + \advance\pgf@ya by \pgf@pt@bb\pgf@y% + \pgfmathsetlength\pgf@x{#3}% + \pgfmathsetlength\pgf@y{#4}% + \pgf@xb=\pgf@pt@aa\pgf@x% + \advance\pgf@xb by \pgf@pt@ba\pgf@y% + \pgf@yb=\pgf@pt@ab\pgf@x% + \advance\pgf@yb by \pgf@pt@bb\pgf@y% + \pgf@x=\pgf@pt@aa\pgf@xc% + \advance\pgf@x by\pgf@pt@ba\pgf@yc% + \advance\pgf@x by\pgf@pt@x% + \pgf@y=\pgf@pt@ab\pgf@xc% + \advance\pgf@y by\pgf@pt@bb\pgf@yc% + \advance\pgf@y by\pgf@pt@y% + \global\pgf@x=\pgf@x% + \global\pgf@y=\pgf@y% + \xdef\pgf@tempaa{\pgf@sys@tonumber{\pgf@xa}}% + \xdef\pgf@tempab{\pgf@sys@tonumber{\pgf@ya}}% + \xdef\pgf@tempba{\pgf@sys@tonumber{\pgf@xb}}% + \xdef\pgf@tempbb{\pgf@sys@tonumber{\pgf@yb}}% + }% + \pgf@pt@x=\pgf@x% + \pgf@pt@y=\pgf@y% + \let\pgf@pt@aa=\pgf@tempaa% + \let\pgf@pt@ba=\pgf@tempba% + \let\pgf@pt@ab=\pgf@tempab% + \let\pgf@pt@bb=\pgf@tempbb% + \edef\pgf@idtest{\pgf@pt@aa,\pgf@pt@ba,\pgf@pt@ab,\pgf@pt@bb}% + \ifx\pgf@idtest\pgf@idmatrixtext% + \pgf@pt@identitytrue% + \else% + \pgf@pt@identityfalse% + \fi% +} + + + + +\def\pgftransformtriangle#1#2#3{% + \pgf@process{#2}% + \pgf@xa=\pgf@x% + \pgf@ya=\pgf@y% + \pgf@process{#3}% + \pgf@xb=\pgf@x% + \pgf@yb=\pgf@y% + \pgf@process{#1}% + \advance\pgf@xa by-\pgf@x% + \advance\pgf@ya by-\pgf@y% + \advance\pgf@xb by-\pgf@x% + \advance\pgf@yb by-\pgf@y% + \pgftransformcm% + {\pgf@sys@tonumber\pgf@xa}{\pgf@sys@tonumber\pgf@ya}% + {\pgf@sys@tonumber\pgf@xb}{\pgf@sys@tonumber\pgf@yb}% + {\pgfpoint{\pgf@x}{\pgf@y}}% +} + + + + +\def\pgftransformreset{% + \pgf@pt@x=0pt% + \pgf@pt@y=0pt% + \let\pgf@pt@aa\pgf@one@text% + \let\pgf@pt@ba\pgf@zero@text% + \let\pgf@pt@ab\pgf@zero@text% + \let\pgf@pt@bb\pgf@one@text% + \pgf@pt@identitytrue% +} + + + +\def\pgftransformresetnontranslations{% + \let\pgf@pt@aa\pgf@one@text% + \let\pgf@pt@ba\pgf@zero@text% + \let\pgf@pt@ab\pgf@zero@text% + \let\pgf@pt@bb\pgf@one@text% + \pgf@pt@identitytrue% +} + + + + + +\def\pgftransformshift#1{\pgftransformcm{1}{0}{0}{1}{#1}} +\def\pgftransformxshift#1{\pgftransformcm{1}{0}{0}{1}{\pgfpoint{#1}{+0pt}}} +\def\pgftransformyshift#1{\pgftransformcm{1}{0}{0}{1}{\pgfpoint{+0pt}{#1}}} + + + + + +\def\pgftransformscale#1{\pgftransformcm{#1}{0}{0}{#1}{\pgfpointorigin}} +\def\pgftransformxscale#1{\pgftransformcm{#1}{0}{0}{1.0}{\pgfpointorigin}} +\def\pgftransformyscale#1{\pgftransformcm{1.0}{0}{0}{#1}{\pgfpointorigin}} + + + + +\def\pgftransformxslant#1{\pgftransformcm{1.0}{0}{#1}{1.0}{\pgfpointorigin}} +\def\pgftransformyslant#1{\pgftransformcm{1.0}{#1}{0}{1.0}{\pgfpointorigin}} + + + + +\def\pgftransformrotate#1{% + \pgfmathparse{#1}% + \let\pgftransform@angle=\pgfmathresult% + \pgfmathsin@{\pgftransform@angle}% + \let\pgftransform@sin=\pgfmathresult% + \pgfmathcos@{\pgftransform@angle}% + \let\pgftransform@cos=\pgfmathresult% + \pgf@x=\pgftransform@sin pt% + \pgf@xa=-\pgf@x% + \pgftransformcm{\pgftransform@cos}{\pgftransform@sin}{\pgf@sys@tonumber{\pgf@xa}}{\pgftransform@cos}{\pgfpointorigin}% +} + + + + +\newif\ifpgfslopedattime +\newif\ifpgfallowupsidedownattime +\newif\ifpgfresetnontranslationattime + + +\def\pgftransformlineattime#1#2#3{% + \pgf@process{#2}% + \pgf@xb=\pgf@x% + \pgf@yb=\pgf@y% + \pgf@process{#3}% + \pgf@xc=\pgf@x% + \pgf@yc=\pgf@y% + \pgftransformshift{\pgfpointlineattime{#1}{\pgfqpoint{\pgf@xb}{\pgf@yb}}{\pgfqpoint{\pgf@xc}{\pgf@yc}}}% + \ifpgfresetnontranslationattime% + \pgftransformresetnontranslations% + \fi% + \ifpgfslopedattime% + \advance\pgf@xc by-\pgf@xb% + \advance\pgf@yc by-\pgf@yb% + \ifpgfallowupsidedownattime% + \else% + \ifdim\pgf@xc<0pt% + \pgf@xc=-\pgf@xc% + \pgf@yc=-\pgf@yc% + \fi% + \fi% + \pgf@x=\pgf@xc% + \pgf@y=\pgf@yc% + \pgfpointnormalised{}% + \pgf@ya=-\pgf@y% + \pgftransformcm% + {\pgf@sys@tonumber{\pgf@x}}{\pgf@sys@tonumber{\pgf@y}}% + {\pgf@sys@tonumber{\pgf@ya}}{\pgf@sys@tonumber{\pgf@x}}{\pgfpointorigin}% + \fi% +} + + + +\def\pgftransformarcaxesattime#1#2#3#4#5#6{% + \pgfpointarcaxesattime{#1}{#2}{#3}{#4}{#5}{#6}% + \pgftransformshift{\pgfqpoint{\pgf@x}{\pgf@y}}% + \ifpgfresetnontranslationattime% + \pgftransformresetnontranslations% + \fi% + \ifpgfslopedattime% + \pgf@x=\pgf@xa% + \pgf@y=\pgf@ya% + \pgf@process{\pgfpointnormalised{}}% + \ifpgfallowupsidedownattime% + \else% + \ifdim\pgf@x<0pt% + \pgf@x=-\pgf@x% + \pgf@y=-\pgf@y% + \fi% + \fi% + \pgf@ya=-\pgf@y% + \pgftransformcm% + {\pgf@sys@tonumber{\pgf@x}}{\pgf@sys@tonumber{\pgf@y}}% + {\pgf@sys@tonumber{\pgf@ya}}{\pgf@sys@tonumber{\pgf@x}}{\pgfpointorigin}% + \fi% +} + + + + +\def\pgftransformcurveattime#1#2#3#4#5{% + \pgfpointcurveattime{#1}{#2}{#3}{#4}{#5}% + \pgftransformshift{\pgfqpoint{\pgf@x}{\pgf@y}}% + \ifpgfresetnontranslationattime% + \pgftransformresetnontranslations% + \fi% + \ifpgfslopedattime% + \pgf@x=\pgf@xa% + \pgf@y=\pgf@ya% + \advance\pgf@x by-\pgf@xb% + \advance\pgf@y by-\pgf@yb% + \ifpgfallowupsidedownattime% + \else% + \ifdim\pgf@x<0pt% + \pgf@x=-\pgf@x% + \pgf@y=-\pgf@y% + \fi% + \fi% + \pgfpointnormalised{}% + \pgf@ya=-\pgf@y% + \pgftransformcm% + {\pgf@sys@tonumber{\pgf@x}}{\pgf@sys@tonumber{\pgf@y}}% + {\pgf@sys@tonumber{\pgf@ya}}{\pgf@sys@tonumber{\pgf@x}}{\pgfpointorigin}% + \fi% +} + + + + + + +\def\pgftransformarrow#1#2{% + \pgf@process{#2}% + \pgftransformshift{}% + \pgf@xa=\pgf@x% + \pgf@ya=\pgf@y% + \pgf@process{#1}% + \advance\pgf@xa by-\pgf@x% + \advance\pgf@ya by-\pgf@y% + \pgf@x=\pgf@xa% + \pgf@y=\pgf@ya% + \pgfpointnormalised{}% + \pgf@ya=-\pgf@y% + \pgftransformcm% + {\pgf@sys@tonumber{\pgf@x}}{\pgf@sys@tonumber{\pgf@y}}% + {\pgf@sys@tonumber{\pgf@ya}}{\pgf@sys@tonumber{\pgf@x}}{\pgfqpoint{0pt}{0pt}}% +} + + + + + +\def\pgftransformationadjustments{% + \ifpgf@pt@identity% + \let\pgfhorizontaltransformationadjustment\pgf@one@text% + \let\pgfverticaltransformationadjustment\pgf@one@text% + \else% + {% + \pgf@pt@x0pt% + \pgf@pt@y0pt% + \pgfpointtransformed{\pgfqpoint{1pt}{0pt}}% + \pgfmathveclen@{\pgf@sys@tonumber\pgf@x}{\pgf@sys@tonumber\pgf@y}% + \pgfmathreciprocal@{\pgfmathresult}% + \global\let\pgf@temp@hori\pgfmathresult + \pgfpointtransformed{\pgfqpoint{0pt}{1pt}}% + \pgfmathveclen@{\pgf@sys@tonumber\pgf@x}{\pgf@sys@tonumber\pgf@y}% + \pgfmathreciprocal@{\pgfmathresult}% + \global\let\pgf@temp@vert\pgfmathresult + \xdef\pgf@marshal@b{\def\noexpand\pgfverticaltransformationadjustment{\pgfmathresult}} + }% + \let\pgfhorizontaltransformationadjustment\pgf@temp@hori% + \let\pgfverticaltransformationadjustment\pgf@temp@vert% + \fi% +} + +\let\pgfhorizontaltransformationadjustment\pgf@one@text +\let\pgfverticaltransformationadjustment\pgf@one@text + + + + +\def\pgflowlevelsynccm{% + \pgfsys@transformcm% + {\pgf@pt@aa}{\pgf@pt@ab}% + {\pgf@pt@ba}{\pgf@pt@bb}% + {\pgf@pt@x}{\pgf@pt@y}% + \pgftransformreset% + \pgf@relevantforpicturesizefalse% +} + + + + +\def\pgflowlevel#1{% + {% + \pgftransformreset% + #1% + \pgflowlevelsynccm% + }% + \pgf@relevantforpicturesizefalse% +} + +\def\pgflowlevelscope#1{\pgfscope\pgflowlevel{#1}} +\def\endpgflowlevelscope{\endpgfscope} + +\long\def\pgflowlevelobj#1#2{\pgfscope{\pgflowlevel{#1}#2}\endpgfscope} + + + + + +\def\pgfviewboxscope#1#2#3#4#5{% + {% + \pgf@process{#1}% + \pgf@pos@transform@glob + \pgf@xa=\pgf@x% + \pgf@ya=\pgf@y% + \pgf@process{#2}% + \pgf@pos@transform@glob + \ifdim\pgf@x<\pgf@xa% + \pgf@xb\pgf@x% + \pgf@xc\pgf@xa% + \else% + \pgf@xb\pgf@xa% + \pgf@xc\pgf@x% + \fi% + \ifdim\pgf@y<\pgf@ya% + \pgf@yb\pgf@y% + \pgf@yc\pgf@ya% + \else% + \pgf@yb\pgf@ya% + \pgf@yc\pgf@y% + \fi% + \pgf@process{#3}% + \pgf@pos@transform@glob + \pgf@xa=\pgf@x% + \pgf@ya=\pgf@y% + \pgf@process{#4}% + \pgf@pos@transform@glob + \ifdim\pgf@x<\pgf@xa% + \else% + \pgfutil@tempdima\pgf@x + \pgf@x\pgf@xa% + \pgf@xa\pgfutil@tempdima% + \fi% + \ifdim\pgf@y<\pgf@ya% + \else% + \pgfutil@tempdima\pgf@y + \pgf@y\pgf@ya% + \pgf@ya\pgfutil@tempdima% + \fi% + \pgfusetype{.view}% + \edef\pgf@marshal{\expandafter\noexpand\csname pgfsys@viewbox#5\endcsname{\the\pgf@x}{\the\pgf@y}{\the\pgf@xa}{\the\pgf@ya}{\the\pgf@xb}{\the\pgf@yb}{\the\pgf@xc}{\the\pgf@yc}}% + \pgf@marshal% + }% + \pgf@relevantforpicturesizefalse% +} + +\def\endpgfviewboxscope{\pgfsys@endviewbox} + +\let\startpgfviewboxscope=\pgfviewboxscope +\let\stoppgfviewboxscope=\endpgfviewboxscope + + +\let\pgfapproximatenonlineartranslation\relax +\let\pgfapproximatenonlineartransformation\relax +\def\pgftransformnonlinear#1{\pgferror{You need to say + \string\usepgfmodule{nonlineartransformations} to use nonlinear transformations}} + + +\endinput diff --git a/preprints/A-polynomial-time-algorithm-for-three-machine-unit-job-scheduling-September-24-2026/build/tex/runtime/generic/pgf/basiclayer/pgfcoretransparency.code.tex b/preprints/A-polynomial-time-algorithm-for-three-machine-unit-job-scheduling-September-24-2026/build/tex/runtime/generic/pgf/basiclayer/pgfcoretransparency.code.tex new file mode 100644 index 0000000000000000000000000000000000000000..9cf7ea4496989baa39be6bd29e4009e96ac2476d --- /dev/null +++ b/preprints/A-polynomial-time-algorithm-for-three-machine-unit-job-scheduling-September-24-2026/build/tex/runtime/generic/pgf/basiclayer/pgfcoretransparency.code.tex @@ -0,0 +1,146 @@ + +\ProvidesFileRCS{pgfcoretransparency.code.tex} + + + +\def\pgfsetstrokeopacity#1{% + \pgfmathparse{#1}% + \expandafter\pgfsys@stroke@opacity\expandafter{\pgfmathresult}} + + + +\def\pgfsetfillopacity#1{% + \pgfmathparse{#1}% + \expandafter\pgfsys@fill@opacity\expandafter{\pgfmathresult}} + + + + +\def\pgfsetblendmode#1{\pgfsys@blend@mode{#1}} + + + +\def\pgfdeclarefading#1#2{% + \setbox\pgfutil@tempboxa=\hbox{{\pgfsys@beginscope\pgfutil@color{white}\ignorespaces#2\pgfsys@endscope}}% + \pgfsys@fadingfrombox{#1}\pgfutil@tempboxa% +} + +\pgfutil@colorlet{pgftransparent}{black} + + + +\def\pgfsetfading#1#2{% + {% + \pgftransformreset% + #2% + \pgfgettransform{\pgftrans@temp}% + \edef\pgf@marshal{\noexpand\pgfsys@usefading{#1}\pgftrans@temp}% + \pgf@marshal% + }% +} + + + + +\def\pgfsetfadingforcurrentpath#1#2{% + \ifdim\pgf@pathminx=16000pt% + \pgfsetfading{#1}{#2}% + \else% + \pgfsetfading{#1}{% + \pgf@xb=.5\pgf@pathmaxx% + \advance\pgf@xb by.5\pgf@pathminx% + \pgf@yb=.5\pgf@pathmaxy% + \advance\pgf@yb by.5\pgf@pathminy% + \pgf@xc=\pgf@pathmaxx% + \advance\pgf@xc by-\pgf@pathminx% + \ifdim\pgf@xc<0.1pt\pgf@xc=0.1pt\fi% + \pgf@yc=\pgf@pathmaxy% + \advance\pgf@yc by-\pgf@pathminy% + \ifdim\pgf@yc<0.1pt\pgf@yc=0.1pt\fi% + \pgf@xc=.02\pgf@xc% + \pgf@yc=.02\pgf@yc% + \pgftransformcm{1}{0}{0}{1}{\pgfqpoint{\pgf@xb}{\pgf@yb}}% + \pgftransformcm{\pgf@sys@tonumber{\pgf@xc}}{0}{0}{\pgf@sys@tonumber{\pgf@yc}}{\pgfpointorigin}% + \pgfsys@clipfading% + #2% + }% + \fi% +} + + + + + +\def\pgfsetfadingforcurrentpathstroked#1#2{% + \ifdim\pgf@pathminx=16000pt% + \pgfsetfading{#1}{#2}% + \else% + \pgfsetfading{#1}{% + \pgf@xb=.5\pgf@pathmaxx% + \advance\pgf@xb by.5\pgf@pathminx% + \pgf@yb=.5\pgf@pathmaxy% + \advance\pgf@yb by.5\pgf@pathminy% + \pgf@xc=\pgf@pathmaxx% + \advance\pgf@xc by-\pgf@pathminx% + \advance\pgf@xc by\pgflinewidth% + \ifdim\pgf@xc<0.1pt\pgf@xc=0.1pt\fi% + \pgf@yc=\pgf@pathmaxy% + \advance\pgf@yc by-\pgf@pathminy% + \advance\pgf@yc by\pgflinewidth% + \ifdim\pgf@yc<0.1pt\pgf@yc=0.1pt\fi% + \pgf@xc=.02\pgf@xc% + \pgf@yc=.02\pgf@yc% + \pgftransformcm{1}{0}{0}{1}{\pgfqpoint{\pgf@xb}{\pgf@yb}}% + \pgftransformcm{\pgf@sys@tonumber{\pgf@xc}}{0}{0}{\pgf@sys@tonumber{\pgf@yc}}{\pgfpointorigin}% + \pgfsys@clipfading% + #2% + }% + \fi% +} + + + +\def\pgftransparencygroup{% + \pgfutil@ifnextchar[{\pgftransparencygroup@}{\pgftransparencygroup@[]}% +} +\def\pgftransparencygroup@[#1]{% + \begingroup% + \pgfkeys{/pgf/transparency group options/.cd,% + isolated=false, + knockout=false, + #1}% + \setbox\pgfutil@tempboxa=\hbox\bgroup% +} +\pgfkeys{ + /pgf/transparency group options/.cd, + isolated/.is if=pgfsys@transparency@group@isolated, + knockout/.is if=pgfsys@transparency@group@knockout +} + +\def\endpgftransparencygroup{% + \egroup% + \ifdim\pgf@picminx=16000pt% + \else% + \advance\pgf@picminx by-1cm% + \advance\pgf@picmaxx by1cm% + \advance\pgf@picminy by-1cm% + \advance\pgf@picmaxy by1cm% + \setbox\pgfutil@tempboxa=\hbox{% + \vrule width0pt height\pgf@picmaxy depth-\pgf@picminy% + \hskip-\pgf@picminx% + \box\pgfutil@tempboxa% + \hskip\pgf@picmaxx% + }% + \pgfsys@transparencygroupfrombox\pgfutil@tempboxa% + \wd\pgfutil@tempboxa=0pt% + \ht\pgfutil@tempboxa=0pt% + \dp\pgfutil@tempboxa=0pt% + \hskip\pgf@picminx\box\pgfutil@tempboxa\hskip-\pgf@picminx% + \fi% + \endgroup% +} + +\let\startpgftransparencygroup=\pgftransparencygroup +\let\stoppgftransparencygroup=\endpgftransparencygroup + +\endinput diff --git a/preprints/A-polynomial-time-algorithm-for-three-machine-unit-job-scheduling-September-24-2026/build/tex/runtime/generic/pgf/frontendlayer/tikz/libraries/tikzlibrarycalc.code.tex b/preprints/A-polynomial-time-algorithm-for-three-machine-unit-job-scheduling-September-24-2026/build/tex/runtime/generic/pgf/frontendlayer/tikz/libraries/tikzlibrarycalc.code.tex new file mode 100644 index 0000000000000000000000000000000000000000..a9d0c5a915737625006897f0ef0279d52168e611 --- /dev/null +++ b/preprints/A-polynomial-time-algorithm-for-three-machine-unit-job-scheduling-September-24-2026/build/tex/runtime/generic/pgf/frontendlayer/tikz/libraries/tikzlibrarycalc.code.tex @@ -0,0 +1,455 @@ + +\ProvidesFileRCS{tikzlibrarycalc.code.tex} + + + + +\def\tikz@let@command et{% + \let\p=\tikz@cc@dop% + \let\x=\tikz@cc@dox% + \let\y=\tikz@cc@doy% + \let\n=\tikz@cc@don% + \pgfutil@ifnextchar i{\tikz@cc@stop@let}{\tikz@cc@handle@line}% +}% +\def\tikz@cc@handle@line{% + \pgfutil@ifnextchar\p{% + \tikz@cc@handle@coor% + }{% + \pgfutil@ifnextchar\n{% + \tikz@cc@handle@num% + }{% + \pgfutil@ifnextchar i{% + \tikz@cc@stop@let + }{% + \tikzerror{``\string\p'' or ``\string\n'' expected}% + }% + }% + }% +}% +\def\tikz@cc@handle@num\n#1#2=#3{% + \pgfmathparse{#3}% + \expandafter\edef\csname tikz@cc@n@#1\endcsname{\pgfmathresult\ifpgfmathunitsdeclared pt\fi} + \pgfutil@ifnextchar,{\tikz@cc@handle@nextline}{\tikz@cc@stop@let}% +}% +\def\tikz@cc@handle@coor\p#1#2={% + \def\tikz@cc@coord@name{#1}% + \tikz@scan@one@point\tikz@cc@dolet% +}% +\def\tikz@cc@dolet#1{% + \pgf@process{#1}% + \expandafter\edef\csname tikz@cc@p@\tikz@cc@coord@name\endcsname{\the\pgf@x,\the\pgf@y}% + \expandafter\edef\csname tikz@cc@x@\tikz@cc@coord@name\endcsname{\the\pgf@x}% + \expandafter\edef\csname tikz@cc@y@\tikz@cc@coord@name\endcsname{\the\pgf@y}% + \pgfutil@ifnextchar,{\tikz@cc@handle@nextline}{\tikz@cc@stop@let}% +}% +\def\tikz@cc@handle@nextline,{% + \tikz@cc@handle@line% +}% +\def\tikz@cc@stop@let in{% + \tikz@scan@next@command% +}% + +\def\tikz@cc@dop#1{\csname tikz@cc@p@#1\endcsname}% +\def\tikz@cc@dox#1{\csname tikz@cc@x@#1\endcsname}% +\def\tikz@cc@doy#1{\csname tikz@cc@y@#1\endcsname}% +\def\tikz@cc@don#1{\csname tikz@cc@n@#1\endcsname}% + + + + +\def\tikz@parse@calculator#1(${% + \def\tikz@cc@command{#1}% + \begingroup% + \pgf@xa=0pt% + \pgf@ya=0pt% + \tikz@cc@parse+% +}% + +\def\tikz@cc@parse{% + \pgfutil@ifnextchar${% + \tikz@cc@end% + } + {\pgfutil@ifnextchar+{% + \tikz@cc@add% + }{% + \pgfutil@ifnextchar-{% + \tikz@cc@sub% + }{% + \tikzerror{+ or - expected}% + \tikz@cc@end$% + }% + }% + }% +}% + +\def\tikz@cc@end$#1){% + \xdef\tikz@marshal{\noexpand\pgfqpoint{\the\pgf@xa}{\the\pgf@ya}}% + \endgroup% + \expandafter\tikz@cc@command\expandafter{\tikz@marshal}% +}% + + +\def\tikz@cc@add+{% + \def\tikz@cc@factor{1}% + \tikz@cc@factororcoordinate% +}% +\def\tikz@cc@sub-{% + \def\tikz@cc@factor{-1}% + \tikz@cc@factororcoordinate% +}% + +\def\tikz@cc@factororcoordinate{% + \pgfutil@ifnextchar({% + \tikz@cc@coordinate% + }{% + \tikz@cc@parse@factor% + }% +}% + +\def\tikz@cc@parse@factor#1*({% + \pgfmathparse{#1*\tikz@cc@factor}% + \let\tikz@cc@factor=\pgfmathresult% + \tikz@cc@coordinate(% +}% + +\def\tikz@cc@coordinate{% + \tikz@scan@absolute\tikz@cc@after@coordinate% +}% +\def\tikz@cc@after@coordinate#1{% + \pgf@process{#1}% + \pgf@xb=\pgf@x% + \pgf@yb=\pgf@y% + \tikz@cc@mid@checks% +}% + + + +\def\tikz@cc@mid@checks{% + \ifnum\the\catcode`\!=\active\relax + \expandafter\tikz@cc@mid@checks@active + \else + \expandafter\tikz@cc@mid@checks@nonactive + \fi +}% + +\def\tikz@cc@mid@checks@nonactive{% + \pgfutil@ifnextchar!{% + \tikz@cc@mid@nonactive% + }{% + \advance\pgf@xa by\tikz@cc@factor\pgf@xb + \advance\pgf@ya by\tikz@cc@factor\pgf@yb + \tikz@cc@parse% + }% +}% + +\def\tikz@cc@mid@nonactive!{% + \pgfutil@ifnextchar({% + \tikz@scan@one@point\tikz@cc@project% + }{% + \tikz@cc@mid@num@nonactive% + }% +}% + +\begingroup + \catcode`\!=\active + \gdef\tikz@cc@mid@checks@active{% + \pgfutil@ifnextchar!{% + \tikz@cc@mid@active% + }{% + \advance\pgf@xa by\tikz@cc@factor\pgf@xb + \advance\pgf@ya by\tikz@cc@factor\pgf@yb + \tikz@cc@parse% + }% + }% + + \gdef\tikz@cc@mid@active!{% + \pgfutil@ifnextchar({% + \tikz@scan@one@point\tikz@cc@project% + }{% + \tikz@cc@mid@num@active% + }% + }% +\endgroup + + +\def\tikz@cc@mid@num@nonactive#1!{\tikz@cc@mid@num{#1}}% + +\begingroup + \catcode`\!=\active + \gdef\tikz@cc@mid@num@active#1!{\tikz@cc@mid@num{#1}}% +\endgroup + +\def\tikz@cc@mid@num#1{% + \pgfmathparse{#1}% + \ifpgfmathunitsdeclared% + \let\tikz@cc@mid@unit=\pgfmathresult% + \expandafter\tikz@cc@scan@rot\expandafter\tikz@cc@after@unit% + \else% + \let\tikz@cc@mid@factor=\pgfmathresult% + \pgfmathparse{1-\tikz@cc@mid@factor}% + \let\tikz@cc@mid@factor@one=\pgfmathresult% + \expandafter\tikz@cc@scan@rot\expandafter\tikz@cc@after@num% + \fi% +}% + +\def\tikz@cc@after@num#1{% + \pgf@process{#1}% + \pgf@xb=\tikz@cc@mid@factor@one\pgf@xb% + \pgf@yb=\tikz@cc@mid@factor@one\pgf@yb% + \advance\pgf@xb by\tikz@cc@mid@factor\pgf@x% + \advance\pgf@yb by\tikz@cc@mid@factor\pgf@y% + \tikz@cc@mid@checks% +}% + + + + +\def\tikz@cc@after@unit#1{% + \pgf@process{#1}% + \advance\pgf@x by-\pgf@xb% + \advance\pgf@y by-\pgf@yb% + \pgf@process{\pgfpointnormalised{}}% + \advance\pgf@xb by\tikz@cc@mid@unit\pgf@x% + \advance\pgf@yb by\tikz@cc@mid@unit\pgf@y% + \tikz@cc@mid@checks% +}% + +\def\tikz@cc@project#1{% + \pgf@process{#1}% + \pgf@xc=\pgf@x% + \pgf@yc=\pgf@y% + \begingroup + \ifnum\the\catcode`\!=\active + \def\tikz@next{% + \endgroup + \expandafter\tikz@cc@scan@rot\expandafter\tikz@cc@after@project + \tikz@cc@scan@ex@active}% + \else + \def\tikz@next{% + \endgroup + \expandafter\tikz@cc@scan@rot\expandafter\tikz@cc@after@project + \tikz@cc@scan@ex@nonactive}% + \fi + \tikz@next% +}% + +\def\tikz@cc@scan@ex@nonactive!{}% + +\begingroup + \catcode`\!=\active + \gdef\tikz@cc@scan@ex@active!{}% +\endgroup + +\def\tikz@cc@after@project#1{% + \pgf@process{#1}% + \advance\pgf@x by-\pgf@xb% + \advance\pgf@y by-\pgf@yb% + \advance\pgf@xc by-\pgf@xb% + \advance\pgf@yc by-\pgf@yb% + \pgf@process{\pgfpointnormalised{}}% + \pgf@xc=\pgf@sys@tonumber{\pgf@xc}\pgf@x% + \advance\pgf@xc by\pgf@sys@tonumber{\pgf@yc}\pgf@y% + \advance\pgf@xb by\pgf@sys@tonumber{\pgf@xc}\pgf@x% + \advance\pgf@yb by\pgf@sys@tonumber{\pgf@xc}\pgf@y% + \tikz@cc@mid@checks% +}% + + +\def\tikz@cc@scan@rot#1{% + \pgfutil@ifnextchar({% + \tikz@scan@one@point#1% + }% + {% + \def\tikz@cc@scan@rot@cmd{#1}% + \ifnum\the\catcode`\:=\active\relax + \expandafter\tikz@cc@scan@one@rot@active% + \else + \expandafter\tikz@cc@scan@one@rot@nonactive% + \fi + }% +}% + +\def\tikz@cc@scan@one@rot@nonactive#1:{% + \def\tikz@cc@scan@rot@angle{#1}% + \tikz@scan@one@point\tikz@cc@handle@rot% +}% + +\begingroup + \catcode`\:=\active + \gdef\tikz@cc@scan@one@rot@active#1:{% + \def\tikz@cc@scan@rot@angle{#1}% + \tikz@scan@one@point\tikz@cc@handle@rot% + }% +\endgroup + +\def\tikz@cc@handle@rot#1{% + \pgf@process{#1}% + {% + \pgftransformreset% + \pgf@xc=\pgf@x% + \pgf@yc=\pgf@y% + \pgftransformshift{\pgfqpoint{\pgf@xb}{\pgf@yb}}% + \pgftransformrotate{\tikz@cc@scan@rot@angle}% + \pgftransformshift{\pgfqpoint{-\pgf@xb}{-\pgf@yb}}% + \pgfpointtransformed{\pgfqpoint{\pgf@xc}{\pgf@yc}}% + \expandafter + }% + \edef\tikz@marshal{\noexpand\tikz@cc@scan@rot@cmd{\noexpand\pgfqpoint{\the\pgf@x}{\the\pgf@y}}}% + \tikz@marshal% +}% + + + + + + +\tikzdeclarecoordinatesystem{tangent} +{% + \tikzset{cs/.cd,#1}% + \expandafter\ifx\csname tikz@tangent@\tikz@cs@type\endcsname\relax% + \tikzerror{I do not know how to compute the tangent to + a \tikz@cs@type}% + \pgfpointorigin% + \else% + \expandafter\tikz@scan@one@point\expandafter\tikz@lib@do@tangent\tikz@cs@point% + \fi% +}% + +\tikzset{cs/node/.code=\tikz@cs@unpack{\tikz@cs@node}{\tikz@cs@type}{#1}}% +\tikzset{cs/point/.store in=\tikz@cs@point}% + +\def\tikz@lib@do@tangent{\csname tikz@tangent@\tikz@cs@type\endcsname}% + +\def\tikz@tangent@coordinate#1{% + \pgfpointanchor{\tikz@cs@node}{center}% +}% + +\def\tikz@tangent@circle#1{% + {% + \pgf@process{\pgfpointtransformed{#1}}% + \pgf@xa=\pgf@x% + \pgf@ya=\pgf@y% + \pgf@process{\pgfpointtransformed{\pgfpointanchor{\tikz@cs@node}{center}}}% + \advance\pgf@xa by-\pgf@x% + \advance\pgf@ya by-\pgf@y% + \pgftransformreset% + \pgftransformshift{\pgfpointanchor{\tikz@cs@node}{center}}% + \pgf@process{\pgfpointanchor{\tikz@cs@node}{east}}% + \pgf@xc=\pgf@x% + \pgfmathparse{veclen(\the\pgf@xa,\the\pgf@ya)} + \pgfmathparse{acos(\the\pgf@xc/\pgfmathresult)} + \ifnum\pgfkeysvalueof{/tikz/cs/solution}>1\relax% + \pgfmathparse{0-\pgfmathresult}% + \fi% + \let\tikz@lib@temp=\pgfmathresult% + \pgf@process{\pgfpointnormalised{\pgfqpoint{\pgf@xa}{\pgf@ya}}}% + \pgf@ya=-\pgf@y% + \pgftransformcm{\pgf@sys@tonumber{\pgf@x}}{\pgf@sys@tonumber{\pgf@y}}{\pgf@sys@tonumber{\pgf@ya}}{\pgf@sys@tonumber{\pgf@x}}{\pgfpointorigin}% + \pgf@process{\pgfpointtransformed{\pgfpointpolar{\tikz@lib@temp}{\the\pgf@xc}}}% + }% + {% + \pgftransforminvert% + \pgf@process{\pgfpointtransformed{}}% + }% +}% + + + + +\def\tikz@intersect@circle@and@circle{% + {% + \pgftransformreset% + \pgf@process{\pgfpointanchor{\tikz@cs@node@a}{center}}% + \pgf@xa=\pgf@x% + \pgf@ya=\pgf@y% + \pgf@process{\pgfpointanchor{\tikz@cs@node@a}{east}}% + \advance\pgf@x by-\pgf@xa% + \pgf@xc=\pgf@x% + \pgf@process{\pgfpointanchor{\tikz@cs@node@b}{center}}% + \pgf@xb=\pgf@x% + \pgf@yb=\pgf@y% + \pgf@process{\pgfpointanchor{\tikz@cs@node@b}{east}}% + \advance\pgf@x by-\pgf@xb% + \pgf@yc=\pgf@x% + \pgf@process{% + \pgfpointintersectionofcircles{\pgfqpoint{\pgf@xa}{\pgf@ya}}{\pgfqpoint{\pgf@xb}{\pgf@yb}}{\pgf@xc}{\pgf@yc}{\pgfkeysvalueof{/tikz/cs/solution}}% + }% + }% + {% + \pgftransforminvert% + \pgf@process{\pgfpointtransformed{}}% + }% +}% + + +\def\tikz@intersect@line@and@circle{% + {% + \expandafter\tikz@scan@one@point\expandafter\tikz@parse@line\tikz@cs@line@a% + \pgf@process{\pgfpointtransformed{}}% + \pgf@xb=\pgf@x% + \pgf@yb=\pgf@y% + \pgf@process{\pgfpointtransformed{\pgfqpoint{\pgf@xc}{\pgf@yc}}}% + \pgf@xa=\pgf@x% + \pgf@ya=\pgf@y% + \pgf@process{\pgfpointtransformed{\pgfpointanchor{\tikz@cs@node@b}{center}}}% + \advance\pgf@xa by-\pgf@x% + \advance\pgf@ya by-\pgf@y% + \advance\pgf@xb by-\pgf@x% + \advance\pgf@yb by-\pgf@y% + \pgftransformreset% + \pgftransformshift{\pgfpointanchor{\tikz@cs@node@b}{center}}% + \pgf@process{\pgfpointanchor{\tikz@cs@node@b}{east}}% + \edef\tikz@lib@saved@radius{\pgf@sys@tonumber{\pgf@x}}% + \pgf@x=\pgf@xb% + \pgf@y=\pgf@yb% + \advance\pgf@x by-\pgf@xa% + \advance\pgf@y by-\pgf@ya% + \pgf@process{\pgfpointnormalised{}}% + \pgf@xc=\pgf@sys@tonumber{\pgf@xa}\pgf@x% + \advance\pgf@xc by\pgf@sys@tonumber{\pgf@ya}\pgf@y% + \pgf@xc=-\pgf@xc% + \advance\pgf@xa by\pgf@sys@tonumber{\pgf@xc}\pgf@x% + \advance\pgf@ya by\pgf@sys@tonumber{\pgf@xc}\pgf@y% + \pgf@xc=\pgf@x% + \pgf@yc=\pgf@y% + \pgf@xb=\tikz@lib@saved@radius pt% + \c@pgf@counta=1\relax% + \loop% + \ifdim\pgf@xb>50pt% + \multiply\c@pgf@counta by2\relax% + \divide\pgf@xa by2\relax% + \divide\pgf@ya by2\relax% + \divide\pgf@xb by2\relax% + \repeat% + \pgf@xb=\pgf@sys@tonumber{\pgf@xb}\pgf@xb% + \pgf@yb=\pgf@sys@tonumber{\pgf@xa}\pgf@xa% + \advance\pgf@xb by-\pgf@yb% + \pgf@yb=\pgf@sys@tonumber{\pgf@ya}\pgf@ya% + \advance\pgf@xb by-\pgf@yb% + \ifdim\pgf@xb<0pt% + \pgf@xb=0pt% + \fi% + \pgfmathsqrt@{\pgf@sys@tonumber{\pgf@xb}}% + \pgfmathmultiply@{\pgfmathresult}{\the\c@pgf@counta}% + \multiply\pgf@xa by\c@pgf@counta\relax% + \multiply\pgf@ya by\c@pgf@counta\relax% + \ifnum\pgfkeysvalueof{/tikz/cs/solution}>1\relax% + \pgfmathsubtract{0}{\pgfmathresult}% + \fi% + \advance\pgf@xa by \pgfmathresult\pgf@xc% + \advance\pgf@ya by \pgfmathresult\pgf@yc% + \pgf@process{\pgfpointtransformed{\pgfqpoint{\pgf@xa}{\pgf@ya}}}% + }% + {% + \pgftransforminvert% + \pgf@process{\pgfpointtransformed{}}% + }% +}% + +\def\tikz@intersect@circle@and@line{% + {% + \let\tikz@cs@node@b=\tikz@cs@node@a% + \let\tikz@cs@line@a=\tikz@cs@line@b% + \tikz@intersect@line@and@circle% + }% +}% diff --git a/preprints/A-polynomial-time-algorithm-for-three-machine-unit-job-scheduling-September-24-2026/build/tex/runtime/generic/pgf/frontendlayer/tikz/libraries/tikzlibrarydecorations.code.tex b/preprints/A-polynomial-time-algorithm-for-three-machine-unit-job-scheduling-September-24-2026/build/tex/runtime/generic/pgf/frontendlayer/tikz/libraries/tikzlibrarydecorations.code.tex new file mode 100644 index 0000000000000000000000000000000000000000..ee19fe154ccd8557b7b3dc4e400e676e21172040 --- /dev/null +++ b/preprints/A-polynomial-time-algorithm-for-three-machine-unit-job-scheduling-September-24-2026/build/tex/runtime/generic/pgf/frontendlayer/tikz/libraries/tikzlibrarydecorations.code.tex @@ -0,0 +1,173 @@ + +\usepgfmodule{decorations}% + + +\tikzset{decorate/.is if=tikz@decoratepath, + /pgf/decoration/name/.store in=\tikz@decoration@name, + /pgf/decoration/.unknown/.code=% + \pgfifdecoration{\pgfkeyscurrentname}{\edef\tikz@decoration@name{\pgfkeyscurrentname}} + {\pgfifmetadecoration{\pgfkeyscurrentname}{\edef\tikz@decoration@name{\pgfkeyscurrentname}} + {\pgfkeys{/errors/unknown + key/.expanded={/pgf/decoration/\pgfkeyscurrentname}{\pgfutil@unexpanded{#1}}}}},% + /pgf/decoration/raise/.code={\def\tikz@dec@shift{\pgftransformyshift{#1}}\tikz@dec@trans}, + /pgf/decoration/mirror/.code={% + \csname if#1\endcsname + \def\tikz@dec@mirror{\pgftransformyscale{-1}}% + \else% + \let\tikz@dec@mirror=\relax% + \fi% + \tikz@dec@trans% + }, + /pgf/decoration/mirror/.default=true, + /pgf/decoration/transform/.code={% + {% + \pgftransformreset% + \def\tikz@transform{}% + \pgfkeysalso{/tikz/.cd, #1}% + \expandafter\gdef\expandafter\tikz@g@temptransform\expandafter{\tikz@transform}% + }% + \let\tikz@dec@user@transform=\tikz@g@temptransform% + \tikz@dec@trans% + },% + /pgf/decoration/pre/.initial=lineto, + /pgf/decoration/post/.initial=lineto, + /pgf/decoration/pre length/.initial=0pt, + /pgf/decoration/post length/.initial=0pt, +}% + +\let\tikz@decoration@name=\tikz@nonetext + +\def\tikz@dec@trans{\pgfsetdecorationsegmenttransformation{\tikz@dec@user@transform\tikz@dec@mirror\tikz@dec@shift}}% +\let\tikz@dec@user@transform=\relax +\let\tikz@dec@shift=\relax +\let\tikz@dec@mirror=\relax + +\pgfdeclaremetadecoration{tikz@internal}{pre}{% + \state{pre}[width=\pgfkeysvalueof{/pgf/decoration/pre length}, next state=main] + { + \tikz@dec@trans + \decoration{\pgfkeysvalueof{/pgf/decoration/pre}} + }% + \state{main}[width=\pgfmetadecoratedremainingdistance-\pgfkeysvalueof{/pgf/decoration/post length}, next state=final] + { + \tikz@dec@trans + \decoration{\tikz@decoration@name} + }% + \state{final} + { + \tikz@dec@trans + \decoration{\pgfkeysvalueof{/pgf/decoration/post}} + }% +}% + + +\newif\iftikz@metadecoration + +\def\tikz@lib@dec@addtomacro#1#2{\expandafter\def\expandafter#1\expandafter{#1#2}}% + +\def\tikz@lib@decoration[#1]{% + \begingroup% + \tikzset{#1}% + \ifx\tikz@decoration@name\tikz@nonetext% + \else + \expandafter\tikz@lib@do@dec% + \fi% + \pgfutil@ifnextchar\bgroup{% + \begingroup% + \tikz@auto@end@pathfalse% + \aftergroup\tikz@enddecoration% + \afterassignment\tikz@scan@next@command% + \let\tikz@lib@next% + }% + {% + \tikzerror{A decoration must begin with a brace}% + \tikz@enddecoration% + }% +}% +\def\tikz@lib@do@dec{% + \pgfifmetadecoration{\tikz@decoration@name}% + {\expandafter\pgfmetadecoration\expandafter{\tikz@decoration@name}}% + {% + \pgfifdecoration{\tikz@decoration@name}% + {\tikz@lib@dec@startdecoration}% + {\tikzerror{Unknown (meta-)decoration '\tikz@decoration@name'. Perhaps you + misspelled it?}}% + }% + \ifx\pgfdecorateexistingpath\pgfutil@empty% + \pgfpathmoveto{\pgfqpoint{\the\tikz@lastxsaved}{\the\tikz@lastysaved}}% + \fi% +}% +\newbox\tikz@lib@dec@box +\def\tikz@enddecoration{% + \global\setbox\tikz@lib@dec@box=\box\tikz@tempbox% + \ifx\tikz@decoration@name\tikz@nonetext% + \else + \iftikz@metadecoration% + \endpgfmetadecoration% + \else% + \tikz@lib@dec@stopdecoration% + \fi% + \fi + \endgroup% + \setbox\tikz@tempbox=\box\tikz@lib@dec@box% +}% + + +\def\tikz@lib@dec@decorate@path{% + \ifx\tikz@decoration@name\tikz@nonetext% + \else% + \pgfgetpath\tikz@lib@dec@currentpath% + \pgfsetpath\pgfutil@empty% + \pgfifmetadecoration{\tikz@decoration@name} + { + \expandafter\pgfmetadecoration\expandafter{\tikz@decoration@name}% + \pgfsetpath\tikz@lib@dec@currentpath% + \endpgfmetadecoration% + } + { + \pgfifdecoration{\tikz@decoration@name} + { + \tikz@lib@dec@startdecoration% + \pgfsetpath\tikz@lib@dec@currentpath% + \tikz@lib@dec@stopdecoration% + } + { + \tikzerror{Unknown (meta-)decoration '\tikz@decoration@name'. Perhaps + you misspelled it?} + } + } + \fi% +}% + +\def\tikz@lib@dec@zpttext{0pt}% + +\newif\iftikz@lib@dec@temp +\def\tikz@lib@dec@startdecoration{% + \pgfkeysgetvalue{/pgf/decoration/pre length}{\tikz@lib@pre}% + \pgfkeysgetvalue{/pgf/decoration/post length}{\tikz@lib@post}% + \tikz@lib@dec@tempfalse% + \ifx\tikz@lib@pre\tikz@lib@dec@zpttext% + \ifx\tikz@lib@post\tikz@lib@dec@zpttext% + \else% + \tikz@lib@dec@temptrue% + \fi% + \else% + \tikz@lib@dec@temptrue% + \fi% + \iftikz@lib@dec@temp% + \pgfmetadecoration{tikz@internal}% + \else% + \expandafter\pgfdecoration\expandafter{\expandafter{\tikz@decoration@name}{\pgfdecoratedpathlength}{}{}}% + \fi% +}% + +\def\tikz@lib@dec@stopdecoration{% + \iftikz@lib@dec@temp% + \endpgfmetadecoration% + \else% + \endpgfdecoration% + \fi% +}% + + +\endinput diff --git a/preprints/A-polynomial-time-algorithm-for-three-machine-unit-job-scheduling-September-24-2026/build/tex/runtime/generic/pgf/frontendlayer/tikz/libraries/tikzlibrarydecorations.pathreplacing.code.tex b/preprints/A-polynomial-time-algorithm-for-three-machine-unit-job-scheduling-September-24-2026/build/tex/runtime/generic/pgf/frontendlayer/tikz/libraries/tikzlibrarydecorations.pathreplacing.code.tex new file mode 100644 index 0000000000000000000000000000000000000000..b357e7a59580ec6c44fcfb401cb240c106c2735b --- /dev/null +++ b/preprints/A-polynomial-time-algorithm-for-three-machine-unit-job-scheduling-September-24-2026/build/tex/runtime/generic/pgf/frontendlayer/tikz/libraries/tikzlibrarydecorations.pathreplacing.code.tex @@ -0,0 +1,25 @@ + +\usetikzlibrary{decorations}% +\usepgflibrary{decorations.pathreplacing}% + + +\def\tikz@lib@dec@installinputsegmentpoints{% + \pgf@process{\pgf@decorate@inputsegment@first}% + \edef\tikzinputsegmentfirst{\the\pgf@x,\the\pgf@y}% + \pgf@process{\pgf@decorate@inputsegment@last}% + \edef\tikzinputsegmentlast{\the\pgf@x,\the\pgf@y}% + \pgf@process{\pgf@decorate@inputsegment@supporta}% + \edef\tikzinputsegmentsupporta{\the\pgf@x,\the\pgf@y}% + \pgf@process{\pgf@decorate@inputsegment@supportb}% + \edef\tikzinputsegmentsupportb{\the\pgf@x,\the\pgf@y}% +}% + +\pgfkeys{ + /pgf/decoration/.cd, + moveto code/.code=\def\pgfdecoratedinputsegmentmovetocode{\tikz@lib@dec@installinputsegmentpoints#1}, + lineto code/.code=\def\pgfdecoratedinputsegmentlinetocode{\tikz@lib@dec@installinputsegmentpoints#1}, + curveto code/.code=\def\pgfdecoratedinputsegmentcurvetocode{\tikz@lib@dec@installinputsegmentpoints#1}, + closepath code/.code=\def\pgfdecoratedinputsegmentclosepathcode{\tikz@lib@dec@installinputsegmentpoints#1} +}% + +\endinput diff --git a/preprints/A-polynomial-time-algorithm-for-three-machine-unit-job-scheduling-September-24-2026/build/tex/runtime/generic/pgf/frontendlayer/tikz/libraries/tikzlibrarypositioning.code.tex b/preprints/A-polynomial-time-algorithm-for-three-machine-unit-job-scheduling-September-24-2026/build/tex/runtime/generic/pgf/frontendlayer/tikz/libraries/tikzlibrarypositioning.code.tex new file mode 100644 index 0000000000000000000000000000000000000000..8836ee58559c690ce7550b42c10ad42e33e2f23f --- /dev/null +++ b/preprints/A-polynomial-time-algorithm-for-three-machine-unit-job-scheduling-September-24-2026/build/tex/runtime/generic/pgf/frontendlayer/tikz/libraries/tikzlibrarypositioning.code.tex @@ -0,0 +1,107 @@ + +\ProvidesFileRCS{tikzlibrarypositioning.code.tex} + + +\tikzset{above/.code=\tikz@lib@place@handle@{#1}{south}{0}{1}{north}{1}}% +\tikzset{above left/.code=\tikz@lib@place@handle@{#1}{south east}{-1}{1}{north west}{0.707106781}}% +\tikzset{above right/.code=\tikz@lib@place@handle@{#1}{south west}{1}{1}{north east}{0.707106781}}% +\tikzset{base left/.code =\tikz@lib@place@handle@{#1}{base east}{-1}{0}{base west}{1}}% +\tikzset{base right/.code=\tikz@lib@place@handle@{#1}{base west}{1}{0}{base east}{1}}% +\tikzset{below/.code=\tikz@lib@place@handle@{#1}{north}{0}{-1}{south}{1}}% +\tikzset{below left/.code=\tikz@lib@place@handle@{#1}{north east}{-1}{-1}{south west}{0.707106781}}% +\tikzset{below right/.code=\tikz@lib@place@handle@{#1}{north west}{1}{-1}{south east}{0.707106781}}% +\tikzset{left/.code =\tikz@lib@place@handle@{#1}{east}{-1}{0}{west}{1}}% +\tikzset{mid left/.code =\tikz@lib@place@handle@{#1}{mid east}{-1}{0}{mid west}{1}}% +\tikzset{mid right/.code=\tikz@lib@place@handle@{#1}{mid west}{1}{0}{mid east}{1}}% +\tikzset{right/.code=\tikz@lib@place@handle@{#1}{west}{1}{0}{east}{1}}% + +\newif\iftikz@lib@ignore@size + +\pgfutil@g@addto@macro\tikz@node@reset@hook{\tikz@addtransform{\tikz@lib@pos@call}\let\tikz@lib@pos@call=\relax}% + +\tikzset{on grid/.is if=tikz@lib@ignore@size}% + +\tikzset{node distance=1cm and 1cm}% + +\def\tikz@lib@place@handle@#1#2#3#4#5#6{% + \def\tikz@anchor{#2}% + \let\tikz@do@auto@anchor=\relax% + \edef\tikz@temp{#1}% + \def\tikz@lib@place@single@factor{#6}% + \expandafter\tikz@lib@place@handle@@\expandafter{\tikz@temp}{#3}{#4}{#5}% +}% +\def\tikz@lib@place@handle@@#1#2#3#4{% + \pgfutil@in@{of }{#1}% + \ifpgfutil@in@% + \tikz@lib@place@of#1\tikz@stop{#4}% + \else% + \edef\tikz@lib@place@nums{#1}% + \fi% + \ifx\tikz@lib@place@nums\pgfutil@empty% + \else% + \expandafter\tikz@lib@place@parse@nums\expandafter{\tikz@lib@place@nums}% + \pgf@x=#2\pgf@x% + \pgf@y=#3\pgf@y% + \edef\tikz@lib@pos@call{\noexpand\pgftransformshift{\noexpand\pgfqpoint{\the\pgf@x}{\the\pgf@y}}}% + \fi% +}% + +\def\tikz@lib@place@parse@nums#1{% + \pgfutil@in@{and}{#1}% + \ifpgfutil@in@% + \tikz@lib@place@parse@nums@#1\tikz@stop% + \else% + \tikz@lib@place@parse@nums@#1and#1\tikz@stop% + \pgf@x=\tikz@lib@place@single@factor\pgf@x% + \pgf@y=\tikz@lib@place@single@factor\pgf@y% + \fi +}% +\def\tikz@lib@place@parse@nums@#1and#2\tikz@stop{% + \pgfmathparse{#2}% + \ifpgfmathunitsdeclared% + \pgf@xa=\pgfmathresult pt% + \pgf@ya=0pt% + \else% + \let\tikz@lib@temp=\pgfmathresult% + \pgf@process{\pgfpointxy{\tikz@lib@temp}{0}}% + \pgf@xa=\pgf@x% + \pgf@ya=\pgf@y% + \fi% + \pgfmathparse{#1}% + \ifpgfmathunitsdeclared% + \advance\pgf@ya by\pgfmathresult pt% + \else% + \let\tikz@lib@temp=\pgfmathresult% + \pgf@process{\pgfpointxy{0}{\tikz@lib@temp}}% + \advance\pgf@xa by\pgf@x% + \advance\pgf@ya by\pgf@y% + \fi% + \pgf@x=\pgf@xa% + \pgf@y=\pgf@ya% +}% + +\def\tikz@lib@place@of#1of #2\tikz@stop#3{% + \def\tikz@temp{#1}% + \ifx\tikz@temp\pgfutil@empty% + \tikz@lib@place@of@{\tikz@node@distance}{#2}{#3}% + \else% + \tikz@lib@place@of@{#1}{#2}{#3}% + \fi% +}% +\def\tikz@lib@place@of@#1#2#3{% + \tikz@scan@one@point\tikz@lib@place@remember(#2)% + \iftikz@shapeborder% + \iftikz@lib@ignore@size% + \edef\tikz@node@at{\noexpand\pgfpointanchor{\tikz@shapeborder@name}{center}}% + \def\tikz@anchor{center}% + \else% + \edef\tikz@node@at{\noexpand\pgfpointanchor{\tikz@shapeborder@name}{#3}}% + \fi% + \fi% + \edef\tikz@lib@place@nums{#1}% +}% +\def\tikz@lib@place@remember#1{\def\tikz@node@at{#1}}% + + + +\endinput diff --git a/preprints/A-polynomial-time-algorithm-for-three-machine-unit-job-scheduling-September-24-2026/build/tex/runtime/generic/pgf/frontendlayer/tikz/libraries/tikzlibrarytopaths.code.tex b/preprints/A-polynomial-time-algorithm-for-three-machine-unit-job-scheduling-September-24-2026/build/tex/runtime/generic/pgf/frontendlayer/tikz/libraries/tikzlibrarytopaths.code.tex new file mode 100644 index 0000000000000000000000000000000000000000..e337077c18fe6c57c751057be8d10c286a65c396 --- /dev/null +++ b/preprints/A-polynomial-time-algorithm-for-three-machine-unit-job-scheduling-September-24-2026/build/tex/runtime/generic/pgf/frontendlayer/tikz/libraries/tikzlibrarytopaths.code.tex @@ -0,0 +1,347 @@ + +\ProvidesFileRCS{tikzlibrarytopaths.code.tex} + + + +\tikzset{move to/.style= {to path={(\tikztotarget) \tikztonodes}}} + + + +\tikzset{line to/.style= {to path={-- (\tikztotarget) \tikztonodes}}} + + + +\tikzset{every curve to/.style= {}} +\tikzset{curve to/.style= {to path=\tikz@to@curve@path}} + +\tikzoption{bend angle}{\pgfmathsetmacro\tikz@to@bend{#1}} + +\tikzoption{bend left}[]{% + \def\pgf@temp{#1}% + \ifx\pgf@temp\pgfutil@empty% + \else% + \pgfmathsetmacro\tikz@to@bend{#1}% + \fi% + \let\tikz@to@out=\tikz@to@bend% + \pgfmathsetmacro\tikz@to@in{180-(\tikz@to@out)}% + \tikz@to@switch@on% + \tikz@to@relativetrue% +} + +\tikzoption{bend right}[]{% + \def\pgf@temp{#1}% + \ifx\pgf@temp\pgfutil@empty% + \else% + \pgfmathsetmacro\tikz@to@bend{#1}% + \fi% + \pgfmathsetmacro\tikz@to@out{-(\tikz@to@bend)} + \pgfmathsetmacro\tikz@to@in{180-(\tikz@to@out)} + \tikz@to@switch@on% + \tikz@to@relativetrue% +} + +\tikzoption{relative}[true]{\csname tikz@to@relative#1\endcsname} +\newif\iftikz@to@relative +\tikz@to@relativefalse + +\tikzoption{in}{\def\tikz@to@in{#1}\tikz@to@switch@on} +\tikzoption{out}{\def\tikz@to@out{#1}\tikz@to@switch@on} + +\tikzoption{in looseness}{\tikz@to@set@in@looseness{#1}} +\tikzoption{out looseness}{\tikz@to@set@out@looseness{#1}} +\tikzoption{looseness}{\tikz@to@set@in@looseness{#1}\tikz@to@set@out@looseness{#1}} + +\tikzoption{in control}{\tikz@to@set@in@control{#1}} +\tikzoption{out control}{\tikz@to@set@out@control{#1}} +\tikzoption{controls}{\tikz@to@parse@controls#1\pgf@stop} + +\tikzoption{in min distance}{\tikz@to@set@distances{#1}{}{}{}} +\tikzoption{in max distance}{\tikz@to@set@distances{}{#1}{}{}} +\tikzoption{in distance}{\tikz@to@set@distances{#1}{#1}{}{}} +\tikzoption{out min distance}{\tikz@to@set@distances{}{}{#1}{}} +\tikzoption{out max distance}{\tikz@to@set@distances{}{}{}{#1}} +\tikzoption{out distance}{\tikz@to@set@distances{}{}{#1}{#1}} +\tikzoption{min distance}{\tikz@to@set@distances{#1}{}{#1}{}} +\tikzoption{max distance}{\tikz@to@set@distances{}{#1}{}{#1}} +\tikzoption{distance}{\tikz@to@set@distances{#1}{#1}{#1}{#1}} + +\def\tikz@to@set@distances#1#2#3#4{% + \tikz@to@setifnotempy{#1}{\tikz@to@in@min}{\let\tikz@to@end@compute=\tikz@to@end@compute@looseness}% + \tikz@to@setifnotempy{#2}{\tikz@to@in@max}{\let\tikz@to@end@compute=\tikz@to@end@compute@looseness}% + \tikz@to@setifnotempy{#3}{\tikz@to@out@min}{\let\tikz@to@start@compute=\tikz@to@start@compute@looseness}% + \tikz@to@setifnotempy{#4}{\tikz@to@out@max}{\let\tikz@to@start@compute=\tikz@to@start@compute@looseness}% + \tikz@to@switch@on% +} + +\def\tikz@to@setifnotempy#1#2#3{% + \def\pgf@temp{#1}% + \ifx\pgf@temp\pgfutil@empty\else\def#2{#1}#3\fi% +} + + +\def\tikz@to@set@in@looseness#1{% + \pgfmathsetmacro\tikz@to@in@looseness{#1}% + \let\tikz@to@end@compute=\tikz@to@end@compute@looseness% + \tikz@to@switch@on% +} +\def\tikz@to@set@out@looseness#1{% + \pgfmathsetmacro\tikz@to@out@looseness{#1}% + \let\tikz@to@start@compute=\tikz@to@start@compute@looseness% + \tikz@to@switch@on% +} + +\def\tikz@to@parse@controls#1and#2\pgf@stop{\tikz@to@set@in@control{#2}\tikz@to@set@out@control{#1}} + +\def\tikz@to@set@in@control#1{% + \def\tikz@to@in@control{#1}% + \let\tikz@to@end@compute=\tikz@to@end@compute@control% + \tikz@to@switch@on% +} +\def\tikz@to@set@out@control#1{% + \def\tikz@to@out@control{#1}% + \let\tikz@to@start@compute=\tikz@to@start@compute@control% + \tikz@to@switch@on% +} + + +\def\tikz@to@bend{30} + +\def\tikz@to@out{45} +\def\tikz@to@in{135} + +\def\tikz@to@out@looseness{1} +\def\tikz@to@in@looseness{1} + +\def\tikz@to@in@min{0pt} +\def\tikz@to@in@max{10000pt} +\def\tikz@to@out@min{0pt} +\def\tikz@to@out@max{10000pt} + +\def\tikz@to@switch@on{\let\tikz@to@path=\tikz@to@curve@path} + +\def\tikz@to@curve@path{% + [every curve to] + \pgfextra{\iftikz@to@relative\tikz@to@compute@relative\else\tikz@to@compute\fi} + \tikz@computed@path + \pgfextra{\tikz@updatenexttrue\tikz@updatecurrenttrue}% + \tikztonodes% +} + +\def\tikz@to@modify#1#2{% + \pgfutil@ifundefined{pgf@sh@ns@#1} + {}% + {\edef#1{#1.#2}} +}% + +\def\tikz@to@compute{% + \let\tikz@tofrom=\tikztostart% + \let\tikz@toto=\tikztotarget% + \tikz@to@modify\tikz@tofrom\tikz@to@out% + \tikz@to@modify\tikz@toto\tikz@to@in% + \ifx\tikz@to@start@compute\tikz@to@start@compute@looseness% + \tikz@to@compute@distance% + \else% + \ifx\tikz@from@start@compute\tikz@to@start@compute@looseness% + \tikz@to@compute@distance% + \fi% + \fi% + \tikz@to@start@compute% + \tikz@to@end@compute% + \edef\tikz@computed@path{.. controls \tikz@computed@start and \tikz@computed@end .. (\tikztotarget)} +} + +\def\tikz@to@compute@distance{\tikz@scan@one@point\tikz@@to@compute@distance(\tikz@tofrom)} +\def\tikz@@to@compute@distance#1{% + \def\tikz@first@point{#1}% + \tikz@scan@one@point\tikz@@@to@compute@distance(\tikz@toto)} +\def\tikz@@@to@compute@distance#1{% + \iftikz@updatecurrent\else + \tikz@updatenextfalse + \fi + \def\tikz@second@point{#1}% + \tikz@to@compute@distance@main% +} +\def\tikz@to@compute@distance@main{% + \pgf@process{\pgfpointdiff{\tikz@first@point}{\tikz@second@point}}% + \ifdim\pgf@x<0pt\pgf@xa=-\pgf@x\else\pgf@xa=\pgf@x\fi% + \ifdim\pgf@y<0pt\pgf@ya=-\pgf@y\else\pgf@ya=\pgf@y\fi% + \pgf@process{\pgfpointnormalised{\pgfqpoint{\pgf@xa}{\pgf@ya}}}% + \ifdim\pgf@x>\pgf@y% + \c@pgf@counta=\pgf@x% + \ifnum\c@pgf@counta=0\relax% + \else% + \divide\c@pgf@counta by 255\relax% + \pgf@xa=16\pgf@xa\relax% + \divide\pgf@xa by\c@pgf@counta% + \pgf@xa=16\pgf@xa\relax% + \fi% + \else% + \c@pgf@counta=\pgf@y% + \ifnum\c@pgf@counta=0\relax% + \else% + \divide\c@pgf@counta by 255\relax% + \pgf@ya=16\pgf@ya\relax% + \divide\pgf@ya by\c@pgf@counta% + \pgf@xa=16\pgf@ya\relax% + \fi% + \fi% + \pgf@x=0.3915\pgf@xa% + \pgf@xa=\tikz@to@out@looseness\pgf@x% + \pgf@xb=\tikz@to@in@looseness\pgf@x% + \pgfmathsetlength{\pgf@ya}{\tikz@to@out@min} + \ifdim\pgf@xa<\pgf@ya% + \pgf@xa=\pgf@ya% + \fi% + \pgfmathsetlength{\pgf@ya}{\tikz@to@out@max} + \ifdim\pgf@xa>\pgf@ya% + \pgf@xa=\pgf@ya% + \fi% + \pgfmathsetlength{\pgf@ya}{\tikz@to@in@min} + \ifdim\pgf@xb<\pgf@ya% + \pgf@xb=\pgf@ya% + \fi% + \pgfmathsetlength{\pgf@ya}{\tikz@to@in@max} + \ifdim\pgf@xb>\pgf@ya% + \pgf@xb=\pgf@ya% + \fi% +} + +\def\tikz@to@start@compute@looseness{% + \edef\tikz@computed@start{([shift=({\tikz@to@out}:\the\pgf@xa)]\tikz@tofrom)}% +} +\def\tikz@to@end@compute@looseness{% + \edef\tikz@computed@end{([shift=({\tikz@to@in}:\the\pgf@xb)]\tikz@toto)}% +} +\def\tikz@to@start@compute@control{% + \let\tikz@computed@start=\tikz@to@out@control% +} +\def\tikz@to@end@compute@control{% + \let\tikz@computed@end=\tikz@to@in@control% +} + +\let\tikz@to@start@compute=\tikz@to@start@compute@looseness% +\let\tikz@to@end@compute=\tikz@to@end@compute@looseness% + + + + + +\def\tikz@to@compute@relative{% + \tikz@scan@one@point\tikz@@to@compute@relative(\tikztostart)% +} +\def\tikz@@to@compute@relative#1{% + \def\tikz@tofrom{#1}% + \tikz@scan@one@point\tikz@@@to@compute@relative(\tikztotarget)% +} +\def\tikz@@@to@compute@relative#1{% + \def\tikz@toto{#1}% + \begingroup + \pgfutil@ifundefined{pgf@sh@ns@\tikztostart} + {% + \let\tikz@first@point=\tikz@tofrom% + \let\tikz@tostart@tikz=\pgfutil@empty + }% + {% + {% + \tikz@tofrom% + \pgf@xc=\pgf@x% + \pgf@yc=\pgf@y% + {% + \pgftransformreset% + \pgftransformshift{\pgfqpoint{\pgf@xc}{\pgf@yc}}% + \pgftransformrotate{\tikz@to@out}% + \pgftransformshift{\pgfqpoint{-\pgf@xc}{-\pgf@yc}}% + \pgf@process{\pgfpointtransformed{\tikz@toto}}% + }% + \pgf@xc=\pgf@x% + \pgf@yc=\pgf@y% + \pgfpointshapeborder{\tikztostart}{\pgfqpoint{\pgf@xc}{\pgf@yc}}% + \xdef\tikz@tofrom@smuggle{\noexpand\pgfqpoint{\the\pgf@x}{\the\pgf@y}} + }% + \let\tikz@first@point=\tikz@tofrom@smuggle% + \tikz@first@point% + \edef\tikz@tostart@tikz{(\the\pgf@x,\the\pgf@y)}% + }% + \pgfutil@ifundefined{pgf@sh@ns@\tikztotarget} + {% + \let\tikz@second@point=\tikz@toto% + }% + {% + {% + \tikz@toto% + \pgf@xc=\pgf@x% + \pgf@yc=\pgf@y% + {% + \pgftransformreset% + \pgftransformshift{\pgfqpoint{\pgf@xc}{\pgf@yc}}% + \pgftransformrotate{180}% + \pgftransformrotate{\tikz@to@in}% + \pgftransformshift{\pgfqpoint{-\pgf@xc}{-\pgf@yc}}% + \pgf@process{\pgfpointtransformed{\tikz@tofrom}}% + }% + \pgf@xc=\pgf@x% + \pgf@yc=\pgf@y% + \pgfpointshapeborder{\tikztotarget}{\pgfqpoint{\pgf@xc}{\pgf@yc}}% + \xdef\tikz@toto@smuggle{\noexpand\pgfqpoint{\the\pgf@x}{\the\pgf@y}} + }% + \let\tikz@second@point=\tikz@toto@smuggle% + }% + \tikz@second@point% + \edef\tikz@totarget@tikz{(\the\pgf@x,\the\pgf@y)}% + \tikz@to@compute@distance@main% + \edef\tikz@to@first@distance{\the\pgf@xa}% + \edef\tikz@to@second@distance{\the\pgf@xb}% + \pgftransformreset% + \pgf@process{\tikz@first@point}% + \pgf@xa=\pgf@x% + \pgf@ya=\pgf@y% + \pgf@process{\tikz@second@point}% + \advance\pgf@x by-\pgf@xa% + \advance\pgf@y by-\pgf@ya% + \pgfpointnormalised{}% + \pgf@xc=\pgf@x% + \pgf@yc=\pgf@y% + \pgf@xb=-\pgf@x% + \pgf@yb=-\pgf@y% + {% + \pgftransformshift{\tikz@first@point}% + \pgftransformcm{\pgf@sys@tonumber\pgf@xc}{\pgf@sys@tonumber\pgf@yc}{\pgf@sys@tonumber\pgf@yb}{\pgf@sys@tonumber\pgf@xc}% + {\pgfpointorigin}% + \pgf@process{\pgfpointtransformed{\pgfpointpolar{\tikz@to@out}{\tikz@to@first@distance}}}% + \xdef\tikz@computed@start{(\the\pgf@x,\the\pgf@y)}% + } + {% + \pgftransformshift{\tikz@second@point}% + \pgftransformcm{\pgf@sys@tonumber\pgf@xc}{\pgf@sys@tonumber\pgf@yc}{\pgf@sys@tonumber\pgf@yb}{\pgf@sys@tonumber\pgf@xc}% + {\pgfpointorigin}% + \pgf@process{\pgfpointtransformed{\pgfpointpolar{\tikz@to@in}{\tikz@to@second@distance}}}% + \xdef\tikz@computed@end{(\the\pgf@x,\the\pgf@y)}% + } + \xdef\tikz@computed@path{ + \tikz@tostart@tikz + .. controls \tikz@computed@start and \tikz@computed@end .. + \tikz@totarget@tikz}% + \endgroup +} + + + + +\tikzset{loop/.style= {to path={ + \pgfextra{\let\tikztotarget=\tikztostart} + [looseness=8,min distance=5mm,every loop] + \tikz@to@curve@path + }}} + +\tikzset{every loop/.style= {->,shorten >=1pt}} + +\tikzset{loop right/.style= {right,out=15,in=-15,loop}} +\tikzset{loop above/.style= {above,out=105,in=75,loop}} +\tikzset{loop left/.style= {left,out=195,in=165,loop}} +\tikzset{loop below/.style= {below,out=285,in=255,loop}} + + + + + +\endinput diff --git a/preprints/A-polynomial-time-algorithm-for-three-machine-unit-job-scheduling-September-24-2026/build/tex/runtime/generic/pgf/frontendlayer/tikz/tikz.code.tex b/preprints/A-polynomial-time-algorithm-for-three-machine-unit-job-scheduling-September-24-2026/build/tex/runtime/generic/pgf/frontendlayer/tikz/tikz.code.tex new file mode 100644 index 0000000000000000000000000000000000000000..db51c43d5d52f8771c1d78f8a4034c57fba1f40e --- /dev/null +++ b/preprints/A-polynomial-time-algorithm-for-three-machine-unit-job-scheduling-September-24-2026/build/tex/runtime/generic/pgf/frontendlayer/tikz/tikz.code.tex @@ -0,0 +1,5119 @@ + +\ProvidesPackageRCS{tikz.code.tex} + + +\def\tikzerror#1{\pgfutil@packageerror{tikz}{#1}{}} + +\usepgflibrary{plothandlers} + +\newdimen\tikz@lastx +\newdimen\tikz@lasty +\newdimen\tikz@lastxsaved +\newdimen\tikz@lastysaved +\newdimen\tikz@lastmovetox +\newdimen\tikz@lastmovetoy + +\newdimen\tikzleveldistance +\newdimen\tikzsiblingdistance + +\newbox\tikz@figbox +\newbox\tikz@figbox@bg +\newbox\tikz@tempbox +\newbox\tikz@tempbox@bg + +\newcount\tikztreelevel +\newcount\tikznumberofchildren +\newcount\tikznumberofcurrentchild + +\newcount\tikz@fig@count + +\newif\iftikz@node@is@a@label +\newif\iftikz@snaked +\newif\iftikz@decoratepath + +\let\tikz@options\pgfutil@empty +\def\tikz@addoption#1{% + \expandafter\def\expandafter\tikz@options\expandafter{\tikz@options#1}}% +\def\tikz@addmode#1{% + \expandafter\def\expandafter\tikz@mode\expandafter{\tikz@mode#1}}% +\def\tikz@addtransform#1{% + \ifx\tikz@transform\relax + #1% + \else + \expandafter\def\expandafter\tikz@transform\expandafter{\tikz@transform#1}% + \fi +}% + + + +\pgfkeys{/tikz/.is family}% + +\def\tikzset{\pgfqkeys{/tikz}}% + +\def\tikzoption#1{% + \pgfutil@ifnextchar[% + {\tikzoption@opt{#1}}{\tikzoption@noopt{#1}}}% +\def\tikzoption@opt#1[#2]#3{% + \pgfkeysdef{/tikz/#1}{#3}% + \pgfkeyssetvalue{/tikz/#1/.@def}{#2}}% +\def\tikzoption@noopt#1#2{% + \pgfkeysdef{/tikz/#1}{#2}% + \pgfkeyssetvalue{/tikz/#1/.@def}{\pgfkeysvaluerequired}}% + +\tikzoption{baseline}[0pt]{% + \pgfutil@ifnextchar(% + {\tikz@baseline@coordinate}{\tikz@baseline@simple}#1\@nil}% +\def\tikz@baseline@simple#1\@nil{\pgfsetbaseline{#1}}% +\def\tikz@baseline@coordinate#1\@nil{% + \pgfsetbaselinepointlater{\tikz@scan@one@point\pgfutil@firstofone#1}}% + +\tikzoption{trim left}[0pt]{\pgfutil@ifnextchar({\tikz@trim@coordinate{left}}{\tikz@trim@simple{left}}#1\@nil}% +\tikzoption{trim right}{\pgfutil@ifnextchar({\tikz@trim@coordinate{right}}{\tikz@trim@simple{right}}#1\@nil}% +\def\tikz@trim@simple#1#2\@nil{\csname pgfsettrim#1\endcsname{#2}}% +\def\tikz@trim@coordinate#1#2\@nil{\csname pgfsettrim#1pointlater\endcsname{\tikz@scan@one@point\pgfutil@firstofone#2}}% + +\tikzoption{line width}{\tikz@semiaddlinewidth{#1}}% + +\def\tikz@semiaddlinewidth#1{\tikz@addoption{\pgfsetlinewidth{#1}}\pgfmathsetlength\pgflinewidth{#1}}% + +\tikzoption{cap}{\tikz@addoption{\csname pgfset#1cap\endcsname}}% +\tikzoption{join}{\tikz@addoption{\csname pgfset#1join\endcsname}}% +\tikzoption{line cap}{\tikz@addoption{\csname pgfset#1cap\endcsname}}% +\tikzoption{line join}{\tikz@addoption{\csname pgfset#1join\endcsname}}% +\tikzoption{miter limit}{\tikz@addoption{\pgfsetmiterlimit{#1}}}% + +\tikzoption{dash pattern}{% + \def\tikz@temp{#1}% + \ifx\tikz@temp\pgfutil@empty% + \def\tikz@dashpattern{}% + \tikz@addoption{\pgfsetdash{}{0pt}}% + \else% + \def\tikz@dashpattern{}% + \expandafter\tikz@scandashon\pgfutil@gobble#1o\@nil% + \edef\tikz@temp{{\tikz@dashpattern}{\noexpand\tikz@dashphase}}% + \expandafter\tikz@addoption\expandafter{\expandafter\pgfsetdash\tikz@temp}% + \fi}% +\tikzoption{dash phase}{% + \def\tikz@dashphase{#1}% + \edef\tikz@temp{{\tikz@dashpattern}{\noexpand\tikz@dashphase}}% + \expandafter\tikz@addoption\expandafter{\expandafter\pgfsetdash\tikz@temp}% +}% +\tikzoption{dash}{\tikz@parse@full@dash#1\pgf@stop}% +\def\tikz@parse@full@dash#1phase#2\pgf@stop{% + \def\tikz@dashphase{#2}% + \def\tikz@temp{#1}% + \ifx\tikz@temp\pgfutil@empty% + \def\tikz@dashpattern{}% + \tikz@addoption{\pgfsetdash{}{0pt}}% + \else% + \def\tikz@dashpattern{}% + \expandafter\tikz@scandashon\pgfutil@gobble#1o\@nil% + \edef\tikz@temp{{\tikz@dashpattern}{\noexpand\tikz@dashphase}}% + \expandafter\tikz@addoption\expandafter{\expandafter\pgfsetdash\tikz@temp}% + \fi% +}% +\def\tikz@dashphase{0pt}% +\def\tikz@dashpattern{}% + +\def\tikz@scandashon n#1o{% + \expandafter\def\expandafter\tikz@dashpattern\expandafter{\tikz@dashpattern{#1}}% + \pgfutil@ifnextchar\@nil{\pgfutil@gobble}{\tikz@scandashoff}}% +\def\tikz@scandashoff ff#1o{% + \expandafter\def\expandafter\tikz@dashpattern\expandafter{\tikz@dashpattern{#1}}% + \pgfutil@ifnextchar\@nil{\pgfutil@gobble}{\tikz@scandashon}}% + +\tikzset{ + dash expand off/.code={% + \ifcsname tikz@library@decorations@loaded\endcsname\else + \tikzerror{You need \string\usetikzlibrary{decorations} for ``dash expand off''}% + \fi + \tikz@addoption{% + \pgfgetpath\currentpath + \pgfprocessround{\currentpath}{\currentpath}% + \pgf@decorate@parsesoftpath{\currentpath}{\currentpath}% + \pgfmathsetmacro\on{\expandafter\pgfutil@firstoftwo\tikz@dashpattern}% + \pgfmathsetmacro\off{\expandafter\pgfutil@secondoftwo\tikz@dashpattern}% + \pgfmathsetmacro\tikz@dashphase{\tikz@dashphase}% + \pgfmathsubtract@{\on}{\tikz@dashphase}% + \pgfmathmax@{\pgfmathresult,0}% + \let\dashphase=\pgfmathresult + \edef\rest{\pgf@sys@tonumber\dimexpr\pgf@decorate@totalpathlength - \on pt + 2\dimexpr\dashphase pt\relax\relax}% + \edef\onoff{\pgf@sys@tonumber\dimexpr\on pt+\off pt\relax}% + \pgfmathdivide@{\rest}{\onoff}% + \pgfmathfloor@{\pgfmathresult}% + \pgfmathmax@{\pgfmathresult,1}% + \pgfmathdivide@{\rest}{\pgfmathresult}% + \pgfmathsubtract@{\pgfmathresult}{\on}% + \pgfmathmax@{\pgfmathresult,\off}% + \edef\tikz@marshal{\noexpand\pgfsetdash{{+\on pt}{+\pgfmathresult pt}}{+\dashphase pt}}% + \tikz@marshal + }% + } +} + +\tikzoption{draw opacity}{\tikz@addoption{\pgfsetstrokeopacity{#1}}}% + +\tikzoption{double}[]{% + \def\tikz@temp{#1}% + \ifx\tikz@temp\tikz@nonetext% + \tikz@addmode{\tikz@mode@doublefalse}% + \else% + \ifx\tikz@temp\pgfutil@empty% + \else% + \pgfsetinnerstrokecolor{#1}% + \fi% + \tikz@addmode{\tikz@mode@doubletrue}% + \tikzset{every double/.try}% + \fi}% +\tikzoption{double distance}{% + \pgfmathsetlength{\pgf@x}{#1}% + \edef\tikz@double@setup{% + \pgf@x=\the\pgf@x% + \advance\pgf@x by2\pgflinewidth% + \pgflinewidth=\pgf@x% + \noexpand\pgfsetlinewidth{\pgflinewidth}% + \noexpand\pgfsetinnerlinewidth{\the\pgf@x}% + }% + \tikzset{double}}% +\def\tikz@double@setup{% + \pgf@x=2\pgflinewidth% + \advance\pgf@x by0.6pt% + \pgflinewidth=\pgf@x% + \pgfsetlinewidth{\pgflinewidth}% + \pgfsetinnerlinewidth{0.6pt}% +}% +\tikzset{double distance between line centers/.code={ + \pgfmathsetlength{\pgf@x}{#1}% + \edef\tikz@double@setup{% + \pgf@x=\pgflinewidth% + \pgf@xa=\the\pgf@x% + \advance\pgf@x by\pgf@xa% + \advance\pgf@xa by-\pgflinewidth% + \pgflinewidth=\pgf@x% + \noexpand\pgfsetlinewidth{\pgflinewidth}% + \noexpand\pgfsetinnerlinewidth{\pgf@xa}% + }% + \tikzset{double}}}% +\tikzset{double equal sign distance/.style={double distance between line centers=0.45ex}}% + + + + + +\tikzoption{even odd rule}[]{\tikz@addoption{\pgfseteorule}}% +\tikzoption{nonzero rule}[]{\tikz@addoption{\pgfsetnonzerorule}}% + +\tikzoption{fill opacity}{\tikz@addoption{\pgfsetfillopacity{#1}}}% + + + +\tikzoption{opacity}{\tikz@addoption{\pgfsetstrokeopacity{#1}\pgfsetfillopacity{#1}}}% + + + +\tikzset{blend mode/.code={\tikz@addoption{\pgfsetblendmode{#1}}}}% + + +\tikzoption{color}{% + \tikz@addoption{% + \ifx\tikz@fillcolor\pgfutil@empty% + \ifx\tikz@strokecolor\pgfutil@empty% + \else% + \pgfsys@color@reset@inorderfalse% + \let\tikz@strokecolor\pgfutil@empty% + \let\tikz@fillcolor\pgfutil@empty% + \fi% + \else% + \pgfsys@color@reset@inorderfalse% + \let\tikz@strokecolor\pgfutil@empty% + \let\tikz@fillcolor\pgfutil@empty% + \fi% + \pgfutil@colorlet{tikz@color}{#1}% + \pgfutil@colorlet{.}{tikz@color}% + \pgfsetcolor{.}% + \pgfsys@color@reset@inordertrue% + }% + \def\tikz@textcolor{#1}}% + + + +\tikzoption{rounded corners}[4pt]{\pgfsetcornersarced{\pgfpoint{#1}{#1}}}% +\tikzoption{sharp corners}[]{\pgfsetcornersarced{\pgfpointorigin}}% + +\tikzset{x radius/.initial=0pt}% +\tikzset{y radius/.initial=0pt}% +\tikzset{% + radius/.code={% + \pgfmathparse{#1}% + \ifpgfmathunitsdeclared + \pgfkeyssetevalue{/tikz/x radius}{\pgfmathresult pt}% + \pgfkeyssetevalue{/tikz/y radius}{\pgfmathresult pt}% + \else + \pgfkeyssetevalue{/tikz/x radius}{\pgfmathresult}% + \pgfkeyssetevalue{/tikz/y radius}{\pgfmathresult}% + \fi + }% +}% +\tikzset{start angle/.initial=}% +\tikzset{end angle/.initial=}% +\tikzset{delta angle/.initial=}% + + +\tikzoption{x}{\tikz@handle@vec{\pgfsetxvec}{\tikz@handle@x}#1\relax}% +\tikzoption{y}{\tikz@handle@vec{\pgfsetyvec}{\tikz@handle@y}#1\relax}% +\tikzoption{z}{\tikz@handle@vec{\pgfsetzvec}{\tikz@handle@z}#1\relax}% + +\def\tikz@handle@vec#1#2{\pgfutil@ifnextchar({\tikz@handle@coordinate#1}{\tikz@handle@single#2}}% +\def\tikz@handle@coordinate#1{\tikz@scan@one@point#1}% +\def\tikz@handle@single#1#2\relax{#1{#2}}% +\def\tikz@handle@x#1{\pgfsetxvec{\pgfpoint{#1}{0pt}}}% +\def\tikz@handle@y#1{\pgfsetyvec{\pgfpoint{0pt}{#1}}}% +\def\tikz@handle@z#1{\pgfsetzvec{\pgfpoint{#1}{#1}}}% + + +\tikzoption{scale}{\tikz@addtransform{\pgftransformscale{#1}}}% +\tikzoption{scale around}{\tikz@addtransform{\def\tikz@aroundaction{\pgftransformscale}\tikz@doaround{#1}}}% +\tikzoption{xscale}{\tikz@addtransform{\pgftransformxscale{#1}}}% +\tikzoption{xslant}{\tikz@addtransform{\pgftransformxslant{#1}}}% +\tikzoption{yscale}{\tikz@addtransform{\pgftransformyscale{#1}}}% +\tikzoption{yslant}{\tikz@addtransform{\pgftransformyslant{#1}}}% +\tikzoption{rotate}{\tikz@addtransform{\pgftransformrotate{#1}}}% +\tikzoption{rotate around}{\tikz@addtransform{\def\tikz@aroundaction{\pgftransformrotate}\tikz@doaround{#1}}}% +\def\tikz@doaround#1{% + \edef\tikz@temp{#1}% + \expandafter\tikz@doparseA\tikz@temp% +}% +\def\tikz@doparseA#1:{% + \def\tikz@temp@rot{#1}% + \tikz@scan@one@point\tikz@doparseB% +}% +\def\tikz@doparseB#1{% + \pgf@process{#1}% + \pgf@xc=\pgf@x% + \pgf@yc=\pgf@y% + \pgftransformshift{\pgfqpoint{\pgf@xc}{\pgf@yc}}% + \tikz@aroundaction{\tikz@temp@rot}% + \pgftransformshift{\pgfqpoint{-\pgf@xc}{-\pgf@yc}}% +}% + +\tikzoption{shift}{\tikz@addtransform{\tikz@scan@one@point\pgftransformshift#1\relax}}% +\tikzoption{xshift}{\tikz@addtransform{\pgftransformxshift{#1}}}% +\tikzoption{yshift}{\tikz@addtransform{\pgftransformyshift{#1}}}% +\tikzoption{cm}{\tikz@addtransform{\tikz@parse@cm#1\relax}}% +\tikzoption{reset cm}[]{\tikz@addtransform{\pgftransformreset}}% +\tikzoption{shift only}[]{\tikz@addtransform{\pgftransformresetnontranslations}}% + +\def\tikz@parse@cm#1,#2,#3,#4,{% + \def\tikz@p@cm{{#1}{#2}{#3}{#4}}% + \tikz@scan@one@point\tikz@parse@cmA}% +\def\tikz@parse@cmA#1{% + \expandafter\pgftransformcm\tikz@p@cm{#1}% +}% + +\tikzset{transform canvas/.code=% + {% + \tikz@addoption + {% + {% + \pgftransformreset% + \let\tikz@transform=\relax% + \tikzset{#1}% + \pgflowlevelsynccm% + }% + \pgf@relevantforpicturesizefalse% + }% + }% +}% + +\tikzset{turn/.code={% + \pgf@x=0pt% + \pgf@y\pgf@x% + \pgf@process{\tikz@tangent}% + \advance\pgf@x by-\tikz@lastx% + \advance\pgf@y by-\tikz@lasty% + \pgfpointnormalised{}% + \pgf@x=-\pgf@x% + \pgf@ya=-\pgf@y% + \pgftransformcm% + {\pgf@sys@tonumber{\pgf@x}}{\pgf@sys@tonumber{\pgf@ya}}% + {\pgf@sys@tonumber{\pgf@y}}{\pgf@sys@tonumber{\pgf@x}}{\pgfqpoint{\tikz@lastx}{\tikz@lasty}}% + }% +}% + +\def\tikz@tangent@lookup{% + \pgfgetpath\tikz@temp% + \pgfprocesspathextractpoints\tikz@temp% + \pgfpointsecondlastonpath% +}% + +\def\tikz@xyz@rotate@let{% + \let\pgf@z=\pgf@yc% + \let\pgf@za=\pgf@xc% +}% + +\def\tikz@xyz@rotate@xyz@xaxis#1#2#3#4{% + \tikz@xyz@rotate@let% + \pgf@x=#1\relax% + \pgf@ya=#2\relax% + \pgf@za=#3\relax% + \pgfmathsin@{#4}\let\tikz@xyz@sin=\pgfmathresult% + \pgfmathcos@{#4}\let\tikz@xyz@cos=\pgfmathresult% + \pgf@y=\tikz@xyz@cos\pgf@ya% + \advance\pgf@y by-\tikz@xyz@sin\pgf@za% + \pgf@z=\tikz@xyz@sin\pgf@ya% + \advance\pgf@z by\tikz@xyz@cos\pgf@za% +}% + +\def\tikz@xyz@rotate@xyz@yaxis#1#2#3#4{% + \tikz@xyz@rotate@let% + \pgf@xa=#1\relax% + \pgf@y=#2\relax% + \pgf@za=#3\relax% + \pgfmathsin@{#4}\let\tikz@xyz@sin=\pgfmathresult% + \pgfmathcos@{#4}\let\tikz@xyz@cos=\pgfmathresult% + \pgf@x=\tikz@xyz@cos\pgf@xa% + \advance\pgf@x by\tikz@xyz@sin\pgf@za% + \pgf@z=-\tikz@xyz@sin\pgf@xa% + \advance\pgf@z by\tikz@xyz@cos\pgf@za% +}% + +\def\tikz@xyz@rotate@xyz@zaxis#1#2#3#4{% + \tikz@xyz@rotate@let% + \pgf@xa=#1\relax% + \pgf@ya=#2\relax% + \pgf@z=#3\relax% + \pgfmathsin@{#4}\let\tikz@xyz@sin=\pgfmathresult% + \pgfmathcos@{#4}\let\tikz@xyz@cos=\pgfmathresult% + \pgf@x=\tikz@xyz@cos\pgf@xa% + \advance\pgf@x by-\tikz@xyz@sin\pgf@ya% + \pgf@y=\tikz@xyz@sin\pgf@xa% + \advance\pgf@y by\tikz@xyz@cos\pgf@ya% +}% + +\tikzset{rotate around x/.code={% + \tikz@xyz@rotate@let% + \pgfmathparse{#1}\let\tikz@xyz@angle=\pgfmathresult% + \tikz@xyz@rotate@xyz@xaxis{0pt}{1pt}{0pt}{\tikz@xyz@angle}% + \pgfextract@process\tikz@xyz@rotate@yvec{\pgfpointxyz{\pgf@sys@tonumber{\pgf@x}}{\pgf@sys@tonumber{\pgf@y}}{\pgf@sys@tonumber{\pgf@z}}}% + \tikz@xyz@rotate@xyz@xaxis{0pt}{0pt}{1pt}{\tikz@xyz@angle}% + \pgfsetzvec{\pgfpointxyz{\pgf@sys@tonumber{\pgf@x}}{\pgf@sys@tonumber{\pgf@y}}{\pgf@sys@tonumber{\pgf@z}}}% + \pgfsetyvec{\tikz@xyz@rotate@yvec}% + }, + rotate around y/.code={% + \tikz@xyz@rotate@let% + \pgfmathparse{#1}\let\tikz@xyz@angle=\pgfmathresult% + \tikz@xyz@rotate@xyz@yaxis{1pt}{0pt}{0pt}{\tikz@xyz@angle}% + \pgfextract@process\tikz@xyz@rotate@xvec{\pgfpointxyz{\pgf@sys@tonumber{\pgf@x}}{\pgf@sys@tonumber{\pgf@y}}{\pgf@sys@tonumber{\pgf@z}}}% + \tikz@xyz@rotate@xyz@yaxis{0pt}{0pt}{1pt}{\tikz@xyz@angle}% + \pgfsetzvec{\pgfpointxyz{\pgf@sys@tonumber{\pgf@x}}{\pgf@sys@tonumber{\pgf@y}}{\pgf@sys@tonumber{\pgf@z}}}% + \pgfsetxvec{\tikz@xyz@rotate@xvec}% + }, + rotate around z/.code={% + \tikz@xyz@rotate@let% + \pgfmathparse{#1}\let\tikz@xyz@angle=\pgfmathresult% + \tikz@xyz@rotate@xyz@zaxis{1pt}{0pt}{0pt}{\tikz@xyz@angle}% + \pgfextract@process\tikz@xyz@rotate@xvec{\pgfpointxyz{\pgf@sys@tonumber{\pgf@x}}{\pgf@sys@tonumber{\pgf@y}}{\pgf@sys@tonumber{\pgf@z}}}% + \tikz@xyz@rotate@xyz@zaxis{0pt}{1pt}{0pt}{\tikz@xyz@angle}% + \pgfsetyvec{\pgfpointxyz{\pgf@sys@tonumber{\pgf@x}}{\pgf@sys@tonumber{\pgf@y}}{\pgf@sys@tonumber{\pgf@z}}}% + \pgfsetxvec{\tikz@xyz@rotate@xvec}% + }, +}% + +\tikzoption{xstep}{\def\tikz@grid@x{#1}}% +\tikzoption{ystep}{\def\tikz@grid@y{#1}}% +\tikzoption{step}{\tikz@handle@vec{\tikz@step@point}{\tikz@step@single}#1\relax}% +\def\tikz@step@single#1{\def\tikz@grid@x{#1}\def\tikz@grid@y{#1}}% +\def\tikz@step@point#1{\pgf@process{#1}\edef\tikz@grid@x{\the\pgf@x}\edef\tikz@grid@y{\the\pgf@y}}% + +\def\tikz@grid@x{1cm}% +\def\tikz@grid@y{1cm}% + + +\newif\iftikz@current@point@local +\tikzset{current point is local/.is if=tikz@current@point@local}% + +\newif\iftikz@mode@double +\newif\iftikz@mode@fill +\newif\iftikz@mode@draw +\newif\iftikz@mode@clip +\newif\iftikz@mode@boundary +\newif\iftikz@mode@shade +\newif\iftikz@mode@fade@path +\newif\iftikz@mode@fade@scope +\let\tikz@mode=\pgfutil@empty + +\def\tikz@nonetext{none}% + +\tikzoption{path only}[]{\let\tikz@mode=\pgfutil@empty}% +\tikzset{ + shade/.is choice, + shade/.default=true, + shade/true/.code=\tikz@addmode{\tikz@mode@shadetrue}, + shade/false/.code=\tikz@addmode{\tikz@mode@shadefalse}, + shade/none/.code=\tikz@addmode{\tikz@mode@shadefalse}, +}% + +\tikzoption{fill}[]{% + \edef\tikz@temp{#1}% + \ifx\tikz@temp\tikz@nonetext% + \tikz@addmode{\tikz@mode@fillfalse}% + \else% + \ifx\tikz@temp\pgfutil@empty% + \else% + \tikz@addoption{\pgfsetfillcolor{#1}}% + \def\tikz@fillcolor{#1}% + \fi% + \tikz@addmode{\tikz@mode@filltrue}% + \fi% +}% +\tikzoption{draw}[]{% + \edef\tikz@temp{#1}% + \ifx\tikz@temp\tikz@nonetext% + \tikz@addmode{\tikz@mode@drawfalse}% + \else% + \ifx\tikz@temp\pgfutil@empty% + \else% + \tikz@addoption{\pgfsetstrokecolor{#1}}% + \def\tikz@strokecolor{#1}% + \fi% + \tikz@addmode{\tikz@mode@drawtrue}% + \fi% +}% +\tikzoption{clip}[]{\tikz@addmode{\tikz@mode@cliptrue}}% +\tikzoption{use as bounding box}[]{\tikz@addmode{\tikz@mode@boundarytrue}}% + +\tikzoption{save path}{\tikz@addmode{\pgfsyssoftpath@getcurrentpath#1\global\let#1=#1}}% +\tikzoption{use path}{\tikz@addmode{\pgfsyssoftpath@setcurrentpath#1}}% + +\let\tikz@fillcolor=\pgfutil@empty +\let\tikz@strokecolor=\pgfutil@empty + +\tikzset{insert path/.code=\tikz@scan@next@command#1\pgf@stop}% + +\tikzset{pattern/.code=\tikzerror{You need to say \string\usetikzlibrary{patterns}}, + pattern color/.style=pattern}% + +\tikzset{path picture/.code=\tikz@addmode{\def\tikz@path@picture{#1}}}% + +\tikzset{path fading/.code={ + \def\tikz@temp{#1}% + \ifx\tikz@temp\tikz@nonetext% + \tikz@addmode{\tikz@mode@fade@pathfalse}% + \else% + \ifx\tikz@temp\pgfutil@empty% + \else% + \def\tikz@path@fading{#1}% + \fi% + \tikz@addmode{\tikz@mode@fade@pathtrue}% + \fi% + }, + path fading/.default=, + scope fading/.code={ + \def\tikz@temp{#1}% + \ifx\tikz@temp\tikz@nonetext% + \tikz@addmode{\tikz@mode@fade@scopefalse}% + \else% + \ifx\tikz@temp\pgfutil@empty% + \else% + \def\tikz@scope@fading{#1}% + \fi% + \tikz@addmode{\tikz@mode@fade@scopetrue}% + \fi% + }, + scope fading/.default=}% +\tikzset{fit fading/.is if=tikz@fade@adjust}% +\tikzset{fading transform/.store in=\tikz@fade@transform}% +\tikzset{fading angle/.style={fading transform={rotate={#1}}}}% + +\newif\iftikz@fade@adjust% +\tikz@fade@adjusttrue% +\let\tikz@fade@transform\pgfutil@empty% + +\pgfutil@colorlet{transparent}{pgftransparent}% +\def\tikz@do@fade@transform{\let\tikz@transform=\relax\expandafter\tikzset\expandafter{\tikz@fade@transform}}% + + + +\newif\iftikz@transparency@group% +\tikzset{/tikz/transparency group/.code=\tikz@transparency@grouptrue\def\tikz@transparency@group@options{isolated=true,#1}\let\tikz@blend@group\pgfutil@empty}% +\tikzset{/tikz/blend group/.code=\tikz@transparency@grouptrue\def\tikz@transparency@group@options{isolated=true}\def\tikz@blend@group{\pgfsetblendmode{#1}}}% + +\let\tikz@blend@group\pgfutil@empty + +\tikzoption{shading}{\def\tikz@shading{#1}\tikz@addmode{\tikz@mode@shadetrue}}% +\tikzoption{shading angle}{\def\tikz@shade@angle{#1}\tikz@addmode{\tikz@mode@shadetrue}}% +\tikzoption{top color}{% + \pgfutil@colorlet{tikz@axis@top}{#1}% + \pgfutil@colorlet{tikz@axis@middle}{tikz@axis@top!50!tikz@axis@bottom}% + \def\tikz@shading{axis}\def\tikz@shade@angle{0}\tikz@addmode{\tikz@mode@shadetrue}}% +\tikzoption{bottom color}{% + \pgfutil@colorlet{tikz@axis@bottom}{#1}% + \pgfutil@colorlet{tikz@axis@middle}{tikz@axis@top!50!tikz@axis@bottom}% + \def\tikz@shading{axis}\def\tikz@shade@angle{0}\tikz@addmode{\tikz@mode@shadetrue}}% +\tikzoption{middle color}{% + \pgfutil@colorlet{tikz@axis@middle}{#1}% + \def\tikz@shading{axis}\tikz@addmode{\tikz@mode@shadetrue}}% +\tikzoption{left color}{% + \pgfutil@colorlet{tikz@axis@top}{#1}% + \pgfutil@colorlet{tikz@axis@middle}{tikz@axis@top!50!tikz@axis@bottom}% + \def\tikz@shading{axis}\def\tikz@shade@angle{90}\tikz@addmode{\tikz@mode@shadetrue}}% +\tikzoption{right color}{% + \pgfutil@colorlet{tikz@axis@bottom}{#1}% + \pgfutil@colorlet{tikz@axis@middle}{tikz@axis@top!50!tikz@axis@bottom}% + \def\tikz@shading{axis}\def\tikz@shade@angle{90}\tikz@addmode{\tikz@mode@shadetrue}}% +\tikzoption{ball color}{\pgfutil@colorlet{tikz@ball}{#1}\def\tikz@shading{ball}\tikz@addmode{\tikz@mode@shadetrue}}% +\tikzoption{inner color}{\pgfutil@colorlet{tikz@radial@inner}{#1}\def\tikz@shading{radial}\tikz@addmode{\tikz@mode@shadetrue}}% +\tikzoption{outer color}{\pgfutil@colorlet{tikz@radial@outer}{#1}\def\tikz@shading{radial}\tikz@addmode{\tikz@mode@shadetrue}}% + +\def\tikz@shading{axis}% +\def\tikz@shade@angle{0}% + +\pgfdeclareverticalshading[tikz@axis@top,tikz@axis@middle,tikz@axis@bottom]{axis}{100bp}{% + color(0bp)=(tikz@axis@bottom); + color(25bp)=(tikz@axis@bottom); + color(50bp)=(tikz@axis@middle); + color(75bp)=(tikz@axis@top); + color(100bp)=(tikz@axis@top)}% + +\pgfutil@colorlet{tikz@axis@top}{gray}% +\pgfutil@colorlet{tikz@axis@middle}{gray!50!white}% +\pgfutil@colorlet{tikz@axis@bottom}{white}% + +\pgfdeclareradialshading[tikz@ball]{ball}{\pgfqpoint{-10bp}{10bp}}{% + color(0bp)=(tikz@ball!15!white); + color(9bp)=(tikz@ball!75!white); + color(18bp)=(tikz@ball!70!black); + color(25bp)=(tikz@ball!50!black); + color(50bp)=(black)}% + +\pgfutil@colorlet{tikz@ball}{blue}% + +\pgfdeclareradialshading[tikz@radial@inner,tikz@radial@outer]{radial}{\pgfpointorigin}{% + color(0bp)=(tikz@radial@inner); + color(25bp)=(tikz@radial@outer); + color(50bp)=(tikz@radial@outer)}% + +\pgfutil@colorlet{tikz@radial@inner}{gray}% +\pgfutil@colorlet{tikz@radial@outer}{white}% + + +\tikzset{ + pin distance/.store in=\tikz@pin@distance, + pin distance=3ex, + pin position/.store in=\tikz@pin@default@pos, + pin position=above, + pin edge/.store in=\tikz@pin@edge@style, + pin edge={}, + tikz@pin@post/.code={\global\let\tikz@pin@edge@style@smuggle=\tikz@pin@edge@style}, + tikz@pre@pin@edge/.code={% + \toks0=\expandafter{\tikz@pin@edge@style@smuggle}% + \edef\pgf@marshal{\noexpand\tikzset{tikz@pin@options/.style={\the\toks0}}}% + \pgf@marshal + },% +}% + +\tikzset{% + pin/.code={% + \begingroup + \ifnum\the\catcode`\:=\active\relax + \def\tikz@next{% + \endgroup + \tikz@parse@pin@active@i{#1}}% + \else + \def\tikz@next{% + \endgroup + \pgfutil@ifnextchar[% + {\tikz@parse@pin@nonactive} + {\tikz@parse@pin@nonactive[]}#1:\pgf@nil}% + \fi + \tikz@next}}% + +\begingroup + \catcode`\:=\active\relax + + \gdef\tikz@parse@pin@active@i#1{% + \pgfutil@ifnextchar[% + {\tikz@parse@pin@active} + {\tikz@parse@pin@active[]}#1:\pgf@nil}% + + \long\gdef\tikz@parse@pin@active[#1]#2:#3\pgf@nil{% + \def\tikz@temp{#3}% + \ifx\tikz@temp\pgfutil@empty + \tikz@@parse@pin@active[#1]\tikz@pin@default@pos:#2:\pgf@nil% + \else + \tikz@@parse@pin@active[#1]#2:#3\pgf@nil% + \fi}% + + \long\gdef\tikz@@parse@pin@active[#1]#2:#3:\pgf@nil{% + \tikzset{% + append after command = {% + \bgroup + [current point is local = true] + \pgfextra{\let\tikz@save@last@node=\tikzlastnode\tikz@node@is@a@labelfalse}% + node [tikz@label@angle = #2, + anchor=@auto, + every pin, + #1, + append after command = {% + (\tikz@save@last@node) + edge [every pin edge, + tikz@pre@pin@edge, + tikz@pin@options] + (\tikzlastnode)}, + tikz@label@post = \tikz@pin@distance, + tikz@pin@post] {#3} + \egroup}}}% +\endgroup + +\long\def\tikz@parse@pin@nonactive[#1]#2:#3\pgf@nil{% + \def\tikz@temp{#3}% + \ifx\tikz@temp\pgfutil@empty + \tikz@@parse@pin@nonactive[#1]\tikz@pin@default@pos:#2:\pgf@nil% + \else + \tikz@@parse@pin@nonactive[#1]#2:#3\pgf@nil% + \fi}% + +\long\def\tikz@@parse@pin@nonactive[#1]#2:#3:\pgf@nil{% + \tikzset{% + append after command = {% + \bgroup + [current point is local = true] + \pgfextra{\let\tikz@save@last@node=\tikzlastnode\tikz@node@is@a@labelfalse}% + node [tikz@label@angle = #2, + anchor=@auto, + every pin, + #1, + append after command = {% + (\tikz@save@last@node) + edge [every pin edge, + tikz@pre@pin@edge, + tikz@pin@options] + (\tikzlastnode)}, + tikz@label@post = \tikz@pin@distance, + tikz@pin@post] {#3} + \egroup}}}% + + +\tikzset{ + label distance/.store in=\tikz@label@distance, + label distance=0pt, + label position/.store in=\tikz@label@default@pos, + label position=above, + absolute/.is if=tikz@absolute, + tikz@label@angle/.store in=\tikz@label@angle +}% + +\newif\iftikz@absolute +\def\tikz@on@text{center}% + +\tikzset{tikz@label@post/.code 2 args={ + \edef\tikz@label@angle{\tikz@label@angle}% + \expandafter\pgfkeys@spdef\expandafter\tikz@label@angle\expandafter{\tikz@label@angle}% + \csname tikz@label@angle@is@\tikz@label@angle\endcsname + \ifx\tikz@label@angle\tikz@on@text% + \def\tikz@node@at{\pgfpointanchor{\tikzlastnode}{center}}% + \def\tikz@anchor{center}% + \else% + \iftikz@absolute% + \pgftransformreset% + \pgf@process{% + \pgfpointshapeborder{\tikzlastnode}% + {\pgfpointadd{\pgfpointanchor{\tikzlastnode}{center}}{\pgfpointpolar{\tikz@label@angle}{1pt}}}}% + \edef\tikz@node@at{\noexpand\pgfqpoint{\the\pgf@x}{\the\pgf@y}}% + \tikz@compute@direction{\tikz@label@angle}% + \tikz@addtransform{\pgftransformshift{\pgfpointpolar{\tikz@label@angle}{#1}}}% + \else% + \pgf@process{\pgfpointanchor{\tikzlastnode}{\tikz@label@angle}}% + \edef\tikz@node@at{\noexpand\pgfqpoint{\the\pgf@x}{\the\pgf@y}}% + \pgf@xb=\pgf@x% + \pgf@yb=\pgf@y% + \pgf@process{\pgfpointanchor{\tikzlastnode}{center}}% + \pgf@xc=\pgf@x% + \pgf@yc=\pgf@y% + \tikz@label@simplefalse% + \ifdim\pgf@xc=\pgf@xb\relax% + \ifdim\pgf@yc=\pgf@yb\relax% + \tikz@label@simpletrue% + \fi% + \fi% + \iftikz@label@simple% + \tikz@compute@direction{\tikz@label@angle}% + \tikz@addtransform{\pgftransformshift{\pgfpointpolar{\tikz@label@angle}{#1}}}% + \else% + \pgf@process{\pgfpointnormalised{% + \pgfpointdiff{\pgfpointtransformed{\pgfqpoint{\pgf@xc}{\pgf@yc}}}{\pgfpointtransformed{\pgfqpoint{\pgf@xb}{\pgf@yb}}}}}% + \edef\pgf@marshal{% + \noexpand\tikz@addtransform{\noexpand\pgftransformshift{\noexpand\pgfpointscale{#1}{ + \noexpand\pgfqpoint{\the\pgf@x}{\the\pgf@y}}}}}% + \pgf@marshal% + \pgf@xc=\pgf@x% + \pgf@yc=\pgf@y% + \pgf@x=\pgf@yc% + \pgf@y=-\pgf@xc% + \ifx\tikz@anchor\tikz@auto@text% + \tikz@auto@anchor% + \fi% + \fi% + \fi% + \fi} +}% + +\newif\iftikz@label@simple% + +\tikzset{% + label/.code={% + \begingroup + \ifnum\the\catcode`\:=\active\relax + \def\tikz@next{% + \endgroup + \tikz@parse@label@active@i{#1}}% + \else + \def\tikz@next{% + \endgroup + \pgfutil@ifnextchar[% + {\tikz@parse@label@nonactive} + {\tikz@parse@label@nonactive[]}#1:\pgf@nil}% + \fi + \tikz@next}}% + +\begingroup + \catcode`\:=\active\relax + + \gdef\tikz@parse@label@active@i#1{% + \pgfutil@ifnextchar[% + {\tikz@parse@label@active} + {\tikz@parse@label@active[]}#1:\pgf@nil}% + + \gdef\tikz@parse@label@active[#1]#2:#3\pgf@nil{% + \def\tikz@temp{#3}% + \ifx\tikz@temp\pgfutil@empty + \tikz@@parse@label@active[#1]\tikz@label@default@pos:#2:\pgf@nil% + \else + \def\tikz@temp{#2}% + \ifx\tikz@temp\pgfutil@empty + \tikz@@parse@label@active[#1]\tikz@label@default@pos:#3\pgf@nil% + \else + \tikz@@parse@label@active[#1]#2:#3\pgf@nil% + \fi + \fi + }% + + \gdef\tikz@@parse@label@active[#1]#2:#3:\pgf@nil{% + \tikzset{% + append after command = {% + \bgroup + [current point is local=true] + \pgfextra{\let\tikz@save@last@fig@name=\tikz@last@fig@name\tikz@node@is@a@labelfalse} + node [tikz@label@angle = #2, + anchor=@auto, + every label, + #1, + tikz@label@post = \tikz@label@distance] {\iftikz@handle@active@nodes\expandafter\scantokens\else\expandafter\pgfutil@firstofone\fi{#3\noexpand}} + \pgfextra{\global\let\tikz@last@fig@name=\tikz@save@last@fig@name} + \egroup}}}% +\endgroup + +\def\tikz@parse@label@nonactive[#1]#2:#3\pgf@nil{% + \def\tikz@temp{#3}% + \ifx\tikz@temp\pgfutil@empty + \tikz@@parse@label@nonactive[#1]\tikz@label@default@pos:#2:\pgf@nil% + \else + \def\tikz@temp{#2}% + \ifx\tikz@temp\pgfutil@empty + \tikz@@parse@label@nonactive[#1]\tikz@label@default@pos:#3\pgf@nil% + \else + \tikz@@parse@label@nonactive[#1]#2:#3\pgf@nil% + \fi + \fi +}% + +\def\tikz@@parse@label@nonactive[#1]#2:#3:\pgf@nil{% + \tikzset{% + append after command = {% + \bgroup + [current point is local=true] + \pgfextra{\let\tikz@save@last@fig@name=\tikz@last@fig@name\tikz@node@is@a@labelfalse} + node [tikz@label@angle = #2, + anchor=@auto, + every label, + #1, + tikz@label@post = \tikz@label@distance] {\iftikz@handle@active@nodes\expandafter\scantokens\else\expandafter\pgfutil@firstofone\fi{#3\noexpand}} + \pgfextra{\global\let\tikz@last@fig@name=\tikz@save@last@fig@name} + \egroup}}}% + +\expandafter\def\csname tikz@label@angle@is@right\endcsname{\def\tikz@label@angle{0}}% +\expandafter\def\csname tikz@label@angle@is@above right\endcsname{\def\tikz@label@angle{45}}% +\expandafter\def\csname tikz@label@angle@is@above\endcsname{\def\tikz@label@angle{90}}% +\expandafter\def\csname tikz@label@angle@is@above left\endcsname{\def\tikz@label@angle{135}}% +\expandafter\def\csname tikz@label@angle@is@left\endcsname{\def\tikz@label@angle{180}}% +\expandafter\def\csname tikz@label@angle@is@below left\endcsname{\def\tikz@label@angle{225}}% +\expandafter\def\csname tikz@label@angle@is@below\endcsname{\def\tikz@label@angle{270}}% +\expandafter\def\csname tikz@label@angle@is@below right\endcsname{\def\tikz@label@angle{315}}% + +\def\tikz@compute@direction#1{% + \ifx\tikz@anchor\tikz@auto@text% + \let\tikz@do@auto@anchor=\relax + \pgfmathsetcount{\c@pgf@counta}{#1}% + \ifnum\c@pgf@counta<0\relax + \advance\c@pgf@counta by 360\relax% + \fi% + \ifnum\c@pgf@counta>359\relax + \advance\c@pgf@counta by-360\relax% + \fi% + \ifnum\c@pgf@counta<4\relax% + \def\tikz@anchor{west}% + \else\ifnum\c@pgf@counta<87\relax% + \def\tikz@anchor{south west}% + \else\ifnum\c@pgf@counta<94\relax% + \def\tikz@anchor{south}% + \else\ifnum\c@pgf@counta<177\relax% + \def\tikz@anchor{south east}% + \else\ifnum\c@pgf@counta<184\relax% + \def\tikz@anchor{east}% + \else\ifnum\c@pgf@counta<267\relax% + \def\tikz@anchor{north east}% + \else\ifnum\c@pgf@counta<274\relax% + \def\tikz@anchor{north}% + \else\ifnum\c@pgf@counta<357\relax% + \def\tikz@anchor{north west}% + \else% + \def\tikz@anchor{west}% + \fi\fi\fi\fi\fi\fi\fi\fi% + \fi% +}% +\def\tikz@auto@text{@auto}% + +\tikzset{ + name/.code={\edef\tikz@fig@name{\tikz@pp@name{#1}}\let\tikz@id@name\tikz@fig@name},% + name prefix/.initial=,% + name suffix/.initial=,% + local bounding box/.style={/pgf/local bounding box/.expanded=\tikz@pp@name{#1}} +}% +\def\tikz@pp@name#1{\csname pgfk@/tikz/name prefix\endcsname#1\csname pgfk@/tikz/name suffix\endcsname}% + + +\tikzset{ + node contents/.code=\def\tikz@node@content{#1}, + pic type/.code=\def\tikz@node@content{#1}, % +}% + +\tikzset{ + behind path/.code=\def\tikz@whichbox{\tikz@figbox@bg}, + in front of path/.code=\def\tikz@whichbox{\tikz@figbox} +}% +\def\tikz@whichbox{\tikz@figbox}% + +\tikzoption{at}{\tikz@scan@one@point\tikz@set@at#1}% +\def\tikz@set@at#1{\def\tikz@node@at{#1}}% + +\tikzoption{shape}{\edef\tikz@shape{#1}}% + +\tikzoption{nodes}{\tikzset{every node/.append style={#1}}}% + +\tikzset{alias/.code={% + \tikz@fig@mustbenamed + \begingroup + \toks0=\expandafter{\tikz@alias}% + \edef\pgf@temp{\noexpand\pgfnodealias{\tikz@pp@name{#1}}{\noexpand\tikz@fig@name}}% + \toks1=\expandafter{\pgf@temp}% + \xdef\pgf@marshal{% + \noexpand\def\noexpand\tikz@alias{\the\toks0 \the\toks1 }% + }% + \endgroup + \pgf@marshal +}}% + +\def\tikzaddafternodepathoption#1{#1\tikzset{prefix after command={\pgfextra{#1}}}}% +\tikzset{after node path/.style={append after command={#1}}}% + + +\tikzset{pic text/.store in=\tikzpictext}% +\let\tikzpictext\relax +\tikzset{pic text options/.store in=\tikzpictextoptions}% +\let\tikzpictextoptions\pgfutil@empty + + + +\tikzoption{anchor}{\def\tikz@anchor{#1}\let\tikz@do@auto@anchor=\relax}% + +\tikzoption{left}[]{\def\tikz@anchor{east}\tikz@possibly@transform{x}{-}{#1}}% +\tikzoption{right}[]{\def\tikz@anchor{west}\tikz@possibly@transform{x}{}{#1}}% +\tikzoption{above}[]{\def\tikz@anchor{south}\tikz@possibly@transform{y}{}{#1}}% +\tikzoption{below}[]{\def\tikz@anchor{north}\tikz@possibly@transform{y}{-}{#1}}% +\tikzoption{above left}[]% + {\def\tikz@anchor{south east}% + \tikz@possibly@transform{x}{-}{#1}\tikz@possibly@transform{y}{}{#1}}% +\tikzoption{above right}[]% + {\def\tikz@anchor{south west}% + \tikz@possibly@transform{x}{}{#1}\tikz@possibly@transform{y}{}{#1}}% +\tikzoption{below left}[]% + {\def\tikz@anchor{north east}% + \tikz@possibly@transform{x}{-}{#1}\tikz@possibly@transform{y}{-}{#1}}% +\tikzoption{below right}[]% + {\def\tikz@anchor{north west}% + \tikz@possibly@transform{x}{}{#1}\tikz@possibly@transform{y}{-}{#1}}% +\tikzset{centered/.code=\def\tikz@anchor{center}}% + +\tikzoption{node distance}{\def\tikz@node@distance{#1}}% +\def\tikz@node@distance{1cm}% + +\tikzoption{above of}{\tikz@of{#1}{90}}% +\tikzoption{below of}{\tikz@of{#1}{-90}}% +\tikzoption{left of}{\tikz@of{#1}{180}}% +\tikzoption{right of}{\tikz@of{#1}{0}}% +\tikzoption{above left of}{\tikz@of{#1}{135}}% +\tikzoption{below left of}{\tikz@of{#1}{-135}}% +\tikzoption{above right of}{\tikz@of{#1}{45}}% +\tikzoption{below right of}{\tikz@of{#1}{-45}}% +\def\tikz@of#1#2{% + \def\tikz@anchor{center}% + \let\tikz@do@auto@anchor=\relax% + \tikz@addtransform{% + \expandafter\tikz@extract@node@dist\tikz@node@distance and\pgf@stop% + \pgftransformshift{\pgfpointpolar{#2}{\tikz@extracted@node@distance}}}% + \def\tikz@node@at{\pgfpointanchor{\tikz@pp@name{#1}}{center}}}% +\def\tikz@extract@node@dist#1and#2\pgf@stop{% + \def\tikz@extracted@node@distance{#1}}% + +\tikzset{ + transform shape nonlinear/.is choice, + transform shape nonlinear/.default=true, + transform shape nonlinear/true/.code=\let\tikz@nlt\relax, + transform shape nonlinear/false/.code=\def\tikz@nlt{\pgfapproximatenonlineartranslation}, + transform shape nonlinear=false, +}% + + +\tikzoption{transform shape}[true]{% + \csname tikz@fullytransformed#1\endcsname% + \iftikz@fullytransformed% + \pgfresetnontranslationattimefalse% + \else% + \pgfresetnontranslationattimetrue% + \fi% +}% + +\newif\iftikz@fullytransformed +\pgfresetnontranslationattimetrue% + +\def\tikz@anchor{center}% +\def\tikz@shape{rectangle}% + +\def\tikz@possibly@transform#1#2#3{% + \let\tikz@do@auto@anchor=\relax% + \def\tikz@test{#3}% + \ifx\tikz@test\pgfutil@empty% + \else% + \pgfmathsetlength{\pgf@x}{#3}% + \pgf@x=#2\pgf@x\relax% + \edef\tikz@marshal{\noexpand\tikz@addtransform{% + \expandafter\noexpand\csname pgftransform#1shift\endcsname{\the\pgf@x}}}% + \tikz@marshal% + \fi% +}% + + +\tikzoption{remember picture}[true]{\csname pgfrememberpicturepositiononpage#1\endcsname} +\tikzset{ + overlay/.is choice, + overlay/true/.code={\pgf@relevantforpicturesizefalse}, + overlay/false/.code={\pgf@relevantforpicturesizetrue}, + overlay/.default=true +} + + + +\tikzoption{sloped}[true]{\csname pgfslopedattime#1\endcsname}% +\tikzoption{allow upside down}[true]{\csname pgfallowupsidedownattime#1\endcsname}% + +\tikzoption{pos}{\edef\tikz@time{#1}\ifx\tikz@time\pgfutil@empty\else\pgfmathsetmacro\tikz@time{\tikz@time}\fi}% + +\tikzoption{auto}[]{\csname tikz@install@auto@anchor@#1\endcsname}% +\tikzoption{swap}[]{% + \def\tikz@temp{left}% + \ifx\tikz@auto@anchor@direction\tikz@temp% + \def\tikz@auto@anchor@direction{right}% + \else% + \def\tikz@auto@anchor@direction{left}% + \fi% +}% +\tikzset{'/.style=swap}% + + +\def\tikz@install@auto@anchor@{\let\tikz@do@auto@anchor=\tikz@auto@anchor@on}% +\def\tikz@install@auto@anchor@false{\let\tikz@do@auto@anchor=\relax}% +\def\tikz@install@auto@anchor@left{\let\tikz@do@auto@anchor=\tikz@auto@anchor@on\def\tikz@auto@anchor@direction{left}}% +\def\tikz@install@auto@anchor@right{\let\tikz@do@auto@anchor=\tikz@auto@anchor@on\def\tikz@auto@anchor@direction{right}}% + +\let\tikz@do@auto@anchor=\relax% + +\def\tikz@auto@anchor@on{\csname tikz@auto@anchor@\tikz@auto@anchor@direction\endcsname} + +\def\tikz@auto@anchor@left{\tikz@auto@pre\tikz@auto@anchor\tikz@auto@post}% +\def\tikz@auto@anchor@right{\tikz@auto@pre\tikz@auto@anchor@prime\tikz@auto@post}% + +\def\tikz@auto@anchor@direction{left}% + +\tikzoption{text}{\def\tikz@textcolor{#1}}% +\tikzoption{font}{\def\tikz@textfont{#1}}% +\tikzoption{node font}{\def\tikz@node@textfont{#1}}% +\tikzoption{text opacity}{\def\tikz@textopacity{#1}}% +\tikzoption{text width}{\def\tikz@text@width{#1}}% +\tikzoption{text height}{\def\tikz@text@height{#1}}% +\tikzoption{text depth}{\def\tikz@text@depth{#1}}% +\tikzoption{text ragged}[]% +{\def\tikz@text@action{\pgfutil@raggedright\rightskip0pt plus2em \spaceskip.3333em \xspaceskip.5em\relax}}% +\tikzoption{text badly ragged}[]{\def\tikz@text@action{\pgfutil@raggedright\relax}}% +\tikzoption{text ragged left}[]% +{\def\tikz@text@action{\pgfutil@raggedleft\leftskip0pt plus2em \spaceskip.3333em \xspaceskip.5em\relax}}% +\tikzoption{text badly ragged left}[]{\def\tikz@text@action{\pgfutil@raggedleft\relax}}% +\tikzoption{text justified}[]{\def\tikz@text@action{\leftskip0pt\rightskip0pt\relax}}% +\tikzoption{text centered}[]{\def\tikz@text@action{% + \leftskip0pt plus2em% + \rightskip0pt plus2em% + \spaceskip.3333em \xspaceskip.5em% + \parfillskip=0pt% + \iftikz@warn@for@narrow@centered\else\hbadness10000\fi% + \let\\=\@centercr% + \relax}}% +\tikzoption{text badly centered}[]% +{\def\tikz@text@action{% + \let\\=\@centercr% + \parfillskip=0pt% + \rightskip\pgfutil@flushglue% + \leftskip\pgfutil@flushglue\relax}}% +\tikzset{badness warnings for centered text/.is if=tikz@warn@for@narrow@centered}% +\newif\iftikz@warn@for@narrow@centered + +\def\tikz@text@reset{% + \let\tikz@text@width=\pgfutil@empty + \let\tikz@text@height=\pgfutil@empty + \let\tikz@text@depth=\pgfutil@empty + \let\tikz@textcolor=\pgfutil@empty + \let\tikz@textfont=\pgfutil@empty + \let\tikz@textopacity=\pgfutil@empty + \let\tikz@node@textfont=\pgfutil@empty + \def\tikz@text@action{\pgfutil@raggedright\rightskip0pt plus2em \spaceskip.3333em \xspaceskip.5em\relax}% +} +\tikz@text@reset + + +\tikzset{ + node halign header/.initial=, + align/.is choice, + align/left/.style ={text ragged,node halign header=\tikz@align@left@header}, + align/flush left/.style ={text badly ragged,node halign header=\tikz@align@left@header}, + align/right/.style ={text ragged left,node halign header=\tikz@align@right@header}, + align/flush right/.style ={text badly ragged left,node halign header=\tikz@align@right@header}, + align/center/.style={text centered,node halign header=\tikz@align@center@header}, + align/flush center/.style={text badly centered,node halign header=\tikz@align@center@header}, + align/justify/.style ={text justified,node halign header=\tikz@align@left@header}, + align/none/.style ={text justified,node halign header=}, +}% +\def\tikz@align@left@header{##\hfil\cr}% +\def\tikz@align@right@header{\hfil##\cr}% +\def\tikz@align@center@header{\hfil##\hfil\cr}% + + + +\tikzoption{arrows}{\tikz@processarrows{#1}}% + + +\tikzoption{>}{\pgfdeclarearrow{name=<->,means={#1}}}% +\pgfdeclarearrow{name=|<->|, means={>[sep=0pt].|}}% + +\tikzoption{shorten <}{\pgfsetshortenstart{#1}}% +\tikzoption{shorten >}{\pgfsetshortenend{#1}}% + +\def\tikz@processarrows#1{% + \def\tikz@current@arrows{#1}% + \def\tikz@temp{#1}% + \ifx\tikz@temp\pgfutil@empty% + \else% + \pgfsetarrows{#1}% + \fi% +}% + +\def\tikz@current@arrows{-}% + +\tikzoption{bend}{\tikz@scan@one@point\tikz@set@parabola@bend#1\relax}% +\tikzoption{bend pos}{\def\tikz@parabola@bend@factor{#1}}% +\tikzoption{parabola height}{% + \def\tikz@parabola@bend@factor{.5}% + \def\tikz@parabola@bend{\pgfpointadd{\pgfpoint{0pt}{#1}}{\tikz@last@position@saved}}}% + +\def\tikz@parabola@bend{\tikz@last@position@saved}% +\def\tikz@parabola@bend@factor{0}% + +\def\tikz@set@parabola@bend#1{\def\tikz@parabola@bend{#1}}% + +\tikzoption{domain}{\edef\tikz@plot@domain{#1}\expandafter\tikz@plot@samples@recalc\tikz@plot@domain\relax}% +\tikzoption{range}{\def\tikz@plot@range{#1}}% +\tikzoption{yrange}{\def\tikz@plot@range{#1}}% +\let\tikz@plot@range=\pgfutil@empty +\tikzoption{xrange}{\def\tikz@plot@xrange{#1}}% +\let\tikz@plot@xrange=\pgfutil@empty + +\tikzoption{smooth}[]{\let\tikz@plot@handler=\pgfplothandlercurveto}% +\tikzoption{smooth cycle}[]{\let\tikz@plot@handler=\pgfplothandlerclosedcurve}% +\tikzoption{sharp plot}[]{\let\tikz@plot@handler\pgfplothandlerlineto}% +\tikzoption{sharp cycle}[]{\let\tikz@plot@handler\pgfplothandlerpolygon}% + +\tikzoption{tension}{\pgfsetplottension{#1}}% + +\tikzoption{xcomb}[]{\let\tikz@plot@handler=\pgfplothandlerxcomb}% +\tikzoption{ycomb}[]{\let\tikz@plot@handler=\pgfplothandlerycomb}% +\tikzoption{polar comb}[]{\let\tikz@plot@handler=\pgfplothandlerpolarcomb}% +\tikzoption{ybar}[]{\let\tikz@plot@handler=\pgfplothandlerybar}% +\tikzoption{ybar interval}[]{\let\tikz@plot@handler=\pgfplothandlerybarinterval}% +\tikzoption{xbar interval}[]{\let\tikz@plot@handler=\pgfplothandlerxbarinterval}% +\tikzoption{xbar}[]{\let\tikz@plot@handler=\pgfplothandlerxbar}% +\tikzoption{const plot}[]{\let\tikz@plot@handler=\pgfplothandlerconstantlineto}% +\tikzoption{const plot mark left}[]{\let\tikz@plot@handler=\pgfplothandlerconstantlineto}% +\tikzoption{const plot mark right}[]{\let\tikz@plot@handler=\pgfplothandlerconstantlinetomarkright}% +\tikzoption{const plot mark mid}[]{\let\tikz@plot@handler=\pgfplothandlerconstantlinetomarkmid}% +\tikzoption{jump mark right}[]{\let\tikz@plot@handler=\pgfplothandlerjumpmarkright}% +\tikzoption{jump mark mid}[]{\let\tikz@plot@handler=\pgfplothandlerjumpmarkmid}% +\tikzoption{jump mark left}[]{\let\tikz@plot@handler=\pgfplothandlerjumpmarkleft}% + +\tikzoption{raw gnuplot}[true]{\csname tikz@plot@raw@gnuplot#1\endcsname}% +\tikzoption{prefix}{\def\tikz@plot@prefix{#1}}% +\tikzoption{id}{\def\tikz@plot@id{#1}}% + +\tikzoption{samples}{\pgfmathsetmacro\tikz@plot@samples{max(2,#1)}\expandafter\tikz@plot@samples@recalc\tikz@plot@domain\relax}% +\tikzoption{samples at}{\def\tikz@plot@samplesat{#1}}% +\tikzoption{parametric}[true]{\csname tikz@plot@parametric#1\endcsname}% + +\tikzoption{variable}{\def\tikz@plot@var{#1}}% + +\tikzoption{only marks}[]{\let\tikz@plot@handler\pgfplothandlerdiscard}% + +\tikzoption{mark}{% + \def\tikz@plot@mark{#1}% + \def\tikz@temp{none}% + \ifx\tikz@temp\tikz@plot@mark + \let\tikz@plot@mark=\pgfutil@empty + \fi +}% +\tikzset{ + no marks/.style={mark=none},% + no markers/.style={mark=none},% + every mark/.style={}, + mark options/.style={% + every mark/.style={#1}% + }}% +\tikzoption{mark size}{\pgfsetplotmarksize{#1}}% + +\tikzoption{mark indices}{\def\tikz@mark@list{#1}}% +\tikzoption{mark phase}{\pgfsetplotmarkphase{#1}}% +\tikzoption{mark repeat}{\pgfsetplotmarkrepeat{#1}}% + +\let\tikz@mark@list=\pgfutil@empty + +\let\tikz@plot@handler=\pgfplothandlerlineto +\let\tikz@plot@mark=\pgfutil@empty + +\def\tikz@plot@samples{25}% +\def\tikz@plot@domain{-5:5}% +\def\tikz@plot@var{\x}% +\def\tikz@plot@samplesat{-5,-4.5833333,...,5}% +\def\tikz@plot@samples@recalc#1:#2\relax{% + \begingroup + \pgfmathparse{#1}% + \let\tikz@temp@start=\pgfmathresult% + \pgfmathparse{#2}% + \let\tikz@temp@end=\pgfmathresult% + \pgfmathsetmacro\tikz@temp@diff{(\tikz@temp@end-\tikz@temp@start)/(\tikz@plot@samples-1)}% + \pgfkeys{/pgf/fpu/output format/fixed/.try}% + \pgfmathsetmacro\tikz@temp@diff@abs{abs(\tikz@temp@diff)}% + \ifdim\tikz@temp@diff@abs pt<0.0001pt\relax% + \edef\tikz@plot@samplesat{\tikz@temp@start,\tikz@temp@end}% + \else% + \pgfmathparse{\tikz@temp@start+\tikz@temp@diff}% + \edef\tikz@plot@samplesat{\tikz@temp@start,\pgfmathresult,...,\tikz@temp@end}% + \fi% + \pgfmath@smuggleone\tikz@plot@samplesat + \endgroup +}% + + +\def\tikz@plot@prefix{\jobname.}% +\def\tikz@plot@id{pgf-plot}% + +\newif\iftikz@plot@parametric +\newif\iftikz@plot@raw@gnuplot + + +\tikzoption{to path}{\def\tikz@to@path{#1}}% + +\def\tikz@to@path{-- (\tikztotarget) \tikztonodes}% + +\tikzset{edge macro/.store in=\tikz@edge@macro}% +\let\tikz@edge@macro\pgfutil@empty + +\tikzset{ + edge node/.code={ + \expandafter\def\expandafter\tikz@tonodes\expandafter{\tikz@tonodes #1} + }, + edge label/.style={/tikz/edge node={node[auto]{#1}}}, + edge label'/.style={/tikz/edge node={node[auto,swap]{#1}}}, +}% + + +\tikzset{ + append after command/.code=\expandafter\def\expandafter\tikz@after@path\expandafter{\tikz@after@path#1}, + prefix after command/.code={% + \def\tikz@temp{#1}% + \expandafter\expandafter\expandafter\def% + \expandafter\expandafter\expandafter\tikz@after@path% + \expandafter\expandafter\expandafter{% + \expandafter\tikz@temp\tikz@after@path}% + }, +}% +\let\tikz@after@path\pgfutil@empty + + +\newif\iftikz@child@missing +\pgfkeys{/tikz/missing/.is if=tikz@child@missing}% + +\tikzset{edge from parent macro/.initial=\tikz@edge@from@parent@macro}% +\def\tikz@edge@from@parent@macro#1#2{ + [style=edge from parent, #1, /utils/exec=\tikz@node@is@a@labeltrue] \tikz@edge@to@parent@path #2}% + +\tikzoption{edge from parent path}{\def\tikz@edge@to@parent@path{#1}}% + +\tikzoption{parent anchor}{\def\tikzparentanchor{.#1}\ifx\tikzparentanchor\tikz@border@text\let\tikzparentanchor\pgfutil@empty\fi}% +\tikzoption{child anchor}{\def\tikzchildanchor{.#1}\ifx\tikzchildanchor\tikz@border@text\let\tikzchildanchor\pgfutil@empty\fi}% + +\tikzoption{level distance}{\pgfmathsetlength\tikzleveldistance{#1}}% +\tikzoption{sibling distance}{\pgfmathsetlength\tikzsiblingdistance{#1}}% + +\tikzoption{growth function}{\let\tikz@grow=#1}% + +\tikzset{grow siblings on line/.style={growth function=\tikz@grow@direction}}% + +\tikzoption{growth parent anchor}{\def\tikz@growth@anchor{#1}}% +\tikzoption{grow}{\tikz@set@growth{#1}\edef\tikz@special@level{\the\tikztreelevel}}% +\tikzoption{grow'}{\tikz@set@growth{#1}\tikz@swap@growth\edef\tikz@special@level{\the\tikztreelevel}}% + +\def\tikz@growth@anchor{center}% + +\def\tikz@special@level{-1}% + +\def\tikz@swap@growth{% + \let\tikz@temp=\tikz@angle@grow@right% + \let\tikz@angle@grow@right=\tikz@angle@grow@left% + \let\tikz@angle@grow@left=\tikz@temp% +}% + +\def\tikz@set@growth#1{% + \let\tikz@grow=\tikz@grow@direction% + \expandafter\ifx\csname tikz@grow@direction@#1\endcsname\relax% + \c@pgf@counta=#1\relax% + \else% + \c@pgf@counta=\csname tikz@grow@direction@#1\endcsname% + \fi% + \edef\tikz@angle@grow{\the\c@pgf@counta}% + \advance\c@pgf@counta by-90\relax% + \edef\tikz@angle@grow@left{\the\c@pgf@counta}% + \advance\c@pgf@counta by180\relax% + \edef\tikz@angle@grow@right{\the\c@pgf@counta}% +}% + +\def\tikz@border@text{.border}% +\let\tikzparentanchor=\pgfutil@empty +\let\tikzchildanchor=\pgfutil@empty +\def\tikz@edge@to@parent@path{(\tikzparentnode\tikzparentanchor) -- (\tikzchildnode\tikzchildanchor)}% + +\tikzleveldistance=15mm% +\tikzsiblingdistance=15mm% + +\def\tikz@grow@direction@down{-90}% +\def\tikz@grow@direction@up{90}% +\def\tikz@grow@direction@left{180}% +\def\tikz@grow@direction@right{0}% + +\def\tikz@grow@direction@south{-90}% +\def\tikz@grow@direction@north{90}% +\def\tikz@grow@direction@west{180}% +\def\tikz@grow@direction@east{0}% + +\expandafter\def\csname tikz@grow@direction@north east\endcsname{45}% +\expandafter\def\csname tikz@grow@direction@north west\endcsname{135}% +\expandafter\def\csname tikz@grow@direction@south east\endcsname{-45}% +\expandafter\def\csname tikz@grow@direction@south west\endcsname{-135}% + +\def\tikz@grow@direction{% + \pgftransformshift{\pgfpointpolar{\tikz@angle@grow}{\tikzleveldistance}}% + \ifnum\tikztreelevel=\tikz@special@level% + \else% + \pgf@xc=.5\tikzsiblingdistance% + \c@pgf@counta=\tikznumberofchildren% + \advance\c@pgf@counta by1\relax% + \pgfutil@tempdima=\c@pgf@counta\pgf@xc% + \pgftransformshift{\pgfpointpolar{\tikz@angle@grow@left}{\pgfutil@tempdima}}% + \pgftransformshift{\pgfpointpolar{\tikz@angle@grow@right}{\tikznumberofcurrentchild\tikzsiblingdistance}}% + \fi% +}% + +\tikzset{grow=down}% + + +\tikzset{snake/.code=\tikzerror{You need to say \string\usetikzlibrary{snakes}}}% + +\tikzset{decorate/.code=\tikzerror{You need to load a decoration library}}% + +\usepgfmodule{matrix}% + +\tikzoption{matrix}[true]{\csname tikz@is@matrix#1\endcsname}% + +\tikzoption{matrix anchor}{\def\tikz@matrix@anchor{#1}}% + +\tikzoption{column sep}{\def\pgfmatrixcolumnsep{#1}}% +\tikzoption{row sep}{\def\pgfmatrixrowsep{#1}}% + +\tikzoption{cells}{\tikzset{every cell/.append style={#1}}}% + +\tikzoption{ampersand replacement}{\def\tikz@ampersand@replacement{#1}}% + +\newif\iftikz@is@matrix +\let\tikz@matrix@anchor=\pgfutil@empty +\let\tikz@ampersand@replacement=\pgfutil@empty + + +\tikzset{% + handle active characters in code/.is if=tikz@handle@active@code, + handle active characters in nodes/.is if=tikz@handle@active@nodes, +}% +\newif\iftikz@handle@active@code +\newif\iftikz@handle@active@nodes + + +\tikzoption{execute at begin picture}{\expandafter\def\expandafter\tikz@atbegin@picture\expandafter{\tikz@atbegin@picture#1}}% +\tikzoption{execute at end picture}{\expandafter\def\expandafter\tikz@atend@picture\expandafter{\tikz@atend@picture#1}}% +\tikzoption{execute at begin scope}{\expandafter\def\expandafter\tikz@atbegin@scope\expandafter{\tikz@atbegin@scope#1}}% +\tikzoption{execute at end scope}{\expandafter\def\expandafter\tikz@atend@scope\expandafter{\tikz@atend@scope#1}}% +\tikzoption{execute at begin to}{\expandafter\def\expandafter\tikz@atbegin@to\expandafter{\tikz@atbegin@to#1}}% +\tikzoption{execute at end to}{\expandafter\def\expandafter\tikz@atend@to\expandafter{\tikz@atend@to#1}}% +\tikzoption{execute at begin node}{\expandafter\def\expandafter\tikz@atbegin@node\expandafter{\tikz@atbegin@node#1}}% +\tikzoption{execute at end node}{\expandafter\def\expandafter\tikz@atend@node\expandafter{\tikz@atend@node#1}}% +\tikzoption{execute at begin matrix}{\expandafter\def\expandafter\tikz@atbegin@matrix\expandafter{\tikz@atbegin@matrix#1}}% +\tikzoption{execute at end matrix}{\expandafter\def\expandafter\tikz@atend@matrix\expandafter{\tikz@atend@matrix#1}}% +\tikzoption{execute at begin cell}{\expandafter\def\expandafter\tikz@atbegin@cell\expandafter{\tikz@atbegin@cell#1}}% +\tikzoption{execute at end cell}{\expandafter\def\expandafter\tikz@atend@cell\expandafter{\tikz@atend@cell#1}}% +\tikzoption{execute at empty cell}{\expandafter\def\expandafter\tikz@at@emptycell\expandafter{\tikz@at@emptycell#1}}% + +\let\tikz@atbegin@picture=\pgfutil@empty +\let\tikz@atend@picture=\pgfutil@empty +\let\tikz@atbegin@scope=\pgfutil@empty +\let\tikz@atend@scope=\pgfutil@empty +\let\tikz@atbegin@to=\pgfutil@empty +\let\tikz@atend@to=\pgfutil@empty +\let\tikz@atbegin@node=\pgfutil@empty +\let\tikz@atend@node=\pgfutil@empty +\let\tikz@atbegin@cell=\pgfutil@empty +\let\tikz@atend@cell=\pgfutil@empty +\let\tikz@at@emptycell=\pgfutil@empty +\let\tikz@atbegin@matrix=\pgfutil@empty +\let\tikz@atend@matrix=\pgfutil@empty + + +\tikzset{preaction/.code=\expandafter\def\expandafter\tikz@preactions\expandafter{\tikz@preactions\tikz@extra@preaction{#1}}}% +\tikzset{postaction/.code=\expandafter\def\expandafter\tikz@postactions\expandafter{\tikz@postactions\tikz@extra@postaction{#1}}}% +\let\tikz@preactions=\pgfutil@empty +\let\tikz@postactions=\pgfutil@empty + +\tikzoption{set style}{\tikzstyle#1}% + +\def\tikzstyle{\pgfutil@ifnextchar\bgroup\tikz@style@parseA\tikz@style@parseB}% +\def\tikz@style@parseB#1={\tikz@style@parseA{#1}=}% +\def\tikz@style@parseA#1#2=#3[#4]{% + \pgfutil@in@[{#2}% + \ifpgfutil@in@% + \tikz@style@parseC{#1}#2={#4}% + \else% + \tikz@style@parseD{#1}#2={#4}% + \fi% +}% + +\def\tikz@style@parseC#1[#2]#3=#4{% + \pgfkeys{/tikz/#1/.default={#2}}% + \pgfutil@in@+{#3}% + \ifpgfutil@in@% + \pgfkeys{/tikz/#1/.append style={#4}}% + \else% + \pgfkeys{/tikz/#1/.style={#4}}% + \fi}% +\def\tikz@style@parseD#1#2=#3{% + \pgfutil@in@+{#2}% + \ifpgfutil@in@% + \pgfkeys{/tikz/#1/.append style={#3}}% + \else% + \pgfkeys{/tikz/#1/.style={#3}}% + \fi}% + + + +\tikzset{help lines/.style= {color=gray,line width=0.2pt}}% + +\tikzset{every picture/.style= {}}% +\tikzset{every path/.style= {}}% +\tikzset{every scope/.style= {}}% +\tikzset{every plot/.style= {}}% +\tikzset{every node/.style= {}}% +\tikzset{every child/.style= {}}% +\tikzset{every child node/.style= {}}% +\tikzset{every to/.style= {}}% +\tikzset{every cell/.style= {}}% +\tikzset{every matrix/.style= {}}% +\tikzset{every edge/.style= {draw}}% +\tikzset{every label/.style= {draw=none,fill=none}}% +\tikzset{every pin/.style= {draw=none,fill=none}}% +\tikzset{every pin edge/.style= {help lines}}% + +\tikzset{ultra thin/.style= {line width=0.1pt}}% +\tikzset{very thin/.style= {line width=0.2pt}}% +\tikzset{thin/.style= {line width=0.4pt}}% +\tikzset{semithick/.style= {line width=0.6pt}}% +\tikzset{thick/.style= {line width=0.8pt}}% +\tikzset{very thick/.style= {line width=1.2pt}}% +\tikzset{ultra thick/.style= {line width=1.6pt}}% + +\tikzset{solid/.style= {dash pattern=}}% +\tikzset{dotted/.style= {dash pattern=on \pgflinewidth off 2pt}}% +\tikzset{densely dotted/.style= {dash pattern=on \pgflinewidth off 1pt}}% +\tikzset{loosely dotted/.style= {dash pattern=on \pgflinewidth off 4pt}}% +\tikzset{dashed/.style= {dash pattern=on 3pt off 3pt}}% +\tikzset{densely dashed/.style= {dash pattern=on 3pt off 2pt}}% +\tikzset{loosely dashed/.style= {dash pattern=on 3pt off 6pt}}% +\tikzset{dashdotted/.style= {dash pattern=on 3pt off 2pt on \the\pgflinewidth off 2pt}}% +\tikzset{dash dot/.style= {dash pattern=on 3pt off 2pt on \the\pgflinewidth off 2pt}}% +\tikzset{densely dashdotted/.style= {dash pattern=on 3pt off 1pt on \the\pgflinewidth off 1pt}}% +\tikzset{densely dash dot/.style= {dash pattern=on 3pt off 1pt on \the\pgflinewidth off 1pt}}% +\tikzset{loosely dashdotted/.style= {dash pattern=on 3pt off 4pt on \the\pgflinewidth off 4pt}}% +\tikzset{loosely dash dot/.style= {dash pattern=on 3pt off 4pt on \the\pgflinewidth off 4pt}}% +\tikzset{dashdotdotted/.style= {dash pattern=on 3pt off 2pt on \the\pgflinewidth off 2pt on \the\pgflinewidth off 2pt}}% +\tikzset{densely dashdotdotted/.style= {dash pattern=on 3pt off 1pt on \the\pgflinewidth off 1pt on \the\pgflinewidth off 1pt}}% +\tikzset{loosely dashdotdotted/.style= {dash pattern=on 3pt off 4pt on \the\pgflinewidth off 4pt on \the\pgflinewidth off 4pt}}% +\tikzset{dash dot dot/.style= {dash pattern=on 3pt off 2pt on \the\pgflinewidth off 2pt on \the\pgflinewidth off 2pt}}% +\tikzset{densely dash dot dot/.style= {dash pattern=on 3pt off 1pt on \the\pgflinewidth off 1pt on \the\pgflinewidth off 1pt}}% +\tikzset{loosely dash dot dot/.style= {dash pattern=on 3pt off 4pt on \the\pgflinewidth off 4pt on \the\pgflinewidth off 4pt}}% + + +\tikzset{transparent/.style= {opacity=0}}% +\tikzset{ultra nearly transparent/.style={opacity=0.05}}% +\tikzset{very nearly transparent/.style= {opacity=0.1}}% +\tikzset{nearly transparent/.style= {opacity=0.25}}% +\tikzset{semitransparent/.style= {opacity=0.5}}% +\tikzset{nearly opaque/.style= {opacity=0.75}}% +\tikzset{very nearly opaque/.style= {opacity=0.9}}% +\tikzset{ultra nearly opaque/.style= {opacity=0.95}}% +\tikzset{opaque/.style= {opacity=1}}% + +\tikzset{at start/.style= {pos=0}}% +\tikzset{very near start/.style= {pos=0.125}}% +\tikzset{near start/.style= {pos=0.25}}% +\tikzset{midway/.style= {pos=0.5}}% +\tikzset{near end/.style= {pos=0.75}}% +\tikzset{very near end/.style= {pos=0.875}}% +\tikzset{at end/.style= {pos=1}}% + +\tikzset{bend at start/.style= {bend pos=0,bend={+(0,0)}}}% +\tikzset{bend at end/.style= {bend pos=1,bend={+(0,0)}}}% + +\tikzset{edge from parent/.style= {draw}}% + + + +\tikzset{ + animate/.code=\tikzerror{You need to say \string\usetikzlibrary{animations} to use animations} +} + +\newif\iftikz@is@node +\let\tikz@id@name\pgfutil@empty +\let\tikz@id@hook\pgfutil@empty +\def\tikz@call@id@hook{\ifx\tikz@id@hook\pgfutil@empty\else\tikz@id@hook\pgfuseid{\tikz@id@name}\fi}% + + +\let\tikz@clear@rdf@options\relax +\let\tikz@do@rdf@post@options\relax +\let\tikz@do@rdf@pre@options\relax + + + +\pgfkeys{/tikz/style/.style={#1}}% + +\pgfkeys{/tikz/.unknown/.code=% + \let\tikz@key\pgfkeyscurrentname% + \pgfkeys{/pgf/\tikz@key/.try={#1}}% + \ifpgfkeyssuccess% + \else% + \expandafter\pgfutil@in@\expandafter!\expandafter{\tikz@key}% + \ifpgfutil@in@% + \expandafter\tikz@addoption\expandafter{\expandafter\tikz@compat@color@set\expandafter{\tikz@key}}% + \edef\tikz@textcolor{\tikz@key}% + \else% + \pgfutil@doifcolorelse{\tikz@key} + {% + \expandafter\tikz@addoption\expandafter{\expandafter\tikz@compat@color@set\expandafter{\tikz@key}}% + \edef\tikz@textcolor{\tikz@key}% + }% + {% + \expandafter\pgfutil@in@\expandafter-\expandafter{\tikz@key}% + \ifpgfutil@in@% + \expandafter\tikz@processarrows\expandafter{\tikz@key}% + \else% + \expandafter\ifx\csname pgf@sh@s@\tikz@key\endcsname\relax% + \pgfkeys{/errors/unknown key/.expand + once=\expandafter{\expandafter/\expandafter t\expandafter i\expandafter k\expandafter z\expandafter/\tikz@key}{#1}}% + \else% + \edef\tikz@shape{\tikz@key}% + \fi% + \fi% + }% + \fi% + \fi% +}% +\def\tikz@compat@color@set#1{% + \pgfutil@color{#1}\pgfutil@colorlet{pgffillcolor}{#1}% + \expandafter\let\expandafter\pgf@temp\csname\string\color@pgffillcolor\endcsname% + \global\let\pgf@strokecolor@global=\pgf@temp + \global\let\pgf@fillcolor@global=\pgf@temp +}% + +\def\tikz@startup@env{% + \ifnum\the\catcode`\;=\active\relax\expandafter\let\expandafter\tikz@origsemi\expandafter=\tikz@activesemicolon\fi% + \ifnum\the\catcode`\:=\active\relax\expandafter\let\expandafter\tikz@origcolon\expandafter=\tikz@activecolon\fi% + \ifnum\the\catcode`\|=\active\relax\expandafter\let\expandafter\tikz@origbar\expandafter=\tikz@activebar\fi% + \tikz@deactivatthings% + \iftikz@handle@active@code% + \tikz@switchoff@shorthands% + \fi% +}% + +\newif\iftikz@inside@picture +\tikz@inside@picturefalse +\def\tikz@check@inside@picture{% + \iftikz@inside@picture% + \pgfwarning{Nesting tikzpictures is NOT supported}% + \fi% + \tikz@inside@picturetrue% +} + +\def\tikzpicture{% + \begingroup% + \tikz@startup@env% + \tikz@collect@scope@anims\tikz@picture}% +\def\tikz@picture[#1]{% + \pgfpicture% + \let\tikz@atbegin@picture=\pgfutil@empty% + \let\tikz@atend@picture=\pgfutil@empty% + \let\tikz@transform=\relax% + \def\tikz@time{.5}% + \tikz@installcommands% + \scope[every picture,#1]% + \iftikz@handle@active@code% + \tikz@switchoff@shorthands% + \fi% + \expandafter\tikz@atbegin@picture% + \tikz@lib@scope@check% +}% +\def\endtikzpicture{% + \tikz@atend@picture% + \global\let\pgf@shift@baseline@smuggle=\pgf@baseline% + \global\let\pgf@trimleft@final@smuggle=\pgf@trimleft% + \global\let\pgf@trimright@final@smuggle=\pgf@trimright% + \global\let\pgf@remember@smuggle=\ifpgfrememberpicturepositiononpage% + \pgf@remember@layerlist@globally + \endscope% + \let\pgf@baseline=\pgf@shift@baseline@smuggle% + \let\pgf@trimleft=\pgf@trimleft@final@smuggle% + \let\pgf@trimright=\pgf@trimright@final@smuggle% + \let\ifpgfrememberpicturepositiononpage=\pgf@remember@smuggle% + \pgf@restore@layerlist@from@global + \endpgfpicture\endgroup}% + + + + +\def\tikz{% + \begingroup% + \tikz@startup@env% + \tikz@collect@scope@anims\tikz@opt}% +\def\tikz@opt[#1]{\tikzpicture[#1]\pgfutil@ifnextchar\bgroup{\tikz@}{\tikz@@single}}% +\def\tikz@{\bgroup\tikz@auto@end@pathtrue\aftergroup\endtikzpicture\aftergroup\endgroup\let\pgf@temp=}% +\def\tikz@@single#1{% + \expandafter\ifx\csname tikz@protected@command\string#1\endcsname\relax% + \expandafter\tikz@@% + \else% + \begingroup\def\tikz@path@do@at@end{\endgroup\endtikzpicture\endgroup}% + \fi% + #1% +}% + +\expandafter\let\csname tikz@protected@command\string\draw\endcsname\pgfutil@empty% +\expandafter\let\csname tikz@protected@command\string\pattern\endcsname\pgfutil@empty% +\expandafter\let\csname tikz@protected@command\string\fill\endcsname\pgfutil@empty% +\expandafter\let\csname tikz@protected@command\string\filldraw\endcsname\pgfutil@empty% +\expandafter\let\csname tikz@protected@command\string\shade\endcsname\pgfutil@empty% +\expandafter\let\csname tikz@protected@command\string\shadedraw\endcsname\pgfutil@empty% +\expandafter\let\csname tikz@protected@command\string\clip\endcsname\pgfutil@empty% +\expandafter\let\csname tikz@protected@command\string\graph\endcsname\pgfutil@empty% +\expandafter\let\csname tikz@protected@command\string\useasboundingbox\endcsname\pgfutil@empty% +\expandafter\let\csname tikz@protected@command\string\node\endcsname\pgfutil@empty% +\expandafter\let\csname tikz@protected@command\string\coordinate\endcsname\pgfutil@empty% +\expandafter\let\csname tikz@protected@command\string\matrix\endcsname\pgfutil@empty% +\expandafter\let\csname tikz@protected@command\string\datavisualization\endcsname\pgfutil@empty% +\expandafter\let\csname tikz@protected@command\string\path\endcsname\pgfutil@empty% +\expandafter\let\csname tikz@protected@command\string\pic\endcsname\pgfutil@empty% + +\def\tikz@@{% + \let\tikz@next=\tikz@collectnormalsemicolon% + \ifnum\the\catcode`\;=\active\relax% + \let\tikz@next=\tikz@collectactivesemicolon% + \fi% + \tikz@next}% +\def\tikz@collectnormalsemicolon#1;{#1;\endtikzpicture\endgroup} +{ + \catcode`\;=\active + \gdef\tikz@collectactivesemicolon#1;{#1;\endtikzpicture\endgroup} +}% + +\def\tikzifinpicture#1#2{% + \pgfutil@ifundefined{filldraw}{#2}{#1}% +}% + + +\def\tikz@collect@scope@anims#1{% + \pgfutil@ifnextchar[#1{#1[]}% +}% + +\def\tikz@scope@env{% + \pgfscope% + \begingroup% + \let\tikz@atbegin@scope=\pgfutil@empty% + \let\tikz@atend@scope=\pgfutil@empty% + \let\tikz@options=\pgfutil@empty% + \tikz@clear@rdf@options% + \let\tikz@mode=\pgfutil@empty% + \let\tikz@id@name=\pgfutil@empty% + \tikz@transparency@groupfalse% + \tikzset{every scope/.try}% + \tikz@collect@scope@anims\tikz@scope@opt% +}% +\def\tikz@scope@opt[#1]{% + \tikzset{#1}% + \tikz@options% + \tikz@do@rdf@pre@options% + \iftikz@transparency@group\expandafter\pgftransparencygroup\expandafter[\tikz@transparency@group@options]\tikz@blend@group\fi% + \tikz@is@nodefalse% + \tikz@call@id@hook% + \pgfidscope% + \tikz@do@rdf@post@options% + \begingroup% + \let\tikz@id@name\pgfutil@empty% + \expandafter\tikz@atbegin@scope% + \expandafter\pgfclearid% + \tikz@lib@scope@check% +}% +\def\endtikz@scope@env{% + \tikz@atend@scope% + \endgroup% + \endpgfidscope% + \iftikz@transparency@group\endpgftransparencygroup\fi% + \endgroup% + \endpgfscope% + \tikz@lib@scope@check% +}% + + +\def\tikz@scoped{\tikz@collect@scope@anims\tikz@scoped@opt}% +\def\tikz@scoped@opt[#1]{\scope[#1]\pgfutil@ifnextchar\bgroup{\tikz@scoped@}{\tikz@scoped@@single}}% +\def\tikz@scoped@{\bgroup\tikz@auto@end@pathtrue\aftergroup\endscope\let\pgf@temp=}% +\def\tikz@scoped@@single#1{% + \expandafter\ifx\csname tikz@scoped@protected@command\string#1\endcsname\relax% + \expandafter\tikz@scoped@@% + \else% + \begingroup\def\tikz@scoped@path@do@at@end{\endgroup\endscope}% + \fi% + #1% +}% +\def\tikz@scoped@@{% + \let\tikz@scoped@next=\tikz@scoped@collectnormalsemicolon% + \ifnum\the\catcode`\;=\active\relax% + \let\tikz@scoped@next=\tikz@scoped@collectactivesemicolon% + \fi% + \tikz@scoped@next}% +\def\tikz@scoped@collectnormalsemicolon#1;{#1;\endscope} +{ + \catcode`\;=\active + \gdef\tikz@scoped@collectactivesemicolon#1;{#1;\endscope} +}% + + +\def\tikzaddtikzonlycommandshortcutlet#1#2{% + \expandafter\def\expandafter\tikz@installcommands\expandafter{\tikz@installcommands + \let#1=#2% + }% +}% + +\def\tikzaddtikzonlycommandshortcutdef#1#2{% + \expandafter\def\expandafter\tikz@installcommands\expandafter{\tikz@installcommands + \def#1{#2}% + }% +}% + +\def\tikz@installcommands{% + \let\tikz@origscope=\scope% + \let\tikz@origscoped=\scoped% + \let\tikz@origendscope=\endscope% + \let\tikz@origstartscope=\startscope% + \let\tikz@origstopscope=\stopscope% + \let\tikz@origpath=\path% + \let\tikz@origagainpath=\againpath% + \let\tikz@origdraw=\draw% + \let\tikz@origpattern=\pattern% + \let\tikz@origfill=\fill% + \let\tikz@origfilldraw=\filldraw% + \let\tikz@origshade=\shade% + \let\tikz@origshadedraw=\shadedraw% + \let\tikz@origclip=\clip% + \let\tikz@origuseasboundingbox=\useasboundingbox% + \let\tikz@orignode=\node% + \let\tikz@origpic=\pic% + \let\tikz@origcoordinate=\coordinate% + \let\tikz@origmatrix=\matrix% + \let\tikz@origcalendar=\calendar% + \let\tikz@origdv=\datavisualization% + \let\tikz@origgraph=\graph% + \let\scope=\tikz@scope@env% + \let\scoped=\tikz@scoped% + \let\endscope=\endtikz@scope@env% + \let\startscope=\scope% + \let\stopscope=\endscope% + \let\path=\tikz@command@path% + \let\againpath=\tikz@command@againpath% + \def\draw{\path[draw]}% + \def\pattern{\path[pattern]}% + \def\fill{\path[fill]}% + \def\filldraw{\path[fill,draw]}% + \def\shade{\path[shade]}% + \def\shadedraw{\path[shade,draw]}% + \def\clip{\path[clip]}% + \def\graph{\path graph}% + \def\useasboundingbox{\path[use as bounding box]}% + \def\node{\tikz@path@overlay{node}}% + \def\pic{\tikz@path@overlay{pic}}% + \def\coordinate{\tikz@path@overlay{coordinate}}% + \def\matrix{\tikz@path@overlay{node[matrix]}}% + \def\calendar{\tikz@lib@cal@calendar}% + \def\datavisualization{\tikz@lib@datavisualization}% +}% +\ifx\tikz@lib@cal@calendar\@undefined +\def\tikz@lib@cal@calendar{\tikzerror{You need to say \string\usetikzlibrary{calendar} to use the \string\calendar{} command}}% +\fi +\ifx\tikz@lib@datavisualization\@undefined +\def\tikz@lib@datavisualization{\tikzerror{You need to say \string\usetikzlibrary{datavisualization} to use the \string\datavisualization{} command}}% +\fi + +\def\tikz@path@overlay#1{% + \let\tikz@signal@path=\tikz@signal@path% + \pgfutil@ifnextchar<{\tikz@path@overlayed{#1}}{\path #1}}% +\def\tikz@path@overlayed#1<#2>{\path<#2> #1}% + +\def\tikz@uninstallcommands{% + \let\scope=\tikz@origscope% + \let\scoped=\tikz@origscoped% + \let\endscope=\tikz@origendscope% + \let\startscope=\tikz@origstartscope% + \let\stopscope=\tikz@origstopscope% + \let\path=\tikz@origpath% + \let\againpath=\tikz@origagainpath% + \let\draw=\tikz@origdraw% + \let\pattern=\tikz@origpattern% + \let\fill=\tikz@origfill% + \let\filldraw=\tikz@origfilldraw% + \let\shade=\tikz@origshade% + \let\shadedraw=\tikz@origshadedraw% + \let\clip=\tikz@origclip% + \let\useasboundingbox=\tikz@origuseasboundingbox% + \let\node=\tikz@orignode% + \let\pic=\tikz@origpic% + \let\coordinate=\tikz@origcoordinate% + \let\matrix=\tikz@origmatrix% + \let\calendar=\tikz@origcalendar% + \let\datavisualization=\tikz@origdv% + \let\graph=\tikz@origgraph% +}% + + +{% + \catcode`\;=12 + \gdef\tikz@nonactivesemicolon{;}% + \catcode`\:=12 + \gdef\tikz@nonactivecolon{:}% + \catcode`\|=12 + \gdef\tikz@nonactivebar{|}% + \catcode`\!=12 + \gdef\tikz@nonactiveexlmark{!}% + \catcode`\;=\active + \catcode`\:=\active + \catcode`\|=\active + \catcode`\"=\active + \catcode`\!=\active + \gdef\tikz@activesemicolon{;}% + \gdef\tikz@activecolon{:}% + \gdef\tikz@activebar{|}% + \gdef\tikz@activequotes{"}% + \global\let\tikz@active@quotes@token="% + \gdef\tikz@activeexlmark{!}% + \gdef\tikz@deactivatthings{% + \def;{\tikz@nonactivesemicolon}% + \def:{\tikz@nonactivecolon}% + \def|{\tikz@nonactivebar}% + \def!{\tikz@nonactiveexlmark}% + }% +}% + +\let\tikz@orig@shorthands\pgfutil@empty +\def\tikz@switchoff@shorthands{% + \ifx\tikz@orig@shorthands\pgfutil@empty% + \edef\tikz@orig@shorthands{% + \catcode\noexpand`\noexpand\;\the\catcode`\;\relax% + \catcode\noexpand`\noexpand\:\the\catcode`\:\relax% + \catcode\noexpand`\noexpand\|\the\catcode`\|\relax% + \catcode\noexpand`\noexpand\!\the\catcode`\!\relax% + \catcode\noexpand`\noexpand\,\the\catcode`\,\relax% + \catcode\noexpand`\noexpand\<\the\catcode`\<\relax% + \catcode\noexpand`\noexpand\>\the\catcode`\>\relax% + \catcode\noexpand`\noexpand\"\the\catcode`\"\relax% + \catcode\noexpand`\noexpand\'\the\catcode`\'\relax% + \catcode\noexpand`\noexpand\-\the\catcode`\-\relax% + \catcode\noexpand`\noexpand\=\the\catcode`\=\relax% + \catcode\noexpand`\noexpand\.\the\catcode`\.\relax% + \catcode\noexpand`\noexpand\$\the\catcode`\$\relax% + }% + \catcode`\;12\relax% + \catcode`\:12\relax% + \catcode`\|12\relax% + \catcode`\!12\relax% + \catcode`\,12\relax% + \catcode`\<12\relax% + \catcode`\>12\relax% + \catcode`\"12\relax% + \catcode`\'12\relax% + \catcode`\-12\relax% + \catcode`\=12\relax% + \catcode`\.12\relax% + \catcode`\$3\relax% + \fi% +}% + + + + +\def\tikz@command@path{% + \let\tikz@signal@path=\tikz@signal@path% + \pgfutil@ifnextchar[{\tikz@check@earg}% + {\pgfutil@ifnextchar<{\tikz@doopt}{\tikz@@command@path}}}% +\pgfutil@protected\def\tikz@signal@path{\tikz@signal@path}% +\def\tikz@check@earg[#1]{% + \pgfutil@ifnextchar<{\tikz@swap@args[#1]}{\tikz@@command@path[#1]}} +\def\tikz@swap@args[#1]<#2>{\tikz@command@path<#2>[#1]}% + +\def\tikz@doopt{% + \let\tikz@next=\tikz@eargnormalsemicolon% + \ifnum\the\catcode`\;=\active\relax% + \let\tikz@next=\tikz@eargactivesemicolon% + \fi% + \tikz@next}% +\long\def\tikz@eargnormalsemicolon<#1>#2;{\alt<#1>{\tikz@@command@path#2;}{\tikz@path@do@at@end}}% +{ + \catcode`\;=\active + \long\global\def\tikz@eargactivesemicolon<#1>#2;{\alt<#1>{\tikz@@command@path#2;}{\tikz@path@do@at@end}}% +} + +\def\tikz@@command@path{% + \edef\tikzscope@linewidth{\the\pgflinewidth}% + \begingroup% + \setbox\tikz@figbox=\box\pgfutil@voidb@x% + \setbox\tikz@figbox@bg=\box\pgfutil@voidb@x% + \let\tikz@path@do@at@end=\tikz@lib@scope@check% + \let\tikz@options=\pgfutil@empty% + \tikz@clear@rdf@options% + \let\tikz@mode=\pgfutil@empty% + \let\tikz@moveto@waiting=\relax% + \let\tikz@timer=\relax% + \let\tikz@tangent=\relax% + \let\tikz@collected@onpath=\pgfutil@empty% + \let\tikz@preactions=\pgfutil@empty% + \let\tikz@postactions=\pgfutil@empty% + \tikz@snakedfalse% + \tikz@decoratepathfalse% + \tikz@node@is@a@labelfalse% + \tikz@resetexpandcount + \pgf@path@lastx=0pt% + \pgf@path@lasty=0pt% + \tikz@lastx=0pt% + \tikz@lasty=0pt% + \tikz@lastxsaved=0pt% + \tikz@lastysaved=0pt% + \tikzset{every path/.try}% + \tikz@scan@next@command% +}% +\def\tikz@scan@next@command{% + \ifx\tikz@collected@onpath\pgfutil@empty% + \else% + \tikz@invoke@collected@onpath% + \fi% + \afterassignment\tikz@handle\let\pgf@let@token=% +}% +\newcount\tikz@expandcount +\def\tikz@resetexpandcount{\tikz@expandcount=100\relax} +\let\tikz@collected@onpath=\pgfutil@empty% + +\edef\tikz@frozen@relax@token{\ifnum0=0\fi} + +\def\tikz@handle{% + \pgfutil@switch\pgfutil@ifx\pgf@let@token{% + {(}{\let\pgfutil@next\tikz@movetoabs}% + {+}{\let\pgfutil@next\tikz@movetorel}% + {-}{\let\pgfutil@next\tikz@lineto}% + {.}{\let\pgfutil@next\tikz@dot}% + {r}{\let\pgfutil@next\tikz@rect}% + {n}{\let\pgfutil@next\tikz@fig}% + {[}{\let\pgfutil@next\tikz@parse@options}% + {c}{\let\pgfutil@next\tikz@cchar}% + {\bgroup}{\let\pgfutil@next\tikz@beginscope}% + {\egroup}{\let\pgfutil@next\tikz@endscope}% + {;}{\let\pgfutil@next\tikz@finish}% + {a}{\let\pgfutil@next\tikz@a@char}% + {e}{\let\pgfutil@next\tikz@e@char}% + {g}{\let\pgfutil@next\tikz@g@char}% + {s}{\let\pgfutil@next\tikz@schar}% + {|}{\let\pgfutil@next\tikz@vh@lineto}% + {p}{\pgfsetmovetofirstplotpoint\let\pgfutil@next\tikz@pchar}% + {t}{\let\pgfutil@next\tikz@to}% + {\pgfextra}{\let\pgfutil@next\tikz@extra}% + {\foreach}{\let\pgfutil@next\tikz@foreach}% + {f}{\let\pgfutil@next\tikz@fchar}% + {\pgf@stop}{\let\pgfutil@next\relax}% + {\par}{\let\pgfutil@next\tikz@scan@next@command}% + {d}{\let\pgfutil@next\tikz@decoration}% + {l}{\let\pgfutil@next\tikz@l@char}% + {:}{\let\pgfutil@next\tikz@colon@char}% + {\relax}{\relax\let\pgfutil@next\tikz@scan@next@command}% + }{\tikz@resetexpandcount\pgfutil@next}{\tikz@expand}% +}% + +\def\tikz@l@char{% + \pgfutil@ifnextchar e{\tikz@let@command}{% + \pgfutil@ifnextchar i{\tikz@lsystem}{% + \pgfutil@ifnextchar-{\tikz@@lsystem}{\tikz@expand}% + }% + }% +}% + +\def\tikz@lsystem{% + \tikzerror{You need to say \string\usetikzlibrary{lindenmayersystems} to draw Lindenmayer systems} +}% + +\def\tikz@@lsystem{% + \tikzerror{You need to say \string\usetikzlibrary{lindenmayersystems} to draw L-systems} +}% + +\def\tikz@pchar{\pgfutil@ifnextchar l{\tikz@plot}{\pgfutil@ifnextchar i{\tikz@subpicture}{\tikz@parabola}}}% +\def\tikz@cchar{% + \pgfutil@ifnextchar i{\tikz@circle}% + {\pgfutil@ifnextchar h{\tikz@children}{\tikz@cochar}}}% +\def\tikz@cochar o{% + \pgfutil@ifnextchar o{\tikz@coordinate}{\tikz@cosine}}% +\def\tikz@e@char{% + \pgfutil@ifnextchar l{\tikz@ellipse}{\tikz@@e@char}}% +\def\tikz@a@char{% + \pgfutil@ifnextchar r{\tikz@arcA}{\tikzerror{Arc expected}}}% +\def\tikz@@e@char dge{% + \pgfutil@ifnextchar f{\tikz@edgetoparent}{\tikz@edge@plain}}% + +\def\tikz@schar{\pgfutil@ifnextchar i{\tikz@sine}{\tikz@svg@path}}% + +\def\tikz@g@char r{\pgfutil@ifnextchar i{\tikz@grid}{\tikz@graph}}% + + +\def\tikz@svg@path{% + \tikzerror{You need to say \string\usetikzlibrary{svg.path} to use the svg path command} +}% + + +\def\tikz@finish{% + \box\tikz@figbox@bg% + \iftikz@decoratepath% + \tikz@lib@dec@decorate@path% + \fi% + \pgfsyssoftpath@getcurrentpath\tikz@actions@path% + \edef\tikz@restorepathsize{% + \global\pgf@pathmaxx=\the\pgf@pathmaxx% + \global\pgf@pathmaxy=\the\pgf@pathmaxy% + \global\pgf@pathminx=\the\pgf@pathminx% + \global\pgf@pathminy=\the\pgf@pathminy% + }% + \tikz@preactions% + \let\tikz@path@picture=\pgfutil@empty% + \tikz@mode@fillfalse% + \tikz@mode@drawfalse% + \tikz@mode@doublefalse% + \tikz@mode@clipfalse% + \tikz@mode@boundaryfalse% + \tikz@mode@fade@pathfalse% + \tikz@mode@fade@scopefalse% + \edef\tikz@pathextend{% + {\noexpand\pgfqpoint{\the\pgf@pathminx}{\the\pgf@pathminy}}% + {\noexpand\pgfqpoint{\the\pgf@pathmaxx}{\the\pgf@pathmaxy}}% + }% + \tikz@mode% + \iftikz@mode@fade@path% + \tikz@addoption{% + \iftikz@fade@adjust% + \iftikz@mode@draw% + \pgfsetfadingforcurrentpathstroked{\tikz@path@fading}{\tikz@do@fade@transform}% + \else% + \pgfsetfadingforcurrentpath{\tikz@path@fading}{\tikz@do@fade@transform}% + \fi% + \else% + \pgfsetfading{\tikz@path@fading}{\tikz@do@fade@transform}% + \fi% + \tikz@mode@fade@pathfalse% + }% + \fi% + \iftikz@mode@fade@scope% + \iftikz@fade@adjust% + \iftikz@mode@draw% + \pgfsetfadingforcurrentpathstroked{\tikz@scope@fading}{\tikz@do@fade@transform}% + \else% + \pgfsetfadingforcurrentpath{\tikz@scope@fading}{\tikz@do@fade@transform}% + \fi% + \else% + \pgfsetfading{\tikz@scope@fading}{\tikz@do@fade@transform}% + \fi% + \tikz@mode@fade@scopefalse% + \fi% + \ifx\tikz@options\pgfutil@empty% + \else% + \pgfsys@beginscope% + \let\pgfscope@stroke@color=\pgf@strokecolor@global% + \let\pgfscope@fill@color=\pgf@fillcolor@global% + \begingroup% + \tikz@options% + \fi% + \tikz@do@rdf@pre@options% + \tikz@is@nodefalse% + \tikz@call@id@hook% + \iftikz@mode@clip\else% + \pgfidscope% + \tikz@do@rdf@post@options% + \begingroup% + \fi% + \iftikz@mode@fill% + \iftikz@mode@shade% + \pgfsyssoftpath@getcurrentpath\tikz@temppath + \pgfprocessround{\tikz@temppath}{\tikz@temppath}% + \pgfsyssoftpath@setcurrentpath\tikz@temppath% + \pgfsyssoftpath@invokecurrentpath% + \pgfpushtype% + \pgfusetype{.path fill}% + \pgfsys@fill% + \pgfpoptype% + \tikz@mode@fillfalse% + \else% + \ifx\tikz@path@picture\pgfutil@empty% + \else% + \pgfsyssoftpath@getcurrentpath\tikz@temppath + \pgfprocessround{\tikz@temppath}{\tikz@temppath}% + \pgfsyssoftpath@setcurrentpath\tikz@temppath% + \pgfsyssoftpath@invokecurrentpath% + \pgfpushtype% + \pgfusetype{.path fill}% + \pgfsys@fill% + \pgfpoptype% + \tikz@mode@fillfalse% + \fi% + \fi% + \fi% + \iftikz@mode@shade% + \pgfsyssoftpath@getcurrentpath\tikz@temppath + \pgfprocessround{\tikz@temppath}{\tikz@temppath}% + \pgfsyssoftpath@setcurrentpath\tikz@temppath% + \pgfpushtype% + \pgfusetype{.path shade}% + \pgfshadepath{\tikz@shading}{\tikz@shade@angle}% + \pgfpoptype% + \tikz@mode@shadefalse% + \fi% + \ifx\tikz@path@picture\pgfutil@empty% + \else% + \begingroup% + \pgfusetype{.path picture}% + \pgfidscope% + \pgfsys@beginscope% + \let\tikz@id@name\pgfutil@empty% + \pgfclearid% + \pgfsyssoftpath@getcurrentpath\tikz@temppath + \pgfprocessround{\tikz@temppath}{\tikz@temppath}% + \pgfsyssoftpath@setcurrentpath\tikz@temppath% + \pgfsyssoftpath@invokecurrentpath% + \pgfsys@clipnext% + \pgfsys@discardpath% + \pgf@relevantforpicturesizefalse% + \expandafter\def\csname pgf@sh@ns@path picture bounding box\endcsname{rectangle} + \expandafter\edef\csname pgf@sh@np@path picture bounding box\endcsname{% + \def\noexpand\southwest{\noexpand\pgfqpoint{\the\pgf@pathminx}{\the\pgf@pathminy}}% + \def\noexpand\northeast{\noexpand\pgfqpoint{\the\pgf@pathmaxx}{\the\pgf@pathmaxy}}% + } + \expandafter\def\csname pgf@sh@nt@path picture bounding box\endcsname{{1}{0}{0}{1}{0pt}{0pt}} + \expandafter\def\csname pgf@sh@pi@path picture bounding box\endcsname{\pgfpictureid} + \pgfinterruptpath% + \tikz@path@picture% + \endpgfinterruptpath% + \pgfsys@endscope% + \endpgfidscope% + \endgroup% + \let\tikz@path@picture=\pgfutil@empty% + \fi% + \iftikz@mode@draw% + \iftikz@mode@double% + \begingroup% + \pgfsys@beginscope% + \tikz@double@setup% + \fi% + \fi% + \pgfpushtype% + \edef\tikz@temp{\noexpand\pgfusepath{% + \iftikz@mode@fill fill,\fi% + \iftikz@mode@draw draw,\fi% + \iftikz@mode@clip clip\fi% + }}% + \pgfusetype{.path}% + \tikz@temp% + \pgfpoptype% + \tikz@mode@fillfalse% + \iftikz@mode@draw% + \iftikz@mode@double% + \pgfsys@endscope% + \endgroup% + \fi% + \fi% + \tikz@mode@drawfalse% + \tikz@postactions% + \box\tikz@figbox% + \iftikz@mode@clip\else\endgroup\endpgfidscope\fi% + \ifx\tikz@options\pgfutil@empty% + \else% + \endgroup% + \global\let\pgf@strokecolor@global=\pgfscope@stroke@color% + \global\let\pgf@fillcolor@global=\pgfscope@fill@color% + \pgfsys@endscope% + \iftikz@mode@clip% + \tikzerror{Extra options not allowed for clipping path command.}% + \fi% + \fi% + \iftikz@mode@clip% + \aftergroup\pgf@relevantforpicturesizefalse% + \fi% + \iftikz@mode@boundary% + \aftergroup\pgf@relevantforpicturesizefalse% + \fi% + \endgroup% + \global\pgflinewidth=\tikzscope@linewidth% + \tikz@path@do@at@end% +}% +\let\tikz@lib@scope@check\pgfutil@empty% +\def\tikz@path@do@at@end{\tikz@lib@scope@check}% +\def\tikz@@pathtext{@path}% + +\def\pgf@outer@auto@adjust@hook{% + {% + \tikz@mode@drawfalse% + \tikz@mode% + \expandafter% + }% + \iftikz@mode@draw\else% + \pgfkeyslet{/pgf/outer xsep}\pgf@zero@text + \pgfkeyslet{/pgf/outer ysep}\pgf@zero@text + \fi% +}% + + +\def\tikz@extra@preaction#1{% + {% + \pgfsys@beginscope% + \setbox\tikz@figbox=\box\pgfutil@voidb@x% + \setbox\tikz@figbox@bg=\box\pgfutil@voidb@x% + \path[#1];% + \pgfsyssoftpath@setcurrentpath\tikz@actions@path% + \tikz@restorepathsize% + \pgfsys@endscope% + }% +}% + +\def\tikz@extra@postaction#1{% + {% + \pgfsys@beginscope% + \setbox\tikz@figbox=\box\pgfutil@voidb@x% + \setbox\tikz@figbox@bg=\box\pgfutil@voidb@x% + \tikz@restorepathsize% + \path[#1]\pgfextra{\pgfsyssoftpath@setcurrentpath\tikz@actions@path};% + \pgf@resetpathsizes% + \pgfsys@endscope% + }% +}% + + + +\def\tikz@skip#1{\tikz@scan@next@command#1}% +\def\tikz@expand{% + \advance\tikz@expandcount by -1 + \ifnum\tikz@expandcount<0\relax% + \expandafter\pgfutil@firstoftwo + \else + \expandafter\pgfutil@secondoftwo + \fi + {% + \tikzerror{Giving up on this path. Did you forget a semicolon?}% + \global\let\tikz@expand@last@token=\pgf@let@token + \tikz@finish% + \expandafter\let\expandafter\tikz@expand@last@token@\csname tikz@expand@last@token\endcsname + \global\let\tikz@expand@last@token=\relax + \tikz@expand@last@token@ + }{% + \tikz@@expand + }% +} + +\def\tikz@@expand{% + \expandafter\tikz@scan@next@command\pgf@let@token}% + + + + +\newif\iftikz@auto@end@path + +\def\tikz@beginscope{\begingroup\tikz@auto@end@pathfalse\tikz@scan@next@command}% +\def\tikz@endscope{% + \iftikz@auto@end@path\expandafter\tikz@finish\expandafter\egroup\else\expandafter\tikz@@endscope\fi% +}% +\def\tikz@@endscope{% + \global\setbox\tikz@tempbox=\box\tikz@figbox% + \global\setbox\tikz@tempbox@bg=\box\tikz@figbox@bg% + \global\let\tikz@tangent@temp\tikz@tangent% + \edef\tikz@marshal{% + \tikz@lastx=\the\tikz@lastx% + \tikz@lasty=\the\tikz@lasty% + \iftikz@current@point@local% + \else% + \tikz@lastxsaved=\the\tikz@lastxsaved% + \tikz@lastysaved=\the\tikz@lastysaved% + \ifx\tikz@moveto@waiting\relax% + \let\tikz@moveto@waiting\relax% + \else% + \def\noexpand\tikz@moveto@waiting{\tikz@moveto@waiting}% + \fi% + \iftikz@shapeborder% + \noexpand\tikz@shapebordertrue% + \def\noexpand\tikz@shapeborder@name{\tikz@shapeborder@name}% + \else% + \noexpand\tikz@shapeborderfalse% + \fi% + \fi% + }% + \expandafter% + \endgroup\tikz@marshal% + \setbox\tikz@figbox=\box\tikz@tempbox% + \setbox\tikz@figbox@bg=\box\tikz@tempbox@bg% + \let\tikz@tangent\tikz@tangent@temp% + \tikz@scan@next@command}% + + + +\def\tikz@extra{\pgfutil@ifnextchar\bgroup\tikz@@extra\relax}% +\long\def\tikz@@extra#1{#1\tikz@scan@next@command}% +\let\endpgfextra=\tikz@scan@next@command + +\def\pgfextra{pgfextra}% + + + +\def\tikz@fchar oreach{\tikz@foreach}% + +\def\tikz@foreach{% + \def\pgffor@beginhook{% + \tikz@lastx=\tikz@foreach@save@lastx% + \tikz@lasty=\tikz@foreach@save@lasty% + \tikz@lastxsaved=\tikz@foreach@save@lastxsaved% + \tikz@lastysaved=\tikz@foreach@save@lastysaved% + \setbox\tikz@figbox=\box\tikz@tempbox% + \setbox\tikz@figbox@bg=\box\tikz@tempbox@bg% + \expandafter\tikz@scan@next@command\pgfutil@firstofone}% + \def\pgffor@endhook{\pgfextra{% + \xdef\tikz@foreach@save@lastx{\the\tikz@lastx}% + \xdef\tikz@foreach@save@lasty{\the\tikz@lasty}% + \xdef\tikz@foreach@save@lastxsaved{\the\tikz@lastxsaved}% + \xdef\tikz@foreach@save@lastysaved{\the\tikz@lastysaved}% + \global\setbox\tikz@tempbox=\box\tikz@figbox% + \global\setbox\tikz@tempbox@bg=\box\tikz@figbox@bg% + \pgfutil@gobble}}% + \def\pgffor@afterhook{% + \tikz@lastx=\tikz@foreach@save@lastx% + \tikz@lasty=\tikz@foreach@save@lasty% + \tikz@lastxsaved=\tikz@foreach@save@lastxsaved% + \tikz@lastysaved=\tikz@foreach@save@lastysaved% + \let\pgffor@beginhook\relax% + \let\pgffor@endhook\relax% + \let\pgffor@afterhook\relax% + \setbox\tikz@figbox=\box\tikz@tempbox% + \setbox\tikz@figbox@bg=\box\tikz@tempbox@bg% + \tikz@scan@next@command}% + \global\setbox\tikz@tempbox=\box\tikz@figbox% + \global\setbox\tikz@tempbox@bg=\box\tikz@figbox@bg% + \xdef\tikz@foreach@save@lastx{\the\tikz@lastx}% + \xdef\tikz@foreach@save@lasty{\the\tikz@lasty}% + \xdef\tikz@foreach@save@lastxsaved{\the\tikz@lastxsaved}% + \xdef\tikz@foreach@save@lastysaved{\the\tikz@lastysaved}% + \foreach}% + + + +\def\tikz@command@againpath#1{% + \pgfextra{% + \pgfsyssoftpath@getcurrentpath\tikz@temp% + \expandafter\pgfutil@g@addto@macro\expandafter\tikz@temp\expandafter{#1}% + \pgfsyssoftpath@setcurrentpath\tikz@temp% + }% +}% + + + +\def\tikz@colon@char#1=#2{% + \tikz@scan@next@command{[animate={myself:{#1}={#2}}]}% +}% + + + + +\newif\iftikz@shapeborder + + +\def\tikz@movetoabs{\tikz@moveto(}% +\def\tikz@movetorel{\tikz@moveto+}% +\def\tikz@moveto{% + \tikz@scan@one@point{\tikz@@moveto}}% +\def\tikz@@moveto#1{% + \tikz@make@last@position{#1}% + \iftikz@shapeborder% + \edef\tikz@moveto@waiting{\tikz@shapeborder@name}% + \else% + \tikz@@movetosave{\tikz@last@position}% + \let\tikz@moveto@waiting=\relax% + \fi% + \tikz@scan@next@command% +}% + +\def\tikz@@movetosave#1{% + {\pgftransformreset + \pgf@process{#1}% + \xdef\tikz@marshal{% + \tikz@lastmovetox=\the\pgf@x\relax% + \tikz@lastmovetoy=\the\pgf@y\relax% + }% + }% + \tikz@marshal + \pgfpathmoveto{#1}% +}% + + +\let\tikz@moveto@waiting=\relax % + +\def\tikz@flush@moveto{% + \ifx\tikz@moveto@waiting\relax% + \else% + \tikz@@movetosave{\tikz@last@position}% + \fi% + \let\tikz@moveto@waiting=\relax% +}% + + +\def\tikz@flush@moveto@toward#1#2#3{% + \ifx\tikz@moveto@waiting\relax% + \else% + \pgf@process{\pgfpointshapeborder{\tikz@moveto@waiting}{#1}}% + #2=\pgf@x% + #3=\pgf@y% + \edef\tikz@timer@start{\noexpand\pgfqpoint{\the\pgf@x}{\the\pgf@y}}% + \tikz@@movetosave{\pgfqpoint{\pgf@x}{\pgf@y}}% + \fi% + \let\tikz@moveto@waiting=\relax% +}% + + + +\def\tikz@collect@coordinate@onpath#1c{% + \pgfutil@ifnextchar y{\tikz@cycle@expander@add#1}{\tikz@collect@coordinate@onpath@{#1}}}% +\def\tikz@collect@coordinate@onpath@#1oordinate{% + \pgfutil@ifnextchar[{\tikz@@collect@coordinate@opt#1}{\tikz@@collect@coordinate@opt#1[]}}% +\def\tikz@@collect@coordinate@opt#1[#2]{% + \pgfutil@ifnextchar({\tikz@@collect@coordinate#1[#2]}{% + \tikz@collect@label@onpath#1node[shape=coordinate,#2]{}}}% +\def\tikz@@collect@coordinate#1[#2](#3){% + \tikz@collect@label@onpath#1node[shape=coordinate,#2](#3){}}% + +\newif\iftikz@collect@pic + +\def\tikz@collect@label@onpath#1node{% + \expandafter\def\expandafter\tikz@collected@onpath\expandafter{\tikz@collected@onpath node}% + \let\tikz@collect@cont#1% + \tikz@collect@picfalse% + \tikz@collect@label@scan}% + +\def\tikz@collect@pic@onpath#1pic{% + \expandafter\def\expandafter\tikz@collected@onpath\expandafter{\tikz@collected@onpath pic}% + \let\tikz@collect@cont#1 + \tikz@collect@pictrue% + \tikz@collect@label@scan}% + +\def\tikz@collect@label@scan{% + \pgfutil@ifnextchar f{\tikz@collect@nodes}{% + \pgfutil@ifnextchar({\tikz@collect@paran}% + {\pgfutil@ifnextchar[{\tikz@collect@options}% + {\pgfutil@ifnextchar:{\tikz@collect@animation}% + {\pgfutil@ifnextchar\bgroup{\tikz@collect@arg}% + {\tikz@collect@cont}}}}}% +}% + +\def\tikz@collect@nodes foreach#1in{% + \expandafter\def\expandafter\tikz@collected@onpath\expandafter{\tikz@collected@onpath foreach#1in}% + \pgfutil@ifnextchar\bgroup\tikz@collect@nodes@group\tikz@collect@nodes@one% +}% +\def\tikz@collect@nodes@one#1{% + \expandafter\def\expandafter\tikz@collected@onpath\expandafter{\tikz@collected@onpath #1}% + \tikz@collect@label@scan% +}% +\def\tikz@collect@nodes@group#1{% + \expandafter\def\expandafter\tikz@collected@onpath\expandafter{\tikz@collected@onpath{#1}}% + \tikz@collect@label@scan% +}% + +\def\tikz@collect@animation#1=#2{% + \expandafter\def\expandafter\tikz@collected@onpath\expandafter{\tikz@collected@onpath#1={#2}}% + \tikz@collect@label@scan% +}% +\def\tikz@collect@paran#1){% + \expandafter\def\expandafter\tikz@collected@onpath\expandafter{\tikz@collected@onpath#1)}% + \tikz@collect@label@scan% +}% +\def\tikz@collect@options#1]{% + \expandafter\def\expandafter\tikz@collected@onpath\expandafter{\tikz@collected@onpath#1]}% + \tikz@collect@label@scan% +}% +\def\tikz@collect@arg#1{% + \iftikz@handle@active@nodes% + \iftikz@collect@pic% + \expandafter\def\expandafter\tikz@collected@onpath\expandafter{\tikz@collected@onpath{#1}}% + \else% + \expandafter\def\expandafter\tikz@collected@onpath\expandafter{\tikz@collected@onpath{\scantokens{#1}}}% + \fi% + \else% + \expandafter\def\expandafter\tikz@collected@onpath\expandafter{\tikz@collected@onpath{#1}}% + \fi% + \tikz@collect@cont% +}% + +\def\tikz@invoke@collected@onpath{% + \tikz@node@is@a@labeltrue% + \let\tikz@temp=\tikz@collected@onpath% + \let\tikz@collected@onpath=\pgfutil@empty% + \expandafter\tikz@scan@next@command\tikz@temp\pgf@stop% + \tikz@node@is@a@labelfalse% +}% + + + +\def\tikz@cycle@expander#1{\pgfutil@ifnextchar c{\tikz@cycle@expander@{#1}}{#1}}% +\def\tikz@cycle@expander@#1c{\pgfutil@ifnextchar y{\tikz@cycle@expander@add{#1}}{#1c}}% +\def\tikz@cycle@expander@add#1ycle{#1(current subpath start)--cycle}% + + + + + +\def\tikz@lineto{% + \pgfutil@ifnextchar |% + {\expandafter\tikz@hv@lineto\pgfutil@gobble}% + {\expandafter\pgfutil@ifnextchar\tikz@activebar{\expandafter\tikz@hv@lineto\pgfutil@gobble}% + {\expandafter\tikz@lineto@mid\pgfutil@gobble}}}% +\def\tikz@lineto@mid{% + \pgfutil@ifnextchar n{\tikz@collect@label@onpath\tikz@lineto@mid}% + {% + \pgfutil@ifnextchar c{\tikz@close}{% + \pgfutil@ifnextchar p{\tikz@lineto@plot@or@pic}{\tikz@scan@one@point{\tikz@@lineto}}}}}% +\def\tikz@lineto@plot@or@pic p{% + \pgfutil@ifnextchar i{\tikz@collect@pic@onpath\tikz@lineto@mid p}{% + \pgfsetlinetofirstplotpoint\tikz@plot}% +}% +\def\tikz@@lineto#1{% + \edef\tikz@timer@start{\noexpand\pgfqpoint{\the\tikz@lastx}{\the\tikz@lasty}} + \iftikz@shapeborder% + \pgf@process{\pgfpointshapeborder{\tikz@shapeborder@name}{\tikz@last@position}}% + \tikz@make@last@position{\pgfqpoint{\pgf@x}{\pgf@y}}% + \tikz@flush@moveto@toward{\tikz@last@position}\pgf@x\pgf@y% + \tikz@path@lineto{\tikz@last@position}% + \edef\tikz@timer@end{\noexpand\pgfqpoint{\the\tikz@lastx}{\the\tikz@lasty}}% + \tikz@make@last@position{#1}% + \edef\tikz@moveto@waiting{\tikz@shapeborder@name}% + \else% + \tikz@make@last@position{#1}% + \tikz@flush@moveto@toward{\tikz@last@position}\pgf@x\pgf@y% + \tikz@path@lineto{\tikz@last@position}% + \edef\tikz@timer@end{\noexpand\pgfqpoint{\the\tikz@lastx}{\the\tikz@lasty}}% + \fi% + \let\tikz@timer=\tikz@timer@line% + \let\tikz@tangent\tikz@timer@start% + \tikz@scan@next@command% +}% + +\def\tikz@path@lineto#1{% + \iftikz@snaked% + { + \pgfsyssoftpathmovetorelevantfalse% + \pgfpathsnakesto{\tikz@presnake,{\tikz@snake}{\tikz@mainsnakelength}{\noexpand\tikz@snake@install@trans}{},\tikz@postsnake}{#1}% + } + \else% + \pgfpathlineto{#1}% + \fi% +}% + +\def\tikz@path@close#1{% + \iftikz@snaked% + {% + \pgftransformreset% + \pgfpathsnakesto{\tikz@presnake,{\tikz@snake}{\tikz@mainsnakelength}{\noexpand\tikz@snake@install@trans}{},\tikz@postsnake}{#1}% + }% + \fi% + \pgfpathclose% +}% + + + +\def\tikz@hv@lineto{% + \pgfutil@ifnextchar n{\tikz@collect@label@onpath\tikz@hv@lineto}{% + \pgfutil@ifnextchar p{\tikz@collect@pic@onpath\tikz@hv@lineto}% + {\pgfutil@ifnextchar c{\tikz@collect@coordinate@onpath\tikz@hv@lineto}% + {\tikz@scan@one@point{\tikz@@hv@lineto}}}}}% +\def\tikz@@hv@lineto#1{% + \edef\tikz@timer@start{\noexpand\pgfqpoint{\the\tikz@lastx}{\the\tikz@lasty}}% + \pgf@yc=\tikz@lasty% + \tikz@make@last@position{#1}% + \edef\tikz@tangent{\noexpand\pgfqpoint{\the\tikz@lastx}{\the\pgf@yc}}% + \tikz@flush@moveto@toward{\pgfqpoint{\tikz@lastx}{\pgf@yc}}\pgf@x\pgf@yc% + \iftikz@shapeborder% + {% + \pgf@process{\pgfpointshapeborder{\tikz@shapeborder@name}{\pgfqpoint{\tikz@lastx}{\pgf@yc}}}% + \tikz@make@last@position{\pgfqpoint{\pgf@x}{\pgf@y}}% + \tikz@path@lineto{\pgfqpoint{\tikz@lastx}{\pgf@yc}}% + \tikz@path@lineto{\tikz@last@position}% + \xdef\tikz@timer@end@temp{\noexpand\pgfqpoint{\the\tikz@lastx}{\the\tikz@lasty}}% + }% + \let\tikz@timer@end=\tikz@timer@end@temp% + \edef\tikz@moveto@waiting{\tikz@shapeborder@name}% + \else% + \tikz@path@lineto{\pgfqpoint{\tikz@lastx}{\pgf@yc}}% + \tikz@path@lineto{\tikz@last@position}% + \edef\tikz@timer@end{\noexpand\pgfqpoint{\the\tikz@lastx}{\the\tikz@lasty}}% + \fi% + \let\tikz@timer=\tikz@timer@hvline% + \tikz@scan@next@command% +}% + + +\def\tikz@vh@lineto-{\tikz@vh@lineto@next}% +\def\tikz@vh@lineto@next{% + \pgfutil@ifnextchar n{\tikz@collect@label@onpath\tikz@vh@lineto@next}{% + \pgfutil@ifnextchar p{\tikz@collect@pic@onpath\tikz@vh@lineto@next}% + {\pgfutil@ifnextchar c{\tikz@collect@coordinate@onpath\tikz@vh@lineto@next}% + {\tikz@scan@one@point\tikz@@vh@lineto}}}}% +\def\tikz@@vh@lineto#1{% + \edef\tikz@timer@start{\noexpand\pgfqpoint{\the\tikz@lastx}{\the\tikz@lasty}}% + \pgf@xc=\tikz@lastx% + \tikz@make@last@position{#1}% + \edef\tikz@tangent{\noexpand\pgfqpoint{\the\pgf@xc}{\the\tikz@lasty}}% + \tikz@flush@moveto@toward{\pgfqpoint{\pgf@xc}{\tikz@lasty}}\pgf@xc\pgf@y% + \iftikz@shapeborder% + {% + \pgf@process{\pgfpointshapeborder{\tikz@shapeborder@name}{\pgfqpoint{\pgf@xc}{\tikz@lasty}}}% + \tikz@make@last@position{\pgfqpoint{\pgf@x}{\pgf@y}}% + \tikz@path@lineto{\pgfqpoint{\pgf@xc}{\tikz@lasty}}% + \tikz@path@lineto{\tikz@last@position}% + \xdef\tikz@timer@end@temp{\noexpand\pgfqpoint{\the\tikz@lastx}{\the\tikz@lasty}}% + }% + \let\tikz@timer@end=\tikz@timer@end@temp% + \edef\tikz@moveto@waiting{\tikz@shapeborder@name}% + \else% + \tikz@path@lineto{\pgfqpoint{\pgf@xc}{\tikz@lasty}}% + \tikz@path@lineto{\tikz@last@position}% + \edef\tikz@timer@end{\noexpand\pgfqpoint{\the\tikz@lastx}{\the\tikz@lasty}}% + \fi% + \let\tikz@timer=\tikz@timer@vhline% + \tikz@scan@next@command% +}% + +\def\tikz@close c{% + \pgfutil@ifnextchar o{\tikz@collect@coordinate@onpath\tikz@lineto@mid c}% + {\tikz@@close c}}% +\def\tikz@@close cycle{% + \tikz@flush@moveto% + \edef\tikz@timer@start{\noexpand\pgfqpoint{\the\tikz@lastx}{\the\tikz@lasty}} + \tikz@make@last@position{\expandafter\pgfpoint\pgfsyssoftpath@lastmoveto}% + \tikz@path@close{\expandafter\pgfpoint\pgfsyssoftpath@lastmoveto}% + \def\pgfstrokehook{}% + \edef\tikz@timer@end{% + \noexpand\pgfqpoint{\the\tikz@lastmovetox}{\the\tikz@lastmovetoy}}% + \let\tikz@timer=\tikz@timer@line% + \let\tikz@tangent\tikz@timer@start% + \tikz@scan@next@command% +}% + + +\def\tikz@parse@options#1]{% + \tikzset{#1}% + \tikz@scan@next@command% +}% + +\def\tikz@edge@plain{% + \begingroup% + \ifx\tikz@to@use@whom\pgfutil@undefined\else\tikz@to@use@whom\fi + \let\tikz@to@or@edge@function=\tikz@do@edge% + \let\tikz@@to@local@options\pgfutil@empty% + \let\tikz@collected@onpath=\pgfutil@empty% + \tikz@to@or@edge}% + +\def\tikz@to o{% + \tikz@to@use@last@coordinate% + \let\tikz@to@or@edge@function=\tikz@do@to% + \let\tikz@@to@local@options\pgfutil@empty% + \let\tikz@collected@onpath=\pgfutil@empty% + \tikz@to@or@edge}% + +\def\tikz@to@or@edge{% + \pgfutil@ifnextchar[{\tikz@to@or@edge@option}{% + \pgfutil@ifnextchar:{\tikz@to@or@edge@animation}{% + \tikz@@to@collect}}% +}% +\def\tikz@to@or@edge@option[#1]{% + \expandafter\def\expandafter\tikz@@to@local@options\expandafter{\tikz@@to@local@options,#1}% + \tikz@to@or@edge% +}% +\def\tikz@to@or@edge@animation:#1=#2{% + \expandafter\def\expandafter\tikz@@to@local@options\expandafter{\tikz@@to@local@options,% + animate={myself:{#1}={#2}}}% + \tikz@to@or@edge% +}% +\def\tikz@@to@collect{% + \pgfutil@ifnextchar(\tikz@@to@or@edge@coordinate% + {\pgfutil@ifnextchar n{\tikz@collect@label@onpath\tikz@@to@collect}% + {\pgfutil@ifnextchar p{\tikz@collect@pic@onpath\tikz@@to@collect}% + {\pgfutil@ifnextchar c{\tikz@collect@coordinate@onpath\tikz@@to@collect}% + {\pgfutil@ifnextchar +{\tikz@scan@one@point\tikz@@to@or@edge@math}% + {\tikzerror{(, +, coordinate, pic, or node expected}% + \tikz@@to@or@edge@coordinate()}}}}}% +}% + +\def\tikz@@to@or@edge@coordinate({% + \pgfutil@ifnextchar${% + \tikz@scan@one@point\tikz@@to@or@edge@math(% + }{% + \pgfutil@ifnextchar[{% + \tikz@scan@one@point\tikz@@to@or@edge@math(% + }{% + \tikz@@to@or@edge@@coordinate(% + }% + }% +}% +\def\tikz@@to@or@edge@math#1{% + \pgf@process{#1}% + \iftikz@updatecurrent\else + \tikz@updatenextfalse + \fi + \edef\tikztotarget{\the\pgf@x,\the\pgf@y}% + \tikz@to@or@edge@function% +}% + +\def\tikz@@to@or@edge@@coordinate(#1){% + \def\tikztotarget{#1}% + \tikz@to@or@edge@function% +}% + +\def\tikz@do@edge{% + \ifx\tikz@edge@macro\pgfutil@empty% + \setbox\tikz@whichbox=\hbox\bgroup% + \unhbox\tikz@whichbox% + \hbox\bgroup + \bgroup% + \pgfinterruptpath% + \pgfscope% + \let\tikz@transform=\pgfutil@empty% + \let\tikz@options=\pgfutil@empty% + \tikz@clear@rdf@options% + \let\tikz@tonodes=\tikz@collected@onpath% + \def\tikztonodes{{\pgfextra{\tikz@node@is@a@labeltrue}\tikz@tonodes}}% + \let\tikz@collected@onpath=\pgfutil@empty% + \tikz@options% + \tikz@do@rdf@pre@options% + \pgfidscope% + \tikz@do@rdf@post@options% + \tikz@transform% + \let\tikz@transform=\relax% + \let\tikz@after@path\pgfutil@empty% + \tikz@atbegin@to% + \tikz@enable@edge@quotes% + \path[style=every edge]\expandafter[\tikz@@to@local@options](\tikztostart)\tikz@to@path + \pgfextra{\global\let\tikz@after@path@smuggle=\tikz@after@path};% + \tikz@atend@to% + \endpgfidscope% + \endpgfscope% + \endpgfinterruptpath% + \egroup + \egroup% + \egroup% + \global\setbox\tikz@tempbox=\box\tikz@whichbox% + \expandafter\endgroup% + \expandafter\setbox\tikz@whichbox=\box\tikz@tempbox% + \else% + \expandafter\expandafter\expandafter\tikz@edge@macro% + \expandafter\expandafter\expandafter{\expandafter\tikz@@to@local@options\expandafter}\expandafter{\tikz@collected@onpath}% + \endgroup% + \let\tikz@after@path@smuggle=\pgfutil@empty% + \fi% + \expandafter\tikz@scan@next@command\tikz@after@path@smuggle% +}% + +\def\tikz@do@to{% + \let\tikz@tonodes=\tikz@collected@onpath% + \def\tikztonodes{{\pgfextra{\tikz@node@is@a@labeltrue}\tikz@tonodes}}% + \let\tikz@collected@onpath=\pgfutil@empty% + \tikz@scan@next@command% + {% + \pgfextra{\let\tikz@after@path\pgfutil@empty}% + \pgfextra{\tikz@atbegin@to}% + \pgfextra{\tikz@enable@edge@quotes}% + [style=every to]\expandafter[\tikz@@to@local@options]\tikz@to@path% + \pgfextra{\tikz@atend@to}% + \pgf@stop% + \expandafter\tikz@scan@next@command\expandafter% + }\tikz@after@path% + \pgfextra{\tikz@updatenexttrue\tikz@updatecurrenttrue}% +}% + + +\def\tikz@to@use@last@coordinate{% + \iftikz@shapeborder% + \edef\tikztostart{\tikz@shapeborder@name}% + \else% + \edef\tikztostart{\the\tikz@lastx,\the\tikz@lasty}% + \fi% +}% +\def\tikz@to@use@last@fig@name{% + \edef\tikztostart{\tikz@to@last@fig@name}% +}% + + + + +\def\tikz@graph aph{\tikz@lib@graph@parser}% + +\def\tikz@lib@graph@parser{\pgfutil@ifnextchar[\tikz@graph@error{\tikz@graph@error[]}}% +\def\tikz@graph@error[#1]#2{% + \tikzerror{You need to say \string\usetikzlibrary{graphs} in order to use the graph syntax}% + \tikz@lib@graph@parser@done% +}% + +\def\tikz@lib@graph@parser@done{% + \tikz@scan@next@command% +}% + + + + +\def\tikz@edgetoparent from parent{\pgfutil@ifnextchar[\tikz@@edgetoparent{\tikz@@edgetoparent[]}}% +\def\tikz@@edgetoparent[#1]{% + \let\tikz@edge@to@parent@needed=\pgfutil@empty% + \def\tikz@edgetoparent@options{#1}% + \begingroup% + \let\tikz@collected@onpath=\pgfutil@empty% + \tikz@edgetoparentcollect% +}% +\def\tikz@edgetoparentcollect{% + \pgfutil@ifnextchar n{\tikz@collect@label@onpath\tikz@edgetoparentcollect}% + {% + \expandafter% + \endgroup% + \expandafter\tikz@edgetoparent@rollout\expandafter{\tikz@collected@onpath}% + }% +}% + +\def\tikz@edgetoparent@rollout#1{% + \pgfkeysgetvalue{/tikz/edge from parent macro}\tikz@etop@temp + \expandafter\tikz@scan@next@command\expandafter\tikz@etop@temp\expandafter{\tikz@edgetoparent@options}{#1}% +}% + + + +\def\tikz@dot.{\tikz@@dot}% +\def\tikz@@dot{% + \pgfutil@ifnextchar n{\tikz@collect@label@onpath\tikz@@dot}{% + \pgfutil@ifnextchar p{\tikz@collect@pic@onpath\tikz@@dot}% + {\pgfutil@ifnextchar c{\tikz@curveto@double}{\tikz@curveto@auto}}}% +}% + +\def\tikz@curveto@double co{% + \pgfutil@ifnextchar o{\tikz@collect@coordinate@onpath\tikz@@dot co} + {\tikz@cureveto@@double}}% +\def\tikz@cureveto@@double ntrols#1{% + \tikz@scan@one@point\tikz@curveA#1% +}% +\def\tikz@curveA#1{% + \edef\tikz@timer@start{\noexpand\pgfqpoint{\the\tikz@lastx}{\the\tikz@lasty}}% + {% + \tikz@lastxsaved=\tikz@lastx% + \tikz@lastysaved=\tikz@lasty% + \tikz@make@last@position{#1}% + \xdef\tikz@curve@first{\noexpand\pgfqpoint{\the\tikz@lastx}{\the\tikz@lasty}}% + }% + \pgfutil@ifnextchar a + {\tikz@curveBand}% + {\let\tikz@curve@second\tikz@curve@first\tikz@curveCdots}% +}% +\def\tikz@curveBand and{% + \tikz@scan@one@point\tikz@curveB% +}% +\def\tikz@curveB#1{% + \def\tikz@curve@second{#1}% + \tikz@curveCdots} +\def\tikz@curveCdots{% + \afterassignment\tikz@curveCdot\let\pgfutil@next=}% +\def\tikz@curveCdot.{% + \ifx\pgfutil@next.% + \else% + \tikzerror{Dot expected}% + \fi% + \iftikz@updatenext + \tikz@updatecurrenttrue% + \fi + \tikz@curveCcheck% +}% +\def\tikz@curveCcheck{% + \pgfutil@ifnextchar n{\tikz@collect@label@onpath\tikz@curveCcheck}{% + \pgfutil@ifnextchar p{\tikz@collect@pic@onpath\tikz@curveCcheck}% + {\pgfutil@ifnextchar c{\tikz@collect@coordinate@onpath\tikz@curveCcheck} + {\tikz@scan@one@point\tikz@curveC}}}% +}% +\def\tikz@curveC#1{% + \tikz@make@last@position{#1}% + \edef\tikz@curve@third{\noexpand\pgfqpoint{\the\tikz@lastx}{\the\tikz@lasty}}% + {% + \tikz@lastxsaved=\tikz@lastx% + \tikz@lastysaved=\tikz@lasty% + \tikz@make@last@position{\tikz@curve@second}% + \xdef\tikz@curve@second{\noexpand\pgfqpoint{\the\tikz@lastx}{\the\tikz@lasty}}% + }% + \ifx\tikz@moveto@waiting\relax% + \else% + \pgf@process{\pgfpointshapeborder{\tikz@moveto@waiting}{\tikz@curve@first}}% + \edef\tikz@timer@start{\noexpand\pgfqpoint{\the\pgf@x}{\the\pgf@y}}% + \tikz@@movetosave{\pgfqpoint{\pgf@x}{\pgf@y}}% + \fi% + \let\tikz@timer@cont@one=\tikz@curve@first% + \let\tikz@timer@cont@two=\tikz@curve@second% + \iftikz@shapeborder% + {% + \pgf@process{\pgfpointshapeborder{\tikz@shapeborder@name}{\tikz@curve@second}}% + \tikz@make@last@position{\pgfqpoint{\pgf@x}{\pgf@y}}% + \edef\tikz@curve@third{\noexpand\pgfqpoint{\the\tikz@lastx}{\the\tikz@lasty}}% + \pgfpathcurveto{\tikz@curve@first}{\tikz@curve@second}{\tikz@curve@third}% + \global\let\tikz@timer@end@temp=\tikz@curve@third% + }% + \let\tikz@timer@end=\tikz@timer@end@temp% + \edef\tikz@moveto@waiting{\tikz@shapeborder@name}% + \else% + \pgfpathcurveto{\tikz@curve@first}{\tikz@curve@second}{\tikz@curve@third}% + \let\tikz@timer@end=\tikz@curve@third + \let\tikz@moveto@waiting=\relax% + \fi% + \let\tikz@timer=\tikz@timer@curve% + \let\tikz@tangent=\tikz@curve@second% + \tikz@scan@next@command% +}% + + +\def\tikz@rect ectangle{% + \tikz@flush@moveto% + \edef\tikz@timer@start{\noexpand\pgfqpoint{\the\tikz@lastx}{\the\tikz@lasty}}% + \tikz@@rect}% +\def\tikz@@rect{% + \pgfutil@ifnextchar n{\tikz@collect@label@onpath\tikz@@rect}{% + \pgfutil@ifnextchar p{\tikz@collect@pic@onpath\tikz@@rect}% + {\pgfutil@ifnextchar c{\tikz@collect@coordinate@onpath\tikz@@rect}% + { + \pgf@xa=\tikz@lastx\relax% + \pgf@ya=\tikz@lasty\relax% + \tikz@scan@one@point\tikz@rectB}}}}% +\def\tikz@rectB#1{% + \tikz@make@last@position{#1}% + \edef\tikz@timer@end{\noexpand\pgfqpoint{\the\tikz@lastx}{\the\tikz@lasty}}% + \let\tikz@timer=\tikz@timer@line% + \tikz@@movetosave{\pgfqpoint{\pgf@xa}{\pgf@ya}}% + \tikz@path@lineto{\pgfqpoint{\pgf@xa}{\tikz@lasty}}% + \tikz@path@lineto{\pgfqpoint{\tikz@lastx}{\tikz@lasty}}% + \tikz@path@lineto{\pgfqpoint{\tikz@lastx}{\pgf@ya}}% + \iftikz@snaked% + \tikz@path@lineto{\pgfqpoint{\pgf@xa}{\pgf@ya}}% + \fi% + \pgfpathclose% + \tikz@@movetosave{\pgfqpoint{\tikz@lastx}{\tikz@lasty}}% + \def\pgfstrokehook{}% + \let\tikz@tangent\relax% + \tikz@scan@next@command% +}% + + + +\def\tikz@grid id{% + \tikz@flush@moveto% + \pgf@xa=\tikz@lastx\relax% + \pgf@ya=\tikz@lasty\relax% + \pgfutil@ifnextchar[{\tikz@gridA}{\tikz@gridA[]}}% +\def\tikz@gridA[#1]{% + \def\tikz@grid@options{#1}% + \tikz@cycle@expander{\tikz@scan@one@point\tikz@gridB}}% +\def\tikz@gridB#1{% + \tikz@make@last@position{#1}% + \let\tikz@tangent\relax% + {% + \let\tikz@after@path\pgfutil@empty% + \expandafter\tikzset\expandafter{\tikz@grid@options} + \tikz@checkunit{\tikz@grid@x}% + \iftikz@isdimension% + \pgf@process{\pgfpoint{\tikz@grid@x}{0pt}}% + \else% + \pgf@process{\pgfpointxy{\tikz@grid@x}{0}}% + \fi% + \pgf@xb=\pgf@x% + \pgf@yb=\pgf@y% + \tikz@checkunit{\tikz@grid@y}% + \iftikz@isdimension% + \pgf@process{\pgfpoint{0pt}{\tikz@grid@y}}% + \else% + \pgf@process{\pgfpointxy{0}{\tikz@grid@y}}% + \fi% + \advance\pgf@xb by\pgf@x% + \advance\pgf@yb by\pgf@y% + \pgfpathgrid[stepx=\pgf@xb,stepy=\pgf@yb]% + {\pgfqpoint{\pgf@xa}{\pgf@ya}}{\pgfqpoint{\tikz@lastx}{\tikz@lasty}}% + \expandafter}% + \expandafter\tikz@scan@next@command\tikz@after@path% +}% + + + +\def\tikz@plot lot{% + \tikz@flush@moveto% + \pgfutil@ifnextchar[{\tikz@@plot}{\tikz@@plot[]}}% +\def\tikz@@plot[#1]{% + \let\tikz@tangent\tikz@tangent@lookup% + \begingroup% + \let\tikz@after@path\pgfutil@empty% + \let\tikz@options=\pgfutil@empty% + \tikzset{every plot/.try}% + \tikzset{#1}% + \pgfutil@ifnextchar f{\tikz@plot@f}% + {\pgfutil@ifnextchar c{\tikz@plot@scan@points}% + {\pgfutil@ifnextchar ({\tikz@plot@expression}{% + \tikzerror{Cannot parse this plotting data}% + \endgroup}}}}% +\def\tikz@plot@f f{\pgfutil@ifnextchar i{\tikz@plot@file}{\tikz@plot@function}}% + +\def\tikz@plot@file ile#1{\def\tikz@plot@data{\pgfplotxyfile{#1}}\tikz@@@plot}% +\def\tikz@plot@scan@points coordinates#1{% + \pgfplothandlerrecord\tikz@plot@data% + \pgfplotstreamstart% + \pgfutil@ifnextchar\pgf@stop{\pgfplotstreamend\expandafter\tikz@@@plot\pgfutil@gobble} + {\tikz@scan@one@point\tikz@plot@next@point}% + #1\pgf@stop% +}% +\def\tikz@plot@next@point#1{% + \pgfplotstreampoint{#1}% + \pgfutil@ifnextchar\pgf@stop{\pgfplotstreamend\expandafter\tikz@@@plot\pgfutil@gobble}% + {\tikz@scan@one@point\tikz@plot@next@point}% +}% +\def\tikz@plot@function unction#1{% + \def\tikz@plot@filename{\tikz@plot@prefix\tikz@plot@id}% + \iftikz@plot@raw@gnuplot% + \def\tikz@plot@data{\pgfplotgnuplot[\tikz@plot@filename]{#1}}% + \else% + \iftikz@plot@parametric% + \def\tikz@plot@data{\pgfplotgnuplot[\tikz@plot@filename]{% + set samples \tikz@plot@samples; + set parametric; + plot [t=\tikz@plot@domain] + [\tikz@plot@xrange] + [\tikz@plot@range] + #1}}% + \else% + \def\tikz@plot@data{\pgfplotgnuplot[\tikz@plot@filename]{% + set samples \tikz@plot@samples; + plot [x=\tikz@plot@domain] + \ifx\tikz@plot@range\pgfutil@empty\else[\tikz@plot@range]\fi + #1}}% + \fi% + \fi% + \tikz@@@plot% +}% + +\def\tikz@plot@no@resample{% + \pgfutil@IfFileExists{\tikz@plot@filename.table}% + {\def\tikz@plot@data{\pgfplotxyfile{\tikz@plot@filename.table}}}% + {}% +}% + +\def\tikz@plot@expression(#1){% + \edef\tikz@plot@data{\noexpand\pgfplotfunction{\expandafter\noexpand\tikz@plot@var}{\tikz@plot@samplesat}}% + \expandafter\def\expandafter\tikz@plot@data\expandafter{\tikz@plot@data{\tikz@scan@one@point\pgfutil@firstofone(#1)}}% + \tikz@@@plot% +}% + +\def\tikz@@@plot{% + \def\pgfplotlastpoint{\pgfpointorigin}% + \tikz@plot@handler% + \tikz@plot@data% + \global\let\tikz@@@temp=\pgfplotlastpoint% + \ifx\tikz@plot@mark\pgfutil@empty% + \else% + \setbox\tikz@whichbox=\hbox{% + \unhbox\tikz@whichbox% + \hbox{{% + \pgfinterruptpath% + \pgfscope% + \let\tikz@options=\pgfutil@empty% + \let\tikz@transform=\pgfutil@empty% + \tikzset{every mark}% + \tikz@options% + \ifx\tikz@mark@list\pgfutil@empty% + \pgfplothandlermark{\tikz@transform\pgfuseplotmark{\tikz@plot@mark}}% + \else + \pgfplothandlermarklisted{\tikz@transform\pgfuseplotmark{\tikz@plot@mark}}{\tikz@mark@list}% + \fi + \tikz@plot@data% + \endpgfscope + \endpgfinterruptpath% + }}% + }% + \fi% + \global\setbox\tikz@tempbox=\box\tikz@whichbox% + \global\let\tikz@after@path@smuggle=\tikz@after@path + \expandafter\endgroup% + \expandafter\setbox\tikz@whichbox=\box\tikz@tempbox% + \tikz@make@last@position{\tikz@@@temp}% + \expandafter\tikz@scan@next@command\tikz@after@path@smuggle% +}% + + +\pgfdeclareplotmark{ball} +{% + \def\tikz@shading{ball}% + \shade (0pt,0pt) circle (\pgfplotmarksize);% +}% + + + + +\def\tikz@cosine s{\tikz@cycle@expander{\tikz@scan@one@point\tikz@@cosine}} +\def\tikz@@cosine#1{% + \let\tikz@tangent\tikz@tangent@lookup% + \tikz@flush@moveto% + \pgf@process{#1}% + \pgf@xc=\pgf@x% + \pgf@yc=\pgf@y% + \advance\pgf@xc by-\tikz@lastx% + \advance\pgf@yc by-\tikz@lasty% + \advance\tikz@lastx by\pgf@xc% + \advance\tikz@lasty by\pgf@yc% + \tikz@lastxsaved=\tikz@lastx% + \tikz@lastysaved=\tikz@lasty% + \tikz@updatecurrenttrue% + \pgfpathcosine{\pgfqpoint{\pgf@xc}{\pgf@yc}}% + \tikz@scan@next@command% +}% + +\def\tikz@sine in{\tikz@cycle@expander{\tikz@scan@one@point\tikz@@sine}} +\def\tikz@@sine#1{% + \let\tikz@tangent\tikz@tangent@lookup% + \tikz@flush@moveto% + \pgf@process{#1}% + \pgf@xc=\pgf@x% + \pgf@yc=\pgf@y% + \advance\pgf@xc by-\tikz@lastx% + \advance\pgf@yc by-\tikz@lasty% + \advance\tikz@lastx by\pgf@xc% + \advance\tikz@lasty by\pgf@yc% + \tikz@lastxsaved=\tikz@lastx% + \tikz@lastysaved=\tikz@lasty% + \tikz@updatecurrenttrue% + \pgfpathsine{\pgfqpoint{\pgf@xc}{\pgf@yc}}% + \tikz@scan@next@command% +}% + +\def\tikz@parabola arabola{% + \let\tikz@tangent\tikz@tangent@lookup% + \pgfutil@ifnextchar[{\tikz@parabola@options}{\tikz@parabola@options[]}}% + +\def\tikz@parabola@options[#1]{% + \def\tikz@parabola@option{#1}% + \pgfutil@ifnextchar b{\tikz@parabola@scan@bend}{\tikz@cycle@expander{\tikz@scan@one@point\tikz@parabola@semifinal}}}% +\def\tikz@parabola@scan@bend bend{\tikz@scan@one@point\tikz@parabola@scan@bendB}% +\def\tikz@parabola@scan@bendB#1{% + \def\tikz@parabola@bend{#1}% + \tikz@cycle@expander{\tikz@scan@one@point\tikz@parabola@semifinal}% +}% +\def\tikz@parabola@semifinal#1{% + \tikz@flush@moveto% + \pgf@xb=\tikz@lastx% + \pgf@yb=\tikz@lasty% + \tikz@make@last@position{#1}% + \pgf@xc=\tikz@lastx% + \pgf@yc=\tikz@lasty% + \begingroup% + \let\tikz@after@path\pgfutil@empty% + \expandafter\tikzset\expandafter{\tikz@parabola@option}% + \tikz@lastxsaved=\tikz@parabola@bend@factor\tikz@lastx% + \tikz@lastysaved=\tikz@parabola@bend@factor\tikz@lasty% + \advance\tikz@lastxsaved by\pgf@xb% + \advance\tikz@lastysaved by\pgf@yb% + \advance\tikz@lastxsaved by-\tikz@parabola@bend@factor\pgf@xb% + \advance\tikz@lastysaved by-\tikz@parabola@bend@factor\pgf@yb% + \expandafter\tikz@make@last@position\expandafter{\tikz@parabola@bend}% + \advance\pgf@xc by-\tikz@lastx% + \advance\pgf@yc by-\tikz@lasty% + \advance\tikz@lastx by-\pgf@xb% + \advance\tikz@lasty by-\pgf@yb% + \xdef\tikz@parabola@b{{\noexpand\pgfqpoint{\the\tikz@lastx}{\the\tikz@lasty}}{\noexpand\pgfqpoint{\the\pgf@xc}{\the\pgf@yc}}}% + \expandafter\endgroup% + \expandafter\expandafter\expandafter\pgfpathparabola\expandafter\tikz@parabola@b% + \expandafter\tikz@scan@next@command\tikz@after@path% +}% + + +\def\tikz@circle ircle{\tikz@flush@moveto\tikz@@circle}% +\def\tikz@ellipse llipse{\tikz@flush@moveto\tikz@@circle}% +\def\tikz@@circle{% + \let\tikz@tangent\relax% + \pgfutil@ifnextchar(\tikz@@@circle + {\pgfutil@ifnextchar[\tikz@circle@opt{% + \advance\tikz@expandcount by -10\relax% + \ifnum\tikz@expandcount<0\relax% + \let\pgfutil@next=\tikz@@circle@normal% + \else% + \let\pgfutil@next=\tikz@@circle@scanexpand% + \fi% + \pgfutil@next% + }}% +}% +\def\tikz@@circle@scanexpand{\expandafter\tikz@@circle}% +\def\tikz@@circle@normal{\tikz@circle@opt[]}% + +\def\tikz@circle@opt[#1]{% + {% + \def\tikz@node@at{\tikz@last@position}% + \let\tikz@transform=\pgfutil@empty% + \tikzset{every circle/.try,#1}% + \pgftransformshift{\tikz@node@at}% + \tikz@transform% + \tikz@do@ellipse{\pgfkeysvalueof{/tikz/x radius}}{\pgfkeysvalueof{/tikz/y radius}} + }% + \tikz@scan@next@command% +}% + +\def\tikz@@@circle(#1){% + {% + \pgftransformshift{\tikz@last@position}% + \pgfutil@in@{ and }{#1}% + \ifpgfutil@in@% + \tikz@@ellipseB(#1)% + \else% + \tikz@do@circle{#1}% + \fi% + }% + \tikz@scan@next@command% +}% +\def\tikz@@ellipseB(#1 and #2){% + \tikz@do@ellipse{#1}{#2}% +}% +\def\tikz@do@circle#1{% + \pgfmathparse{#1}% + \let\tikz@ellipse@x=\pgfmathresult + \ifpgfmathunitsdeclared + \pgfpathellipse{\pgfpointorigin}% + {\pgfqpoint{\tikz@ellipse@x pt}{0pt}}% + {\pgfpoint{0pt}{\tikz@ellipse@x pt}}% + \else + \pgfpathellipse{\pgfpointorigin}% + {\pgfpointxy{\tikz@ellipse@x}{0}}% + {\pgfpointxy{0}{\tikz@ellipse@x}}% + \fi +} +\def\tikz@do@ellipse#1#2{ + \pgfmathparse{#1}% + \let\tikz@ellipse@x=\pgfmathresult% + \ifpgfmathunitsdeclared% + \pgfmathparse{#2}% + \let\tikz@ellipse@y=\pgfmathresult% + \ifpgfmathunitsdeclared% + \pgfpathellipse{\pgfpointorigin}{% + \pgfqpoint{\tikz@ellipse@x pt}{0pt}}{\pgfpoint{0pt}{\tikz@ellipse@y pt}}% + \else% + \tikzerror{You cannot mix dimensions and dimensionless values in an ellipse}% + \fi% + \else% + \pgfmathparse{#2}% + \let\tikz@ellipse@y=\pgfmathresult% + \ifpgfmathunitsdeclared% + \tikzerror{You cannot mix dimensions and dimensionless values in an ellipse}% + \else% + \pgfpathellipse{\pgfpointorigin}{% + \pgfpointxy{\tikz@ellipse@x}{0}}{\pgfpointxy{0}{\tikz@ellipse@y}}% + \fi% + \fi% +}% + +\def\tikz@arcA rc{\tikz@flush@moveto\tikz@arc@cont}% +\def\tikz@arc@cont{% + \pgfutil@ifnextchar(% + {\tikz@@arcto}{% + \pgfutil@ifnextchar[% + {\tikz@arc@opt}% + {% + \advance\tikz@expandcount by -10\relax% + \ifnum\tikz@expandcount<0\relax% + \let\pgfutil@next=\tikz@@arc@normal% + \else% + \let\pgfutil@next=\tikz@@arc@scanexpand% + \fi% + \pgfutil@next% + }% + }% +}% +\def\tikz@@arc@scanexpand{\expandafter\tikz@arc@cont}% +\def\tikz@@arc@normal{\tikz@arc@opt[]}% + + +\def\tikz@arc@opt[#1]{% + {% + \tikzset{every arc/.try,#1}% + \pgfkeysgetvalue{/tikz/start angle}\tikz@s + \pgfkeysgetvalue{/tikz/end angle}\tikz@e + \pgfkeysgetvalue{/tikz/delta angle}\tikz@d + \ifx\tikz@s\pgfutil@empty% + \pgfmathsetmacro\tikz@s{\tikz@e-\tikz@d} + \else + \ifx\tikz@e\pgfutil@empty% + \pgfmathsetmacro\tikz@e{\tikz@s+\tikz@d} + \fi% + \fi% + \xdef\pgf@marshal{\noexpand% + \tikz@do@arc{\tikz@s}{\tikz@e} + {\pgfkeysvalueof{/tikz/x radius}} + {\pgfkeysvalueof{/tikz/y radius}}}% + }% + \pgf@marshal% + \tikz@arcfinal% +}% + +\def\tikz@@arcto(#1){% + \edef\tikz@temp{(#1)}% + \expandafter\tikz@@@arcto@check@slashand\tikz@temp% +}% + +\def\tikz@@@arcto@check@slashand(#1:#2:#3){% + \pgfutil@in@{ and }{#3}% + \ifpgfutil@in@% + \tikz@parse@arc@and(#1:#2:#3)% + \else% + \tikz@parse@arc@and(#1:#2:{#3} and {#3})% + \fi% + \tikz@arcfinal% +}% + +\def\tikz@parse@arc@and(#1:#2:#3 and #4){% + \tikz@do@arc{#1}{#2}{#3}{#4}% +}% +\def\tikz@do@arc#1#2#3#4{% + \let\tikz@tangent\tikz@tangent@lookup% + \edef\tikz@timer@start{\noexpand\pgfqpoint{\the\tikz@lastx}{\the\tikz@lasty}}% + \pgfmathsetmacro\tikz@timer@start@angle{#1}% + \pgfmathsetmacro\tikz@timer@end@angle{#2}% + \pgfmathparse{#3}% + \let\tikz@arc@x=\pgfmathresult% + \ifpgfmathunitsdeclared% + \pgfmathparse{#4}% + \let\tikz@arc@y=\pgfmathresult% + \ifpgfmathunitsdeclared% + \tikz@@@arcfinal{\pgfpatharc{\tikz@timer@start@angle}{\tikz@timer@end@angle}{\tikz@arc@x pt and \tikz@arc@y pt}} + {\pgfpointpolar{\tikz@timer@start@angle}{\tikz@arc@x pt and \tikz@arc@y pt}} + {\pgfpointpolar{\tikz@timer@end@angle}{\tikz@arc@x pt and \tikz@arc@y pt}}% + \edef\tikz@timer@zero@axis{\noexpand\pgfqpoint{\tikz@arc@x pt}{0pt}} + \edef\tikz@timer@ninety@axis{\noexpand\pgfqpoint{0pt}{\tikz@arc@y pt}} + \else% + \tikzerror{You cannot mix dimensions and dimensionless values in an arc}% + \fi% + \else% + \pgfmathparse{#4}% + \let\tikz@arc@y=\pgfmathresult% + \ifpgfmathunitsdeclared% + \tikzerror{You cannot mix dimensions and dimensionless values in an arc}% + \else% + \tikz@@@arcfinal{\pgfpatharcaxes{\tikz@timer@start@angle}{\tikz@timer@end@angle}{\pgfpointxy{\tikz@arc@x}{0}}{\pgfpointxy{0}{\tikz@arc@y}}} + {\pgfpointpolarxy{\tikz@timer@start@angle}{\tikz@arc@x and \tikz@arc@y}}{\pgfpointpolarxy{\tikz@timer@end@angle}{\tikz@arc@x and \tikz@arc@y}}% + \pgf@process{\pgfpointxy{\tikz@arc@x}{0}} + \edef\tikz@timer@zero@axis{\noexpand\pgfqpoint{\the\pgf@x}{\the\pgf@y}} + \pgf@process{\pgfpointxy{0}{\tikz@arc@y}} + \edef\tikz@timer@ninety@axis{\noexpand\pgfqpoint{\the\pgf@x}{\the\pgf@y}} + \fi% + \fi% +}% + +\def\tikz@@@arcfinal#1#2#3{% + #1% + \pgf@process{#2} + \xdef\tikz@arc@save@first{\pgfqpoint{\the\pgf@x}{\the\pgf@y}}% + \pgf@process{#3} + \xdef\tikz@arc@save@second{\pgfqpoint{\the\pgf@x}{\the\pgf@y}}% +}% + +\def\tikz@arcfinal{% + \pgf@process{\tikz@arc@save@first}% + \advance\tikz@lastx by-\pgf@x% + \advance\tikz@lasty by-\pgf@y% + \pgf@process{\tikz@arc@save@second}% + \advance\tikz@lastx by\pgf@x% + \advance\tikz@lasty by\pgf@y% + \tikz@lastxsaved=\tikz@lastx% + \tikz@lastysaved=\tikz@lasty% + \let\tikz@timer=\tikz@timer@arc% + \tikz@scan@next@command% +}% + + +\def\tikz@coordinate ordinate{% + \pgfutil@ifnextchar[{\tikz@@coordinate@opt}{\tikz@@coordinate@opt[]}}% +\def\tikz@@coordinate@opt[#1]{% + \pgfutil@ifnextchar({\tikz@@coordinate[#1]} + {\tikz@fig ode[shape=coordinate,#1]{}}}% +\def\tikz@@coordinate[#1](#2){% + \pgfutil@ifnextchar a{\tikz@@coordinate@at[#1](#2)} + {\tikz@fig ode[shape=coordinate,#1](#2){}}}% +\def\tikz@@coordinate@at[#1](#2)a{% + \pgfutil@ifnextchar t{\tikz@@coordinate@@at[#1](#2)a}% + {\tikz@fig ode[shape=coordinate,#1](#2){}a}% +}% +\def\tikz@@coordinate@@at[#1](#2)at#3({% + \def\tikz@coordinate@caller{\tikz@fig ode[shape=coordinate,#1](#2)at}% + \tikz@scan@one@point\tikz@@coordinate@at@math(% +}% +\def\tikz@@coordinate@at@math#1{% + \pgf@process{#1}% + \edef\tikz@temp{(\the\pgf@x,\the\pgf@y)}% + \expandafter\tikz@coordinate@caller\tikz@temp{}% +}% + + + +\def\tikz@fig ode{% + \pgfutil@ifnextchar a\tikz@test@also{% + \pgfutil@ifnextchar f{\tikz@nodes@start}\tikz@normal@fig}}% +\def\tikz@test@also a{\pgfutil@ifnextchar l\tikz@node@also{\tikz@normal@fig a}}% +\def\tikz@normal@fig{% + \edef\tikz@save@line@width{\the\pgflinewidth}% + \begingroup% + \let\tikz@fig@name=\pgfutil@empty% + \begingroup% + \tikz@is@matrixfalse% + \let\nodepart=\tikz@nodepart% + \let\tikz@atbegin@scope=\pgfutil@empty% + \let\tikz@atend@scope=\pgfutil@empty% + \let\tikz@do@after@node=\tikz@scan@next@command% + \let\tikz@options=\pgfutil@empty% + \tikz@clear@rdf@options% + \let\tikz@id@name=\pgfutil@empty% + \let\tikz@after@path=\pgfutil@empty% + \let\tikz@transform=\pgfutil@empty% + \let\tikz@mode=\pgfutil@empty% + \tikz@decoratepathfalse% + \let\tikz@preactions=\pgfutil@empty% + \let\tikz@postactions=\pgfutil@empty% + \let\tikz@alias=\pgfutil@empty% + \def\tikz@node@at{\pgfqpoint{\the\tikz@lastx}{\the\tikz@lasty}}% + \let\tikz@time@for@matrix\tikz@time% + \let\tikz@node@content\relax% + \pgfgetpath\tikzpathuptonow% + \iftikz@node@is@a@label% + \else% + \let\tikz@time\pgfutil@empty% + \fi% + \tikz@node@reset@hook% + \tikzset{every node/.try}% + \tikz@@scan@fig}% +\def\tikz@@scan@fig{% + \pgfutil@ifnextchar a{\tikz@fig@scan@at} + {\pgfutil@ifnextchar({\tikz@fig@scan@name} + {\pgfutil@ifnextchar[{\tikz@fig@scan@options}% + {\pgfutil@ifnextchar:{\tikz@fig@scan@animation}% + {\pgfutil@ifnextchar\bgroup{\tikz@fig@main}% + {\tikzerror{A node must have a (possibly empty) label text}% + \tikz@fig@main{}}}}}}}% +\def\tikz@fig@scan@at at{% + \tikz@scan@one@point\tikz@@fig@scan@at}% +\def\tikz@@fig@scan@at#1{% + \def\tikz@node@at{#1}\tikz@@scan@fig}% +\def\tikz@fig@scan@name(#1){% + \pgfkeysvalueof{/tikz/name/.@cmd}#1\pgfeov% + \tikz@@scan@fig}% +\long\def\tikz@fig@scan@options[#1]{\iftikz@node@is@pic\tikz@enable@pic@quotes\else\tikz@enable@node@quotes\fi\tikzset{#1}\ifx\tikz@node@content\relax\expandafter\tikz@@scan@fig\else\tikz@expand@node@contents\fi}% +\def\tikz@fig@scan@animation:#1=#2{\tikzset{animate={myself:{#1}={#2}}}\tikz@@scan@fig}% +\def\tikz@expand@node@contents{% + \expandafter\tikz@@scan@fig\expandafter{\tikz@node@content}% +}% +\let\tikz@node@reset@hook=\pgfutil@empty% +\let\tikz@node@begin@hook=\pgfutil@empty% +\def\tikz@fig@main{% + \iftikz@node@is@pic% + \tikz@node@is@picfalse% + \expandafter\tikz@subpicture@handle% + \else% + \afterassignment\tikz@@fig@main\expandafter\let\expandafter\next\expandafter=% + \fi}% +\def\tikz@@fig@main{% + \pgfutil@ifundefined{pgf@sh@s@\tikz@shape}% + {\tikzerror{Unknown shape ``\tikz@shape.'' Using ``rectangle'' instead}% + \def\tikz@shape{rectangle}}% + {}% + \expandafter\xdef\csname tikz@dcl@coord@\tikz@fig@name\endcsname{% + \csname tikz@scan@point@coordinate\endcsname}% + \tikzset{every \tikz@shape\space node/.try}% + \tikz@node@textfont% + \tikz@node@begin@hook% + \iftikz@is@matrix% + \let\tikz@next=\tikz@do@matrix% + \else% + \let\tikz@next=\tikz@do@fig% + \fi% + \tikz@next% +}% +\let\tikz@nodepart@list\pgfutil@empty +\def\tikz@do@fig{% + \pgfutil@for\tikz@temp:=\tikz@nodepart@list\do{% + \expandafter\setbox\csname pgfnodepart\tikz@temp box\endcsname=\box\pgfutil@voidb@x% + }% + \setbox\pgfnodeparttextbox=\hbox% + \bgroup% + \pgfinterruptpicture% + \pgfsys@begin@text% + \pgfsys@text@to@black@hook% + \tikzset{every text node part/.try}% + \ifx\tikz@textopacity\pgfutil@empty% + \else% + \pgfsetfillopacity{\tikz@textopacity}% + \pgfsetstrokeopacity{\tikz@textopacity}% + \fi% + \ifx\tikz@text@width\pgfutil@empty% + \tikz@textfont% + \else% + \begingroup% + \pgfmathsetlength{\pgf@x}{\tikz@text@width}% + \pgfutil@minipage[t]{\pgf@x}\leavevmode\hbox{}% + \tikz@textfont% + \tikz@text@action% + \fi% + \tikz@atbegin@node% + \bgroup% + \aftergroup\unskip% + \ifx\tikz@textcolor\pgfutil@empty% + \else% + \pgfutil@colorlet{.}{\tikz@textcolor}% + \pgfutil@color{\tikz@textcolor}% + \fi% + \setbox\tikz@figbox=\box\pgfutil@voidb@x% + \setbox\tikz@figbox@bg=\box\pgfutil@voidb@x% + \tikz@uninstallcommands% + \iftikz@handle@active@code% + \tikz@orig@shorthands% + \let\tikz@orig@shorthands\pgfutil@empty% + \fi% + \ifnum\the\catcode`\;=\active\relax\expandafter\let\tikz@activesemicolon=\tikz@origsemi\fi% + \ifnum\the\catcode`\:=\active\relax\expandafter\let\tikz@activecolon=\tikz@origcolon\fi% + \ifnum\the\catcode`\|=\active\relax\expandafter\let\tikz@activebar=\tikz@origbar\fi% + \aftergroup\tikz@fig@collectresetcolor% + \tikz@signal@halign@check% + \tikz@text@reset% + \tikz@halign@check% + \ignorespaces% +}% +\def\tikz@fig@collectresetcolor{% + \pgfutil@ifnextchar\reset@color% + {\reset@color\afterassignment\tikz@fig@collectresetcolor\let\tikz@temp=}\tikz@fig@boxdone% +}% +\def\tikz@fig@boxdone{% + \tikz@atend@node% + \ifx\tikz@text@width\pgfutil@empty% + \else% + \pgfutil@endminipage% + \endgroup% + \fi% + \pgfsys@end@text% + \endpgfinterruptpicture% + \egroup% + \pgfutil@ifnextchar c{\tikz@fig@mustbenamed\tikz@fig@continue}% + {\pgfutil@ifnextchar[{\tikz@fig@mustbenamed\tikz@fig@continue}% + {\pgfutil@ifnextchar t{\tikz@fig@mustbenamed\tikz@fig@continue} + {\pgfutil@ifnextchar e{\tikz@fig@mustbenamed\tikz@fig@continue} + {\ifx\tikz@after@path\pgfutil@empty\expandafter\tikz@fig@continue\else\expandafter\tikz@fig@mustbenamed\expandafter\tikz@fig@continue\fi}}}}}% + +\tikzset{ + matrix/inner style/every cell/.code={% + \tikzset{every cell/.try={\the\pgfmatrixcurrentrow}{\the\pgfmatrixcurrentcolumn}}% + }, + matrix/inner style/column/.code={% + \tikzset{column \the\pgfmatrixcurrentcolumn/.try}% + }, + matrix/inner style/even odd column/.code={ + \ifodd\pgfmatrixcurrentcolumn% + \tikzset{every odd column/.try}% + \else% + \tikzset{every even column/.try}% + \fi + }, + matrix/inner style/row/.code={% + \tikzset{row \the\pgfmatrixcurrentrow/.try}% + }, + matrix/inner style/even odd row/.code={% + \ifodd\pgfmatrixcurrentrow% + \tikzset{every odd row/.try}% + \else% + \tikzset{every even row/.try}% + \fi + }, + matrix/inner style/cell/.code={% + \tikzset{row \the\pgfmatrixcurrentrow\space column \the\pgfmatrixcurrentcolumn/.try}% + }, + matrix/inner style order/.store in=\tikz@common@matrix@code@styleorder, + matrix/inner style order={ + every cell, + column, + even odd column, + row, + even odd row, + cell + }, +}% + +\def\tikz@do@matrix{% + \tikzset{every matrix/.try}% + \tikz@node@transformations% + \tikz@fig@mustbenamed% + \setbox\tikz@whichbox=\hbox\bgroup% + \unhbox\tikz@whichbox% + \hbox\bgroup\bgroup% + \pgfinterruptpath% + \pgfscope% + \ifx\tikz@time\pgfutil@empty\let\tikz@time\tikz@time@for@matrix\fi% + \tikz@options% + \tikz@do@rdf@pre@options% + \tikz@is@nodefalse% + \tikz@call@id@hook% + \pgfidscope% + \tikz@do@rdf@post@options% + \begingroup% + \let\tikz@id@name\pgfutil@empty% + \pgfclearid% + \setbox\tikz@figbox=\box\pgfutil@voidb@x% + \setbox\tikz@figbox@bg=\box\pgfutil@voidb@x% + \let\tikzmatrixname=\tikz@fig@name% + \edef\tikz@m@anchor{\ifx\tikz@matrix@anchor\pgfutil@empty\tikz@anchor\else\tikz@matrix@anchor\fi}% + \expandafter\pgfutil@in@\expandafter{\expandafter.\expandafter}\expandafter{\tikz@m@anchor}% + \ifpgfutil@in@% + \expandafter\tikz@matrix@split\tikz@m@anchor\relax% + \else% + \def\tikz@matrix@shift{\pgfpointorigin}% + \fi% + \let\tikz@transform=\relax% + \pgfmatrixbeforeassemblenode{\tikzset{every outer matrix/.try}}% + \pgfmatrix% + {\tikz@shape}% + {\tikz@m@anchor}% + {\tikz@fig@name}% + {% + \pgfutil@tempdima=\pgflinewidth% + {\begingroup\tikz@finish}% + \global\pgflinewidth=\pgfutil@tempdima% + }% + {\tikz@matrix@shift}% + {% + \tikz@matrix@make@active@ampersand% + \def\pgfmatrixbegincode{% + \pgfsys@beginscope% + \tikz@common@matrix@code% + \tikz@atbegin@cell% + }% + \def\tikz@common@matrix@code{% + \let\tikz@options=\pgfutil@empty% + \let\tikz@mode=\pgfutil@empty% + \pgfutil@for\pgf@temp:=\tikz@common@matrix@code@styleorder\do{% + \toks0=\expandafter{\romannumeral-`0\expandafter\pgfutil@trimspaces\expandafter{\pgf@temp}}% + \def\pgf@marshal{}% + \pgfutil@ifxempty\pgf@temp{}{% + \edef\pgf@marshal{\noexpand\tikzset{matrix/inner style/.cd,\the\toks0}}% + }% + \pgf@marshal + }% + \tikz@options% + }% + \def\pgfmatrixendcode{% + \tikz@atend@cell% + \pgfsys@endscope% + }% + \def\pgfmatrixemptycode{% + \pgfsys@beginscope% + \tikz@common@matrix@code% + \tikz@at@emptycell% + \pgfsys@endscope% + }% + \tikz@atbegin@matrix% + \aftergroup\tikz@do@matrix@cont}% + \bgroup% +}% +\def\tikz@do@matrix@cont{% + \tikz@atend@matrix% + \endgroup% + \endpgfidscope% + \endpgfscope + \endpgfinterruptpath% + \egroup\egroup% + \egroup% + \tikz@node@finish% +}% +{% + \catcode`\&=13 + \gdef\tikz@matrix@make@active@ampersand{% + \ifx\tikz@ampersand@replacement\pgfutil@empty% + \catcode`\&=13% + \let&=\pgfmatrixnextcell% + \else% + \expandafter\let\tikz@ampersand@replacement=\pgfmatrixnextcell% + \fi% + }% +}% + + +\def\tikz@matrix@split#1.#2\relax{% + \def\tikz@m@anchor{text}% + \def\tikz@matrix@shift{\pgfpointanchor{#1}{#2}}% +}% + +\def\tikz@fig@continue{% + \ifx\tikz@text@width\pgfutil@empty% + \else% + \pgfmathsetlength{\pgf@x}{\tikz@text@width}% + \wd\pgfnodeparttextbox=\pgf@x% + \fi% + \ifx\tikz@text@height\pgfutil@empty% + \else% + \pgfmathsetlength{\pgf@x}{\tikz@text@height}% + \ht\pgfnodeparttextbox=\pgf@x% + \fi% + \ifx\tikz@text@depth\pgfutil@empty% + \else% + \pgfmathsetlength{\pgf@x}{\tikz@text@depth}% + \dp\pgfnodeparttextbox=\pgf@x% + \fi% + \tikz@node@transformations% + \tikz@nlt% + \setbox\tikz@whichbox=\hbox{% + \unhbox\tikz@whichbox% + \hbox{{% + \pgfinterruptpath% + \pgfscope% + \tikz@options% + \tikz@do@rdf@pre@options% + \tikz@is@nodetrue% + \tikz@call@id@hook% + \pgfidscope% + \tikz@do@rdf@post@options% + \let\tikz@id@name\pgfutil@empty% + \setbox\tikz@figbox=\box\pgfutil@voidb@x% + \setbox\tikz@figbox@bg=\box\pgfutil@voidb@x% + \setbox\pgfnodeparttextbox=\hbox{{% + \pgfsys@begin@text% + \ifx\tikz@textcolor\pgfutil@empty% + \else% + \pgfutil@colorlet{.}{\tikz@textcolor}% + \fi% + \pgfsetcolor{.}% + \pgfusetype{.text}% + \pgfidscope% + \box\pgfnodeparttextbox% + \endpgfidscope% + \pgfsys@end@text% + }}% + \pgfmultipartnode{\tikz@shape}{\tikz@anchor}{\tikz@fig@name}{% + \pgfutil@tempdima=\pgflinewidth% + {\begingroup\tikz@finish}% + \global\pgflinewidth=\pgfutil@tempdima% + }% + \endpgfidscope% + \endpgfscope% + \endpgfinterruptpath% + }}% + }% + \tikz@alias% + \tikz@node@finish% +}% + + +\def\tikz@fig@mustbenamed{% + \ifx\tikz@fig@name\pgfutil@empty% + \global\advance\tikz@fig@count by1\relax + \edef\tikz@fig@name{tikz@f@\the\tikz@fig@count}% + \let\tikz@id@name\tikz@fig@name% + \fi% +}% + +\def\tikz@node@transformations{% + \ifx\tikz@time\pgfutil@empty% + \pgftransformshift{\tikz@node@at}% + \iftikz@fullytransformed% + \else% + \pgftransformresetnontranslations% + \fi% + \else% + \tikz@do@auto@anchor% + \tikz@timer% + \fi% + \tikz@transform% +}% + +\def\tikz@node@finish{% + \global\let\tikz@last@fig@name=\tikz@fig@name% + \global\let\tikz@after@path@smuggle=\tikz@after@path% + \global\setbox\tikz@tempbox=\box\tikz@figbox% + \global\setbox\tikz@tempbox@bg=\box\tikz@figbox@bg% + \endgroup\endgroup% + \setbox\tikz@figbox=\box\tikz@tempbox% + \setbox\tikz@figbox@bg=\box\tikz@tempbox@bg% + \global\pgflinewidth=\tikz@save@line@width% + \tikz@do@after@path@smuggle% + \tikz@node@is@picfalse + \tikz@do@after@node% +}% +\let\tikz@fig@continue@orig=\tikz@fig@continue + +\def\tikz@do@after@node{\tikz@scan@next@command}% + +\def\tikz@do@after@path@smuggle{% + \let\tikz@to@last@fig@name=\tikz@last@fig@name% + \let\tikz@to@use@whom=\tikz@to@use@last@fig@name% + \let\tikzlastnode=\tikz@last@fig@name% + \ifx\tikz@after@path@smuggle\pgfutil@empty% + \else% + \ifpgflatenodepositioning% + \expandafter\expandafter\expandafter\tikz@call@late% + \expandafter\expandafter\expandafter{\expandafter\tikz@last@fig@name\expandafter}\expandafter{\tikz@after@path@smuggle}% + \else% + \tikz@scan@next@command{\tikz@after@path@smuggle}\pgf@stop% + \fi% + \fi% +}% + +\def\tikz@call@late#1#2{\pgfnodepostsetupcode{#1}{\path[late options={name={#1},append after command={#2}}];}}% + +\newif\iftikz@do@align + +\def\tikz@signal@halign@check{% + \tikz@do@alignfalse + \ifx\tikz@text@width\pgfutil@empty% + \pgfkeysgetvalue{/tikz/node halign header}\tikz@align@header% + \ifx\tikz@align@header\pgfutil@empty% + \else% + \tikz@do@aligntrue% + \fi% + \fi% +} +\def\tikz@halign@check{% + \iftikz@do@align% + \setbox\tikz@align@aligned@box=\box\pgfutil@voidb@x% + \let\\=\tikz@align@newline% + \expandafter\tikz@start@align% + \fi% +}% +\def\tikz@align@newline{\pgfutil@protect\tikz@align@newline@}% +\def\tikz@align@newline@{\unskip\pgfutil@ifnextchar[\tikz@@align@newline{\tikz@@align@newline[0pt]}}% +\def\tikz@@align@newline[#1]{\egroup\tikz@align@continue\pgfmathparse{#1}\let\tikz@align@temp=\pgfmathresult\tikz@start@align}% +\let\tikz@align@aligned@box=\pgfnodeparttextbox +\let\tikz@align@line@box=\tikz@figbox + +\def\tikz@start@align{% + \setbox\tikz@align@line@box=\hbox\bgroup\bgroup% + \aftergroup\tikz@align@collectresetcolor\ignorespaces% +}% +\def\tikz@align@collectresetcolor{% + \pgfutil@ifnextchar\reset@color% + {\reset@color\afterassignment\tikz@align@collectresetcolor\let\tikz@temp=}% + {\tikz@align@end@check}% +}% +\def\tikz@align@end@check{% + \egroup% + \ifvoid\tikz@align@aligned@box% + \setbox\tikz@align@aligned@box=\box\tikz@align@line@box% + \else% + \setbox\tikz@align@aligned@box=\vbox{% + \expandafter\expandafter\expandafter\halign\expandafter\expandafter\expandafter{\tikz@align@header% + \box\tikz@align@aligned@box\cr% + \noalign{\vskip\tikz@align@temp pt}% + \unhbox\tikz@align@line@box\unskip\cr}}% + \fi% + \pgfutil@ifnextchar\tikz@align@continue{} + {% + \box\tikz@align@aligned@box% + \egroup% + }% +}% +\def\tikz@align@continue{\tikz@@align@continue}% +\let\tikz@@align@continue=\pgfutil@empty + + +\def\tikz@node@also lso{\pgfutil@ifnextchar[\tikz@node@also@opt{\tikz@node@also@opt[]}}% +\def\tikz@node@also@opt[#1]{ + \pgfutil@ifnextchar(% + {\tikz@node@also@opt@cont[#1]}% + {\tikzerror{Syntax error in node also: ``('' expected.}% + \tikz@scan@next@command}% +}% +\def\tikz@node@also@opt@cont[#1](#2){\tikzset{late options={name=#2,#1}}\tikz@scan@next@command}% + + + + +\def\tikz@nodepart{\pgfutil@ifnextchar[\tikz@@nodepart{\tikz@@nodepart[]}}% +\def\tikz@@nodepart[#1]#2{% + \tikz@atend@node% + \unskip% + \gdef\tikz@nodepart@options{#1}% + \gdef\tikz@nodepart@name{#2}% + \global\let\tikz@fig@continue=\tikz@nodepart@continue% + \pgfutil@ifnextchar x{\egroup\relax}{\egroup\relax}% +}% +\def\tikz@nodepart@continue{% + \global\let\tikz@fig@continue=\tikz@fig@continue@orig% + \ifx\tikz@nodepart@list\pgfutil@empty% + \let\tikz@nodepart@list\tikz@nodepart@name% + \else% + \edef\tikz@nodepart@list{\tikz@nodepart@list,\tikz@nodepart@name}% + \fi% + \expandafter\setbox\csname pgfnodepart\tikz@nodepart@name box\endcsname=\hbox% + \bgroup% + \pgfinterruptpicture% + \pgfsys@begin@text% + \pgfsys@text@to@black@hook% + \tikzset{every \tikz@nodepart@name\space node part/.try}% + \expandafter\tikzset\expandafter{\tikz@nodepart@options}% + \ifx\tikz@textopacity\pgfutil@empty% + \else% + \pgfsetfillopacity{\tikz@textopacity}% + \pgfsetstrokeopacity{\tikz@textopacity}% + \fi% + \ifx\tikz@textcolor\pgfutil@empty% + \else% + \pgfutil@colorlet{.}{\tikz@textcolor}% + \fi% + \pgfsetcolor{.}% + \ifx\tikz@text@width\pgfutil@empty% + \tikz@textfont% + \else% + \begingroup% + \pgfmathsetlength{\pgf@x}{\tikz@text@width}% + \pgfutil@minipage[t]{\pgf@x}\leavevmode\hbox{}% + \tikz@textfont% + \tikz@text@action% + \fi% + \bgroup% + \aftergroup\unskip% + \setbox\tikz@figbox=\box\pgfutil@voidb@x% + \setbox\tikz@figbox@bg=\box\pgfutil@voidb@x% + \tikz@uninstallcommands% + \iftikz@handle@active@code% + \tikz@orig@shorthands% + \let\tikz@orig@shorthands\pgfutil@empty% + \fi% + \ifnum\the\catcode`\;=\active\relax\expandafter\let\tikz@activesemicolon=\tikz@origsemi\fi% + \ifnum\the\catcode`\:=\active\relax\expandafter\let\tikz@activecolon=\tikz@origcolon\fi% + \ifnum\the\catcode`\|=\active\relax\expandafter\let\tikz@activebar=\tikz@origbar\fi% + \tikz@atbegin@node% + \aftergroup\tikz@fig@collectresetcolor% + \tikz@signal@halign@check% + \tikz@text@reset% + \tikz@halign@check% + \ignorespaces% +}% + + + +\def\tikz@nodes@start{% + \let\tikz@nodes@list\pgfutil@empty% + \iftikz@node@is@pic% + \def\tikz@nodes@collect{pic }% + \else% + \def\tikz@nodes@collect{node }% + \fi% + \tikz@nodes% +}% +\def\tikz@nodes foreach{\pgfutil@ifnextchar x\tikz@nodes@\tikz@nodes@}% +\def\tikz@nodes@#1in{% + \expandafter\def\expandafter\tikz@nodes@list\expandafter{\tikz@nodes@list\foreach#1in}% + \pgfutil@ifnextchar\bgroup\tikz@nodes@group\tikz@nodes@one% +}% +\def\tikz@nodes@one#1{% + \expandafter\def\expandafter\tikz@nodes@list\expandafter{\tikz@nodes@list#1}% + \pgfutil@ifnextchar f\tikz@nodes\tikz@nodes@scan% +}% +\def\tikz@nodes@group#1{% + \expandafter\def\expandafter\tikz@nodes@list\expandafter{\tikz@nodes@list{#1}}% + \pgfutil@ifnextchar f\tikz@nodes\tikz@nodes@scan% +}% +\def\tikz@nodes@scan{% + \pgfutil@ifnextchar a{\tikz@nodes@at}% + {\pgfutil@ifnextchar({\tikz@nodes@name}% + {\pgfutil@ifnextchar[{\tikz@nodes@opt}% + {\pgfutil@ifnextchar\bgroup{\tikz@nodes@main}% + {\tikzerror{Nodes must have a (possibly empty) label text}% + \tikz@fig@main{}}}}}}% + +\def\tikz@nodes@at at{\pgfutil@ifnextchar({\tikz@nodes@at@}{\tikz@nodes@at@@}}% +\def\tikz@nodes@at@#1){% + \expandafter\def\expandafter\tikz@nodes@collect\expandafter{\tikz@nodes@collect at#1)}% + \tikz@nodes@scan}% +\def\tikz@nodes@at@@#1{% + \expandafter\def\expandafter\tikz@nodes@collect\expandafter{\tikz@nodes@collect at#1}% + \tikz@nodes@scan}% + +\def\tikz@nodes@name#1){% + \expandafter\def\expandafter\tikz@nodes@collect\expandafter{\tikz@nodes@collect#1)}% + \tikz@nodes@scan}% +\def\tikz@nodes@opt#1]{% + \expandafter\def\expandafter\tikz@nodes@collect\expandafter{\tikz@nodes@collect#1]}% + \tikz@nodes@scan}% +\def\tikz@nodes@main#1{% + \iftikz@handle@active@nodes% + \iftikz@node@is@pic% + \expandafter\def\expandafter\tikz@nodes@collect\expandafter{\tikz@nodes@collect{#1}}% + \else% + \expandafter\def\expandafter\tikz@nodes@collect\expandafter{\tikz@nodes@collect{\scantokens{#1}}}% + \fi% + \else% + \expandafter\def\expandafter\tikz@nodes@collect\expandafter{\tikz@nodes@collect{#1}}% + \fi% + \global\setbox\tikz@tempbox=\box\tikz@figbox% + \global\setbox\tikz@tempbox@bg=\box\tikz@figbox@bg% + \tikz@nodes@list{% + \setbox\tikz@figbox=\box\tikz@tempbox% + \setbox\tikz@figbox@bg=\box\tikz@tempbox@bg% + \expandafter\tikz@scan@next@command\tikz@nodes@collect\pgfextra\relax% + \global\setbox\tikz@tempbox=\box\tikz@figbox% + \global\setbox\tikz@tempbox@bg=\box\tikz@figbox@bg% + }% + \setbox\tikz@figbox=\box\tikz@tempbox% + \setbox\tikz@figbox@bg=\box\tikz@tempbox@bg% + \tikz@scan@next@command% +}% + + + +\tikzset{late options/.code=\tikz@late@options{#1}}% +\def\tikz@late@options#1{% + \begingroup% + \iftikz@shapeborder% + \let\tikz@fig@name=\tikz@shapeborder@name% + \else% + \let\tikz@fig@name=\pgfutil@empty% + \fi% + \tikz@is@matrixfalse% + \let\tikz@options=\pgfutil@empty% + \tikz@clear@rdf@options% + \let\tikz@after@path=\pgfutil@empty% + \let\tikz@afternodepathoptions=\pgfutil@empty% + \let\tikz@alias=\pgfutil@empty% + \let\tikz@transform=\pgfutil@empty% + \tikz@decoratepathfalse% + \tikz@node@reset@hook% + \tikz@enable@node@quotes% + \tikzset{every node/.try,#1}% + \ifx\tikz@fig@name\pgfutil@empty% + \tikzerror{Late options must reference some existing node}% + \fi% + \tikz@node@begin@hook% + \tikz@alias% + \tikzgdlatenodeoptionacallback{\tikz@fig@name}% + \global\let\tikz@last@fig@name=\tikz@fig@name% + \global\let\tikz@after@path@smuggle=\tikz@after@path% + \endgroup% + \tikz@do@after@path@smuggle% +}% + + + +\def\tikz@auto@pre{% + \begingroup + \pgfresetnontranslationattimefalse + \ifpgfslopedattime + \pgfslopedattimefalse% + \else + \pgfslopedattimetrue% + \fi + \pgfallowupsidedownattimetrue% + \tikz@timer% + \pgf@x=\pgf@pt@aa pt% + \pgf@y=\pgf@pt@ab pt% + \pgfpointnormalised{}% +}% + +\def\tikz@auto@post{% + \global\let\tikz@anchor@smuggle=\tikz@anchor% + \endgroup% + \let\tikz@anchor=\tikz@anchor@smuggle% +}% + +\def\tikz@auto@anchor{% + \ifdim\pgf@x>0.05pt% + \ifdim\pgf@y>0.05pt% + \def\tikz@anchor{south east}% + \else\ifdim\pgf@y<-0.05pt% + \def\tikz@anchor{south west}% + \else + \def\tikz@anchor{south}% + \fi\fi% + \else\ifdim\pgf@x<-0.05pt% + \ifdim\pgf@y>0.05pt% + \def\tikz@anchor{north east}% + \else\ifdim\pgf@y<-0.05pt% + \def\tikz@anchor{north west}% + \else + \def\tikz@anchor{north}% + \fi\fi% + \else% + \ifdim\pgf@y>0pt% + \def\tikz@anchor{east}% + \else% + \def\tikz@anchor{west}% + \fi% + \fi\fi% +}% + +\def\tikz@auto@anchor@prime{% + \ifdim\pgf@x>0.05pt% + \ifdim\pgf@y>0.05pt% + \def\tikz@anchor{north west}% + \else\ifdim\pgf@y<-0.05pt% + \def\tikz@anchor{north east}% + \else + \def\tikz@anchor{north}% + \fi\fi% + \else\ifdim\pgf@x<-0.05pt% + \ifdim\pgf@y>0.05pt% + \def\tikz@anchor{south west}% + \else\ifdim\pgf@y<-0.05pt% + \def\tikz@anchor{south east}% + \else + \def\tikz@anchor{south}% + \fi\fi% + \else% + \ifdim\pgf@y>0pt% + \def\tikz@anchor{west}% + \else% + \def\tikz@anchor{east}% + \fi% + \fi\fi% +}% + + +\def\tikzgdeventcallback#1#2{}% +\def\tikzgdeventgroupcallback#1{}% +\def\tikzgdlatenodeoptionacallback#1{}% + + +\def\tikz@children{% + \let\tikz@children@list=\pgfutil@empty% + \tikznumberofchildren=0\relax% + \tikz@collect@children c}% + +\def\tikz@collect@children{\pgfutil@ifnextchar c{\tikz@collect@children@cchar}{\tikz@children@collected}}% +\def\tikz@collect@children@cchar c{\pgfutil@ifnextchar h{\tikz@collect@child}{\tikz@children@collected c}}% +\def\tikz@collect@child hild{\pgfutil@ifnextchar[{\tikz@collect@childA}{\tikz@collect@childA[]}}% +\def\tikz@collect@childA[#1]{\pgfutil@ifnextchar f{\tikz@collect@children@foreach[#1]}{\tikz@collect@childB[#1]}}% +\def\tikz@collect@childB[#1]{% + \advance\tikznumberofchildren by1\relax + \expandafter\def\expandafter\tikz@children@list\expandafter{\tikz@children@list \tikz@childnode[#1]}% + \pgfutil@ifnextchar\bgroup{\tikz@collect@child@code}{\tikz@collect@child@code{}}}% +\def\tikz@collect@child@code#1{% + \expandafter\def\expandafter\tikz@children@list\expandafter{\tikz@children@list{#1}}% + \tikz@collect@children% +}% +\def\tikz@collect@children@foreach[#1]foreach#2in#3{% + \pgfutil@ifnextchar\bgroup{\tikz@collect@children@foreachA{#1}{#2}{#3}}{\tikz@collect@children@foreachA{#1}{#2}{#3}{}}}% +\def\tikz@collect@children@foreachA#1#2#3#4{% + \expandafter\def\expandafter\tikz@children@list\expandafter + {\tikz@children@list\tikz@childrennodes[#1]{#2}{#3}{#4}}% + \c@pgf@counta=\tikznumberofchildren% + \foreach#2in{#3}% + {% + \global\advance\c@pgf@counta by1\relax% + }% + \tikznumberofchildren=\c@pgf@counta% + \tikz@collect@children% +}% +\long\def\tikz@children@collected{% + \begingroup% + \advance\tikztreelevel by 1\relax% + \tikzgdeventgroupcallback{descendants}% + \let\tikz@options=\pgfutil@empty% + \tikz@clear@rdf@options% + \let\tikz@transform=\pgfutil@empty% + \tikzset{level/.try=\the\tikztreelevel,level \the\tikztreelevel/.try}% + \tikz@transform% + \let\tikz@transform=\relax% + \let\tikzparentnode=\tikz@last@fig@name% + \ifx\tikz@grow\relax\else% + \pgftransformshift{\pgfpointanchor{\tikzparentnode}{\tikz@growth@anchor}}% + \fi% + \tikznumberofcurrentchild=0\relax% + \tikz@children@list% + \global\setbox\tikz@tempbox=\box\tikz@figbox% + \global\setbox\tikz@tempbox@bg=\box\tikz@figbox@bg% + \endgroup% + \setbox\tikz@figbox=\box\tikz@tempbox% + \setbox\tikz@figbox@bg=\box\tikz@tempbox@bg% + \tikz@scan@next@command% +}% + +\def\tikz@childrennodes[#1]#2#3#4{% + \c@pgf@counta=\tikznumberofcurrentchild\relax% + \setbox\tikz@tempbox=\box\tikz@figbox% + \setbox\tikz@tempbox@bg=\box\tikz@figbox@bg% + \foreach#2in{#3}{% + \tikznumberofcurrentchild=\c@pgf@counta\relax% + \setbox\tikz@figbox=\box\tikz@tempbox% + \setbox\tikz@figbox@bg=\box\tikz@tempbox@bg% + \tikz@childnode[#1]{#4}% + \global\c@pgf@counta=\tikznumberofcurrentchild\relax% + \global\setbox\tikz@tempbox=\box\tikz@figbox% + \global\setbox\tikz@tempbox@bg=\box\tikz@figbox@bg% + }% + \tikznumberofcurrentchild=\c@pgf@counta\relax% + \setbox\tikz@figbox=\box\tikz@tempbox% + \setbox\tikz@figbox@bg=\box\tikz@tempbox@bg% +}% + + + +\def\tikz@childnode[#1]#2{% + \advance\tikznumberofcurrentchild by1\relax% + {\tikzset{every child/.try,#1}\expandafter}% + \iftikz@child@missing% + \tikzgdeventcallback{node}{}% + \else% + \setbox\tikz@whichbox=\hbox\bgroup% + \unhbox\tikz@whichbox% + \hbox\bgroup\bgroup% + \pgfinterruptpath% + \pgfscope% + \let\tikz@transform=\pgfutil@empty% + \tikzset{every child/.try,#1}% + \tikz@options% + \tikz@transform% + \let\tikz@transform=\relax% + \tikz@grow% + \edef\tikz@parent@node@name{[name=\tikzparentnode-\the\tikznumberofcurrentchild,style=every child node]}% + \def\tikz@child@node@text{[shape=coordinate]{}} + \tikz@parse@child@node#2\pgf@stop% + \expandafter\expandafter\expandafter\node + \expandafter\tikz@parent@node@name + \tikz@child@node@text + \pgfextra{\global\let\tikz@childnode@name=\tikz@last@fig@name};% + \let\tikzchildnode=\tikz@childnode@name% + {% + \def\tikz@edge@to@parent@needed{edge from parent} + \ifx\tikz@child@node@rest\pgfutil@empty% + \path edge from parent;% + \else% + \path \tikz@child@node@rest \tikz@edge@to@parent@needed;% + \fi% + }% + \endpgfscope% + \endpgfinterruptpath% + \egroup\egroup% + \egroup% + \fi% +}% + +\def\tikz@parse@child@node{% + \pgfutil@ifnextchar n{\tikz@parse@child@node@n}% + {\pgfutil@ifnextchar c{\tikz@parse@child@node@c}% + {\pgfutil@ifnextchar\pgf@stop\tikz@parse@child@node@rest\tikz@parse@child@node@expand}}}% +\def\tikz@parse@child@node@expand{% + \advance\tikz@expandcount by-1\relax% + \ifnum\tikz@expandcount<0\relax% + \expandafter\tikz@parse@child@node@rest% + \else% + \expandafter\expandafter\expandafter\tikz@parse@child@node% + \fi% +}% +\def\tikz@parse@child@node@rest#1\pgf@stop{\tikz@resetexpandcount\def\tikz@child@node@rest{#1}}% +\def\tikz@parse@child@node@c c{\tikz@resetexpandcount\pgfutil@ifnextchar o{\tikz@parse@child@node@co}{\tikz@parse@child@node@rest c}}% +\def\tikz@parse@child@node@co o{\pgfutil@ifnextchar o{\tikz@parse@child@node@coordinate}{\tikz@parse@child@node@rest co}}% +\def\tikz@parse@child@node@coordinate ordinate{% + \pgfutil@ifnextchar ({\tikz@@parse@child@node@coordinate}{% + \def\tikz@child@node@text{[shape=coordinate]{}}% + \tikz@parse@child@node@rest}}% +\def\tikz@@parse@child@node@coordinate(#1){% + \pgfutil@ifnextchar a{\tikz@p@c@n@c@at(#1)}{% + \def\tikz@child@node@text{[shape=coordinate,name=#1]{}}% + \tikz@parse@child@node@rest}}% +\def\tikz@p@c@n@c@at(#1)at#2({% + \def\tikz@child@node@text@pre{[shape=coordinate,name=#1]at}% + \tikz@scan@one@point\tikz@p@c@n@c@at@math(% +}% +\def\tikz@p@c@n@c@at@math#1{% + \pgf@process{#1}% + \edef\tikz@marshal{(\the\pgf@x,\the\pgf@y){}}% + \expandafter\expandafter\expandafter\def% + \expandafter\expandafter\expandafter\tikz@child@node@text% + \expandafter\expandafter\expandafter{\expandafter\tikz@child@node@text@pre\tikz@marshal}% + \tikz@parse@child@node@rest% +}% +\def\tikz@parse@child@node@n node{\tikz@resetexpandcount% + \let\tikz@child@node@text=\pgfutil@empty% + \tikz@p@c@s}% +\def\tikz@p@c@s{% + \pgfutil@ifnextchar a{\tikz@p@c@s@at} + {\pgfutil@ifnextchar ({\tikz@p@c@s@paran} + {\pgfutil@ifnextchar [{\tikz@p@c@s@bra} + {\pgfutil@ifnextchar \bgroup{\tikz@p@c@s@group} + {\tikzerror{Cannot parse this node}}}}}}% +\def\tikz@p@c@s@at at#1({% + \tikz@scan@one@point\tikz@p@c@s@at@math(% +}% +\def\tikz@p@c@s@at@math#1{% + \pgf@process{#1}% + \edef\tikz@marshal{ at(\the\pgf@x,\the\pgf@y)}% + \expandafter\expandafter\expandafter\def% + \expandafter\expandafter\expandafter\tikz@child@node@text% + \expandafter\expandafter\expandafter{\expandafter\tikz@child@node@text\tikz@marshal} + \tikz@p@c@s}% +\def\tikz@p@c@s@paran(#1){% + \expandafter\def\expandafter\tikz@child@node@text\expandafter{\tikz@child@node@text(#1)} + \tikz@p@c@s}% +\def\tikz@p@c@s@bra[#1]{% + \expandafter\def\expandafter\tikz@child@node@text\expandafter{\tikz@child@node@text[#1]} + \tikz@p@c@s}% +\def\tikz@p@c@s@group#1{% + \iftikz@handle@active@nodes% + \expandafter\def\expandafter\tikz@child@node@text\expandafter{\tikz@child@node@text{\scantokens{#1}}}% + \else% + \expandafter\def\expandafter\tikz@child@node@text\expandafter{\tikz@child@node@text{#1}} + \fi% + \tikz@parse@child@node@rest% +}% + + + +\def\tikz@decoration ecorate{% + \pgfutil@ifnextchar[{\tikz@lib@decoration}{\tikz@lib@decoration[]}% +}% + +\def\tikz@lib@decoration[#1]#2{\tikzerror{You need to load a decoration library}}% + +\def\tikz@lib@dec@decorate@path{\tikzerror{You need to load a decoration library}}% + + + +\def\tikz@let@command et#1in{% + \tikzerror{You need to say \string\usetikzlibrary{calc} to use the let command}% + \tikz@scan@next@command% +}% + + +\newif\iftikz@node@is@pic +\def\tikz@subpicture ic{\tikz@node@is@pictrue\tikz@scan@next@command node}% +\def\tikz@subpicture@handle#1{% + \pgfkeys@spdef\tikz@temp{#1}% + \expandafter\tikz@subpicture@handle@\expandafter{\tikz@temp}% +}% +\def\tikz@subpicture@handle@#1{ + \pgfkeys{/tikz/pics/.cd,#1}% + \tikz@node@transformations% + \let\tikz@transform=\relax% + \let\tikz@picmode\tikz@mode% + \tikzset{name prefix ../.style/.expanded={/tikz/name prefix=\pgfkeysvalueof{/tikz/name prefix}}}% + \ifx\tikz@fig@name\pgfutil@empty\else% + \tikzset{name prefix/.expanded=\tikz@fig@name}% + \fi% + \pgfkeysvalueof{/tikz/pics/setup code}% + \pgfkeysgetvalue{/tikz/pics/code}{\tikz@pic@code} + \ifx\tikz@pic@code\pgfutil@empty\else% + \setbox\tikz@whichbox=\hbox\bgroup% + \unhbox\tikz@whichbox% + \hbox\bgroup + \bgroup% + \pgfinterruptpath% + \pgfscope% + \tikz@options% + \setbox\tikz@figbox=\box\pgfutil@voidb@x% + \setbox\tikz@figbox@bg=\box\pgfutil@voidb@x% + \tikz@atbegin@scope% + \scope[every pic/.try]% + \tikz@pic@code% + \endscope% + \tikz@atend@scope% + \endpgfscope% + \endpgfinterruptpath% + \egroup + \egroup% + \egroup% + \fi% + \pgfkeysgetvalue{/tikz/pics/foreground code}{\tikz@pic@code} + \ifx\tikz@pic@code\pgfutil@empty\else% + \setbox\tikz@figbox=\hbox\bgroup% + \unhbox\tikz@figbox% + \hbox\bgroup + \bgroup% + \pgfinterruptpath% + \pgfscope% + \tikz@options% + \setbox\tikz@figbox=\box\pgfutil@voidb@x% + \setbox\tikz@figbox@bg=\box\pgfutil@voidb@x% + \tikz@atbegin@scope% + \scope[every front pic/.try]% + \tikz@pic@code% + \endscope% + \tikz@atend@scope% + \endpgfscope% + \endpgfinterruptpath% + \egroup + \egroup% + \egroup% + \fi% + \pgfkeysgetvalue{/tikz/pics/background code}{\tikz@pic@code} + \ifx\tikz@pic@code\pgfutil@empty\else% + \setbox\tikz@figbox@bg=\hbox\bgroup% + \unhbox\tikz@figbox@bg% + \hbox\bgroup + \bgroup% + \pgfinterruptpath% + \pgfscope% + \tikz@options% + \setbox\tikz@figbox=\box\pgfutil@voidb@x% + \setbox\tikz@figbox@bg=\box\pgfutil@voidb@x% + \tikz@atbegin@scope% + \scope[every behind pic/.try]% + \tikz@pic@code% + \endscope% + \tikz@atend@scope% + \endpgfscope% + \endpgfinterruptpath% + \egroup + \egroup% + \egroup% + \fi% + \tikz@node@finish% +}% +\tikzset{ + pic actions/.code=\tikz@addmode{\tikz@picmode} +}% + +\tikzset{ + pics/setup code/.initial=, + pics/code/.initial=, + pics/background code/.initial=, + pics/foreground code/.initial= +}% + + +\def\tikzdeclarepic#1#2{\pgfkeys{/tikz/#1/.cd,#2}}% + +\pgfkeysdef{/handlers/.pic}{% + \edef\pgf@temp{\pgfkeyscurrentpath}% + \edef\pgf@temp{\expandafter\tikz@smuggle@pics@in\pgf@temp\pgf@stop}% + \expandafter\pgfkeys\expandafter{\pgf@temp/.style={code={#1}}}% +}% +\def\tikz@smuggle@pics@in/tikz/#1\pgf@stop{/tikz/pics/#1}% + + +\def\tikz@timer@line{% + \pgftransformlineattime{\tikz@time}{\tikz@timer@start}{\tikz@timer@end}% +}% + +\def\tikz@timer@vhline{% + \ifdim\tikz@time pt<0.5pt% + \pgf@process{\tikz@timer@start}% + \pgf@xa=\pgf@x% + \pgf@ya=\pgf@y% + \pgf@process{\tikz@timer@end}% + \pgf@xb=\tikz@time pt% + \pgf@xb=2\pgf@xb% + \edef\tikz@marshal{\noexpand\pgftransformlineattime{\pgf@sys@tonumber{\pgf@xb}}{\noexpand\tikz@timer@start}{% + \noexpand\pgfqpoint{\the\pgf@xa}{\the\pgf@y}}}% + \tikz@marshal% + \else% + \pgf@process{\tikz@timer@start}% + \pgf@xa=\pgf@x% + \pgf@ya=\pgf@y% + \pgf@process{\tikz@timer@end}% + \pgf@xb=\tikz@time pt% + \pgf@xb=2\pgf@xb% + \advance\pgf@xb by-1pt% + \edef\tikz@marshal{\noexpand\pgftransformlineattime{\pgf@sys@tonumber{\pgf@xb}}% + {\noexpand\pgfqpoint{\the\pgf@xa}{\the\pgf@y}}{\noexpand\tikz@timer@end}}% + \tikz@marshal% + \fi% +}% + +\def\tikz@timer@hvline{% + \ifdim\tikz@time pt<0.5pt% + \pgf@process{\tikz@timer@start}% + \pgf@xa=\pgf@x% + \pgf@ya=\pgf@y% + \pgf@process{\tikz@timer@end}% + \pgf@xb=\tikz@time pt% + \pgf@xb=2\pgf@xb% + \edef\tikz@marshal{\noexpand\pgftransformlineattime{\pgf@sys@tonumber{\pgf@xb}}{\noexpand\tikz@timer@start}{% + \noexpand\pgfqpoint{\the\pgf@x}{\the\pgf@ya}}}% + \tikz@marshal% + \else% + \pgf@process{\tikz@timer@start}% + \pgf@xa=\pgf@x% + \pgf@ya=\pgf@y% + \pgf@process{\tikz@timer@end}% + \pgf@xb=\tikz@time pt% + \pgf@xb=2\pgf@xb% + \advance\pgf@xb by-1pt% + \edef\tikz@marshal{\noexpand\pgftransformlineattime{\pgf@sys@tonumber{\pgf@xb}}% + {\noexpand\pgfqpoint{\the\pgf@x}{\the\pgf@ya}}{\noexpand\tikz@timer@end}}% + \tikz@marshal% + \fi% +}% + +\def\tikz@timer@curve{% + \pgftransformcurveattime{\tikz@time}{\tikz@timer@start}{\tikz@timer@cont@one}{\tikz@timer@cont@two}{\tikz@timer@end}% +}% + + +\def\tikz@timer@arc{% + \pgfmathcos@{\tikz@timer@start@angle}% + \let\tikz@angle@cos\pgfmathresult% + \pgfmathsin@{\tikz@timer@start@angle}% + \let\tikz@angle@sin\pgfmathresult% + \pgftransformarcaxesattime{\tikz@time}{% + \pgfpointdiff{% + \pgfpointadd{% + \pgfpointscale{\tikz@angle@cos}{\tikz@timer@zero@axis}% + }{% + \pgfpointscale{\tikz@angle@sin}{\tikz@timer@ninety@axis}% + }% + }% + {\tikz@timer@start}% + }% + {\tikz@timer@zero@axis}% + {\tikz@timer@ninety@axis}% + {\tikz@timer@start@angle}{\tikz@timer@end@angle}% +}% + + + + +\def\tikzdeclarecoordinatesystem#1#2{% + \expandafter\def\csname tikz@parse@cs@#1\endcsname(##1){% + \pgf@process{% + #2% + \global\let\tikz@smubble@b=\tikz@shapeborder@name% + }% + \let\tikz@shapeborder@name=\tikz@smubble@b% + \edef\tikz@return@coordinate{\noexpand\pgfqpoint{\the\pgf@x}{\the\pgf@y}}}% +}% +\def\tikzaliascoordinatesystem#1#2{% + \edef\pgf@marshal{\noexpand\let\expandafter\noexpand\csname + tikz@parse@cs@#1\endcsname=\expandafter\noexpand\csname + tikz@parse@cs@#2\endcsname}% + \pgf@marshal% +}% + + + +\tikzdeclarecoordinatesystem{canvas} +{% + \tikzset{cs/.cd,x=0pt,y=0pt,#1}% + \pgfpoint{\tikz@cs@x}{\tikz@cs@y}% +}% + +\tikzdeclarecoordinatesystem{canvas polar} +{% + \tikzset{cs/.cd,angle=0,radius=0cm,#1}% + \pgfpointpolar{\tikz@cs@angle}{\tikz@cs@xradius and \tikz@cs@yradius}% +}% + +\tikzdeclarecoordinatesystem{xyz} +{% + \tikzset{cs/.cd,x=0,y=0,z=0,#1}% + \pgfpointxyz{\tikz@cs@x}{\tikz@cs@y}{\tikz@cs@z}% +}% + +\tikzdeclarecoordinatesystem{xyz polar} +{% + \tikzset{cs/.cd,angle=0,radius=0,#1}% + \pgfpointpolarxy{\tikz@cs@angle}{\tikz@cs@xradius and \tikz@cs@yradius}% +}% +\tikzaliascoordinatesystem{xy polar}{xyz polar}% + + +\tikzdeclarecoordinatesystem{node} +{% + \tikzset{cs/.cd,name=,anchor=none,angle=none,#1}% + \ifx\tikz@cs@anchor\tikz@nonetext% + \ifx\tikz@cs@angle\tikz@nonetext% + \expandafter\ifx\csname pgf@sh@ns@\tikz@cs@node\endcsname\tikz@coordinate@text% + \else + \aftergroup\tikz@shapebordertrue% + \edef\tikz@shapeborder@name{\tikz@pp@name{\tikz@cs@node}}% + \fi% + \pgfpointanchor{\tikz@pp@name{\tikz@cs@node}}{center}% + \else% + \pgfpointanchor{\tikz@pp@name{\tikz@cs@node}}{\tikz@cs@angle}% + \fi% + \else% + \pgfpointanchor{\tikz@pp@name{\tikz@cs@node}}{\tikz@cs@anchor}% + \fi% +}% + +\tikzset{cs/first line/.code=\def\tikz@cs@line@a{#1}\def\tikz@cs@type@a{line}}% +\tikzset{cs/second line/.code=\def\tikz@cs@line@b{#1}\def\tikz@cs@type@b{line}}% + +\tikzset{cs/first node/.code=\tikz@cs@unpack{\tikz@cs@node@a}{\tikz@cs@type@a}{#1}}% +\tikzset{cs/second node/.code=\tikz@cs@unpack{\tikz@cs@node@b}{\tikz@cs@type@b}{#1}}% + +\def\tikz@cs@unpack#1#2#3{% + \expandafter\ifx\csname pgf@sh@ns@#3\endcsname\relax% + \tikzerror{Undefined node ``#3''}% + \else% + \def#1{#3}% + \edef#2{\csname pgf@sh@ns@#3\endcsname}% + \fi% +}% + +\tikzset{cs/solution/.initial=1}% + +\tikzset{cs/horizontal line through/.store in=\tikz@cs@hori@line}% +\tikzset{cs/vertical line through/.store in=\tikz@cs@vert@line}% + +\tikzdeclarecoordinatesystem{intersection} +{% + \tikzset{cs/.cd,#1}% + \expandafter\ifx\csname tikz@intersect@\tikz@cs@type@a @and@\tikz@cs@type@b\endcsname\relax% + \tikzerror{I do not know how to compute the intersection + of a \tikz@cs@type@a and a \tikz@cs@type@b. Try saying + \string\usetikzlibrary{calc}}% + \pgfpointorigin% + \else% + \csname tikz@intersect@\tikz@cs@type@a @and@\tikz@cs@type@b\endcsname% + \fi% +}% + +\def\tikz@intersect@line@and@line{% + \expandafter\tikz@scan@one@point\expandafter\tikz@parse@line\tikz@cs@line@a% + \pgf@xa=\pgf@xc% + \pgf@ya=\pgf@yc% + \pgf@xb=\pgf@x% + \pgf@yb=\pgf@y% + \expandafter\tikz@scan@one@point\expandafter\tikz@parse@line\tikz@cs@line@b% + \edef\pgf@marshal{% + {\noexpand\pgfpointintersectionoflines% + {\noexpand\pgfqpoint{\the\pgf@xa}{\the\pgf@ya}}% + {\noexpand\pgfqpoint{\the\pgf@xb}{\the\pgf@yb}}% + {\noexpand\pgfqpoint{\the\pgf@xc}{\the\pgf@yc}}% + {\noexpand\pgfqpoint{\the\pgf@x}{\the\pgf@y}}}}% + \pgf@marshal% +}% + +\def\tikz@parse@line#1--{% + \pgf@process{#1}% + \pgf@xc=\pgf@x% + \pgf@yc=\pgf@y% + \tikz@scan@one@point\pgf@process% +}% + + +\tikzdeclarecoordinatesystem{perpendicular} +{% + \tikzset{cs/.cd,#1}% + \expandafter\tikz@scan@one@point\expandafter\tikz@parse@intersection@a\tikz@cs@hori@line% + \expandafter\tikz@scan@one@point\expandafter\tikz@parse@intersection@b\tikz@cs@vert@line% + \pgfqpoint{\the\pgf@xb}{\the\pgf@ya} +}% + +\tikzdeclarecoordinatesystem{barycentric} +{% + {% + \pgf@xa=0pt% + \pgf@ya=0pt% + \pgf@xb=0pt% + \tikz@bary@dolist#1,=,% + \pgfmathparse{1/\the\pgf@xb}% + \global\pgf@x=\pgfmathresult\pgf@xa% + \global\pgf@y=\pgfmathresult\pgf@ya% + }% +}% + +\def\tikz@bary@dolist#1=#2,{% + \def\tikz@temp{#1}% + \ifx\tikz@temp\pgfutil@empty% + \else + \pgf@process{\pgfpointanchor{#1}{center}}% + \pgfmathparse{#2}% + \advance\pgf@xa by\pgfmathresult\pgf@x% + \advance\pgf@ya by\pgfmathresult\pgf@y% + \advance\pgf@xb by\pgfmathresult pt% + \expandafter\tikz@bary@dolist% + \fi% +}% + +\tikzset{cs/x/.store in=\tikz@cs@x}% +\tikzset{cs/y/.store in=\tikz@cs@y}% +\tikzset{cs/z/.store in=\tikz@cs@z}% +\tikzset{cs/angle/.store in=\tikz@cs@angle}% +\tikzset{cs/x radius/.store in=\tikz@cs@xradius}% +\tikzset{cs/y radius/.store in=\tikz@cs@yradius}% +\tikzset{cs/radius/.style={/tikz/cs/x radius={#1},/tikz/cs/y radius={#1}}}% +\tikzset{cs/name/.store in=\tikz@cs@node}% +\tikzset{cs/anchor/.store in=\tikz@cs@anchor}% + + + + + +\def\tikz@last@position{\pgfqpoint{\tikz@lastx}{\tikz@lasty}}% +\def\tikz@last@position@saved{\pgfqpoint{\tikz@lastxsaved}{\tikz@lastysaved}}% + +\def\tikz@make@last@position#1{% + \pgf@process{#1}% + \tikz@lastx=\pgf@x\relax% + \tikz@lasty=\pgf@y\relax% + \iftikz@updatecurrent% + \tikz@lastxsaved=\pgf@x\relax% + \tikz@lastysaved=\pgf@y\relax% + \fi% + \iftikz@updatenext + \tikz@updatecurrenttrue% + \fi +}% + +\newif\iftikz@updatecurrent +\tikz@updatecurrenttrue +\newif\iftikz@updatenext +\tikz@updatenexttrue + + + + +\def\tikz@scan@one@point#1{% + \let\tikz@to@use@whom=\tikz@to@use@last@coordinate% + \tikz@shapeborderfalse% + \pgfutil@ifnextchar+{\tikz@scan@relative#1}{\tikz@scan@absolute#1}}% +\def\tikz@scan@absolute#1{% + \pgfutil@ifnextchar({\tikz@scan@@absolute#1}% + {% + \advance\tikz@expandcount by -1 + \ifnum\tikz@expandcount<0\relax% + \let\pgfutil@next=\tikz@@scangiveup% + \else% + \let\pgfutil@next=\tikz@@scanexpand% + \fi% + \pgfutil@next{#1}% + }% +}% +\def\tikz@@scanexpand#1{\expandafter\tikz@scan@one@point\expandafter#1}% +\def\tikz@@scangiveup#1{\tikzerror{Cannot parse this coordinate}#1{\pgfpointorigin}}% +\def\tikz@scan@@absolute#1({% + \pgfutil@ifnextchar[% + {\def\tikz@scan@point@recall{#1}\tikz@scan@options}% + {\tikz@@@scan@@absolute#1(}% +}% + +\def\tikz@scan@options[#1]#2{% + \def\tikz@scan@point@options{#1}% + \tikz@@@scan@@absolute\tikz@scan@handle@options(#2% +}% + +\def\tikz@scan@handle@options#1{% + {% + \pgftransformreset% + \let\tikz@transform=\pgfutil@empty% + \expandafter\tikzset\expandafter{\tikz@scan@point@options}% + \tikz@transform% + \pgf@process{\pgfpointtransformed{#1}}% + \xdef\tikz@marshal{\expandafter\noexpand\tikz@scan@point@recall{\noexpand\pgfqpoint{\the\pgf@x}{\the\pgf@y}}}% + }% + \tikz@marshal% +}% + +\def\tikz@@@scan@@absolute#1({% + \pgfutil@ifnextchar{$}% + {\tikz@parse@calculator#1(} + {\tikz@scan@no@calculator#1(}% +}% +\def\tikz@scan@no@calculator#1(#2){% + \edef\tikz@scan@point@coordinate{(#2)}% + \expandafter\tikz@@scan@@no@calculator\expandafter#1\tikz@scan@point@coordinate% +}% +\def\tikz@@scan@@no@calculator#1(#2){% + \pgfutil@in@{cs:}{#2}% + \ifpgfutil@in@% + \let\pgfutil@next\tikz@parse@coordinatesystem% + \else% + \pgfutil@in@{intersection }{#2}% + \ifpgfutil@in@% + \let\pgfutil@next\tikz@parse@intersection% + \else% + \pgfutil@in@|{#2}% + \ifpgfutil@in@ + \pgfutil@in@{-|}{#2}% + \ifpgfutil@in@ + \let\pgfutil@next\tikz@parse@hv% + \else% + \let\pgfutil@next\tikz@parse@vh% + \fi% + \else% + \pgfutil@in@:{#2}% + \ifpgfutil@in@ + \let\pgfutil@next\tikz@parse@polar% + \else% + \pgfutil@in@,{#2}% + \ifpgfutil@in@% + \let\pgfutil@next\tikz@parse@regular% + \else% + \let\pgfutil@next\tikz@parse@node% + \fi% + \fi% + \fi% + \fi% + \fi% + \pgfutil@next#1(#2)% +}% + +\def\tikz@parse@calculator#1($#2$){% + \tikzerror{You need to say \string\usetikzlibrary{calc} for coordinate calculation}% + #1{\pgfpointorigin}% +}% + +\def\tikz@parse@coordinatesystem#1(#2 cs:#3){% + \let\tikz@return@coordinate=\pgfpointorigin% + \pgfutil@ifundefined{tikz@parse@cs@#2} + {\tikzerror{Unknown coordinate system '#2'}} + {\csname tikz@parse@cs@#2\endcsname(#3)}% + \expandafter#1\expandafter{\tikz@return@coordinate}% +}% + + +\newif\iftikz@isdimension +\def\tikz@checkunit#1{% + \pgfmathparse{#1}% + \let\iftikz@isdimension=\ifpgfmathunitsdeclared% +}% + +\def\tikz@parse@polar#1(#2:#3){% + \pgfutil@ifundefined{tikz@polar@dir@#2} + {\tikz@@parse@polar#1({#2}:{#3})} + {\tikz@@parse@polar#1(\csname tikz@polar@dir@#2\endcsname:{#3})}% +}% +\def\tikz@@parse@polar#1(#2:#3){% + \pgfutil@in@{ and }{#3}% + \ifpgfutil@in@% + \edef\tikz@args{({#2}:#3)}% + \else% + \edef\tikz@args{({#2}:{#3} and {#3})}% + \fi% + \expandafter\tikz@@@parse@polar\expandafter#1\tikz@args% +}% +\def\tikz@@@parse@polar#1(#2:#3 and #4){% + \tikz@checkunit{#3}% + \iftikz@isdimension% + \tikz@checkunit{#4}% + \iftikz@isdimension% + \def\tikz@next{#1{\pgfpointpolar{#2}{#3 and #4}}}% + \else% + \tikzerror{You cannot mix dimension and dimensionless values for polar coordinates} + \def\tikz@next{#1{\pgfpointorigin}}% + \fi% + \else% + \tikz@checkunit{#4}% + \iftikz@isdimension% + \tikzerror{You cannot mix dimension and dimensionless values for polar coordinates} + \def\tikz@next{#1{\pgfpointorigin}}% + \else% + \def\tikz@next{#1{\pgfpointpolarxy{#2}{#3 and #4}}}% + \fi% + \fi% + \tikz@next% +}% +\def\tikz@polar@dir@up{90}% +\def\tikz@polar@dir@down{-90}% +\def\tikz@polar@dir@left{180}% +\def\tikz@polar@dir@right{0}% +\def\tikz@polar@dir@north{90}% +\def\tikz@polar@dir@south{-90}% +\def\tikz@polar@dir@east{0}% +\def\tikz@polar@dir@west{180}% +\expandafter\def\csname tikz@polar@dir@north east\endcsname{45}% +\expandafter\def\csname tikz@polar@dir@north west\endcsname{135}% +\expandafter\def\csname tikz@polar@dir@south east\endcsname{-45}% +\expandafter\def\csname tikz@polar@dir@south west\endcsname{-135}% + + +\def\tikz@parse@regular#1(#2,{% + \pgfutil@ifnextchar\bgroup{\tikz@@parse@regular#1{#2}}{\tikz@@@parse@regular#1{#2}}% +}% +\def\tikz@@parse@regular#1#2#3{% + \pgfutil@ifnextchar[{% + \tikz@@@parse@regular#1{#2}{#3}}% + {\tikz@@@parse@regular#1{#2}{{#3}}}}% + +\def\tikz@@@parse@regular#1#2#3){% + \pgfutil@in@,{#3}% + \ifpgfutil@in@% + \tikz@parse@splitxyz{#1}{#2}#3,% + \else% + \tikz@checkunit{#2}% + \iftikz@isdimension% + \tikz@checkunit{#3}% + \iftikz@isdimension% + \def\pgfutil@next{#1{\pgfpoint{#2}{#3}}}% + \else% + \def\pgfutil@next{#1{\pgfpointadd{\pgfpoint{#2}{0pt}}{\pgfpointxy{0}{#3}}}}% + \fi% + \else% + \tikz@checkunit{#3}% + \iftikz@isdimension% + \def\pgfutil@next{#1{\pgfpointadd{\pgfpoint{0pt}{#3}}{\pgfpointxy{#2}{0}}}}% + \else% + \def\pgfutil@next{#1{\pgfpointxy{#2}{#3}}}% + \fi% + \fi% + \fi% + \pgfutil@next% +}% + +\def\tikz@parse@splitxyz#1#2#3,#4,{% + \def\pgfutil@next{#1{\pgfpointxyz{#2}{#3}{#4}}}% +}% + +\def\tikz@coordinate@text{coordinate}% + +\def\tikz@parse@node#1(#2){% + \pgfutil@in@.{#2}% + \ifpgfutil@in@ + \tikz@calc@anchor#2\tikz@stop% + \else% + \tikz@calc@anchor#2.center\tikz@stop% + \ifcsname pgf@sh@ns@\tikz@pp@name{#2}\endcsname + \expandafter\ifx\csname pgf@sh@ns@\tikz@pp@name{#2}\endcsname\tikz@coordinate@text% + \else + \tikz@shapebordertrue% + \def\tikz@shapeborder@name{\tikz@pp@name{#2}}% + \fi% + \else\ifcsname pgf@sh@ns@#2\endcsname + \expandafter\ifx\csname pgf@sh@ns@#2\endcsname\tikz@coordinate@text% + \else + \tikz@shapebordertrue% + \def\tikz@shapeborder@name{#2}% + \fi% + \fi\fi + \fi% + \edef\tikz@marshal{\noexpand#1{\noexpand\pgfqpoint{\the\pgf@x}{\the\pgf@y}}}% + \tikz@marshal% +}% + +\def\tikz@calc@anchor#1.#2\tikz@stop{% + \ifcsname pgf@sh@ns@\tikz@pp@name{#1}\endcsname% + \pgfpointanchor{\tikz@pp@name{#1}}{#2}% + \else + \pgfpointanchor{#1}{#2}% + \fi +}% + + +\def\tikz@parse@hv#1(#2){% + \pgfutil@in@{ -| }{#2}% + \ifpgfutil@in@% + \let\tikz@next=\tikz@parse@hvboth% + \else% + \pgfutil@in@{ -|}{#2}% + \ifpgfutil@in@% + \let\tikz@next=\tikz@parse@hvleft% + \else% + \pgfutil@in@{-| }{#2}% + \ifpgfutil@in@% + \let\tikz@next=\tikz@parse@hvright% + \else% + \let\tikz@next=\tikz@parse@hvdone% + \fi% + \fi% + \fi% + \tikz@next#1(#2)}% +\def\tikz@parse@hvboth#1(#2 -| #3){\tikz@parse@vhdone#1({#3}|-{#2})}% +\def\tikz@parse@hvleft#1(#2 -|#3){\tikz@parse@vhdone#1({#3}|-{#2})}% +\def\tikz@parse@hvright#1(#2-| #3){\tikz@parse@vhdone#1({#3}|-{#2})}% +\def\tikz@parse@hvdone#1(#2-|#3){\tikz@parse@vhdone#1({#3}|-{#2})}% + +\def\tikz@parse@vh#1(#2){% + \pgfutil@in@{ |- }{#2}% + \ifpgfutil@in@% + \let\tikz@next=\tikz@parse@vhboth% + \else% + \pgfutil@in@{ |-}{#2}% + \ifpgfutil@in@% + \let\tikz@next=\tikz@parse@vhleft% + \else% + \pgfutil@in@{|- }{#2}% + \ifpgfutil@in@% + \let\tikz@next=\tikz@parse@vhright% + \else% + \let\tikz@next=\tikz@parse@vhdone% + \fi% + \fi% + \fi% + \tikz@next#1(#2)}% +\def\tikz@parse@vhboth#1(#2 |- #3){\tikz@parse@vhdone#1({#2}|-{#3})}% +\def\tikz@parse@vhleft#1(#2 |-#3){\tikz@parse@vhdone#1({#2}|-{#3})}% +\def\tikz@parse@vhright#1(#2|- #3){\tikz@parse@vhdone#1({#2}|-{#3})}% +\def\tikz@parse@vhdone#1(#2|-#3){% + {% + \tikz@@@scan@@absolute\tikz@parse@vh@mid(#2)% + \tikz@@@scan@@absolute\tikz@parse@vh@end(#3)% + \xdef\tikz@marshal{\noexpand#1{\noexpand\pgfqpoint{\the\pgf@xa}{\the\pgf@ya}}}% + }% + \tikz@marshal% +}% +\def\tikz@parse@vh@mid#1{\pgf@process{#1}\pgf@xa=\pgf@x}% +\def\tikz@parse@vh@end#1{\pgf@process{#1}\pgf@ya=\pgf@y}% + +\def\tikz@parse@intersection#1(intersection{% + \pgfutil@ifnextchar o{% + \tikz@parse@main@intersection#1 1% + }{% + \tikz@parse@main@intersection#1% + }% +}% +\def\tikz@parse@main@intersection#1#2of #3 and #4){% + \tikzset{cs/solution={#2}}% + \pgfutil@in@{--}{#3}% + \ifpgfutil@in@% + \tikz@reparse@line{first}#3\pgf@stop% + \else% + \tikzset{cs/first node={#3}}% + \fi% + \pgfutil@in@{--}{#4}% + \ifpgfutil@in@% + \tikz@reparse@line{second}#4\pgf@stop% + \else% + \tikzset{cs/second node={#4}}% + \fi% + \tikz@parse@cs@intersection()% + \edef\pgf@marshal{\noexpand#1{\noexpand\pgfqpoint{\the\pgf@x}{\the\pgf@y}}}% + \pgf@marshal% +}% +\def\tikz@reparse@line#1#2--#3\pgf@stop{% + \tikzset{cs/#1 line={(#2)--(#3)}}% +}% + + +\def\tikz@parse@intersection@a#1{\pgf@process{#1}\pgf@xa=\pgf@x\pgf@ya=\pgf@y}% +\def\tikz@parse@intersection@b#1{\pgf@process{#1}\pgf@xb=\pgf@x\pgf@yb=\pgf@y}% + +\def\tikz@scan@relative#1+{% + \pgfutil@ifnextchar+{\tikz@scan@plusplus#1}{\tikz@scan@oneplus#1}}% + +\def\tikz@scan@plusplus#1+{% + \def\tikz@doafter{#1}% + \tikz@scan@absolute\tikz@add% +}% +\def\tikz@add#1{% + \tikz@doafter{\pgfpointadd{#1}{\tikz@last@position@saved}}% +}% +\def\tikz@scan@oneplus#1{% + \def\tikz@doafter{#1}% + \tikz@updatecurrentfalse% + \tikz@scan@absolute\tikz@add% +}% + + + + +\let\tikz@enable@node@quotes\relax +\let\tikz@enable@edge@quotes\relax +\let\tikz@enable@pic@quotes\relax + + + + + +\def\usetikzlibrary{\pgfutil@ifnextchar[{\use@tikzlibrary}{\use@@tikzlibrary}}% +\def\use@tikzlibrary[#1]{\use@@tikzlibrary{#1}}% +\def\use@@tikzlibrary#1{% + \edef\pgf@list{#1}% + \pgfutil@for\pgf@temp:=\pgf@list\do{% + \expandafter\pgfkeys@spdef\expandafter\pgf@temp\expandafter{\pgf@temp}% + \ifx\pgf@temp\pgfutil@empty + \else + \expandafter\ifx\csname tikz@library@\pgf@temp @loaded\endcsname\relax% + \expandafter\global\expandafter\let\csname tikz@library@\pgf@temp @loaded\endcsname=\pgfutil@empty% + \expandafter\edef\csname tikz@library@#1@atcode\endcsname{\the\catcode`\@} + \expandafter\edef\csname tikz@library@#1@barcode\endcsname{\the\catcode`\|} + \expandafter\edef\csname tikz@library@#1@dollarcode\endcsname{\the\catcode`\$} + \catcode`\@=11 + \catcode`\|=12 + \catcode`\$=3 + \pgfutil@InputIfFileExists{tikzlibrary\pgf@temp.code.tex}{}{ + \pgfutil@IfFileExists{pgflibrary\pgf@temp.code.tex}{% + \expandafter\usepgflibrary\expandafter{\pgf@temp}% + }{% + \tikzerror{I did not find the tikz library + '\pgf@temp'. I looked for files named + tikzlibrary\pgf@temp.code.tex and + pgflibrary\pgf@temp.code.tex, but neither + could be found in the current texmf trees.} + }}% + \catcode`\@=\csname tikz@library@#1@atcode\endcsname + \catcode`\|=\csname tikz@library@#1@barcode\endcsname + \catcode`\$=\csname tikz@library@#1@dollarcode\endcsname + \fi% + \fi + }% +}% + + + +\usetikzlibrary{topaths}% + + + + +\endinput diff --git a/preprints/A-polynomial-time-algorithm-for-three-machine-unit-job-scheduling-September-24-2026/build/tex/runtime/generic/pgf/libraries/decorations/pgflibrarydecorations.pathreplacing.code.tex b/preprints/A-polynomial-time-algorithm-for-three-machine-unit-job-scheduling-September-24-2026/build/tex/runtime/generic/pgf/libraries/decorations/pgflibrarydecorations.pathreplacing.code.tex new file mode 100644 index 0000000000000000000000000000000000000000..2292b281c09b3998987f3120e415d44d96c7c0b4 --- /dev/null +++ b/preprints/A-polynomial-time-algorithm-for-three-machine-unit-job-scheduling-September-24-2026/build/tex/runtime/generic/pgf/libraries/decorations/pgflibrarydecorations.pathreplacing.code.tex @@ -0,0 +1,203 @@ + +\usepgfmodule{decorations}% + + + + + + + + +\pgfdeclaredecoration{ticks}{ticks} +{% + \state{ticks}[width=+\pgfdecorationsegmentlength] + { + \pgfpathmoveto{\pgfqpoint{0pt}{\pgfdecorationsegmentamplitude}} + \pgfpathlineto{\pgfqpoint{0pt}{-\pgfdecorationsegmentamplitude}} + }% + \state{final} + { + \pgfpathmoveto{\pgfqpoint{0pt}{\pgfdecorationsegmentamplitude}} + \pgfpathlineto{\pgfqpoint{0pt}{-\pgfdecorationsegmentamplitude}} + \pgfpathmoveto{\pgfpointdecoratedpathlast} + }% +}% + + + + +\pgfdeclaredecoration{expanding waves}{initial} +{% + \state{initial}[width=+\pgfdecorationsegmentlength,next state=wave] + {}% + + \state{wave}[switch if less than=+\pgfdecorationsegmentlength to last, + width=+\pgfdecorationsegmentlength] + { + \pgfpathmoveto{ + \pgfpointadd + {\pgfqpoint{-\pgfdecoratedcompleteddistance}{0pt}}% + {\pgfpointpolar{\pgfdecorationsegmentangle}{+\pgfdecoratedcompleteddistance}}}% + \pgfpatharc{\pgfdecorationsegmentangle}{-\pgfdecorationsegmentangle}{+\pgfdecoratedcompleteddistance}% + }% + \state{last}[width=+0pt,next state=final] + { + \pgfpathmoveto{ + \pgfpointadd + {\pgfqpoint{-\pgfdecoratedcompleteddistance}{0pt}}% + {\pgfpointpolar{\pgfdecorationsegmentangle}{+\pgfdecoratedcompleteddistance}}}% + \pgfpatharc{\pgfdecorationsegmentangle}{-\pgfdecorationsegmentangle}{+\pgfdecoratedcompleteddistance}% + }% + \state{final} + { + \pgfpathmoveto{\pgfpointdecoratedpathlast} + }% +}% + + + + +\pgfdeclaredecoration{waves}{wave} +{% + \state{wave}[width=\pgfdecorationsegmentlength] + { + \pgftransformxshift{+\pgfdecorationsegmentlength} + \pgfpathmoveto{ + \pgfpointadd + {\pgfqpoint{-\pgfkeysvalueof{/pgf/decoration/start radius}}{0pt}}% + {\pgfpointpolar{\pgfdecorationsegmentangle}{\pgfkeysvalueof{/pgf/decoration/start radius}}}}% + \pgfpatharc{\pgfdecorationsegmentangle}{-\pgfdecorationsegmentangle}{\pgfkeysvalueof{/pgf/decoration/start radius}}% + }% + \state{final} + { + \pgfpathmoveto{\pgfpointdecoratedpathlast} + }% +}% + + + + +\pgfdeclaredecoration{border}{tick} +{% + \state{tick}[switch if less than=+\pgfdecorationsegmentlength to last, + width=+\pgfdecorationsegmentlength] + { + \pgfpathmoveto{\pgfpointorigin} + \pgfpathlineto{\pgfpointpolar{\pgfdecorationsegmentangle}{+\pgfdecorationsegmentamplitude}} + }% + \state{last}[width=+\pgfdecorationsegmentamplitude,next state=final] + { + \pgfpathmoveto{\pgfpointorigin} + \pgfpathlineto{\pgfpointpolar{\pgfdecorationsegmentangle}{+\pgfdecorationsegmentamplitude}} + }% + \state{final} + { + \pgfpathmoveto{\pgfpointdecoratedpathlast} + }% +}% + + + + + + + +\pgfdeclaredecoration{brace}{brace} +{% + \state{brace}[width=+\pgfdecoratedremainingdistance,next state=final] + { + \pgf@yc=\pgfdecorationsegmentaspect\pgfdecoratedremainingdistance + \ifdim2\pgfdecorationsegmentamplitude>\pgf@yc + \pgf@yc=0.5\pgf@yc + \else + \pgf@yc=\pgfdecorationsegmentamplitude + \fi + \pgf@xc=\pgfdecorationsegmentaspect\pgfdecoratedremainingdistance + \advance\pgf@xc-\pgfdecoratedremainingdistance + \ifdim-2\pgfdecorationsegmentamplitude<\pgf@xc + \pgf@xc=-0.5\pgf@xc + \else + \pgf@xc=\pgfdecorationsegmentamplitude + \fi + \pgfpathmoveto{\pgfpointorigin} + \pgfpathcurveto + {\pgfqpoint{.15\pgf@yc}{.3\pgfdecorationsegmentamplitude}} + {\pgfqpoint{.5\pgf@yc}{.5\pgfdecorationsegmentamplitude}} + {\pgfqpoint{\pgf@yc}{.5\pgfdecorationsegmentamplitude}} + { + \pgftransformxshift{+\pgfdecorationsegmentaspect\pgfdecoratedremainingdistance} + \pgfpathlineto{\pgfqpoint{-\pgf@yc}{.5\pgfdecorationsegmentamplitude}} + \pgfpathcurveto + {\pgfqpoint{-.5\pgf@yc}{.5\pgfdecorationsegmentamplitude}} + {\pgfqpoint{-.15\pgf@yc}{.7\pgfdecorationsegmentamplitude}} + {\pgfqpoint{0\pgf@yc}{1\pgfdecorationsegmentamplitude}} + \pgfpathcurveto + {\pgfqpoint{.15\pgf@xc}{.7\pgfdecorationsegmentamplitude}} + {\pgfqpoint{.5\pgf@xc}{.5\pgfdecorationsegmentamplitude}} + {\pgfqpoint{\pgf@xc}{.5\pgfdecorationsegmentamplitude}} + } + { + \pgftransformxshift{+\pgfdecoratedremainingdistance} + \pgfpathlineto{\pgfqpoint{-\pgf@xc}{.5\pgfdecorationsegmentamplitude}} + \pgfpathcurveto + {\pgfqpoint{-.5\pgf@xc}{.5\pgfdecorationsegmentamplitude}} + {\pgfqpoint{-.15\pgf@xc}{.3\pgfdecorationsegmentamplitude}} + {\pgfqpoint{0pt}{0pt}} + } + }% + \state{final} + {}% +}% + + + + +\pgfdeclaredecoration{show path construction}{check for moveto}{% + \state{check for moveto}[width=0pt, next state=segment, + persistent precomputation={% + \begingroup% + \pgf@lib@decoraions@installinputsegmentpoints% + \ifx\pgfdecorationpreviousinputsegment\pgfdecorationinputsegmentmoveto% + \pgfdecoratedinputsegmentmovetocode% + \fi% + \endgroup% + }]{}% + \state{segment}[width=\pgfdecoratedinputsegmentremainingdistance, next state=check for moveto, + persistent precomputation={% + \begingroup% + \pgf@lib@decoraions@installinputsegmentpoints% + \csname pgfdecoratedinputsegment\pgfdecorationcurrentinputsegment code\endcsname% + \endgroup% + }]{}% + \state{final}{% + \egroup% + \begingroup% + \pgf@lib@decoraions@installinputsegmentpoints% + \csname pgfdecoratedinputsegment\pgfdecorationcurrentinputsegment code\endcsname% + \endgroup% + \bgroup% + }% +}% + +\pgfkeys{ + /pgf/decoration/.cd, + moveto code/.code=\def\pgfdecoratedinputsegmentmovetocode{#1}, + lineto code/.code=\def\pgfdecoratedinputsegmentlinetocode{#1}, + curveto code/.code=\def\pgfdecoratedinputsegmentcurvetocode{#1}, + closepath code/.code=\def\pgfdecoratedinputsegmentclosepathcode{#1} +}% + +\let\pgfdecoratedinputsegmentmovetocode\pgfutil@empty% +\let\pgfdecoratedinputsegmentlinetocode\pgfutil@empty% +\let\pgfdecoratedinputsegmentcurvetocode\pgfutil@empty% +\let\pgfdecoratedinputsegmentclosepathcode\pgfutil@empty% + +\def\pgf@lib@decoraions@installinputsegmentpoints{% + \let\pgfpointdecoratedinputsegmentfirst\pgf@decorate@inputsegment@first% + \let\pgfpointdecoratedinputsegmentlast\pgf@decorate@inputsegment@last% + \let\pgfpointdecoratedinputsegmentsupporta\pgf@decorate@inputsegment@supporta% + \let\pgfpointdecoratedinputsegmentsupportb\pgf@decorate@inputsegment@supportb% +}% + + +\endinput diff --git a/preprints/A-polynomial-time-algorithm-for-three-machine-unit-job-scheduling-September-24-2026/build/tex/runtime/generic/pgf/libraries/pgflibraryarrows.meta.code.tex b/preprints/A-polynomial-time-algorithm-for-three-machine-unit-job-scheduling-September-24-2026/build/tex/runtime/generic/pgf/libraries/pgflibraryarrows.meta.code.tex new file mode 100644 index 0000000000000000000000000000000000000000..12652d298bc990ffd195ad807a38da14e00ebbcb --- /dev/null +++ b/preprints/A-polynomial-time-algorithm-for-three-machine-unit-job-scheduling-September-24-2026/build/tex/runtime/generic/pgf/libraries/pgflibraryarrows.meta.code.tex @@ -0,0 +1,1436 @@ + +\ProvidesFileRCS{pgflibraryarrows.meta.code.tex} + + + + + +\newdimen\pgfarrowinset +\newdimen\pgfarrowlength +\newdimen\pgfarrowwidth +\newdimen\pgfarrowlinewidth +\newif\ifpgfarrowroundcap +\newif\ifpgfarrowroundjoin + +\pgfarrowsaddtolengthscalelist\pgfarrowlength +\pgfarrowsaddtolengthscalelist\pgfarrowinset +\pgfarrowsaddtowidthscalelist\pgfarrowwidth + + +\pgfkeys{ + /pgf/arrow keys/.cd, + length/.code={% + \pgfarrowsthreeparameters{#1}% + \expandafter\pgfarrowsaddtooptions\expandafter{\expandafter\pgfarrowslinewidthdependent\pgfarrowstheparameters\pgfarrowlength\pgf@x}% + }, + inset/.code={% + \pgfarrowsthreeparameters{#1}% + \expandafter\pgfarrowsaddtolateoptions\expandafter{\expandafter\pgfarrowslinewidthdependent\pgfarrowstheparameters\pgfarrowinset\pgf@x}% + }, + inset'/.code={ + \pgfarrowsthreeparameters{#1}% + \expandafter\pgfarrowsaddtolateoptions\expandafter{\expandafter\pgfarrowslengthdependent\pgfarrowstheparameters\pgfarrowinset}% + }, + line width/.code={% + \pgfarrowsthreeparameters{#1}% + \expandafter\pgfarrowsaddtolateoptions\expandafter{\expandafter\pgfarrowslinewidthdependent\pgfarrowstheparameters\pgfarrowlinewidth\pgf@x}% + }, + line width'/.code={ + \pgfarrowsthreeparameters{#1}% + \expandafter\pgfarrowsaddtolateoptions\expandafter{\expandafter\pgfarrowslengthdependent\pgfarrowstheparameters\pgfarrowlinewidth}% + }, + width/.code={% + \pgfarrowsthreeparameters{#1}% + \expandafter\pgfarrowsaddtolateoptions\expandafter{\expandafter\pgfarrowslinewidthdependent\pgfarrowstheparameters\pgfarrowwidth\pgf@x}% + }, + width'/.code={% + \pgfarrowsthreeparameters{#1}% + \expandafter\pgfarrowsaddtolateoptions\expandafter{\expandafter\pgfarrowslengthdependent\pgfarrowstheparameters\pgfarrowwidth}% + }, + angle/.code={% + \pgfarrowsfourparameters{#1}% + \expandafter\pgfarrowsaddtooptions\expandafter{\expandafter\pgfarrows@angle@setup\pgfarrowstheparameters}% + }, + angle'/.code={ + \pgfarrowsaddtolateoptions{\pgfarrows@angle@setup@prime{#1}}% + }, + line cap/.is choice, + line cap/butt/.code=\pgfarrowsaddtooptions{\pgfarrowroundcapfalse}, + line cap/round/.code=\pgfarrowsaddtooptions{\pgfarrowroundcaptrue}, + line join/.is choice, + line join/miter/.code=\pgfarrowsaddtooptions{\pgfarrowroundjoinfalse}, + line join/round/.code=\pgfarrowsaddtooptions{\pgfarrowroundjointrue}, + round/.style={line cap=round,line join=round}, + sharp/.style={line cap=butt,line join=miter}, +}% + + + + + + +\pgfdeclarearrow{ + name = Straight Barb, + defaults = { + length = +1.5pt 2, + width' = +0pt 2, + line width = +0pt 1 1, + }, + setup code = { + \pgfmathdivide@{\pgf@sys@tonumber\pgfarrowlength}{\pgf@sys@tonumber\pgfarrowwidth}% + \let\pgf@temp@quot\pgfmathresult% + \pgf@x\pgfmathresult pt% + \pgf@x\pgfmathresult\pgf@x% + \pgf@x4\pgf@x% + \advance\pgf@x by1pt% + \pgfmathsqrt@{\pgf@sys@tonumber\pgf@x}% + \pgf@xc\pgfmathresult\pgfarrowlinewidth% + \pgf@xc.5\pgf@xc + \pgf@xa\pgf@temp@quot\pgfarrowlinewidth% + \ifpgfarrowroundjoin + \pgfarrowssettipend{\pgfarrowlength\advance\pgf@x by.5\pgfarrowlinewidth} + \else + \pgfarrowssettipend{\pgfarrowlength\advance\pgf@x by\pgf@xc\ifpgfarrowharpoon\advance\pgf@x by\pgf@xa\fi} + \fi + \pgfarrowssetvisualbackend{\pgfarrowlength\advance\pgf@x by.5\pgfarrowlinewidth} + \pgfarrowssetbackend{-.5\pgfarrowlinewidth} + \ifpgfarrowreversed + \pgfarrowssetlineend{\pgfarrowlength\ifpgfarrowharpoon\advance\pgf@x by.5\pgfarrowlinewidth\fi} + \else + \pgfarrowssetlineend{\pgfarrowlength\advance\pgf@x by-.5\pgfarrowlinewidth} + \fi + \ifpgfarrowroundjoin + \pgfarrowshullpoint{\pgfarrowlength\advance\pgf@x by.5\pgfarrowlinewidth}{\ifpgfarrowharpoon-.5\pgfarrowlinewidth\else0pt\fi}% + \else% + \ifpgfarrowharpoon + \pgfarrowshullpoint{\pgfarrowlength\advance\pgf@x by\pgf@xc\advance\pgf@x by\pgf@xa}{-.5\pgfarrowlinewidth}% + \else + \pgfarrowshullpoint{\pgfarrowlength\advance\pgf@x by\pgf@xc\advance\pgf@x by\pgf@xa}{0pt}% + \fi% + \fi + \pgfarrowsupperhullpoint{.5\pgfarrowlinewidth}{.5\pgfarrowwidth\advance\pgf@y by.5\pgfarrowlinewidth}% + \pgfarrowsupperhullpoint{-.5\pgfarrowlinewidth}{.5\pgfarrowwidth\advance\pgf@y by.5\pgfarrowlinewidth}% + \ifpgfarrowharpoon + \pgfarrowshullpoint{-.5\pgfarrowlinewidth}{-.5\pgfarrowlinewidth}% + \fi + \pgfarrowssavethe\pgfarrowlinewidth + \pgfarrowssavethe\pgfarrowlength + \pgfarrowssavethe\pgfarrowwidth + }, + drawing code = { + \pgfsetdash{}{+0pt} + \ifpgfarrowroundjoin\pgfsetroundjoin\else\pgfsetmiterjoin\fi + \ifpgfarrowroundcap\pgfsetroundcap\else\pgfsetbuttcap\fi + \ifdim\pgfarrowlinewidth=\pgflinewidth\else\pgfsetlinewidth{+\pgfarrowlinewidth}\fi + \pgfpathmoveto{\pgfqpoint{0pt}{.5\pgfarrowwidth}} + \pgfpathlineto{\pgfqpoint{\pgfarrowlength}{0pt}} + \ifpgfarrowharpoon + \pgfpathlineto{\pgfqpoint{\pgfarrowlength\advance\pgf@x by\ifpgfarrowreversed\else-\fi\pgfarrowlinewidth}{0pt}} + \else + \pgfpathlineto{\pgfqpoint{0pt}{-.5\pgfarrowwidth}} + \fi + \pgfusepathqstroke + }, + parameters = { + \the\pgfarrowlinewidth,% + \the\pgfarrowlength,% + \the\pgfarrowwidth,% + \ifpgfarrowharpoon h\fi% + \ifpgfarrowroundjoin j\fi% + \ifpgfarrowroundcap c\fi% + }, +}% + +\pgfkeys{ + /pgf/arrow keys/.cd, + arc/.code={% + \pgfmathparse{#1}% + \expandafter\pgfarrowsaddtooptions\expandafter{\expandafter\def\expandafter\pgfarrowarc\expandafter{\pgfmathresult}} + } +}% + +\def\pgfarrowarc{180}% + +\pgfdeclarearrow{ + name = Hooks, + defaults = { + length = +0.75pt 1.25, + width' = +0pt 4 -1, + line width = +0pt 1 1, + }, + setup code = { + \ifpgfarrowharpoon + \pgfarrowshullpoint{.5\pgfarrowlength}{-.5\pgfarrowlinewidth} + \pgfarrowshullpoint{\pgfarrowlength}{-.5\pgfarrowlinewidth\advance\pgf@y by.125\pgfarrowwidth} + \fi + \ifdim\pgfarrowarc pt<90.1pt + \pgfarrowsupperhullpoint{0pt}{.25\pgfarrowwidth\ifpgfarrowroundcap\else\advance\pgf@y by-.25\pgfarrowlinewidth\fi} + \pgfarrowsupperhullpoint{\pgfarrowlength}{.25\pgfarrowwidth\ifpgfarrowroundcap\else\advance\pgf@y by-.25\pgfarrowlinewidth\fi} + \else% + \pgfarrowsupperhullpoint{\pgfarrowlength}{.375\pgfarrowwidth} + \pgfarrowsupperhullpoint{.5\pgfarrowlength}{.5\pgfarrowwidth} + \ifdim\pgfarrowarc pt>180pt + \pgfarrowsupperhullpoint{-.5\pgfarrowlength}{.5\pgfarrowwidth} + \pgfarrowsupperhullpoint{-\pgfarrowlength}{.375\pgfarrowwidth} + \pgfarrowsupperhullpoint{-\pgfarrowlength}{.25\pgfarrowwidth} + \else + \pgfarrowsupperhullpoint{\ifpgfarrowroundcap-.5\pgfarrowlinewidth\else0pt\fi}{.25\pgfarrowwidth} + \pgfarrowsupperhullpoint{\ifpgfarrowroundcap-.5\pgfarrowlinewidth\else0pt\fi}{.5\pgfarrowwidth} + \fi + \fi + \advance\pgfarrowlength by-.5\pgfarrowlinewidth + \advance\pgfarrowwidth by-\pgfarrowlinewidth + \ifpgfarrowreversed + \ifpgfarrowroundjoin + \pgfarrowssetbackend{-.5\pgfarrowlinewidth} + \fi + \fi + \ifdim\pgfarrowarc pt<90pt% + \pgfmathsin@{\pgfarrowarc} + \pgfarrowssettipend{\pgfmathresult\pgfarrowlength\advance\pgf@x by.5\pgfarrowlinewidth}% + \else\ifdim\pgfarrowarc pt<180pt% + \pgfarrowssettipend{\pgfarrowlength\advance\pgf@x by.5\pgfarrowlinewidth}% + \ifpgfarrowroundcap\pgfarrowssetbackend{-.5\pgfarrowlinewidth}\fi% + \else\ifdim\pgfarrowarc pt<270pt% + \pgfarrowssettipend{\pgfarrowlength\advance\pgf@x by.5\pgfarrowlinewidth}% + \ifdim\pgfarrowarc pt>180pt + \pgfmathsin@{\pgfarrowarc} + \pgfarrowssetbackend{\pgfmathresult\pgfarrowlength\advance\pgf@x by-.5\pgfarrowlinewidth}% + \else + \ifpgfarrowroundcap\pgfarrowssetbackend{-.5\pgfarrowlinewidth}\fi% + \fi% + \else% + \pgfarrowssettipend{\pgfarrowlength\advance\pgf@x by.5\pgfarrowlinewidth}% + \pgfarrowssetbackend{-\pgfarrowlength\advance\pgf@x by-.5\pgfarrowlinewidth}% + \fi\fi\fi% + \ifpgfarrowreversed + \pgfarrowssetlineend{.5\pgfarrowlinewidth} + \else% + \ifpgfarrowharpoon + \pgfarrowssetlineend{0pt} + \else + \pgfarrowssetlineend{.25\pgfarrowlinewidth} + \fi + \fi + \pgf@x\pgfarrowarc pt% + \advance\pgf@x by-90pt% + \edef\pgfarrowarc{\pgf@sys@tonumber\pgf@x}% + \pgfarrowssavethe\pgfarrowlinewidth + \pgfarrowssavethe\pgfarrowlength + \pgfarrowssavethe\pgfarrowwidth + \pgfarrowssave\pgfarrowarc + }, + drawing code = { + \pgfsetdash{}{+0pt} + \ifpgfarrowroundjoin\pgfsetroundjoin\else\pgfsetmiterjoin\fi + \ifpgfarrowroundcap\pgfsetroundcap\else\pgfsetbuttcap\fi + \ifdim\pgfarrowlinewidth=\pgflinewidth\else\pgfsetlinewidth{+\pgfarrowlinewidth}\fi + {% + \pgftransformxscale{+\pgfarrowlength} + \pgftransformyscale{+.25\pgfarrowwidth} + \pgfpathmoveto{\pgfpointpolar{+\pgfarrowarc}{+1pt}\advance\pgf@y by1pt} + \pgfpatharc{\pgfarrowarc}{+-90}{+1pt} + \ifpgfarrowharpoon + \else + \pgfpatharc{+90}{+-\pgfarrowarc}{+1pt} + \fi + } + \ifpgfarrowharpoon\ifpgfarrowreversed + \pgfpathlineto{\pgfqpoint{\pgflinewidth}{0pt}} + \fi\fi + \pgfusepathqstroke + }, + parameters = { + \the\pgfarrowlinewidth,% + \the\pgfarrowlength,% + \the\pgfarrowwidth,% + \pgfarrowarc,% + \ifpgfarrowharpoon h\fi% + \ifpgfarrowroundjoin j\fi% + \ifpgfarrowroundcap c\fi% + }, +}% + + +\pgfdeclarearrow{ + name = Arc Barb, + defaults = { + length = +1.5pt 2, + width' = +0pt 2, + line width = +0pt 1 1, + }, + setup code = { + \pgf@x\pgfarrowarc pt% + \pgf@x.5\pgf@x% + \edef\pgfarrowarc{\pgf@sys@tonumber\pgf@x}% + \pgfarrowssettipend{\pgfarrowlength} + \pgfarrowssetvisualbackend{\pgfarrowlength} + \pgfarrowssetlineend{\pgfarrowlength\advance\pgf@x by-.5\pgfarrowlinewidth} + \pgfarrowsupperhullpoint{\pgfarrowlength}{.25\pgfarrowwidth} + \ifpgfarrowharpoon\pgfarrowshullpoint{\pgfarrowlength}{-.5\pgflinewidth}\fi + \pgfmathsincos@{\pgfarrowarc}% + \pgf@xa.5\pgfarrowwidth% + \pgfarrowsupperhullpoint{\pgfmathresultx\pgfarrowlength}{\pgfmathresulty\pgf@xa} + \ifdim\pgfarrowarc pt>60pt + \pgfarrowsupperhullpoint{.5\pgfarrowlength}{\ifdim\pgfarrowarc pt<90pt\pgfmathresulty\pgf@xa\else.5\pgfarrowwidth\fi} + \fi + \ifdim\pgfarrowarc pt>90pt + \ifdim\pgfarrowarc pt<120pt + \pgfarrowsupperhullpoint{\pgfmathresultx\pgfarrowlength}{.5\pgfarrowwidth} + \else + \pgfarrowsupperhullpoint{-.5\pgfarrowlength}{.5\pgfarrowwidth} + \ifdim\pgfarrowarc pt>150pt + \pgfarrowsupperhullpoint{-\pgfarrowlength}{.25\pgfarrowwidth} + \fi + \fi + \fi + \pgfutil@tempswafalse + \ifdim\pgfarrowarc pt<90pt + \pgfutil@tempswatrue + \fi + \ifpgfarrowharpoon + \pgfutil@tempswatrue + \fi + \ifpgfutil@tempswa + \pgf@xa.5\pgfarrowwidth\advance\pgf@xa by-\pgfarrowlinewidth% + \pgf@ya\pgfarrowlength\advance\pgf@ya by-\pgfarrowlinewidth% + \pgfarrowsupperhullpoint{\pgfmathresultx\pgf@ya}{\pgfmathresulty\pgf@xa} + \fi + \ifpgfarrowroundcap + \pgf@y\pgfarrowlength + \advance\pgf@y by-.5\pgfarrowlinewidth + \pgf@x\pgfmathresultx\pgf@y% + \else + \ifdim\pgfarrowarc pt<90pt + \pgf@y\pgfarrowlength + \advance\pgf@y by-\pgfarrowlinewidth + \pgf@x\pgfmathresultx\pgf@y% + \else + \pgf@x\pgfmathresultx\pgfarrowlength% + \fi + \fi + \pgfarrowssetbackend{\pgf@x\ifpgfarrowroundcap\advance\pgf@x by-.5\pgfarrowlinewidth\fi} + \advance\pgfarrowlength by-.5\pgfarrowlinewidth + \advance\pgfarrowwidth by-\pgfarrowlinewidth + \pgfarrowssavethe\pgfarrowlinewidth + \pgfarrowssavethe\pgfarrowlength + \pgfarrowssavethe\pgfarrowwidth + \pgfarrowssave\pgfarrowarc + }, + drawing code = { + \pgfsetdash{}{+0pt} + \ifpgfarrowroundcap\pgfsetroundcap\else\pgfsetbuttcap\fi + \ifpgfarrowroundjoin\pgfsetroundjoin\else\pgfsetmiterjoin\fi + \ifdim\pgfarrowlinewidth=\pgflinewidth\else\pgfsetlinewidth{+\pgfarrowlinewidth}\fi + {% + \pgftransformxscale{+\pgfarrowlength} + \pgftransformyscale{+.5\pgfarrowwidth} + \pgfpathmoveto{\pgfpointpolar{+\pgfarrowarc}{+1pt}} + \ifpgfarrowharpoon + \pgfpatharc{+\pgfarrowarc}{0pt}{+1pt} + \else + \pgfpatharc{+\pgfarrowarc}{+-\pgfarrowarc}{+1pt} + \fi + } + \ifpgfarrowharpoon + \pgfpathlineto{\pgfqpoint{\pgfarrowlength\advance\pgf@x by\ifpgfarrowreversed\else-\fi.5\pgflinewidth}{0pt}} + \fi + \pgfusepathqstroke + }, + parameters = { + \the\pgfarrowlinewidth,% + \the\pgfarrowlength,% + \the\pgfarrowwidth,% + \pgfarrowarc,% + \ifpgfarrowharpoon h\fi% + \ifpgfarrowroundcap c\fi% + \ifpgfarrowroundjoin j\fi% + }, +}% + + + + +\pgfdeclarearrow{ + name = Tee Barb, + defaults = { + length = +1.5pt 2, + width = +3pt 4, + inset' = +0pt .5, + line width = +0pt 1 1, + }, + bending mode=polar, + setup code = { + \pgfutil@tempswafalse% + \pgfutil@tempswbfalse% + \pgfutil@tempdima\pgfarrowlength\advance\pgfutil@tempdima by-\pgfarrowinset\ifdim\pgfutil@tempdima<.5\pgfarrowlinewidth\pgfutil@tempdima.5\pgfarrowlinewidth\pgfutil@tempswatrue\fi% + \pgfutil@tempdimb-\pgfarrowinset\ifdim\pgfutil@tempdimb>-.5\pgfarrowlinewidth\pgfutil@tempdimb-.5\pgfarrowlinewidth\pgfutil@tempswbtrue\fi + \pgfarrowssavethe\pgfarrowwidth + \pgfarrowssavethe\pgfutil@tempdima + \pgfarrowssavethe\pgfutil@tempdimb + \pgfarrowssettipend{\pgfutil@tempdima\ifpgfarrowroundcap\ifpgfutil@tempswa\else\advance\pgf@x by.5\pgfarrowlinewidth\fi\fi} + \pgfarrowssetvisualbackend{.5\pgfarrowlinewidth} + \pgfarrowssetbackend{\pgfutil@tempdimb\ifpgfarrowroundcap\ifpgfutil@tempswb\else\advance\pgf@x by-.5\pgfarrowlinewidth\fi\fi} + \pgfarrowssetlineend{\ifpgfarrowreversed\else-\fi.25\pgfarrowlinewidth}% + \pgfarrowsupperhullpoint{\pgfutil@tempdima\ifpgfarrowroundcap\ifpgfutil@tempswa\else\advance\pgf@x by.5\pgfarrowlinewidth\fi\fi}{.5\pgfarrowwidth}% + \pgfarrowsupperhullpoint{\pgfutil@tempdimb\ifpgfarrowroundcap\ifpgfutil@tempswb\else\advance\pgf@x by-.5\pgfarrowlinewidth\fi\fi}{.5\pgfarrowwidth}% + \ifpgfarrowharpoon + \pgfarrowshullpoint{.5\pgfarrowlinewidth}{-.5\pgfarrowlinewidth}% + \fi + }, + drawing code = { + \pgfsetdash{}{+0pt} + \pgf@yc\pgflinewidth% + \ifdim\pgflinewidth=\pgfarrowlinewidth\else\pgfsetlinewidth{+\pgfarrowlinewidth}\fi + \ifpgfarrowroundcap\pgfsetroundcap\else\pgfsetbuttcap\fi + \ifpgfarrowroundjoin\pgfsetroundjoin\else\pgfsetmiterjoin\fi + \pgf@x\pgfutil@tempdima\advance\pgf@x by-.5\pgfarrowlinewidth% + \pgf@y\pgfutil@tempdimb\advance\pgf@y by.5\pgfarrowlinewidth% + \ifdim\pgf@x=\pgf@y% + \pgfpathmoveto{\pgfqpoint{0pt}{.5\pgfarrowwidth}} + \pgfpathlineto{\pgfqpoint{0pt}{\ifpgfarrowharpoon-.5\pgfarrowlinewidth\else-.5\pgfarrowwidth\fi}} + \else + \ifdim\pgfutil@tempdima=.5\pgfarrowlinewidth% + \pgfpathmoveto{\pgfqpoint{\pgfutil@tempdimb}{.5\pgfarrowwidth\advance\pgf@y by-.5\pgfarrowlinewidth}} + \pgfpathlineto{\pgfqpoint{0pt}{.5\pgfarrowwidth\advance\pgf@y by-.5\pgfarrowlinewidth}} + \ifpgfarrowharpoon% + \pgfpathlineto{\pgfqpoint{0pt}{-.5\pgf@yc}}% + \else% + \pgfpathlineto{\pgfqpoint{0pt}{-.5\pgfarrowwidth\advance\pgf@y by.5\pgfarrowlinewidth}} + \pgfpathlineto{\pgfqpoint{\pgfutil@tempdimb}{-.5\pgfarrowwidth\advance\pgf@y by.5\pgfarrowlinewidth}} + \fi + \else% + \pgfpathmoveto{\pgfqpoint{\pgfutil@tempdimb}{.5\pgfarrowwidth\advance\pgf@y by-.5\pgfarrowlinewidth}} + \pgfpathlineto{\pgfqpoint{\pgfutil@tempdima}{.5\pgfarrowwidth\advance\pgf@y by-.5\pgfarrowlinewidth}} + \pgfpathmoveto{\pgfqpoint{0pt}{.5\pgfarrowwidth\advance\pgf@y by-.5\pgfarrowlinewidth}} + \ifpgfarrowharpoon% + \pgfpathlineto{\pgfqpoint{0pt}{-.5\pgf@yc}}% + \else% + \pgfpathlineto{\pgfqpoint{0pt}{-.5\pgfarrowwidth\advance\pgf@y by.5\pgfarrowlinewidth}} + \pgfpathmoveto{\pgfqpoint{\pgfutil@tempdimb}{-.5\pgfarrowwidth\advance\pgf@y by.5\pgfarrowlinewidth}} + \pgfpathlineto{\pgfqpoint{\pgfutil@tempdima}{-.5\pgfarrowwidth\advance\pgf@y by.5\pgfarrowlinewidth}} + \fi + \fi + \fi + \pgfusepathqstroke + }, + parameters = { + \the\pgfarrowwidth,% + \the\pgfarrowlength,% + \the\pgfarrowinset,% + \the\pgfarrowlinewidth,% + \ifpgfarrowharpoon h\fi% + \ifpgfarrowroundjoin j\fi% + \ifpgfarrowroundcap c\fi% + }, +}% + + + + + + +\pgfdeclarearrow{ + name = Classical TikZ Rightarrow, + defaults = { + length = +1.05pt 1.925, + width' = +0pt 1.9237, + line width = 0pt 0.8 1, + round + }, + setup code = + { + \ifpgfarrowharpoon\pgfarrowroundjointrue\fi + \pgfutil@tempdima\pgfarrowlength + \advance\pgfutil@tempdima by-\pgfarrowlinewidth + \pgfutil@tempdimb\pgfarrowwidth + \advance\pgfutil@tempdimb by-\pgfarrowlinewidth + \ifpgfarrowroundjoin% + \else% + \pgfmathdivide@{\pgf@sys@tonumber\pgfutil@tempdima}{\pgf@sys@tonumber\pgfutil@tempdimb}% + \let\pgf@temp@quot\pgfmathresult% + \pgf@x\pgfmathresult pt% + \pgf@x\pgfmathresult\pgf@x% + \pgf@x40.96\pgf@x% + \advance\pgf@x by1pt% + \pgfmathsqrt@{\pgf@sys@tonumber\pgf@x}% + \pgf@xc\pgfmathresult\pgfarrowlinewidth% + \pgf@xc.5\pgf@xc + \fi% + \pgfarrowssavethe\pgfutil@tempdima + \pgfarrowssavethe\pgfutil@tempdimb + \pgfarrowssettipend{\pgfutil@tempdima\ifpgfarrowroundjoin\advance\pgf@x by.5\pgfarrowlinewidth\else\advance\pgf@x by\pgf@xc\fi}% + \pgfarrowssetvisualbackend{\pgfutil@tempdima\advance\pgf@x by.5\pgfarrowlinewidth}% + \ifpgfarrowreversed% + \ifpgfarrowharpoon% + \pgfarrowssetlineend{\pgfutil@tempdima\advance\pgf@x by.5\pgflinewidth}% + \else% + \pgfarrowssetlineend{\pgfutil@tempdima\advance\pgf@x by-.25\pgflinewidth}% + \fi% + \else% + \pgfarrowssetlineend{\pgfutil@tempdima\advance\pgf@x by-.6\pgflinewidth}% + \fi% + \pgfarrowssetbackend{-.5\pgfarrowlinewidth}% + \pgfarrowshullpoint{\pgfutil@tempdima\ifpgfarrowroundjoin\advance\pgf@x by.5\pgfarrowlinewidth\else\advance\pgf@x by\pgf@xc\fi}{.5\pgfarrowlinewidth}% + \pgfarrowshullpoint{\pgfutil@tempdima\ifpgfarrowroundjoin\advance\pgf@x by.5\pgfarrowlinewidth\else\advance\pgf@x by\pgf@xc\fi}{-.5\pgfarrowlinewidth}% + \pgfarrowsupperhullpoint{.5\pgfarrowlinewidth}{.5\pgfutil@tempdimb\advance\pgf@y by.5\pgfarrowlinewidth}% + \pgfarrowsupperhullpoint{-.5\pgfarrowlinewidth}{.5\pgfutil@tempdimb\advance\pgf@y by.5\pgfarrowlinewidth}% + \ifpgfarrowharpoon + \pgfarrowshullpoint{\pgfutil@tempdima\advance\pgf@x by-\pgflinewidth}{-.5\pgflinewidth}% + \pgfarrowshullpoint{-.5\pgfarrowlinewidth}{.5\pgfutil@tempdimb\advance\pgf@y by-.5\pgflinewidth}% + \fi + }, + drawing code = + { + \pgfsetdash{}{+0pt} + \edef\pgf@orig@linewidth{\the\pgflinewidth} + \ifdim\pgfarrowlinewidth=\pgflinewidth\else\pgfsetlinewidth{+\pgfarrowlinewidth}\fi + \ifpgfarrowharpoon\pgfarrowroundjointrue\fi + \ifpgfarrowroundcap\pgfsetroundcap\else\pgfsetbuttcap\fi + \ifpgfarrowroundjoin\pgfsetroundjoin\else\pgfsetmiterjoin\fi + \pgfpathmoveto + {\pgfqpoint{0pt}{.5\pgfutil@tempdimb}} + \ifpgfarrowharpoon + \pgf@arrows@old@tikz@harpoon + \else + \pgfpathcurveto + {\pgfqpoint{0.066666\pgfutil@tempdima}{0.3125\pgfutil@tempdimb}} + {\pgfqpoint{.8\pgfutil@tempdima}{0.03125\pgfutil@tempdimb}} + {\pgfqpoint{\pgfutil@tempdima}{0pt}} + \pgfpathcurveto + {\pgfqpoint{.8\pgfutil@tempdima}{-.03125\pgfutil@tempdimb}} + {\pgfqpoint{0.066666\pgfutil@tempdima}{-.3125\pgfutil@tempdimb}} + {\pgfqpoint{0pt}{-.5\pgfutil@tempdimb}} + \pgfsetlinewidth{+\pgfarrowlinewidth} + \pgfusepathqstroke + \fi + }, + parameters = { + \the\pgfarrowlinewidth,% + \the\pgfarrowlength,% + \the\pgfarrowwidth,% + \ifpgfarrowharpoon h\fi% + \ifpgfarrowroundjoin j\fi% + \ifpgfarrowroundcap c\fi% + }, +}% + +\def\pgf@arrows@old@tikz@harpoon{ + \ifpgfarrowreversed + \pgf@arrows@old@tikz@harpoon@reversed + \else + \pgfpathcurveto + {\pgfqpoint{0.066666\pgfutil@tempdima}{0.3125\pgfutil@tempdimb}} + {\pgfqpoint{.8\pgfutil@tempdima}{0.03125\pgfutil@tempdimb}} + {\pgfqpoint{\pgfutil@tempdima}{0pt}} + \pgfpathcurveto + {\pgfqpoint{0.95\pgfutil@tempdima}{-0.125\pgflinewidth}} + {\pgfqpoint{0.933333\pgfutil@tempdima}{-0.125\pgflinewidth}} + {\pgfqpoint{0.933333\pgfutil@tempdima}{-0.125\pgflinewidth}} + \pgfpathlineto + {\pgfqpoint{0.8\pgfutil@tempdima\advance\pgf@x by-.5\pgflinewidth}{-0.125\pgflinewidth}} + \pgfusepathqstroke + \fi +}% +\def\pgf@arrows@old@tikz@harpoon@reversed{ + \pgfpathcurveto + {\pgfqpoint{0.066666\pgfutil@tempdima}{0.3125\pgfutil@tempdimb}} + {\pgfqpoint{.8\pgfutil@tempdima}{0.03125\pgfutil@tempdimb}} + {\pgfqpoint{\pgfutil@tempdima}{0.125\pgflinewidth}} + \pgfpathmoveto + {\pgfqpoint{0pt}{.5\pgfutil@tempdimb}} + \pgfpathcurveto + {\pgfqpoint{0.066666\pgfutil@tempdima}{0.3125\pgfutil@tempdimb}} + {\pgfqpoint{.8\pgfutil@tempdima}{0.03125\pgfutil@tempdimb}} + {\pgfqpoint{\pgfutil@tempdima}{-0.125\pgflinewidth}} + \pgfusepathqstroke + \pgfsetlinewidth{+\pgf@orig@linewidth} + \pgfsetroundcap + \pgfpathmoveto{\pgfqpoint{\pgfutil@tempdima\advance\pgf@x by0.6\pgflinewidth}{0pt}} + \pgfpathlineto{\pgfqpoint{\pgfutil@tempdima}{0pt}} + \pgfusepathqstroke +}% + + + +\pgfdeclarearrow{ + name = Computer Modern Rightarrow, + defaults = { + length = +1.6pt 2.2, % + width' = +0pt 2.096774, + line width = 0pt 1 1, + round + }, + setup code = + { + \pgfutil@tempdima\pgfarrowlength + \advance\pgfutil@tempdima by-\pgfarrowlinewidth + \pgfutil@tempdimb\pgfarrowwidth + \advance\pgfutil@tempdimb by-\pgfarrowlinewidth + \pgfarrowssavethe\pgfutil@tempdima + \pgfarrowssavethe\pgfutil@tempdimb + \ifpgfarrowroundjoin% + \else% + \pgfmathdivide@{\pgf@sys@tonumber\pgfutil@tempdima}{\pgf@sys@tonumber\pgfutil@tempdimb}% + \let\pgf@temp@quot\pgfmathresult% + \pgf@x\pgfmathresult pt% + \pgf@x\pgfmathresult\pgf@x% + \pgf@x49.44662\pgf@x% + \advance\pgf@x by1pt% + \pgfmathsqrt@{\pgf@sys@tonumber\pgf@x}% + \pgf@xc\pgfmathresult\pgfarrowlinewidth% + \pgf@xc.5\pgf@xc + \pgf@xa\pgf@temp@quot\pgfarrowlinewidth% + \pgf@xa3.51591\pgf@xa% + \fi% + \pgfarrowssettipend{\ifpgfarrowroundjoin.5\pgfarrowlinewidth\else\pgf@xc\ifpgfarrowharpoon\advance\pgf@x by\pgf@xa\fi\fi}% + \pgfarrowssetvisualbackend{.5\pgfarrowlinewidth}% + \ifpgfarrowreversed% + \ifpgfarrowharpoon% + \pgfarrowssetlineend{.5\pgfarrowlinewidth}% + \else% + \pgfarrowssetlineend{-.5\pgfarrowlinewidth}% + \fi% + \else% + \pgfarrowssetlineend{-.5\pgfarrowlinewidth}% + \fi% + \pgfarrowssetbackend{-\pgfutil@tempdima\advance\pgf@x by-.5\pgfarrowlinewidth}% + \pgfarrowshullpoint{\ifpgfarrowroundjoin.5\pgfarrowlinewidth\else\pgf@xc\ifpgfarrowharpoon\advance\pgf@x by\pgf@xa\fi\fi}{.5\pgfarrowlinewidth}% + \pgfarrowshullpoint{\ifpgfarrowroundjoin.5\pgfarrowlinewidth\else\pgf@xc\ifpgfarrowharpoon\advance\pgf@x by\pgf@xa\fi\fi}{-.5\pgfarrowlinewidth}% + \pgfarrowsupperhullpoint{-\pgfutil@tempdima\advance\pgf@x by.5\pgfarrowlinewidth}{.5\pgfutil@tempdimb\advance\pgf@y by.5\pgfarrowlinewidth}% + \pgfarrowsupperhullpoint{-\pgfutil@tempdima\advance\pgf@x by-.5\pgfarrowlinewidth}{.5\pgfutil@tempdimb\advance\pgf@y by.5\pgfarrowlinewidth}% + \ifpgfarrowharpoon + \pgfarrowshullpoint{-\pgfarrowlinewidth}{-.5\pgfarrowlinewidth}% + \pgfarrowshullpoint{-\pgfutil@tempdima\advance\pgf@x by-.5\pgfarrowlinewidth}{.5\pgfutil@tempdimb\advance\pgf@y by-.5\pgfarrowlinewidth}% + \fi + }, + drawing code = + { + \pgfsetdash{}{+0pt} + \ifpgfarrowroundcap\pgfsetroundcap\else\pgfsetbuttcap\fi + \ifpgfarrowroundjoin\pgfsetroundjoin\else\pgfsetmiterjoin\fi + \ifdim\pgfarrowlinewidth=\pgflinewidth\else\pgfsetlinewidth{+\pgfarrowlinewidth}\fi + \pgfpathmoveto + {\pgfqpoint{-\pgfutil@tempdima}{.5\pgfutil@tempdimb}} + \pgfpathcurveto + {\pgfqpoint{-0.81731\pgfutil@tempdima}{.2\pgfutil@tempdimb}} + {\pgfqpoint{-0.41019\pgfutil@tempdima}{0.05833333\pgfutil@tempdimb}} + {\pgfpointorigin} + \ifpgfarrowharpoon + \pgfpathlineto + {\pgfqpoint{\ifpgfarrowreversed.5\else-\fi\pgfarrowlinewidth}{0pt}} + \else + \pgfpathcurveto + {\pgfqpoint{-0.41019\pgfutil@tempdima}{-0.05833333\pgfutil@tempdimb}} + {\pgfqpoint{-0.81731\pgfutil@tempdima}{-.2\pgfutil@tempdimb}} + {\pgfqpoint{-\pgfutil@tempdima}{-.5\pgfutil@tempdimb}} + \fi + \pgfusepathqstroke + }, + parameters = { + \the\pgfarrowlinewidth,% + \the\pgfarrowlength,% + \the\pgfarrowwidth,% + \ifpgfarrowharpoon h\fi + \ifpgfarrowharpoon r\fi + \ifpgfarrowroundjoin j\fi% + \ifpgfarrowroundcap c\fi% + }, +}% + + + + +\pgfdeclarearrow{ + name = Implies, + setup code = { + \pgf@xa\pgfinnerlinewidth + \pgfutil@tempdima.25\pgflinewidth + \advance\pgfutil@tempdima by.25\pgf@xa% + \pgfutil@tempdimb.5\pgflinewidth + \advance\pgfutil@tempdimb by-.5\pgf@xa% + \pgfarrowssettipend{2.06\pgfutil@tempdima\advance\pgf@x by.5\pgfutil@tempdimb} + \pgfarrowssetbackend{-1.36\pgfutil@tempdima\advance\pgf@x by.5\pgfutil@tempdimb} + \pgfarrowsupperhullpoint{-1.4\pgfutil@tempdima}{2.8\pgfutil@tempdima} + \pgfarrowsupperhullpoint{2\pgfutil@tempdima}{.5\pgfutil@tempdimb} + \pgfarrowssavethe\pgfutil@tempdima + \pgfarrowssavethe\pgfutil@tempdimb + }, + drawing code = { + \pgftransformxshift{.06\pgfutil@tempdima} + \pgfsetlinewidth{\pgfutil@tempdimb} + \pgfsetdash{}{+0pt} + \pgfsetroundcap + \pgfsetroundjoin + \pgfpathmoveto{\pgfpoint{-1.4\pgfutil@tempdima}{2.65\pgfutil@tempdima}} + \pgfpathcurveto + {\pgfpoint{-0.75\pgfutil@tempdima}{1.25\pgfutil@tempdima}} + {\pgfpoint{1\pgfutil@tempdima}{0.05\pgfutil@tempdima}} + {\pgfpoint{2\pgfutil@tempdima}{0pt}} + \pgfpathcurveto + {\pgfpoint{1\pgfutil@tempdima}{-0.05\pgfutil@tempdima}} + {\pgfpoint{-.75\pgfutil@tempdima}{-1.25\pgfutil@tempdima}} + {\pgfpoint{-1.4\pgfutil@tempdima}{-2.65\pgfutil@tempdima}} + \pgfusepathqstroke + }, + parameters = {\the\pgfarrowlength} +}% + + + + + + + +\pgfdeclarearrow{ + name = Latex, + defaults = { + length = +3pt 4.5 .8,% + width' = +0pt .75, + line width = +0pt 1 1, + }, + setup code = { + \pgf@x.2\pgfarrowlength + \ifdim\pgf@x<\pgfarrowlinewidth + \pgfarrowlinewidth\pgf@x + \fi + \pgfarrowssavethe\pgfarrowlinewidth% + \pgfmathdivide@{\pgf@sys@tonumber\pgfarrowlength}{\pgf@sys@tonumber\pgfarrowwidth}% + \let\pgf@temp@quot\pgfmathresult% + \pgf@x\pgfmathresult pt% + \pgf@x\pgfmathresult\pgf@x% + \pgf@x9\pgf@x% + \advance\pgf@x by1pt% + \pgfmathsqrt@{\pgf@sys@tonumber\pgf@x}% + \pgf@xc\pgfmathresult\pgfarrowlinewidth% + \pgf@xa\pgf@temp@quot\pgfarrowlinewidth + \pgfutil@tempdima\pgfarrowlength% + \advance\pgfutil@tempdima by-.5\pgf@xc% + \advance\pgfutil@tempdima by-.5\pgfarrowlinewidth% + \pgfarrowssetbackend{-.5\pgfarrowlinewidth} + \ifpgfarrowreversed + \ifdim\pgfinnerlinewidth>0pt + \pgfarrowssetlineend{0pt} + \else + \pgfarrowssetlineend{\pgfutil@tempdima\advance\pgf@x by-.5\pgflinewidth} + \fi + \else + \pgfarrowssetlineend{0pt} + \fi + \ifpgfarrowroundjoin + \pgfarrowssettipend{\pgfutil@tempdima\advance\pgf@x by.5\pgfarrowlinewidth} + \else + \pgfarrowssettipend{\pgfarrowlength\advance\pgf@x by-.5\pgfarrowlinewidth\ifpgfarrowharpoon\advance\pgf@x by1.5\pgf@xa\fi} + \fi + \pgfarrowssavethe\pgfutil@tempdima + \pgf@process{\pgfpointnormalised{\pgfqpoint{.3\pgfarrowlength}{.2333333\pgfarrowwidth}}}% + \advance\pgf@y by1pt% + \pgf@yc\pgf@y\pgf@xc\pgf@x% + \pgfmathdivide@{\pgf@sys@tonumber\pgf@yc}{\pgf@sys@tonumber\pgf@xc}% + \pgfutil@tempdimb\pgfmathresult\pgfarrowlinewidth% + \pgfutil@tempdimb-.5\pgfutil@tempdimb% + \advance\pgfutil@tempdimb by.5\pgfarrowwidth% + \pgfarrowssavethe\pgfutil@tempdimb% + \pgfarrowshullpoint{\pgfarrowlength\advance\pgf@x by-.5\pgfarrowlinewidth\ifpgfarrowroundjoin\else\ifpgfarrowharpoon\advance\pgf@x by1.5\pgf@xa\fi\fi}{\ifpgfarrowharpoon-.5\pgfarrowlinewidth\else0pt\fi}% + \pgfarrowshullpoint{-.5\pgfarrowlinewidth}{.5\pgfarrowwidth\ifpgfarrowharpoon\advance\pgf@y by-.5\pgfarrowlinewidth\fi}% + \pgfarrowshullpoint{-.5\pgfarrowlinewidth}{\ifpgfarrowharpoon-.5\pgfarrowlinewidth\else-.5\pgfarrowwidth\fi}% + }, + drawing code = { + \pgfsetdash{}{+0pt} + \ifpgfarrowroundjoin\pgfsetroundjoin\else\pgfsetmiterjoin\fi + \ifdim\pgfarrowlinewidth=\pgflinewidth\else\pgfsetlinewidth{+\pgfarrowlinewidth}\fi + \pgfpathmoveto {\pgfqpoint{\pgfutil@tempdima}{0pt}} + \pgfpathcurveto {\pgfqpoint{.877192\pgfutil@tempdima}{.077922\pgfutil@tempdimb}} + {\pgfqpoint{.337381\pgfutil@tempdima}{.519480\pgfutil@tempdimb}} + {\pgfqpoint{0pt}{\pgfutil@tempdimb}} + \ifpgfarrowharpoon + \pgfpathlineto {\pgfpointorigin} + \else + \pgfpathlineto {\pgfqpoint{0pt}{-\pgfutil@tempdimb}} + \pgfpathcurveto {\pgfqpoint{.337381\pgfutil@tempdima}{-.519480\pgfutil@tempdimb}} + {\pgfqpoint{.877192\pgfutil@tempdima}{-.077922\pgfutil@tempdimb}} + {\pgfqpoint{\pgfutil@tempdima}{0pt}} + \fi + \pgfpathclose + \ifpgfarrowopen\pgfusepathqstroke\else\ifdim\pgfarrowlinewidth>0pt\pgfusepathqfillstroke\else\pgfusepathqfill\fi\fi + }, + parameters = { + \the\pgfarrowlinewidth,% + \the\pgfarrowlength,% + \the\pgfarrowwidth,% + \ifpgfarrowharpoon h\fi% + \ifpgfarrowopen o\fi% + \ifpgfarrowroundjoin j\fi% + }, +}% + + + + +\pgfdeclarearrow{ + name = Stealth, + defaults = { + length = +3pt 4.5 .8, + width' = +0pt .75, + inset' = +0pt 0.325, + line width = +0pt 1 1, + }, + setup code = { + \pgf@x\pgfarrowlength + \advance\pgf@x by-\pgfarrowinset + \pgf@x.25\pgf@x + \ifdim\pgf@x<\pgfarrowlinewidth + \pgfarrowlinewidth\pgf@x + \fi + \pgfmathdivide@{\pgf@sys@tonumber\pgfarrowlength}{\pgf@sys@tonumber\pgfarrowwidth}% + \let\pgf@temp@quot\pgfmathresult% + \pgf@x\pgfmathresult pt% + \pgf@x\pgfmathresult\pgf@x% + \pgf@x4\pgf@x% + \advance\pgf@x by1pt% + \pgfmathsqrt@{\pgf@sys@tonumber\pgf@x}% + \pgf@xc\pgfmathresult\pgfarrowlinewidth% + \pgf@xc.5\pgf@xc + \pgf@xa\pgf@temp@quot\pgfarrowlinewidth% + \pgf@ya.5\pgfarrowwidth% + \csname pgfmathatan2@\endcsname{\pgfmath@tonumber\pgfarrowlength}{\pgfmath@tonumber\pgf@ya}% + \pgf@yb\pgfmathresult pt% + \csname pgfmathatan2@\endcsname{\pgfmath@tonumber\pgfarrowinset}{\pgfmath@tonumber\pgf@ya}% + \pgf@ya\pgfmathresult pt% + \advance\pgf@yb by-\pgf@ya% + \pgf@yb.5\pgf@yb% + \pgfmathtan@{\pgf@sys@tonumber\pgf@yb}% + \pgfmathreciprocal@{\pgfmathresult}% + \pgf@yc\pgfmathresult\pgfarrowlinewidth% + \pgf@yc.5\pgf@yc% + \advance\pgf@ya by\pgf@yb% + \pgfmathsincos@{\pgf@sys@tonumber\pgf@ya}% + \pgf@ya\pgfmathresulty\pgf@yc% + \pgf@yb\pgfmathresultx\pgf@yc% + \ifdim\pgfarrowinset=0pt% + \pgf@ya.5\pgfarrowlinewidth% + \fi + \pgfmathdivide@{\pgf@sys@tonumber\pgfarrowinset}{\pgf@sys@tonumber\pgfarrowwidth}% + \let\pgf@temp@quot\pgfmathresult% + \pgf@x\pgfmathresult pt% + \pgf@x\pgfmathresult\pgf@x% + \pgf@x4\pgf@x% + \advance\pgf@x by1pt% + \pgfmathsqrt@{\pgf@sys@tonumber\pgf@x}% + \pgf@yc\pgfmathresult\pgfarrowlinewidth% + \pgf@yc.5\pgf@yc% + \pgfutil@tempdima\pgfarrowlength% + \advance\pgfutil@tempdima by-\pgf@xc% + \advance\pgfutil@tempdima by-\pgf@ya% + \pgfutil@tempdimb.5\pgfarrowwidth% + \advance\pgfutil@tempdimb by-\pgf@yb% + \ifpgfarrowroundjoin + \pgfarrowssetbackend{\pgf@ya\advance\pgf@x by-.5\pgfarrowlinewidth} + \else + \pgfarrowssetbackend{0pt} + \fi + \ifpgfarrowharpoon + \pgfarrowssetlineend{\pgfarrowinset\advance\pgf@x + by\pgf@yc\advance\pgf@x by.5\pgfarrowlinewidth} + \else + \pgfarrowssetlineend{\pgfarrowinset\advance\pgf@x by\pgf@yc\advance\pgf@x by-.25\pgfarrowlinewidth} + \ifpgfarrowreversed + \ifdim\pgfinnerlinewidth>0pt + \pgfarrowssetlineend{\pgfarrowinset} + \else + \pgfarrowssetlineend{\pgfutil@tempdima\advance\pgf@x by\pgf@ya\advance\pgf@x by-.25\pgfarrowlinewidth} + \fi + \fi + \fi + \ifpgfarrowroundjoin + \pgfarrowssettipend{\pgfutil@tempdima\advance\pgf@x by\pgf@ya\advance\pgf@x by.5\pgfarrowlinewidth} + \else + \pgfarrowssettipend{\pgfarrowlength\ifpgfarrowharpoon\advance\pgf@x by\pgf@xa\fi} + \fi + \pgfarrowshullpoint{\pgfarrowlength\ifpgfarrowroundjoin\else\ifpgfarrowharpoon\advance\pgf@x by\pgf@xa\fi\fi}{\ifpgfarrowharpoon-.5\pgfarrowlinewidth\else0pt\fi}% + \pgfarrowsupperhullpoint{0pt}{.5\pgfarrowwidth}% + \pgfarrowshullpoint{\pgfarrowinset}{\ifpgfarrowharpoon-.5\pgfarrowlinewidth\else 0pt\fi}% + \pgfarrowssetvisualbackend{\pgfarrowinset} + \advance\pgfarrowinset by\pgf@yc% + \pgfarrowssavethe\pgfutil@tempdima + \pgfarrowssavethe\pgfutil@tempdimb + \pgfarrowssavethe\pgfarrowlinewidth + \pgfarrowssavethe\pgf@ya + \pgfarrowssavethe\pgfarrowinset + }, + drawing code = { + \pgfsetdash{}{+0pt} + \ifpgfarrowroundjoin\pgfsetroundjoin\else\pgfsetmiterjoin\fi + \ifdim\pgfarrowlinewidth=\pgflinewidth\else\pgfsetlinewidth{+\pgfarrowlinewidth}\fi + \pgfpathmoveto{\pgfqpoint{\pgfutil@tempdima\advance\pgf@x by\pgf@ya}{0pt}} + \pgfpathlineto{\pgfqpoint{\pgf@ya}{\pgfutil@tempdimb}} + \pgfpathlineto{\pgfqpoint{\pgfarrowinset}{0pt}} + \ifpgfarrowharpoon \else + \pgfpathlineto{\pgfqpoint{\pgf@ya}{-\pgfutil@tempdimb}} + \fi + \pgfpathclose + \ifpgfarrowopen\pgfusepathqstroke\else\ifdim\pgfarrowlinewidth>0pt\pgfusepathqfillstroke\else\pgfusepathqfill\fi\fi + }, + parameters = { + \the\pgfarrowlinewidth,% + \the\pgfarrowlength,% + \the\pgfarrowwidth,% + \the\pgfarrowinset,% + \ifpgfarrowharpoon h\fi% + \ifpgfarrowopen o\fi% + \ifpgfarrowroundjoin j\fi% + }, +}% + + + + +\pgfdeclarearrow{ + name = Kite, + defaults = { + length = +3.6pt +5.4, + width' = +0pt +0.5, + inset' = +0pt 0.25, + line width = +0pt 1 1, + }, + setup code = { + \pgf@x.4\pgfarrowlength + \ifdim\pgf@x<\pgfarrowlinewidth + \pgfarrowlinewidth\pgf@x + \fi + \pgf@x.4\pgfarrowwidth + \ifdim\pgf@x<\pgfarrowlinewidth + \pgfarrowlinewidth\pgf@x + \fi + \pgf@xa\pgfarrowlength% + \advance\pgf@xa by-\pgfarrowinset% + \pgfmathdivide@{\pgf@sys@tonumber\pgf@xa}{\pgf@sys@tonumber\pgfarrowwidth}% + \let\pgf@temp@quot\pgfmathresult% + \pgf@x\pgfmathresult pt% + \pgf@x\pgfmathresult\pgf@x% + \pgf@x4\pgf@x% + \advance\pgf@x by1pt% + \pgfmathsqrt@{\pgf@sys@tonumber\pgf@x}% + \pgf@xc\pgfmathresult\pgfarrowlinewidth% + \pgf@xc.5\pgf@xc + \pgf@xa\pgf@temp@quot\pgfarrowlinewidth% + \pgfmathdivide@{\pgf@sys@tonumber\pgfarrowinset}{\pgf@sys@tonumber\pgfarrowwidth}% + \let\pgf@temp@quot\pgfmathresult% + \pgf@x\pgfmathresult pt% + \pgf@x\pgfmathresult\pgf@x% + \pgf@x4\pgf@x% + \advance\pgf@x by1pt% + \pgfmathsqrt@{\pgf@sys@tonumber\pgf@x}% + \pgf@yc\pgfmathresult\pgfarrowlinewidth% + \pgf@yc.5\pgf@yc + \pgf@ya\pgf@temp@quot\pgfarrowlinewidth% + {% + \pgfutil@tempdimb.5\pgfarrowwidth% + \pgfutil@tempdima\pgfarrowlength\advance\pgfutil@tempdima by-\pgfarrowinset% + \csname pgfmathatan2@\endcsname{\pgfmath@tonumber\pgfutil@tempdima}{\pgfmath@tonumber\pgfutil@tempdimb}% + \pgf@yb\pgfmathresult pt% + \csname pgfmathatan2@\endcsname{\pgfmath@tonumber\pgfarrowinset}{\pgfmath@tonumber\pgfutil@tempdimb}% + \pgf@ya\pgfmathresult pt% + \advance\pgf@yb by\pgf@ya% + \pgf@yb.5\pgf@yb% + \ifdim\pgf@yb=45pt% + \def\pgfmathresult{1.414213}% + \else% + \pgfmathsin@{\pgf@sys@tonumber\pgf@yb}% + \pgfmathreciprocal@{\pgfmathresult}% + \fi% + \pgf@yc\pgfmathresult\pgfarrowlinewidth% + \pgf@yc.5\pgf@yc% + \pgf@ya-\pgf@ya% + \advance\pgf@ya by-90pt% + \advance\pgf@ya by\pgf@yb% + \pgfmathsincos@{\pgf@sys@tonumber\pgf@ya}% + \pgf@xb\pgfmathresultx\pgf@yc% + \pgf@yb\pgfmathresulty\pgf@yc% + \expandafter\expandafter\expandafter% + }\expandafter\expandafter\expandafter\pgf@xb\expandafter\the\expandafter\pgf@xb\expandafter\pgf@yb\the\pgf@yb% + \pgfutil@tempdima\pgfarrowlength% + \advance\pgfutil@tempdima by-\pgf@xc% + \pgfutil@tempdimb.5\pgfarrowwidth% + \advance\pgfutil@tempdimb by\pgf@yb% + \ifpgfarrowroundjoin + \pgfarrowssetbackend{\pgf@ya\advance\pgf@x by-.5\pgfarrowlinewidth} + \else + \pgfarrowssetbackend{0pt} + \fi + \ifpgfarrowreversed + \pgfarrowssetlineend{\pgfutil@tempdima\advance\pgf@x by-\pgflinewidth\advance\pgf@x by\pgfarrowlinewidth} + \else + \pgfarrowssetlineend{\pgf@yc\advance\pgf@x by\pgflinewidth\advance\pgf@x by-\pgfarrowlinewidth} + \fi + \ifpgfarrowroundjoin + \pgfarrowssettipend{\pgfutil@tempdima\advance\pgf@x by.5\pgfarrowlinewidth} + \else + \pgfarrowssettipend{\pgfarrowlength\ifpgfarrowharpoon\advance\pgf@x by\pgf@xa\fi} + \fi + \pgfarrowshullpoint{\pgfarrowlength\ifpgfarrowroundjoin\else\ifpgfarrowharpoon\advance\pgf@x by\pgf@xa\fi\fi}{\ifpgfarrowharpoon-.5\pgfarrowlinewidth\else0pt\fi}% + \pgfarrowsupperhullpoint{\pgfarrowinset}{.5\pgfarrowwidth}% + \pgfarrowshullpoint{0pt}{\ifpgfarrowharpoon-.5\pgfarrowlinewidth\else0pt\fi}% + \advance\pgfarrowinset by\pgf@xb% + \pgfarrowssavethe\pgfutil@tempdima + \pgfarrowssavethe\pgfutil@tempdimb + \pgfarrowssavethe\pgfarrowlinewidth + \pgfarrowssavethe\pgf@yc + \pgfarrowssavethe\pgfarrowinset + }, + drawing code = { + \pgfsetdash{}{+0pt} + \ifpgfarrowroundjoin\pgfsetroundjoin\else\pgfsetmiterjoin\fi + \ifdim\pgfarrowlinewidth=\pgflinewidth\else\pgfsetlinewidth{\pgfarrowlinewidth}\fi + \pgfpathmoveto{\pgfqpoint{\pgfutil@tempdima}{0pt}} + \pgfpathlineto{\pgfqpoint{\pgfarrowinset}{\pgfutil@tempdimb}} + \pgfpathlineto{\pgfqpoint{\pgf@yc}{0pt}} + \ifpgfarrowharpoon \else + \pgfpathlineto{\pgfqpoint{\pgfarrowinset}{-\pgfutil@tempdimb}} + \fi + \pgfpathclose + \ifpgfarrowopen\pgfusepathqstroke\else\ifdim\pgfarrowlinewidth>0pt\pgfusepathqfillstroke\else\pgfusepathqfill\fi\fi + }, + parameters = { + \the\pgfarrowlinewidth,% + \the\pgfarrowlength,% + \the\pgfarrowwidth,% + \the\pgfarrowinset,% + \ifpgfarrowharpoon h\fi% + \ifpgfarrowopen o\fi% + \ifpgfarrowroundjoin j\fi% + }, +}% + + + + + +\pgfdeclarearrow{ + name = Square, + defaults = { + length = +2.12132pt +2.828427, + width' = +0pt +1, + line width = +0pt +1 +1, + }, + setup code = { + \pgf@x.5\pgfarrowlength + \ifdim\pgf@x<\pgfarrowlinewidth + \pgfarrowlinewidth\pgf@x + \fi + \ifpgfarrowreversed + \pgfarrowssetlineend{\pgfarrowlength\advance\pgf@x by-.5\pgfarrowlinewidth} + \else + \pgfarrowssetlineend{.5\pgfarrowlinewidth} + \fi + \pgfarrowssettipend{\pgfarrowlength} + \pgfarrowshullpoint{\pgfarrowlength}{.5\pgfarrowwidth} + \pgfarrowshullpoint{0pt}{.5\pgfarrowwidth} + \pgfarrowshullpoint{\pgfarrowlength}{\ifpgfarrowharpoon-.5\pgfarrowlinewidth\else-.5\pgfarrowwidth\fi} + \pgfarrowshullpoint{0pt}{\ifpgfarrowharpoon-.5\pgfarrowlinewidth\else-.5\pgfarrowwidth\fi} + \pgfarrowssavethe\pgfarrowlinewidth + \pgfarrowssavethe\pgfarrowlength + \pgfarrowssavethe\pgfarrowwidth + }, + drawing code = { + \pgfsetdash{}{+0pt} + \ifpgfarrowroundjoin\pgfsetroundjoin\else\pgfsetmiterjoin\fi + \ifdim\pgfarrowlinewidth=\pgflinewidth\else\pgfsetlinewidth{+\pgfarrowlinewidth}\fi + \pgfpathmoveto{\pgfqpoint{\pgfarrowlength\advance\pgf@x by-.5\pgfarrowlinewidth}{.5\pgfarrowwidth\advance\pgf@y by-.5\pgfarrowlinewidth}} + \pgfpathlineto{\pgfqpoint{.5\pgfarrowlinewidth}{.5\pgfarrowwidth\advance\pgf@y by-.5\pgfarrowlinewidth}} + \pgfpathlineto{\pgfqpoint{.5\pgfarrowlinewidth}{\ifpgfarrowharpoon0pt\else-.5\pgfarrowwidth\advance\pgf@y by.5\pgfarrowlinewidth\fi}} + \pgfpathlineto{\pgfqpoint{\pgfarrowlength\advance\pgf@x by-.5\pgfarrowlinewidth}{\ifpgfarrowharpoon0pt\else-.5\pgfarrowwidth\advance\pgf@y by.5\pgfarrowlinewidth\fi}} + \pgfpathclose + \ifpgfarrowopen\pgfusepathqstroke\else\ifdim\pgfarrowlinewidth>0pt\pgfusepathqfillstroke\else\pgfusepathqfill\fi\fi + }, + parameters = { + \the\pgfarrowlinewidth,% + \the\pgfarrowlength,% + \the\pgfarrowwidth,% + \ifpgfarrowharpoon h\fi% + \ifpgfarrowopen o\fi% + \ifpgfarrowroundjoin j\fi% + }, +}% + + + +\pgfdeclarearrow{ + name = Circle, + defaults = { + length = +2.39365pt +3.191538, + width' = +0pt 1, + line width = +0pt 1 1, + }, + setup code = { + \pgf@x.5\pgfarrowlength + \ifdim\pgf@x<\pgfarrowlinewidth + \pgfarrowlinewidth\pgf@x + \fi + \ifpgfarrowreversed + \pgfarrowssetlineend{\pgfarrowlength\advance\pgf@x by-.5\pgfarrowlinewidth} + \else + \pgfarrowssetlineend{.5\pgfarrowlinewidth} + \fi + \pgfarrowssettipend{\pgfarrowlength} + \pgfarrowsupperhullpoint{\pgfarrowlength}{.25\pgfarrowwidth} + \pgfarrowsupperhullpoint{.75\pgfarrowlength}{.5\pgfarrowwidth} + \pgfarrowsupperhullpoint{.25\pgfarrowlength}{.5\pgfarrowwidth} + \pgfarrowsupperhullpoint{0pt}{.25\pgfarrowwidth} + \ifpgfarrowharpoon% + \pgfarrowshullpoint{0pt}{-.5\pgfarrowlinewidth} + \pgfarrowshullpoint{\pgfarrowlength}{-.5\pgfarrowlinewidth} + \fi% + \pgfarrowssavethe\pgfarrowlinewidth + \pgfarrowssavethe\pgfarrowlength + \pgfarrowssavethe\pgfarrowwidth + }, + drawing code = { + \pgfsetdash{}{+0pt} + \ifpgfarrowroundjoin\pgfsetroundjoin\else\pgfsetmiterjoin\fi + \ifdim\pgfarrowlinewidth=\pgflinewidth\else\pgfsetlinewidth{+\pgfarrowlinewidth}\fi + \ifpgfarrowharpoon + \pgfpathmoveto{\pgfqpoint{\pgfarrowlength\advance\pgf@x + by-.5\pgfarrowlinewidth}{0pt}} + {% + \pgf@xa.5\pgfarrowlength + \advance\pgf@xa by-.5\pgfarrowlinewidth + \pgftransformxscale{+\the\pgf@xa} + \pgf@xa.5\pgfarrowwidth + \advance\pgf@xa by-.5\pgfarrowlinewidth + \pgftransformyscale{+\the\pgf@xa} + \pgfpatharc{0}{180}{1pt} + } + \pgfpathclose + \else% + \pgfpathellipse{\pgfqpoint{.5\pgfarrowlength}{0pt}}{\pgfqpoint{.5\pgfarrowlength\advance\pgf@x + by-.5\pgfarrowlinewidth}{0pt}}{\pgfqpoint{0pt}{.5\pgfarrowwidth\advance\pgf@y + by-.5\pgfarrowlinewidth}} + \fi + \ifpgfarrowopen\pgfusepathqstroke\else\ifdim\pgfarrowlinewidth>0pt\pgfusepathqfillstroke\else\pgfusepathqfill\fi\fi + }, + parameters = { + \the\pgfarrowlinewidth,% + \the\pgfarrowlength,% + \the\pgfarrowwidth,% + \ifpgfarrowharpoon h\fi% + \ifpgfarrowopen o\fi% + \ifpgfarrowroundjoin j\fi% + }, +}% + + +\pgfdeclarearrow{ + name = Round Cap, + defaults = { + length = +0pt .5 + }, + setup code = { + \ifpgfarrowreversed + \pgfarrowssettipend{\pgfarrowlength} + \pgfarrowssetlineend{\pgfarrowlength\advance\pgf@x by.5\pgflinewidth} + \else + \pgfarrowssettipend{\pgfarrowlength} + \pgfarrowssetlineend{-.5\pgflinewidth} + \fi + \pgfarrowssavethe\pgfarrowlength + \pgfarrowsupperhullpoint{0pt}{.5\pgflinewidth} + \pgfarrowsupperhullpoint{\pgfarrowlength}{.5\pgflinewidth} + }, + drawing code = { + \ifpgfarrowreversed + \pgftransformxscale{-1} + \pgftransformxshift{-\pgfarrowlength} + \fi + \pgfutil@tempdima.5\pgflinewidth + \pgfpathmoveto {\pgfqpoint{-0.75\pgflinewidth}{\pgfutil@tempdima}} + \pgfpathlineto {\pgfqpoint{0pt}{\pgfutil@tempdima}} + \ifpgfarrowreversed + \pgfpathlineto {\pgfqpoint{\pgfarrowlength}{\pgfutil@tempdima}} + \begingroup + \pgftransformxshift{\pgfarrowlength} + \pgftransformxscale{-1} + \fi + \pgfpathcurveto {\pgfqpoint{0.55228475\pgfarrowlength}{\pgfutil@tempdima}} + {\pgfqpoint{\pgfarrowlength}{0.55228475\pgfutil@tempdima}} + {\pgfqpoint{\pgfarrowlength}{0pt}} + \pgfpathcurveto {\pgfqpoint{\pgfarrowlength}{-0.55228475\pgfutil@tempdima}} + {\pgfqpoint{0.55228475\pgfarrowlength}{-\pgfutil@tempdima}} + {\pgfqpoint{0pt}{-\pgfutil@tempdima}} + \ifpgfarrowreversed + \endgroup + \fi + \pgfpathlineto {\pgfqpoint{-.75\pgflinewidth}{-\pgfutil@tempdima}} + \pgfpathclose + \pgfusepathqfill + }, + parameters = {\the\pgfarrowlength} +}% + + + +\pgfdeclarearrow{ + name = Butt Cap, + defaults = { + length = +0pt .5 + }, + setup code = { + \pgfarrowssettipend{\pgfarrowlength} + \pgfarrowssetlineend{-.5\pgflinewidth} + \pgfarrowssavethe\pgfarrowlength + \pgfarrowsupperhullpoint{0pt}{.5\pgflinewidth} + \pgfarrowsupperhullpoint{\pgfarrowlength}{.5\pgflinewidth} + }, + drawing code = { + \pgfutil@tempdima.5\pgflinewidth + \pgfpathmoveto {\pgfqpoint{-0.75\pgflinewidth}{\pgfutil@tempdima}} + \pgfpathlineto {\pgfqpoint{0pt}{\pgfutil@tempdima}} + \pgfpathlineto {\pgfqpoint{\pgfarrowlength}{\pgfutil@tempdima}} + \pgfpathlineto {\pgfqpoint{\pgfarrowlength}{-\pgfutil@tempdima}} + \pgfpathlineto {\pgfqpoint{-.75\pgflinewidth}{-\pgfutil@tempdima}} + \pgfpathclose + \pgfusepathqfill + }, + parameters = {\the\pgfarrowlength} +}% + + + +\pgfkeys{ + /pgf/arrow keys/.cd, + cap angle/.code={% + \pgfmathsetlength\pgf@x{#1}% + \pgf@x.5\pgf@x + \pgfmathtan@{\pgf@sys@tonumber\pgf@x}% + \pgfmathreciprocal@{\pgfmathresult}% + \pgf@x\pgfmathresult pt% + \pgf@x.5\pgf@x% + \edef\pgf@marshal{\noexpand\pgfarrowsaddtooptions{\noexpand\pgfarrowslinewidthdependent{+0pt}{\pgf@sys@tonumber\pgf@x}{0}\pgfarrowlength\pgf@x}} + \pgf@marshal + } +}% + + +\pgfdeclarearrow{ + name = Triangle Cap, + defaults = { + length = +0pt .5 + }, + setup code = { + \ifpgfarrowreversed + \pgfarrowssettipend{\pgfarrowlength} + \pgfarrowssetlineend{\pgfarrowlength\advance\pgf@x by.5\pgflinewidth} + \else + \pgfarrowssettipend{\pgfarrowlength} + \pgfarrowssetlineend{-.5\pgflinewidth} + \fi + \pgfarrowssavethe\pgfarrowlength + \pgfarrowsupperhullpoint{0pt}{.5\pgflinewidth} + \pgfarrowshullpoint{\pgfarrowlength}{0pt} + }, + drawing code = { + \ifpgfarrowreversed + \pgftransformxscale{-1} + \pgftransformxshift{-\pgfarrowlength} + \fi + \pgfutil@tempdima.5\pgflinewidth + \pgfpathmoveto {\pgfqpoint{-0.75\pgflinewidth}{\pgfutil@tempdima}} + \pgfpathlineto {\pgfqpoint{0pt}{\pgfutil@tempdima}} + \ifpgfarrowreversed + \pgfpathlineto {\pgfqpoint{\pgfarrowlength}{\pgfutil@tempdima}} + \begingroup + \pgftransformxshift{\pgfarrowlength} + \pgftransformxscale{-1} + \fi + \pgfpathlineto {\pgfqpoint{\pgfarrowlength}{0pt}} + \ifpgfarrowreversed + \endgroup + \pgfpathlineto {\pgfqpoint{\pgfarrowlength}{-\pgfutil@tempdima}} + \fi + \pgfpathlineto {\pgfqpoint{0pt}{-\pgfutil@tempdima}} + \pgfpathlineto {\pgfqpoint{-.75\pgflinewidth}{-\pgfutil@tempdima}} + \pgfpathclose + \pgfusepathqfill + }, + parameters = {\the\pgfarrowlength} +}% + + + +\pgfdeclarearrow{ + name = Fast Triangle, + defaults = { + length = +0pt .5, + inset = +0pt .5, + }, + setup code = { + \pgfarrowssettipend{\pgfarrowlength\advance\pgf@x by\pgfarrowinset} + \ifpgfarrowreversed\pgfarrowssetlineend{\pgfarrowlength\advance\pgf@x by\pgfarrowinset}\fi + \pgfarrowssavethe\pgfarrowlength + \pgfarrowssavethe\pgfarrowinset + \pgfarrowsupperhullpoint{0pt}{.5\pgflinewidth} + \pgfarrowsupperhullpoint{\pgfarrowinset}{.5\pgflinewidth} + \pgfarrowshullpoint{\pgfarrowlength\advance\pgf@x by\pgfarrowinset}{0pt} + }, + drawing code = { + \pgfutil@tempdima.5\pgflinewidth + \pgfpathmoveto {\pgfqpoint{0pt}{\pgfutil@tempdima}} + \pgfpathlineto {\pgfqpoint{\pgfarrowinset}{\pgfutil@tempdima}} + \pgfpathlineto {\pgfqpoint{\pgfarrowinset\advance\pgf@x by\pgfarrowlength}{0pt}} + \pgfpathlineto {\pgfqpoint{\pgfarrowinset}{-\pgfutil@tempdima}} + \pgfpathlineto {\pgfqpoint{0pt}{-\pgfutil@tempdima}} + \pgfpathlineto {\pgfqpoint{\pgfarrowlength}{0pt}} + \pgfpathclose + \pgfusepathqfill + }, + parameters = {\the\pgfarrowlength,\the\pgfarrowinset} +}% + + +\pgfdeclarearrow{ + name = Fast Round, + defaults = { + length = +0pt .5, + inset = +0pt .5, + }, + setup code = { + \pgfarrowssettipend{\pgfarrowlength\advance\pgf@x by\pgfarrowinset} + \ifpgfarrowreversed\pgfarrowssetlineend{\pgfarrowlength\advance\pgf@x by\pgfarrowinset}\fi + \pgfarrowssavethe\pgfarrowlength + \pgfarrowssavethe\pgfarrowinset + \pgfarrowsupperhullpoint{0pt}{.5\pgflinewidth} + \pgfarrowsupperhullpoint{\pgfarrowlength\advance\pgf@x by\pgfarrowinset}{.5\pgflinewidth} + }, + drawing code = { + \pgfutil@tempdima.5\pgflinewidth + \pgfpathmoveto {\pgfqpoint{0pt}{\pgfutil@tempdima}} + \pgfpathlineto {\pgfqpoint{\pgfarrowinset}{\pgfutil@tempdima}} + { + \pgftransformxshift{\pgfarrowinset} + \pgfpathcurveto{\pgfqpoint{0.55228475\pgfarrowlength}{\pgfutil@tempdima}} + {\pgfqpoint{\pgfarrowlength}{0.55228475\pgfutil@tempdima}} + {\pgfqpoint{\pgfarrowlength}{0pt}} + \pgfpathcurveto{\pgfqpoint{\pgfarrowlength}{-0.55228475\pgfutil@tempdima}} + {\pgfqpoint{0.55228475\pgfarrowlength}{-\pgfutil@tempdima}} + {\pgfqpoint{0pt}{-\pgfutil@tempdima}} + } + \pgfpathlineto {\pgfqpoint{0pt}{-\pgfutil@tempdima}} + \pgfpathcurveto {\pgfqpoint{0.55228475\pgfarrowlength}{-\pgfutil@tempdima}} + {\pgfqpoint{\pgfarrowlength}{-0.55228475\pgfutil@tempdima}} + {\pgfqpoint{\pgfarrowlength}{0pt}} + \pgfpathcurveto {\pgfqpoint{\pgfarrowlength}{0.55228475\pgfutil@tempdima}} + {\pgfqpoint{0.55228475\pgfarrowlength}{\pgfutil@tempdima}} + {\pgfqpoint{0pt}{\pgfutil@tempdima}} + \pgfpathclose + \pgfusepathqfill + }, + parameters = {\the\pgfarrowlength,\the\pgfarrowinset} +}% + + + + + + +\pgfkeys{ + /pgf/arrow keys/.cd, + n/.code={% + \pgfmathparse{#1}% + \expandafter\pgf@lib@meta@strip@dot\pgfmathresult.\relax% + } +}% +\def\pgf@lib@meta@strip@dot#1.#2\relax{% + \pgfarrowsaddtooptions{\def\pgfarrown{#1}}% +}% + +\def\pgfarrown{4}% + +\pgfdeclarearrow{ + name = Rays, + defaults = { + length = +3pt 4, + width' = +0pt 1, + line width = +0pt 1 1, + }, + setup code = { + \pgfmathdivide@{360}{\pgfarrown}% + \let\pgfarrow@inc\pgfmathresult% + \pgf@xa\pgfarrow@inc pt% + \ifodd\pgfarrown\pgf@ya.25\pgf@xa\else\pgf@ya.5\pgf@xa\fi% + \pgfmathsincos@{\pgf@sys@tonumber\pgf@ya}% + \pgf@x.5\pgfarrowlength% + \pgf@xa\pgfmathresultx\pgf@x% + \ifpgfarrowroundcap + \advance\pgf@xa by.5\pgfarrowlinewidth + \else + \pgf@process{\pgfpointnormalised{\pgfqpoint{\pgfmathresultx\pgfarrowlength}{\pgfmathresulty\pgfarrowwidth}}}% + \pgf@x.5\pgfarrowlinewidth% + \advance\pgf@xa by\pgf@sys@tonumber{\pgf@y}\pgf@x% + \fi% + \pgfarrowssettipend{\pgf@xa} + \pgfarrowssetbackend{-\pgf@xa} + \pgfarrowsupperhullpoint{\pgf@xa}{.25\pgfarrowwidth} + \pgfarrowsupperhullpoint{.25\pgfarrowlength}{.5\pgfarrowwidth\ifpgfarrowroundcap\advance\pgf@y by.5\pgfarrowlinewidth\fi} + \pgfarrowsupperhullpoint{-.25\pgfarrowlength}{.5\pgfarrowwidth\ifpgfarrowroundcap\advance\pgf@y by.5\pgfarrowlinewidth\fi} + \pgfarrowsupperhullpoint{-\pgf@xa}{.25\pgfarrowwidth} + \ifpgfarrowharpoon + \pgfarrowshullpoint{-.5\pgfarrowlength\ifpgfarrowroundcap\advance\pgf@x by-.5\pgfarrowlinewidth\fi}{-.5\pgfarrowlinewidth} + \pgfarrowshullpoint{.5\pgfarrowlength\ifpgfarrowroundcap\advance\pgf@x by.5\pgfarrowlinewidth\fi}{-.5\pgfarrowlinewidth} + \fi + \pgfarrowssavethe\pgfarrowlinewidth + \pgfarrowssavethe\pgfarrowlength + \pgfarrowssavethe\pgfarrowwidth + \pgfarrowssave\pgfarrow@inc + \pgfarrowssave\pgfarrown + }, + drawing code = { + \pgfsetdash{}{+0pt} + \ifpgfarrowroundcap\pgfsetroundcap\else\pgfsetbuttcap\fi + \ifdim\pgfarrowlinewidth=\pgflinewidth\else\pgfsetlinewidth{+\pgfarrowlinewidth}\fi + {% + \pgfgettransform\pgf@temp@trans + \pgftransformxscale{+.5\pgfarrowlength} + \pgftransformyscale{+.5\pgfarrowwidth} + \pgf@xa\pgfarrow@inc pt% + \ifodd\pgfarrown\pgf@ya.25\pgf@xa\else\pgf@ya.5\pgf@xa\fi% + \ifpgfarrowharpoon\pgf@yb180pt\else\pgf@yb360pt\fi + \pgfutil@loop% + \ifdim\pgf@ya<\pgf@yb% + \pgfpathmoveto{\pgfpointpolar{+\pgf@ya}{+1pt}} + \pgfpathlineto{\pgfpointorigin} + \ifpgfarrowharpoon\ifdim\pgf@ya>\pgf@xa\else{\pgfsettransform\pgf@temp@trans\pgfpathlineto{\pgfqpoint{-.5\pgflinewidth}{0pt}}}\fi\fi + \advance\pgf@ya by\pgf@xa\relax% + \pgfutil@repeat% + } + \pgfusepathqstroke + }, + parameters = { + \the\pgfarrowlinewidth,% + \the\pgfarrowlength,% + \the\pgfarrowwidth,% + \pgfarrown,% + \ifpgfarrowharpoon h\fi% + \ifpgfarrowroundcap c\fi% + } +}% + + + + +\pgfkeys{ + LaTeX /.tip = {Latex}, + Triangle /.tip = {Stealth[inset=+0pt, angle=+60:+2.7pt +3.6]}, + Rectangle /.tip = {Square[length=+3pt +4.5,width'=+0pt +.5]}, + Ellipse /.tip = {Circle[length=+3.3pt +4.95, width'=+0pt +0.5]}, + Diamond /.tip = {Kite[inset'=+0pt .5]}, + Turned Square /.tip = {Kite[length=+3pt 4,width'= +0pt 1,inset'= +0pt 0.5]}, + To /.tip = {Computer Modern Rightarrow}, + Bar /.tip = {Tee Barb[length=+0pt]}, + Bracket /.tip = {Tee Barb[inset'=+0pt +1,length=+0.75pt +1]}, + Parenthesis /.tip = {Arc Barb[arc=+120,length=+1.725pt +2.3]} +}% + + + + +\endinput diff --git a/preprints/A-polynomial-time-algorithm-for-three-machine-unit-job-scheduling-September-24-2026/build/tex/runtime/generic/pgf/libraries/pgflibraryplothandlers.code.tex b/preprints/A-polynomial-time-algorithm-for-three-machine-unit-job-scheduling-September-24-2026/build/tex/runtime/generic/pgf/libraries/pgflibraryplothandlers.code.tex new file mode 100644 index 0000000000000000000000000000000000000000..dc578cc1feed49ddfad9021c5d332e6140796e9f --- /dev/null +++ b/preprints/A-polynomial-time-algorithm-for-three-machine-unit-job-scheduling-September-24-2026/build/tex/runtime/generic/pgf/libraries/pgflibraryplothandlers.code.tex @@ -0,0 +1,773 @@ + +\ProvidesFileRCS{pgflibraryplothandlers.code.tex} + + + +\pgfdeclareplothandler{\pgfplothandlercurveto}{}{% + point macro=\pgf@plot@curveto@handler@initial, + jump macro=\pgf@plot@smooth@next@moveto, + end macro=\pgf@plot@curveto@handler@finish +}% + +\def\pgf@plot@smooth@next@moveto{% + \pgf@plot@curveto@handler@finish% + \global\pgf@plot@startedfalse% + \global\let\pgf@plotstreampoint\pgf@plot@curveto@handler@initial% +}% + +\def\pgf@plot@curveto@handler@initial#1{% + \pgf@process{#1}% + \pgf@xa=\pgf@x% + \pgf@ya=\pgf@y% + \pgf@plot@first@action{\pgfqpoint{\pgf@xa}{\pgf@ya}}% + \xdef\pgf@plot@curveto@first{\noexpand\pgfqpoint{\the\pgf@xa}{\the\pgf@ya}}% + \global\let\pgf@plot@curveto@first@support=\pgf@plot@curveto@first% + \global\let\pgf@plotstreampoint=\pgf@plot@curveto@handler@second% +}% + +\def\pgf@plot@curveto@handler@second#1{% + \pgf@process{#1}% + \xdef\pgf@plot@curveto@second{\noexpand\pgfqpoint{\the\pgf@x}{\the\pgf@y}}% + \global\let\pgf@plotstreampoint=\pgf@plot@curveto@handler@third% + \global\pgf@plot@startedtrue% +}% + +\def\pgf@plot@curveto@handler@third#1{% + \pgf@process{#1}% + \xdef\pgf@plot@curveto@current{\noexpand\pgfqpoint{\the\pgf@x}{\the\pgf@y}}% + \pgf@xa=\pgf@x% + \pgf@ya=\pgf@y% + \pgf@process{\pgf@plot@curveto@first} + \advance\pgf@xa by-\pgf@x% + \advance\pgf@ya by-\pgf@y% + \pgf@xa=\pgf@plottension\pgf@xa% + \pgf@ya=\pgf@plottension\pgf@ya% + \pgf@process{\pgf@plot@curveto@second}% + \pgf@xb=\pgf@x% + \pgf@yb=\pgf@y% + \pgf@xc=\pgf@x% + \pgf@yc=\pgf@y% + \advance\pgf@xb by-\pgf@xa% + \advance\pgf@yb by-\pgf@ya% + \advance\pgf@xc by\pgf@xa% + \advance\pgf@yc by\pgf@ya% + \edef\pgf@marshal{\noexpand\pgfpathcurveto{\noexpand\pgf@plot@curveto@first@support}% + {\noexpand\pgfqpoint{\the\pgf@xb}{\the\pgf@yb}}{\noexpand\pgf@plot@curveto@second}}% + {\pgf@marshal}% + \global\let\pgf@plot@curveto@first=\pgf@plot@curveto@second% + \global\let\pgf@plot@curveto@second=\pgf@plot@curveto@current% + \xdef\pgf@plot@curveto@first@support{\noexpand\pgfqpoint{\the\pgf@xc}{\the\pgf@yc}}% +}% + +\def\pgf@plot@curveto@handler@finish{% + \ifpgf@plot@started% + \pgfpathcurveto{\pgf@plot@curveto@first@support}{\pgf@plot@curveto@second}{\pgf@plot@curveto@second}% + \fi% +}% + + + +\def\pgfsetplottension#1{% + \pgf@x=#1pt\relax% + \pgf@x=0.2775\pgf@x\relax% + \edef\pgf@plottension{\pgf@sys@tonumber\pgf@x}}% +\pgfsetplottension{0.5}% + + + +\pgfdeclareplothandler{\pgfplothandlerclosedcurve}{}{% + point macro=\pgf@plot@closedcurve@handler@initial, + end macro=\pgf@plot@closedcurve@handler@finish, + jump macro=\pgf@plot@closedsmooth@next@moveto +}% + +\def\pgf@plot@closedsmooth@next@moveto{% + \pgf@plot@closedcurve@handler@finish% + \global\pgf@plot@startedfalse% + \global\let\pgf@plotstreampoint\pgf@plot@closedcurve@handler@initial% +}% + +\def\pgf@plot@closedcurve@handler@initial#1{% + \pgf@process{#1}% + \pgf@xa=\pgf@x% + \pgf@ya=\pgf@y% + \xdef\pgf@plot@closedcurve@initial{\noexpand\pgfqpoint{\the\pgf@xa}{\the\pgf@ya}}% + \global\let\pgf@plotstreampoint=\pgf@plot@closedcurve@handler@second% + \global\pgf@plot@startedfalse% +}% + +\def\pgf@plot@closedcurve@handler@second#1{% + \pgf@process{#1}% + \xdef\pgf@plot@closedcurve@after@initial{\noexpand\pgfqpoint{\the\pgf@x}{\the\pgf@y}}% + {\pgfpathmoveto{}}% + \global\let\pgf@plotstreampoint=\pgf@plot@closedcurve@handler@third% +}% + +\def\pgf@plot@closedcurve@handler@third#1{% + \global\pgf@plot@startedtrue% + \pgf@process{#1}% + \xdef\pgf@plot@closedcurve@current{\noexpand\pgfqpoint{\the\pgf@x}{\the\pgf@y}}% + \pgf@xa=\pgf@x% + \pgf@ya=\pgf@y% + \pgf@process{\pgf@plot@closedcurve@initial} + \advance\pgf@xa by-\pgf@x% + \advance\pgf@ya by-\pgf@y% + \pgf@xa=\pgf@plottension\pgf@xa% + \pgf@ya=\pgf@plottension\pgf@ya% + \pgf@process{\pgf@plot@closedcurve@after@initial}% + \pgf@xb=\pgf@x% + \pgf@yb=\pgf@y% + \pgf@xc=\pgf@x% + \pgf@yc=\pgf@y% + \advance\pgf@xb by-\pgf@xa% + \advance\pgf@yb by-\pgf@ya% + \advance\pgf@xc by\pgf@xa% + \advance\pgf@yc by\pgf@ya% + \global\let\pgf@plot@closedcurve@first=\pgf@plot@closedcurve@after@initial% + \global\let\pgf@plot@closedcurve@second=\pgf@plot@closedcurve@current% + \xdef\pgf@plot@closedcurve@after@initial@presupport{\noexpand\pgfqpoint{\the\pgf@xb}{\the\pgf@yb}}% + \xdef\pgf@plot@closedcurve@first@support{\noexpand\pgfqpoint{\the\pgf@xc}{\the\pgf@yc}}% + \global\let\pgf@plotstreampoint=\pgf@plot@closedcurve@handler@fourth% +}% + +\def\pgf@plot@closedcurve@handler@fourth#1{% + \pgf@process{#1}% + \xdef\pgf@plot@closedcurve@current{\noexpand\pgfqpoint{\the\pgf@x}{\the\pgf@y}}% + \pgf@xa=\pgf@x% + \pgf@ya=\pgf@y% + \pgf@process{\pgf@plot@closedcurve@first} + \advance\pgf@xa by-\pgf@x% + \advance\pgf@ya by-\pgf@y% + \pgf@xa=\pgf@plottension\pgf@xa% + \pgf@ya=\pgf@plottension\pgf@ya% + \pgf@process{\pgf@plot@closedcurve@second}% + \pgf@xb=\pgf@x% + \pgf@yb=\pgf@y% + \pgf@xc=\pgf@x% + \pgf@yc=\pgf@y% + \advance\pgf@xb by-\pgf@xa% + \advance\pgf@yb by-\pgf@ya% + \advance\pgf@xc by\pgf@xa% + \advance\pgf@yc by\pgf@ya% + \edef\pgf@marshal{\noexpand\pgfpathcurveto{\noexpand\pgf@plot@closedcurve@first@support}% + {\noexpand\pgfqpoint{\the\pgf@xb}{\the\pgf@yb}}{\noexpand\pgf@plot@closedcurve@second}}% + {\pgf@marshal}% + \global\let\pgf@plot@closedcurve@first=\pgf@plot@closedcurve@second% + \global\let\pgf@plot@closedcurve@second=\pgf@plot@closedcurve@current% + \xdef\pgf@plot@closedcurve@first@support{\noexpand\pgfqpoint{\the\pgf@xc}{\the\pgf@yc}}% +}% + +\def\pgf@plot@closedcurve@handler@finish{% + \ifpgf@plot@started + \pgf@process{\pgf@plot@closedcurve@initial}% + \pgf@xa=\pgf@x% + \pgf@ya=\pgf@y% + \pgf@process{\pgf@plot@closedcurve@first}% + \advance\pgf@xa by-\pgf@x% + \advance\pgf@ya by-\pgf@y% + \pgf@xa=\pgf@plottension\pgf@xa% + \pgf@ya=\pgf@plottension\pgf@ya% + \pgf@process{\pgf@plot@closedcurve@second}% + \pgf@xb=\pgf@x% + \pgf@yb=\pgf@y% + \pgf@xc=\pgf@x% + \pgf@yc=\pgf@y% + \advance\pgf@xb by-\pgf@xa% + \advance\pgf@yb by-\pgf@ya% + \advance\pgf@xc by\pgf@xa% + \advance\pgf@yc by\pgf@ya% + \edef\pgf@marshal{\noexpand\pgfpathcurveto{\noexpand\pgf@plot@closedcurve@first@support}% + {\noexpand\pgfqpoint{\the\pgf@xb}{\the\pgf@yb}}{\noexpand\pgf@plot@closedcurve@second}}% + {\pgf@marshal}% + \xdef\pgf@plot@closedcurve@first@support{\noexpand\pgfqpoint{\the\pgf@xc}{\the\pgf@yc}}% + \pgf@process{\pgf@plot@closedcurve@after@initial}% + \pgf@xa=\pgf@x% + \pgf@ya=\pgf@y% + \pgf@process{\pgf@plot@closedcurve@second}% + \advance\pgf@xa by-\pgf@x% + \advance\pgf@ya by-\pgf@y% + \pgf@xa=\pgf@plottension\pgf@xa% + \pgf@ya=\pgf@plottension\pgf@ya% + \pgf@process{\pgf@plot@closedcurve@initial}% + \pgf@xb=\pgf@x% + \pgf@yb=\pgf@y% + \pgf@xc=\pgf@x% + \pgf@yc=\pgf@y% + \advance\pgf@xb by-\pgf@xa% + \advance\pgf@yb by-\pgf@ya% + \advance\pgf@xc by\pgf@xa% + \advance\pgf@yc by\pgf@ya% + \edef\pgf@marshal{\noexpand\pgfpathcurveto{\noexpand\pgf@plot@closedcurve@first@support}% + {\noexpand\pgfqpoint{\the\pgf@xb}{\the\pgf@yb}}{\noexpand\pgf@plot@closedcurve@initial}}% + {\pgf@marshal}% + \edef\pgf@marshal{\noexpand\pgfpathcurveto{\noexpand\pgfqpoint{\the\pgf@xc}{\the\pgf@yc}}% + {\noexpand\pgf@plot@closedcurve@after@initial@presupport}{\noexpand\pgf@plot@closedcurve@after@initial}}% + {\pgf@marshal}% + \pgfpathclose% + \fi% +}% + + + + + + +\pgfdeclareplothandler{\pgfplothandlerxcomb}{}{% + start macro=\pgfplotxzerolevelstreamstart, + point macro=\pgf@plot@xcomb@handler, + end macro=\pgfplotxzerolevelstreamend, +}% + +\def\pgf@plot@xcomb@handler#1{% + \pgf@process{#1}% + \pgf@xa=\pgf@x% + \pgf@ya=\pgf@y% + \begingroup + \pgfplotxzerolevelstreamnext + \endgroup + \pgf@yb=\pgf@x + \pgfpathmoveto{\pgfqpoint{\pgf@yb}{\pgf@ya}}% + \pgfpathlineto{\pgfqpoint{\pgf@xa}{\pgf@ya}}% +}% + + + +\pgfdeclareplothandler{\pgfplothandlerycomb}{}{% + start macro=\pgfplotyzerolevelstreamstart, + point macro=\pgf@plot@ycomb@handler, + end macro=\pgfplotyzerolevelstreamend +}% + +\def\pgf@plot@ycomb@handler#1{% + \pgf@process{#1}% + \pgf@xa=\pgf@x% + \pgf@ya=\pgf@y% + \begingroup + \pgfplotyzerolevelstreamnext + \endgroup + \pgf@yb=\pgf@x + \pgfpathmoveto{\pgfqpoint{\pgf@xa}{\pgf@yb}}% + \pgfpathlineto{\pgfqpoint{\pgf@xa}{\pgf@ya}}% +}% + +\def\pgfplotxzerolevelstreamstart{\pgf@plotxzerolevelstreamstart}% +\def\pgfplotxzerolevelstreamend{\pgf@plotxzerolevelstreamend}% +\def\pgfplotxzerolevelstreamnext{\pgf@plotxzerolevelstreamnext} +\def\pgfplotyzerolevelstreamstart{\pgf@plotyzerolevelstreamstart}% +\def\pgfplotyzerolevelstreamend{\pgf@plotyzerolevelstreamend}% +\def\pgfplotyzerolevelstreamnext{\pgf@plotyzerolevelstreamnext}% + +\def\pgfplotxzerolevelstreamconstant#1{% + \edef\pgfplotxzerolevelstreamconstant@val{#1}% + \def\pgf@plotxzerolevelstreamstart{% + \global\let\pgf@plotxzerolevelstreamend=\relax + \gdef\pgf@plotxzerolevelstreamnext{\global\pgf@x=\pgfplotxzerolevelstreamconstant@val\relax}% + }% +}% +\pgfplotxzerolevelstreamconstant{0pt}% + +\def\pgfplotyzerolevelstreamconstant#1{% + \edef\pgfplotyzerolevelstreamconstant@val{#1}% + \def\pgf@plotyzerolevelstreamstart{% + \global\let\pgf@plotyzerolevelstreamend=\relax + \gdef\pgf@plotyzerolevelstreamnext{\global\pgf@x=\pgfplotyzerolevelstreamconstant@val\relax}% + }% +}% +\pgfplotyzerolevelstreamconstant{0pt}% + +\def\pgfplotbarwidth{\pgfkeysvalueof{/pgf/bar width}}% +\def\pgfplotbarshift{\pgfkeysvalueof{/pgf/bar shift}}% + +\pgfqkeys{/pgf}{% + bar width/.initial=10pt, + bar shift/.initial=0pt, + bar interval width/.initial=1, + bar interval shift/.initial=0.5, + at begin bar/.initial=, + at end bar/.initial=, +}% + + +\pgfdeclareplothandler{\pgfplothandlerybar}{}{% + start={ + \pgfmathparse{\pgfplotbarwidth}% + \xdef\pgfplotbarwidth@{\pgfmathresult pt}% + \pgfmathparse{\pgfplotbarshift}% + \xdef\pgfplotbarshift@{\pgfmathresult pt}% + \pgfplotyzerolevelstreamstart + }, + end macro=\pgfplotyzerolevelstreamend, + point macro=\pgf@plot@ybar@handler +}% + +\def\pgf@plot@ybar@handler#1{% + \pgfkeysvalueof{/pgf/at begin bar}% + \pgf@process{#1}% + \pgf@ya=\pgf@y + \expandafter\pgf@xb\pgfplotbarwidth@\relax + \pgf@xc=\pgf@x + \advance\pgf@xc by-.5\pgf@xb + \advance\pgf@xc by\pgfplotbarshift@\relax + \begingroup + \pgfplotyzerolevelstreamnext + \endgroup + \pgf@yb=\pgf@x + \advance\pgf@ya by-\pgf@yb + \pgfpathrectangle + {\pgfqpoint{\pgf@xc}{\pgf@yb}}% + {\pgfqpoint{\pgf@xb}{\pgf@ya}}% + \pgfkeysvalueof{/pgf/at end bar}% +}% + +\pgfdeclareplothandler{\pgfplothandlerxbar}{}{% + start={ + \pgfmathparse{\pgfplotbarwidth}% + \xdef\pgfplotbarwidth@{\pgfmathresult pt}% + \pgfmathparse{\pgfplotbarshift}% + \xdef\pgfplotbarshift@{\pgfmathresult pt}% + \pgfplotxzerolevelstreamstart + }, + end macro=\pgfplotxzerolevelstreamend, + point macro=\pgf@plot@xbar@handler +}% + +\def\pgf@plot@xbar@handler#1{% + \pgfkeysvalueof{/pgf/at begin bar}% + \pgf@process{#1}% + \pgf@ya=\pgf@x + \expandafter\pgf@xb\pgfplotbarwidth@\relax + \pgf@xc=\pgf@y + \advance\pgf@xc by-.5\pgf@xb + \advance\pgf@xc by\pgfplotbarshift@\relax + \begingroup + \pgfplotxzerolevelstreamnext + \endgroup + \pgf@yb=\pgf@x + \advance\pgf@ya by-\pgf@yb + \pgfpathrectangle + {\pgfqpoint{\pgf@yb}{\pgf@xc}}% + {\pgfqpoint{\pgf@ya}{\pgf@xb}}% + \pgfkeysvalueof{/pgf/at end bar}% +}% + +\pgfdeclareplothandler{\pgfplothandlerybarinterval}{}{% + start={ + \pgfmathparse{\pgfkeysvalueof{/pgf/bar interval width}}% + \xdef\pgfplotbarintervalwidth@{\pgfmathresult}% + \pgfmathparse{\pgfkeysvalueof{/pgf/bar interval shift}}% + \xdef\pgfplotbarintervalshift@{\pgfmathresult}% + \pgfplotyzerolevelstreamstart + }, + end macro=\pgfplotyzerolevelstreamend, + point macro=\pgf@plot@ybarinterval@handler@first +}% + +\def\pgf@plot@ybarinterval@handler@first#1{% + \pgf@process{#1}% + \xdef\pgf@plot@barinterval@intervalstart{\the\pgf@x}% + \xdef\pgf@plot@barinterval@bar{\the\pgf@y}% + \global\let\pgf@plotstreampoint=\pgf@plot@ybarinterval@handler% +}% +\def\pgf@plot@ybarinterval@handler#1{% + \pgfkeysvalueof{/pgf/at begin bar}% + \pgf@process{#1}% + \pgf@ya=\pgf@plot@barinterval@bar + \xdef\pgf@plot@barinterval@bar{\the\pgf@y}% + \pgf@xc=\pgf@plot@barinterval@intervalstart\relax + \xdef\pgf@plot@barinterval@intervalstart{\the\pgf@x}% + \pgf@xb=\pgf@x + \advance\pgf@xb by-\pgf@xc + \advance\pgf@xc by\pgfplotbarintervalshift@\pgf@xb + \pgf@xb=\pgfplotbarintervalwidth@\pgf@xb + \advance\pgf@xc by-.5\pgf@xb% + \begingroup + \pgfplotyzerolevelstreamnext + \endgroup + \pgf@yb=\pgf@x + \advance\pgf@ya by-\pgf@yb + \pgfpathrectangle + {\pgfqpoint{\pgf@xc}{\pgf@yb}}% + {\pgfqpoint{\pgf@xb}{\pgf@ya}}% + \pgfkeysvalueof{/pgf/at end bar}% +}% + +\pgfdeclareplothandler{\pgfplothandlerxbarinterval}{}{% + start={ + \pgfmathparse{\pgfkeysvalueof{/pgf/bar interval width}}% + \xdef\pgfplotbarintervalwidth@{\pgfmathresult}% + \pgfmathparse{\pgfkeysvalueof{/pgf/bar interval shift}}% + \xdef\pgfplotbarintervalshift@{\pgfmathresult}% + \pgfplotxzerolevelstreamstart + }, + end macro=\pgfplotxzerolevelstreamend, + point macro=\pgf@plot@xbarinterval@handler@first +}% + +\def\pgf@plot@xbarinterval@handler@first#1{% + \pgf@process{#1}% + \xdef\pgf@plot@barinterval@intervalstart{\the\pgf@y}% + \xdef\pgf@plot@barinterval@bar{\the\pgf@x}% + \global\let\pgf@plotstreampoint=\pgf@plot@xbarinterval@handler% +}% +\def\pgf@plot@xbarinterval@handler#1{% + \pgfkeysvalueof{/pgf/at begin bar}% + \pgf@process{#1}% + \pgf@ya=\pgf@plot@barinterval@bar + \xdef\pgf@plot@barinterval@bar{\the\pgf@x}% + \pgf@xc=\pgf@plot@barinterval@intervalstart\relax + \xdef\pgf@plot@barinterval@intervalstart{\the\pgf@y}% + \pgf@xb=\pgf@y + \advance\pgf@xb by-\pgf@xc + \advance\pgf@xc by\pgfplotbarintervalshift@\pgf@xb + \pgf@xb=\pgfplotbarintervalwidth@\pgf@xb + \advance\pgf@xc by-.5\pgf@xb% + \begingroup + \pgfplotxzerolevelstreamnext + \endgroup + \pgf@yb=\pgf@x + \advance\pgf@ya by-\pgf@yb + \pgfpathrectangle + {\pgfqpoint{\pgf@yb}{\pgf@xc}}% + {\pgfqpoint{\pgf@ya}{\pgf@xb}}% + \pgfkeysvalueof{/pgf/at end bar}% +}% + + + +\pgfdeclareplothandler{\pgfplothandlerconstantlineto}{}{% + point macro=\pgf@plot@const@line@handler +}% + +\def\pgf@plot@const@line@handler#1{% + \pgf@process{#1}% + \xdef\pgf@plot@const@line@handler@last{\the\pgf@y}% + \pgf@plot@first@action{}% + \global\let\pgf@plotstreampoint=\pgf@plot@const@line@handler@@% +}% +\def\pgf@plot@const@line@handler@@#1{% + \pgf@process{#1}% + \pgf@xa=\pgf@x + \pgf@ya=\pgf@y + \pgf@yb\pgf@plot@const@line@handler@last\relax + \pgfpathlineto{\pgfqpoint{\pgf@xa}{\pgf@yb}}% + \pgfpathlineto{\pgfqpoint{\pgf@xa}{\pgf@ya}}% + \xdef\pgf@plot@const@line@handler@last{\the\pgf@ya}% +}% + + +\pgfdeclareplothandler{\pgfplothandlerconstantlinetomarkright}{}{% + point macro=\pgf@plot@const@line@mark@right@handler +}% + +\def\pgf@plot@const@line@mark@right@handler#1{% + \pgf@process{#1}% + \xdef\pgf@plot@const@line@handler@last{\the\pgf@x}% + \pgf@plot@first@action{}% + \global\let\pgf@plotstreampoint=\pgf@plot@const@line@mark@right@handler@@% +}% +\def\pgf@plot@const@line@mark@right@handler@@#1{% + \pgf@process{#1}% + \pgf@xa=\pgf@x + \pgf@ya=\pgf@y + \pgf@yb\pgf@plot@const@line@handler@last\relax + \pgfpathlineto{\pgfqpoint{\pgf@yb}{\pgf@ya}}% + \pgfpathlineto{\pgfqpoint{\pgf@xa}{\pgf@ya}}% + \xdef\pgf@plot@const@line@handler@last{\the\pgf@xa}% +}% + + +\pgfdeclareplothandler{\pgfplothandlerconstantlinetomarkmid}{}{% + point macro=\pgf@plot@const@line@mark@mid@handler +}% + +\def\pgf@plot@const@line@mark@mid@handler#1{% + \pgf@process{#1}% + \xdef\pgf@plot@const@line@handler@last{\global\pgf@x=\the\pgf@x\space\global\pgf@y=\the\pgf@y\space}% + \pgf@plot@first@action{}% + \global\let\pgf@plotstreampoint=\pgf@plot@const@line@mark@mid@handler@@% +}% +\def\pgf@plot@const@line@mark@mid@handler@@#1{% + \pgf@process{#1}% + \pgf@xa=\pgf@x + \pgf@ya=\pgf@y + \pgf@plot@const@line@handler@last + \pgf@xc=.5\pgf@xa + \advance\pgf@xc by.5\pgf@x + \pgfpathlineto{\pgfqpoint{\pgf@xc}{\pgf@y}}% + \pgfpathlineto{\pgfqpoint{\pgf@xc}{\pgf@ya}}% + \pgfpathlineto{\pgfqpoint{\pgf@xa}{\pgf@ya}}% + \xdef\pgf@plot@const@line@handler@last{\global\pgf@x=\the\pgf@x\space\global\pgf@y=\the\pgf@y\space}% +}% + + +\pgfdeclareplothandler{\pgfplothandlerjumpmarkright}{}{% + point macro=\pgf@plot@jumpmarkright@handler +}% + +\def\pgf@plot@jumpmarkright@handler#1{% + \pgf@process{#1}% + \xdef\pgf@plot@const@line@handler@last{\the\pgf@x}% + \pgf@plot@first@action{}% + \global\let\pgf@plotstreampoint=\pgf@plot@jumpmarkright@handler@@% +}% +\def\pgf@plot@jumpmarkright@handler@@#1{% + \pgf@process{#1}% + \pgf@xa=\pgf@x + \pgf@ya=\pgf@y + \pgf@yb\pgf@plot@const@line@handler@last\relax + \pgfpathmoveto{\pgfqpoint{\pgf@yb}{\pgf@ya}}% + \pgfpathlineto{\pgfqpoint{\pgf@xa}{\pgf@ya}}% + \xdef\pgf@plot@const@line@handler@last{\the\pgf@xa}% +}% + + +\pgfdeclareplothandler{\pgfplothandlerjumpmarkleft}{}{% + point macro=\pgf@plot@jumpmarkleft@handler +}% +\def\pgf@plot@jumpmarkleft@handler#1{% + \pgf@process{#1}% + \xdef\pgf@plot@const@line@handler@last{\the\pgf@y}% + \pgf@plot@first@action{}% + \global\let\pgf@plotstreampoint=\pgf@plot@jumpmarkleft@handler@@% +}% +\def\pgf@plot@jumpmarkleft@handler@@#1{% + \pgf@process{#1}% + \pgf@xa=\pgf@x + \pgf@ya=\pgf@y + \pgf@yb\pgf@plot@const@line@handler@last\relax + \pgfpathlineto{\pgfqpoint{\pgf@xa}{\pgf@yb}}% + \pgfpathmoveto{\pgfqpoint{\pgf@xa}{\pgf@ya}}% + \xdef\pgf@plot@const@line@handler@last{\the\pgf@ya}% +}% + + +\pgfdeclareplothandler{\pgfplothandlerjumpmarkmid}{}{% + point macro=\pgf@plot@jump@mark@mid@handler +}% + +\def\pgf@plot@jump@mark@mid@handler#1{% + \pgf@process{#1}% + \xdef\pgf@plot@const@line@handler@last{\global\pgf@x=\the\pgf@x\space\global\pgf@y=\the\pgf@y\space}% + \pgf@plot@first@action{}% + \global\let\pgf@plotstreampoint=\pgf@plot@jump@mark@mid@handler@@% +}% +\def\pgf@plot@jump@mark@mid@handler@@#1{% + \pgf@process{#1}% + \pgf@xa=\pgf@x + \pgf@ya=\pgf@y + \pgf@plot@const@line@handler@last + \pgf@xc=.5\pgf@xa + \advance\pgf@xc by.5\pgf@x + \pgfpathlineto{\pgfqpoint{\pgf@xc}{\pgf@y}}% + \pgfpathmoveto{\pgfqpoint{\pgf@xc}{\pgf@ya}}% + \pgfpathlineto{\pgfqpoint{\pgf@xa}{\pgf@ya}}% + \xdef\pgf@plot@const@line@handler@last{\global\pgf@x=\the\pgf@x\space\global\pgf@y=\the\pgf@y\space}% +}% + + + +\pgfdeclareplothandler{\pgfplothandlerpolarcomb}{}{% + point macro=\pgf@plot@polarcomb@handler +}% + +\def\pgf@plot@polarcomb@handler#1{% + \pgf@process{#1}% + \pgf@xa=\pgf@x% + \pgf@ya=\pgf@y% + \pgfpathmoveto{\pgfpointorigin}% + \pgfpathlineto{\pgfqpoint{\pgf@xa}{\pgf@ya}}% +}% + + + + + +\pgfdeclareplothandler{\pgfplothandlermark}{#1}{% + start={ + \def\pgf@plot@mark{% + \ifnum\pgf@plot@mark@count<\pgf@plot@mark@repeat\relax% + \else% + \global\pgf@plot@mark@count=0\relax% + #1% + \fi% + }% + \global\pgf@plot@mark@count=\pgf@plot@mark@repeat\relax% + \global\advance\pgf@plot@mark@count by-\pgf@plot@mark@phase\relax% + }, + point macro=\pgf@plot@mark@handler +}% + +\newcount\pgf@plot@mark@count +\def\pgf@plot@mark@phase{0}% + +\def\pgf@plot@mark@handler#1{% + \global\advance\pgf@plot@mark@count by1\relax% + {\pgftransformshift{#1}\pgf@plot@mark}% +}% + + + +\def\pgfsetplotmarkrepeat#1{\def\pgf@plot@mark@repeat{#1}}% +\pgfsetplotmarkrepeat{1}% + + + +\def\pgfsetplotmarkphase#1{\def\pgf@plot@mark@phase{#1}}% +\pgfsetplotmarkphase{1}% + + + + +\pgfdeclareplothandler{\pgfplothandlermarklisted}{#1#2}{% + start={ + \let\pgf@plot@mark@list=\pgfutil@empty% + \edef\pgf@marshal{\noexpand\foreach\noexpand\pgf@temp in{#2}} + \pgf@marshal{\xdef\pgf@plot@mark@list{\pgf@plot@mark@list(\pgf@temp)}}% + \def\pgf@plot@mark{% + \edef\pgf@marshal{\noexpand\pgfutil@in@{(\the\pgf@plot@mark@count)}{\pgf@plot@mark@list}}% + \pgf@marshal% + \ifpgfutil@in@#1\fi% + }% + \global\pgf@plot@mark@count=\pgf@plot@mark@repeat\relax% + \global\advance\pgf@plot@mark@count by-\pgf@plot@mark@phase\relax% + }, + point macro=\pgf@plot@mark@handler +}% + + + + +\def\pgfdeclareplotmark#1#2{\expandafter\def\csname pgf@plot@mark@#1\endcsname{#2}}% + + + +\def\pgfsetplotmarksize#1{\pgfmathsetlength\pgfplotmarksize{#1}}% + +\newdimen\pgfplotmarksize +\pgfplotmarksize=2pt + + + +\def\pgfuseplotmark#1{\csname pgf@plot@mark@#1\endcsname}% + + + +\pgfdeclareplotmark{*} +{% + \pgfpathellipse{\pgfpointorigin}{\pgfqpoint{\pgfplotmarksize}{0sp}}{\pgfqpoint{0sp}{\pgfplotmarksize}}% + \pgfusepathqfillstroke +}% + + + +\pgfdeclareplotmark{+} +{% + \pgfpathmoveto{\pgfqpoint{-\pgfplotmarksize}{0pt}}% + \pgfpathlineto{\pgfqpoint{\pgfplotmarksize}{0pt}}% + \pgfpathmoveto{\pgfqpoint{0pt}{\pgfplotmarksize}}% + \pgfpathlineto{\pgfqpoint{0pt}{-\pgfplotmarksize}}% + \pgfusepathqstroke +}% + + + +\pgfdeclareplotmark{x} +{% + \pgfpathmoveto{\pgfqpoint{-.70710678\pgfplotmarksize}{-.70710678\pgfplotmarksize}}% + \pgfpathlineto{\pgfqpoint{.70710678\pgfplotmarksize}{.70710678\pgfplotmarksize}}% + \pgfpathmoveto{\pgfqpoint{-.70710678\pgfplotmarksize}{.70710678\pgfplotmarksize}}% + \pgfpathlineto{\pgfqpoint{.70710678\pgfplotmarksize}{-.70710678\pgfplotmarksize}}% + \pgfusepathqstroke +}% + + + + + + +\pgfkeys{/pgf/gap around stream point/.initial=1.5pt}% + +\pgfdeclareplothandler{\pgfplothandlergaplineto}{}{% + point macro=\pgf@plot@gap@lineto@handler@initial, + jump macro=\pgf@plot@gap@next@moveto +}% + +\def\pgf@plot@gap@next@moveto{% + \global\let\pgf@plotstreampoint=\pgf@plot@gap@lineto@handler@initial% +}% + +\def\pgf@plot@gap@lineto@handler@initial#1{% + \pgf@process{#1}% + \pgf@xa=\pgf@x% + \pgf@ya=\pgf@y% + \xdef\pgf@plot@gap@lineto@last{\noexpand\pgfqpoint{\the\pgf@xa}{\the\pgf@ya}}% + \global\let\pgf@plotstreampoint=\pgf@plot@gap@lineto@handler% +}% + +\def\pgf@plot@gap@lineto@handler#1{% + \pgf@process{#1}% + \pgf@xa=\pgf@x% + \pgf@ya=\pgf@y% + \xdef\pgf@plot@gap@lineto@next{\noexpand\pgfqpoint{\the\pgf@xa}{\the\pgf@ya}}% + \pgf@process{\pgfpointnormalised{\pgfpointdiff{\pgf@plot@gap@lineto@last}{\pgf@plot@gap@lineto@next}}}% + \pgf@xc=\pgf@x% + \pgf@yc=\pgf@y% + \pgfpathmoveto{\pgfpointadd{\pgfpointscale{\pgfkeysvalueof{/pgf/gap + around stream point}}{\pgfqpoint{\pgf@xc}{\pgf@yc}}}{\pgf@plot@gap@lineto@last}}% + \pgfpathlineto{\pgfpointadd{\pgfpointscale{\pgfkeysvalueof{/pgf/gap + around stream point}}{\pgfqpoint{-\pgf@xc}{-\pgf@yc}}}{\pgf@plot@gap@lineto@next}}% + \global\let\pgf@plot@gap@lineto@last=\pgf@plot@gap@lineto@next% +}% + + + + + +\pgfdeclareplothandler{\pgfplothandlergapcycle}{}{% + point macro=\pgf@plot@gap@cycle@handler@initial, + jump macro=\pgf@plot@gapcycle@next@moveto, + end macro=\pgf@plot@gap@cycle@handler@finish, + start={\global\let\pgf@plot@gap@cycle@first\relax} +}% + +\def\pgf@plot@gapcycle@next@moveto{% + \pgf@plot@gap@cycle@handler@finish% + \global\let\pgf@plot@gap@cycle@first=\relax% + \global\let\pgf@plotstreampoint=\pgf@plot@gap@cycle@handler@initial% +}% + +\def\pgf@plot@gap@cycle@handler@initial#1{% + \pgf@process{#1}% + \pgf@xa=\pgf@x% + \pgf@ya=\pgf@y% + \xdef\pgf@plot@gap@cycle@last{\noexpand\pgfqpoint{\the\pgf@xa}{\the\pgf@ya}}% + \global\let\pgf@plot@gap@cycle@first=\pgf@plot@gap@cycle@last% + \global\let\pgf@plotstreampoint=\pgf@plot@gap@cycle@handler% +}% + +\def\pgf@plot@gap@cycle@handler#1{% + \pgf@process{#1}% + \pgf@xa=\pgf@x% + \pgf@ya=\pgf@y% + \xdef\pgf@plot@gap@cycle@next{\noexpand\pgfqpoint{\the\pgf@xa}{\the\pgf@ya}}% + \pgf@process{\pgfpointnormalised{\pgfpointdiff{\pgf@plot@gap@cycle@last}{\pgf@plot@gap@cycle@next}}}% + \pgf@xc=\pgf@x% + \pgf@yc=\pgf@y% + \pgfpathmoveto{\pgfpointadd{\pgfpointscale{\pgfkeysvalueof{/pgf/gap + around stream point}}{\pgfqpoint{\pgf@xc}{\pgf@yc}}}{\pgf@plot@gap@cycle@last}}% + \pgfpathlineto{\pgfpointadd{\pgfpointscale{\pgfkeysvalueof{/pgf/gap + around stream point}}{\pgfqpoint{-\pgf@xc}{-\pgf@yc}}}{\pgf@plot@gap@cycle@next}}% + \global\let\pgf@plot@gap@cycle@last=\pgf@plot@gap@cycle@next% +}% + +\def\pgf@plot@gap@cycle@handler@finish{% + \ifx\pgf@plot@gap@cycle@first\relax% + \else + \pgf@plot@gap@cycle@handler{\pgf@plot@gap@cycle@first}% + \fi +}% + + + +\endinput + diff --git a/preprints/A-polynomial-time-algorithm-for-three-machine-unit-job-scheduling-September-24-2026/build/tex/runtime/generic/pgf/math/pgfint.code.tex b/preprints/A-polynomial-time-algorithm-for-three-machine-unit-job-scheduling-September-24-2026/build/tex/runtime/generic/pgf/math/pgfint.code.tex new file mode 100644 index 0000000000000000000000000000000000000000..0562146de73644d7a6765b5014ccb7bad4a8d0c4 --- /dev/null +++ b/preprints/A-polynomial-time-algorithm-for-three-machine-unit-job-scheduling-September-24-2026/build/tex/runtime/generic/pgf/math/pgfint.code.tex @@ -0,0 +1,94 @@ + + +\ifx\pgfintloaded\pgfutil@undefined + \let\pgfintloaded=\relax +\else + \expandafter\endinput +\fi + + + +\def\pgfinteval#1{\number\numexpr#1\relax} + +\def\pgfintabs#1{\number\expandafter\pgfint@abs\number\numexpr#1\relax} +\def\pgfint@abs#1{\ifx-#1\else\expandafter#1\fi} + +\def\pgfintmax#1#2{% + \number\expandafter\pgfint@minmax + \number\numexpr#1\expandafter;% + \number\numexpr#2;% + > +} +\def\pgfintmin#1#2{% + \number\expandafter\pgfint@minmax + \number\numexpr#1\expandafter;% + \number\numexpr#2;% + < +} +\def\pgfint@minmax#1;#2;#3{% + \ifnum#1#3#2 + #1% + \else + #2% + \fi +} + +\def\pgfintdivtruncate#1#2{% + \number\numexpr\expandafter\pgfint@divtruncate + \number\numexpr#1\expandafter;% + \number\numexpr#2;% + \relax +} +\def\pgfint@divtruncate#1#2;#3#4;{% + \ifx0#1 + 0% + \else + (#1#2% + \ifx-#1 +\else-\fi + (\ifx-#3 -\fi#3#4-1)/2% + )% + \fi + /#3#4% +} + +\def\pgfintdivfloor#1#2{% + \number\numexpr\expandafter\pgfint@divfloor + \number\numexpr#1\expandafter;% + \number\numexpr#2;% + \relax +} +\def\pgfint@divfloor#1#2;#3#4;{% + \ifx0#1 + 0% + \else + \ifnum + 0\ifx-#1\ifx#3-\else1\fi\fi + 0\ifx-#3\ifx#1-\else1\fi\fi + >0 + \pgfint@@divfloor#1#2;#3#4;-+% + \else + \pgfint@@divfloor#1#2;#3#4;+-% + \fi + \fi + /#3#4% +} +\def\pgfint@@divfloor#1#2;#3#4;#5#6{% + (#1#2% + \ifx-#1 #5\else#6\fi + (\ifx-#3 -\fi#3#4#61)/2% + )% +} + +\def\pgfintdivround#1#2{\number\numexpr(#1)/(#2)\relax} + +\def\pgfintmod#1#2{% + \number\numexpr\expandafter\pgfint@mod% + \number\numexpr#1\expandafter;% + \number\numexpr#2;% + \relax +} +\def\pgfint@mod#1;#2;{#1-(\pgfint@divfloor#1;#2;)*#2} + +\def\pgfintset#1#2{#1 \numexpr#2\relax} + +\endinput diff --git a/preprints/A-polynomial-time-algorithm-for-three-machine-unit-job-scheduling-September-24-2026/build/tex/runtime/generic/pgf/math/pgfmath.code.tex b/preprints/A-polynomial-time-algorithm-for-three-machine-unit-job-scheduling-September-24-2026/build/tex/runtime/generic/pgf/math/pgfmath.code.tex new file mode 100644 index 0000000000000000000000000000000000000000..ca11896e2fd1e67d11473210615e17189afdea12 --- /dev/null +++ b/preprints/A-polynomial-time-algorithm-for-three-machine-unit-job-scheduling-September-24-2026/build/tex/runtime/generic/pgf/math/pgfmath.code.tex @@ -0,0 +1,27 @@ + + +\ifx\pgfmathloaded\pgfutil@undefined + \let\pgfmathloaded=\relax +\else + \expandafter\endinput +\fi + +\ifx\pgfkeysloaded\pgfutil@undefined + \input pgfkeys.code.tex +\fi + +\input pgfmathutil.code.tex +\input pgfmathparser.code.tex +\input pgfmathfunctions.code.tex +\input pgfmathfunctions.basic.code.tex +\input pgfmathfunctions.trigonometric.code.tex +\input pgfmathfunctions.random.code.tex +\input pgfmathfunctions.comparison.code.tex +\input pgfmathfunctions.base.code.tex +\input pgfmathfunctions.round.code.tex +\input pgfmathfunctions.misc.code.tex +\input pgfmathfunctions.integerarithmetics.code.tex +\input pgfmathcalc.code.tex +\input pgfmathfloat.code.tex + +\endinput diff --git a/preprints/A-polynomial-time-algorithm-for-three-machine-unit-job-scheduling-September-24-2026/build/tex/runtime/generic/pgf/math/pgfmathcalc.code.tex b/preprints/A-polynomial-time-algorithm-for-three-machine-unit-job-scheduling-September-24-2026/build/tex/runtime/generic/pgf/math/pgfmathcalc.code.tex new file mode 100644 index 0000000000000000000000000000000000000000..dbbf3bae40e35b77649468126110e6b1013f4ba9 --- /dev/null +++ b/preprints/A-polynomial-time-algorithm-for-three-machine-unit-job-scheduling-September-24-2026/build/tex/runtime/generic/pgf/math/pgfmathcalc.code.tex @@ -0,0 +1,353 @@ + + + +\def\pgfmathsetlength#1#2{% + \expandafter\pgfmath@onquick#2\pgfmath@% + {% + \begingroup% + \pgfmath@selectfont% + #1#2\unskip% + \expandafter\endgroup% + \expandafter#1\the#1\relax% + }% + {% + \pgfmathparse{#2}% + \ifpgfmathmathunitsdeclared% + #1\pgfmathresult mu\relax% + \else% + #1\pgfmathresult pt\relax% + \fi% + }% + \ignorespaces% +} + +\def\pgfmathaddtolength#1#2{% + \expandafter\pgfmath@onquick#2\pgfmath@% + {% + \begingroup% + \pgfmath@selectfont% + \advance#1#2\unskip% + \expandafter% + \endgroup% + \expandafter#1\the#1\relax% + }% + {% + \pgfmathparse{#2}% + \ifpgfmathmathunitsdeclared% + \advance#1\pgfmathresult mu\relax% + \else% + \advance#1\pgfmathresult pt\relax% + \fi% + }% + \ignorespaces% +} + +\def\pgfmathsetcount#1#2{% + \expandafter\pgfmath@onquick#2\pgfmath@% + {% + \afterassignment\pgfmath@gobbletilpgfmath@% + #1#2\relax\pgfmath@% + }% + {% + \pgfmathparse{#2}% + \afterassignment\pgfmath@gobbletilpgfmath@% + #1\pgfmathresult\relax\pgfmath@% + }% + \ignorespaces% +} + +\def\pgfmathaddtocount#1#2{% + \expandafter\pgfmath@onquick#2\pgfmath@% + {% + \edef\pgfmath@addtocounttemp{\the#1}% + \afterassignment\pgfmath@gobbletilpgfmath@% + #1#2\relax\pgfmath@% + \advance#1\pgfmath@addtocounttemp\relax% + }% + {% + \edef\pgfmath@addtocounttemp{\the#1}% + \pgfmathparse{#2}% + \afterassignment\pgfmath@gobbletilpgfmath@% + #1\pgfmathresult\relax\pgfmath@% + \advance#1\pgfmath@addtocounttemp\relax% + }% + \ignorespaces% +} + +\def\pgfmathnewcounter#1{% + \expandafter\ifx\csname c@#1\endcsname\relax% + \def\pgfmath@marshal{\csname newcount\endcsname}% + \expandafter\pgfmath@marshal\csname c@#1\endcsname% + \expandafter\def\csname the#1\endcsname{\expandafter\the\csname c@#1\endcsname}% + \fi% +} + +\def\pgfmathsetcounter#1#2{% + \expandafter\pgfmathsetcount\csname c@#1\endcsname{#2}% +} + +\def\pgfmathaddtocounter#1#2{% + \expandafter\pgfmathaddtocount\csname c@#1\endcsname{#2}% +} + +\def\pgfmath@pgftext{pgf} +\def\pgfmath@tikztext{tikz} +\def\pgfmathmakecounterglobal#1{% + \pgfmath@ifundefined{c@#1}{}{% + \expandafter\pgfmath@in@\expandafter{\pgfmath@pgftext}{#1}% + \ifpgfmath@in@% + \else% + \expandafter\pgfmath@in@\expandafter{\pgfmath@tikztext}{#1}% + \ifpgfmath@in@% + \else% + \expandafter\global\csname c@#1\endcsname\csname c@#1\endcsname\relax% + \fi% + \fi% + }% +} + +\def\pgfmathsetmacro#1#2{% + \begingroup% + \pgfmathparse{#2}% + \let#1=\pgfmathresult + \pgfmath@smuggleone{#1}% + \endgroup% +} + +\def\pgfmathsetlengthmacro#1#2{% + \begingroup% + \pgfmathsetlength\pgfmath@x{#2}% + \edef#1{\the\pgfmath@x}% + \pgfmath@smuggleone{#1}% + \endgroup% +} + +\def\pgfmathtruncatemacro#1#2{% + \begingroup% + \pgfmathsetcount\c@pgfmath@counta{#2}% + \edef#1{\the\c@pgfmath@counta}% + \pgfmath@smuggleone{#1}% + \endgroup% +} + + +\def\pgfmath@onquick{% + \futurelet\pgfmath@next% + \pgfmath@@onquick} +\def\pgfmath@@onquick#1\pgfmath@{% + \ifx\pgfmath@next+% + \let\pgfmath@next=\pgfmath@firstoftwo% + \else% + \let\pgfmath@next=\pgfmath@secondoftwo% + \fi% + \pgfmath@next% +} + + +\def\pgfmathanglebetweenpoints#1#2{% + \begingroup% + \pgf@process{\pgfpointdiff{#1}{#2}}% + \edef\pgf@marshall{\expandafter\noexpand\csname pgfmathatan2@\endcsname{\expandafter\Pgf@geT\the\pgf@y}{\expandafter\Pgf@geT\the\pgf@x}}% + \pgf@marshall% + \ifdim\pgfmathresult pt<0pt% + \pgfmathparse{\pgfmathresult+360}% + \fi% + \expandafter% + \endgroup\expandafter\def\expandafter\pgfmathresult\expandafter{\pgfmathresult}% +} + +\def\pgfmathrotatepointaround#1#2#3{% + \pgf@process{% + \pgf@process{#1}% + \pgf@xc=\pgf@x% + \pgf@yc=\pgf@y% + \pgf@process{#2}% + \pgf@xa\pgf@x% + \pgf@ya\pgf@y% + \pgf@xb\pgf@x% + \pgf@yb\pgf@y% + \pgf@x=\pgf@xc% + \pgf@y=\pgf@yc% + \advance\pgf@x-\pgf@xa% + \advance\pgf@y-\pgf@ya% + \pgfmathsetmacro\angle{#3}% + \pgfmathsin@{\angle}% + \let\sineangle\pgfmathresult% + \pgfmathcos@{\angle}% + \let\cosineangle\pgfmathresult% + \pgf@xa\cosineangle\pgf@x% + \advance\pgf@xa-\sineangle\pgf@y% + \pgf@ya\sineangle\pgf@x% + \advance\pgf@ya\cosineangle\pgf@y% + \pgf@x\pgf@xb% + \pgf@y\pgf@yb% + \advance\pgf@x\pgf@xa% + \advance\pgf@y\pgf@ya% + }% +} + +\def\pgfmathanglebetweenlines#1#2#3#4{% + \begingroup% + \pgfmathanglebetweenpoints{#1}{#2}% + \let\firstangle\pgfmathresult% + \pgfmathanglebetweenpoints{#3}{#4}% + \let\secondangle\pgfmathresult% + \ifdim\firstangle pt>\secondangle pt\relax% + \pgfmathadd@{\secondangle}{360}% + \let\secondangle\pgfmathresult% + \fi% + \pgfmathsubtract@{\secondangle}{\firstangle}% + \pgfmath@smuggleone\pgfmathresult% + \endgroup% +} + +\def\pgfmathreflectpointalongaxis#1#2#3{% + \pgf@process{% + \pgfmathanglebetweenpoints{#2}{#1}% + \pgfmath@tempdima\pgfmathresult pt\relax% + \pgfmathparse{#3}% + \advance\pgfmath@tempdima-\pgfmathresult pt\relax% + \pgfmath@tempdima-2.0\pgfmath@tempdima% + \pgfmathrotatepointaround{#1}{#2}{\pgfmath@tonumber{\pgfmath@tempdima}}% + }% +} + + + +\def\pgfmathpointintersectionoflineandarc#1#2#3#4#5#6{% + \pgf@process{% + \pgfmathanglebetweenpoints{#2}{#1}% + \let\x\pgfmathresult% + \pgfmath@in@{and }{#6}% + \ifpgfmath@in@% + \pgf@polar@#6\@@% + \else% + \pgf@polar@#6 and #6\@@% + \fi% + \edef\xarcradius{\the\pgf@x}% + \edef\yarcradius{\the\pgf@y}% + \pgfmathsetmacro\s{#4}% + \pgfmathsetmacro\e{#5}% + \pgfmathmod@{\s}{360}% + \ifdim\pgfmathresult pt<0pt\relax% + \pgfmathadd@{\pgfmathresult}{360}% + \fi% + \let\ss\pgfmathresult% + \pgfmathmod@{\e}{360}% + \ifdim\pgfmathresult pt<0pt\relax% + \pgfmathadd@{\pgfmathresult}{360}% + \fi% + \let\ee\pgfmathresult% + \ifdim\ee pt<\ss pt\relax% + \pgfmathadd@{\x}{180}% + \pgfmathmod@{\pgfmathresult}{360}% + \let\x\pgfmathresult% + \fi% + \def\m{360}% + \pgfmathadd@{\s}{\e}% + \pgfmathdivide@{\pgfmathresult}{2}% + \let\n\pgfmathresult% + \pgfmathloop% + \pgfmathadd@{\s}{\e}% + \pgfmathdivide@{\pgfmathresult}{2}% + \let\p\pgfmathresult% + \ifdim\p pt=\s pt\relax% + \else% + \pgfmathanglebetweenpoints{#2}{% + \pgfpointadd{#3}{% + \pgf@x\xarcradius\relax% + \pgfmathcos@{\p}% + \pgf@x\pgfmathresult\pgf@x% + \pgf@y\yarcradius\relax% + \pgfmathsin@{\p}% + \pgf@y\pgfmathresult\pgf@y% + }% + }% + \ifdim\ee pt<\ss pt\relax% + \pgfmathadd@{\pgfmathresult}{180}% + \pgfmathmod@{\pgfmathresult}{360}% + \fi% + \let\q\pgfmathresult% + \ifdim\x pt>335pt\relax% + \ifdim\q pt<45pt\relax% + \pgfmathadd@{\q}{360}% + \let\q\pgfmathresult% + \fi% + \fi% + \ifdim\x pt=\q pt% + \pgfmathbreakloop% + \else + \ifdim\x pt<\q pt\relax% + \let\e\p% + \else% + \let\s\p% + \fi% + \fi% + \pgfmathsubtract@{\x}{\q}% + \pgfmathabs@{\pgfmathresult}% + \ifdim\pgfmathresult pt<\m pt\relax% + \let\m\pgfmathresult% + \let\n\p% + \fi% + \repeatpgfmathloop% + \pgfpointadd{#3}{\pgfpointpolar{\n}{\xarcradius and \yarcradius}}% + }% +} + +\def\pgfmathangleonellipse#1#2{% + \begingroup% + \pgfmath@in@{and }{#2}% + \ifpgfmath@in@% + \pgf@polar@#2\@@% + \else% + \pgf@polar@#2 and #2\@@% + \fi% + \pgf@xa\pgf@x% + \pgf@ya\pgf@y% + \pgf@process{#1}% + \ifdim\pgf@x=0pt\relax% + \pgfutil@tempdima1pt\relax% + \else% + \pgfutil@tempdima\pgf@x% + \pgfmathdivide@{\pgfmath@tonumber{\pgf@xa}}{\pgfmath@tonumber{\pgfutil@tempdima}}% + \pgfutil@tempdima\pgfmathresult pt\relax% + \fi% + \ifdim\pgf@y=0pt\relax% + \pgfutil@tempdima1pt\relax% + \else% + \pgfmathdivide@{\pgfmath@tonumber{\pgf@y}}{\pgfmath@tonumber{\pgf@ya}}% + \pgfutil@tempdima\pgfmathresult\pgfutil@tempdima% + \pgfmathatan@{\pgfmath@tonumber{\pgfutil@tempdima}}% + \fi% + \pgfutil@tempdima\pgfmathresult pt\relax% + \ifdim\pgfutil@tempdima<0pt\relax% + \advance\pgfutil@tempdima360pt\relax% + \fi% + \ifdim\pgf@x<0pt\relax% + \ifdim\pgf@y=0pt\relax% + \pgfutil@tempdima180pt\relax% + \else% + \ifdim\pgf@y<0pt\relax% + \advance\pgfutil@tempdima180pt\relax% + \else% + \advance\pgfutil@tempdima-180pt\relax% + \fi% + \fi% + \else% + \ifdim\pgf@x=0pt\relax% + \ifdim\pgf@y<0pt\relax% + \pgfutil@tempdima270pt\relax% + \else% + \pgfutil@tempdima90pt\relax% + \fi% + \else% + \ifdim\pgf@y=0pt\relax% + \pgfutil@tempdima0pt\relax% + \fi% + \fi% + \fi% + \pgfmath@returnone\pgfutil@tempdima% + \endgroup% +} + + diff --git a/preprints/A-polynomial-time-algorithm-for-three-machine-unit-job-scheduling-September-24-2026/build/tex/runtime/generic/pgf/math/pgfmathfloat.code.tex b/preprints/A-polynomial-time-algorithm-for-three-machine-unit-job-scheduling-September-24-2026/build/tex/runtime/generic/pgf/math/pgfmathfloat.code.tex new file mode 100644 index 0000000000000000000000000000000000000000..abec5f7cad737d7a495a8b951d0e3f730682e879 --- /dev/null +++ b/preprints/A-polynomial-time-algorithm-for-three-machine-unit-job-scheduling-September-24-2026/build/tex/runtime/generic/pgf/math/pgfmathfloat.code.tex @@ -0,0 +1,2156 @@ + + +\pgfkeys{ + /pgf/compat/number printer/.is choice, + /pgf/compat/number printer/2.10/.code={% + \def\pgfmathprintnumber@begingroup{\bgroup}% + \def\pgfmathprintnumber@endgroup{\egroup}% + }, + /pgf/compat/number printer/next after 2.10/.code={% + \def\pgfmathprintnumber@begingroup{\begingroup}% + \def\pgfmathprintnumber@endgroup{\endgroup}% + }, + /pgf/compat/number printer/next after 2.10,% +} + + +\newif\ifpgfmathfloatcomparison +\newif\ifpgfmathfloatroundhasperiod +\newif\ifpgfmathprintnumberskipzeroperiod + + +\toksdef\pgfmathfloat@tmptoks=1 +\newif\ifpgfmathfloat@fixed@digits@after@period +\newif\ifpgfmathfloatroundmayneedrenormalize +\toksdef\pgfmathfloat@a@Mtok=0 + +\let\pgfmathfloat@a@S=\c@pgf@counta +\let\pgfmathfloat@a@M=\pgf@xa +\let\pgfmathfloat@a@E=\c@pgf@countb +\let\pgfmathfloat@b@S=\c@pgf@countc +\let\pgfmathfloat@b@M=\pgf@xb +\let\pgfmathfloat@b@E=\c@pgf@countd + +\def\pgfmathfloat@round@precision{2} + +\def\pgfmathfloat@round@precision@orig{\pgfmathfloat@round@precision}% + +\gdef\pgfmathfloat@glob@TMP{} + + +\def\pgfmathfloatparsenumber#1{% + \begingroup + \edef\pgfmathresult{#1}% + \expandafter\pgfflt@impl\pgfmathresult\pgfflt@EOI + \ifpgfmathfloatparsenumberpendingperiod + \pgfmathfloat@a@Mtok=\expandafter{\the\pgfmathfloat@a@Mtok.0}% + \fi + \pgfmathfloatcreate{\the\pgfmathfloat@a@S}{\the\pgfmathfloat@a@Mtok}{\the\pgfmathfloat@a@E}% + \pgfmath@smuggleone\pgfmathresult + \endgroup +} + +\let\pgfmathfloatqparsenumber=\pgfmathfloatparsenumber + +\def\pgfmathfloattoregisters#1#2#3#4{% + \expandafter\pgfmathfloat@decompose#1\relax{#2}{#3}{#4}% +} + +\def\pgfmathfloattoregisterstok#1#2#3#4{% + \expandafter\pgfmathfloat@decompose@tok#1\relax{#2}{#3}{#4}% +} + +\def\pgfmathfloatgetflags#1#2{% + \expandafter\pgfmathfloat@decompose@F#1\relax{#2}% +} +\def\pgfmathfloatgetflagstomacro#1#2{% + \expandafter\pgfmathfloat@decompose@Fmacro#1\relax{#2}% +}% + +\def\pgfmathfloatgetmantissa#1#2{% + \expandafter\pgfmathfloat@decompose@M#1\relax{#2}% +} +\let\pgfmathfloatgetmantisse=\pgfmathfloatgetmantissa +\def\pgfmathfloatgetmantissatok#1#2{% + \expandafter\pgfmathfloat@decompose@Mtok#1\relax{#2}% +} +\let\pgfmathfloatgetmantissetok=\pgfmathfloatgetmantissatok +\def\pgfmathfloatgetexponent#1#2{% + \expandafter\pgfmathfloat@decompose@E#1\relax{#2}% +} + + +\def\pgfmathfloat@to@FMeE@style#1{\expandafter\pgfmathfloat@to@FMeE@style@#1\relax\relax}% +\def\pgfmathfloat@to@FMeE@style@#1#2{% + \ifx#2Y% + \expandafter\pgfmathfloat@to@FMeE@style@@% + \else + \expandafter\pgfmathfloat@to@FMeE@style@error% + \fi + #1#2% +}% +\def\pgfmathfloat@to@FMeE@style@@#1Y#2e#3]\relax\relax{\def\pgfmathresult{#1#2e#3}}% +\def\pgfmathfloat@to@FMeE@style@error#1\relax\relax{% + \begingroup + \pgfmathfloatparsenumber@handleerror{wrong lowlevel format}{#1}{#1}% + \xdef\pgfmathfloat@decompose@TMP{% + \noexpand\pgfmathfloatcreate{\the\pgfmathfloat@a@S}{\the\pgfmathfloat@a@Mtok}{\the\pgfmathfloat@a@E}% + }% + \endgroup + \pgfmathfloat@decompose@TMP +}% + + +\def\pgfmathfloat@decompose#1#2{% + \ifx#2Y% + \expandafter\pgfmathfloat@decompose@% + \else + \expandafter\pgfmathfloat@decompose@error% + \fi + #1#2% +} +\def\pgfmathfloat@decompose@#1Y#2e#3]\relax#4#5#6{% + #4=#1\relax + #5=#2pt % + #6=#3\relax% +} + +\def\pgfmathfloat@decompose@tok#1#2{% + \ifx#2Y% + \expandafter\pgfmathfloat@decompose@tok@% + \else + \expandafter\pgfmathfloat@decompose@tok@error% + \fi + #1#2% +}% +\def\pgfmathfloat@decompose@tok@#1Y#2e#3]\relax#4#5#6{% + #4=#1\relax + #5={#2}% + #6=#3\relax% +} +\def\pgfmathfloat@decompose@F#1#2{% + \ifx#2Y% + \expandafter\pgfmathfloat@decompose@F@% + \else + \expandafter\pgfmathfloat@decompose@F@error% + \fi + #1#2% +}% +\def\pgfmathfloat@decompose@Fmacro#1#2{% + \ifx#2Y% + \expandafter\pgfmathfloat@decompose@Fmacro@% + \else + \expandafter\pgfmathfloat@decompose@F@error% + \fi + #1#2% +}% +\def\pgfmathfloat@decompose@M#1#2{% + \ifx#2Y% + \expandafter\pgfmathfloat@decompose@M@% + \else + \expandafter\pgfmathfloat@decompose@M@error% + \fi + #1#2% +}% +\def\pgfmathfloat@decompose@E#1#2{% + \ifx#2Y% + \expandafter\pgfmathfloat@decompose@E@% + \else + \expandafter\pgfmathfloat@decompose@E@error% + \fi + #1#2% +}% +\def\pgfmathfloat@decompose@Mtok#1#2{% + \ifx#2Y% + \expandafter\pgfmathfloat@decompose@Mtok@% + \else + \expandafter\pgfmathfloat@decompose@Mtok@error% + \fi + #1#2% +}% +\def\pgfmathfloat@decompose@F@#1Y#2e#3]\relax#4{#4=#1\relax}% +\def\pgfmathfloat@decompose@Fmacro@#1Y#2e#3]\relax#4{\def#4{#1}}% +\def\pgfmathfloat@decompose@M@#1Y#2e#3]\relax#4{#4=#2pt}% +\def\pgfmathfloat@decompose@Mtok@#1Y#2e#3]\relax#4{#4={#2}}% +\def\pgfmathfloat@decompose@E@#1Y#2e#3]\relax#4{#4=#3\relax}% + +\def\pgfmathfloat@decompose@error#1\relax#2#3#4{% + \begingroup + \pgfmathfloatparsenumber@handleerror{wrong lowlevel format}{#1}{#1}% + \xdef\pgfmathfloat@decompose@TMP{% + \noexpand#2=\the\pgfmathfloat@a@S\space + \noexpand#3=\the\pgfmathfloat@a@Mtok pt + \noexpand#4=\the\pgfmathfloat@a@E\space + }% + \endgroup + \pgfmathfloat@decompose@TMP +}% +\def\pgfmathfloat@decompose@tok@error#1\relax#2#3#4{% + \begingroup + \pgfmathfloatparsenumber@handleerror{wrong lowlevel format}{#1}{#1}% + \xdef\pgfmathfloat@decompose@TMP{% + \noexpand#2=\the\pgfmathfloat@a@S\space + \noexpand#3={\the\pgfmathfloat@a@Mtok}% + \noexpand#4=\the\pgfmathfloat@a@E\space + }% + \endgroup + \pgfmathfloat@decompose@TMP +}% +\def\pgfmathfloat@decompose@F@error#1\relax#2{% + \begingroup + \pgfmathfloatparsenumber@handleerror{wrong lowlevel format}{#1}{#1}% + \xdef\pgfmathfloat@decompose@TMP{\noexpand#2=\the\pgfmathfloat@a@S\space}% + \endgroup + \pgfmathfloat@decompose@TMP +}% +\def\pgfmathfloat@decompose@M@error#1\relax#2{% + \begingroup + \pgfmathfloatparsenumber@handleerror{wrong lowlevel format}{#1}{#1}% + \xdef\pgfmathfloat@decompose@TMP{\noexpand#2=\the\pgfmathfloat@a@Mtok pt }% + \endgroup + \pgfmathfloat@decompose@TMP +}% +\def\pgfmathfloat@decompose@Mtok@error#1\relax#2{% + \begingroup + \pgfmathfloatparsenumber@handleerror{wrong lowlevel format}{#1}{#1}% + \xdef\pgfmathfloat@decompose@TMP{\noexpand#2={\the\pgfmathfloat@a@Mtok}}% + \endgroup + \pgfmathfloat@decompose@TMP +}% +\def\pgfmathfloat@decompose@E@error#1\relax#2{% + \begingroup + \pgfmathfloatparsenumber@handleerror{wrong lowlevel format}{#1}{#1}% + \xdef\pgfmathfloat@decompose@TMP{\noexpand#2=\the\pgfmathfloat@a@E\space}% + \endgroup + \pgfmathfloat@decompose@TMP +}% +\def\pgfmathfloattomacro#1#2#3#4{% + \begingroup + \expandafter\pgfmathfloat@decompose@tok#1\relax\pgfmathfloat@a@S\pgfmathfloat@a@Mtok\pgfmathfloat@a@E + \xdef\pgfmathfloat@glob@TMP{% + \noexpand\def\noexpand#2{\the\pgfmathfloat@a@S}% + \noexpand\def\noexpand#3{\the\pgfmathfloat@a@Mtok}% + \noexpand\def\noexpand#4{\the\pgfmathfloat@a@E}% + }% + \endgroup + \pgfmathfloat@glob@TMP +} + + +\def\pgfmathfloatcreate#1#2#3{% + \edef\pgfmathresult{#1Y#2e#3]}% +}% +\def\pgfmathfloat@POSTFLAGSCHAR{Y} + + + +\def\pgfmathfloattofixed@#1{% + \begingroup + \expandafter\pgfmathfloat@decompose@tok#1\relax\pgfmathfloat@a@S\pgfmathfloat@a@Mtok\pgfmathfloat@a@E + \ifcase\pgfmathfloat@a@S + \def\pgfmathresult{0.0}% + \or + \expandafter\pgfmathfloattofixed@impl\the\pgfmathfloat@a@Mtok\relax + \or + \expandafter\pgfmathfloattofixed@impl\the\pgfmathfloat@a@Mtok\relax + \pgfmathfloat@a@Mtok=\expandafter{\pgfmathresult}% + \edef\pgfmathresult{-\the\pgfmathfloat@a@Mtok}% + \or\def\pgfmathresult{nan}% + \or\def\pgfmathresult{inf}% + \or\def\pgfmathresult{-inf}% + \fi + \pgfmath@smuggleone\pgfmathresult + \endgroup +} +\let\pgfmathfloattofixed=\pgfmathfloattofixed@ + +\def\pgfmathfloattoint#1{% + \pgfmathfloattofixed{#1}% + \expandafter\pgfmathfloattoint@@\pgfmathresult\relax +}% +\def\pgfmathfloattoint@@#1.#2\relax{% + \def\pgfmathresult{#1}% +}% + +\def\pgfmathfloattosci@#1{% + \begingroup + \expandafter\pgfmathfloat@decompose@tok#1\relax\pgfmathfloat@a@S\pgfmathfloat@a@Mtok\pgfmathfloat@a@E + \ifcase\pgfmathfloat@a@S + \def\pgfmathresult{0.0e0}% + \or + \edef\pgfmathresult{\the\pgfmathfloat@a@Mtok e\the\pgfmathfloat@a@E}% + \or + \edef\pgfmathresult{-\the\pgfmathfloat@a@Mtok e\the\pgfmathfloat@a@E}% + \or\def\pgfmathresult{nan}% + \or\def\pgfmathresult{inf}% + \or\def\pgfmathresult{-inf}% + \fi + \pgfmath@smuggleone\pgfmathresult + \endgroup +}% +\let\pgfmathfloattosci=\pgfmathfloattosci@ + +\def\pgfmathfloatvalueof#1{\expandafter\pgfmathfloatvalueof@@#1@@}% + +\def\pgfmathfloatvalueof@@#1#2{% + \if#2Y% + \expandafter\pgfmathfloatvalueof@@@\expandafter#1\expandafter#2% + \else + \expandafter\pgfmathfloatvalueof@abort\expandafter#1\expandafter#2% + \fi +} +\def\pgfmathfloatvalueof@@@#1Y#2e#3]@@{% + \ifcase#1 % + 0.0e0% + \or + #2e#3% + \or + -#2e#3% + \or nan% + \or inf% + \or -inf% + \fi +}% +\def\pgfmathfloatvalueof@abort#1@@{#1}% + + + +\def\pgfmathroundto#1{% + \pgfmathfloatroundhasperiodtrue + \begingroup + \pgfmathfloat@fixed@digits@after@periodfalse + \pgfmathroundto@impl{#1}% + \pgfmath@smuggleone\pgfmathresult + \endgroup +} + + +\def\pgfmathroundtozerofill#1{% + \pgfmathfloatroundhasperiodtrue + \begingroup + \pgfmathfloat@fixed@digits@after@periodtrue + \pgfmathroundto@impl{#1}% + \pgfmath@smuggleone\pgfmathresult + \endgroup +} + +\def\pgfmathprintnumber@fixed@styleDEFAULT#1#2#3e#4\relax{% + \begingroup + \pgfkeysgetvalue{/pgf/number format/set decimal separator}\pgfmathprintnumber@fixed@styleDEFAULT@DEC@SEP + \pgfkeysgetvalue{/pgf/number format/@dec sep mark}\pgfmathprintnumber@fixed@styleDEFAULT@DEC@SEP@MARK + \pgfkeysgetvalue{/pgf/number format/set thousands separator}\pgfmathprintnumber@fixed@styleDEFAULT@THOUSAND@SEP + \c@pgf@counta=#4\relax % + \ifnum#2=0 + \c@pgf@counta=0 + \fi + \def\pgfmathresult{}% + \ifpgfmathfloatroundhasperiod + \expandafter\pgfmathprintnumber@fixed@styleDEFAULT@impl@period#1\pgfmathfloat@EOI + \else + \expandafter\pgfmathprintnumber@fixed@styleDEFAULT@impl@noperiod#1\pgfmathfloat@EOI\pgfmathfloat@EOI\pgfmathfloat@EOI% + \begingroup + \toks0=\expandafter{\pgfmathresult}% + \toks1=\expandafter{\pgfmathprintnumber@fixed@styleDEFAULT@DEC@SEP@MARK}% + \xdef\pgfmathfloat@glob@TMP{\the\toks0 \the\toks1 }% + \endgroup + \let\pgfmathresult=\pgfmathfloat@glob@TMP + \fi + \pgfmath@smuggleone\pgfmathresult + \endgroup +} +\def\pgfmathprintnumber@fixed@styleDEFAULT@impl@period#1.#2\pgfmathfloat@EOI{% + \ifpgfmathprintnumberskipzeroperiod + \def\pgfmathfloat@loc@TMPb{#1}% + \ifx\pgfmathfloatparsenumber@tok@ZERO\pgfmathfloat@loc@TMPb + \else + \def\pgfmathfloat@loc@TMPc{-0}% + \ifx\pgfmathfloat@loc@TMPc\pgfmathfloat@loc@TMPb + \def\pgfmathresult{-}% + \else + \def\pgfmathfloat@loc@TMPc{+0}% + \ifx\pgfmathfloat@loc@TMPc\pgfmathfloat@loc@TMPb + \def\pgfmathresult{+}% + \else + \pgfmathprintnumber@fixed@styleDEFAULT@impl@noperiod#1\pgfmathfloat@EOI\pgfmathfloat@EOI\pgfmathfloat@EOI + \fi + \fi + \fi + \else + \pgfmathprintnumber@fixed@styleDEFAULT@impl@noperiod#1\pgfmathfloat@EOI\pgfmathfloat@EOI\pgfmathfloat@EOI + \fi + \ifpgfmathprintnumber@thousand@sep@in@fractional + \let\pgfmathresultfractional\pgfutil@empty% + \pgfmathprintnumber@thousand@sep@in@fractional#2MMMM\@@ + \else + \def\pgfmathresultfractional{#2}% + \fi + \begingroup + \toks0=\expandafter{\pgfmathresult}% + \toks1=\expandafter{\pgfmathprintnumber@fixed@styleDEFAULT@DEC@SEP@MARK}% + \toks2=\expandafter{\pgfmathprintnumber@fixed@styleDEFAULT@DEC@SEP}% + \toks3=\expandafter{\pgfmathresultfractional}% + \xdef\pgfmathfloat@glob@TMP{\the\toks0 \the\toks1 \the\toks2 \the\toks3 }% + \endgroup + \let\pgfmathresult=\pgfmathfloat@glob@TMP +}% + +\def\pgfmathprintnumber@fixed@styleDEFAULT@impl@noperiod{% + \ifx\pgfmathprintnumber@fixed@styleDEFAULT@THOUSAND@SEP\pgfutil@empty + \def\pgfmathprintnumber@fixed@styleDEFAULT@impl@noperiod@NEXT{% + \pgfmathprintnumber@fixed@styleDEFAULT@impl@noperiod@printall}% + \else + \ifnum\c@pgf@counta<0\relax % + \def\pgfmathprintnumber@fixed@styleDEFAULT@impl@noperiod@NEXT{% + \pgfmathprintnumber@fixed@styleDEFAULT@impl@noperiod@printall}% + \else + \ifnum\c@pgf@counta<\pgfkeysvalueof{/pgf/number format/min exponent for 1000 sep}\relax + \def\pgfmathprintnumber@fixed@styleDEFAULT@impl@noperiod@NEXT{% + \pgfmathprintnumber@fixed@styleDEFAULT@impl@noperiod@printall}% + \else + \advance\c@pgf@counta by1 % + \c@pgf@countb=\c@pgf@counta + \divide\c@pgf@countb by3 % + \c@pgf@countc=\c@pgf@countb + \multiply\c@pgf@countc by3 + \advance\c@pgf@counta by-\c@pgf@countc% + \def\pgfmathprintnumber@fixed@styleDEFAULT@impl@noperiod@NEXT{% + \pgfmathprintnumber@fixed@styleDEFAULT@impl@noperiod@printsign + }% + \fi + \fi + \fi + \pgfmathprintnumber@fixed@styleDEFAULT@impl@noperiod@NEXT +} +\def\pgfmathprintnumber@fixed@styleDEFAULT@impl@noperiod@printsign#1{% + \def\pgfmathfloat@loc@TMPb{#1}% + \ifx\pgfmathfloat@loc@TMPb\pgfmathfloatparsenumber@tok@MINUS + \expandafter\def\expandafter\pgfmathresult\expandafter{\pgfmathresult #1}% + \let\pgfmathfloat@loc@TMPb=\pgfutil@empty + \else + \ifx\pgfmathfloat@loc@TMPb\pgfmathfloatparsenumber@tok@PLUS + \expandafter\def\expandafter\pgfmathresult\expandafter{\pgfmathresult #1}% + \let\pgfmathfloat@loc@TMPb=\pgfutil@empty + \else + \ifpgfmathprintnumber@showpositive + \expandafter\def\expandafter\pgfmathresult\expandafter{\pgfmathresult +}% + \fi + \fi + \fi + \ifnum\c@pgf@counta>0 % + \def\pgfmathprintnumber@fixed@styleDEFAULT@impl@noperiod@NEXT{% + \expandafter\pgfmathprintnumber@fixed@styleDEFAULT@impl@noperiod@printtrailingdigits\pgfmathfloat@loc@TMPb + }% + \else + \def\pgfmathprintnumber@fixed@styleDEFAULT@impl@noperiod@NEXT{% + \expandafter\pgfmathprintnumber@fixed@styleDEFAULT@impl@noperiod@counteverythird\pgfmathfloat@loc@TMPb + }% + \fi + \pgfmathprintnumber@fixed@styleDEFAULT@impl@noperiod@NEXT +} +\def\pgfmathprintnumber@fixed@styleDEFAULT@impl@noperiod@printtrailingdigits#1#2{% + \ifcase\c@pgf@counta % + \or + \expandafter\def\expandafter\pgfmathresult\expandafter{\pgfmathresult #1}% + \def\pgfmathprintnumber@fixed@styleDEFAULT@impl@noperiod@NEXT{% + \pgfmathprintnumber@fixed@styleDEFAULT@impl@noperiod@counteverythird#2% + }% + \or + \expandafter\def\expandafter\pgfmathresult\expandafter{\pgfmathresult #1#2}% + \def\pgfmathprintnumber@fixed@styleDEFAULT@impl@noperiod@NEXT{% + \pgfmathprintnumber@fixed@styleDEFAULT@impl@noperiod@counteverythird% + }% + \fi + \ifnum\c@pgf@countb>0\relax % + \begingroup + \toks0=\expandafter{\pgfmathresult}% + \toks1=\expandafter{\pgfmathprintnumber@fixed@styleDEFAULT@THOUSAND@SEP}% + \xdef\pgfmathfloat@glob@TMP{\the\toks0 \the\toks1 }% + \endgroup + \let\pgfmathresult=\pgfmathfloat@glob@TMP + \fi + \pgfmathprintnumber@fixed@styleDEFAULT@impl@noperiod@NEXT +} +\def\pgfmathprintnumber@fixed@styleDEFAULT@impl@noperiod@counteverythird#1#2#3{% + \ifnum\c@pgf@countb>0\relax + \expandafter\def\expandafter\pgfmathresult\expandafter{\pgfmathresult #1#2#3}% + \advance\c@pgf@countb by-1 + \ifnum\c@pgf@countb>0\relax + \begingroup + \toks0=\expandafter{\pgfmathresult}% + \toks1=\expandafter{\pgfmathprintnumber@fixed@styleDEFAULT@THOUSAND@SEP}% + \xdef\pgfmathfloat@glob@TMP{\the\toks0 \the\toks1 }% + \endgroup + \let\pgfmathresult=\pgfmathfloat@glob@TMP + \fi + \def\pgfmathprintnumber@fixed@styleDEFAULT@impl@noperiod@NEXT{% + \pgfmathprintnumber@fixed@styleDEFAULT@impl@noperiod@counteverythird% + }% + \else + \let\pgfmathprintnumber@fixed@styleDEFAULT@impl@noperiod@NEXT=\relax + \fi + \pgfmathprintnumber@fixed@styleDEFAULT@impl@noperiod@NEXT +} + +\def\pgfmathprintnumber@thousand@sep@in@fractional#1#2#3#4{% + \if#4M% + \def\pgfmathprint@next{\pgfmathprintnumber@thousand@sep@in@fractional@finish#1#2#3#4}% + \else + \begingroup + \toks0=\expandafter{\pgfmathresultfractional}% + \toks1={#1#2#3}% + \toks2=\expandafter{\pgfmathprintnumber@fixed@styleDEFAULT@THOUSAND@SEP}% + \xdef\pgfmathfloat@glob@TMP{\the\toks0 \the\toks1 \the\toks2 }% + \endgroup + \let\pgfmathresultfractional=\pgfmathfloat@glob@TMP + \def\pgfmathprint@next{\pgfmathprintnumber@thousand@sep@in@fractional#4}% + \fi + \pgfmathprint@next +}% +\def\pgfmathprintnumber@thousand@sep@in@fractional@finish#1M{% + \expandafter\def\expandafter\pgfmathresultfractional\expandafter{\pgfmathresultfractional #1}% + \pgfmathprintnumber@thousand@sep@in@fractional@finish@ +}% +\def\pgfmathprintnumber@thousand@sep@in@fractional@finish@#1\@@{}% + +\def\pgfmathprintnumber@fixed@styleDEFAULT@impl@noperiod@printall#1{% + \def\pgfmathfloat@loc@TMPb{#1}% + \let\pgfmathfloat@loc@TMPc=\pgfutil@empty + \ifx\pgfmathfloat@loc@TMPb\pgfmathfloatparsenumber@tok@MINUS + \else + \ifx\pgfmathfloat@loc@TMPb\pgfmathfloatparsenumber@tok@PLUS + \else + \ifpgfmathprintnumber@showpositive + \def\pgfmathfloat@loc@TMPc{+}% + \fi + \fi + \fi + \expandafter\pgfmathprintnumber@fixed@styleDEFAULT@impl@noperiod@printall@\pgfmathfloat@loc@TMPc#1% +} +\def\pgfmathprintnumber@fixed@styleDEFAULT@impl@noperiod@printall@#1\pgfmathfloat@EOI\pgfmathfloat@EOI\pgfmathfloat@EOI{% + \expandafter\def\expandafter\pgfmathresult\expandafter{\pgfmathresult #1}% +}% + +\def\pgfmathprintnumber@fixed@stylePERIOD#1#2#3e#4\relax{% + \def\pgfmathresult{#1}% +} + +\def\pgfmathprintnumber@fixed@styleCOMMA#1#2#3e#4\relax{% + \ifpgfmathfloatroundhasperiod + \expandafter\pgfmathprintnumber@fixed@styleCOMMA@impl#1\pgfmathfloat@EOI + \else + \def\pgfmathresult{#1}% + \fi +} +\def\pgfmathprintnumber@fixed@styleCOMMA@impl#1.#2\pgfmathfloat@EOI{\def\pgfmathresult{#1{,}#2}} + + +\let\pgfmathprintnumber@fixed@style=\pgfmathprintnumber@fixed@styleDEFAULT + + +\def\pgfmathfloatround#1{% + \pgfmathfloatroundhasperiodtrue + \begingroup + \pgfmathfloat@fixed@digits@after@periodfalse + \expandafter\pgfmathfloat@decompose@tok#1\relax\pgfmathfloat@a@S\pgfmathfloat@a@Mtok\pgfmathfloat@a@E + \pgfmathfloatround@impl + \pgfmath@smuggleone\pgfmathresult + \endgroup +} + +\def\pgfmathfloatroundzerofill#1{% + \pgfmathfloatroundhasperiodtrue + \begingroup + \pgfmathfloat@fixed@digits@after@periodtrue + \expandafter\pgfmathfloat@decompose@tok#1\relax\pgfmathfloat@a@S\pgfmathfloat@a@Mtok\pgfmathfloat@a@E + \pgfmathfloatround@impl + \pgfmath@smuggleone\pgfmathresult + \endgroup +} + +\newif\ifpgfmathfloatround@allow@empty@mantissa +\def\pgfmathfloatround@mantissa@ONE{1pt }% + +\def\pgfmathfloatrounddisplaystyle@shared@impl#1#2e#3\relax#4#5#6{% + \pgfkeysgetvalue{/pgf/number format/@sci exponent mark}\pgfmathfloatrounddisplaystyle@e@mark + \ifcase#1\relax + \pgfmathprintnumber@fixed@style{#2}#1#2e0\relax% + \expandafter\pgfmathfloatrounddisplaystyle@shared@impl@\expandafter{\pgfmathresult}{#4#6}% + \or + \def\pgfmathfloat@loc@TMPa{1}% + \ifpgfmathfloatround@allow@empty@mantissa + \else + \ifdim#2pt=\pgfmathfloatround@mantissa@ONE + \def\pgfmathfloat@loc@TMPa{0}% + \fi + \fi + \if1\pgfmathfloat@loc@TMPa + \pgfmathprintnumber@fixed@style{#2}#1#2e0\relax% + \expandafter\pgfmathfloatrounddisplaystyle@shared@impl@\expandafter{\pgfmathresult}{#4#6}% + \else + \def\pgfmathfloat@loc@TMPa{0}% + \pgfmathfloatrounddisplaystyle@shared@impl@{#5}{#6}% + \fi + \or + \def\pgfmathfloat@loc@TMPa{1}% + \ifpgfmathfloatround@allow@empty@mantissa + \else + \ifdim#2pt=\pgfmathfloatround@mantissa@ONE + \def\pgfmathfloat@loc@TMPa{0}% + \fi + \fi + \if1\pgfmathfloat@loc@TMPa + \pgfmathprintnumber@fixed@style{-#2}#1#2e0\relax% + \expandafter\pgfmathfloatrounddisplaystyle@shared@impl@\expandafter{\pgfmathresult}{#4#6}% + \else + \def\pgfmathfloat@loc@TMPa{0}% + \pgfmathfloatrounddisplaystyle@shared@impl@{-#5}{#6}% + \fi + \or + \pgfmathfloatrounddisplaystyle@shared@impl@@{\hbox{NaN}}{}% + \or + \ifpgfmathprintnumber@showpositive + \pgfmathfloatrounddisplaystyle@shared@impl@@{+\infty}{}% + \else + \pgfmathfloatrounddisplaystyle@shared@impl@@{\infty}{}% + \fi + \or + \pgfmathfloatrounddisplaystyle@shared@impl@@{-\infty}{}% + \fi +} + +\def\pgfmathfloatrounddisplaystyle@shared@impl@#1#2{% + \pgfmathprintnumber@begingroup + \toks0={#1}% + \toks1=\expandafter{\pgfmathfloatrounddisplaystyle@e@mark #2}% + \xdef\pgfmathfloat@glob@TMP{\the\toks0 \the\toks1 }% + \pgfmathprintnumber@endgroup + \let\pgfmathresult=\pgfmathfloat@glob@TMP +}% +\def\pgfmathfloatrounddisplaystyle@shared@impl@@#1#2{% + \pgfmathprintnumber@begingroup + \toks0={#1}% + \toks1=\expandafter{\pgfmathfloatrounddisplaystyle@e@mark #2}% + \pgfkeysgetvalue{/pgf/number format/@dec sep mark}\pgfmathprintnumber@fixed@styleDEFAULT@DEC@SEP@MARK + \toks2=\expandafter{\pgfmathprintnumber@fixed@styleDEFAULT@DEC@SEP@MARK}% + \xdef\pgfmathfloat@glob@TMP{\the\toks0 \the\toks1 \the\toks2 }% + \pgfmathprintnumber@endgroup + \let\pgfmathresult=\pgfmathfloat@glob@TMP +} + +\def\pgfmathfloatrounddisplaystyle@std#1#2e#3\relax{% + \pgfmathfloatrounddisplaystyle@shared@impl#1#2e#3\relax{\cdot}{}{10^{#3}}% +} +\def\pgfmathfloatrounddisplaystyle@subscript#1#2e#3\relax{% + \pgfmathfloatrounddisplaystyle@shared@impl#1#2e#3\relax{}{1}{_{#3}}% +} +\def\pgfmathfloatrounddisplaystyle@superscript#1#2e#3\relax{% + \pgfmathfloatrounddisplaystyle@shared@impl#1#2e#3\relax{}{1}{^{#3}}% +} +\def\pgfmathfloatrounddisplaystyle@e#1#2e#3\relax{% + \ifnum#3<0\relax + \pgfmathprintnumber@begingroup + \count0=#3\relax + \multiply\count0 by-1 + \xdef\pgfmathfloat@glob@TMP{e{-}\the\count0}% + \pgfmathprintnumber@endgroup% + \let\pgfmathresult=\pgfmathfloat@glob@TMP + \else + \def\pgfmathresult{e{+}#3}% + \fi + \def\pgfmathfloat@loc@TMPb{\pgfmathfloatrounddisplaystyle@shared@impl#1#2e#3\relax{}{1}}% + \expandafter\pgfmathfloat@loc@TMPb\expandafter{\pgfmathresult}% +} +\def\pgfmathfloatrounddisplaystyle@E#1#2e#3\relax{% + \ifnum#3<0\relax + \pgfmathprintnumber@begingroup + \count0=#3\relax + \multiply\count0 by-1 + \xdef\pgfmathfloat@glob@TMP{E{-}\the\count0}% + \pgfmathprintnumber@endgroup% + \let\pgfmathresult=\pgfmathfloat@glob@TMP + \else + \def\pgfmathresult{E{+}#3}% + \fi + \def\pgfmathfloat@loc@TMPb{\pgfmathfloatrounddisplaystyle@shared@impl#1#2e#3\relax{}{1}}% + \expandafter\pgfmathfloat@loc@TMPb\expandafter{\pgfmathresult}% +} + +\def\pgfmathfloatrounddisplaystyle@generic#1#2e#3\relax{% + \begingroup + \pgfkeysvalueof {/pgf/number format/sci generic/@/.@cmd}{#3}{#1}{#2}\pgfeov + \pgfkeysgetvalue{/pgf/number format/sci generic/mantissa sep}\pgfmathdisplay@aftermantissa + \pgfkeysgetvalue{/pgf/number format/sci generic/empty mantissa sep}\pgfmathdisplay@nomantissa + \pgfkeysgetvalue{/pgf/number format/sci generic/exponent}\pgfmathdisplay@e + \toks0=\expandafter{\pgfmathdisplay@aftermantissa}% + \toks1=\expandafter{\pgfmathdisplay@nomantissa}% + \toks2=\expandafter{\pgfmathdisplay@e}% + \xdef\pgfmathfloat@glob@TMP{\noexpand\pgfmathfloatrounddisplaystyle@shared@impl#1#2e#3\noexpand\relax{\the\toks0}{\the\toks1}{\the\toks2}}% + \endgroup + \pgfmathfloat@glob@TMP +} + +\let\pgfmathfloatrounddisplaystyle=\pgfmathfloatrounddisplaystyle@std +\newif\ifpgfmathfloat@usezerofill@sci +\newif\ifpgfmathfloat@usezerofill@fixed +\newif\ifpgfmathprintnumber@assumemathmode +\newif\ifpgfmathprintnumber@showpositive +\newif\ifpgfmathprintnumber@frac@warn +\newif\ifpgfmathprintnumber@frac@whole +\newif\ifpgfmathparsenumber@comma@as@period + +\long\def\pgfmathfloatfrac@verbatim#1#2{#1/#2}% + +\let\pgfmathround@@orig=\pgfmathround@ + +\def\pgfmathprintnumber@RELATIVE@choice@roundtofixed{2}% + +\newif\ifpgfmathprintnumber@thousand@sep@in@fractional +\pgfkeys{% + /pgf/number format/.is family, + /pgf/number format, + fixed/.code= \pgfmath@set@number@printer{pgfmathprintnumber@FIXED}, + sci/.code= \pgfmath@set@number@printer{pgfmathprintnumber@SCI}, + std/.code= {\pgfmath@set@number@printer{pgfmathprintnumber@STD}\pgfmathprintnumber@STD@setparam{#1}}, + std/.default=,% + relative round mode/.is choice, + relative round mode/fixed/.code={\def\pgfmathprintnumber@RELATIVE@choice@roundtofixed{1}}, + relative round mode/mantissa/.code={\def\pgfmathprintnumber@RELATIVE@choice@roundtofixed{0}}, + relative round mode/compatible/.code={\def\pgfmathprintnumber@RELATIVE@choice@roundtofixed{2}}, + relative/.code={% + \pgfmath@set@number@printer{pgfmathprintnumber@RELATIVE}% + \pgfmathround@@orig{#1}% + \expandafter\pgfmathfloattoint@@\pgfmathresult\relax + \let\pgfmathprintnumber@RELATIVE@param=\pgfmathresult + },% + relative/.value required, + relative*/.style={/pgf/number format/relative={#1},/pgf/number format/relative round mode=fixed}, + every relative/.style=std, + relative style/.style={/pgf/number format/every relative/.append style={#1}}, + fixed relative/.code= \pgfmath@set@number@printer{pgfmathprintnumber@FIXED@RELATIVE}, + int detect/.code= \pgfmath@set@number@printer{pgfmathprintnumber@INT@DETECT}, + int trunc/.code= \pgfmath@set@number@printer{pgfmathprintnumber@INT@TRUNC}, + frac/.code= \pgfmath@set@number@printer{pgfmathprintnumber@frac},% + frac TeX/.initial=\frac, + frac whole format/.code={\def\pgfmathresult{#1}}, + frac denom/.initial=, + frac whole/.is if=pgfmathprintnumber@frac@whole, + frac whole=true, + frac shift/.initial=4, + frac warning/.is if=pgfmathprintnumber@frac@warn, + frac warning=true, + assume math mode/.is if=pgfmathprintnumber@assumemathmode, + assume math mode/.default=true, + fixed zerofill/.is if= pgfmathfloat@usezerofill@fixed, + fixed zerofill/.default=true, + sci zerofill/.is if= pgfmathfloat@usezerofill@sci, + sci zerofill/.default=true, + zerofill/.style= {/pgf/number format/fixed zerofill={#1},/pgf/number format/sci zerofill={#1}}, + zerofill/.default= true, + precision/.store in= \pgfmathfloat@round@precision, + sci precision/.code={% + \edef\pgfmathfloat@loc@TMPa{#1}% + \ifx\pgfmathfloat@loc@TMPa\pgfutil@empty + \def\pgfmathprintnumber@sci@precision{\pgfmathfloat@round@precision}% + \else + \let\pgfmathprintnumber@sci@precision=\pgfmathfloat@loc@TMPa + \fi + },% + sci precision/.default=, + sci precision=, + fixed default/.code= {\let\pgfmathprintnumber@fixed@style=\pgfmathprintnumber@fixed@styleDEFAULT}, + set decimal separator/.initial=, + dec sep/.style={/pgf/number format/set decimal separator={#1}}, + @dec sep mark/.initial=, + @sci exponent mark/.initial=, + set thousands separator/.initial=, + 1000 sep in fractionals/.is if=pgfmathprintnumber@thousand@sep@in@fractional, + 1000 sep in fractionals/.default=true, + 1000 sep/.style={/pgf/number format/set thousands separator={#1}}, + min exponent for 1000 sep/.initial=0, + use period/.style= {/pgf/number format/set decimal separator={.},/pgf/number format/set thousands separator={{{,}}}}, + use comma/.style= {/pgf/number format/set decimal separator={{{,}}},/pgf/number format/set thousands separator={.}}, + showpos/.is if=pgfmathprintnumber@showpositive, + showpos/.default=true, + print sign/.is if=pgfmathprintnumber@showpositive, + print sign/.default=true, + read comma as period/.is if=pgfmathparsenumber@comma@as@period, + read comma as period/.default=true, + skip 0./.is if=pgfmathprintnumberskipzeroperiod, + skip 0./.default=true, + skip 0.= false, + use period, + sci 10^e/.code= {\let\pgfmathfloatrounddisplaystyle=\pgfmathfloatrounddisplaystyle@std}, + sci 10e/.code= {\let\pgfmathfloatrounddisplaystyle=\pgfmathfloatrounddisplaystyle@std}, + sci e/.code= {\let\pgfmathfloatrounddisplaystyle=\pgfmathfloatrounddisplaystyle@e}, + sci E/.code= {\let\pgfmathfloatrounddisplaystyle=\pgfmathfloatrounddisplaystyle@E}, + sci subscript/.code= {\let\pgfmathfloatrounddisplaystyle=\pgfmathfloatrounddisplaystyle@subscript}, + sci superscript/.code= {\let\pgfmathfloatrounddisplaystyle=\pgfmathfloatrounddisplaystyle@superscript}, + sci generic/.code= {% + \let\pgfmathfloatrounddisplaystyle=\pgfmathfloatrounddisplaystyle@generic + \pgfkeysdefargs{/pgf/number format/sci generic/@}% + {##1##2##3}% + {\pgfqkeys{/pgf/number format/sci generic}{#1}}% + }, + sci generic/mantissa sep/.initial=,% + sci generic/empty mantissa sep/.initial=,% + sci generic/mantisse sep/.style={/pgf/number format/sci generic/mantissa sep={#1}},% + sci generic/empty mantisse sep/.style={/pgf/number format/sci generic/empty mantissa sep={#1}},% + sci generic/exponent/.initial=,% + sci generic/.value required, + verbatim/.code={% + \pgfqkeys{/pgf/number format}{% + sci generic={mantissa sep=,exponent={e##1}}, + 1000 sep=, + skip 0.=false, + print sign=false, + dec sep=., + assume math mode, + frac TeX=\pgfmathfloatfrac@verbatim, + frac whole format/.code={\def\pgfmathresult{##1 }}, + @dec sep mark=, + @sci exponent mark=, + }% + }, + retain unit mantissa/.is if=pgfmathfloatround@allow@empty@mantissa, + retain unit mantissa/.default=true, + retain unit mantissa=true, + implementation/.code={\def\pgfmathprintnumber@protected{#1}}, +} + + + +\def\pgfmathprintnumber@STD#1{% + \pgfmathfloatparsenumber{#1}% + \pgfmathfloat@to@FMeE@style\pgfmathresult + \expandafter\pgfmathprintnumber@STD@issci\pgfmathresult\relax +} + +\def\pgfmathprintnumber@STD@setparam#1{% + \edef\pgfmathfloat@loc@TMPb{#1}% + \ifx\pgfmathfloat@loc@TMPb\pgfutil@empty + \def\pgfmathprintnumber@STD@getlowerbound{% + \c@pgf@counta=\pgfmathfloat@round@precision@orig\relax + \divide\c@pgf@counta by-2\relax + }% + \def\pgfmathprintnumber@STD@upperbound{4}% + \else + \expandafter\pgfutil@in@\expandafter:\expandafter{\pgfmathfloat@loc@TMPb}% + \ifpgfutil@in@ + \def\pgfmathfloat@loc@TMPa##1:##2\relax{% + \edef\pgfmathprintnumber@STD@getlowerbound{% + \noexpand\c@pgf@counta=##1\space + }% + \edef\pgfmathprintnumber@STD@upperbound{##2}% + }% + \expandafter\pgfmathfloat@loc@TMPa\pgfmathfloat@loc@TMPb\relax + \else + \def\pgfmathprintnumber@STD@upperbound{4}% + \edef\pgfmathprintnumber@STD@getlowerbound{% + \noexpand\c@pgf@counta=\pgfmathfloat@loc@TMPb\space + }% + \fi + \fi +}% + +\def\pgfmathprintnumber@STD@issci#1#2e#3\relax{% + \expandafter\ifnum#1<3 + \expandafter\ifnum#3>\pgfmathprintnumber@STD@upperbound\relax + \pgfmathprintnumber@SCI@issci#1#2e#3\relax% + \else + \begingroup + \pgfmathprintnumber@STD@getlowerbound + \ifnum#3<\c@pgf@counta + \pgfmathprintnumber@SCI@issci#1#2e#3\relax% + \else + \pgfmathprintnumber@FIXED@issci#1#2e#3\relax% + \fi + \pgfmath@smuggleone\pgfmathresult + \endgroup + \fi + \else% + \pgfmathfloatrounddisplaystyle#1#2e#3\relax% + \fi +} + +\def\pgfmathprintnumber@RELATIVE#1{% + \pgfmathfloatparsenumber{#1}% + \pgfmathfloat@to@FMeE@style\pgfmathresult + \expandafter\pgfmathprintnumber@RELATIVE@issci\pgfmathresult\relax +} + +\def\pgfmathprintnumber@RELATIVE@choice@roundtofixed@warn{% + \pgf@typeout{! Package pgf warning: the use of /pgf/number format/relative is discouraged (buggy). Please use 'relative*' instead or type \string\pgfkeys{/pgf/number format/relative round mode=fixed} into your preamble.}% + \global\let\pgfmathprintnumber@RELATIVE@choice@roundtofixed@warn=\relax +} + +\def\pgfmathprintnumber@RELATIVE@issci#1#2e#3\relax{% + \begingroup + \c@pgf@counta=#3\relax + \advance\c@pgf@counta by-\pgfmathprintnumber@RELATIVE@param\relax + \pgfmathfloatcreate{#1}{#2}{\the\c@pgf@counta}% + \if0\pgfmathprintnumber@RELATIVE@choice@roundtofixed + \def\b@round@fixed{0}% + \else + \if1\pgfmathprintnumber@RELATIVE@choice@roundtofixed + \def\b@round@fixed{1}% + \else + \pgfmathprintnumber@RELATIVE@choice@roundtofixed@warn + \ifnum\c@pgf@counta>0 + \def\b@round@fixed{0}% + \else + \def\b@round@fixed{1}% + \fi + \fi + \fi + \if1\b@round@fixed + \pgfmathfloattofixed\pgfmathresult + \expandafter\pgfmathroundto\expandafter{\pgfmathresult}% + \pgfmathfloatparsenumber{\pgfmathresult}% + \pgfmathfloatroundhasperiodtrue% + \else + \expandafter\pgfmathfloatround\expandafter{\pgfmathresult}% + \fi + \pgfmathfloattoregisterstok\pgfmathresult\pgfmathfloat@a@S\pgfmathfloat@a@Mtok\pgfmathfloat@a@E + \ifnum\pgfmathfloat@a@S=0 + \pgfmathfloat@a@E=\pgfmathfloat@a@S % + \else + \advance\pgfmathfloat@a@E by+\pgfmathprintnumber@RELATIVE@param\relax + \fi + \edef\pgfmathresultX{% + \the\pgfmathfloat@a@S + \the\pgfmathfloat@a@Mtok + \ifpgfmathfloatroundhasperiod\else.0\fi + e\the\pgfmathfloat@a@E}% + \pgfqkeys{/pgf/number format}{/pgf/number format/every relative, + fixed zerofill=false,% + sci zerofill=false}% + \ifx\pgfmathprintnumber@issci\pgfmathprintnumber@RELATIVE@issci + \pgfmath@error{The '/pgf/number format/every relative' style should set a valid display style}{}% + \fi + \let\pgfmathfloat@round@precision@orig=\pgfmathfloat@round@precision + \def\pgfmathfloat@round@precision{9999}% + \expandafter\pgfmathprintnumber@issci\pgfmathresultX\relax% + \pgfmath@smuggleone\pgfmathresult + \endgroup +} + +\def\pgfmathprintnumber@FIXED@RELATIVE#1{% + \pgfmathfloatparsenumber{#1}% + \pgfmathfloat@to@FMeE@style\pgfmathresult + \expandafter\pgfmathprintnumber@FIXED@RELATIVE@issci\pgfmathresult\relax +} + +\def\pgfmathprintnumber@FIXED@RELATIVE@issci#1#2e#3\relax{% + \begingroup + \pgfmathfloatcreate{#1}{#2}{#3}% + \c@pgf@counta=\pgfmathfloat@round@precision\relax + \advance\c@pgf@counta by-1 + \edef\pgfmathfloat@round@precision{\the\c@pgf@counta}% + \pgfmathfloatround{\pgfmathresult}% + \pgfmathfloattoregisterstok\pgfmathresult\pgfmathfloat@a@S\pgfmathfloat@a@Mtok\pgfmathfloat@a@E + \ifnum\pgfmathfloat@a@S=0 + \pgfmathfloat@a@E=\pgfmathfloat@a@S % + \fi + \edef\pgfmathresultX{% + \the\pgfmathfloat@a@S + \the\pgfmathfloat@a@Mtok + \ifpgfmathfloatroundhasperiod\else.0\fi + e\the\pgfmathfloat@a@E}% + \let\pgfmathfloat@round@precision@orig=\pgfmathfloat@round@precision + \pgfmathfloat@usezerofill@fixedfalse + \def\pgfmathfloat@round@precision{9999}% + \expandafter\pgfmathprintnumber@FIXED@issci\pgfmathresultX\relax% + \pgfmath@smuggleone\pgfmathresult + \endgroup +} + +\def\pgfmathprintnumber@frac#1{% + \pgfmathfloatparsenumber{#1}% + \pgfmathfloatgetfrac{\pgfmathresult}% + \expandafter\pgfmathprintnumber@frac@formatresult\pgfmathresult +} +\def\pgfmathprintnumber@frac@issci#1#2e#3\relax{% + \pgfmathfloatcreate{#1}{#2}{#3}% + \pgfmathfloatgetfrac{\pgfmathresult}% + \expandafter\pgfmathprintnumber@frac@formatresult\pgfmathresult +} +\def\pgfmathprintnumber@frac@formatresult#1#2#3{% + \begingroup + \pgfkeysgetvalue{/pgf/number format/frac TeX}\pgfmathresult + \toks0=\expandafter{\pgfmathresult}% + \def\pgfmathfloat@loc@TMPa{#1}% + \ifx\pgfmathfloat@loc@TMPa\pgfutil@empty + \ifpgfmathprintnumber@showpositive + \def\pgfmathfloat@loc@TMPa{+}% + \fi + \else + \ifx\pgfmathfloat@loc@TMPa-% + \else + \ifx\pgfmathfloat@loc@TMPa+% + \else + \ifpgfmathprintnumber@showpositive + \edef\pgfmathfloat@loc@TMPa{+\pgfmathfloat@loc@TMPa}% + \fi + \def\pgfmathfloat@loc@TMPb{\pgfkeysvalueof{/pgf/number format/frac whole format/.@cmd}}% + \expandafter\pgfmathfloat@loc@TMPb\pgfmathfloat@loc@TMPa\pgfeov + \let\pgfmathfloat@loc@TMPa=\pgfmathresult + \fi + \fi + \fi + \toks1=\expandafter{\pgfmathfloat@loc@TMPa}% + \edef\pgfmathresult{% + \the\toks1 \ifnum#2=0 \else\the\toks0 {#2}{#3}\fi% + }% + \pgfmath@smuggleone\pgfmathresult + \endgroup +}% + +\def\pgfmathfloatgetfrac#1{% + \pgfutil@ifundefined{pgfmathfloatmultiply@}{% + \pgfmath@error{Sorry, the number format 'frac' requires '\string\usetikzlibrary{fpu}' (and, optionally, \string\usepackage{fp}) in order to work correctly}{}% + \edef\pgfmathresult{{#1}{0}{1}}% + }{% + \pgfmathfloatgetfrac@{#1}% + }% +}% +\def\pgfmathfloatgetfrac@#1{% + \begingroup + \edef\pgfmathfloat@arg{#1}% + \expandafter\pgfmathfloat@decompose@tok\pgfmathfloat@arg\relax\pgfmathfloat@a@S\pgfmathfloat@a@Mtok\pgfmathfloat@a@E + \def\pgfmathfloat@loc@TMPa{0}% + \ifcase\pgfmathfloat@a@S + \def\pgfmathresult{{0}{0}{1}}% + \or% + \pgfmathfloat@a@S=1 + \def\pgfmathfloat@loc@TMPa{1}% + \or% + \pgfmathfloat@a@S=-1 + \def\pgfmathfloat@loc@TMPa{1}% + \or + \edef\pgfmathresult{{}{NaN}{1}}% + \or + \edef\pgfmathresult{{}{\infty}{1}}% + \or + \edef\pgfmathresult{{-}{\infty}{1}}% + \fi + \if1\pgfmathfloat@loc@TMPa + \ifnum\pgfmathfloat@a@E<0 + \def\pgfmathfloat@wholenumber{}% + \else + \pgfmathfloatcreate{1}{\the\pgfmathfloat@a@Mtok}{\the\pgfmathfloat@a@E}% + \pgfmathfloattofixed\pgfmathresult% + \def\pgfmathfloat@loc@TMPa##1.##2\relax{% + \def\pgfmathfloat@wholenumber{##1}% + \pgfmathfloatparsenumber{0.##2}% + }% + \expandafter\pgfmathfloat@loc@TMPa\pgfmathresult\relax + \expandafter\pgfmathfloat@decompose@tok\pgfmathresult\relax\pgfmathfloat@b@S\pgfmathfloat@a@Mtok\pgfmathfloat@a@E + \fi + \def\pgfmathfloat@loc@TMPa##1.##2\relax{% + \def\pgfmathfloat@mantissa@first{##1}% + \def\pgfmathfloat@mantissa@ltone{0.##2}% + }% + \expandafter\pgfmathfloat@loc@TMPa\the\pgfmathfloat@a@Mtok\relax + \def\pgfmathfloat@loc@@to@int##1.##2\relax##3{% + ##3=##1 + \ifdim0.##2pt>0.5pt + \advance##3 by1 + \fi + }% + \let\pgfmathresultnumerator=\pgfmathfloat@b@S + \let\pgfmathresultdenom=\pgfmathfloat@b@E + \pgfkeysgetvalue{/pgf/number format/frac denom}\pgfmath@target@denominator + \ifx\pgfmath@target@denominator\pgfutil@empty + \pgfmathfloat@a@E=-\pgfmathfloat@a@E + \ifdim\pgfmathfloat@mantissa@ltone pt=0pt + \def\pgfmathfloat@factor{1}% + \edef\pgfmathfloat@scaled@numerator{\the\pgfmathfloat@a@Mtok}% + \else + \pgfutil@ifundefined{FPdiv}{% + \pgfmathfloatparsenumber{\pgfmathfloat@mantissa@ltone}% + \pgfmathfloatreciprocal@{\pgfmathresult}% + }{% + \FPmessagesfalse% + \FPdebugfalse% + \FPdiv\pgfmathresult{1}{\pgfmathfloat@mantissa@ltone}% + \pgfmathfloatparsenumber{\pgfmathresult}% + }% + \let\pgfmathfloat@inv=\pgfmathresult + \pgfmathfloatcreate{1}{1.0}{\pgfkeysvalueof{/pgf/number format/frac shift}}% + \let\pgfmathfloat@scalebaseten=\pgfmathresult + \pgfmathfloatmultiply@{\pgfmathfloat@inv}{\pgfmathfloat@scalebaseten}% + \let\pgfmathfloat@factor=\pgfmathresult + \expandafter\pgfmathfloatparsenumber\expandafter{\the\pgfmathfloat@a@Mtok}% + \let\pgfmathfloat@loc@TMPa=\pgfmathresult + \pgfmathfloatmultiply@{\pgfmathfloat@factor}{\pgfmathfloat@loc@TMPa}% + \pgfmathfloattofixed\pgfmathresult + \let\pgfmathfloat@scaled@numerator=\pgfmathresult + \pgfmathfloattofixed\pgfmathfloat@factor + \expandafter\pgfmathfloat@loc@@to@int\pgfmathresult\relax{\pgfmathfloat@b@S}% + \edef\pgfmathfloat@factor{\the\pgfmathfloat@b@S}% + \fi + \pgfmathresultdenom=\pgfmathfloat@factor\relax + \pgfmathfloat@multiply@ten@to@the@E\pgfmathresultdenom + \expandafter\pgfmathfloat@loc@@to@int\pgfmathfloat@scaled@numerator\relax{\pgfmathresultnumerator}% + \pgfmathgreatestcommondivisor{\pgfmathresultnumerator}{\pgfmathresultdenom}% + \divide\pgfmathresultnumerator by\pgfmathresult\relax + \divide\pgfmathresultdenom by\pgfmathresult\relax + \ifpgfmathprintnumber@frac@warn + \ifnum\pgfmathresultdenom>1000 + \pgfutil@ifundefined{FPdiv}{% + \pgfmathfloattosci@\pgfmathfloat@arg + \pgf@typeout{! Package pgf /pgf/number format/frac warning=true: /pgf/number format/frac of `\pgfmathresult' = \the\pgfmathresultnumerator\space / \the\pgfmathresultdenom\space might be large due to instabilities. Try \string\usepackage{fp} to improve accuracy.}% + }{}% + \fi + \fi + \else + \pgfmathresultdenom=\pgfmath@target@denominator\relax + \pgfmathfloatcreate{1}{\the\pgfmathfloat@a@Mtok}{\the\pgfmathfloat@a@E}% + \pgfmathfloattofixed\pgfmathresult + \pgf@xa=\pgfmathresult pt + \multiply\pgf@xa by\pgfmathresultdenom + \edef\pgfmathfloat@scaled@numerator{\pgfmath@tonumber\pgf@xa}% + \expandafter\pgfmathfloat@loc@@to@int\pgfmathfloat@scaled@numerator\relax{\pgfmathresultnumerator}% + \fi + \ifpgfmathprintnumber@frac@whole + \else + \ifx\pgfmathfloat@wholenumber\pgfutil@empty + \else + \count0=\pgfmathfloat@wholenumber\relax + \multiply\count0 by\pgfmathresultdenom + \advance\pgfmathresultnumerator by\count0 + \def\pgfmathfloat@wholenumber{}% + \fi + \fi + \ifnum\pgfmathfloat@a@S<0 + \edef\pgfmathresult{{-\pgfmathfloat@wholenumber}{\the\pgfmathresultnumerator}{\the\pgfmathresultdenom}}% + \else + \edef\pgfmathresult{{\pgfmathfloat@wholenumber}{\the\pgfmathresultnumerator}{\the\pgfmathresultdenom}}% + \fi + \fi + \pgfmath@smuggleone\pgfmathresult + \endgroup +}% +\def\pgfmathfloat@gobble@until@relax#1\relax{} + +\def\pgfmathgreatestcommondivisor#1#2{% + \begingroup + \count0=#1\relax + \count1=#2\relax + \ifnum\count0=0 + \edef\pgfmathresult{\the\count1}% + \else + \pgfmathgreatestcommondivisor@loop + \edef\pgfmathresult{\the\count0}% + \fi + \pgfmath@smuggleone\pgfmathresult + \endgroup +} +\def\pgfmathgreatestcommondivisor@loop{% + \ifnum\count1=0 + \else + \ifnum\count0>\count1 + \advance\count0 by-\count1 + \else + \advance\count1 by-\count0 + \fi + \expandafter\pgfmathgreatestcommondivisor@loop + \fi +}% + +\def\pgfmathfloat@multiply@ten@to@the@E#1{% + \ifnum\pgfmathfloat@a@E>0 + \multiply#1 by10 + \advance\pgfmathfloat@a@E by-1 + \def\pgfmathfloat@multiply@ten@to@the@E@next{\pgfmathfloat@multiply@ten@to@the@E{#1}}% + \else + \let\pgfmathfloat@multiply@ten@to@the@E@next=\relax% + \fi + \pgfmathfloat@multiply@ten@to@the@E@next +}% + +\def\pgfmathprintnumber@INT@TRUNC#1{% + \pgfmathfloatparsenumber{#1}% + \pgfmathfloat@to@FMeE@style\pgfmathresult + \expandafter\pgfmathprintnumber@INT@TRUNC@issci\pgfmathresult\relax +} + +\def\pgfmathprintnumber@INT@TRUNC@impl#1.#2\relax#3#4e#5\relax{% + \pgfmathfloatroundhasperiodfalse + \pgfmathprintnumber@fixed@style{#1}#3#4e#5\relax% +} +\def\pgfmathprintnumber@INT@TRUNC@issci#1#2e#3\relax{% + \ifnum#1<3\relax + \pgfmathfloatcreate{#1}{#2}{#3}% + \pgfmathfloattofixed{\pgfmathresult}% + \expandafter\pgfmathprintnumber@INT@TRUNC@impl\pgfmathresult\relax#1#2e#3\relax + \else + \pgfmathfloatrounddisplaystyle#1#2e#3\relax% + \fi +} + +\def\pgfmathifisint#1#2#3{% + \gdef\pgfmathfloatisint@@{0}% + \begingroup + \pgfmathfloatparsenumber{#1}% + \let\pgfretval\pgfmathresult + \pgfmathfloat@to@FMeE@style\pgfmathresult + \expandafter\pgfmathfloatisint@\pgfmathresult\relax + \pgfmath@smuggleone\pgfretval + \endgroup + \if1\pgfmathfloatisint@@ #2\else #3\fi +}% +\def\pgfmathfloatisint@#1#2e#3\relax{% + \ifnum#1<3\relax + \pgfmathfloatcreate{#1}{#2}{#3}% + \pgfmathfloattofixed{\pgfmathresult}% + \def\pgfmathfloat@round@precision{6}% + \expandafter\pgfmathroundto\expandafter{\pgfmathresult}% + \ifpgfmathfloatroundhasperiod + \else + \let\pgfretval\pgfmathresult% + \gdef\pgfmathfloatisint@@{1}% + \fi + \fi +} + +\def\pgfmathprintnumber@INT@DETECT#1{% + \pgfmathfloatparsenumber{#1}% + \pgfmathfloat@to@FMeE@style\pgfmathresult + \expandafter\pgfmathprintnumber@INT@DETECT@issci\pgfmathresult\relax +} + +\def\pgfmathprintnumber@INT@DETECT@issci#1#2e#3\relax{% + \begingroup + \ifnum#1<3\relax + \pgfmathfloatcreate{#1}{#2}{#3}% + \pgfmathfloattofixed{\pgfmathresult}% + \let\pgfmathfloat@round@precision@=\pgfmathfloat@round@precision + \def\pgfmathfloat@round@precision{6}% + \expandafter\pgfmathroundto\expandafter{\pgfmathresult}% + \let\pgfmathfloat@round@precision=\pgfmathfloat@round@precision@ + \ifpgfmathfloatroundhasperiod + \pgfmathprintnumber@SCI@issci#1#2e#3\relax + \else + \expandafter\pgfmathprintnumber@fixed@style\expandafter{\pgfmathresult}#1#2e#3\relax + \fi + \else + \pgfmathfloatrounddisplaystyle#1#2e#3\relax% + \fi + \pgfmath@smuggleone\pgfmathresult + \endgroup +} + +\def\pgfmathprintnumber@FIXED#1{% + \pgfmathfloatparsenumber{#1}% + \pgfmathfloat@to@FMeE@style\pgfmathresult + \expandafter\pgfmathprintnumber@FIXED@issci\pgfmathresult\relax +} + +\def\pgfmathprintnumber@FIXED@issci#1#2e#3\relax{% + \begingroup + \ifnum#1<3 + \pgfmathfloatcreate{#1}{#2}{#3}% + \pgfmathfloattofixed{\pgfmathresult}% + \ifpgfmathfloat@usezerofill@fixed + \expandafter\pgfmathroundtozerofill\expandafter{\pgfmathresult}% + \else + \expandafter\pgfmathroundto\expandafter{\pgfmathresult}% + \fi + \ifpgfmathfloatroundmayneedrenormalize + \pgfmathfloat@a@E=#3\relax + \advance\pgfmathfloat@a@E by1 + \edef\pgfmathfloat@loc@TMPb{\noexpand\pgfmathprintnumber@fixed@style{\pgfmathresult}#1#2e\the\pgfmathfloat@a@E}% + \pgfmathfloat@loc@TMPb\relax% + \else + \expandafter\pgfmathprintnumber@fixed@style\expandafter{\pgfmathresult}#1#2e#3\relax% + \fi + \else% + \pgfmathfloatrounddisplaystyle#1#2e#3\relax% + \fi + \pgfmath@smuggleone\pgfmathresult + \endgroup +} + + +\def\pgfmathprintnumber@SCI#1{% + \pgfmathfloatparsenumber{#1}% + \pgfmathfloat@to@FMeE@style\pgfmathresult + \expandafter\pgfmathprintnumber@SCI@issci\pgfmathresult\relax +} + +\def\pgfmathprintnumber@SCI@issci#1#2e#3\relax{% + \begingroup + \pgfmathfloatcreate{#1}{#2}{#3}% + \edef\pgfmathfloat@round@precision{\pgfmathprintnumber@sci@precision}% + \ifpgfmathfloat@usezerofill@sci + \pgfmathfloatroundzerofill{\pgfmathresult}% + \else + \pgfmathfloatround{\pgfmathresult}% + \fi + \pgfmathfloat@to@FMeE@style\pgfmathresult + \expandafter\pgfmathfloatrounddisplaystyle\pgfmathresult\relax + \pgfmath@smuggleone\pgfmathresult + \endgroup +} + +\pgfutil@protected\def\pgfmathprintnumber{% + \pgfutil@ifnextchar[% + {\pgfmathprintnumber@OPT}% + {\pgfmathprintnumber@noopt}% +} + +\def\pgfmathprintnumber@noopt#1{% + \pgfmathprintnumber@{#1}% + \ifpgfmathprintnumber@assumemathmode + \pgfmathresult + \else + \pgfutilensuremath{\pgfmathresult}% + \fi +}% +\def\pgfmathprintnumber@OPT[#1]#2{% + \begingroup + \pgfqkeys{/pgf/number format}{#1}% + \pgfmathprintnumber@{#2}% + \ifpgfmathprintnumber@assumemathmode + \pgfmathresult + \else + \pgfutilensuremath{\pgfmathresult}% + \fi + \endgroup +}% + +\def\pgfmathprintnumberto{% + \pgfutil@ifnextchar[% + {\pgfmathprintnumberto@OPT}% + {\pgfmathprintnumberto@noopt}% +} + +\def\pgfmathprintnumberto@noopt#1#2{% + \begingroup + \pgfmathprintnumber@{#1}% + \ifpgfmathprintnumber@assumemathmode + \global\let\pgfmathfloat@glob@TMP=\pgfmathresult + \else + \toks0=\expandafter{\pgfmathresult}% + \xdef\pgfmathfloat@glob@TMP{\noexpand\pgfutilensuremath{\the\toks0 }}% + \fi + \endgroup + \let#2=\pgfmathfloat@glob@TMP +}% +\def\pgfmathprintnumberto@OPT[#1]#2#3{% + \begingroup + \pgfqkeys{/pgf/number format}{#1}% + \pgfmathprintnumber@{#2}% + \ifpgfmathprintnumber@assumemathmode + \global\let\pgfmathfloat@glob@TMP=\pgfmathresult + \else + \toks0=\expandafter{\pgfmathresult}% + \xdef\pgfmathfloat@glob@TMP{\noexpand\pgfutilensuremath{\the\toks0 }}% + \fi + \endgroup + \let#3=\pgfmathfloat@glob@TMP +}% + + +\def\pgfmath@set@number@printer#1{% + \expandafter\let\expandafter\pgfmathprintnumber@\csname #1\endcsname + \expandafter\let\expandafter\pgfmathprintnumber@issci\csname #1@issci\endcsname +} + +\pgfkeys{/pgf/number format/std} + + +\def\pgfmathfloat@EOI{\pgfmathfloat@EOI}% + +\let\c@pgfmathroundto@prec=\pgfmathfloat@b@S% +\let\c@pgfmathroundto@offsetbehindperiod=\pgfmathfloat@b@E +\newcount\c@pgfmathroundto@lastzeros + +\newif\ifpgfmathround@impl@PREPERIOD@is@negative@zero + +\def\pgfmathroundto@impl#1{% + \edef\pgfmathround@input{#1}% + \global\pgfmathfloatroundmayneedrenormalizefalse + \pgfmathfloat@tmptoks={}% + \let\pgfmathround@next=\pgfutil@empty + \let\pgfmathround@cur=\pgfutil@empty + \let\pgfmathresult=\pgfutil@empty + \expandafter\c@pgfmathroundto@prec\pgfmathfloat@round@precision\relax + \c@pgfmathroundto@lastzeros=0 + \c@pgfmathroundto@offsetbehindperiod=-1 % + \pgfmathround@impl@PREPERIOD@is@negative@zerotrue + \expandafter\pgfmathroundto@impl@ITERATE@NODOT@firstcall\pgfmathround@input\pgfmathfloat@EOI +} + + +\def\pgfmathroundto@impl@discard@period#1.#2\pgfmathfloat@EOI{% + \pgfmathfloatroundhasperiodfalse + \aftergroup\pgfmathfloatroundhasperiodfalse + \def\pgfmathresult{#1}% +} + +\def\pgfmathroundto@impl@gobble@rest@and@start#1\pgfmathfloat@EOI{% + \pgfmathroundto@impl@start +} +\def\pgfmathroundto@impl@gobble@and@start\pgfmathfloat@EOI{% + \pgfmathroundto@impl@start +} +\def\pgfmathroundto@impl@gobble\pgfmathfloat@EOI{}% +\def\pgfmathroundto@impl@gobble@rest#1\pgfmathfloat@EOI{}% + +\def\pgfmathroundto@impl@start{% + \ifx\pgfmathround@next\pgfutil@empty + \ifnum\c@pgfmathroundto@offsetbehindperiod<0 + \ifpgfmathfloat@fixed@digits@after@period + \ifnum\c@pgfmathroundto@prec=0\relax + \pgfmathfloatroundhasperiodfalse + \aftergroup\pgfmathfloatroundhasperiodfalse + \edef\pgfmathresult{\pgfmathround@input}% + \else + \pgfmathfloat@tmptoks=\expandafter{\pgfmathround@input.}% + \c@pgfmathroundto@offsetbehindperiod=\c@pgfmathroundto@prec + \pgfmathroundto@impl@append@zeros + \edef\pgfmathresult{\the\pgfmathfloat@tmptoks}% + \fi + \else + \pgfmathfloatroundhasperiodfalse + \aftergroup\pgfmathfloatroundhasperiodfalse + \edef\pgfmathresult{\pgfmathround@input}% + \fi + \else + \pgfmathroundto@impl@finish@with@truncation + \fi + \else + \expandafter\ifnum\pgfmathround@next<5\relax + \pgfmathroundto@impl@finish@with@truncation + \else + \multiply\c@pgfmathroundto@offsetbehindperiod by-1 + \expandafter\pgfmathroundto@impl@ADD@ONE\the\pgfmathfloat@tmptoks\pgfmathfloat@EOI + \fi + \fi +} + +\def\pgfmathroundto@impl@finish@with@truncation{% + \ifpgfmathfloat@fixed@digits@after@period + \expandafter\pgfmathroundto@impl@REVERSE\the\pgfmathfloat@tmptoks\pgfmathfloat@EOI + \ifnum\c@pgfmathroundto@lastzeros=\c@pgfmathroundto@offsetbehindperiod + \ifpgfmathround@impl@PREPERIOD@is@negative@zero + \expandafter\pgfmathroundto@impl@discard@minus\the\pgfmathfloat@tmptoks\pgfmathfloat@EOI + \fi + \fi + \ifnum\c@pgfmathroundto@prec=0\relax + \expandafter\pgfmathroundto@impl@discard@period\the\pgfmathfloat@tmptoks\pgfmathfloat@EOI + \else + \advance\c@pgfmathroundto@prec by-\c@pgfmathroundto@offsetbehindperiod + \c@pgfmathroundto@offsetbehindperiod=\c@pgfmathroundto@prec + \pgfmathroundto@impl@append@zeros + \edef\pgfmathresult{\the\pgfmathfloat@tmptoks}% + \fi + \else + \pgfmathroundto@impl@discard@suffix@zeros + \fi +} + +\def\pgfmathroundto@impl@discard@minus-#1\pgfmathfloat@EOI{\pgfmathfloat@tmptoks={#1}}% + +\def\pgfmathroundto@impl@append@zeros{% + \ifnum\c@pgfmathroundto@offsetbehindperiod>0 + \pgfmathfloat@tmptoks=\expandafter{\the\pgfmathfloat@tmptoks0}% + \advance\c@pgfmathroundto@offsetbehindperiod by-1 + \pgfmathroundto@impl@append@zeros + \fi +} + +\def\pgfmathroundto@impl@ADD@ONE{% + \pgfmathfloat@tmptoks={}% + \pgfmathroundto@impl@ADD@ONE@ITERATE +} +\def\pgfmathroundto@impl@ADD@ONE@ITERATE{% + \pgfutil@ifnextchar\pgfmathfloat@EOI{% + \global\pgfmathfloatroundmayneedrenormalizetrue + \edef\pgfmathresult{1\the\pgfmathfloat@tmptoks}% + \pgfmathroundto@impl@gobble + }{% + \pgfutil@ifnextchar.{% + \ifnum\c@pgfmathroundto@prec=0 + \def\pgfmathround@nextcmd{\pgfmathroundto@impl@ADD@ONE@ITERATE@gobble@dot}% + \else + \def\pgfmathround@nextcmd{\pgfmathroundto@impl@ADD@ONE@NEXT@COLLECT}% + \fi + \pgfmathround@nextcmd + }{% + \pgfutil@ifnextchar+{% + \global\pgfmathfloatroundmayneedrenormalizetrue + \edef\pgfmathresult{1\the\pgfmathfloat@tmptoks}% + \pgfmathroundto@impl@gobble@rest + }{% + \pgfutil@ifnextchar-{% + \global\pgfmathfloatroundmayneedrenormalizetrue + \edef\pgfmathresult{-1\the\pgfmathfloat@tmptoks}% + \pgfmathroundto@impl@gobble@rest + }{% + \pgfmathroundto@impl@ADD@ONE@NEXT + }% + }% + }% + }% +} + +\def\pgfmathroundto@impl@ADD@ONE@ITERATE@gobble@dot.{% + \pgfmathfloatroundhasperiodfalse + \aftergroup\pgfmathfloatroundhasperiodfalse + \pgfmathroundto@impl@ADD@ONE@ITERATE +} + +\def\pgfmathroundto@impl@ADD@ONE@NEXT@COLLECT#1{% + \pgfmathfloat@tmptoks=\expandafter{\expandafter#1\the\pgfmathfloat@tmptoks}% + \pgfmathroundto@impl@ADD@ONE@ITERATE +} +\def\pgfmathroundto@impl@ADD@ONE@NEXT#1{% + \ifnum#1=9 + \ifnum\c@pgfmathroundto@offsetbehindperiod<0 + \ifpgfmathfloat@fixed@digits@after@period + \pgfmathfloat@tmptoks=\expandafter{\expandafter0\the\pgfmathfloat@tmptoks}% + \else + \fi + \else + \pgfmathfloat@tmptoks=\expandafter{\expandafter0\the\pgfmathfloat@tmptoks}% + \fi + \advance\c@pgfmathroundto@offsetbehindperiod by1 + \ifnum\c@pgfmathroundto@offsetbehindperiod=0 + \ifpgfmathfloat@fixed@digits@after@period + \def\pgfmathround@nextcmd{\pgfmathroundto@impl@ADD@ONE@ITERATE}% + \else + \def\pgfmathround@nextcmd{\pgfmathroundto@impl@ADD@ONE@ITERATE@gobble@dot}% + \fi + \else + \def\pgfmathround@nextcmd{\pgfmathroundto@impl@ADD@ONE@ITERATE}% + \fi + \else + \c@pgfmathroundto@lastzeros=#1 + \advance\c@pgfmathroundto@lastzeros by1 + \edef\pgfmathresult{\the\c@pgfmathroundto@lastzeros\the\pgfmathfloat@tmptoks}% + \pgfmathfloat@tmptoks=\expandafter{\pgfmathresult}% + \def\pgfmathround@nextcmd{\pgfmathroundto@impl@REVERSE@ITERATE}% + \fi + \pgfmathround@nextcmd +} + + +\def\pgfmathroundto@impl@discard@suffix@zeros{% + \ifnum\c@pgfmathroundto@lastzeros=\c@pgfmathroundto@offsetbehindperiod + \ifpgfmathround@impl@PREPERIOD@is@negative@zero + \pgfmathfloatroundhasperiodfalse + \aftergroup\pgfmathfloatroundhasperiodfalse + \def\pgfmathresult{0}% + \else + \expandafter\pgfmathroundto@impl@discard@period\pgfmathround@input\pgfmathfloat@EOI + \fi + \else + \ifnum\c@pgfmathroundto@lastzeros=0 + \expandafter\pgfmathroundto@impl@REVERSE\the\pgfmathfloat@tmptoks\pgfmathfloat@EOI + \else + \expandafter\pgfmathroundto@impl@discard@suffix@zeros@ITERATE\the\pgfmathfloat@tmptoks\pgfmathfloat@EOI + \fi + \fi +} + +\def\pgfmathroundto@impl@discard@suffix@zeros@ITERATE#1{% + \advance\c@pgfmathroundto@lastzeros by-1 + \ifnum\c@pgfmathroundto@lastzeros=0 + \def\pgfmathround@nextcmd{\pgfmathroundto@impl@REVERSE}% + \else + \def\pgfmathround@nextcmd{\pgfmathroundto@impl@discard@suffix@zeros@ITERATE}% + \fi + \pgfmathround@nextcmd +} + +\def\pgfmathroundto@impl@REVERSE{% + \pgfmathfloat@tmptoks={}% + \pgfmathroundto@impl@REVERSE@ITERATE +} + +\def\pgfmathroundto@impl@REVERSE@ITERATE{% + \pgfutil@ifnextchar\pgfmathfloat@EOI{% + \edef\pgfmathresult{\the\pgfmathfloat@tmptoks}% + \pgfmathroundto@impl@gobble + }{% + \pgfmathroundto@impl@REVERSE@NEXT + }% +} + +\def\pgfmathroundto@impl@REVERSE@NEXT#1{% + \pgfmathfloat@tmptoks=\expandafter{\expandafter#1\the\pgfmathfloat@tmptoks}% + \pgfmathroundto@impl@REVERSE@ITERATE +} + +\def\pgfmathroundto@impl@BEGIN@DOT.{% + \pgfmathfloat@tmptoks=\expandafter{\expandafter.\the\pgfmathfloat@tmptoks}% + \c@pgfmathroundto@offsetbehindperiod=0 + \pgfmathroundto@impl@ITERATE@DOT +} +\def\pgfmathroundto@impl@ITERATE@DOT{% + \pgfutil@ifnextchar\pgfmathfloat@EOI{% + \pgfmathroundto@impl@gobble@and@start + }{% + \pgfmathroundto@impl@NEXT@DOT + }% +} +\def\pgfmathroundto@impl@NEXT@DOT#1{% + \ifnum\c@pgfmathroundto@offsetbehindperiod=\c@pgfmathroundto@prec + \def\pgfmathround@next{#1}% + \def\pgfmathround@nextcmd{\pgfmathroundto@impl@gobble@rest@and@start}% + \else + \advance\c@pgfmathroundto@offsetbehindperiod by1 + \ifnum#1=0 + \advance\c@pgfmathroundto@lastzeros by1 + \else + \c@pgfmathroundto@lastzeros=0 + \fi + \pgfmathfloat@tmptoks=\expandafter{\expandafter#1\the\pgfmathfloat@tmptoks}% + \def\pgfmathround@nextcmd{\pgfmathroundto@impl@ITERATE@DOT}% + \fi + \pgfmathround@nextcmd +}% + +\def\pgfmathroundto@impl@ITERATE@NODOT@firstcall#1{% + \def\pgfmathround@cur{#1}% + \ifx\pgfmathround@cur\pgfmathfloat@EOI + \pgfmathround@impl@PREPERIOD@is@negative@zerofalse + \def\pgfmathround@nextcmd{\pgfmathroundto@impl@start}% + \else + \ifx\pgfmathround@cur\pgfmathfloatparsenumber@tok@PERIOD + \pgfmathround@impl@PREPERIOD@is@negative@zerofalse + \def\pgfmathround@nextcmd{\pgfmathroundto@impl@BEGIN@DOT.}% + \else + \ifx\pgfmathround@cur\pgfmathfloatparsenumber@tok@MINUS + \else + \pgfmathround@impl@PREPERIOD@is@negative@zerofalse + \fi + \pgfmathfloat@tmptoks=\expandafter{\expandafter#1\the\pgfmathfloat@tmptoks}% + \def\pgfmathround@nextcmd{\pgfmathroundto@impl@ITERATE@NODOT}% + \fi + \fi + \pgfmathround@nextcmd +} + +\def\pgfmathroundto@impl@ITERATE@NODOT#1{% + \def\pgfmathround@cur{#1}% + \ifx\pgfmathround@cur\pgfmathfloat@EOI + \def\pgfmathround@nextcmd{\pgfmathroundto@impl@start}% + \else + \ifx\pgfmathround@cur\pgfmathfloatparsenumber@tok@PERIOD + \def\pgfmathround@nextcmd{\pgfmathroundto@impl@BEGIN@DOT.}% + \else + \ifx\pgfmathround@cur\pgfmathfloatparsenumber@tok@ZERO + \else + \pgfmathround@impl@PREPERIOD@is@negative@zerofalse + \fi + \pgfmathfloat@tmptoks=\expandafter{\expandafter#1\the\pgfmathfloat@tmptoks}% + \def\pgfmathround@nextcmd{\pgfmathroundto@impl@ITERATE@NODOT}% + \fi + \fi + \pgfmathround@nextcmd +} + + + +\def\pgfmathfloatround@impl{% + \expandafter\pgfmathroundto@impl\expandafter{\the\pgfmathfloat@a@Mtok}% + \ifpgfmathfloatroundmayneedrenormalize + \ifpgfmathfloatroundhasperiod + \expandafter\pgfmathfloatround@impl@renormalize\pgfmathresult\pgfmathfloat@EOI% + \else + \expandafter\pgfmathfloatround@impl@renormalize\pgfmathresult.\pgfmathfloat@EOI% + \fi + \advance\pgfmathfloat@a@E by1 + \fi + \pgfmathfloatcreate{\the\pgfmathfloat@a@S}{\pgfmathresult}{\the\pgfmathfloat@a@E}% +} + +\def\pgfmathfloatround@impl@renormalize#1#2.#3\pgfmathfloat@EOI{% + \pgfmathroundto@impl{#1.#2#3}% +} + + + + + +\def\pgfmathfloattofixed@impl#1.#2\relax{% + \ifnum\pgfmathfloat@a@E<0 + \ifcase-\pgfmathfloat@a@E + \or% + \pgfmathfloat@a@Mtok={0.}% + \or% + \pgfmathfloat@a@Mtok={0.0}% + \or% + \pgfmathfloat@a@Mtok={0.00}% + \or% + \pgfmathfloat@a@Mtok={0.000}% + \or% + \pgfmathfloat@a@Mtok={0.0000}% + \or% + \pgfmathfloat@a@Mtok={0.00000}% + \or% + \pgfmathfloat@a@Mtok={0.000000}% + \or% + \pgfmathfloat@a@Mtok={0.0000000}% + \or% + \pgfmathfloat@a@Mtok={0.00000000}% + \or% + \pgfmathfloat@a@Mtok={0.000000000}% + \else% + \pgfmathfloat@a@Mtok={0.0000000000}% + \advance\pgfmathfloat@a@E by10 + \pgfutil@loop + \ifnum\pgfmathfloat@a@E<-1 + \pgfmathfloat@a@Mtok=\expandafter{\the\pgfmathfloat@a@Mtok 0}% + \advance\pgfmathfloat@a@E by1 + \pgfutil@repeat + \fi + \def\pgfmathfloat@loc@TMPb{#2}% + \def\pgfmathfloat@loc@TMPc{0}% + \ifx\pgfmathfloat@loc@TMPb\pgfmathfloat@loc@TMPc + \edef\pgfmathresult{\the\pgfmathfloat@a@Mtok #1}% + \else + \edef\pgfmathresult{\the\pgfmathfloat@a@Mtok #1#2}% + \fi + \else + \pgfmathfloat@a@Mtok{#1}% + \pgfmathfloattofixed@impl@pos#2000000000\pgfmathfloat@EOI + \edef\pgfmathresult{\the\pgfmathfloat@a@Mtok}% + \fi +} + +\def\pgfmathfloattofixed@impl@pos#1#2#3#4#5#6#7#8#9\pgfmathfloat@EOI{% + \ifcase\pgfmathfloat@a@E + \pgfmathfloat@a@Mtok\expandafter{\the\pgfmathfloat@a@Mtok .#1#2#3#4#5#6#7#8#9}% + \or% + \pgfmathfloat@a@Mtok\expandafter{\the\pgfmathfloat@a@Mtok #1.#2#3#4#5#6#7#8#9}% + \or% + \pgfmathfloat@a@Mtok\expandafter{\the\pgfmathfloat@a@Mtok #1#2.#3#4#5#6#7#8#9}% + \or% + \pgfmathfloat@a@Mtok\expandafter{\the\pgfmathfloat@a@Mtok #1#2#3.#4#5#6#7#8#9}% + \or% + \pgfmathfloat@a@Mtok\expandafter{\the\pgfmathfloat@a@Mtok #1#2#3#4.#5#6#7#8#9}% + \or% + \pgfmathfloat@a@Mtok\expandafter{\the\pgfmathfloat@a@Mtok #1#2#3#4#5.#6#7#8#9}% + \or% + \pgfmathfloat@a@Mtok\expandafter{\the\pgfmathfloat@a@Mtok #1#2#3#4#5#6.#7#8#9}% + \or% + \pgfmathfloat@a@Mtok\expandafter{\the\pgfmathfloat@a@Mtok #1#2#3#4#5#6#7.#8#9}% + \or% + \pgfmathfloat@a@Mtok\expandafter{\the\pgfmathfloat@a@Mtok #1#2#3#4#5#6#7#8.#9}% + \else% + \pgfmathfloat@a@Mtok\expandafter{\the\pgfmathfloat@a@Mtok #1#2#3#4#5#6#7#8}% + \advance\pgfmathfloat@a@E by-8 + \pgfmathfloattofixed@impl@collectmantissa#9\count\pgfmathfloat@a@E + \fi +}% + +\def\pgfmathfloattofixed@impl@collectmantissa#1#2\count#3{% + \pgfmathfloat@a@Mtok=\expandafter{\the\pgfmathfloat@a@Mtok #1}% + \advance\pgfmathfloat@a@E by-1% + \def\pgfmathfloat@loc@TMPb{#2}% + \ifx\pgfmathfloat@loc@TMPb\pgfutil@empty + \pgfutil@loop + \ifnum\pgfmathfloat@a@E>0% + \pgfmathfloat@a@Mtok=\expandafter{\the\pgfmathfloat@a@Mtok 0}% + \advance\pgfmathfloat@a@E by-1% + \pgfutil@repeat + \pgfmathfloat@a@Mtok=\expandafter{\the\pgfmathfloat@a@Mtok .0}% + \else + \ifnum\pgfmathfloat@a@E=0 + \pgfmathfloat@a@Mtok=\expandafter{\the\pgfmathfloat@a@Mtok .#2}% + \else + \pgfmathfloattofixed@impl@collectmantissa#2\count#3% + \fi + \fi +} + + + +\def\pgfmathfloatparsenumber@tok@ZERO{0} +\def\pgfmathfloatparsenumber@tok@PLUS{+} +\def\pgfmathfloatparsenumber@tok@MINUS{-} +\def\pgfmathfloatparsenumber@tok@PERIOD{.} + +\newif\ifpgfmathfloatparsenumberpendingperiod + +\def\pgfflt@EOI{p@EOI}% +\def\pgfflt@EOI@unexpanded{\pgfflt@EOI}% + +\def\pgfflt@impl#1{% + \pgfmathfloatparsenumberpendingperiodfalse + \pgfmathfloat@tmptoks={}% + \pgfmathfloat@a@E=0 + \expandafter\ifx\csname pgfflt@#1\endcsname\relax + \pgfmathfloat@a@S=1 + \expandafter\pgfflt@init\expandafter#1% + \else + \expandafter\expandafter\csname pgfflt@#1\endcsname + \fi +} +\expandafter\def\csname pgfflt@+\endcsname{% + \pgfmathfloat@a@S=1 % + \pgfflt@init}% +\expandafter\def\csname pgfflt@-\endcsname{% + \pgfmathfloat@a@S=2 % + \pgfflt@init}% +\expandafter\def\csname pgfflt@\pgfflt@EOI\endcsname{% + \pgfmathfloatparsenumber@handleerror{empty number}{}{}}% + +\pgfkeys{ + /pgf/fpu/handlers/empty number/.style 2 args={% + /pgf/fpu/handlers/invalid number={#1}{#2}% + }, + /pgf/fpu/empty number is zero/.style={% + /pgf/fpu/handlers/empty number/.code 2 args={% + \pgfmathfloatcreate{0}{0.0}{0}% + }% + },% + /pgf/fpu/handlers/invalid separator/.code 2 args={% + \pgfmath@error{Could not parse input '#1' as a floating point number, sorry. The unreadable part was near the decimal separator '#2'. Do you need the option 'read comma as period'?}{}% + }, + /pgf/fpu/handlers/invalid number/.code 2 args={% + \pgfmath@error{Could not parse input '#1' as a floating point number, sorry. The unreadable part was near '#2'.}{}% + }, + /pgf/fpu/handlers/wrong lowlevel format/.code 2 args={% + \pgfmath@error{Sorry, an internal routine of the floating point unit got an ill-formatted floating point number `#1'. The unreadable part was near '#2'.}{}% + \let\pgfmathresult=\pgfutil@empty + }, +} + +\def\pgfmathfloatparsenumber@handleerror#1#2#3{% + \let\pgfmathresult=\pgfutil@empty + \pgfmathfloatparsenumberpendingperiodfalse + \pgfkeys{/pgf/fpu/handlers/#1={#2}{#3}}% + \ifx\pgfmathresult\pgfutil@empty + \pgfmathfloat@a@S=3 + \pgfmathfloat@a@Mtok={0.0}% + \pgfmathfloat@a@E=0 + \else + \expandafter\pgfmathfloat@decompose@tok\pgfmathresult\relax\pgfmathfloat@a@S\pgfmathfloat@a@Mtok\pgfmathfloat@a@E + \fi +} +\def\pgfflt@error#1\pgfflt@EOI{% + \begingroup + \pgfmathfloat@a@Mtok={#1}% + \xdef\pgfmathfloat@glob@TMP{% + \noexpand\pgfmathfloatparsenumber@handleerror + {invalid number}{\pgfmathresult}{\the\pgfmathfloat@a@Mtok}}% + \endgroup + \pgfmathfloat@glob@TMP +}% + + + +\def\pgfflt@init#1{% + \expandafter\ifx\csname pgffltA@#1\endcsname\relax + \expandafter\pgfflt@error\expandafter#1% + \else + \expandafter\expandafter\csname pgffltA@#1\endcsname + \fi} +\expandafter\def\csname pgffltA@0\endcsname{\pgfflt@leadingzero}% +\expandafter\def\csname pgffltA@1\endcsname{\pgfflt@nonzerodigit1}% +\expandafter\def\csname pgffltA@2\endcsname{\pgfflt@nonzerodigit2}% +\expandafter\def\csname pgffltA@3\endcsname{\pgfflt@nonzerodigit3}% +\expandafter\def\csname pgffltA@4\endcsname{\pgfflt@nonzerodigit4}% +\expandafter\def\csname pgffltA@5\endcsname{\pgfflt@nonzerodigit5}% +\expandafter\def\csname pgffltA@6\endcsname{\pgfflt@nonzerodigit6}% +\expandafter\def\csname pgffltA@7\endcsname{\pgfflt@nonzerodigit7}% +\expandafter\def\csname pgffltA@8\endcsname{\pgfflt@nonzerodigit8}% +\expandafter\def\csname pgffltA@9\endcsname{\pgfflt@nonzerodigit9}% +\def\pgfflt@nonzerodigit#1{% + \pgfmathfloat@a@Mtok={#1}% + \pgfmathfloatparsenumberpendingperiodtrue + \pgfflt@positiveexp}% +\expandafter\def\csname pgffltA@n\endcsname{\pgfflt@readnan}% +\expandafter\def\csname pgffltA@N\endcsname{\pgfflt@readnan}% +\expandafter\def\csname pgffltA@i\endcsname{\pgfflt@readinf}% +\expandafter\def\csname pgffltA@I\endcsname{\pgfflt@readinf}% +\expandafter\def\csname pgffltA@.\endcsname{% + \pgfmathfloat@a@E=-1 + \pgfflt@leadingzero@foundperiod}% +\expandafter\def\csname pgffltA@,\endcsname{% + \pgfmathfloat@frenchinput{% + \pgfmathfloat@a@E=-1 + \pgfflt@leadingzero@foundperiod + }% +}% + +\def\pgfmathfloat@frenchinput#1{% + \ifpgfmathparsenumber@comma@as@period + \def\pgf@marshal{#1}% + \expandafter\pgf@marshal + \else + \expandafter\pgfflt@error@separator\expandafter,% + \fi +}% +\def\pgfflt@error@separator#1\pgfflt@EOI{% + \begingroup + \pgfmathfloat@a@Mtok={#1}% + \xdef\pgfmathfloat@glob@TMP{% + \noexpand\pgfmathfloatparsenumber@handleerror + {invalid separator}{\pgfmathresult}{\the\pgfmathfloat@a@Mtok}}% + \endgroup + \pgfmathfloat@glob@TMP +}% + +\def\pgfflt@finish#1\pgfflt@EOI{} + +\def\pgfflt@readnan #1#2{% + \def\pgfflt@readnan@ok{1}% + \if#1a\else\if#1A\else\def\pgfflt@readnan@ok{0}\fi\fi + \if#2n\else\if#2N\else\def\pgfflt@readnan@ok{0}\fi\fi + \if\pgfflt@readnan@ok1% + \pgfmathfloat@a@S=3\relax% + \pgfmathfloat@a@Mtok={0.0}% + \pgfmathfloat@a@E=0% + \expandafter\pgfflt@finish + \else + \def\pgfflt@readnan@{\pgfflt@error #1#2}% + \expandafter\pgfflt@readnan@ + \fi +} +\def\pgfflt@readinf #1#2{% + \def\pgfflt@readinf@ok{1}% + \if#1n\else\if#1N\else\def\pgfflt@readinf@ok{0}\fi\fi + \if#2f\else\if#2F\else\def\pgfflt@readinf@ok{0}\fi\fi + \if\pgfflt@readinf@ok1% + \ifnum\pgfmathfloat@a@S=1 % + \pgfmathfloat@a@S=4 % + \pgfmathfloat@a@Mtok={0.0}% + \else + \pgfmathfloat@a@S=5 % + \pgfmathfloat@a@Mtok={0.0}% + \fi + \pgfmathfloat@a@E=0 % + \expandafter\pgfflt@finish + \else + \def\pgfflt@readinf@{\pgfflt@error #1#2}% + \expandafter\pgfflt@readinf@ + \fi +} +\def\pgfflt@leadingzero#1{% + \expandafter\ifx\csname pgffltB@#1\endcsname\relax + \expandafter\pgfflt@error\expandafter#1% + \else + \expandafter\expandafter\csname pgffltB@#1\endcsname + \fi} +\expandafter\def\csname pgffltB@0\endcsname{\pgfflt@leadingzero}% +\expandafter\def\csname pgffltB@1\endcsname{\pgfmathfloat@a@Mtok={1}\pgfmathfloatparsenumberpendingperiodtrue \pgfflt@positiveexp} +\expandafter\def\csname pgffltB@2\endcsname{\pgfmathfloat@a@Mtok={2}\pgfmathfloatparsenumberpendingperiodtrue \pgfflt@positiveexp} +\expandafter\def\csname pgffltB@3\endcsname{\pgfmathfloat@a@Mtok={3}\pgfmathfloatparsenumberpendingperiodtrue \pgfflt@positiveexp} +\expandafter\def\csname pgffltB@4\endcsname{\pgfmathfloat@a@Mtok={4}\pgfmathfloatparsenumberpendingperiodtrue \pgfflt@positiveexp} +\expandafter\def\csname pgffltB@5\endcsname{\pgfmathfloat@a@Mtok={5}\pgfmathfloatparsenumberpendingperiodtrue \pgfflt@positiveexp} +\expandafter\def\csname pgffltB@6\endcsname{\pgfmathfloat@a@Mtok={6}\pgfmathfloatparsenumberpendingperiodtrue \pgfflt@positiveexp} +\expandafter\def\csname pgffltB@7\endcsname{\pgfmathfloat@a@Mtok={7}\pgfmathfloatparsenumberpendingperiodtrue \pgfflt@positiveexp} +\expandafter\def\csname pgffltB@8\endcsname{\pgfmathfloat@a@Mtok={8}\pgfmathfloatparsenumberpendingperiodtrue \pgfflt@positiveexp} +\expandafter\def\csname pgffltB@9\endcsname{\pgfmathfloat@a@Mtok={9}\pgfmathfloatparsenumberpendingperiodtrue \pgfflt@positiveexp} +\expandafter\def\csname pgffltB@\pgfmathfloat@POSTFLAGSCHAR\endcsname{\pgfmathfloat@a@Mtok={0}\pgfflt@readlowlevelfloat}% +\expandafter\def\csname pgffltB@.\endcsname{% + \pgfmathfloat@a@E=-1 + \pgfflt@leadingzero@foundperiod}% +\expandafter\def\csname pgffltB@,\endcsname{% + \pgfmathfloat@frenchinput{% + \pgfmathfloat@a@E=-1 + \pgfflt@leadingzero@foundperiod + }% +}% +\expandafter\def\csname pgffltB@e\endcsname{% + \pgfmathfloat@a@S=0\relax% + \pgfmathfloat@a@Mtok={0.0}% + \pgfmathfloat@a@E=0% + \pgfflt@readexponent}% +\let\pgffltB@E=\pgffltB@e +\expandafter\def\csname pgffltB@\pgfflt@EOI\endcsname{% + \pgfmathfloat@a@S=0\relax% + \pgfmathfloat@a@Mtok={0.0}% + \pgfmathfloat@a@E=0}% + +\def\pgfflt@leadingzero@foundperiod#1{% + \expandafter\ifx\csname pgffltC@#1\endcsname\relax + \expandafter\pgfflt@error\expandafter#1% + \else + \expandafter\expandafter\csname pgffltC@#1\endcsname + \fi} +\expandafter\def\csname pgffltC@0\endcsname{% + \advance\pgfmathfloat@a@E by-1 + \pgfflt@leadingzero@foundperiod}% +\expandafter\def\csname pgffltC@1\endcsname{\pgfmathfloat@a@Mtok={1}\pgfmathfloatparsenumberpendingperiodtrue\pgfflt@finish@number@after@period}% +\expandafter\def\csname pgffltC@2\endcsname{\pgfmathfloat@a@Mtok={2}\pgfmathfloatparsenumberpendingperiodtrue\pgfflt@finish@number@after@period}% +\expandafter\def\csname pgffltC@3\endcsname{\pgfmathfloat@a@Mtok={3}\pgfmathfloatparsenumberpendingperiodtrue\pgfflt@finish@number@after@period}% +\expandafter\def\csname pgffltC@4\endcsname{\pgfmathfloat@a@Mtok={4}\pgfmathfloatparsenumberpendingperiodtrue\pgfflt@finish@number@after@period}% +\expandafter\def\csname pgffltC@5\endcsname{\pgfmathfloat@a@Mtok={5}\pgfmathfloatparsenumberpendingperiodtrue\pgfflt@finish@number@after@period}% +\expandafter\def\csname pgffltC@6\endcsname{\pgfmathfloat@a@Mtok={6}\pgfmathfloatparsenumberpendingperiodtrue\pgfflt@finish@number@after@period}% +\expandafter\def\csname pgffltC@7\endcsname{\pgfmathfloat@a@Mtok={7}\pgfmathfloatparsenumberpendingperiodtrue\pgfflt@finish@number@after@period}% +\expandafter\def\csname pgffltC@8\endcsname{\pgfmathfloat@a@Mtok={8}\pgfmathfloatparsenumberpendingperiodtrue\pgfflt@finish@number@after@period}% +\expandafter\def\csname pgffltC@9\endcsname{\pgfmathfloat@a@Mtok={9}\pgfmathfloatparsenumberpendingperiodtrue\pgfflt@finish@number@after@period}% +\expandafter\def\csname pgffltC@e\endcsname{% + \pgfmathfloat@a@S=0\relax% + \pgfmathfloat@a@Mtok={0.0}% + \pgfmathfloat@a@E=0% + \pgfflt@readexponent}% +\let\pgffltC@E=\pgffltC@e +\expandafter\def\csname pgffltC@\pgfflt@EOI\endcsname{% + \pgfmathfloat@a@S=0\relax% + \pgfmathfloat@a@Mtok={0.0}% + \pgfmathfloat@a@E=0}% + + +\def\pgfflt@positiveexp#1{% + \expandafter\ifx\csname pgffltD@#1\endcsname\relax + \expandafter\pgfflt@error\expandafter#1% + \else + \expandafter\expandafter\csname pgffltD@#1\endcsname + \fi} +\expandafter\def\csname pgffltD@0\endcsname{% + \advance\pgfmathfloat@a@E by1 + \pgfmathfloat@tmptoks=\expandafter{\the\pgfmathfloat@tmptoks0}% + \pgfflt@pendingzeros@positiveexp}% +\expandafter\def\csname pgffltD@1\endcsname{\pgffltD@nonzerodigit1}% +\expandafter\def\csname pgffltD@2\endcsname{\pgffltD@nonzerodigit2}% +\expandafter\def\csname pgffltD@3\endcsname{\pgffltD@nonzerodigit3}% +\expandafter\def\csname pgffltD@4\endcsname{\pgffltD@nonzerodigit4}% +\expandafter\def\csname pgffltD@5\endcsname{\pgffltD@nonzerodigit5}% +\expandafter\def\csname pgffltD@6\endcsname{\pgffltD@nonzerodigit6}% +\expandafter\def\csname pgffltD@7\endcsname{\pgffltD@nonzerodigit7}% +\expandafter\def\csname pgffltD@8\endcsname{\pgffltD@nonzerodigit8}% +\expandafter\def\csname pgffltD@9\endcsname{\pgffltD@nonzerodigit9}% +\expandafter\def\csname pgffltD@\pgfmathfloat@POSTFLAGSCHAR\endcsname{\pgfflt@readlowlevelfloat}% +\expandafter\def\csname pgffltD@.\endcsname{\pgfflt@finish@number@after@period}% +\expandafter\def\csname pgffltD@,\endcsname{\pgfmathfloat@frenchinput{\pgfflt@finish@number@after@period}}% +\expandafter\def\csname pgffltD@e\endcsname{\pgfflt@readexponent}% +\let\pgffltD@E=\pgffltD@e +\expandafter\def\csname pgffltD@\pgfflt@EOI\endcsname{}% +\def\pgffltD@nonzerodigit#1{% + \advance\pgfmathfloat@a@E by1 + \ifpgfmathfloatparsenumberpendingperiod + \pgfmathfloat@a@Mtok=\expandafter{\the\pgfmathfloat@a@Mtok.}% + \pgfmathfloatparsenumberpendingperiodfalse + \fi + \pgfmathfloat@a@Mtok=\expandafter{\the\pgfmathfloat@a@Mtok#1}% + \pgfflt@positiveexp}% + +\def\pgfflt@readlowlevelfloat#1e#2]\pgfflt@EOI{% + \pgfmathfloatparsenumberpendingperiodfalse + \pgfmathfloat@a@S=\the\pgfmathfloat@a@Mtok\relax + \pgfmathfloat@a@Mtok={#1}% + \pgfmathfloat@a@E=#2 +}% + +\def\pgfflt@readexponent#1\pgfflt@EOI{% + \afterassignment\pgfflt@readexponent@ + \pgfmathfloat@b@E=#1 \pgfflt@EOI% +} +\def\pgfflt@readexponent@#1{% + \advance\pgfmathfloat@a@E by\pgfmathfloat@b@E\relax + \def\pgfmathfloat@loc@TMPb{#1}% + \ifx\pgfmathfloat@loc@TMPb\pgfflt@EOI@unexpanded + \else + \def\pgfmathfloat@loc@TMPb{\pgfflt@error#1}% + \expandafter\pgfmathfloat@loc@TMPb + \fi +} + +\def\pgfflt@pendingzeros@positiveexp#1{% + \expandafter\ifx\csname pgffltE@#1\endcsname\relax + \expandafter\pgfflt@error\expandafter#1% + \else + \expandafter\expandafter\csname pgffltE@#1\endcsname + \fi} +\expandafter\def\csname pgffltE@0\endcsname{% + \pgfmathfloat@tmptoks=\expandafter{\the\pgfmathfloat@tmptoks0}% + \advance\pgfmathfloat@a@E by1 + \pgfflt@pendingzeros@positiveexp}% +\expandafter\def\csname pgffltE@1\endcsname{\pgffltE@nonzerodigit1}% +\expandafter\def\csname pgffltE@2\endcsname{\pgffltE@nonzerodigit2}% +\expandafter\def\csname pgffltE@3\endcsname{\pgffltE@nonzerodigit3}% +\expandafter\def\csname pgffltE@4\endcsname{\pgffltE@nonzerodigit4}% +\expandafter\def\csname pgffltE@5\endcsname{\pgffltE@nonzerodigit5}% +\expandafter\def\csname pgffltE@6\endcsname{\pgffltE@nonzerodigit6}% +\expandafter\def\csname pgffltE@7\endcsname{\pgffltE@nonzerodigit7}% +\expandafter\def\csname pgffltE@8\endcsname{\pgffltE@nonzerodigit8}% +\expandafter\def\csname pgffltE@9\endcsname{\pgffltE@nonzerodigit9}% +\expandafter\def\csname pgffltE@.\endcsname{\pgfflt@finish@number@after@period}% +\expandafter\def\csname pgffltE@,\endcsname{\pgfmathfloat@frenchinput{\pgfflt@finish@number@after@period}}% +\def\pgffltE@e{\pgfflt@readexponent}% +\let\pgffltE@E=\pgffltE@e +\expandafter\def\csname pgffltE@\pgfflt@EOI\endcsname{}% +\def\pgffltE@nonzerodigit#1{% + \advance\pgfmathfloat@a@E by1 + \ifpgfmathfloatparsenumberpendingperiod + \pgfmathfloat@a@Mtok=\expandafter{\the\pgfmathfloat@a@Mtok.}% + \pgfmathfloatparsenumberpendingperiodfalse + \fi + \pgfmathfloat@a@Mtok=\expandafter\expandafter\expandafter{\expandafter\the\expandafter\pgfmathfloat@a@Mtok\the\pgfmathfloat@tmptoks}% + \pgfmathfloat@tmptoks={}% + \pgfmathfloat@a@Mtok=\expandafter{\the\pgfmathfloat@a@Mtok#1}% + \pgfflt@positiveexp}% + +\def\pgfflt@finish@number@after@period#1{% + \expandafter\ifx\csname pgffltF@#1\endcsname\relax + \expandafter\pgfflt@error\expandafter#1% + \else + \expandafter\expandafter\csname pgffltF@#1\endcsname + \fi} +\expandafter\def\csname pgffltF@0\endcsname{% + \pgfmathfloat@tmptoks=\expandafter{\the\pgfmathfloat@tmptoks0}% + \pgfflt@pendingzeros}% +\expandafter\def\csname pgffltF@1\endcsname{\pgffltF@nonzerodigit1}% +\expandafter\def\csname pgffltF@2\endcsname{\pgffltF@nonzerodigit2}% +\expandafter\def\csname pgffltF@3\endcsname{\pgffltF@nonzerodigit3}% +\expandafter\def\csname pgffltF@4\endcsname{\pgffltF@nonzerodigit4}% +\expandafter\def\csname pgffltF@5\endcsname{\pgffltF@nonzerodigit5}% +\expandafter\def\csname pgffltF@6\endcsname{\pgffltF@nonzerodigit6}% +\expandafter\def\csname pgffltF@7\endcsname{\pgffltF@nonzerodigit7}% +\expandafter\def\csname pgffltF@8\endcsname{\pgffltF@nonzerodigit8}% +\expandafter\def\csname pgffltF@9\endcsname{\pgffltF@nonzerodigit9}% +\def\pgffltF@e{\pgfflt@readexponent}% +\let\pgffltF@E=\pgffltF@e +\expandafter\def\csname pgffltF@\pgfflt@EOI\endcsname{}% +\def\pgffltF@nonzerodigit#1{% + \ifpgfmathfloatparsenumberpendingperiod + \pgfmathfloat@a@Mtok=\expandafter{\the\pgfmathfloat@a@Mtok.}% + \pgfmathfloatparsenumberpendingperiodfalse + \fi + \pgfmathfloat@a@Mtok=\expandafter\expandafter\expandafter{\expandafter\the\expandafter\pgfmathfloat@a@Mtok\the\pgfmathfloat@tmptoks}% + \pgfmathfloat@tmptoks={}% + \pgfmathfloat@a@Mtok=\expandafter{\the\pgfmathfloat@a@Mtok#1}% + \pgfflt@finish@number@after@period}% + + +\def\pgfflt@pendingzeros#1{% + \expandafter\ifx\csname pgffltG@#1\endcsname\relax + \expandafter\pgfflt@error\expandafter#1% + \else + \expandafter\expandafter\csname pgffltG@#1\endcsname + \fi} +\expandafter\def\csname pgffltG@0\endcsname{% + \pgfmathfloat@tmptoks=\expandafter{\the\pgfmathfloat@tmptoks0}% + \pgfflt@pendingzeros}% +\expandafter\def\csname pgffltG@1\endcsname{\pgffltG@nonzerodigit1}% +\expandafter\def\csname pgffltG@2\endcsname{\pgffltG@nonzerodigit2}% +\expandafter\def\csname pgffltG@3\endcsname{\pgffltG@nonzerodigit3}% +\expandafter\def\csname pgffltG@4\endcsname{\pgffltG@nonzerodigit4}% +\expandafter\def\csname pgffltG@5\endcsname{\pgffltG@nonzerodigit5}% +\expandafter\def\csname pgffltG@6\endcsname{\pgffltG@nonzerodigit6}% +\expandafter\def\csname pgffltG@7\endcsname{\pgffltG@nonzerodigit7}% +\expandafter\def\csname pgffltG@8\endcsname{\pgffltG@nonzerodigit8}% +\expandafter\def\csname pgffltG@9\endcsname{\pgffltG@nonzerodigit9}% +\def\pgffltG@e{\pgfflt@readexponent}% +\let\pgffltG@E=\pgffltG@e +\expandafter\def\csname pgffltG@\pgfflt@EOI\endcsname{}% +\def\pgffltG@nonzerodigit#1{% + \ifpgfmathfloatparsenumberpendingperiod + \pgfmathfloat@a@Mtok=\expandafter{\the\pgfmathfloat@a@Mtok.}% + \pgfmathfloatparsenumberpendingperiodfalse + \fi + \pgfmathfloat@a@Mtok=\expandafter\expandafter\expandafter{\expandafter\the\expandafter\pgfmathfloat@a@Mtok\the\pgfmathfloat@tmptoks}% + \pgfmathfloat@tmptoks={}% + \pgfmathfloat@a@Mtok=\expandafter{\the\pgfmathfloat@a@Mtok#1}% + \pgfflt@finish@number@after@period}% + +\endinput diff --git a/preprints/A-polynomial-time-algorithm-for-three-machine-unit-job-scheduling-September-24-2026/build/tex/runtime/generic/pgf/math/pgfmathfunctions.base.code.tex b/preprints/A-polynomial-time-algorithm-for-three-machine-unit-job-scheduling-September-24-2026/build/tex/runtime/generic/pgf/math/pgfmathfunctions.base.code.tex new file mode 100644 index 0000000000000000000000000000000000000000..8fb945b74d100dd15e088c35f1e8d3f1effec25e --- /dev/null +++ b/preprints/A-polynomial-time-algorithm-for-three-machine-unit-job-scheduling-September-24-2026/build/tex/runtime/generic/pgf/math/pgfmathfunctions.base.code.tex @@ -0,0 +1,211 @@ + +\pgfmathdeclarefunction{bin}{1}{% + \begingroup% + \afterassignment\pgfmath@gobbletilpgfmath@% + \c@pgfmath@counta=#1\relax\pgfmath@% + \expandafter\pgfmathbasetobase\expandafter\pgfmathresult\expandafter{\the\c@pgfmath@counta}{10}{2}% + \pgfmath@smuggleone\pgfmathresult% + \endgroup% +} +\pgfmathdeclarefunction{hex}{1}{% + \begingroup% + \afterassignment\pgfmath@gobbletilpgfmath@% + \c@pgfmath@counta=#1\relax\pgfmath@% + \expandafter\pgfmathbasetobase\expandafter\pgfmathresult\expandafter{\the\c@pgfmath@counta}{10}{16}% + \pgfmath@smuggleone\pgfmathresult% + \endgroup% +} + +\pgfmathdeclarefunction{Hex}{1}{% + \begingroup% + \afterassignment\pgfmath@gobbletilpgfmath@% + \c@pgfmath@counta=#1\relax\pgfmath@% + \expandafter\pgfmathbasetoBase\expandafter\pgfmathresult\expandafter{\the\c@pgfmath@counta}{10}{16}% + \pgfmath@smuggleone\pgfmathresult% + \endgroup% +} + +\pgfmathdeclarefunction{oct}{1}{% + \begingroup% + \afterassignment\pgfmath@gobbletilpgfmath@% + \c@pgfmath@counta=#1\relax\pgfmath@% + \expandafter\pgfmathbasetobase\expandafter\pgfmathresult\expandafter{\the\c@pgfmath@counta}{10}{8}% + \pgfmath@smuggleone\pgfmathresult% + \endgroup% +} + + +\def\pgfmathbasetodec#1#2#3{% + \pgfmath@checknumber{#2}% + \pgfmath@checkbase{#3}% + \def\pgfmath@base{#3}% + \def\pgfmath@base@macro{#1}% + \c@pgfmath@counta=1\relax% + \let\pgfmath@reversed=\pgfmath@empty% + \expandafter\pgfmathbasetodec@#2\pgfmath@base@stop} + +\def\pgfmathbasetodec@#1{% + \ifx#1\pgfmath@base@stop% + \c@pgfmath@countb=0% + \divide\c@pgfmath@counta by\pgfmath@base\relax% + \expandafter\pgfmathbasetodec@@% + \else% + \edef\pgfmath@reversed{\pgfmath@reversed#1}% + \expandafter\multiply\expandafter\c@pgfmath@counta\pgfmath@base\relax% + \expandafter\pgfmathbasetodec@% + \fi} + +\def\pgfmathbasetodec@@{\expandafter\pgfmathbasetodec@@@\pgfmath@reversed\pgfmathbasetodec} + +\def\pgfmathbasetodec@@@#1{% + \ifx\pgfmathbasetodec#1\relax% + \expandafter\edef\pgfmath@base@macro{\the\c@pgfmath@countb}% + \let\pgfmath@next=\pgfmathbasetodec@@@@% + \else% + \chardef\pgfmath@charnum`#1\relax% + \c@pgfmath@countc=\pgfmath@charnum\relax% + \ifnum\c@pgfmath@countc>96\relax% + \advance\c@pgfmath@countc by-87\relax% + \else + \ifnum\c@pgfmath@countc>64\relax% + \advance\c@pgfmath@countc by-55\relax% + \else + \advance\c@pgfmath@countc by-48\relax% + \fi% + \fi% + \ifnum\c@pgfmath@countc<\pgfmath@base\relax% + \multiply\c@pgfmath@countc by\c@pgfmath@counta% + \advance\c@pgfmath@countb by\c@pgfmath@countc% + \divide\c@pgfmath@counta by\pgfmath@base\relax% + \let\pgfmath@next\pgfmathbasetodec@@@% + \else + \pgfmath@error{Digit `#1' invalid for base \pgfmath@base}{}% + \let\pgfmath@next=\relax% + \fi% + \fi% + \pgfmath@next} +\def\pgfmathbasetodec@@@@{% + \expandafter\pgfmath@ensurenumberlength\expandafter{\pgfmath@base@macro}% +} + +\def\pgfmathdectobase#1#2#3{% + \pgfmath@checknumber{#2}% + \pgfmath@checkbase{#3}% + \c@pgfmath@counta=#2\relax% + \let#1=\pgfmath@empty% + \pgfmathloop + \ifnum\c@pgfmath@counta>0\relax% + \c@pgfmath@countb=\c@pgfmath@counta% + \divide\c@pgfmath@countb by#3\relax% + \multiply\c@pgfmath@countb by-#3\relax% + \advance\c@pgfmath@countb by\c@pgfmath@counta% + \edef#1{\csname pgfmath@lowercase digit@\the\c@pgfmath@countb\endcsname#1}% + \divide\c@pgfmath@counta by#3\relax% + \repeatpgfmathloop% + \pgfmath@ensurenumberlength{#1}} + +\def\pgfmathdectoBase#1#2#3{% + \pgfmath@checkbase{#3}% + \pgfmath@checknumber{#2}% + \c@pgfmath@counta=#2\relax% + \let#1=\pgfmath@empty% + \pgfmathloop + \ifnum\c@pgfmath@counta>0\relax% + \c@pgfmath@countb=\c@pgfmath@counta% + \divide\c@pgfmath@countb by#3\relax% + \multiply\c@pgfmath@countb by-#3\relax% + \advance\c@pgfmath@countb by\c@pgfmath@counta% + \edef#1{\csname pgfmath@uppercase digit@\the\c@pgfmath@countb\endcsname#1}% + \divide\c@pgfmath@counta by#3\relax% + \repeatpgfmathloop% + \pgfmath@ensurenumberlength{#1}} + +\def\pgfmath@base@digits@create{% + \def\pgfmath@base@digit@style{lowercase digit}% + \c@pgfmath@counta=0\relax% + \pgfmath@base@digits@create@0123456789abcdefghijklmnopqrstuvwxyz\pgfmath@base@digits@create% + \def\pgfmath@base@digit@style{uppercase digit}% + \c@pgfmath@counta=0\relax% + \pgfmath@base@digits@create@0123456789ABCDEFGHIJKLMNOPQRSTUVWXYZ\pgfmath@base@digits@create} + +\def\pgfmath@base@digits@create@#1{% + \ifx\pgfmath@base@digits@create#1\relax% + \else% + \expandafter\pgfmath@def\expandafter{\pgfmath@base@digit@style}{\the\c@pgfmath@counta}{#1}% + \advance\c@pgfmath@counta by1\relax% + \expandafter\pgfmath@base@digits@create@% + \fi} +\pgfmath@base@digits@create + +\def\pgfmathbasetobase#1#2#3#4{% + \pgfmathbasetodec{\pgfmath@base@temp}{#2}{#3}% + \pgfmathdectobase{#1}{\pgfmath@base@temp}{#4}% +} + +\def\pgfmathbasetoBase#1#2#3#4{% + \pgfmathbasetodec{\pgfmath@base@temp}{#2}{#3}% + \pgfmathdectoBase{#1}{\pgfmath@base@temp}{#4}% +} + +\def\pgfmath@checkbase#1{% + \ifnum#1<2\relax% + \pgfmath@error{Cannot process numbers in base `#1'.}{}% + \else% + \ifnum#1>36\relax% + \pgfmath@error{Cannot process numbers in base `#1'.}{}% + \fi% + \fi} + +\def\pgfmath@checknumber#1{% + \expandafter\pgfmath@checknumber@#1\pgfmath@} +\def\pgfmath@checknumber@#1#2\pgfmath@{% + \ifx#1-% + \pgfmath@error{Cannot process negative numbers.}{}% + \fi} + +\def\pgfmath@ensurenumberlength#1{% + \def\pgfmath@base@tempa{#1}% + \expandafter\c@pgfmath@counta\pgfmath@basenumberlength\relax% + \expandafter\pgfmath@ensurenumberlength@#1\pgfmath@ensurenumberlength} +\def\pgfmath@ensurenumberlength@#1{% + \ifx\pgfmath@ensurenumberlength#1\relax% + \expandafter\pgfmath@ensurenumberlength@@% + \else% + \advance\c@pgfmath@counta by-1\relax% + \expandafter\pgfmath@ensurenumberlength@ + \fi} + +\def\pgfmath@ensurenumberlength@@{% + \edef\pgfmath@base@tempb{\pgfmath@base@tempa}% + \pgfmath@ensurenumberlength@@@} +\def\pgfmath@ensurenumberlength@@@{% + \ifnum\c@pgfmath@counta>0\relax% + \advance\c@pgfmath@counta by-1\relax% + \edef\pgfmath@base@tempb{0\pgfmath@base@tempb}% + \expandafter\pgfmath@ensurenumberlength@@@% + \fi% + \expandafter\edef\pgfmath@base@tempa{\pgfmath@base@tempb}} + +\def\pgfmathsetbasenumberlength#1{\def\pgfmath@basenumberlength{#1}} +\pgfmathsetbasenumberlength{1}% + +\def\pgfmathtodigitlist#1#2{% + \def\pgfmath@temp{#1}% + \let\pgfmath@digitlist=\pgfutil@empty% + \edef\pgfmath@@tmp{#2}% + \expandafter\pgfmath@todigitlist\pgfmath@@tmp @% +} +\def\pgfmath@base@atchar{@} +\def\pgfmath@todigitlist#1{% + \def\pgfmath@digit{#1}% + \ifx\pgfmath@digit\pgfmath@base@atchar% + \expandafter\let\pgfmath@temp=\pgfmath@digitlist% + \else% + \ifx\pgfmath@digitlist\pgfutil@empty% + \edef\pgfmath@digitlist{#1}% + \else% + \edef\pgfmath@digitlist{\pgfmath@digitlist,#1}% + \fi% + \expandafter\pgfmath@todigitlist% + \fi% +} diff --git a/preprints/A-polynomial-time-algorithm-for-three-machine-unit-job-scheduling-September-24-2026/build/tex/runtime/generic/pgf/math/pgfmathfunctions.basic.code.tex b/preprints/A-polynomial-time-algorithm-for-three-machine-unit-job-scheduling-September-24-2026/build/tex/runtime/generic/pgf/math/pgfmathfunctions.basic.code.tex new file mode 100644 index 0000000000000000000000000000000000000000..f4bdd66875e0ef196152ce416ddf322c3274c97c --- /dev/null +++ b/preprints/A-polynomial-time-algorithm-for-three-machine-unit-job-scheduling-September-24-2026/build/tex/runtime/generic/pgf/math/pgfmathfunctions.basic.code.tex @@ -0,0 +1,726 @@ + + +\pgfmathdeclarefunction{add}{2}{% + \begingroup% + \pgfmath@x=#1pt\relax% + \pgfmath@y=#2pt\relax% + \advance\pgfmath@x by\pgfmath@y% + \pgfmath@returnone\pgfmath@x% + \endgroup% +} + + +\pgfmathdeclarefunction{subtract}{2}{% + \begingroup% + \pgfmath@x=#1pt\relax% + \pgfmath@y=#2pt\relax% + \advance\pgfmath@x by-\pgfmath@y% + \pgfmath@returnone\pgfmath@x% + \endgroup% +} + +\pgfmathdeclarefunction{neg}{1}{% + \begingroup% + \pgfmath@x=-#1pt\relax% + \edef\pgfmathresult{\pgfmath@tonumber{\pgfmath@x}}% + \pgfmath@smuggleone\pgfmathresult% + \endgroup% +} +\def\pgfmath@operation@neg@precedence{500} + +\pgfmathdeclarefunction{multiply}{2}{% + \begingroup% + \pgfmath@x=#1pt\relax% + \pgfmath@y=#2pt\relax% + \pgfmath@x=\pgfmath@tonumber{\pgfmath@y}\pgfmath@x% + \pgfmath@returnone\pgfmath@x% + \endgroup% +} + +\newif\ifpgfmath@divide@period + +\pgfmathdeclarefunction{divide}{2}{% + \begingroup% + \pgfmath@x=#1pt\relax% + \pgfmath@y=#2pt\relax% + \let\pgfmath@sign=\pgfmath@empty% + \ifdim0pt=\pgfmath@y% + \pgfmath@error{You've asked me to divide `#1' by `#2', % + but I cannot divide any number by `#2'}{}% + \fi% + \afterassignment\pgfmath@xa% + \c@pgfmath@counta\the\pgfmath@y\relax% + \ifdim0pt=\pgfmath@xa% + \divide\pgfmath@x by\c@pgfmath@counta% + \else% + \ifdim0pt>\pgfmath@x% + \def\pgfmath@sign{-}% + \pgfmath@x=-\pgfmath@x% + \fi% + \ifdim0pt>\pgfmath@y% + \expandafter\def\expandafter\pgfmath@sign\expandafter{\pgfmath@sign-}% + \pgfmath@y=-\pgfmath@y% + \fi% + \ifdim1pt>\pgfmath@y% + \pgfmathreciprocal@{\pgfmath@tonumber{\pgfmath@y}}% + \pgfmath@x=\pgfmath@sign\pgfmathresult\pgfmath@x% + \else% + \def\pgfmathresult{0}% + \pgfmath@divide@periodtrue% + \c@pgfmath@counta=0\relax% + \pgfmathdivide@@% + \pgfmath@x=\pgfmath@sign\pgfmathresult pt\relax% + \fi% + \fi% + \pgfmath@returnone\pgfmath@x% + \endgroup% +} + +\def\pgfmath@small@number{0.00002} + +\def\pgfmathdivide@@{% + \let\pgfmath@next=\relax% + \ifdim\pgfmath@small@number pt<\pgfmath@x% + \ifdim\pgfmath@small@number pt<\pgfmath@y% + \ifdim\pgfmath@y>\pgfmath@x% + \ifpgfmath@divide@period% + \expandafter\def\expandafter\pgfmathresult\expandafter{\pgfmathresult.}% + \pgfmath@divide@periodfalse% + \fi% + \pgfmathdivide@dimenbyten\pgfmath@y% + \ifdim\pgfmath@y>\pgfmath@x% + \expandafter\def\expandafter\pgfmathresult\expandafter{\pgfmathresult0}% + \fi% + \else% + \c@pgfmath@counta=\pgfmath@x% + \c@pgfmath@countb=\pgfmath@y% + \divide\c@pgfmath@counta by\c@pgfmath@countb% + \pgfmath@ya=\c@pgfmath@counta\pgfmath@y% + \advance\pgfmath@x by-\pgfmath@ya% + \def\pgfmath@next{% + \toks0=\expandafter{\pgfmathresult}% + \edef\pgfmathresult{\the\toks0 \the\c@pgfmath@counta}% + }% + \ifpgfmath@divide@period + \else + \ifnum\c@pgfmath@counta>9 % + \expandafter\pgfmathdivide@advance@last@digit\pgfmathresult CCCCC\@@ + \advance\c@pgfmath@counta by-10 % + \ifnum\c@pgfmath@counta=0 + \let\pgfmath@next=\relax + \fi + \fi + \fi + \pgfmath@next + \fi% + \let\pgfmath@next=\pgfmathdivide@@% + \fi% + \fi% + \pgfmath@next% +} + +\def\pgfmathdivide@advance@last@digit#1.#2#3#4#5#6#7\@@{% + \pgfmath@ya=\pgfmathresult pt % + \if#2C% + \pgfmath@xa=1pt % + \else + \if#3C% + \pgfmath@xa=0.1pt % + \else + \if#4C% + \pgfmath@xa=0.01pt % + \else + \if#5C% + \pgfmath@xa=0.001pt % + \else + \if#6C% + \pgfmath@xa=0.0001pt % + \else + \pgfmath@xa=0.00001pt % + \fi + \fi + \fi + \fi + \fi + \advance\pgfmath@ya by\pgfmath@xa + \edef\pgfmathresult{\pgfmath@tonumber@notrailingzero\pgfmath@ya}% +}% + +{ +\catcode`\p=12 +\catcode`\t=12 +\gdef\Pgf@geT@NO@TRAILING@ZERO#1.#2pt{% + #1.% + \ifnum#2=0 \else #2\fi +} +} +\def\pgfmath@tonumber@notrailingzero#1{\expandafter\Pgf@geT@NO@TRAILING@ZERO\the#1} + + +\def\pgfmathdivide@dimenbyten#1{% + \edef\pgfmath@temp{\pgfmath@tonumber{#1}}% + \expandafter\pgfmathdivide@@dimenbyten\pgfmath@temp\@@#1\@@} +\def\pgfmathdivide@@dimenbyten#1.#2\@@#3\@@{% + \pgfmath@tempcnta=#1\relax% + \divide\pgfmath@tempcnta by10\relax% + \pgfmath@tempcntb=\pgfmath@tempcnta% + \multiply\pgfmath@tempcnta by-10\relax% + \advance\pgfmath@tempcnta by#1\relax% + #3=\the\pgfmath@tempcntb.\the\pgfmath@tempcnta#2pt\relax% +} + + +\pgfmathdeclarefunction{reciprocal}{1}{% + \begingroup% + \expandafter\pgfmath@x#1pt\relax% + \ifdim\pgfmath@x=0pt\relax% + \pgfmath@error{You asked me to calculate `1/#1', but I cannot divide any number by zero}{}% + \fi% + \edef\pgfmath@reciprocaltemp{\pgfmath@tonumber{\pgfmath@x}}% + \expandafter\pgfmathreciprocal@@\pgfmath@reciprocaltemp0000000\pgfmath@} +\def\pgfmathreciprocal@@#1.#2#3#4#5#6#7\pgfmath@{% + \c@pgfmath@counta#2#3#4#5#6\relax% + \ifnum\c@pgfmath@counta=0\relax% + \pgfmath@x1pt\relax% + \divide\pgfmath@x#1\relax% + \else% + \ifnum#1>100\relax% + \c@pgfmath@counta#1#2#3#4\relax% + \c@pgfmath@countb1000000000\relax% + \divide\c@pgfmath@countb\c@pgfmath@counta% + \c@pgfmath@counta\c@pgfmath@countb% + \divide\c@pgfmath@counta10000\relax% + \pgfmath@x\c@pgfmath@counta pt\relax% + \multiply\c@pgfmath@counta-10000\relax% + \advance\c@pgfmath@countb\c@pgfmath@counta% + \pgfmath@y\c@pgfmath@countb pt\relax% + \divide\pgfmath@y1000000\relax% + \advance\pgfmath@x\pgfmath@y% + \else% + \c@pgfmath@counta#1#2#3#4#5#6\relax% + \c@pgfmath@countb1000000000\relax% + \divide\c@pgfmath@countb\c@pgfmath@counta% + \c@pgfmath@counta\c@pgfmath@countb% + \divide\c@pgfmath@counta10000\relax% + \pgfmath@x\c@pgfmath@counta pt\relax% + \multiply\c@pgfmath@counta-10000\relax% + \advance\c@pgfmath@countb\c@pgfmath@counta% + \pgfmath@y\c@pgfmath@countb pt\relax% + \pgfmath@y.1\pgfmath@y% + \pgfmath@y.1\pgfmath@y% + \pgfmath@y.1\pgfmath@y% + \pgfmath@y.1\pgfmath@y% + \advance\pgfmath@x\pgfmath@y% + \fi% + \fi% + \pgfmath@returnone\pgfmath@x% + \endgroup% +} + +\pgfmathdeclarefunction{div}{2}{% + \begingroup% + \pgfmathdivide@{#1}{#2}% + \ifdim\pgfmathresult pt<0pt\relax% + \pgfmathceil@{\pgfmathresult}% + \else% + \pgfmathfloor@{\pgfmathresult}% + \fi% + \afterassignment\pgfmath@gobbletilpgfmath@% + \c@pgfmath@counta=\pgfmathresult\relax\pgfmath@% + \edef\pgfmathresult{\the\c@pgfmath@counta}% + \pgfmath@smuggleone\pgfmathresult% + \endgroup% +} + +\pgfmathdeclarefunction{mod}{2}{% + \begingroup% + \pgfmath@xa#1pt% + \pgfmath@xb#2pt% + \pgfmath@x\pgfmath@xa% + \c@pgfmath@counta\pgfmath@xa% + \c@pgfmath@countb\pgfmath@xb% + \divide\c@pgfmath@counta\c@pgfmath@countb% + \multiply\pgfmath@xb\c@pgfmath@counta% + \advance\pgfmath@x-\pgfmath@xb% + \pgfmath@returnone\pgfmath@x% + \endgroup% +} + +\pgfmathdeclarefunction{Mod}{2}{% + \begingroup% + \pgfmathmod@{#1}{#2}% + \pgfmath@x=\pgfmathresult pt\relax% + \ifdim\pgfmath@x<0pt\relax% + \advance\pgfmath@x by#2pt\relax% + \fi% + \edef\pgfmathresult{\pgfmath@tonumber{\pgfmath@x}}% + \pgfmath@smuggleone\pgfmathresult% + \endgroup% +} + +\pgfmathdeclarefunction{abs}{1}{% + \begingroup% + \expandafter\pgfmath@x#1pt\relax% + \ifdim\pgfmath@x<0pt\relax% + \pgfmath@x-\pgfmath@x% + \fi% + \pgfmath@returnone\pgfmath@x% + \endgroup% +} + +\pgfmathdeclarefunction{sign}{1}{% + \begingroup + \ifdim#1pt>0sp % + \def\pgfmathresult{1}% + \else + \ifdim#1pt=0sp % + \def\pgfmathresult{0}% + \else + \def\pgfmathresult{-1}% + \fi + \fi + \pgfmath@smuggleone\pgfmathresult + \endgroup +} + +\pgfmathdeclarefunction{e}{0}{\def\pgfmathresult{2.718281828}} + +\pgfmathdeclarefunction{ln}{1}{% + \begingroup% + \expandafter\pgfmath@x#1pt\relax% + \ifdim\pgfmath@x>0pt\else% + \pgfmath@error{I cannot calculate the logarithm of `#1'}{}% + \fi% + \c@pgfmath@counta=0\relax% + \ifdim\pgfmath@x>2pt\relax% + \ifdim\pgfmath@x<128pt\relax% + \ifdim\pgfmath@x<8pt\relax% + \ifdim\pgfmath@x<4pt\relax% + \divide\pgfmath@x by2\relax% + \c@pgfmath@counta=1\relax% + \else% + \divide\pgfmath@x by4\relax% + \c@pgfmath@counta=2\relax% + \fi% + \else% + \ifdim\pgfmath@x<32pt\relax% + \ifdim\pgfmath@x<16pt\relax% + \divide\pgfmath@x by8\relax% + \c@pgfmath@counta=3\relax% + \else% + \divide\pgfmath@x by16\relax% + \c@pgfmath@counta=4\relax% + \fi% + \else% + \ifdim\pgfmath@x<64pt\relax% + \divide\pgfmath@x by32\relax% + \c@pgfmath@counta=5\relax% + \else% + \divide\pgfmath@x by64\relax% + \c@pgfmath@counta=6\relax% + \fi% + \fi% + \fi% + \else% + \ifdim\pgfmath@x<2048pt\relax% + \ifdim\pgfmath@x<512pt\relax% + \ifdim\pgfmath@x<256pt\relax% + \divide\pgfmath@x by128\relax% + \c@pgfmath@counta=7\relax% + \else% + \divide\pgfmath@x by256\relax% + \c@pgfmath@counta=8\relax% + \fi% + \else% + \ifdim\pgfmath@x<1024pt\relax% + \divide\pgfmath@x by512\relax% + \c@pgfmath@counta=9\relax% + \else% + \divide\pgfmath@x by1024\relax% + \c@pgfmath@counta=10\relax% + \fi% + \fi% + \else% + \ifdim\pgfmath@x<8192pt\relax% + \ifdim\pgfmath@x<4096pt\relax% + \divide\pgfmath@x by2048\relax% + \c@pgfmath@counta=11\relax% + \else% + \divide\pgfmath@x by4096\relax% + \c@pgfmath@counta =12\relax% + \fi% + \else% + \divide\pgfmath@x by8192\relax% + \c@pgfmath@counta=13\relax% + \fi% + \fi% + \fi% + \else% + \ifdim\pgfmath@x<1pt\relax% + \ifdim\pgfmath@x>0.0078125pt\relax% + \ifdim\pgfmath@x>0.125pt\relax% + \ifdim\pgfmath@x>0.5pt\relax% + \multiply\pgfmath@x by2\relax% + \c@pgfmath@counta=-1\relax% + \else% + \multiply\pgfmath@x by4\relax% + \c@pgfmath@counta=-2\relax% + \fi% + \else% + \ifdim\pgfmath@x>0.03125pt\relax% + \ifdim\pgfmath@x>0.0625pt\relax% + \multiply\pgfmath@x by8\relax% + \c@pgfmath@counta=-3\relax% + \else% + \multiply\pgfmath@x by16\relax% + \c@pgfmath@counta=-4\relax% + \fi% + \else% + \ifdim\pgfmath@x>0.015625pt\relax% + \multiply\pgfmath@x by32\relax% + \c@pgfmath@counta=-5\relax% + \else% + \multiply\pgfmath@x by64\relax% + \c@pgfmath@counta=-6\relax% + \fi% + \fi% + \fi% + \else% + \ifdim\pgfmath@x>0.00048828125pt\relax% + \ifdim\pgfmath@x>0.001953125pt\relax% + \ifdim\pgfmath@x>0.00390625pt\relax% + \multiply\pgfmath@x by128\relax% + \c@pgfmath@counta=-7\relax% + \else% + \multiply\pgfmath@x by256\relax% + \c@pgfmath@counta=-8\relax% + \fi% + \else% + \ifdim\pgfmath@x>0.0009765625pt\relax% + \multiply\pgfmath@x by512\relax% + \c@pgfmath@counta=-9\relax% + \else% + \multiply\pgfmath@x by1024\relax% + \c@pgfmath@counta=-10\relax% + \fi% + \fi% + \else% + \ifdim\pgfmath@x>0.0001220703125pt\relax% + \ifdim\pgfmath@x>0.0002441406256pt\relax% + \multiply\pgfmath@x by2048\relax% + \c@pgfmath@counta=-11\relax% + \else% + \multiply\pgfmath@x by4096\relax% + \c@pgfmath@counta=-12\relax% + \fi% + \else% + \multiply\pgfmath@x by8192\relax% + \c@pgfmath@counta=-13\relax% + \fi% + \fi% + \fi% + \fi% + \fi% + \edef\pgfmath@temp{\pgfmath@tonumber{\pgfmath@x}}% + \pgfmath@x=0.04300521312\pgfmath@x% + \advance\pgfmath@x by-.4025153233pt\relax% + \pgfmath@x=\pgfmath@temp\pgfmath@x% + \advance\pgfmath@x by1.589480044pt\relax% + \pgfmath@x=\pgfmath@temp\pgfmath@x% + \advance\pgfmath@x by-3.489886985pt\relax% + \pgfmath@x=\pgfmath@temp\pgfmath@x% + \advance\pgfmath@x by5.047861502pt\relax% + \pgfmath@x=\pgfmath@temp\pgfmath@x% + \advance\pgfmath@x by-2.787927935pt\relax% + \advance\pgfmath@x by\c@pgfmath@counta pt\relax% + \pgfmath@x=0.6931471806\pgfmath@x% + \pgfmath@returnone\pgfmath@x% + \endgroup% +} + +\pgfmathdeclarefunction{log2}{1}{% + \begingroup% + \pgfmathln@{#1}% + \pgfmath@x=\pgfmathresult pt\relax% + \pgfmath@x=1.442695041\pgfmath@x% + \pgfmath@returnone\pgfmath@x% + \endgroup% +} + +\pgfmathdeclarefunction{log10}{1}{% + \begingroup% + \pgfmathln@{#1}% + \pgfmath@x=\pgfmathresult pt\relax% + \pgfmath@x=0.43429441\pgfmath@x% + \pgfmath@returnone\pgfmath@x% + \endgroup% +} + +\expandafter\let\expandafter\pgfmathlogtwo\csname pgfmathlog2\endcsname +\expandafter\let\expandafter\pgfmathlogtwo@\csname pgfmathlog2@\endcsname +\expandafter\let\expandafter\pgfmathlogten\csname pgfmathlog10\endcsname +\expandafter\let\expandafter\pgfmathlogten@\csname pgfmathlog10@\endcsname + +\pgfmathdeclarefunction{exp}{1}{% + \begingroup + \pgfmath@xc=#1pt\relax + \pgfmath@yc=#1pt\relax + \ifdim\pgfmath@xc<-9pt + \pgfmath@x=1sp\relax + \else + \ifdim\pgfmath@xc<0pt + \pgfmath@xc=-\pgfmath@xc + \fi + \pgfmath@x=1pt\relax + \pgfmath@xa=1pt\relax + \pgfmath@xb=\pgfmath@x + \pgfmathloop% + \divide\pgfmath@xa by\pgfmathcounter + \pgfmath@xa=\pgfmath@tonumber\pgfmath@xc\pgfmath@xa% + \advance\pgfmath@x by\pgfmath@xa + \ifdim\pgfmath@x=\pgfmath@xb + \else + \pgfmath@xb=\pgfmath@x + \repeatpgfmathloop% + \ifdim\pgfmath@yc<0pt + \pgfmathreciprocal@{\pgfmath@tonumber\pgfmath@x}% + \pgfmath@x=\pgfmathresult pt\relax + \fi + \fi + \pgfmath@returnone\pgfmath@x% + \endgroup +} + +\pgfmathdeclarefunction{sqrt}{1}{% + \begingroup% + \expandafter\pgfmath@x#1pt\relax% + \ifdim\pgfmath@x<0pt\relax% + \pgfmath@error{I cannot calculate the square-root of the negative number `#1'}{}% + \else% + \ifdim\pgfmath@x<10pt\relax% + \def\pgfmath@zeros{0}% + \def\pgfmath@targetiterations{1}% + \else% + \ifdim\pgfmath@x<100pt\relax% + \def\pgfmath@zeros{}% + \def\pgfmath@targetiterations{1}% + \else% + \ifdim\pgfmath@x<1000pt\relax% + \def\pgfmath@zeros{0}% + \def\pgfmath@targetiterations{2}% + \else% + \ifdim\pgfmath@x<10000pt\relax% + \def\pgfmath@zeros{}% + \def\pgfmath@targetiterations{2}% + \else% + \def\pgfmath@zeros{0}% + \def\pgfmath@targetiterations{3}% + \fi\fi\fi\fi\fi% + \edef\pgfmath@temp{\pgfmath@zeros\pgfmath@tonumber{\pgfmath@x}}% + \expandafter\pgfmath@sqrt@\pgfmath@temp\pgfmath@% +} +\def\pgfmath@sqrt@#1.#2\pgfmath@{\pgfmath@@sqrt@#1#2\pgfmath@empty\pgfmath@empty\pgfmath@} + +\def\pgfmath@@sqrt@#1#2{% + \c@pgfmath@countb=#1#2\relax% + \ifnum\c@pgfmath@countb>35\relax% + \ifnum\c@pgfmath@countb>63\relax% + \ifnum\c@pgfmath@countb>80\relax% + \c@pgfmath@counta=9\relax% + \else% + \c@pgfmath@counta=8\relax% + \fi% + \else% + \ifnum\c@pgfmath@countb>48\relax% + \c@pgfmath@counta=7\relax% + \else% + \c@pgfmath@counta=6\relax% + \fi% + \fi% + \else% + \ifnum\c@pgfmath@countb>15\relax% + \ifnum\c@pgfmath@countb>24\relax% + \c@pgfmath@counta5\relax% + \else% + \c@pgfmath@counta=4\relax% + \fi% + \else% + \ifnum\c@pgfmath@countb>3\relax% + \ifnum\c@pgfmath@countb>8\relax% + \c@pgfmath@counta=3\relax% + \else% + \c@pgfmath@counta=2\relax% + \fi% + \else% + \ifnum\c@pgfmath@countb>0\relax% + \c@pgfmath@counta=1\relax% + \else% + \c@pgfmath@counta=0\relax% + \fi% + \fi% + \fi% + \fi% + \edef\pgfmath@root{\the\c@pgfmath@counta}% + \edef\pgfmath@rootspecial{\the\c@pgfmath@counta}% + \multiply\c@pgfmath@counta by-\c@pgfmath@counta\relax% + \advance\c@pgfmath@counta by#1#2\relax% + \edef\pgfmath@remainder{\the\c@pgfmath@counta}% + \pgfmath@@@sqrt@% +} + +\def\pgfmath@@@sqrt@#1#2{% + \ifx\pgfmath@empty#1% + \edef\pgfmath@remainder{\pgfmath@remainder00}% + \def\pgfmath@tokens{\pgfmath@empty\pgfmath@empty}% + \else% + \ifx\pgfmath@empty#2% + \edef\pgfmath@remainder{\pgfmath@remainder#10}% + \def\pgfmath@tokens{\pgfmath@empty\pgfmath@empty}% + \else% + \edef\pgfmath@remainder{\pgfmath@remainder#1#2}% + \def\pgfmath@tokens{}% + \fi\fi% + \c@pgfmath@counta=\pgfmath@rootspecial\relax% + \multiply\c@pgfmath@counta by20\relax% + \c@pgfmath@countb=\c@pgfmath@counta% + \multiply\c@pgfmath@countb by6\relax% + \advance\c@pgfmath@countb by36\relax% + \c@pgfmath@countc=\c@pgfmath@counta% + \multiply\c@pgfmath@countc by2\relax% + \ifnum\c@pgfmath@countb>\pgfmath@remainder\relax% + \advance\c@pgfmath@countb by-\c@pgfmath@countc% + \advance\c@pgfmath@countb by-20\relax% + \ifnum\c@pgfmath@countb>\pgfmath@remainder\relax% + \advance\c@pgfmath@countb by-\c@pgfmath@countc% + \advance\c@pgfmath@countb by-12\relax% + \ifnum\c@pgfmath@countb>\pgfmath@remainder\relax% + \advance\c@pgfmath@countb by-\c@pgfmath@counta% + \advance\c@pgfmath@countb by-3\relax% + \ifnum\c@pgfmath@countb>\pgfmath@remainder\relax% + \def\pgfmath@digit{0}% + \else% + \def\pgfmath@digit{1}% + \fi% + \else% + \advance\c@pgfmath@countb by\c@pgfmath@counta% + \advance\c@pgfmath@countb by5\relax% + \ifnum\c@pgfmath@countb>\pgfmath@remainder\relax% + \def\pgfmath@digit{2}% + \else% + \def\pgfmath@digit{3}% + \fi% + \fi% + \else% + \advance\c@pgfmath@countb by\c@pgfmath@counta% + \advance\c@pgfmath@countb by9\relax% + \ifnum\c@pgfmath@countb>\pgfmath@remainder\relax% + \def\pgfmath@digit{4}% + \else% + \def\pgfmath@digit{5}% + \fi% + \fi% + \else% + \advance\c@pgfmath@countb by\c@pgfmath@countc% + \advance\c@pgfmath@countb by28\relax% + \ifnum\c@pgfmath@countb>\pgfmath@remainder\relax% + \advance\c@pgfmath@countb by-\c@pgfmath@counta% + \advance\c@pgfmath@countb by-15\relax% + \ifnum\c@pgfmath@countb>\pgfmath@remainder\relax% + \def\pgfmath@digit{6}% + \else% + \def\pgfmath@digit{7}% + \fi% + \else% + \advance\c@pgfmath@countb by\c@pgfmath@counta% + \advance\c@pgfmath@countb by17\relax% + \ifnum\c@pgfmath@countb>\pgfmath@remainder\relax% + \def\pgfmath@digit{8}% + \else% + \def\pgfmath@digit{9}% + \fi% + \fi% + \fi% + \edef\pgfmath@rootspecial{\pgfmath@rootspecial\pgfmath@digit}% + \advance\c@pgfmath@counta by\pgfmath@digit\relax% + \multiply\c@pgfmath@counta by-\pgfmath@digit\relax% + \advance\c@pgfmath@counta by\pgfmath@remainder\relax% + \edef\pgfmath@remainder{\the\c@pgfmath@counta}% + \c@pgfmath@counta=\pgfmath@targetiterations\relax% + \advance\c@pgfmath@counta by-1\relax% + \edef\pgfmath@targetiterations{\the\c@pgfmath@counta}% + \ifnum\c@pgfmath@counta=0\relax% + \edef\pgfmath@root{\pgfmath@root.\pgfmath@digit}% + \else% + \edef\pgfmath@root{\pgfmath@root\pgfmath@digit}% + \fi% + \ifnum\c@pgfmath@counta=-4\relax% + \let\pgfmath@next\pgfmath@sqrt@end% + \else% + \let\pgfmath@next\pgfmath@@@sqrt@% + \fi% + \expandafter\pgfmath@next\pgfmath@tokens% +} + +\def\pgfmath@sqrt@end#1\pgfmath@{% + \edef\pgfmathresult{\pgfmath@root}% + \pgfmath@smuggleone\pgfmathresult% + \endgroup} + +\pgfmathdeclarefunction{pow}{2}{% + \begingroup% + \pgfmath@xa=#1pt% + \pgfmath@xb=#2pt% + \ifdim\pgfmath@xa=0pt\relax% + \ifdim\pgfmath@xb=0pt\relax% + \pgfmath@x=1pt\relax% + \else% + \pgfmath@x=0pt\relax% + \fi% + \else% + \afterassignment\pgfmath@x% + \expandafter\c@pgfmath@counta\the\pgfmath@xb\relax% + \ifnum\c@pgfmath@counta<0\relax% + \c@pgfmath@counta=-\c@pgfmath@counta% + \pgfmathreciprocal@{#1}% + \pgfmath@xa=\pgfmathresult pt\relax% + \fi + \ifdim\pgfmath@x=0pt\relax% + \pgfmath@x=1pt\relax% + \pgfmathloop% + \ifnum\c@pgfmath@counta>0\relax% + \ifodd\c@pgfmath@counta% + \pgfmath@x=\pgfmath@tonumber{\pgfmath@x}\pgfmath@xa% + \fi + \ifnum\c@pgfmath@counta>1\relax% + \pgfmath@xa=\pgfmath@tonumber{\pgfmath@xa}\pgfmath@xa% + \fi% + \divide\c@pgfmath@counta by2\relax% + \repeatpgfmathloop% + \else% + \pgfmathln@{#1}% + \pgfmath@x=\pgfmathresult pt\relax% + \pgfmath@x=\pgfmath@tonumber{\pgfmath@xb}\pgfmath@x% + \pgfmathexp@{\pgfmath@tonumber{\pgfmath@x}}% + \pgfmath@x=\pgfmathresult pt\relax% + \fi% + \fi + \pgfmath@returnone\pgf@x% + \endgroup% +} + +\pgfmathdeclarefunction{factorial}{1}{% + \begingroup% + \afterassignment\pgfmath@gobbletilpgfmath@% + \c@pgfmath@counta=#1\relax\pgfmath@% + \pgfmath@x=1pt\relax% + \pgfmathloop% + \ifnum\c@pgfmath@counta<2\relax% + \else% + \multiply\pgfmath@x by\c@pgfmath@counta% + \advance\c@pgfmath@counta by-1\relax% + \repeatpgfmathloop% + \edef\pgfmathresult{\pgfmath@tonumber{\pgfmath@x}}% + \pgfmath@smuggleone\pgfmathresult% + \endgroup% +} + diff --git a/preprints/A-polynomial-time-algorithm-for-three-machine-unit-job-scheduling-September-24-2026/build/tex/runtime/generic/pgf/math/pgfmathfunctions.code.tex b/preprints/A-polynomial-time-algorithm-for-three-machine-unit-job-scheduling-September-24-2026/build/tex/runtime/generic/pgf/math/pgfmathfunctions.code.tex new file mode 100644 index 0000000000000000000000000000000000000000..08d0b00c7d7966072c28863ff009ae77ad342da1 --- /dev/null +++ b/preprints/A-polynomial-time-algorithm-for-three-machine-unit-job-scheduling-September-24-2026/build/tex/runtime/generic/pgf/math/pgfmathfunctions.code.tex @@ -0,0 +1,235 @@ + +\def\pgfmathdeclarefunction{\pgfmath@ifnextchar*\pgfmathdeclarefunction@force\pgfmathdeclarefunction@check} +\def\pgfmathdeclarefunction@check#1{% + \expandafter\ifx\csname pgfmath@function@#1\endcsname\relax\else% + \pgfmath@error{The function `#1' already exists}{}% + \fi% + \pgfmath@declarefunction{#1}% +} +\def\pgfmathdeclarefunction@force*{\pgfmath@declarefunction}% + +\def\pgfmathredeclarefunction#1#2{% + \expandafter\ifx\csname pgfmath@function@#1\endcsname\relax% + \pgfmath@error{The function `#1' does not exist}{}% + \else% + \edef\pgf@marshal{\noexpand\pgfmathdeclarefunction*{#1}{\csname pgfmath@operation@#1@arity\endcsname}}% + \pgf@marshal{#2}% + \fi% +} + +\def\pgfmath@declarefunction#1#2{% + \pgfmath@namedef{pgfmath@function@#1}{#1}% + \pgfmath@namedef{pgfmath@operation@#1@type}{function}% + \pgfmath@namedef{pgfmath@operation@#1@arity}{#2}% + \pgfmath@namedef{pgfmath@operation@#1@infix}{#1}% + \pgfmath@namedef{pgfmath@operation@#1@precedence}{1000}% + \expandafter\ifx\csname pgfmath@operation@#1@arity\endcsname\pgfmath@dots% + \pgfmath@namedef{pgfmath@operation@#1@arity}{1}% + \else% + \ifnum\csname pgfmath@operation@#1@arity\endcsname>9\relax% + \pgfmath@namedef{pgfmath@operation@#1@arity}{1}% + \fi% + \fi% + \edef\pgfmath@marshal{% + \noexpand\pgfmath@@declarefunction% + {\expandafter\noexpand\csname pgfmath#1\endcsname}% + {\expandafter\noexpand\csname pgfmath#1@\endcsname}% + {\csname pgfmath@operation@#1@arity\endcsname}% + }% + \pgfkeysvalueof{/pgf/declare function/execute at begin function}% + \pgfmath@marshal% +} + +\def\pgfmath@@declarefunction#1#2#3#4{% + \pgfkeysvalueof{/pgf/declare function/execute at end function}% + \ifcase#3\relax% + \def#1{#2}% + \def\pgfmath@@head{\def#2}% + \or + \def#1##1{\pgfmathparse{##1}\expandafter#2\expandafter{\pgfmathresult}}% + \def\pgfmath@@head{\def#2####1}% + \else% + \pgfmath@toks={}% + \pgfmathloop% + \ifnum\pgfmathcounter>#3\relax% + \else% + \expandafter\pgfmath@addto@toks\expandafter{\pgfmath@char@hash}% + \expandafter\pgfmath@addto@toks\expandafter{\pgfmathcounter}% + \repeatpgfmathloop% + \edef\pgfmath@head{\noexpand\def\noexpand#1\the\pgfmath@toks}% + \edef\pgfmath@@head{\noexpand\def\noexpand#2\the\pgfmath@toks}% + \pgfmath@toks={}% + \def\pgfmath@arguments{}% + \pgfmathloop% + \ifnum\pgfmathcounter>#3\relax% + \else% + \pgfmath@addto@toks{\pgfmathparse}% + \expandafter\expandafter\expandafter\pgfmath@addto@toks% + \expandafter\expandafter\expandafter{\expandafter\expandafter\expandafter% + {\expandafter\pgfmath@char@hash\pgfmathcounter}}% + \pgfmath@addto@toks{\let}% + \expandafter\pgfmath@addto@toks\expandafter{% + \csname pgfmath@argument@\pgfmathcounter\endcsname=\pgfmathresult}% + \expandafter\expandafter\expandafter\def\expandafter\expandafter\expandafter% + \pgfmath@arguments\expandafter\expandafter\expandafter{\expandafter% + \pgfmath@arguments\expandafter{\csname pgfmath@argument@\pgfmathcounter\endcsname}}% + \repeatpgfmathloop% + \expandafter\pgfmath@addto@toks\expandafter{\expandafter#2\pgfmath@arguments}% + \edef\pgfmath@body{{\the\pgfmath@toks}}% + \expandafter\pgfmath@head\pgfmath@body% + \fi% + \pgfmath@@head{#4}% +} + +\def\pgfmath@addto@toks#1{% + \edef\pgfmath@toks@expanded{\the\pgfmath@toks}% + \expandafter\pgfmath@toks\expandafter{\pgfmath@toks@expanded#1}} + +\pgfkeys{% + /pgf/declare function/.code={% + \pgfmath@local@functions#1@=@;% + }, + /pgf/declare function/execute at begin function/.initial={}, + /pgf/declare function/execute at end function/.initial={}, + /pgf/declare function/ignore spaces/.is choice, + /pgf/declare function/ignore spaces/.default=true, + /pgf/declare function/ignore spaces/true/.style={% + /pgf/declare function/execute at begin function={% + \begingroup + \catcode`\^^I=9\relax + \catcode`\ =9\relax + \catcode`\~=10\relax + \endlinechar=`\ \relax + }, + /pgf/declare function/execute at end function={% + \endgroup + }, + }, + /pgf/declare function/ignore spaces/false/.style={% + /pgf/declare function/execute at begin function={}, + /pgf/declare function/execute at end function={}, + }, +} + +\def\pgfmath@local@gobbleone#1{} +\def\pgfmath@local@at{@}% + +\def\pgfmath@local@functions{% + \pgfutil@ifnextchar x{\pgfmath@local@@functions}{\pgfmath@local@@functions}% +} + +\def\pgfmath@local@@functions#1=#2;{% + \def\pgfmath@local@temp{#1}% + \ifx\pgfmath@local@temp\pgfmath@local@at% + \let\pgfmath@local@next=\relax% + \else% + \pgfmath@local@function#1=#2;% + \let\pgfmath@local@next=\pgfmath@local@functions% + \fi% + \pgfmath@local@next% +} + +\def\pgfmath@local@function{% + \let\pgfmath@local@name=\pgfutil@empty% + \let\pgfmath@local@args=\pgfutil@empty% + \let\pgfmath@local@body=\pgfutil@empty% + \pgfmath@local@@function} + +\def\pgfmath@local@@function#1{% + \if#1=% + \let\pgfmath@local@next=\pgfmath@local@function@body% + \else% + \if#1(% + \let\pgfmath@local@next=\pgfmath@local@function@args% + \else% + \expandafter\def\expandafter\pgfmath@local@name\expandafter{\pgfmath@local@name#1}% + \let\pgfmath@local@next=\pgfmath@local@@function% + \fi% + \fi% + \pgfmath@local@next% +} + +\def\pgfmath@local@function@args#1){% + \def\pgfmath@local@args{#1}% + \pgfmath@local@@function} + +\def\pgfmath@local@function@body#1;{% + \def\pgfmath@local@body{#1}% + \begingroup% + \c@pgf@counta=0\relax% + \ifx\pgfmath@local@args\pgfmath@empty% + \expandafter\pgfmath@toks\expandafter=\expandafter{\pgfmath@local@body}% + \else% + \pgfmath@toks={}% + \expandafter\pgfmath@local@function@@body\pgfmath@local@args,,% + \fi% + \xdef\pgfmath@local@temp{% + \noexpand\pgfmathnotifynewdeclarefunction{\pgfmath@local@name}{\the\c@pgf@counta}% + {\the\pgfmath@toks}% + }% + \endgroup% + \pgfmath@local@temp% +} + +\def\pgfmathnotifynewdeclarefunction#1#2#3{% + \pgfmathdeclarefunction{#1}{#2}{\pgfmathparse{#3}}% +}% + +\def\pgfmath@local@function@@body#1,{% + \def\pgfmath@local@test{#1}% + \ifx\pgfmath@local@test\pgfutil@empty% + \let\pgfmath@local@next=\relax% + \else% + \edef\pgfmath@local@var@string{@\expandafter\pgfmath@local@gobbleone\string#1}% + \advance\c@pgf@counta by1\relax% + \expandafter\expandafter\expandafter\pgfmath@toks\expandafter\expandafter\expandafter=% + \expandafter\expandafter\expandafter{\expandafter\pgfmath@char@hash\the\c@pgf@counta}% + \edef#1{\the\pgfmath@toks}% + \pgfmath@toks={}% + \expandafter\pgfmath@local@function@@@body\pgfmath@local@body @% + \edef\pgfmath@local@body{\the\pgfmath@toks}% + \let\pgfmath@local@next=\pgfmath@local@function@@body% + \fi% + \pgfmath@local@next% +} + +\def\pgfmath@local@function@@@body{% + \pgfutil@ifnextchar\bgroup{\pgfmath@local@function@@@body@bgroup}% + {\pgfmath@local@function@@@body@nobgroup}}% + +\def\pgfmath@local@function@@@body@bgroup#1{% + \begingroup% + \pgfmath@toks={}% + \pgfmath@local@function@@@body#1 @% + \xdef\pgfmath@local@temp{{\the\pgfmath@toks}}% + \endgroup% + \expandafter\pgfmath@addto@toks\expandafter{\pgfmath@local@temp}% + \pgfmath@local@function@@@body% +} + +\def\pgfmath@local@function@@@body@nobgroup#1{% + \ifx#1@% + \let\pgfmath@local@next=\relax% + \else% + \let\pgfmath@local@next=\pgfmath@local@function@@@body% + \edef\pgfmath@local@var@@string{@\expandafter\pgfmath@local@gobbleone\string#1}% + \ifx\pgfmath@local@var@@string\pgfmath@local@var@string% + \expandafter\pgfmath@addto@toks\expandafter{#1}% + \else% + \edef\pgfmath@local@temp{\the\pgfmath@toks}% + \pgfmath@toks={#1}% + \edef\pgfmath@local@@temp{\the\pgfmath@toks}% + \expandafter\pgfmath@toks\expandafter=\expandafter{\pgfmath@local@temp}% + \expandafter\pgfmath@addto@toks\expandafter{\pgfmath@local@@temp}% + \fi% + \fi% + \pgfmath@local@next% +} + +\def\pgfmathdeclarepseudoconstant#1#2{% + \pgfmathdeclarefunction*{#1}{0}{#2}% +}% +\def\pgfmathredeclarepseudoconstant#1#2{% + \pgfmathredeclarefunction{#1}{#2}% +}% + diff --git a/preprints/A-polynomial-time-algorithm-for-three-machine-unit-job-scheduling-September-24-2026/build/tex/runtime/generic/pgf/math/pgfmathfunctions.comparison.code.tex b/preprints/A-polynomial-time-algorithm-for-three-machine-unit-job-scheduling-September-24-2026/build/tex/runtime/generic/pgf/math/pgfmathfunctions.comparison.code.tex new file mode 100644 index 0000000000000000000000000000000000000000..717c5412b6c9a559627f8e25f964886f6f9117a6 --- /dev/null +++ b/preprints/A-polynomial-time-algorithm-for-three-machine-unit-job-scheduling-September-24-2026/build/tex/runtime/generic/pgf/math/pgfmathfunctions.comparison.code.tex @@ -0,0 +1,148 @@ + + +\pgfmathdeclarefunction{greater}{2}{% + \begingroup% + \ifdim#1pt>#2pt\relax% + \def\pgfmathresult{1}% + \else% + \def\pgfmathresult{0}% + \fi% + \pgfmath@smuggleone\pgfmathresult% + \endgroup% +} +\let\pgfmathgreaterthan=\pgfmathgreater + +\pgfmathdeclarefunction{less}{2}{% + \begingroup% + \ifdim#1pt<#2pt\relax% + \def\pgfmathresult{1}% + \else% + \def\pgfmathresult{0}% + \fi% + \pgfmath@smuggleone\pgfmathresult% + \endgroup% +} +\let\pgfmathlessthan=\pgfmathless + +\pgfmathdeclarefunction{equal}{2}{% + \begingroup% + \ifdim#1pt=#2pt\relax% + \def\pgfmathresult{1}% + \else% + \def\pgfmathresult{0}% + \fi% + \pgfmath@smuggleone\pgfmathresult% + \endgroup% +} +\let\pgfmathequalto=\pgfmathequal + + +\newif\ifpgfmathcomparison +\pgfmathdeclarefunction{approxequalto}{2}{% + \begingroup% + \pgfmath@x=#1pt\relax% + \pgfmath@y=#2pt\relax% + \advance\pgfmath@x by-\pgfmath@y% + \ifdim\pgfmath@x<0pt\relax% + \multiply\pgfmath@x by-1\relax% + \fi + \ifdim\pgfmath@x<0.0001pt\relax% + \def\pgfmathresult{1.0}% + \global\pgfmathcomparisontrue% + \else% + \def\pgfmathresult{0.0}% + \global\pgfmathcomparisonfalse% + \fi% + \pgfmath@smuggleone\pgfmathresult% + \endgroup% +} + + + +\pgfmathdeclarefunction{ifthenelse}{3}{% + \begingroup% + \ifdim#1pt=0.0pt\relax% + \expandafter\def\expandafter\pgfmathresult\expandafter{#3}% + \else% + \expandafter\def\expandafter\pgfmathresult\expandafter{#2}% + \fi% + \pgfmath@smuggleone\pgfmathresult% + \endgroup% +} + +\pgfmathdeclarefunction{notequal}{2}{% + \begingroup% + \ifdim#1pt=#2pt\relax% + \def\pgfmathresult{0}% + \else% + \def\pgfmathresult{1}% + \fi% + \pgfmath@smuggleone\pgfmathresult% + \endgroup% +} + +\pgfmathdeclarefunction{notless}{2}{% + \begingroup% + \ifdim#1pt<#2pt\relax% + \def\pgfmathresult{0}% + \else% + \def\pgfmathresult{1}% + \fi% + \pgfmath@smuggleone\pgfmathresult% + \endgroup% +} + +\pgfmathdeclarefunction{notgreater}{2}{% + \begingroup% + \ifdim#1pt>#2pt\relax% + \def\pgfmathresult{0}% + \else% + \def\pgfmathresult{1}% + \fi% + \pgfmath@smuggleone\pgfmathresult% + \endgroup% +} + +\pgfmathdeclarefunction{and}{2}{% + \begingroup% + \def\pgfmathresult{1}% + \ifdim#1pt=0pt\relax% + \def\pgfmathresult{0}% + \else% + \ifdim#2pt=0pt\relax% + \def\pgfmathresult{0}% + \fi% + \fi% + \pgfmath@smuggleone\pgfmathresult% + \endgroup% +} + +\pgfmathdeclarefunction{or}{2}{% + \begingroup% + \def\pgfmathresult{0}% + \ifdim#1pt=0pt\relax% + \ifdim#2pt=0pt\relax% + \else% + \def\pgfmathresult{1}% + \fi% + \else% + \def\pgfmathresult{1}% + \fi% + \pgfmath@smuggleone\pgfmathresult% + \endgroup% +} + +\pgfmathdeclarefunction{not}{1}{% + \begingroup% + \ifdim#1pt=0pt\relax% + \def\pgfmathresult{1}% + \else% + \def\pgfmathresult{0}% + \fi% + \pgfmath@smuggleone\pgfmathresult% + \endgroup% +} + +\pgfmathdeclarefunction{true}{0}{\def\pgfmathresult{1}} + +\pgfmathdeclarefunction{false}{0}{\def\pgfmathresult{0}} diff --git a/preprints/A-polynomial-time-algorithm-for-three-machine-unit-job-scheduling-September-24-2026/build/tex/runtime/generic/pgf/math/pgfmathfunctions.integerarithmetics.code.tex b/preprints/A-polynomial-time-algorithm-for-three-machine-unit-job-scheduling-September-24-2026/build/tex/runtime/generic/pgf/math/pgfmathfunctions.integerarithmetics.code.tex new file mode 100644 index 0000000000000000000000000000000000000000..56eca18d07fe29b82e1391d63688aa8c6f2f7f43 --- /dev/null +++ b/preprints/A-polynomial-time-algorithm-for-three-machine-unit-job-scheduling-September-24-2026/build/tex/runtime/generic/pgf/math/pgfmathfunctions.integerarithmetics.code.tex @@ -0,0 +1,112 @@ + +\pgfmathdeclarefunction{gcd}{2}{% + \begingroup + \pgfmathcontinuelooptrue + \pgfmath@xa=#1pt + \pgfmath@xb=#2pt + \ifdim\pgfmath@xa=0pt\relax + \pgfmathcontinueloopfalse + \pgfmath@xa=\pgfmath@xb + \fi + \ifdim\pgfmath@xb=0pt\relax + \pgfmathcontinueloopfalse + \pgfmath@xb=\pgfmath@xa + \fi + \ifdim\pgfmath@xa<0pt\relax + \pgfmath@xa=-\pgfmath@xa + \fi + \ifdim\pgfmath@xb<0pt\relax + \pgfmath@xb=-\pgfmath@xb + \fi + \loop + \ifpgfmathcontinueloop + \ifdim\pgfmath@xa=\pgfmath@xb + \pgfmathcontinueloopfalse + \else + \ifdim\pgfmath@xa>\pgfmath@xb + \advance\pgfmath@xa by-\pgfmath@xb + \else + \advance\pgfmath@xb by-\pgfmath@xa + \fi + \fi + \repeat + \pgfmathparse{int(\pgfmath@xa)}% + \pgfmath@smuggleone\pgfmathresult + \endgroup} + +\newif\ifpgfmath@isprime@unknown +\newif\ifpgfmath@isprime + +\pgfmathdeclarefunction{isprime}{1}{% + \begingroup + \pgfmath@isprime@unknowntrue + \pgfmathsetcount\c@pgfmath@counta{abs(int(#1))}% + \ifcase\c@pgfmath@counta\relax + \pgfmath@isprimefalse + \pgfmath@isprime@unknownfalse + \or + \pgfmath@isprimefalse + \pgfmath@isprime@unknownfalse + \or + \pgfmath@isprimetrue + \pgfmath@isprime@unknownfalse + \or + \pgfmath@isprimetrue + \pgfmath@isprime@unknownfalse + \else + \ifodd\c@pgfmath@counta\relax + \else + \pgfmath@isprimefalse + \pgfmath@isprime@unknownfalse + \fi + \fi + \ifpgfmath@isprime@unknown + \c@pgfmath@countd=3 + \pgfmath@isprimetrue + \pgfmathloop + \c@pgfmath@countb=\c@pgfmath@counta + \divide\c@pgfmath@countb by\c@pgfmath@countd + \ifnum\c@pgfmath@countb>\c@pgfmath@countd + \pgfmath@isprime@unknowntrue + \else + \pgfmath@isprime@unknownfalse + \fi + \multiply\c@pgfmath@countb by\c@pgfmath@countd + \ifnum\c@pgfmath@countb=\c@pgfmath@counta + \global\pgfmath@isprimefalse + \pgfmath@isprime@unknownfalse + \fi + \ifpgfmath@isprime@unknown\advance\c@pgfmath@countd by 2 + \repeatpgfmathloop + \fi + \ifpgfmath@isprime + \def\pgfmathresult{1}% + \else + \def\pgfmathresult{0}% + \fi + \pgfmath@smuggleone\pgfmathresult + \endgroup} + +\pgfmathdeclarefunction{isodd}{1}{% + \begingroup + \pgfmathsetcount\c@pgfmath@counta{abs(int(#1))}% + \ifodd\c@pgfmath@counta + \def\pgfmathresult{1}% + \else + \def\pgfmathresult{0}% + \fi + \pgfmath@smuggleone\pgfmathresult + \endgroup} + +\pgfmathdeclarefunction{iseven}{1}{% + \begingroup + \pgfmathsetcount\c@pgfmath@counta{abs(int(#1))}% + \ifodd\c@pgfmath@counta + \def\pgfmathresult{0}% + \else + \def\pgfmathresult{1}% + \fi + \pgfmath@smuggleone\pgfmathresult + \endgroup} + +\endinput diff --git a/preprints/A-polynomial-time-algorithm-for-three-machine-unit-job-scheduling-September-24-2026/build/tex/runtime/generic/pgf/math/pgfmathfunctions.misc.code.tex b/preprints/A-polynomial-time-algorithm-for-three-machine-unit-job-scheduling-September-24-2026/build/tex/runtime/generic/pgf/math/pgfmathfunctions.misc.code.tex new file mode 100644 index 0000000000000000000000000000000000000000..89ce863a178d51d6acd4db105a8a0cead7991a6e --- /dev/null +++ b/preprints/A-polynomial-time-algorithm-for-three-machine-unit-job-scheduling-September-24-2026/build/tex/runtime/generic/pgf/math/pgfmathfunctions.misc.code.tex @@ -0,0 +1,292 @@ + + +\pgfmathdeclarefunction{veclen}{2}{% + \begingroup% + \pgfmath@x#1pt\relax% + \pgfmath@y#2pt\relax% + \ifdim\pgfmath@x<0pt\relax% + \pgfmath@x-\pgfmath@x% + \fi% + \ifdim\pgfmath@y<0pt\relax% + \pgfmath@y-\pgfmath@y% + \fi% + \ifdim\pgfmath@x=0pt\relax% + \pgfmath@x\pgfmath@y% + \else% + \ifdim\pgfmath@y=0pt\relax% + \else% + \ifdim\pgfmath@x>\pgfmath@y% + \pgfmath@xa\pgfmath@x% + \pgfmath@x\pgfmath@y% + \pgfmath@y\pgfmath@xa% + \fi% + \ifdim\pgfmath@y>10000pt\relax% + \c@pgfmath@counta1500\relax% + \else% + \ifdim\pgfmath@y>1000pt\relax% + \c@pgfmath@counta150\relax% + \else% + \ifdim\pgfmath@y>100pt\relax% + \c@pgfmath@counta50\relax% + \else% + \c@pgfmath@counta1\relax% + \fi% + \fi% + \fi% + \divide\pgfmath@x\c@pgfmath@counta\relax% + \divide\pgfmath@y\c@pgfmath@counta\relax% + \pgfmathreciprocal@{\pgfmath@tonumber{\pgfmath@y}}% + \pgfmath@x\pgfmathresult\pgfmath@x% + \pgfmath@xa\pgfmath@tonumber{\pgfmath@x}\pgfmath@x% + \edef\pgfmath@temp{\pgfmath@tonumber{\pgfmath@xa}}% + \pgfmath@x-0.01019210944\pgfmath@xa% + \advance\pgfmath@x0.04453994279pt\relax% + \pgfmath@x\pgfmath@temp\pgfmath@x% + \advance\pgfmath@x-0.1195159052pt\relax% + \pgfmath@x\pgfmath@temp\pgfmath@x% + \advance\pgfmath@x0.4993615349pt\relax% + \pgfmath@x\pgfmath@temp\pgfmath@x% + \advance\pgfmath@x1.000012594pt\relax% + \ifdim\pgfmath@y<0pt\relax% + \pgfmath@y-\pgfmath@y% + \fi% + \pgfmath@x\pgfmath@tonumber{\pgfmath@y}\pgfmath@x% + \multiply\pgfmath@x\c@pgfmath@counta\relax% + \fi% + \fi% + \pgfmath@returnone\pgfmath@x% + \endgroup% +} + +\pgfmathdeclarefunction{cosh}{1}{% + \begingroup + \edef\pgfmath@arg{#1}% + \pgfmathexp@{\pgfmath@arg}% + \pgfmath@x=\pgfmathresult pt\relax% + \pgfmathexp@{-\pgfmath@arg}% + \advance\pgfmath@x by\pgfmathresult pt\relax% + \divide\pgfmath@x by2\relax% + \pgfmath@returnone\pgfmath@x% + \endgroup} + +\pgfmathdeclarefunction{sinh}{1}{% + \begingroup + \edef\pgfmath@arg{#1}% + \pgfmathexp@{\pgfmath@arg}% + \pgfmath@x=\pgfmathresult pt\relax% + \pgfmathexp@{-\pgfmath@arg}% + \advance\pgfmath@x by-\pgfmathresult pt\relax% + \divide\pgfmath@x by2\relax% + \pgfmath@returnone\pgfmath@x% + \endgroup +}% + +\pgfmathdeclarefunction{tanh}{1}{% + \begingroup + \edef\pgfmath@arg{#1}% + \pgfmathsinh@{\pgfmath@arg}% + \pgfmath@x=\pgfmathresult pt\relax% + \pgfmathcosh@{\pgfmath@arg}% + \pgfmath@y=\pgfmathresult pt\relax% + \pgfmathdivide@{\pgfmath@tonumber{\pgfmath@x}}{\pgfmath@tonumber{\pgfmath@y}}% + \pgfmath@x=\pgfmathresult pt\relax% + \pgfmath@returnone\pgfmath@x% + \endgroup +}% + +\def\pgfmathscientific#1#2{% + \begingroup% + \edef\pgfmath@sci@exponent{#2}% + \pgfmath@x#1pt\relax% + \ifdim\pgfmath@x<0pt\relax% + \pgfmath@x-\pgfmath@x% + \edef\pgfmath@sci@sign{-}% + \expandafter\edef\expandafter\pgfmath@temp\expandafter{\expandafter\pgfmath@gobbleone#1}% + \else% + \edef\pgfmath@sci@sign{+}% + \edef\pgfmath@temp{#1}% + \fi% + \expandafter\pgfmath@in@\expandafter.\expandafter{#1}% + \ifpgfmath@in@% + \else% + \edef\pgfmath@temp{\pgfmath@temp.0}% + \fi% + \expandafter\pgfmath@scientific@@\pgfmath@temp00000000000\pgfmath@} + +\def\pgfmath@scientific@@#1.#2#3#4#5#6{% + \edef\pgfmath@sci@int{#1}% + \edef\pgfmath@sci@mantissaA{#2#3#4#5#6}% + \pgfmath@scientific@@@} + +\def\pgfmath@scientific@@@#1#2#3#4#5#6\pgfmath@{% + \edef\pgfmath@sci@mantissaB{#1#2#3#4#5}% + \c@pgfmath@counta\pgfmath@sci@exponent\relax% + \ifnum\c@pgfmath@counta<-4\relax% + \pgfmath@x0pt\relax% + \else% + \c@pgfmath@countb\c@pgfmath@counta% + \ifnum\c@pgfmath@counta<0\relax% + \c@pgfmath@counta-\c@pgfmath@counta% + \fi% + \pgfmathpow@{10}{\the\c@pgfmath@counta}% + \afterassignment\pgfmath@gobbletilpgfmath@ + \c@pgfmath@countc\pgfmathresult\relax\pgfmath@ + \edef\pgfmath@sci@factor{\the\c@pgfmath@countc}% + \ifnum\c@pgfmath@countb<0\relax% + \pgfmath@x\pgfmath@sci@int.\pgfmath@sci@mantissaA pt\relax% + \divide\pgfmath@x\pgfmath@sci@factor\relax% + \else% + \c@pgfmath@counta\pgfmath@sci@int% + \c@pgfmath@countb\pgfmath@sci@mantissaA% + \multiply\c@pgfmath@counta\pgfmath@sci@factor\relax% + \multiply\c@pgfmath@countb\pgfmath@sci@factor\relax% + \c@pgfmath@countc\c@pgfmath@countb% + \divide\c@pgfmath@countb100000\relax% + \advance\c@pgfmath@counta\c@pgfmath@countb% + \multiply\c@pgfmath@countb100000\relax% + \advance\c@pgfmath@countc-\c@pgfmath@countb% + \c@pgfmath@countb\pgfmath@sci@mantissaB\relax% + \multiply\c@pgfmath@countb\pgfmath@sci@factor\relax% + \divide\c@pgfmath@countb100000\relax% + \advance\c@pgfmath@countc\c@pgfmath@countb% + \advance\c@pgfmath@countc100000\relax% + \edef\pgfmath@sci@result{\pgfmath@sci@sign\the\c@pgfmath@counta.\expandafter\pgfmath@gobbleone\the\c@pgfmath@countc pt}% + \pgfmath@x\pgfmath@sci@result\relax% + \fi\fi% + \pgfmath@returnone\pgfmath@x% + \endgroup} + + + +\pgfmathdeclarefunction{width}{1}{% + \begingroup% + \setbox\pgfmath@box=\hbox{{\pgfmath@selectfont{#1}}}% + \pgfmath@x=\wd\pgfmath@box% + \edef\pgfmathresult{\pgfmath@tonumber{\pgfmath@x}}% + \pgfmath@smuggleone\pgfmathresult% + \endgroup% +} + + +\pgfmathdeclarefunction{height}{1}{% + \begingroup% + \setbox\pgfmath@box=\hbox{{\pgfmath@selectfont{#1}}}% + \pgfmath@x=\ht\pgfmath@box% + \edef\pgfmathresult{\pgfmath@tonumber{\pgfmath@x}}% + \pgfmath@smuggleone\pgfmathresult% + \endgroup% +} + +\pgfmathdeclarefunction{depth}{1}{% + \begingroup% + \setbox\pgfmath@box=\hbox{{\pgfmath@selectfont{#1}}}% + \pgfmath@x=\dp\pgfmath@box% + \edef\pgfmathresult{\pgfmath@tonumber{\pgfmath@x}}% + \pgfmath@smuggleone\pgfmathresult% + \endgroup% +} + + +\pgfmathdeclarefunction{array}{2}{% + \begingroup% + \afterassignment\pgfmath@gobbletilpgfmath@\pgfmath@count=#2\relax\pgfmath@% + \expandafter\pgfmatharray@@#1\pgfmath@stop% +} + +\def\pgfmath@token@stop{\pgfmath@stop}% +\def\pgfmatharray@@#1{% + \def\pgfmath@temp{#1}% + \advance\pgfmath@count by-1\relax% + \ifx\pgfmath@temp\pgfmath@token@stop% + \pgfmath@error{Array index out of bounds.}{}% + \def\pgfmathresult{0}% + \pgfmath@smuggleone\pgfmathresult\endgroup% + \let\pgfmath@next=\relax% + \else% + \ifnum\pgfmath@count=-1\relax% + \def\pgfmathresult{#1}% + \let\pgfmath@next=\pgfmatharray@@@% + \else% + \let\pgfmath@next=\pgfmatharray@@% + \fi% + \fi% + \pgfmath@next} + +\def\pgfmatharray@@@#1\pgfmath@stop{% + \pgfmath@smuggleone\pgfmathresult\endgroup% +} + +\pgfmathdeclarefunction{dim}{1}{% + \begingroup + \pgfmath@count=0\relax + \expandafter\pgfmath@dim@i\pgfutil@firstofone#1{\pgfmath@token@stop}% + \edef\pgfmathresult{\the\pgfmath@count}% + \pgfmath@smuggleone\pgfmathresult% + \endgroup} + +\def\pgfmath@dim@i#1#{% + \pgfutil@ifempty{#1}{}{\advance\pgfmath@count by 1}% + \pgfmath@dim@ii{#1}} + +\def\pgfmath@dim@ii#1#2{% + \ifx\pgfmath@token@stop#2% + \else + \advance\pgfmath@count by 1\relax + \expandafter\pgfmath@dim@i + \fi} + + +\pgfmathdeclarefunction{max}{...}{% + \begingroup% + \pgfmath@x=-16383pt\relax% + \pgfmathmax@@#1{}} + +\def\pgfmathmax@@#1{% + \def\pgfmath@temp{#1}% + \ifx\pgfmath@temp\pgfmath@empty% + \let\pgfmath@next=\pgfmathmax@@@% + \else% + \ifdim#1pt>\pgfmath@x% + \pgfmath@x=#1pt\relax% + \fi% + \let\pgfmath@next=\pgfmathmax@@% + \fi% + \pgfmath@next} + +\def\pgfmathmax@@@{\pgfmath@returnone\pgfmath@x\endgroup}% + +\def\pgfmathmax#1#2{% + \pgfmathparse{#1,#2}% + \expandafter\pgfmathmax@\expandafter{\pgfmathresult}% +} + +\pgfmathdeclarefunction{min}{...}{% + \begingroup% + \pgfmath@x=16383pt\relax% + \pgfmathmin@@#1{}} + +\def\pgfmathmin@@#1{% + \def\pgfmath@temp{#1}% + \ifx\pgfmath@temp\pgfmath@empty% + \let\pgfmath@next=\pgfmathmin@@@% + \else% + \ifdim#1pt<\pgfmath@x% + \pgfmath@x=#1pt\relax% + \fi% + \let\pgfmath@next=\pgfmathmin@@% + \fi% + \pgfmath@next} + +\def\pgfmathmin@@@{\pgfmath@returnone\pgfmath@x\endgroup}% + +\def\pgfmathmin#1#2{% + \pgfmathparse{#1,#2}% + \expandafter\pgfmathmin@\expandafter{\pgfmathresult}% +} + + + +\pgfmathdeclarefunction{scalar}{1}{% + \def\pgfmathresult{#1}% + \global\pgfmathunitsdeclaredfalse% +} diff --git a/preprints/A-polynomial-time-algorithm-for-three-machine-unit-job-scheduling-September-24-2026/build/tex/runtime/generic/pgf/math/pgfmathfunctions.random.code.tex b/preprints/A-polynomial-time-algorithm-for-three-machine-unit-job-scheduling-September-24-2026/build/tex/runtime/generic/pgf/math/pgfmathfunctions.random.code.tex new file mode 100644 index 0000000000000000000000000000000000000000..3898e99b826ad1af20cb9b5ca8d29b1416897492 --- /dev/null +++ b/preprints/A-polynomial-time-algorithm-for-three-machine-unit-job-scheduling-September-24-2026/build/tex/runtime/generic/pgf/math/pgfmathfunctions.random.code.tex @@ -0,0 +1,168 @@ + + +\def\pgfmath@rnd@m{2147483647}% + +\begingroup +\c@pgfmath@counta=\time% +\multiply\c@pgfmath@counta by\year% +\ifnum\c@pgfmath@counta=0 + \c@pgfmath@counta=1 +\fi +\xdef\pgfmath@rnd@z{\the\c@pgfmath@counta}% +\endgroup + +\def\pgfmathsetseed#1{% + \pgfmathparse{#1}\expandafter\pgfmath@setseed\pgfmathresult.0\pgfmath@stop +}% +\def\pgfmath@setseed#1.#2\pgfmath@stop{\def\pgfmath@rnd@z{#1}}% + +\def\pgfmath@rnd@a{69621} +\def\pgfmath@rnd@r{23902} +\def\pgfmath@rnd@q{30845} + +\def\pgfmathgeneratepseudorandomnumber{% + \begingroup% + \c@pgfmath@counta=\pgfmath@rnd@z% + \c@pgfmath@countb=\pgfmath@rnd@z% + \c@pgfmath@countc=\pgfmath@rnd@q% + \divide\c@pgfmath@counta by\c@pgfmath@countc% + \multiply\c@pgfmath@counta by-\c@pgfmath@countc% + \advance\c@pgfmath@counta by\c@pgfmath@countb + \c@pgfmath@countc=\pgfmath@rnd@a% + \multiply\c@pgfmath@counta by\c@pgfmath@countc% + \c@pgfmath@countc=\pgfmath@rnd@q% + \divide\c@pgfmath@countb by\c@pgfmath@countc% + \c@pgfmath@countc=\pgfmath@rnd@r% + \multiply\c@pgfmath@countb by\c@pgfmath@countc% + \advance\c@pgfmath@counta by-\c@pgfmath@countb% + \ifnum\c@pgfmath@counta<0\relax% + \c@pgfmath@countb=\pgfmath@rnd@m% + \advance\c@pgfmath@counta by\c@pgfmath@countb% + \fi% + \xdef\pgfmath@rnd@z{\the\c@pgfmath@counta}% + \endgroup% + \edef\pgfmathresult{\pgfmath@rnd@z}% +} + +\pgfmathdeclarefunction{rnd}{0}{% + \begingroup% + \pgfmathgeneratepseudorandomnumber% + \c@pgfmath@counta\pgfmathresult% + \c@pgfmath@countb\c@pgfmath@counta% + \divide\c@pgfmath@countb100001\relax% + \multiply\c@pgfmath@countb-100001\relax% + \advance\c@pgfmath@countb\c@pgfmath@counta% + \advance\c@pgfmath@countb1000000\relax% + \expandafter\pgfmathrnd@@\the\c@pgfmath@countb\pgfmath@% + \pgfmath@returnone\pgfmath@x% + \endgroup% +}% + +\def\pgfmathrnd@@#1#2#3\pgfmath@{% + \edef\pgfmath@temp{#2.#3}% + \pgfmath@x=\pgfmath@temp pt\relax% +}% + +\pgfmathdeclarefunction{rand}{0}{% + \begingroup% + \pgfmathgeneratepseudorandomnumber% + \c@pgfmath@counta\pgfmathresult% + \c@pgfmath@countb\c@pgfmath@counta% + \divide\c@pgfmath@countb200001\relax% + \multiply\c@pgfmath@countb-200001\relax% + \advance\c@pgfmath@countb\c@pgfmath@counta% + \advance\c@pgfmath@countb-100000\relax% + \ifnum\c@pgfmath@countb<0\relax% + \advance\c@pgfmath@countb-1000000\relax% + \else% + \advance\c@pgfmath@countb1000000\relax% + \fi% + \expandafter\pgfmathrand@@\the\c@pgfmath@countb\pgfmath@% + \pgfmath@returnone\pgfmath@x% + \endgroup% +}% + +\def\pgfmathrand@@#1#2#3#4\pgfmath@{% + \ifx#1-% + \edef\pgfmath@temp{-#3.#4}% + \else% + \edef\pgfmath@temp{#2.#3#4}% + \fi% + \pgfmath@x=\pgfmath@temp pt\relax% +}% + +\pgfmathdeclarefunction{random}{...}{% + \begingroup% + \def\pgfmath@temp{#1}% + \ifx\pgfmath@temp\pgfmath@empty% + \pgfmathrnd@% + \else% + \expandafter\pgfmathrandom@@#1\pgfmath@stop% + \fi% + \pgfmath@smuggleone\pgfmathresult% + \endgroup} + +\def\pgfmathrandom@@{\futurelet\pgfmath@token\pgfmathrandom@@@} +\def\pgfmathrandom@@@{% + \ifx\pgfmath@token\bgroup% + \expandafter\pgfmath@random@@@@% + \else% + \expandafter\pgfmath@random@@@@@% + \fi% +} +\def\pgfmath@random@@@@@#1\pgfmath@stop{% + \pgfmathrandominteger\pgfmathresult{1}{#1}% +} +\def\pgfmath@random@@@@#1#2\pgfmath@stop{% + \pgfmathrandominteger\pgfmathresult{#1}{#2}% +} + +\def\pgfmathrandominteger#1#2#3{% + \begingroup% + \pgfmathsetcount\c@pgfmath@counta{#2}% + \pgfmathsetcount\c@pgfmath@countb{#3}% + \c@pgfmath@countc\c@pgfmath@countb% + \ifnum\c@pgfmath@counta>\c@pgfmath@countb\relax% + \c@pgfmath@countc\c@pgfmath@counta% + \c@pgfmath@counta\c@pgfmath@countb% + \c@pgfmath@countb\c@pgfmath@countc% + \fi% + \c@pgfmath@countd\c@pgfmath@counta + \advance\c@pgfmath@countc1\relax% + \advance\c@pgfmath@countc-\c@pgfmath@counta% + \pgfmathgeneratepseudorandomnumber% + \c@pgfmath@counta\pgfmathresult\relax% + \c@pgfmath@countb\c@pgfmath@counta% + \divide\c@pgfmath@countb\c@pgfmath@countc% + \multiply\c@pgfmath@countb-\c@pgfmath@countc% + \advance\c@pgfmath@counta\c@pgfmath@countb% + \advance\c@pgfmath@counta\c@pgfmath@countd + \edef\pgfmathresult{\the\c@pgfmath@counta}% + \pgfmath@smuggleone{\pgfmathresult}% + \endgroup% + \edef#1{\pgfmathresult}% +} + +\def\pgfmathdeclarerandomlist#1#2{% + \def\pgfmath@randomlistname{#1}% + \begingroup% + \c@pgfmath@counta=1\relax% + \def\pgfmath@empty{pgfmath@empty}% + \expandafter\pgfmath@scanrandomlist#2{pgfmath@empty}} +\def\pgfmath@scanrandomlist#1{% + \def\pgfmath@scanneditem{#1}% + \ifx\pgfmath@scanneditem\pgfmath@empty% + \advance\c@pgfmath@counta-1\relax% + \expandafter\xdef\csname pgfmath@randomlist@\pgfmath@randomlistname\endcsname{\the\c@pgfmath@counta}% + \endgroup% + \else% + \expandafter\gdef\csname pgfmath@randomlist@\pgfmath@randomlistname @\the\c@pgfmath@counta\endcsname{#1}% + \advance\c@pgfmath@counta1\relax% + \expandafter\pgfmath@scanrandomlist% + \fi} + +\def\pgfmathrandomitem#1#2{% + \pgfmath@ifundefined{pgfmath@randomlist@#2}{\pgfmath@error{Unknown random list `#2'}{}}{% + \edef\pgfmath@randomlistlength{\csname pgfmath@randomlist@#2\endcsname}% + \pgfmathrandominteger{\pgfmath@randomtemp}{1}{\pgfmath@randomlistlength}% + \expandafter\def\expandafter#1\expandafter{\csname pgfmath@randomlist@#2@\pgfmath@randomtemp\endcsname}}} diff --git a/preprints/A-polynomial-time-algorithm-for-three-machine-unit-job-scheduling-September-24-2026/build/tex/runtime/generic/pgf/math/pgfmathfunctions.round.code.tex b/preprints/A-polynomial-time-algorithm-for-three-machine-unit-job-scheduling-September-24-2026/build/tex/runtime/generic/pgf/math/pgfmathfunctions.round.code.tex new file mode 100644 index 0000000000000000000000000000000000000000..ac3db56976080428c15f00add47e74cae35da702 --- /dev/null +++ b/preprints/A-polynomial-time-algorithm-for-three-machine-unit-job-scheduling-September-24-2026/build/tex/runtime/generic/pgf/math/pgfmathfunctions.round.code.tex @@ -0,0 +1,99 @@ + + +\pgfmathdeclarefunction{round}{1}{% + \begingroup% + \expandafter\pgfmath@x#1pt\relax% + \afterassignment\pgfmath@xa% + \expandafter\c@pgfmath@counta\the\pgfmath@x\relax% + \pgfmath@xb\pgfmath@x% + \ifdim\pgfmath@xb<0pt\relax% + \ifdim\pgfmath@xa<0.5pt\relax% + \else% + \advance\c@pgfmath@counta-1\relax% + \fi% + \else% + \ifdim\pgfmath@xa<0.5pt\relax% + \else% + \advance\c@pgfmath@counta1\relax% + \fi% + \fi% + \pgfmath@returnone\the\c@pgfmath@counta pt% + \endgroup% +}% + +\pgfmathdeclarefunction{floor}{1}{% + \begingroup% + \expandafter\pgfmath@x#1pt\relax% + \afterassignment\pgfmathfloor@collect@remainder% + \expandafter\c@pgfmath@counta\the\pgfmath@x\pgfmath@% + \expandafter\pgfmath@x#1pt\relax% + \ifdim\pgfmath@x<0pt\relax% + \ifdim\pgfmathfloor@remainder>0pt + \advance\c@pgfmath@counta by -1\relax% + \else + \fi + \fi% + \pgfmath@x\c@pgfmath@counta pt\relax% + \pgfmath@returnone\pgfmath@x% + \endgroup% +}% + +\def\pgfmathfloor@collect@remainder#1\pgfmath@{% + \def\pgfmathfloor@remainder{#1}% + \ifx\pgfmathfloor@remainder\pgfutil@empty + \def\pgfmathfloor@remainder{0}% + \fi +}% + +\pgfmathdeclarefunction{ceil}{1}{% + \begingroup% + \expandafter\pgfmath@x#1pt\relax% + \afterassignment\pgfmath@gobbletilpgfmath@% + \expandafter\c@pgfmath@counta\the\pgfmath@x\relax\pgfmath@% + \pgfmath@y\pgfmath@x% + \advance\pgfmath@y-\c@pgfmath@counta pt\relax% + \pgfmath@x\c@pgfmath@counta pt\relax% + \ifdim\pgfmath@y>0pt\relax% + \advance\pgfmath@x1pt\relax% + \fi% + \pgfmath@returnone\pgfmath@x% + \endgroup% +}% + +\pgfmathdeclarefunction{int}{1}{% + \begingroup% + \expandafter\pgfmathint@@#1.\pgfmath@stop + \pgfmath@smuggleone\pgfmathresult% + \endgroup} + +\def\pgfmathint@@#1.#2\pgfmath@stop{% + \def\pgfmath@temp{#1}% + \ifx\pgfmath@temp\pgfmath@empty% + \def\pgfmathresult{0}% + \else% + \pgfmath@in@{#1}{+-+0-0}% + \ifpgfmath@in@ + \def\pgfmathresult{0}% + \else% + \def\pgfmathresult{#1}% + \fi% + \fi} + +\pgfmathdeclarefunction{frac}{1}{% + \begingroup% + \expandafter\pgfmathfrac@@#1.\pgfmath@stop% + \pgfmath@smuggleone\pgfmathresult% + \endgroup} + +\def\pgfmathfrac@@#1.{% + \pgfmath@ifnextchar\pgfmath@stop{\def\pgfmathresult{0.0}\pgfmath@gobbleone}{\pgfmathfrac@@@}% +} +\def\pgfmathfrac@@@#1.\pgfmath@stop{\def\pgfmathresult{0.#1}} + +\pgfmathdeclarefunction{real}{1}{% + \begingroup% + \pgfmath@x=#1pt\relax% + \edef\pgfmathresult{\pgfmath@tonumber{\pgfmath@x}}% + \pgfmath@smuggleone\pgfmathresult% + \endgroup% +} diff --git a/preprints/A-polynomial-time-algorithm-for-three-machine-unit-job-scheduling-September-24-2026/build/tex/runtime/generic/pgf/math/pgfmathfunctions.trigonometric.code.tex b/preprints/A-polynomial-time-algorithm-for-three-machine-unit-job-scheduling-September-24-2026/build/tex/runtime/generic/pgf/math/pgfmathfunctions.trigonometric.code.tex new file mode 100644 index 0000000000000000000000000000000000000000..a8e2fa5efcd42a5773e11a0337ad8300fc06c2bf --- /dev/null +++ b/preprints/A-polynomial-time-algorithm-for-three-machine-unit-job-scheduling-September-24-2026/build/tex/runtime/generic/pgf/math/pgfmathfunctions.trigonometric.code.tex @@ -0,0 +1,1463 @@ + + +\pgfmathdeclarefunction{pi}{0}{\def\pgfmathresult{3.141592654}} + +\pgfkeys{ + /pgf/trig format/.is choice, + /pgf/trig format/deg/.code={\def\pgfmath@trig@format@choice{0}}, + /pgf/trig format/rad/.code={\def\pgfmath@trig@format@choice{1}}, + /pgf/trig format/deg, +} + +\def\pgfmathiftrigonometricusesdeg#1#2{% + \if0\pgfmath@trig@format@choice + #1% + \else + #2% + \fi +}% + +\pgfmathdeclarefunction{radians}{1}{% + \begingroup% + \expandafter\pgfmath@x#1pt\relax% + \pgfmath@x57.29577\pgfmath@x\relax% + \pgfmath@returnone\pgfmath@x% + \endgroup% +}% + +\pgfmathdeclarefunction{deg}{1}{% + \begingroup% + \expandafter\pgfmath@x#1pt\relax% + \pgfmath@scale@rad@to@deg\pgfmath@x + \pgfmath@returnone\pgfmath@x% + \endgroup% +}% + +\def\pgfmath@scale@rad@to@deg#1{% + #1=57.29577#1\relax% +}% + +\pgfmathdeclarefunction{rad}{1}{% + \begingroup% + \expandafter\pgfmath@x#1pt\relax% + \pgfmath@scale@deg@to@rad\pgfmath@x + \pgfmath@returnone\pgfmath@x% + \endgroup% +}% + +\def\pgfmath@scale@deg@to@rad#1{% + #1=3.14159#1% + \divide#1180\relax% +}% + +\pgfmathdeclarefunction{sin}{1}{% + \begingroup% + \expandafter\pgfmath@x#1pt\relax% + \if1\pgfmath@trig@format@choice + \pgfmath@scale@rad@to@deg\pgfmath@x + \fi + \advance\pgfmath@x-90pt\relax% + \afterassignment\pgfmath@gobbletilpgfmath@ + \expandafter\c@pgfmath@counta\the\pgfmath@x\relax\pgfmath@ + \divide\c@pgfmath@counta360\relax% + \multiply\c@pgfmath@counta-360\relax% + \advance\pgfmath@x\c@pgfmath@counta pt\relax% + \ifdim\pgfmath@x<0pt\relax\pgfmath@x-\pgfmath@x\fi% + \ifdim\pgfmath@x<180pt\relax% + \else + \pgfmath@x-\pgfmath@x% + \advance\pgfmath@x360pt\relax% + \fi% + \afterassignment\pgfmath@xa% + \expandafter\c@pgfmath@counta\the\pgfmath@x\relax% + \ifdim\pgfmath@xa=0pt% + \expandafter\pgfmath@x\csname pgfmath@cos@\the\c@pgfmath@counta\endcsname pt\relax% + \else% + \pgfmath@x-\pgfmath@xa% + \advance\pgfmath@x1pt\relax% + \expandafter\pgfmath@x\csname pgfmath@cos@\the\c@pgfmath@counta\endcsname\pgfmath@x% + \advance\c@pgfmath@counta1\relax% + \ifnum\c@pgfmath@counta=181\relax\c@pgfmath@counta179\relax\fi% + \expandafter\advance\expandafter\pgfmath@x\csname pgfmath@cos@\the\c@pgfmath@counta\endcsname\pgfmath@xa% + \fi% + \pgfmath@returnone\pgfmath@x% + \endgroup +} + +\pgfmathdeclarefunction{cos}{1}{% + \begingroup% + \expandafter\pgfmath@x#1pt\relax% + \if1\pgfmath@trig@format@choice + \pgfmath@scale@rad@to@deg\pgfmath@x + \fi + \afterassignment\pgfmath@gobbletilpgfmath@% + \expandafter\c@pgfmath@counta\the\pgfmath@x\relax\pgfmath@% + \divide\c@pgfmath@counta360\relax% + \multiply\c@pgfmath@counta-360\relax% + \advance\pgfmath@x\c@pgfmath@counta pt\relax% + \ifdim\pgfmath@x<0pt\relax\pgfmath@x-\pgfmath@x\fi% + \ifdim\pgfmath@x<180pt\relax% + \else% + \pgfmath@x-\pgfmath@x% + \advance\pgfmath@x360pt\relax% + \fi% + \afterassignment\pgfmath@xa% + \expandafter\c@pgfmath@counta\the\pgfmath@x\relax% + \ifdim\pgfmath@xa=0pt% + \expandafter\pgfmath@x\csname pgfmath@cos@\the\c@pgfmath@counta\endcsname pt% + \else% + \pgfmath@x-\pgfmath@xa% + \advance\pgfmath@x1pt\relax% + \expandafter\pgfmath@x\csname pgfmath@cos@\the\c@pgfmath@counta\endcsname\pgfmath@x% + \advance\c@pgfmath@counta1\relax% + \ifnum\c@pgfmath@counta=181\relax\c@pgfmath@counta179\relax\fi% + \expandafter\advance\expandafter\pgfmath@x\csname pgfmath@cos@\the\c@pgfmath@counta\endcsname\pgfmath@xa% + \fi% + \pgfmath@returnone\pgfmath@x% + \endgroup% +} + +\def\pgfmathsincos#1{% + \pgfmathparse{#1}% + \expandafter\pgfmathsincos@\expandafter{\pgfmathresult}} +\def\pgfmathsincos@#1{% + \edef\pgfmath@temparg{#1}% + \pgfmathsin@{\pgfmath@temparg}\edef\pgfmathresulty{\pgfmathresult}% + \pgfmathcos@{\pgfmath@temparg}\edef\pgfmathresultx{\pgfmathresult}% +} + +\pgfmathdeclarefunction{tan}{1}{% + \begingroup% + \edef\pgfmath@temparg{#1}% + \pgfmathcos@{\pgfmath@temparg}% + \expandafter\pgfmathreciprocal@\expandafter{\pgfmathresult}% + \edef\pgfmath@tantemp{\pgfmathresult}% + \pgfmathsin@{\pgfmath@temparg}% + \pgfmath@x\pgfmathresult pt\relax% + \pgfmath@x\pgfmath@tantemp\pgfmath@x% + \pgfmath@returnone\pgfmath@x% + \endgroup% +} + +\pgfmathdeclarefunction{cosec}{1}{% + \expandafter\pgfmathsin@\expandafter{#1}% + \pgfmathreciprocal@{\pgfmathresult}% +} + +\pgfmathdeclarefunction{sec}{1}{% + \expandafter\pgfmathcos@\expandafter{#1}% + \pgfmathreciprocal@{\pgfmathresult}% +} + +\pgfmathdeclarefunction{cot}{1}{% + \begingroup% + \edef\pgfmath@temparg{#1}% + \pgfmathsin@{\pgfmath@temparg}% + \expandafter\pgfmathreciprocal@\expandafter{\pgfmathresult}% + \edef\pgfmath@tantemp{\pgfmathresult}% + \pgfmathcos@{\pgfmath@temparg}% + \pgfmath@x\pgfmathresult pt\relax% + \pgfmath@x\pgfmath@tantemp\pgfmath@x% + \pgfmath@returnone\pgfmath@x% + \endgroup% +} + +\def\pgfmath@multiply@thousand#1.#2#3#4#5\relax{% + \edef\pgfmathresult{#1#2#3#4.#5}% +}% + +\pgfmathdeclarefunction{asin}{1}{% + \begingroup% + \pgfmath@x=#1pt % + \pgfmath@xa\pgfmath@x% + \edef\pgf@temp{\pgfmath@tonumber\pgfmath@x}% + \expandafter\pgfmath@multiply@thousand\pgf@temp 000\relax + \pgfmath@x=\pgfmathresult pt % + \ifdim\pgfmath@x<0pt\relax% + \pgfmath@x-\pgfmath@x% + \fi% + \pgfmath@table@lookup{\pgfmath@x}{pgfmath@acos@}{1001}% + \pgfmath@x=-\pgfmath@x + \advance\pgfmath@x90pt\relax% + \ifdim\pgfmath@xa<0pt\relax% + \pgfmath@x-\pgfmath@x% + \fi% + \if1\pgfmath@trig@format@choice + \pgfmath@scale@deg@to@rad\pgfmath@x + \fi + \pgfmath@returnone\pgfmath@x% + \endgroup% +} + +\def\pgfmath@table@lookup#1#2#3{% + \pgfmath@x=#1\relax + \expandafter\pgfmath@table@lookup@\the\pgfmath@x\relax{#2}{#3}% +} +\def\pgfmath@table@lookup@#1.#2\relax#3#4{% + \edef\pgf@temp{\Pgf@geT#2}% + \pgfmath@x=\csname #3#1\endcsname pt % + \c@pgfmath@countc=#1 % + \advance\c@pgfmath@countc by1 % + \ifnum\c@pgfmath@countc<#4 % + \pgfmath@ya=\csname #3\the\c@pgfmath@countc\endcsname pt % + \advance\pgfmath@ya by-\pgfmath@x % + \advance\pgfmath@x by0.\pgf@temp\pgfmath@ya + \fi +}% + +\pgfmathdeclarefunction{acos}{1}{% + \begingroup% + \pgfmath@x#1pt\relax% + \pgfmath@xa\pgfmath@x% + \edef\pgf@temp{\pgfmath@tonumber\pgfmath@x}% + \expandafter\pgfmath@multiply@thousand\pgf@temp 000\relax + \pgfmath@x=\pgfmathresult pt % + \ifdim\pgfmath@x<0pt\relax% + \pgfmath@x-\pgfmath@x% + \fi% + \pgfmath@table@lookup{\pgfmath@x}{pgfmath@acos@}{1001}% + \ifdim\pgfmath@xa<0pt\relax% + \pgfmath@x-\pgfmath@x% + \advance\pgfmath@x by180pt\relax% + \fi% + \if1\pgfmath@trig@format@choice + \pgfmath@scale@deg@to@rad\pgfmath@x + \fi + \pgfmath@returnone\pgfmath@x% + \endgroup% +} + +\pgfmathdeclarefunction{atan}{1}{% + \begingroup% + \expandafter\pgfmath@x#1pt\relax% + \pgfmath@xa\pgfmath@x% + \ifdim\pgfmath@x<0pt\relax% + \pgfmath@x-\pgfmath@x% + \fi% + \pgfmath@xb\pgfmath@x% + \ifdim\pgfmath@x>1pt\relax% + \edef\pgfmath@temp{\pgfmath@tonumber{\pgfmath@x}}% + \pgfmathreciprocal@{\pgfmath@temp}% + \pgfmath@x\pgfmathresult pt\relax% + \fi% + \edef\pgf@temp{\pgfmath@tonumber\pgfmath@x}% + \expandafter\pgfmath@multiply@thousand\pgf@temp 000\relax + \pgfmath@x=\pgfmathresult pt % + \pgfmath@table@lookup{\pgfmath@x}{pgfmath@atan@}{1001}% + \ifdim\pgfmath@xb>1pt\relax% + \pgfmath@x=-\pgfmath@x% + \advance\pgfmath@x90pt% + \fi% + \ifdim\pgfmath@xa<0pt\relax% + \pgfmath@x-\pgfmath@x% + \fi% + \if1\pgfmath@trig@format@choice + \pgfmath@scale@deg@to@rad\pgfmath@x + \fi + \pgfmath@returnone\pgfmath@x% + \endgroup% +} + +\pgfmathdeclarefunction{atan2}{2}{% + \begingroup% + \let\pgfmath@trig@format@choice@@=\pgfmath@trig@format@choice + \def\pgfmath@trig@format@choice{0}% + \pgfmath@y=#1pt\relax% + \ifdim\pgfmath@y<0pt\relax% + \pgfmath@y=-\pgfmath@y% + \fi% + \ifdim\pgfmath@y<0.001pt\relax% + \ifdim#2pt<0pt\relax% + \pgfmath@x=+180pt\relax% + \else% + \ifdim#2pt>0pt\relax% + \pgfmath@x=0pt\relax% + \else% + \ifdim#1pt<0pt% + \pgfmath@x=-90pt\relax% + \else% + \pgfmath@x=90pt\relax% + \fi% + \fi% + \fi% + \else% + \pgfmathabs@{#1}\let\pgfmath@tempa\pgfmathresult% + \pgfmathabs@{#2}\let\pgfmath@tempb\pgfmathresult% + \ifdim\pgfmath@tempa pt>\pgfmath@tempb pt% + \pgfmathdivide@{#2}{\pgfmath@tempa}% + \expandafter\pgfmathatan@\expandafter{\pgfmathresult}% + \pgfmath@x=90pt% + \advance\pgfmath@x by-\pgfmathresult pt\relax% + \else% + \pgfmathdivide@{\pgfmath@tempa}{#2}% + \expandafter\pgfmathatan@\expandafter{\pgfmathresult}% + \pgfmath@x=\pgfmathresult pt\relax% + \ifdim#2pt<0pt% + \advance\pgfmath@x by180pt\relax% + \fi% + \fi% + \ifdim#1pt<0pt\relax% + \pgfmath@x=-\pgfmath@x% + \fi% + \fi% + \if1\pgfmath@trig@format@choice@@ + \pgfmath@scale@deg@to@rad\pgfmath@x + \fi + \pgfmath@returnone\pgfmath@x% + \endgroup% +} + +\expandafter\let\expandafter\pgfmathatantwo\csname pgfmathatan2\endcsname +\expandafter\let\expandafter\pgfmathatantwo@\csname pgfmathatan2@\endcsname + +\def\pgfmath@def#1#2#3{\expandafter\def\csname pgfmath@#1@#2\endcsname{#3}} +\pgfmath@def{cos}{0}{1.00000} \pgfmath@def{cos}{1}{0.99985} +\pgfmath@def{cos}{2}{0.99939} \pgfmath@def{cos}{3}{0.99863} +\pgfmath@def{cos}{4}{0.99756} \pgfmath@def{cos}{5}{0.99619} +\pgfmath@def{cos}{6}{0.99452} \pgfmath@def{cos}{7}{0.99255} +\pgfmath@def{cos}{8}{0.99027} \pgfmath@def{cos}{9}{0.98769} +\pgfmath@def{cos}{10}{0.98481} \pgfmath@def{cos}{11}{0.98163} +\pgfmath@def{cos}{12}{0.97815} \pgfmath@def{cos}{13}{0.97437} +\pgfmath@def{cos}{14}{0.97030} \pgfmath@def{cos}{15}{0.96593} +\pgfmath@def{cos}{16}{0.96126} \pgfmath@def{cos}{17}{0.95630} +\pgfmath@def{cos}{18}{0.95106} \pgfmath@def{cos}{19}{0.94552} +\pgfmath@def{cos}{20}{0.93969} \pgfmath@def{cos}{21}{0.93358} +\pgfmath@def{cos}{22}{0.92718} \pgfmath@def{cos}{23}{0.92050} +\pgfmath@def{cos}{24}{0.91355} \pgfmath@def{cos}{25}{0.90631} +\pgfmath@def{cos}{26}{0.89879} \pgfmath@def{cos}{27}{0.89101} +\pgfmath@def{cos}{28}{0.88295} \pgfmath@def{cos}{29}{0.87462} +\pgfmath@def{cos}{30}{0.86603} \pgfmath@def{cos}{31}{0.85717} +\pgfmath@def{cos}{32}{0.84805} \pgfmath@def{cos}{33}{0.83867} +\pgfmath@def{cos}{34}{0.82904} \pgfmath@def{cos}{35}{0.81915} +\pgfmath@def{cos}{36}{0.80902} \pgfmath@def{cos}{37}{0.79864} +\pgfmath@def{cos}{38}{0.78801} \pgfmath@def{cos}{39}{0.77715} +\pgfmath@def{cos}{40}{0.76604} \pgfmath@def{cos}{41}{0.75471} +\pgfmath@def{cos}{42}{0.74314} \pgfmath@def{cos}{43}{0.73135} +\pgfmath@def{cos}{44}{0.71934} \pgfmath@def{cos}{45}{0.70711} +\pgfmath@def{cos}{46}{0.69466} \pgfmath@def{cos}{47}{0.68200} +\pgfmath@def{cos}{48}{0.66913} \pgfmath@def{cos}{49}{0.65606} 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\pgfmath@def{atan}{29}{1.66111} +\pgfmath@def{atan}{30}{1.71835} \pgfmath@def{atan}{31}{1.77560} +\pgfmath@def{atan}{32}{1.83284} \pgfmath@def{atan}{33}{1.89007} +\pgfmath@def{atan}{34}{1.94730} \pgfmath@def{atan}{35}{2.00453} +\pgfmath@def{atan}{36}{2.06175} \pgfmath@def{atan}{37}{2.11897} +\pgfmath@def{atan}{38}{2.17619} \pgfmath@def{atan}{39}{2.23340} +\pgfmath@def{atan}{40}{2.29061} \pgfmath@def{atan}{41}{2.34781} +\pgfmath@def{atan}{42}{2.40500} \pgfmath@def{atan}{43}{2.46220} +\pgfmath@def{atan}{44}{2.51938} \pgfmath@def{atan}{45}{2.57657} +\pgfmath@def{atan}{46}{2.63374} \pgfmath@def{atan}{47}{2.69092} +\pgfmath@def{atan}{48}{2.74808} \pgfmath@def{atan}{49}{2.80524} +\pgfmath@def{atan}{50}{2.86240} \pgfmath@def{atan}{51}{2.91955} +\pgfmath@def{atan}{52}{2.97669} \pgfmath@def{atan}{53}{3.03383} +\pgfmath@def{atan}{54}{3.09097} \pgfmath@def{atan}{55}{3.14809} +\pgfmath@def{atan}{56}{3.20521} \pgfmath@def{atan}{57}{3.26232} +\pgfmath@def{atan}{58}{3.31943} 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\pgfmath@def{acos}{887}{27.50134} +\pgfmath@def{acos}{888}{27.37700} \pgfmath@def{acos}{889}{27.25214} +\pgfmath@def{acos}{890}{27.12675} \pgfmath@def{acos}{891}{27.00082} +\pgfmath@def{acos}{892}{26.87434} \pgfmath@def{acos}{893}{26.74731} +\pgfmath@def{acos}{894}{26.61973} \pgfmath@def{acos}{895}{26.49157} +\pgfmath@def{acos}{896}{26.36283} \pgfmath@def{acos}{897}{26.23351} +\pgfmath@def{acos}{898}{26.10359} \pgfmath@def{acos}{899}{25.97306} +\pgfmath@def{acos}{900}{25.84193} \pgfmath@def{acos}{901}{25.71017} +\pgfmath@def{acos}{902}{25.57778} \pgfmath@def{acos}{903}{25.44475} +\pgfmath@def{acos}{904}{25.31106} \pgfmath@def{acos}{905}{25.17671} +\pgfmath@def{acos}{906}{25.04169} \pgfmath@def{acos}{907}{24.90598} +\pgfmath@def{acos}{908}{24.76958} \pgfmath@def{acos}{909}{24.63247} +\pgfmath@def{acos}{910}{24.49464} \pgfmath@def{acos}{911}{24.35608} +\pgfmath@def{acos}{912}{24.21678} \pgfmath@def{acos}{913}{24.07672} +\pgfmath@def{acos}{914}{23.93588} \pgfmath@def{acos}{915}{23.79427} +\pgfmath@def{acos}{916}{23.65185} \pgfmath@def{acos}{917}{23.50863} +\pgfmath@def{acos}{918}{23.36457} \pgfmath@def{acos}{919}{23.21967} +\pgfmath@def{acos}{920}{23.07391} \pgfmath@def{acos}{921}{22.92728} +\pgfmath@def{acos}{922}{22.77975} \pgfmath@def{acos}{923}{22.63132} +\pgfmath@def{acos}{924}{22.48195} \pgfmath@def{acos}{925}{22.33164} +\pgfmath@def{acos}{926}{22.18036} \pgfmath@def{acos}{927}{22.02810} +\pgfmath@def{acos}{928}{21.87483} \pgfmath@def{acos}{929}{21.72053} +\pgfmath@def{acos}{930}{21.56518} \pgfmath@def{acos}{931}{21.40876} +\pgfmath@def{acos}{932}{21.25124} \pgfmath@def{acos}{933}{21.09260} +\pgfmath@def{acos}{934}{20.93281} \pgfmath@def{acos}{935}{20.77185} +\pgfmath@def{acos}{936}{20.60969} \pgfmath@def{acos}{937}{20.44630} +\pgfmath@def{acos}{938}{20.28165} \pgfmath@def{acos}{939}{20.11570} +\pgfmath@def{acos}{940}{19.94844} \pgfmath@def{acos}{941}{19.77982} +\pgfmath@def{acos}{942}{19.60981} \pgfmath@def{acos}{943}{19.43837} +\pgfmath@def{acos}{944}{19.26546} \pgfmath@def{acos}{945}{19.09105} +\pgfmath@def{acos}{946}{18.91509} \pgfmath@def{acos}{947}{18.73754} +\pgfmath@def{acos}{948}{18.55835} \pgfmath@def{acos}{949}{18.37748} +\pgfmath@def{acos}{950}{18.19487} \pgfmath@def{acos}{951}{18.01047} +\pgfmath@def{acos}{952}{17.82423} \pgfmath@def{acos}{953}{17.63609} +\pgfmath@def{acos}{954}{17.44598} \pgfmath@def{acos}{955}{17.25385} +\pgfmath@def{acos}{956}{17.05962} \pgfmath@def{acos}{957}{16.86322} +\pgfmath@def{acos}{958}{16.66457} \pgfmath@def{acos}{959}{16.46359} +\pgfmath@def{acos}{960}{16.26020} \pgfmath@def{acos}{961}{16.05430} +\pgfmath@def{acos}{962}{15.84580} \pgfmath@def{acos}{963}{15.63459} +\pgfmath@def{acos}{964}{15.42057} \pgfmath@def{acos}{965}{15.20360} +\pgfmath@def{acos}{966}{14.98357} \pgfmath@def{acos}{967}{14.76033} +\pgfmath@def{acos}{968}{14.53374} \pgfmath@def{acos}{969}{14.30364} +\pgfmath@def{acos}{970}{14.06986} \pgfmath@def{acos}{971}{13.83221} +\pgfmath@def{acos}{972}{13.59049} \pgfmath@def{acos}{973}{13.34447} +\pgfmath@def{acos}{974}{13.09392} \pgfmath@def{acos}{975}{12.83856} +\pgfmath@def{acos}{976}{12.57811} \pgfmath@def{acos}{977}{12.31225} +\pgfmath@def{acos}{978}{12.04060} \pgfmath@def{acos}{979}{11.76278} +\pgfmath@def{acos}{980}{11.47834} \pgfmath@def{acos}{981}{11.18676} +\pgfmath@def{acos}{982}{10.88748} \pgfmath@def{acos}{983}{10.57984} +\pgfmath@def{acos}{984}{10.26309} \pgfmath@def{acos}{985}{9.93636} +\pgfmath@def{acos}{986}{9.59863} \pgfmath@def{acos}{987}{9.24870} +\pgfmath@def{acos}{988}{8.88512} \pgfmath@def{acos}{989}{8.50614} +\pgfmath@def{acos}{990}{8.10961} \pgfmath@def{acos}{991}{7.69281} +\pgfmath@def{acos}{992}{7.25224} \pgfmath@def{acos}{993}{6.78328} +\pgfmath@def{acos}{994}{6.27958} \pgfmath@def{acos}{995}{5.73196} +\pgfmath@def{acos}{996}{5.12640} \pgfmath@def{acos}{997}{4.43922} +\pgfmath@def{acos}{998}{3.62430} \pgfmath@def{acos}{999}{2.56255} +\pgfmath@def{acos}{1000}{0.00000} + + + diff --git a/preprints/A-polynomial-time-algorithm-for-three-machine-unit-job-scheduling-September-24-2026/build/tex/runtime/generic/pgf/math/pgfmathparser.code.tex b/preprints/A-polynomial-time-algorithm-for-three-machine-unit-job-scheduling-September-24-2026/build/tex/runtime/generic/pgf/math/pgfmathparser.code.tex new file mode 100644 index 0000000000000000000000000000000000000000..20a5e6c8a7ded81b01fcdaf03c5640be8cca15e7 --- /dev/null +++ b/preprints/A-polynomial-time-algorithm-for-three-machine-unit-job-scheduling-September-24-2026/build/tex/runtime/generic/pgf/math/pgfmathparser.code.tex @@ -0,0 +1,859 @@ + +\newdimen\pgfmath@dimen +\newcount\pgfmath@count +\newbox\pgfmath@box +\newtoks\pgfmath@toks + +\newif\ifpgfmath@quickparse +\newif\ifpgfmathfloat +\newif\ifpgfmathunitsdeclared +\newif\ifpgfmathmathunitsdeclared +\newif\ifpgfmathignoreunitscale + +\let\pgfmath@protected@edef=\edef +\ifcsname protected@edef\endcsname + \let\pgfmath@protected@edef=\protected@edef +\fi + +\def\pgfmathprint#1{\pgfmathparse{#1}\pgfmathresult} + +\def\pgfmathparse{% + \begingroup + \pgfmath@catcodes + \pgfmath@quickparsefalse + \ifpgfmathfloat + \let\pgfmath@parse@next=\pgfmathfloatparsenumber + \else + \let\pgfmath@parse@next=\pgfmathparse@ + \fi + \pgfmath@parse@next} + +\begingroup + \lccode`\~=`\" + \lowercase{\endgroup + \def\pgfmath@catcodes{% + \catcode`\==12 % + \catcode`\,=12 % + \catcode`\|=12 % + \catcode`\&=12 % + \let~\pgfmath@char@quote + }}% + +\def\pgfmathqparse{% + \begingroup + \pgfmath@catcodes + \pgfmath@quickparsetrue + \pgfmathparse@} + +\def\pgfmathparse@#1{% + \global\pgfmathunitsdeclaredfalse + \global\pgfmathmathunitsdeclaredfalse + \pgfmath@protected@edef\pgfmath@expression{#1}% + \expandafter\pgfmathparse@trynumber@loop\pgfmath@expression\pgfmath@parse@stop +} + +\def\pgfmath@parse@stop{\pgfmath@parse@stop}% + +\def\pgfmathparse@trynumber@token{numeric} +\begingroup +\def\\{\global\let\pgf@let@space@token= } \\ % +\endgroup + +\def\pgfmathparse@trynumber@loop{% + \futurelet\pgfmath@token@let\pgfmathparse@trynumber@loop@ +}% +\def\pgfmathparse@trynumber@loop@#1{% + \ifx\pgfmath@token@let\pgf@let@space@token + \def\pgfmath@parse@next{\pgfmathparse@@#1}% + \else + \ifx\pgfmath@token@let\bgroup% + \let\pgfmath@parse@next=\pgfmathparse@@ + \else + \ifx\pgfmath@token@let\pgfmath@parse@stop + \let\pgfmath@parse@next=\pgfmathparse@expression@is@number + \else + \expandafter\ifx\csname pgfmath@token@\pgfmathparse@trynumber@token @\string#1\endcsname\relax% + \let\pgfmath@parse@next=\pgfmathparse@@ + \else + \let\pgfmath@parse@next=\pgfmathparse@trynumber@loop + \fi + \fi + \fi + \fi + \pgfmath@parse@next +}% + +\def\pgfmathparse@expression@is@number{% + \let\pgfmathresult=\pgfmath@expression + \pgfmath@smuggleone\pgfmathresult% + \endgroup + \ignorespaces +}% +\def\pgfmathparse@@#1\pgfmath@parse@stop{% + \let\real=\pgfmath@calc@real + \let\minof=\pgfmath@calc@minof + \let\maxof=\pgfmath@calc@maxof + \let\ratio=\pgfmath@calc@ratio + \let\widthof=\pgfmath@calc@widthof + \let\heightof=\pgfmath@calc@heightof + \let\depthof=\pgfmath@calc@depthof + \pgfmath@selectfont + \pgfmath@stack@operation={{}{}{}}% + \pgfmath@stack@operand={{}{}{}}% + \let\pgfmath@stack@operation@top=\pgfmath@stack@empty + \let\pgfmath@stack@operand@top=\pgfmath@stack@empty + \expandafter\pgfmath@bgroup@strip\expandafter{\pgfmath@expression}% + \expandafter\pgfmath@parse@@operand\pgfmath@bgroup@stripped @@@@\pgfmath@parse@stop% +} + +\def\pgfmath@parse@end#1\pgfmath@parse@stop{% + \pgfmathpostparse + \pgfmath@smuggleone\pgfmathresult% + \endgroup + \ignorespaces} + +\let\pgfmathpostparse=\relax + +\def\pgfmathscaleresult{% + \ifpgfmathunitsdeclared + \else + \ifpgfmathignoreunitscale + \else + \afterassignment\pgfmath@gobbletilpgfmath@% + \pgfmath@x\pgfmathresultunitscale pt\relax\pgfmath@% + \expandafter\pgfmath@x\pgfmathresult\pgfmath@x% + \edef\pgfmathresult{\pgfmath@tonumber{\pgfmath@x}}% + \fi + \fi +} + +\def\pgfmathsetresultunitscale#1{\def\pgfmathresultunitscale{#1}} + +\def\pgfmathresultunitscale{1} + + +\def\pgfmathifexpression#1#2#3{% + \pgfmathparse{#1}% + \pgfmath@iftrue{#2}{#3}% +}% +\def\pgfmath@iftrue{% + \ifdim\pgfmathresult pt=1.0pt % + \let\pgfmath@next=\pgfutil@firstoftwo + \else + \let\pgfmath@next=\pgfutil@secondoftwo + \fi + \pgfmath@next% +}% + +\def\pgfmath@calc@real#1{#1} +\def\pgfmath@calc@minof#1#2{min(#1,#2)} +\def\pgfmath@calc@maxof#1#2{max(#1,#2)} +\def\pgfmath@calc@ratio#1#2{#1/#2} +\def\pgfmath@calc@widthof#1{width("#1")} +\def\pgfmath@calc@heightof#1{height("#1")} +\def\pgfmath@calc@depthof#1{depth("#1")} + +\def\pgfmath@bgroup{\string\pgfmath@bgroup} +\def\pgfmath@egroup{\string\pgfmath@egroup} +\def\pgfmath@egroup@token{\pgfmath@egroup} + +\def\pgfmath@text@@{@} +\def\pgfmath@text@operator{operator} +\def\pgfmath@text@function{function} +\def\pgfmath@dots{...} +\def\pgfmath@char@zero{0} +\def\pgfmath@char@quote{"} +\def\pgfmath@char@exclamation{!} +\def\pgfmath@char@plus{+} +\def\pgfmath@char@minus{-} +\def\pgfmath@char@period{.} +\def\pgfmath@char@leftangle{<} +\def\pgfmath@char@leftbracket{[} +\pgfmath@toks={#} +\edef\pgfmath@char@hash{\the\pgfmath@toks} + +\def\pgfmath@tokens@make#1#2{% + \def\pgfmath@prefix{#1}% + \expandafter\pgfmath@tokens@@make#2{}} + +\def\pgfmath@tokens@@make#1{% + \def\pgfmath@token{#1}% + \ifx\pgfmath@token\pgfmath@empty + \else + \pgfmath@namedef{pgfmath@token@\pgfmath@prefix @\string#1}{#1}% + \expandafter\pgfmath@tokens@@make% + \fi} + +\pgfmath@tokens@make{box}{\wd\ht\dp} +\ifx\dimexpr\@undefined\else % + \pgfmath@tokens@make{dimexpr}{\dimexpr\glueexpr} +\fi +\pgfmath@tokens@make{unit}{{bp}{cc}{cm}{dd}{em}{ex}{in}{mm}{pc}{pt}{sp}} +\ifx\pdftexversion\@undefined\else % + \pgfmath@tokens@make{unit}{{nc}{nd}{px}} +\fi +\ifx\directlua\@undefined\else % + \pgfmath@tokens@make{unit}{{nc}{nd}{px}} +\fi +\ifx\kanjiskip\@undefined\else % + \pgfmath@tokens@make{unit}{{H}{Q}{zh}{zw}} +\fi +\pgfmath@tokens@make{mathunit}{{mu}} +\pgfmath@tokens@make{numeric}{.0123456789} +\pgfmath@tokens@make{number}{0123456789} +\pgfmath@tokens@make{functional}{_abcdefghijklmnopqrstuvwxyzABCDEFGHIJKLMNOPQRSTUVWXYZ} +\pgfmath@tokens@make{exponent}{eE} +\pgfmath@tokens@make{group}{{()}{[]}{\pgfmath@bgroup\pgfmath@egroup}} +\pgfmath@tokens@make{specials}{@)]\pgfmath@egroup} +\pgfmath@tokens@make{hexadecimal}{0123456789ABCDEFabcdef} +\pgfmath@tokens@make{octal}{01234567} +\pgfmath@tokens@make{binary}{01} + +\pgfmath@namedef{pgfmath@base@0x}{16} +\pgfmath@namedef{pgfmath@base@0X}{16} +\pgfmath@namedef{pgfmath@base@0b}{2} +\pgfmath@namedef{pgfmath@base@0B}{2} + +\newif\ifpgfmath@dimen@ +\def\pgfmath@dimen@#1{% + \begingroup + \afterassignment\pgfmath@dimen@@% + #1=0.0pt\relax\pgfmath@} +\def\pgfmath@dimen@@#1#2\pgfmath@{% + \endgroup + \ifx#1.% + \pgfmath@dimen@false + \else + \pgfmath@dimen@true + \fi} + +\def\pgfmath@parse@ifbgroup#1#2{% + \let\pgfmath@parse@bgroup@after=#1% + \let\pgfmath@parse@nobgroup@after=#2% + \futurelet\pgfmath@token@let% + \pgfmath@parse@@ifbgroup} + +\def\pgfmath@parse@@ifbgroup{% + \ifx\pgfmath@token@let\bgroup + \let\pgfmath@parse@next=\pgfmath@parse@@@ifbgroup + \else + \ifx\pgfmath@token@let\pgfmath@sptoken + \let\pgfmath@parse@next=\pgfmath@parse@@@@ifbgroup + \else + \let\pgfmath@parse@next=\pgfmath@parse@nobgroup@after + \fi + \fi + \pgfmath@parse@next} + +\newif\ifpgfmath@octalparsing +\pgfmath@octalparsingtrue + +\def\pgfmath@parse@@@ifbgroup#1{% + \pgfmath@parse@bgroup@after\pgfmath@bgroup#1\pgfmath@egroup} + +{ +\def\:{\pgfmath@parse@@@@ifbgroup} +\expandafter\gdef\: {\futurelet\pgfmath@token@let\pgfmath@parse@@ifbgroup} +} + +\def\pgfmath@parse@operand{% + \expandafter\pgfmath@parse@@operand\pgfmath@token@next} + +\def\pgfmath@parse@@operand{% + \pgfmath@parse@ifbgroup{\pgfmath@parse@@operand}{\pgfmath@parse@@operand@}} + +\def\pgfmath@parse@@operand@#1{% + \def\pgfmath@token{#1}% + \ifx\pgfmath@token\pgfmath@char@plus + \let\pgfmath@parse@next=\pgfmath@parse@@operand% + \else + \ifx\pgfmath@token\pgfmath@char@minus + \pgfmath@stack@push@operation{neg}% + \let\pgfmath@parse@next=\pgfmath@parse@@operand + \else + \expandafter\ifx\csname pgfmath@operation@\expandafter\string\pgfmath@token @prefix\endcsname\pgfmath@token + \let\pgfmath@parse@next=\pgfmath@parse@prefix@operator + \else + \ifpgfmath@quickparse + \let\pgfmath@parse@next=\pgfmath@qparse@operand + \else + \let\pgfmath@number=\pgfmath@empty + \expandafter\ifx\csname pgfmath@token@functional@\string#1\endcsname\relax% + \let\pgfmath@base=\pgfmath@empty% + \let\pgfmath@token@next=\pgfmath@token% + \if#10% + \ifpgfmath@octalparsing% + \def\pgfmath@base{8}% + \fi% + \fi% + \let\pgfmath@parse@next=\pgfmath@parse@number% + \else% + \let\pgfmath@function=\pgfmath@token% + \let\pgfmath@parse@next=\pgfmath@parse@function% + \fi% + \fi% + \fi% + \fi% + \fi% + \pgfmath@parse@next% +} + +\def\pgfmath@parse@prefix@operator{% + \ifx\pgfmath@token\pgfmath@char@quote% + \let\pgfmath@parse@next=\pgfmath@parse@operand@quote% + \else% + \ifx\pgfmath@token\pgfmath@char@exclamation% + \pgfmath@stack@push@operation{not}% + \let\pgfmath@parse@next=\pgfmath@parse@@operand% + \else% + \expandafter\pgfmath@stack@push@operation\expandafter{\pgfmath@token}% + \let\pgfmath@parse@next=\pgfmath@parse@@operand% + \fi% + \fi% + \pgfmath@parse@next% +}% + + +\def\pgfmath@parse@operand@quote#1"{% + \def\pgfmathresult{#1}% + \expandafter\pgfmath@stack@push@operand\expandafter{\pgfmathresult}% + \pgfmath@parse@@operator% +} + +\def\pgfmath@qparse@operand{% + \afterassignment\pgfmath@qparse@@operand% + \pgfmath@dimen\pgfmath@token} +\def\pgfmath@qparse@@operand{% + \edef\pgfmathresult{\pgfmath@tonumber{\pgfmath@dimen}}% + \expandafter\pgfmath@stack@push@operand\expandafter{\pgfmathresult}% + \pgfmath@parse@@operator% +} + +\def\pgfmath@parse@number{\pgfmath@parse@ifbgroup{\pgfmath@parse@number}{\pgfmath@parse@number@}} +\def\pgfmath@parse@number@#1{% + \let\pgfmath@token=\pgfmath@token@next% + \def\pgfmath@token@next{#1}% + \pgfmath@parse@@number} + +\def\pgfmath@parse@@number{% + \ifx\pgfmath@token\pgfmath@egroup@token% + \expandafter\pgfmath@stack@push@operand\expandafter{\pgfmath@number}% + \let\pgfmath@parse@next=\pgfmath@parse@@@operator% + \else% + \expandafter\ifcat\pgfmath@token\relax% + \expandafter\ifx\csname pgfmath@token@dimexpr@\expandafter\string\pgfmath@token\endcsname\relax% + \expandafter\ifx\pgfmath@token\numexpr% + \let\pgfmath@parse@next=\pgfmath@parse@numexpr% + \else% + \expandafter\ifx\pgfmath@token\muexpr% + \let\pgfmath@parse@next=\pgfmath@parse@muexpr% + \else% + \expandafter\ifx\csname pgfmath@token@box@\expandafter\string\pgfmath@token\endcsname\relax% + \expandafter\pgfmath@dimen@\pgfmath@token% + \ifpgfmath@dimen@% + \global\pgfmathunitsdeclaredtrue% + \pgfmath@dimen=\pgfmath@number\pgfmath@token% + \edef\pgfmathresult{\pgfmath@tonumber{\pgfmath@dimen}}% + \expandafter\pgfmath@stack@push@operand\expandafter{\pgfmathresult}% + \let\pgfmath@parse@next=\pgfmath@parse@operator% + \else% + \pgfmath@count=\pgfmath@token% + \expandafter\def\expandafter\pgfmath@number\expandafter{\the\pgfmath@count}% + \expandafter\ifcat\pgfmath@token@next\relax% + \let\pgfmath@parse@next=\pgfmath@parse@number% + \else% + \let\pgfmathresult=\pgfmath@number% + \expandafter\pgfmath@stack@push@operand\expandafter{\pgfmathresult}% + \let\pgfmath@parse@next=\pgfmath@parse@operator% + \fi% + \fi% + \else% + \global\pgfmathunitsdeclaredtrue% + \pgfmath@dimen=\pgfmath@number\pgfmath@token\pgfmath@token@next% + \edef\pgfmathresult{\pgfmath@tonumber{\pgfmath@dimen}}% + \expandafter\pgfmath@stack@push@operand\expandafter{\pgfmathresult}% + \let\pgfmath@parse@next=\pgfmath@parse@@operator% + \fi% + \fi% + \fi% + \else% + \global\pgfmathunitsdeclaredtrue% + \let\pgfmath@parse@next=\pgfmath@parse@dimexpr% + \fi% + \else% + \expandafter\ifx\csname pgfmath@token@numeric@\expandafter\string\pgfmath@token\endcsname\relax% + \expandafter\ifx\csname pgfmath@token@unit@\pgfmath@token\expandafter\string\pgfmath@token@next\endcsname\relax% + \expandafter\ifx\csname pgfmath@token@unit@\pgfmath@token\endcsname\relax% + \expandafter\ifx\csname pgfmath@token@exponent@\pgfmath@token\endcsname\relax% + \expandafter\ifx\csname pgfmath@base@\pgfmath@number\pgfmath@token\endcsname\relax% + \expandafter\ifx\csname pgfmath@token@mathunit@\pgfmath@token\expandafter\string\pgfmath@token@next\endcsname\relax% + \ifx\pgfmath@base\pgfmath@empty% + \let\pgfmathresult=\pgfmath@number% + \else% + \ifx\pgfmath@number\pgfmath@char@zero% + \let\pgfmathresult=\pgfmath@number% + \else% + \expandafter\pgfmathbasetobase\expandafter\pgfmathresult\expandafter{\pgfmath@number}{8}{10}% + \fi% + \fi% + \expandafter\pgfmath@stack@push@operand\expandafter{\pgfmathresult}% + \let\pgfmath@parse@next=\pgfmath@parse@@@operator% + \else% + \global\pgfmathunitsdeclaredtrue% + \global\pgfmathmathunitsdeclaredtrue% + \pgfmath@dimen=\pgfmath@number pt\relax% + \edef\pgfmathresult{\pgfmath@tonumber{\pgfmath@dimen}}% + \expandafter\pgfmath@stack@push@operand\expandafter{\pgfmathresult}% + \let\pgfmath@parse@next=\pgfmath@parse@@operator% + \fi% + \else% + \expandafter\let\expandafter\pgfmath@base\expandafter=% + \csname pgfmath@base@\pgfmath@number\pgfmath@token\endcsname% + \let\pgfmath@parse@next=\pgfmath@parse@base% + \fi% + \else% + \let\pgfmath@parse@next=\pgfmath@parse@exponent% + \fi% + \else% + \global\pgfmathunitsdeclaredtrue% + \pgfmath@dimen=\pgfmath@number\pgfmath@token\relax% + \edef\pgfmathresult{\pgfmath@tonumber{\pgfmath@dimen}}% + \expandafter\pgfmath@stack@push@operand\expandafter{\pgfmathresult}% + \let\pgfmath@parse@next=\pgfmath@parse@@operator% + \fi% + \else% + \global\pgfmathunitsdeclaredtrue% + \pgfmath@dimen=\pgfmath@number\pgfmath@token\pgfmath@token@next\relax% + \edef\pgfmathresult{\pgfmath@tonumber{\pgfmath@dimen}}% + \expandafter\pgfmath@stack@push@operand\expandafter{\pgfmathresult}% + \let\pgfmath@parse@next=\pgfmath@parse@@operator% + \fi% + \else% + \expandafter\expandafter\expandafter\def\expandafter\expandafter\expandafter\pgfmath@number% + \expandafter\expandafter\expandafter{\expandafter\pgfmath@number\pgfmath@token}% + \ifx\pgfmath@token\pgfmath@char@period% + \let\pgfmath@base=\pgfmath@empty% + \fi% + \let\pgfmath@parse@next=\pgfmath@parse@number% + \fi% + \fi% + \fi% + \pgfmath@parse@next% +} + + +\def\pgfmath@parse@dimexpr{% + \afterassignment\pgfmath@parse@dimexpr@% + \expandafter\expandafter\expandafter\pgfmath@dimen\expandafter\pgfmath@token\pgfmath@token@next% +} +\def\pgfmath@parse@dimexpr@{% + \edef\pgfmathresult{\pgfmath@tonumber{\pgfmath@dimen}}% + \expandafter\pgfmath@stack@push@operand\expandafter{\pgfmathresult}% + \pgfmath@parse@@operator% +} + + +\def\pgfmath@parse@muexpr{% + \begingroup% + \afterassignment\pgfmath@parse@muexpr@% + \expandafter\expandafter\expandafter\muskip\expandafter\expandafter\expandafter0\expandafter\pgfmath@token\expandafter\pgfmath@token@next% +} +\def\pgfmath@parse@muexpr@{% + \expandafter\endgroup% + \expandafter\edef\expandafter\pgfmathresult\expandafter{\expandafter\pgfMATH@STRIPMU\the\muskip0}% + \expandafter\pgfmath@stack@push@operand\expandafter{\pgfmathresult}% + \pgfmath@parse@@operator% +} + +{ +\catcode`\m=12 +\catcode`\u=12 +\gdef\pgfMATH@STRIPMU#1mu{#1} +} + + +\def\pgfmath@parse@numexpr{% + \afterassignment\pgfmath@parse@numexpr@% + \expandafter\expandafter\expandafter\pgfmath@count\expandafter\pgfmath@token\pgfmath@token@next% +} +\def\pgfmath@parse@numexpr@#1{% + \expandafter\def\expandafter\pgfmath@number\expandafter{\the\pgfmath@count}% + \def\pgfmath@token@next{#1}% + \expandafter\ifcat\pgfmath@token@next\relax% + \let\pgfmath@parse@next=\pgfmath@parse@number% + \else% + \let\pgfmathresult=\pgfmath@number% + \expandafter\pgfmath@stack@push@operand\expandafter{\pgfmathresult}% + \let\pgfmath@parse@next=\pgfmath@parse@operator% + \fi% + \pgfmath@parse@next% +} + + +\def\pgfmath@parse@exponent{% + \afterassignment\pgfmath@parse@@exponent% + \pgfmath@count\pgfmath@token@next} +\def\pgfmath@parse@@exponent{% + \pgfmathscientific{\pgfmath@number}{\the\pgfmath@count}% + \expandafter\pgfmath@stack@push@operand\expandafter{\pgfmathresult}% + \pgfmath@parse@@operator% +} + +\def\pgfmath@parse@base{% + \ifnum\pgfmath@base=2\relax% + \let\pgfmath@parse@next=\pgfmath@parse@binary% + \else% + \ifnum\pgfmath@base=16\relax% + \let\pgfmath@parse@next=\pgfmath@parse@hexadecimal% + \else% + \pgfmath@error{Unexpected numerical base `\pgfmath@base'}{}% + \def\pgfmathresult{0}% + \let\pgfmath@parse@next=\pgfmath@parse@end% + \fi% + \fi% + \let\pgfmath@number=\pgfmath@empty% + \expandafter\pgfmath@parse@next\pgfmath@token@next% +} + +\def\pgfmath@parse@binary#1{% + \expandafter\ifx\csname pgfmath@token@binary@#1\endcsname\relax% + \expandafter\pgfmathbasetobase\expandafter\pgfmathresult\expandafter{\pgfmath@number}{2}{10}% + \expandafter\pgfmath@stack@push@operand\expandafter{\pgfmathresult}% + \def\pgfmath@token@next{#1}% + \let\pgfmath@parse@next=\pgfmath@parse@operator% + \else% + \expandafter\def\expandafter\pgfmath@number\expandafter{\pgfmath@number#1}% + \let\pgfmath@parse@next=\pgfmath@parse@binary% + \fi% + \pgfmath@parse@next% +} + +\def\pgfmath@parse@hexadecimal#1{% + \expandafter\ifx\csname pgfmath@token@hexadecimal@#1\endcsname\relax% + \expandafter\pgfmathbasetobase\expandafter\pgfmathresult\expandafter{\pgfmath@number}{16}{10}% + \expandafter\pgfmath@stack@push@operand\expandafter{\pgfmathresult}% + \def\pgfmath@token@next{#1}% + \let\pgfmath@parse@next=\pgfmath@parse@operator% + \else% + \expandafter\def\expandafter\pgfmath@number\expandafter{\pgfmath@number#1}% + \let\pgfmath@parse@next=\pgfmath@parse@hexadecimal% + \fi% + \pgfmath@parse@next% +} + +\def\pgfmath@parse@function{\pgfmath@parse@ifbgroup{\pgfmath@parse@function}{\pgfmath@parse@@function}} +\def\pgfmath@parse@function@{\futurelet\pgfmath@token@let\pgfmath@parse@@function} +\def\pgfmath@parse@@function{% + \ifx\pgfmath@token@let\pgfmath@sptoken% + \let\pgfmath@token=\pgfmath@empty% + \let\pgfmath@parse@next=\pgfmath@parse@@@@function% + \else% + \let\pgfmath@parse@next=\pgfmath@parse@@@function% + \fi% + \pgfmath@parse@next} + +\def\pgfmath@parse@@@function#1{% + \expandafter\ifx\csname pgfmath@token@functional@\expandafter\string#1\endcsname\relax% + \expandafter\ifx\csname pgfmath@token@number@\expandafter\string#1\endcsname\relax% + \def\pgfmath@token{#1}% + \let\pgfmath@parse@next=\pgfmath@parse@@@@function% + \else% + \expandafter\def\expandafter\pgfmath@function\expandafter{\pgfmath@function#1}% + \let\pgfmath@parse@next=\pgfmath@parse@function@% + \fi% + \else% + \edef\pgfmath@function{\pgfmath@function#1}% + \let\pgfmath@parse@next=\pgfmath@parse@function@% + \fi% + \pgfmath@parse@next} + +\def\pgfmath@parse@@@@function{% + \expandafter\ifx\csname pgfmath@function@\pgfmath@function\endcsname\relax% + \pgfmath@error{Unknown function `\pgfmath@function'}{}% + \def\pgfmathresult{0}% + \let\pgfmath@parse@next=\pgfmath@parse@end% + \else% + \expandafter\ifnum\csname pgfmath@operation@\pgfmath@function @arity\endcsname=0\relax% + \csname pgfmath\pgfmath@function @\endcsname% + \expandafter\pgfmath@stack@push@operand\expandafter{\pgfmathresult}% + \let\pgfmath@parse@next=\pgfmath@parse@@operator% + \else% + \expandafter\pgfmath@stack@push@operation\expandafter{\pgfmath@function}% + \let\pgfmath@parse@next=\pgfmath@parse@@operand% + \fi% + \fi% + \expandafter\pgfmath@parse@next\pgfmath@token} + +\def\pgfmath@parse@operator{\pgfmath@parse@ifbgroup{\pgfmath@parse@operator}{\pgfmath@parse@operator@}}% +\def\pgfmath@parse@operator@#1{% + \let\pgfmath@token=\pgfmath@token@next% + \def\pgfmath@token@next{#1}% + \pgfmath@parse@@@operator} + +\def\pgfmath@parse@@operator{\pgfmath@parse@ifbgroup{\pgfmath@parse@@operator}{\pgfmath@parse@@operator@}}% +\def\pgfmath@parse@@operator@#1{% + \def\pgfmath@token{#1}% + \pgfmath@parse@ifbgroup{\pgfmath@parse@@operator@@}{\pgfmath@parse@@operator@@}}% +\def\pgfmath@parse@@operator@@#1{% + \def\pgfmath@token@next{#1}% + \pgfmath@parse@@@operator} + +\def\pgfmath@parse@@@operator{% + \expandafter\ifx\csname pgfmath@operator@\pgfmath@token\expandafter\string\pgfmath@token@next\endcsname\relax% + \expandafter\ifx\csname pgfmath@operator@\pgfmath@token\endcsname\relax% + \pgfmath@error{Unknown operator `\pgfmath@token' or `\pgfmath@token\pgfmath@token@next'}{}% + \def\pgfmathresult{0}% + \let\pgfmath@parse@next=\pgfmath@parse@end% + \else% + \expandafter\pgfmath@operation@process\expandafter{\pgfmath@token}% + \ifx\pgfmath@token\pgfmath@text@@% + \let\pgfmath@parse@next=\pgfmath@parse@end% + \else% + \expandafter\ifx\csname pgfmath@operation@\pgfmath@token @postfix\endcsname\pgfmath@token% + \ifx\pgfmath@token@next\pgfmath@char@leftbracket% + \let\pgfmath@parse@next=\pgfmath@parse@operand% + \else% + \let\pgfmath@parse@next=\pgfmath@parse@operator% + \fi% + \else% + \let\pgfmath@parse@next=\pgfmath@parse@operand% + \fi% + \fi% + \fi% + \else% + \expandafter\expandafter\expandafter\def\expandafter\expandafter\expandafter% + \pgfmath@token\expandafter\expandafter\expandafter% + {\expandafter\pgfmath@token\pgfmath@token@next}% + \expandafter\pgfmath@operation@process\expandafter{\pgfmath@token}% + \let\pgfmath@parse@next=\pgfmath@parse@@operand% + \fi% + \pgfmath@parse@next% +} + + +\def\pgfmath@operation@process#1{% + \def\pgfmath@operator{#1}% + \ifx\pgfmath@stack@operation@top\pgfmath@stack@empty% + \else% + \pgfmath@operation@@process% + \fi% + \expandafter\ifx\csname pgfmath@token@specials@#1\endcsname\relax% + \pgfmath@stack@push@operation{#1}% + \fi% +} + +\def\pgfmath@operation@@process{% + \ifx\pgfmath@stack@operation@top\pgfmath@stack@empty% + \let\pgfmath@parse@next=\relax% + \else% + \ifnum\csname pgfmath@operation@\pgfmath@operator @precedence\endcsname>% + \csname pgfmath@operation@\pgfmath@stack@operation@top @precedence\endcsname\relax% + \let\pgfmath@parse@next=\relax% + \else% + \let\pgfmath@stack@operation@last=\pgfmath@stack@operation@top% + \expandafter\ifx\csname pgfmath@operator@\pgfmath@stack@operation@top\endcsname\relax% + \pgfmath@function@@@process% + \else% + \pgfmath@operator@@@process% + \fi% + \expandafter\pgfmath@stack@push@operand\expandafter{\pgfmathresult}% + \expandafter\ifx\csname pgfmath@token@group@\pgfmath@stack@operation@last\pgfmath@operator\endcsname\relax% + \let\pgfmath@parse@next=\pgfmath@operation@@process% + \else% + \let\pgfmath@parse@next=\relax% + \fi% + \fi% + \fi% + \pgfmath@parse@next% +} + +\def\pgfmath@operator@@@process{% + \pgfmath@stack@pop@operation\pgfmath@@operator% + \pgfmath@stack@pop@operand\pgfmath@operand@a% + \ifcase\csname pgfmath@operation@\pgfmath@@operator @arity\endcsname\relax% + \or% + \pgfmath@operator@@@@process{\pgfmath@operand@a}% + \or% + \pgfmath@stack@pop@operand\pgfmath@operand@b% + \pgfmath@operator@@@@process{\pgfmath@operand@b}{\pgfmath@operand@a}% + \or% + \pgfmath@stack@pop@operand\pgfmath@operand@b% + \expandafter\pgfmath@operator@@@@process\expandafter% + {\expandafter\pgfmath@operand@b\expandafter}\pgfmath@operand@a% + \else% + \fi% +} + +\def\pgfmath@operator@@@@process{% + \csname pgfmath% + \csname pgfmath@operation@\pgfmath@@operator @name\endcsname% + @\endcsname} + +\def\pgfmath@function@@@process{% + \ifnum\csname pgfmath@operation@\pgfmath@stack@operation@top @arity\endcsname>0\relax% + \pgfmath@stack@pop@operation\pgfmath@@function% + \pgfmath@stack@pop@operand\pgfmath@@operand% + \ifnum\csname pgfmath@operation@\pgfmath@@function @arity\endcsname>1\relax% + \ifnum\csname pgfmath@operation@\pgfmath@@function @arity\endcsname>9\relax% + \csname pgfmath\pgfmath@@function @\endcsname{\pgfmath@@operand}% + \else% + \expandafter\pgfmath@function@@@@process\pgfmath@@operand% + \fi + \else% + \expandafter\pgfmath@function@@@@process\expandafter{\pgfmath@@operand}% + \fi% + \fi% +} + +\def\pgfmath@function@@@@process{\csname pgfmath\pgfmath@@function @\endcsname} + +\newtoks\pgfmath@stack@operand +\newtoks\pgfmath@stack@operation +\let\pgfmath@stack@empty=\pgfmath@empty + +\def\pgfmath@stack@push#1#2{% + \def\pgfmath@stack@top{#2}% + \expandafter\expandafter\expandafter#1% + \expandafter\expandafter\expandafter{\expandafter\expandafter\expandafter% + {\expandafter\pgfmath@stack@top\expandafter}\the#1}}% + +\def\pgfmath@stack@pop#1#2{\expandafter\pgfmath@stack@@pop\the#1\pgfmath@stack@stop{#1}{#2}} +\def\pgfmath@stack@@pop#1#2#3\pgfmath@stack@stop#4#5{\def#5{#1}#4={{#2}#3}\def\pgfmath@stack@top{#2}} + +\def\pgfmath@stack@push@operation#1{% + \pgfmath@stack@push\pgfmath@stack@operation{#1}% + \let\pgfmath@stack@operation@top=\pgfmath@stack@top} + +\def\pgfmath@stack@pop@operation#1{% + \pgfmath@stack@pop\pgfmath@stack@operation{#1}% + \let\pgfmath@stack@operation@top=\pgfmath@stack@top} + +\def\pgfmath@stack@push@operand#1{% + \pgfmath@stack@push\pgfmath@stack@operand{#1}% + \let\pgfmath@stack@operand@top=\pgfmath@stack@top} + +\def\pgfmath@stack@pop@operand#1{% + \pgfmath@stack@pop\pgfmath@stack@operand{#1}% + \let\pgfmath@stack@operand@top=\pgfmath@stack@top} + + +\def\pgfmathdeclareoperator#1#2#3#4#5{% + \ifx\pgfmath@operatorlist\pgfutil@empty + \edef\pgfmath@operatorlist{{\string#1}}% + \else + \edef\pgfmath@operatorlist{\pgfmath@operatorlist,{\string#1}}% + \fi + \pgfmath@namedef{pgfmath@operator@\string#1}{#1}% + \pgfmath@namedef{pgfmath@operation@\string#1@name}{#2}% + \pgfmath@namedef{pgfmath@operation@\string#1@arity}{#3}% + \pgfmath@namedef{pgfmath@operation@\string#1@#4}{#1}% + \pgfmath@namedef{pgfmath@operation@\string#1@precedence}{#5}% +} + +\def\pgfmath@operatorlist{} + +\pgfmathdeclareoperator{+}{add} {2}{infix}{500} +\pgfmathdeclareoperator{-}{subtract}{2}{infix}{500} +\pgfmathdeclareoperator{*}{multiply}{2}{infix}{700} +\pgfmathdeclareoperator{/}{divide} {2}{infix}{700} +\pgfmathdeclareoperator{^}{pow} {2}{infix}{900} + +\pgfmathdeclareoperator{<} {} {1}{prefix} {1} + +\pgfmathdeclareoperator{>} {greater} {2}{infix} {250} +\pgfmathdeclareoperator{<} {less} {2}{infix} {250} +\pgfmathdeclareoperator{==}{equal} {2}{infix} {250} +\pgfmathdeclareoperator{>=}{notless} {2}{infix} {250} +\pgfmathdeclareoperator{<=}{notgreater}{2}{infix} {250} +\pgfmathdeclareoperator{&&}{and} {2}{infix} {200} +\pgfmathdeclareoperator{||}{or} {2}{infix} {200} +\pgfmathdeclareoperator{!=}{notequal} {2}{infix} {250} +\pgfmathdeclareoperator{!} {not} {1}{prefix}{975} +\pgfmathdeclareoperator{?} {ifthenelse}{3}{infix} {100} +\pgfmathdeclareoperator{:} {@@collect} {2}{infix}{101} + +\pgfmathdeclareoperator{!}{factorial}{1}{postfix}{800}% +\pgfmathdeclareoperator{r}{deg} {1}{postfix}{600} + +\pgfmathdeclareoperator{,}{@collect} {2}{infix} {10} +\pgfmathdeclareoperator{[}{@startindex}{2}{prefix} {7} +\pgfmathdeclareoperator{]}{@endindex} {0}{postfix}{6} +\pgfmathdeclareoperator{(}{@startgroup}{1}{prefix} {5} +\pgfmathdeclareoperator{)}{@endgroup} {0}{postfix}{4} + +\pgfmathdeclareoperator{\pgfmath@bgroup}{@startarray}{1}{prefix} {3} +\pgfmathdeclareoperator{\pgfmath@egroup}{@endarray} {1}{postfix}{2}% + +\pgfmathdeclareoperator{"}{}{1}{prefix}{1} +\pgfmathdeclareoperator{@}{}{0}{postfix}{0} + + +\def\pgfmath@startgroup@#1{\pgfmath@protected@edef\pgfmathresult{#1}} +\def\pgfmath@endgroup@{}% + +\def\pgfmath@startindex@#1#2{% + \expandafter\pgfmatharray@#1{#2}% + \expandafter\pgfmath@bgroup@strip\expandafter{\pgfmathresult}% + \ifx\pgfmath@token@let\bgroup% + \expandafter\def\expandafter\pgfmathresult\expandafter{\expandafter{\pgfmathresult}}% + \fi} +\def\pgfmath@endindex@{}% + +\def\pgfmath@startarray@#1{\expandafter\def\expandafter\pgfmathresult\expandafter{\expandafter{#1}}} +\def\pgfmath@endarray@#1{\expandafter\def\expandafter\pgfmathresult\expandafter{#1}}% + + +\def\pgfmath@@collect@#1#2{% + \expandafter\expandafter\expandafter\def\expandafter\expandafter\expandafter% + \pgfmathresult\expandafter\expandafter\expandafter{\expandafter\expandafter\expandafter% + {\expandafter#1\expandafter}\expandafter{#2}}% +} +\def\pgfmath@collect@#1#2{% + \expandafter\def\expandafter\pgfmath@collect@original\expandafter{#1}% + \expandafter\pgfmath@bgroup@strip\expandafter{#1}% + \ifx\pgfmath@token@let\bgroup% + \ifx\pgfmath@collect@original\pgfmath@bgroup@stripped% + \expandafter\def\expandafter\pgfmathresult\expandafter{\pgfmath@bgroup@stripped}% + \else% + \expandafter\def\expandafter\pgfmathresult\expandafter{\expandafter{\pgfmath@bgroup@stripped}}% + \fi% + \else% + \expandafter\def\expandafter\pgfmathresult\expandafter{\expandafter{\pgfmath@bgroup@stripped}}% + \fi% + \expandafter\pgfmath@bgroup@strip\expandafter{#2}% + \expandafter\expandafter\expandafter\def\expandafter\expandafter\expandafter% + \pgfmathresult\expandafter\expandafter\expandafter% + {\expandafter\pgfmathresult\expandafter{\pgfmath@bgroup@stripped}}% +} + +\def\pgfmath@bgroup@strip#1{% + \def\pgfmath@bgroup@strip@last{#1}% + \pgfmath@bgroup@@strip{#1}\pgfmath@\pgfmath@@\pgfmath@@@} + +\def\pgfmath@bgroup@strip@#1{% + \def\pgfmath@bgroup@strip@last{#1}% + \pgfmath@bgroup@@strip#1\pgfmath@\pgfmath@@\pgfmath@@@} + +\def\pgfmath@bgroup@pgfmath@{\pgfmath@} + +\def\pgfmath@bgroup@@strip{% + \futurelet\pgfmath@token@let\pgfmath@bgroup@@@strip} + +\def\pgfmath@bgroup@@@strip#1#2#3\pgfmath@@@{% + \ifx\pgfmath@token@let\bgroup + \def\pgfmath@bgroup@test{#2}% + \ifx\pgfmath@bgroup@test\pgfmath@bgroup@pgfmath@ + \def\pgfmath@next{\pgfmath@bgroup@strip@{#1}}% + \else + \let\pgfmath@bgroup@stripped=\pgfmath@bgroup@strip@last + \let\pgfmath@next=\relax + \fi + \else + \pgfmath@bgroup@@@@strip#1#2#3% + \let\pgfmath@next=\relax + \fi + \pgfmath@next} + +\def\pgfmath@bgroup@@@@strip#1\pgfmath@\pgfmath@@{% + \def\pgfmath@bgroup@stripped{#1}} + diff --git a/preprints/A-polynomial-time-algorithm-for-three-machine-unit-job-scheduling-September-24-2026/build/tex/runtime/generic/pgf/math/pgfmathutil.code.tex b/preprints/A-polynomial-time-algorithm-for-three-machine-unit-job-scheduling-September-24-2026/build/tex/runtime/generic/pgf/math/pgfmathutil.code.tex new file mode 100644 index 0000000000000000000000000000000000000000..77d9e65d46cbdc0a0ed54d63bb92f9aecfe570fd --- /dev/null +++ b/preprints/A-polynomial-time-algorithm-for-three-machine-unit-job-scheduling-September-24-2026/build/tex/runtime/generic/pgf/math/pgfmathutil.code.tex @@ -0,0 +1,209 @@ + + +\def\pgfmathincluded{} + +\def\pgfmath@grabv#1#2#3#4#5#6#7\pgfmath@grabv{#2#3#4#5\ifx#6!!\fi} +\def\pgfmath@ifskip@extract#1{\expandafter\pgfmath@grabv\meaning#1!!!!!!\pgfmath@grabv} +\def\pgfmath@iftoks@extract#1{\expandafter\pgfmath@grabv\meaning#1!!!!!!\pgfmath@grabv} + +\def\pgfmath@grabvi#1#2#3#4#5#6#7#8\pgfmath@grabvi{#2#3#4#5#6\ifx#7!!\fi} +\def\pgfmath@ifdimen@extract#1{\expandafter\pgfmath@grabvi\meaning#1!!!!!!\pgfmath@grabvi} +\def\pgfmath@ifcount@extract#1{\expandafter\pgfmath@grabvi\meaning#1!!!!!!\pgfmath@grabvi} + +\def\pgfmath@grabvii#1#2#3#4#5#6#7#8#9\pgfmath@grabvii{#2#3#4#5#6#7\ifx#8!!\fi} +\def\pgfmath@ifmuskip@extract#1{\expandafter\pgfmath@grabvii\meaning#1!!!!!!\pgfmath@grabvii} + +\def\pgfmath@ifregister@unguarded#1#2{% + \expandafter\ifx\csname\csname pgfmath@if#1@extract\endcsname{#2}\expandafter\endcsname\csname #1\endcsname + \expandafter\pgfutil@firstoftwo + \else + \expandafter\pgfutil@secondoftwo + \fi} + +\def\pgfmath@ifregister#1#2{% + \pgfutil@IfUndefined{pgfmath@if#1@extract}% + {\errmessage{Package PGF Math Error: Cannot test for #1 register}}% + {\pgfutil@IfUndefined{\csname pgfmath@if#1@extract\endcsname{#2}}% + {\pgfutil@secondoftwo}}% + {\pgfmath@ifregister@unguarded{#1}{#2}}} + +\def\pgfmath@ensureregister#1#2{% + \pgfmath@ifregister{#1}{#2}% + {}% + {\csname new#1\endcsname#2}% +} + +\pgfmath@ensureregister{dimen}{\pgf@x} +\pgfmath@ensureregister{dimen}{\pgf@xa} +\pgfmath@ensureregister{dimen}{\pgf@xb} +\pgfmath@ensureregister{dimen}{\pgf@xc} + +\pgfmath@ensureregister{dimen}{\pgf@y} +\pgfmath@ensureregister{dimen}{\pgf@ya} +\pgfmath@ensureregister{dimen}{\pgf@yb} +\pgfmath@ensureregister{dimen}{\pgf@yc} + +\pgfmath@ensureregister{dimen}{\pgfutil@tempdima} +\pgfmath@ensureregister{dimen}{\pgfutil@tempdimb} + +\pgfmath@ensureregister{count}{\c@pgf@counta} +\pgfmath@ensureregister{count}{\c@pgf@countb} +\pgfmath@ensureregister{count}{\c@pgf@countc} +\pgfmath@ensureregister{count}{\c@pgf@countd} + +\pgfmath@ensureregister{count}{\pgfutil@tempcnta} +\pgfmath@ensureregister{count}{\pgfutil@tempcntb} + +\let\pgfmath@x\pgf@x +\let\pgfmath@xa\pgf@xa +\let\pgfmath@xb\pgf@xb +\let\pgfmath@xc\pgf@xc + +\let\pgfmath@y\pgf@y +\let\pgfmath@ya\pgf@ya +\let\pgfmath@yb\pgf@yb +\let\pgfmath@yc\pgf@yc + +\let\pgfmath@tempdima\pgfutil@tempdima +\let\pgfmath@tempdimb\pgfutil@tempdimb + +\let\c@pgfmath@counta\c@pgf@counta +\let\c@pgfmath@countb\c@pgf@countb +\let\c@pgfmath@countc\c@pgf@countc +\let\c@pgfmath@countd\c@pgf@countd + +\let\pgfmath@tempcnta\pgfutil@tempcnta +\let\pgfmath@tempcntb\pgfutil@tempcntb + + +\ifx\pgfutil@ifnextchar\pgfmath@undefined% + \long\def\pgfmath@ifnextchar#1#2#3{% + \let\pgfmath@reserved@d#1% + \def\pgfmath@reserved@a{#2}% + \def\pgfmath@reserved@b{#3}% + \futurelet\pgfmath@let@token\pgfmath@ifnch} + \def\pgfmath@ifnch{% + \ifx\pgfmath@let@token\pgfmath@sptoken% + \let\pgfmath@reserved@c\pgfmath@xifnch% + \else% + \ifx\pgfmath@let@token\pgfmath@reserved@d% + \let\pgfmath@reserved@c\pgfmath@reserved@a% + \else% + \let\pgfmath@reserved@c\pgfmath@reserved@b% + \fi% + \fi% + \pgfmath@reserved@c} + {% + \def\:{\global\let\pgfmath@sptoken= } \: + \def\:{\pgfmath@xifnch} \expandafter\gdef\: {\futurelet\pgfmath@let@token\pgfmath@ifnch} + }% +\else% + \let\pgfmath@ifnextchar\pgfutil@ifnextchar% + {% + \def\:{\global\let\pgfmath@sptoken= } \: + }% +\fi% + +\ifx\pgfutil@ifundefined\pgfmath@undefined% + \long\def\pgfmath@ifundefined#1{% + \expandafter\ifx\csname#1\endcsname\relax% + \expandafter\pgfmath@firstoftwo% + \else% + \expandafter\pgfmath@secondoftwo% + \fi}% +\else% + \let\pgfmath@ifundefined\pgfutil@ifundefined% +\fi% + +\ifx\pgfutil@selectfont\pgfmath@undefined% + \ifx\selectfont\pgfmath@undefined% + \let\pgfmath@selectfont\rm% + \else% + \let\pgfmath@selectfont\selectfont% + \fi% +\else% + \let\pgfmath@selectfont\pgfutil@selectfont% +\fi% + +\newif\ifpgfmath@in@ +\def\pgfmath@in@#1#2{% + \def\pgfmath@in@@##1#1##2##3\pgfmath@in@@{% + \ifx\pgfmath@in@##2\pgfmath@in@false\else\pgfmath@in@true\fi}% + \pgfmath@in@@#2#1\pgfmath@in@\pgfmath@in@@} + + +\ifx\pgf@sys@tonumber\pgfmath@undefined% + {\catcode`\p=12\catcode`\t=12\gdef\Pgf@geT#1pt{#1}} + \def\pgfmath@tonumber#1{\expandafter\Pgf@geT\the#1} +\else% + \let\pgfmath@tonumber\pgf@sys@tonumber% +\fi% + + +\ifx\PackageWarning\pgfmath@undefined + \def\pgfmath@PackageWarning#1#2{\immediate\write-1{Package #1: Warning! #2.}}% +\else% + \let\pgfmath@PackageWarning\PackageWarning% +\fi% +\ifx\PackageError\pgfmath@undefined + \def\pgfmath@PackageError#1#2#3{\errmessage{Package #1 Error: #2.}}% +\else% + \let\pgfmath@PackageError\PackageError% +\fi% + +\def\pgfmath@error#1#2{% + \pgfmath@PackageError{PGF Math}{% + #1% + \pgfutil@ifundefined{pgfmath@expression}{}{ (in '\pgfmath@expression')}% + }{#2}% +} +\def\pgfmath@warning#1{% + \pgfmath@PackageWarning{PGF Math}{% + #1% + \pgfutil@ifundefined{pgfmath@expression}{}{ (in '\pgfmath@expression')}% + }% +} + +{\catcode`\p=12\catcode`\t=12\gdef\PgfmaTh@PT{pt}} +\edef\pgfmath@pt{\PgfmaTh@PT}% + +\def\pgfmath@empty{} + +\def\pgfmath@firstoftwo#1#2{#1} +\def\pgfmath@secondoftwo#1#2{#2} + +\def\pgfmath@gobbletilpgfmath@#1\pgfmath@{} +\def\pgfmath@gobbleone#1{}% + + +\def\pgfmath@namedef#1{\expandafter\def\csname#1\endcsname} +\def\pgfmath@nameedef#1{\expandafter\edef\csname#1\endcsname} + +\newif\ifpgfmathcontinueloop +\def\pgfmathloop#1\repeatpgfmathloop{% + \def\pgfmathcounter{1}% + \def\pgfmath@iterate{% + #1\relax% + {% + \c@pgfmath@counta\pgfmathcounter% + \advance\c@pgfmath@counta1\relax% + \xdef\pgfmathloop@temp{\the\c@pgfmath@counta}% + }% + \edef\pgfmathcounter{\pgfmathloop@temp}% + \expandafter\pgfmath@iterate\fi}% + \pgfmath@iterate\let\pgfmath@iterate\relax} +\let\repeatpgfmathloop\fi +\def\pgfmathbreakloop{\let\pgfmath@iterate\relax}% + +\def\pgfmath@returnone#1\endgroup{% + \pgfmath@x#1% + \edef\pgfmath@temp{\pgfmath@tonumber{\pgfmath@x}}% + \expandafter\endgroup\expandafter\def\expandafter\pgfmathresult\expandafter{\pgfmath@temp}% +} + +\let\pgfmathreturn=\pgfmath@returnone + +\def\pgfmath@smuggleone#1\endgroup{% + \expandafter\endgroup\expandafter\def\expandafter#1\expandafter{#1}} + +\let\pgfmathsmuggle=\pgfmath@smuggleone diff --git a/preprints/A-polynomial-time-algorithm-for-three-machine-unit-job-scheduling-September-24-2026/build/tex/runtime/generic/pgf/modules/pgfmoduledecorations.code.tex b/preprints/A-polynomial-time-algorithm-for-three-machine-unit-job-scheduling-September-24-2026/build/tex/runtime/generic/pgf/modules/pgfmoduledecorations.code.tex new file mode 100644 index 0000000000000000000000000000000000000000..490e160f02285c12bd3912ca8064466c86ea477b --- /dev/null +++ b/preprints/A-polynomial-time-algorithm-for-three-machine-unit-job-scheduling-September-24-2026/build/tex/runtime/generic/pgf/modules/pgfmoduledecorations.code.tex @@ -0,0 +1,1449 @@ + + + + + +\def\pgfgetpath#1{\pgfsyssoftpath@getcurrentpath{#1}}% +\def\pgfsetpath#1{\pgfsyssoftpath@setcurrentpath{#1}}% + + +\newdimen\pgfdecoratedcompleteddistance +\newdimen\pgfdecoratedremainingdistance +\newdimen\pgfdecoratedinputsegmentcompleteddistance +\newdimen\pgfdecoratedinputsegmentremainingdistance +\newdimen\pgf@decorate@distancetomove + +\newcount\pgf@decorate@repeatstate + + + +\newdimen\pgfdecorationsegmentamplitude +\newdimen\pgfdecorationsegmentlength +\pgfdecorationsegmentamplitude2.5pt +\pgfdecorationsegmentlength10pt +\def\pgfdecorationsegmentangle{45}% +\def\pgfdecorationsegmentaspect{0.5}% +\def\pgfmetadecorationsegmentamplitude{2.5pt}% +\def\pgfmetadecorationsegmentlength{1cm}% + +\pgfkeys{% + /pgf/decoration/.code={\pgfkeys{/pgf/decoration/.cd,#1}}, + /pgf/decoration/.cd, + amplitude/.code={\pgfmathsetlength\pgfdecorationsegmentamplitude{#1}}, + meta-amplitude/.store in=\pgfmetadecorationsegmentamplitude, + segment length/.code={\pgfmathsetlength\pgfdecorationsegmentlength{#1}}, + meta-segment length/.store in=\pgfmetadecorationsegmentlength, + angle/.code={\pgfmathparse{#1}\let\pgfdecorationsegmentangle\pgfmathresult}, + aspect/.code={\pgfmathparse{#1}\let\pgfdecorationsegmentaspect\pgfmathresult}, + start radius/.initial=2.5pt, + end radius/.initial=2.5pt, + radius/.style={start radius={#1},end radius={#1}}, + path has corners/.is if=pgfdecoratepathhascorners, + reverse path/.is if=pgf@decorate@inputsegmentobjects@reverse, +}% +\newif\ifpgfdecoratepathhascorners +\newif\ifpgf@decorate@inputsegmentobjects@reverse + + +\long\def\pgfdeclaredecoration#1#2#3{% + \let\pgf@mod@dec@next=\pgf@mod@dec@firstofone% + \pgfifmetadecoration{#1}{% + \pgferror{You cannot declare a decoration of the same + name as a meta-decoration}% + \let\pgf@mod@dec@next=\pgfutil@gobble% + }{}% + \pgf@mod@dec@next + {% + \def\pgf@decorate@name{#1}% + \pgfutil@namedef{pgf@decorate@@#1@initial}{#2}% + \let\pgf@orig@state\state% + \let\state\pgf@decorate@state + #3% + \let\state\pgf@orig@state% + }% +}% + +\def\pgf@decorate@state#1{\pgfutil@ifnextchar[{\pgf@decorate@@start#1}{\pgf@decorate@@start#1[]}}% +\def\pgf@decorate@@start#1[#2]#3{% + \pgfutil@namedef{pgf@decorate@@\pgf@decorate@name @#1@options}{#2}% + \pgfutil@namedef{pgf@decorate@@\pgf@decorate@name @#1@code}{#3}% +}% +\long\def\pgf@mod@dec@firstofone#1{#1}% + + + +\def\pgfifdecoration#1#2#3{\pgfutil@ifundefined{pgf@decorate@@#1@initial}{#3}{#2}}% + + + + +\long\def\pgfdeclaremetadecoration#1#2#3{% + \let\pgf@mod@dec@next=\pgf@mod@dec@firstofone% + \pgfifdecoration{#1}{% + \pgferror{You cannot declare a meta-decoration of the + same name as a decoration}% + \let\pgf@mod@dec@next=\pgfutil@gobble% + }{}% + \pgf@mod@dec@next{% + \def\pgf@metadecoration@name{#1}% + \pgfutil@namedef{pgf@metadecoration@@#1@initial}{#2}% + \let\pgf@orig@state\state% + \let\state\pgf@metadecoration@state + #3% + \let\state\pgf@orig@state% + }% +}% + +\def\pgf@metadecoration@state#1{\pgfutil@ifnextchar[{\pgf@metadecoration@@state#1}{\pgf@metadecoration@@state#1[]}}% +\def\pgf@metadecoration@@state#1[#2]#3{% + \pgfutil@namedef{pgf@metadecoration@@\pgf@metadecoration@name @#1@options}{#2}% + \pgfutil@namedef{pgf@metadecoration@@\pgf@metadecoration@name @#1@code}{#3}% +}% + + + +\def\pgfifmetadecoration#1#2#3{\pgfutil@ifundefined{pgf@metadecoration@@#1@initial}{#3}{#2}}% + + + + + +\let\pgfdecoratebeforecode\pgfutil@empty +\let\pgfdecorateaftercode\pgfutil@empty + +\long\def\pgfdecoratepath#1#2{% + \pgfdecoration{{#1}{\pgfdecoratedpathlength}{\pgfdecoratebeforecode}{\pgfdecorateaftercode}}% + #2% + \endpgfdecoration}% + + + + +\def\pgfdecoratecurrentpath#1{% + \pgfgetpath\pgf@decorate@currentpath% + \pgfsetpath\pgfutil@empty% + \pgfdecoration{{#1}{\pgfdecoratedpathlength}{\pgfdecoratebeforecode}{\pgfdecorateaftercode}}% + \pgfsetpath\pgf@decorate@currentpath% + \endpgfdecoration}% + + + + +\def\pgfdecoration#1{% + \begingroup% + \def\pgf@decorate@decorationlist{#1}% + \pgf@decoration@env% +}% + +\def\endpgfdecoration{% + \pgf@decoration@endenv% + \ifx\pgfdecoratedpath\pgfutil@empty% + \else% + \pgf@decorate@for\pgf@temp:=\pgf@decorate@decorationlist\do{% + \ifx\pgf@temp\pgfutil@empty% + \else% + \expandafter\pgf@decorate@invoke\expandafter{\pgf@temp}% + \fi% + }% + \fi% + \pgfgetpath\pgfdecorationpath% + \global\let\pgf@decorate@decorationpathtemp\pgfdecorationpath% + \global\let\pgf@decorate@decoratedpathtemp\pgfdecoratedpath% + \global\let\pgf@decorate@existingpathtemp\pgfdecorateexistingpath% + \global\let\pgfpoint@decorated@pathlasttemp\pgfpoint@decorated@pathlast% + \endgroup% + \pgfutil@ifnextchar\@checkend{\aftergroup\pgf@decorate@installmacrosatend}% + {\pgf@decorate@installmacrosatend}% +}% + + + +\def\pgfmetadecoration#1{% + \begingroup% + \let\decoration\pgf@metadecoration@decoration% + \let\beforedecoration\pgf@metadecoration@beforedecoration% + \let\afterdecoration\pgf@metadecoration@afterdecoration% + \def\pgf@metadecoration@name{#1}% + \pgf@decoration@env% +}% + +\def\endpgfmetadecoration{% + \pgf@decoration@endenv% + \ifx\pgfdecoratedpath\pgfutil@empty% + \else% + \let\pgfmetadecoratedpathlength\pgf@decorate@totalpathlength% + \def\pgfmetadecoratedinputsegmentremainingdistance{\pgfdecoratedremainingdistance}% + \def\pgfmetadecoratedinputsegmentcompleteddistance{\pgfdecoratedcompleteddistance}% + \expandafter\let\expandafter\pgf@metadecoration@current@state% + \csname pgf@metadecoration@@\pgf@metadecoration@name @initial\endcsname% + \pgf@metadecoration@run% + \let\pgf@decorate@tempname\pgfutil@empty% + \let\pgf@decorate@tempbefore\pgfutil@empty% + \let\pgf@decorate@tempafter\pgfutil@empty% + \csname pgf@metadecoration@@\pgf@metadecoration@name @\pgf@metadecoration@current@state @code\endcsname% + \ifx\pgf@decorate@tempname\pgfutil@empty% + \def\pgf@decorate@tempname{moveto} + \fi% + \pgf@decorate@invoke{% + {\pgf@decorate@tempname}{\pgfdecoratedremainingdistance}% + {\pgf@decorate@tempbefore}{\pgf@decorate@tempafter}% + }% + \fi% + \pgfgetpath\pgfdecorationpath% + \global\let\pgf@decorate@decorationpathtemp\pgfdecorationpath% + \global\let\pgf@decorate@decoratedpathtemp\pgfdecoratedpath% + \global\let\pgf@decorate@existingpathtemp\pgfdecorateexistingpath% + \global\let\pgfpoint@decorated@pathlasttemp\pgfpoint@decorated@pathlast% + \endgroup% + \pgfutil@ifnextchar\@checkend{\aftergroup\pgf@decorate@installmacrosatend}% + {\pgf@decorate@installmacrosatend}% +}% + + + + + + +\def\pgf@decoration@env{% + \pgfgetpath\pgfdecorateexistingpath% + \pgfsetpath\pgfutil@empty% + \let\pgfdecorationpath\pgfutil@empty% + \let\pgfdecoratedpath\pgfutil@empty% + \let\pgfpoint@decorated@pathlast\pgfpointorigin% + \edef\pgfpoint@decorate@existingpathlast{\pgf@x\the\pgf@path@lastx\pgf@y\the\pgf@path@lasty}% + \bgroup% +}% + +\def\pgf@decorate@path@check@moveto#1{% + \expandafter\pgf@decorate@path@@check@moveto#1\pgf@decorate@stop\pgf@decorate@@stop}% + +\def\pgf@decorate@token@stop{\pgf@decorate@stop}% +\def\pgf@decorate@path@@check@moveto#1#2#3#4\pgf@decorate@@stop#5#6{% + \def\pgf@decorate@temp{#4}% + \pgf@x=#2\relax% + \pgf@y=#3\relax% + \ifx\pgf@decorate@temp\pgf@decorate@token@stop% + #5% + \else% + #6% + \fi% +}% + +\def\pgf@decoration@endenv{% + \egroup% + \pgftransformreset% + \pgfgetpath\pgfdecoratedpath% + \pgfsetpath\pgfdecorateexistingpath% + \ifx\pgfdecoratedpath\pgfutil@empty% + \pgferror{I cannot decorate an empty path}% + \else% + \pgf@decorate@path@check@moveto\pgfdecoratedpath{% + \advance\pgf@x by0.0001pt\relax% + \edef\pgfdecoratedpath{% + \expandafter\noexpand\pgfdecoratedpath% + \noexpand\pgfsyssoftpath@linetotoken{\the\pgf@x}{\the\pgf@y}% + }% + }% + {}% + \pgfprocessround{\pgfdecoratedpath}{\pgfdecoratedpath}% + \pgf@decorate@parsesoftpath{\pgfdecoratedpath}{\pgf@decorate@inputsegmentobjects}% + \pgfkeys{/pgf/every decoration/.try}% + \ifpgf@decorate@inputsegmentobjects@reverse% + \pgf@decorate@inputsegmentobjects@reverse{\pgf@decorate@inputsegmentobjects}{\pgf@decorate@inputsegmentobjects}% + \fi% + \let\pgf@decorated@remainingdistance\pgf@decorate@totalpathlength% + \let\pgfpoint@decorated@totalpathfirst\pgfpoint@decorated@firstparsed% + \let\pgfpoint@decorated@totalpathlast\pgfpoint@decorate@lastnonmovetoparsed% + \let\pgfpoint@decorated@pathfirst\pgfpoint@decorated@totalpathfirst + \let\pgfpoint@decorated@pathlast\pgfpoint@decorated@totalpathlast% + \let\pgf@decorate@currentinputsegmentobjects\pgf@decorate@inputsegmentobjects% + \let\pgf@decorate@transformtoinputsegment\pgfutil@empty% + \pgf@decorate@getnextinputsegmentobject\pgf@decorate@nextinputsegmentobject% + \pgf@decorate@processnextinputsegmentobject% + \pgf@decorate@distancetomove0pt\relax% + \fi% +}% + + + + +\def\pgf@decorate@installmacrosatend{% + \let\pgfdecorationpath\pgf@decorate@decorationpathtemp% + \let\pgfdecoratedpath\pgf@decorate@decoratedpathtemp% + \let\pgfdecorateexistingpath\pgf@decorate@existingpathtemp% + \let\pgfpoint@decorated@pathlast\pgfpoint@decorated@pathlasttemp% +}% + +\let\pgfdecorationpath\pgfutil@empty +\let\pgfdecoratedpath\pgfutil@empty +\let\pgfdecorateexisitingpath\pgfutil@empty + +\let\pgfpoint@decorated@pathlast\pgfpointorigin + + + +\def\pgf@decorate@for#1:=#2\do#3{% + \def\pgf@decorate@for@var{#1}% + \def\pgf@decorate@for@action{#3}% + \expandafter\pgf@decorate@@for#2,\pgf@stop,}% + +\def\pgf@decorate@@for#1,{% + \ifx#1\pgf@stop% + \expandafter\let\pgf@decorate@for@var\pgfutil@empty% + \else% + \expandafter\def\pgf@decorate@for@var{#1}% + \pgf@decorate@for@action% + \expandafter\pgf@decorate@@for% + \fi% +}% + + +\def\pgf@decorate@invoke#1{% + \pgf@decorate@@invoke#1\pgf@decorate@invoke\pgf@decorate@invoke\pgf@decorate@invoke\pgf@stop}% + +\def\pgf@decorate@@invoke#1#2#3#4#5\pgf@stop{% + \pgfutil@ifundefined{pgf@decorate@@#1@initial}% + {% + \pgferror{Unknown decoration `#1'}% + }% + {% + \def\pgf@decorate@name{#1}% + \ifx\pgf@decorate@invoke#3% + \let\pgf@decorate@beforecode\pgfutil@empty% + \else% + \def\pgf@decorate@beforecode{#3}% + \fi% + \ifx\pgf@decorate@invoke#4% + \let\pgf@decorate@aftercode\pgfutil@empty% + \else% + \def\pgf@decorate@aftercode{#4}% + \fi% + \pgfdecoratedremainingdistance\pgf@decorated@remainingdistance\relax% + \let\pgfdecoratedpathlength\pgf@decorate@totalpathlength% + \pgfmathsetlength\pgf@xa{#2}% + \ifdim\pgf@xa>\pgf@decorated@remainingdistance\relax% + \pgf@xa\pgf@decorated@remainingdistance\relax% + \fi% + \edef\pgf@decorate@currentpathlength{\the\pgf@xa}% + \pgf@xa-\pgf@decorate@currentpathlength\relax% + \advance\pgf@xa\pgf@decorated@remainingdistance\relax% + \edef\pgf@decorated@remainingdistance{\the\pgf@xa}% + \let\pgfdecoratedpathlength\pgf@decorate@currentpathlength% + \pgfdecoratedremainingdistance\pgfdecoratedpathlength\relax% + \pgfdecoratedcompleteddistance0pt\relax% + \pgf@decorate@beforecode% + \expandafter\let\expandafter\pgf@decorate@current@state% + \csname pgf@decorate@@\pgf@decorate@name @initial\endcsname% + \pgf@decorate@run% + \ifdim\pgf@decorated@remainingdistance<1pt\relax% + \let\pgfpoint@decorated@pathlast\pgfpoint@decorated@totalpathlast% + \else% + {% + \pgf@decorate@movealonginputsegment{\the\pgfdecoratedremainingdistance}% + \pgf@decorate@transformtoinputsegment% + \pgfpointorigin% + \pgf@pos@transform@glob% + \global\pgf@x\pgf@x% + \global\pgf@y\pgf@y% + }% + \edef\pgfpoint@decorated@pathlast{\pgf@x\the\pgf@x\pgf@y\the\pgf@y}% + \fi% + {% + \pgftransformreset% + \pgf@decorate@transformtoinputsegment% + \pgf@decorate@additionaltransform% + \csname pgf@decorate@@\pgf@decorate@name @final@code\endcsname% + }% + \pgf@decorate@movealonginputsegment{\the\pgfdecoratedremainingdistance}% + \pgf@decorate@aftercode% + \let\pgf@decorate@additionaltransform\pgfutil@empty% + \let\pgf@decorate@inputsegmentobjects\pgf@decorate@inputsegmentobjects@aftersplit% + \let\pgfpoint@decorated@pathfirst\pgfpoint@decorated@pathlast% + }% +}% + + + +\let\pgf@decorate@transformtoinputsegment\pgfutil@empty% +\let\pgf@decorate@additionaltransform\pgfutil@empty% + + +\def\pgfpointdecoratedpathfirst{% + {% + \pgftransforminvert% + \pgfpoint@decorated@pathfirst% + \pgf@pos@transform@glob% + \global\pgf@x\pgf@x% + \global\pgf@y\pgf@y% + }% +}% + +\def\pgfpointdecoratedpathlast{% + {% + \pgftransforminvert% + \pgfpoint@decorated@pathlast% + \pgf@pos@transform@glob% + \global\pgf@x\pgf@x% + \global\pgf@y\pgf@y% + }% +}% + +\def\pgfpointdecoratedinputsegmentfirst{% + {% + \pgftransforminvert% + \pgf@decorate@inputsegment@first% + \pgf@pos@transform@glob% + \global\pgf@x\pgf@x% + \global\pgf@y\pgf@y% + }% +}% + +\def\pgfpointdecoratedinputsegmentlast{% + {% + \pgftransforminvert% + \pgf@decorate@inputsegment@last% + \pgf@pos@transform@glob% + \global\pgf@x\pgf@x% + \global\pgf@y\pgf@y% + }% +}% + + +\def\pgf@final@text{final}% + +\def\pgf@decorate@run{% + \let\pgf@decorate@next\pgf@decorate@do@state% + \ifx\pgf@decorate@current@state\pgf@final@text% + \let\pgf@decorate@next\relax% + \fi% + \pgf@decorate@next% +}% + +\def\pgf@decorate@do@state{% + \let\pgf@decorate@next\relax% + \let\pgf@decorate@next@state\pgf@decorate@current@state% + \let\pgf@decorate@persistent@pre=\relax% + \let\pgf@decorate@persistent@post=\relax% + \expandafter\expandafter\expandafter\pgf@decorate@setter + \expandafter\expandafter\expandafter{\csname pgf@decorate@@\pgf@decorate@name @\pgf@decorate@current@state @options\endcsname}% + \ifx\pgf@decorate@next\relax% + \let\pgf@decorate@next\pgf@decorate@do@code% + \fi% + \pgf@decorate@next% +}% + +\def\pgf@decorate@setter{\pgfqkeys{/pgf/decoration automaton}}% + +\pgf@decorate@repeatstate-1\relax + +\def\pgf@decorate@do@code{% + \pgf@decorate@persistent@pre% + {% + \pgftransformreset% + \pgf@decorate@transformtoinputsegment% + \pgf@decorate@additionaltransform% + \csname pgf@decorate@@\pgf@decorate@name @\pgf@decorate@current@state @code\endcsname% + }% + \pgf@decorate@persistent@post% + \pgf@decorate@movealongpath{\pgf@decorate@width}% + \ifnum\pgf@decorate@repeatstate>0\relax% + \advance\pgf@decorate@repeatstate-1\relax% + \else% + \pgf@decorate@repeatstate-1\relax% + \let\pgf@decorate@current@state\pgf@decorate@next@state% + \fi% + \pgf@decorate@run% +}% + +\pgfkeys{ + /pgf/decoration automaton/width/.code=\def\pgf@decorate@width{#1}\pgf@decorate@switch@if#1 to final\pgf@stop,% + /pgf/decoration automaton/switch if less than/.code=\pgf@decorate@switch@if#1\pgf@stop,% + /pgf/decoration automaton/switch if input segment less than/.code=\pgf@decorate@switch@ifinputsegment#1\pgf@stop,% + /pgf/decoration automaton/next state/.store in=\pgf@decorate@next@state,% + /pgf/decoration automaton/persistent precomputation/.store in=\pgf@decorate@persistent@pre,% + /pgf/decoration automaton/persistent postcomputation/.store in=\pgf@decorate@persistent@post,% + /pgf/decoration automaton/repeat state/.code={% + \ifnum\pgf@decorate@repeatstate=-1\relax% + \pgfmathsetcount\pgf@decorate@repeatstate{#1}% + \fi% + },% + /pgf/decoration automaton/if input segment is closepath/.code={% + \ifpgf@decorate@is@closepath@% + \pgfkeysalso{#1}% + \fi% + }, + /pgf/decoration automaton/auto end on length/.code=\pgf@decorate@auto@end{#1}, + /pgf/decoration automaton/auto corner on length/.code=\pgf@decorate@auto@corner{#1}, +}% + +\def\pgf@decorate@auto@end#1{% + \ifx\pgf@decorate@next\relax% + \pgfmathsetlength\pgf@x{#1}% + \ifpgf@decorate@is@closepath@% + \ifdim\pgfdecoratedremainingdistance>\pgf@x% + \ifdim\pgfdecoratedinputsegmentremainingdistance>\pgf@x% + \else% + \pgfpathclose% + \pgf@decorate@movealongpath{\pgfdecoratedinputsegmentremainingdistance}% + \pgf@decorate@repeatstate-1\relax% + \let\pgf@decorate@next\pgf@decorate@run% + \fi% + \else% + \pgfpathclose% + \def\pgf@decorate@current@state{final}% + \def\pgf@decorate@width{\pgfdecoratedremainingdistance} + \pgf@decorate@repeatstate-1\relax% + \let\pgf@decorate@next\pgf@decorate@run% + \fi% + \else% + \ifdim\pgfdecoratedremainingdistance>\pgf@x% + \else% + {% + \pgftransformreset% + \pgf@decorate@transformtoinputsegment% + \pgf@decorate@additionaltransform% + \pgfpathlineto{\pgfpoint{\pgfdecoratedremainingdistance}{0pt}} + }% + \def\pgf@decorate@current@state{final}% + \def\pgf@decorate@width{\pgfdecoratedremainingdistance} + \pgf@decorate@repeatstate-1\relax% + \let\pgf@decorate@next\pgf@decorate@run% + \fi% + \fi% + \fi% +}% + +\def\pgf@decorate@auto@corner#1{% + \ifpgfdecoratepathhascorners% + \ifx\pgf@decorate@next\relax% + \pgfmathsetlength\pgf@x{#1}% + \ifdim\pgfdecoratedinputsegmentremainingdistance>\pgf@x% + \else% + {% + \pgftransformreset% + \pgf@decorate@transformtoinputsegment% + \pgf@decorate@additionaltransform% + \pgfpathlineto{\pgfqpoint{\pgfdecoratedinputsegmentremainingdistance}{0pt}} + }% + \pgf@decorate@movealongpath{\pgfdecoratedinputsegmentremainingdistance}% + \pgf@decorate@repeatstate-1\relax% + \let\pgf@decorate@next\pgf@decorate@run% + \fi% + \fi% + \fi% +}% + +\def\pgf@decorate@switch@if#1to #2\pgf@stop{% + \ifx\pgf@decorate@next\relax% + \ifdim\pgfdecoratedremainingdistance=0pt\relax% + \def\pgf@decorate@current@state{#2}% + \pgf@decorate@repeatstate-1\relax% + \let\pgf@decorate@next\pgf@decorate@run% + \else% + \pgfmathsetlength\pgf@x{#1}% + \ifdim\pgfdecoratedremainingdistance<\pgf@x% + \def\pgf@decorate@current@state{#2}% + \pgf@decorate@repeatstate-1\relax% + \let\pgf@decorate@next\pgf@decorate@run% + \fi% + \fi% + \fi% +}% + +\def\pgf@decorate@switch@ifinputsegment#1to #2\pgf@stop{% + \ifx\pgf@decorate@next\relax% + \pgfmathsetlength\pgf@x{#1}% + \ifdim\pgfdecoratedinputsegmentremainingdistance<\pgf@x% + \def\pgf@decorate@current@state{#2}% + \pgf@decorate@repeatstate-1\relax% + \let\pgf@decorate@next\pgf@decorate@run% + \fi% + \fi% +}% + + + + +\def\pgfsetdecorationsegmenttransformation#1{\def\pgf@decorate@additionaltransform{#1}}% +\let\pgf@decorate@additionaltransform\pgfutil@empty% + + + + +\def\pgf@metadecoration@decoration#1{% + \edef\pgf@decorate@tempname{#1}% +}% +\def\pgf@metadecoration@beforedecoration#1{% + \def\pgf@decorate@tempbefore{#1}% +}% +\def\pgf@metadecoration@afterdecoration#1{% + \def\pgf@decorate@tempafter{#1}% +}% + + +\def\pgf@final@text{final}% + +\def\pgf@metadecoration@run{% + \let\pgf@metadecoration@next\pgf@metadecoration@do@state% + \ifx\pgf@metadecoration@current@state\pgf@final@text% + \let\pgf@metadecoration@next\relax% + \fi% + \pgf@metadecoration@next% +}% + +\def\pgf@metadecoration@do@state{% + \let\pgf@metadecoration@next\relax% + \let\pgf@metadecoration@next@state\pgf@metadecoration@current@state% + \let\pgfmetadecoratedremainingdistance\pgf@decorated@remainingdistance% + \pgf@x-\pgfmetadecoratedremainingdistance\relax% + \advance\pgf@x\pgf@decorate@totalpathlength\relax% + \edef\pgfmetadecoratedcompleteddistance{\the\pgf@x}% + \expandafter\expandafter\expandafter\pgf@metadecoration@setter + \expandafter\expandafter\expandafter{\csname pgf@metadecoration@@\pgf@metadecoration@name @\pgf@metadecoration@current@state @options\endcsname}% + \ifx\pgf@metadecoration@next\relax% + \let\pgf@metadecoration@next\pgf@metadecoration@do@code% + \fi% + \pgf@metadecoration@next% +}% + +\def\pgf@metadecoration@do@code{% + \let\pgf@decorate@tempname\pgfutil@empty% + \let\pgf@decorate@tempbefore\pgfutil@empty% + \let\pgf@decorate@tempafter\pgfutil@empty% + \let\pgfpointmetadecoratedpathfirst\pgfpointdecoratedpathfirst% + \let\pgfpointmetadecoratedpathlast\pgfpointdecoratedpathlast% + \csname pgf@metadecoration@@\pgf@metadecoration@name @\pgf@metadecoration@current@state @code\endcsname% + \ifx\pgf@decorate@tempname\pgfutil@empty% + \def\pgf@decorate@tempname{moveto} + \fi% + \pgf@decorate@invoke{% + {\pgf@decorate@tempname}{\pgf@metadecoration@width}% + {\pgf@decorate@tempbefore}{\pgf@decorate@tempafter}% + }% + \let\pgf@metadecoration@current@state\pgf@metadecoration@next@state% + \pgf@metadecoration@run% +}% + +\def\pgf@metadecoration@setter{\pgfqkeys{/pgf/meta-decoration automaton}}% + +\pgfkeys{ + /pgf/meta-decoration automaton/width/.code=\def\pgf@metadecoration@width{#1}\pgf@metadecoration@switch@if#1 to final\pgf@stop,% + /pgf/meta-decoration automaton/switch if less than/.code=\pgf@metadecoration@switch@if#1\pgf@stop,% + /pgf/meta-decoration automaton/switch if input segment less than/.code=\pgf@metadecoration@switch@ifinputsegment#1\pgf@stop,% + /pgf/meta-decoration automaton/next state/.store in=\pgf@metadecoration@next@state,% +}% + +\def\pgf@metadecoration@switch@if#1to #2\pgf@stop{% + \ifx\pgf@metadecoration@next\relax% + \pgfmathsetlength\pgf@x{#1}% + \ifdim\pgf@decorated@remainingdistance<\pgf@x% + \def\pgf@metadecoration@current@state{#2}% + \let\pgf@metadecoration@next\pgf@metadecoration@run% + \fi% + \fi% +}% + +\def\pgf@metadecoration@switch@ifinputsegment#1to #2\pgf@stop{% + \ifx\pgf@metadecoration@next\relax% + \pgfmathsetlength\pgf@x{#1}% + \ifdim\pgfdecoratedinputsegmentremainingdistance<\pgf@x% + \def\pgf@metadecoration@current@state{#2}% + \let\pgf@metadecoration@next\pgf@metadecoration@run% + \fi% + \fi% +}% + + + +\def\pgf@decorate@movealongpath#1{% + \pgfmathsetlength\pgf@decorate@distancetomove{#1}% + \advance\pgfdecoratedcompleteddistance\pgf@decorate@distancetomove% + \advance\pgfdecoratedremainingdistance-\pgf@decorate@distancetomove% + \pgf@decorate@@movealongpath% +}% +\def\pgf@decorate@@movealongpath{% + \advance\pgfdecoratedinputsegmentcompleteddistance\pgf@decorate@distancetomove% + \advance\pgfdecoratedinputsegmentremainingdistance-\pgf@decorate@distancetomove% + \ifdim\pgfdecoratedinputsegmentremainingdistance>0pt\relax% + \let\pgf@next\pgf@decorate@@@movealongpath% + \else% + \pgf@decorate@distancetomove-\pgfdecoratedinputsegmentremainingdistance% + \pgf@decorate@processnextinputsegmentobject% + \ifx\pgf@decorate@currentinputsegmentobjects\pgfutil@empty% + \pgfdecoratedremainingdistance0pt\relax% + \let\pgf@next\relax% + \else% + \let\pgf@next\pgf@decorate@@movealongpath% + \fi% + \fi% + \pgf@next% +}% + +\def\pgf@decorate@@@movealongpath{% + \pgf@decorate@movealonginputsegment{\the\pgf@decorate@distancetomove}% + \pgf@decorate@distancetomove0pt\relax% + \ifdim\pgf@decorate@inputsegmenttime pt>1pt\relax% + \let\pgf@decorate@inputsegmenttimetemp\pgf@decorate@inputsegmenttime% + \pgf@decorate@processnextinputsegmentobject% + \pgf@x\pgf@decorate@inputsegmenttimetemp pt\relax% + \advance\pgf@x-1pt\relax% + \edef\pgf@decorate@inputsegmenttime{\pgfmath@tonumber{\pgf@x}}% + \ifx\pgf@decorate@currentinputsegmentobjects\pgfutil@empty% + \pgfdecoratedremainingdistance0pt\relax% + \fi% + \fi% +}% + +\def\pgf@decorate@getnextinputsegmentobject#1{% + \ifx\pgf@decorate@currentinputsegmentobjects\pgfutil@empty% + \let\pgf@next\relax% + \else% + \def\pgf@decorate@temp{#1}% + \let\pgf@next\pgf@decorate@@getnextinputsegmentobject% + \fi% + \pgf@next% +}% +\def\pgf@decorate@@getnextinputsegmentobject{% + \expandafter\pgf@decorate@@@getnextinputsegmentobject\pgf@decorate@currentinputsegmentobjects\pgf@stop}% +\def\pgf@decorate@@@getnextinputsegmentobject#1#2\pgf@stop{% + \expandafter\def\pgf@decorate@temp{#1}% + \def\pgf@decorate@currentinputsegmentobjects{#2}}% + +\def\pgf@decorate@processnextinputsegmentobject{% + \let\pgfdecorationpreviousinputsegment\pgfdecoratecurrentinputsegment% + \let\pgf@decorate@currentinputsegmentobject\pgf@decorate@nextinputsegmentobject% + \pgf@decorate@getnextinputsegmentobject\pgf@decorate@nextinputsegmentobject% + \pgf@decorate@is@closepath@false% + \pgf@decorate@currentinputsegmentobject% + \ifx\pgfdecorationcurrentinputsegment\pgfdecorationinputsegmentmoveto% + \pgfpathmoveto{\pgf@decorate@inputsegment@first}% + \let\pgfdecorationpreviousinputsegment\pgfdecorationcurrentinputsegment% + \let\pgf@decorate@currentinputsegmentobject\pgf@decorate@nextinputsegmentobject% + \pgf@decorate@getnextinputsegmentobject\pgf@decorate@nextinputsegmentobject% + \pgf@decorate@is@closepath@false% + \pgf@decorate@currentinputsegmentobject% + \fi% + \ifdim\pgfdecoratedinputsegmentlength=0pt\relax% + \def\pgfdecoratedinputsegmentlength{0.05pt}% + \fi% + \pgfdecoratedinputsegmentremainingdistance\pgfdecoratedinputsegmentlength\relax% + \pgfdecoratedinputsegmentcompleteddistance0pt\relax% + \def\pgf@decorate@inputsegmenttime{0}% + \ifx\pgfdecorationcurrentinputsegment\pgfdecorationinputsegmentcurveto% + \pgfmathanglebetweenpoints{\pgf@decorate@inputsegment@first}{\pgf@decorate@inputsegment@supporta}% + \let\pgfdecoratedangle\pgfmathresult% + \let\pgfdecoratedinputsegmentstartangle\pgfmathresult% + \pgfmathanglebetweenpoints{\pgf@decorate@inputsegment@supportb}{\pgf@decorate@inputsegment@last}% + \let\pgfdecoratedinputsegmentendangle\pgfmathresult% + \else% + \pgfmathanglebetweenpoints{\pgf@decorate@inputsegment@first}{\pgf@decorate@inputsegment@last}% + \let\pgfdecoratedangle\pgfmathresult% + \let\pgfdecoratedinputsegmentstartangle\pgfmathresult% + \let\pgfdecoratedinputsegmentendangle\pgfmathresult% + \fi% + \begingroup% + \pgf@decorate@nextinputsegmentobject% + \global\let\pgf@decorate@temp\pgfdecorationcurrentinputsegment% + \ifx\pgfdecorationcurrentinputsegment\pgfdecorationinputsegmentmoveto% + \expandafter\pgf@decorate@currentinputsegmentobject% + \fi% + \ifx\pgfdecorationcurrentinputsegment\pgfdecorationinputsegmentcurveto% + \pgfmathanglebetweenpoints{\pgf@decorate@inputsegment@first}{\pgf@decorate@inputsegment@supporta}% + \else% + \pgfmathanglebetweenpoints{\pgf@decorate@inputsegment@first}{\pgf@decorate@inputsegment@last}% + \fi% + \pgf@x-\pgfdecoratedangle pt\relax% + \advance\pgf@x\pgfmathresult pt\relax% + \edef\pgfdecoratedangletonextinputsegment{\the\pgf@x}% + \pgfmath@smuggleone\pgfdecoratedangletonextinputsegment% + \endgroup% + \let\pgfdecorationnextinputsegmentobject\pgf@decorate@temp% +}% + + + +\newif\ifpgf@decorate@is@closepath@% + +\def\pgfdecorationinputsegmentmoveto{moveto}% +\def\pgfdecorationinputsegmentlineto{lineto}% +\def\pgfdecorationinputsegmentcurveto{curveto}% +\def\pgfdecorationinputsegmentclosepath{closepath}% +\def\pgfdecorationinputsegmentlast{last}% + +\def\pgf@decorate@inputsegmentobject@moveto#1{% + \def\pgf@decorate@inputsegment@first{#1}% + \def\pgf@decorate@inputsegment@supporta{#1}% + \def\pgf@decorate@inputsegment@supportb{#1}% + \def\pgf@decorate@inputsegment@last{#1}% + \edef\pgf@decorate@lastmoveto{#1}% + \def\pgfdecoratedinputsegmentlength{0pt}% + \let\pgfdecorationcurrentinputsegment\pgfdecorationinputsegmentmoveto% +}% + +\def\pgf@decorate@inputsegmentobject@lineto#1#2#3{% + \def\pgfdecoratedinputsegmentlength{#1}% + \def\pgf@decorate@inputsegment@first{#2}% + \def\pgf@decorate@inputsegment@last{#3}% + \pgfpointdiff{#2}{#3}% + \pgf@xa\pgf@x% + \pgf@ya\pgf@y% + \pgfextract@process\pgf@decorate@inputsegment@supporta{% + \pgf@process{#2}% + \advance\pgf@x0.333333\pgf@xa% + \advance\pgf@y0.333333\pgf@ya% + }% + \pgfextract@process\pgf@decorate@inputsegment@supportb{% + \pgf@process{#2}% + \advance\pgf@x0.666666\pgf@xa% + \advance\pgf@y0.666666\pgf@ya% + }% + \let\pgf@decorate@movealonginputsegment\pgf@decorate@movealonginputsegment@line% + \let\pgf@decorate@transformtoinputsegment\pgf@decorate@transformtoinputsegment@line% + \let\pgfdecorationcurrentinputsegment\pgfdecorationinputsegmentlineto% +}% + +\def\pgf@decorate@inputsegmentobject@curveto#1#2#3#4#5{% + \def\pgfdecoratedinputsegmentlength{#1}% + \def\pgf@decorate@inputsegment@first{#2}% + \def\pgf@decorate@inputsegment@supporta{#3}% + \def\pgf@decorate@inputsegment@supportb{#4}% + \def\pgf@decorate@inputsegment@last{#5}% + \let\pgf@decorate@movealonginputsegment\pgf@decorate@movealonginputsegment@curve% + \let\pgf@decorate@transformtoinputsegment\pgf@decorate@transformtoinputsegment@curve% + \let\pgfdecorationcurrentinputsegment\pgfdecorationinputsegmentcurveto% +}% + +\def\pgf@decorate@inputsegmentobject@closepath#1#2#3{% + \pgf@decorate@is@closepath@true% + \def\pgfdecoratedinputsegmentlength{#1}% + \def\pgf@decorate@inputsegment@first{#2}% + \def\pgf@decorate@inputsegment@last{#3}% + \pgfpointdiff{#2}{#3}% + \pgf@xa\pgf@x% + \pgf@ya\pgf@y% + \pgfextract@process\pgf@decorate@inputsegment@supporta{% + \pgf@process{#2}% + \advance\pgf@x0.333333\pgf@xa% + \advance\pgf@y0.333333\pgf@ya% + }% + \pgfextract@process\pgf@decorate@inputsegment@supportb{% + \pgf@process{#2}% + \advance\pgf@x0.666666\pgf@xa% + \advance\pgf@y0.666666\pgf@ya% + }% + \let\pgf@decorate@movealonginputsegment\pgf@decorate@movealonginputsegment@line% + \let\pgf@decorate@transformtoinputsegment\pgf@decorate@transformtoinputsegment@line% + \let\pgfdecorationcurrentinputsegment\pgfdecorationinputsegmentclosepath% +}% + + +\def\pgf@decorate@inputsegmentobject@endofinputsegments{% + \let\pgfdecorationcurrentinputsegment\pgfdecorationinputsegmentlast% +}% + + + + +\def\pgf@decorate@movealonginputsegment@line#1{}% + +\def\pgf@decorate@movealonginputsegment@curve#1{% + \ifdim#1=0pt\relax% + \else% + \pgfpointcurveattime{\pgf@decorate@inputsegmenttime}% + {\pgf@decorate@inputsegment@first}{\pgf@decorate@inputsegment@supporta}% + {\pgf@decorate@inputsegment@supportb}{\pgf@decorate@inputsegment@last}% + \pgf@xa\pgf@x% + \pgf@ya\pgf@y% + \pgf@xb\pgf@decorate@inputsegmenttime pt\relax% + \ifdim\pgf@decorate@currentpathlength<128pt\relax% + \pgf@yb.03125pt\relax% + \else% + \ifdim\pgf@decorate@currentpathlength<512pt\relax% + \pgf@yb.015625pt\relax% + \else% + \ifdim\pgf@decorate@currentpathlength<2048pt\relax% + \pgf@yb.00390625pt\relax% + \else% + \pgf@yb.0009765625pt\relax% + \fi% + \fi% + \fi% + \c@pgf@counta1\relax% + \pgfutil@tempdima0pt\relax% + \pgfmathloop% + \advance\pgf@xb\c@pgf@counta\pgf@yb% + \edef\pgf@decorate@temp{% + \pgf@xa\the\pgf@xa% + \pgf@xb\the\pgf@xb% + \pgf@ya\the\pgf@ya% + \pgf@yb\the\pgf@yb% + }% + \pgfpoint@decorate@curveattime{\pgfmath@tonumber{\pgf@xb}}% + {\pgf@decorate@inputsegment@first}{\pgf@decorate@inputsegment@supporta}% + {\pgf@decorate@inputsegment@supportb}{\pgf@decorate@inputsegment@last}% + \pgf@decorate@temp% + \pgf@xc\pgf@xa% + \pgf@yc\pgf@ya% + \pgf@xa\pgf@x% + \pgf@ya\pgf@y% + \advance\pgf@x-\pgf@xc% + \advance\pgf@y-\pgf@yc% + \pgfmathveclen@{\pgfmath@tonumber{\pgf@x}}{\pgfmath@tonumber{\pgf@y}}% + \pgfutil@tempdimb\pgfmathresult pt\relax% + \advance\pgfutil@tempdima\c@pgf@counta\pgfutil@tempdimb% + \ifnum\c@pgf@counta>0\relax% + \ifdim\pgfutil@tempdima>#1\relax% + \c@pgf@counta-\c@pgf@counta% + \pgf@yb.5\pgf@yb% + \fi% + \else% + \ifdim\pgfutil@tempdima<#1\relax% + \c@pgf@counta-\c@pgf@counta% + \pgf@yb.5\pgf@yb% + \fi% + \fi% + \ifdim\pgf@yb=0pt\relax% + \else% + \repeatpgfmathloop% + \edef\pgf@decorate@inputsegmenttime{\pgfmath@tonumber{\pgf@xb}}% + \pgfpointcurveattime{\pgf@decorate@inputsegmenttime}% + {\pgf@decorate@inputsegment@first}{\pgf@decorate@inputsegment@supporta}% + {\pgf@decorate@inputsegment@supportb}{\pgf@decorate@inputsegment@last}% + \edef\pgf@marshal{% + \noexpand\pgfmathanglebetweenpoints% + {\pgf@x\the\pgf@xb\pgf@y\the\pgf@yb}% + {\pgf@x\the\pgf@xa\pgf@y\the\pgf@ya}% + }% + \pgf@marshal% + \let\pgfdecoratedangle\pgfmathresult% + \fi% +}% + +\def\pgfpoint@decorate@curveattime#1#2#3#4#5{% + \edef\pgf@time@s{#1}% + \pgf@x-\pgf@time@s pt\relax% + \advance\pgf@x1pt\relax% + \edef\pgf@time@t{\pgf@sys@tonumber{\pgf@x}}% + #5\relax% + \pgf@xc=\pgf@x% + \pgf@yc=\pgf@y% + #4\relax% + \pgf@xb=\pgf@x% + \pgf@yb=\pgf@y% + #3\relax% + \pgf@xa=\pgf@x% + \pgf@ya=\pgf@y% + #2\relax% + \pgf@x=\pgf@time@t\pgf@x\advance\pgf@x by\pgf@time@s\pgf@xa% + \pgf@y=\pgf@time@t\pgf@y\advance\pgf@y by\pgf@time@s\pgf@ya% + \pgf@xa=\pgf@time@t\pgf@xa\advance\pgf@xa by\pgf@time@s\pgf@xb% + \pgf@ya=\pgf@time@t\pgf@ya\advance\pgf@ya by\pgf@time@s\pgf@yb% + \pgf@xb=\pgf@time@t\pgf@xb\advance\pgf@xb by\pgf@time@s\pgf@xc% + \pgf@yb=\pgf@time@t\pgf@yb\advance\pgf@yb by\pgf@time@s\pgf@yc% + \pgf@x=\pgf@time@t\pgf@x\advance\pgf@x by\pgf@time@s\pgf@xa% + \pgf@y=\pgf@time@t\pgf@y\advance\pgf@y by\pgf@time@s\pgf@ya% + \pgf@xa=\pgf@time@t\pgf@xa\advance\pgf@xa by\pgf@time@s\pgf@xb% + \pgf@ya=\pgf@time@t\pgf@ya\advance\pgf@ya by\pgf@time@s\pgf@yb% + \pgf@xb=\pgf@x% + \pgf@yb=\pgf@y% + \pgf@x=\pgf@time@t\pgf@x\advance\pgf@x by\pgf@time@s\pgf@xa% + \pgf@y=\pgf@time@t\pgf@y\advance\pgf@y by\pgf@time@s\pgf@ya% +}% + + +\def\pgf@decorate@transformtoinputsegment@line{% + \pgftransformshift{% + \pgfpointlineatdistance{\pgfdecoratedinputsegmentcompleteddistance}{\pgf@decorate@inputsegment@first}{\pgf@decorate@inputsegment@last} + }% + \pgftransformrotate{\pgfdecoratedangle}% +}% + +\def\pgf@decorate@transformtoinputsegment@curve{% + \pgfslopedattimetrue% + \pgfallowupsidedownattimetrue% + \pgftransformcurveattime{\pgf@decorate@inputsegmenttime}% + {\pgf@decorate@inputsegment@first}{\pgf@decorate@inputsegment@supporta}% + {\pgf@decorate@inputsegment@supportb}{\pgf@decorate@inputsegment@last}% +}% + + + + + + +\def\pgf@decorate@parsesoftpath#1#2{% + \def\pgf@decorate@inputsegmentobjectsmacro{#2}% + \let\pgf@decorate@inputsegmentobjects\pgfutil@empty% + \pgfutil@tempdima0pt\relax% + \let\pgfpoint@decorate@lastparsed\pgfpoint@decorate@existingpathlast% + \let\pgfpoint@decorate@lastnonmovetoparsed\pgfpoint@origin% + \let\pgf@decorate@queueinputsegmentobject\pgfutil@empty% + \let\pgfpoint@decorated@firstparsed\pgfutil@empty% + \expandafter\pgf@decorate@@parsesoftpath#1\pgf@stop% +}% + +\def\pgf@decorate@@parsesoftpath#1{% + \ifx#1\pgf@stop% + \let\pgf@decorate@queueinputsegmentobject\pgfutil@empty% + \pgf@decorate@addtoinputsegmentobjects{\pgf@decorate@inputsegmentobject@endofinputsegments}% + \pgf@decorate@addtoinputsegmentobjects{\pgf@decorate@inputsegmentobject@endofinputsegments}% + \expandafter\let\pgf@decorate@inputsegmentobjectsmacro\pgf@decorate@inputsegmentobjects% + \edef\pgf@decorate@totalpathlength{\the\pgfutil@tempdima}% + \let\pgf@next\relax% + \else% + \ifx#1\pgfsyssoftpath@movetotoken% + \let\pgf@next\pgf@decorate@parsemoveto% + \else% + \ifx#1\pgfsyssoftpath@linetotoken% + \let\pgf@next\pgf@decorate@parselineto% + \else% + \ifx#1\pgfsyssoftpath@curvetosupportatoken% + \let\pgf@next\pgf@decorate@parsecurveto% + \else% + \ifx#1\pgfsyssoftpath@closepathtoken% + \let\pgf@next\pgf@decorate@parseclosepath% + \else% + \ifx#1\pgfsyssoftpath@rectcornertoken% + \let\pgf@next\pgf@decorate@parserect% + \else% + \pgferror{Unrecognised soft path token `#1'}% + \fi% + \fi% + \fi% + \fi% + \fi% + \fi% + \pgf@next}% + + +\def\pgf@decorate@addtoinputsegmentobjects#1{% + \ifx\pgf@decorate@queueinputsegmentobject\pgfutil@empty% + \else% + \let\pgf@decorate@temp\pgf@decorate@queueinputsegmentobject% + \let\pgf@decorate@queueinputsegmentobject\pgfutil@empty% + \expandafter\pgf@decorate@addtoinputsegmentobjects\expandafter{\pgf@decorate@temp}% + \fi% + \ifx\pgfpoint@decorated@firstparsed\pgfutil@empty% + #1% + \let\pgfpoint@decorated@firstparsed\pgf@decorate@inputsegment@first% + \fi% + \expandafter\def\expandafter\pgf@decorate@inputsegmentobjects\expandafter% + {\pgf@decorate@inputsegmentobjects{#1}}% +}% + +\def\pgf@decorate@parsemoveto#1#2{% + \def\pgf@decorate@queueinputsegmentobject{\pgf@decorate@inputsegmentobject@moveto{\pgf@x#1\pgf@y#2}}% + \def\pgfpoint@decorate@lastparsed{\pgf@x#1\pgf@y#2}% + \pgf@decorate@@parsesoftpath% +}% + +\def\pgf@decorate@parselineto#1#2{% + \pgf@decorate@linelength{\pgfpoint@decorate@lastparsed}{\pgf@x#1\pgf@y#2}% + \advance\pgfutil@tempdima\pgfmathresult pt\relax% + \edef\pgf@decorate@temp{% + \noexpand\pgf@decorate@inputsegmentobject@lineto{\pgfmathresult pt}{\pgfpoint@decorate@lastparsed}{\pgf@x#1\pgf@y#2}% + }% + \edef\pgfpoint@decorate@lastparsed{\pgf@x#1\pgf@y#2}% + \let\pgfpoint@decorate@lastnonmovetoparsed\pgfpoint@decorate@lastparsed% + \expandafter\pgf@decorate@addtoinputsegmentobjects\expandafter{\pgf@decorate@temp}% + \pgf@decorate@@parsesoftpath% +}% + +\def\pgf@decorate@parsecurveto#1#2\pgfsyssoftpath@curvetosupportbtoken#3#4\pgfsyssoftpath@curvetotoken#5#6{% + \pgf@decorate@curvelength{\pgfpoint@decorate@lastparsed}{\pgf@x#1\pgf@y#2}{\pgf@x#3\pgf@y#4}{\pgf@x#5\pgf@y#6}% + \advance\pgfutil@tempdima\pgfmathresult pt\relax% + \edef\pgf@decorate@temp{% + \noexpand\pgf@decorate@inputsegmentobject@curveto{\pgfmathresult pt}{\pgfpoint@decorate@lastparsed}% + {\pgf@x#1\pgf@y#2}{\pgf@x#3\pgf@y#4}{\pgf@x#5\pgf@y#6}% + }% + \expandafter\pgf@decorate@addtoinputsegmentobjects\expandafter{\pgf@decorate@temp}% + \edef\pgfpoint@decorate@lastparsed{\pgf@x#5\pgf@y#6}% + \let\pgfpoint@decorate@lastnonmovetoparsed\pgfpoint@decorate@lastparsed% + \pgf@decorate@@parsesoftpath% +}% + + +\def\pgf@decorate@parseclosepath#1#2{% + \pgf@decorate@linelength{\pgfpoint@decorate@lastparsed}{\pgf@x#1\pgf@y#2}% + \advance\pgfutil@tempdima\pgfmathresult pt\relax% + \edef\pgf@decorate@temp{% + \noexpand\pgf@decorate@inputsegmentobject@closepath{\pgfmathresult pt}{\pgfpoint@decorate@lastparsed}{\pgf@x#1\pgf@y#2}% + }% + \expandafter\pgf@decorate@addtoinputsegmentobjects\expandafter{\pgf@decorate@temp}% + \edef\pgfpoint@decorate@lastparsed{\pgf@x#1\pgf@y#2}% + \let\pgfpoint@decorate@lastnonmovetoparsed\pgfpoint@decorate@lastparsed% + \pgf@decorate@@parsesoftpath% +}% + +\newif\ifpgfdecoraterectangleclockwise +\def\pgf@decorate@parserect#1#2\pgfsyssoftpath@rectsizetoken#3#4{% + \let\pgf@decorate@orig@@parsesoftpath\pgf@decorate@@parsesoftpath% + \let\pgf@decorate@@parsesoftpath\relax% + \pgf@decorate@parsemoveto{#1}{#2}% + \pgf@xa#1\relax% + \pgf@ya#2\relax% + \ifpgfdecoraterectangleclockwise% + \advance\pgf@ya#4% + \pgf@yb\pgf@ya% + \edef\pgf@temp{{\the\pgf@xa}{\the\pgf@ya}}% + \expandafter\pgf@decorate@parselineto\pgf@temp% + \advance\pgf@xa#3% + \pgf@xb\pgf@xa% + \edef\pgf@temp{{\the\pgf@xa}{\the\pgf@ya}}% + \expandafter\pgf@decorate@parselineto\pgf@temp% + \advance\pgf@ya-#4% + \edef\pgf@temp{{\the\pgf@xa}{\the\pgf@ya}}% + \expandafter\pgf@decorate@parselineto\pgf@temp% + \advance\pgf@xa-#3% + \edef\pgf@temp{{\the\pgf@xa}{\the\pgf@ya}}% + \expandafter\pgf@decorate@parseclosepath\pgf@temp% + \else% + \advance\pgf@xa#3% + \edef\pgf@temp{{\the\pgf@xa}{\the\pgf@ya}}% + \pgf@xb\pgf@xa% + \expandafter\pgf@decorate@parselineto\pgf@temp% + \advance\pgf@ya#4% + \pgf@yb\pgf@ya% + \edef\pgf@temp{{\the\pgf@xa}{\the\pgf@ya}}% + \expandafter\pgf@decorate@parselineto\pgf@temp% + \advance\pgf@xa-#3% + \edef\pgf@temp{{\the\pgf@xa}{\the\pgf@ya}}% + \expandafter\pgf@decorate@parselineto\pgf@temp% + \advance\pgf@ya-#4% + \edef\pgf@temp{{\the\pgf@xa}{\the\pgf@ya}}% + \expandafter\pgf@decorate@parseclosepath\pgf@temp% + \fi% + \let\pgf@decorate@@parsesoftpath\pgf@decorate@orig@@parsesoftpath% + \edef\pgf@marshal{\noexpand\pgf@decorate@parsemoveto{\the\pgf@xb}{\the\pgf@yb}}% + \pgf@marshal% +}% + + + + +\def\pgf@decorate@linelength#1#2{% + \pgf@process{\pgfpointdiff{#1}{#2}}% + \pgfmathveclen@{\pgfmath@tonumber{\pgf@x}}{\pgfmath@tonumber{\pgf@y}}% +}% + +\newif\ifpgf@decorate@subdivide +\def\pgf@decorate@curvelength#1#2#3#4{% + \begingroup% + \pgf@decorate@recursive@subdividecurve@left{#1}{#2}{#3}{#4}% + \pgfutil@tempdimb\pgfmathresult pt\relax% + \pgf@decorate@recursive@subdividecurve@right{#1}{#2}{#3}{#4}% + \advance\pgfutil@tempdimb\pgfmathresult pt\relax% + \pgfmath@returnone\pgfutil@tempdimb% + \endgroup% +}% + +\def\pgf@decorate@@curvelength#1#2#3#4{% + \begingroup% + \pgf@decorate@curvecordlength{#1}{#2}{#3}{#4}% + \ifpgf@decorate@subdivide% + \pgf@decorate@recursive@subdividecurve@left{#1}{#2}{#3}{#4}% + \pgfutil@tempdimb\pgfmathresult pt\relax% + \pgf@decorate@recursive@subdividecurve@right{#1}{#2}{#3}{#4}% + \advance\pgfutil@tempdimb\pgfmathresult pt\relax% + \expandafter\pgfmath@returnone\expandafter\pgfutil@tempdimb% + \else% + \expandafter\pgfmath@returnone\expandafter\pgfmathresult\expandafter p\expandafter t\expandafter% + \fi% + \endgroup% +}% + +\def\pgf@decorate@curvelength@tolerance{1pt}% + +\def\pgf@decorate@curvecordlength#1#2#3#4{% + #4\relax% + \pgf@xa\pgf@x\pgf@ya\pgf@y% + #1\relax% + \advance\pgf@xa-\pgf@x% + \advance\pgf@ya-\pgf@y% + \ifdim\pgf@xa<0pt\relax\pgf@xa-\pgf@xa\fi% + \ifdim\pgf@ya<0pt\relax\pgf@ya-\pgf@ya\fi% + \pgf@decorate@subdividetrue% + \ifdim\pgf@xa<\pgf@decorate@curvelength@tolerance\relax% + \ifdim\pgf@ya<\pgf@decorate@curvelength@tolerance\relax% + \pgfmathveclen@{\pgfmath@tonumber{\pgf@xa}}{\pgfmath@tonumber{\pgf@ya}}% + \pgf@decorate@subdividefalse% + \fi% + \fi% +}% + +\def\pgf@decorate@recursive@subdividecurve@left#1#2#3#4{% + {% + #1\relax% + \pgfutil@tempdima\pgf@x% + \pgfutil@tempdimb\pgf@y% + \pgf@xa.5\pgf@x\pgf@ya.5\pgf@y% + \pgf@xb.25\pgf@x\pgf@yb.25\pgf@y% + \pgf@xc.125\pgf@x\pgf@yc.125\pgf@y% + #2\relax% + \advance\pgf@xa.5\pgf@x\advance\pgf@ya.5\pgf@y% + \advance\pgf@xb.5\pgf@x\advance\pgf@yb.5\pgf@y% + \advance\pgf@xc.375\pgf@x\advance\pgf@yc.375\pgf@y% + #3\relax% + \advance\pgf@xb.25\pgf@x\advance\pgf@yb.25\pgf@y% + \advance\pgf@xc.375\pgf@x\advance\pgf@yc.375\pgf@y% + #4\relax% + \advance\pgf@xc.125\pgf@x\advance\pgf@yc.125\pgf@y% + \xdef\pgf@marshal{% + \noexpand\pgf@decorate@@curvelength% + {\pgf@x\the\pgfutil@tempdima\pgf@y\the\pgfutil@tempdimb}% + {\pgf@x\the\pgf@xa\pgf@y\the\pgf@ya}{\pgf@x\the\pgf@xb\pgf@y\the\pgf@yb} + {\pgf@x\the\pgf@xc\pgf@y\the\pgf@yc}% + }% + }% + \pgf@marshal% +}% + +\def\pgf@decorate@recursive@subdividecurve@right#1#2#3#4{% + {% + #1\relax% + \pgfutil@tempdima.125\pgf@x\pgfutil@tempdimb.125\pgf@y% + #2\relax% + \advance\pgfutil@tempdima.375\pgf@x\advance\pgfutil@tempdimb.375\pgf@y% + \pgf@xa.25\pgf@x\pgf@ya.25\pgf@y% + #3\relax% + \advance\pgfutil@tempdima.375\pgf@x\advance\pgfutil@tempdimb.375\pgf@y% + \advance\pgf@xa.5\pgf@x\advance\pgf@ya.5\pgf@y% + \pgf@xb.5\pgf@x\pgf@yb.5\pgf@y% + #4\relax% + \advance\pgfutil@tempdima.125\pgf@x\advance\pgfutil@tempdimb.125\pgf@y% + \advance\pgf@xa.25\pgf@x\advance\pgf@ya.25\pgf@y% + \advance\pgf@xb.5\pgf@x\advance\pgf@yb.5\pgf@y% + \pgf@xc\pgf@x\pgf@yc\pgf@y% + \xdef\pgf@marshal{% + \noexpand\pgf@decorate@@curvelength% + {\pgf@x\the\pgfutil@tempdima\pgf@y\the\pgfutil@tempdimb}% + {\pgf@x\the\pgf@xa\pgf@y\the\pgf@ya}{\pgf@x\the\pgf@xb\pgf@y\the\pgf@yb} + {\pgf@x\the\pgf@xc\pgf@y\the\pgf@yc}% + }% + }% + \pgf@marshal% +}% + + +\def\pgf@decorate@inputsegmentobjects@reverse#1#2{% + \def\pgf@decorate@path@storein{#2}% + \def\pgf@decorate@inputsegments@@temp{}% + \expandafter\pgf@decorate@inputsegments@@reverse@pass@first#1\pgf@stop}% + +\def\pgf@decorate@inputsegments@@reverse@pass@first#1#2\pgf@stop{% + \def\pgf@decorate@temp{#1}% + \expandafter\expandafter\expandafter\def\expandafter\expandafter\expandafter% + \pgf@decorate@inputsegments@@temp\expandafter\expandafter\expandafter% + {\expandafter\pgf@decorate@temp\pgf@decorate@inputsegments@@temp}% + \def\pgf@decorate@test{#2} + \ifx\pgf@decorate@test\pgfutil@empty% + \def\pgf@decorate@next{% + \expandafter\pgf@decorate@inputsegmentobjects@reverse@pass@second\pgf@decorate@inputsegments@@temp\pgf@stop% + }% + \else% + \def\pgf@decorate@next{\pgf@decorate@inputsegments@@reverse@pass@first#2\pgf@stop}% + \fi% + \pgf@decorate@next% +}% +\def\pgf@decorate@inputsegmentobjects@reverse@pass@second#1#2#3\pgf@stop{% + \def\pgf@decorate@inputsegments@temp{#3}% + \def\pgf@decorate@inputsegments@@temp{}% + \let\pgf@decorate@current@inputsegmentobject=\pgfutil@empty + \let\pgf@decorate@last@object=\pgfutil@empty + \pgf@decorate@inputsegmentobjects@reverse@moveto@waitingtrue% + \pgf@decorate@inputsegmentobjects@reverse@closepath@waitingfalse% + \expandafter\pgf@decorate@path@@reverse@pass@second\pgf@decorate@inputsegments@temp\pgf@stop}% + + +\def\pgf@decorate@path@@reverse@pass@second#1{% + \ifx#1\pgf@stop% + \expandafter\def\expandafter\pgf@decorate@inputsegments@@temp\expandafter{% + \pgf@decorate@inputsegments@@temp% + {\pgf@decorate@inputsegmentobject@endofinputsegments}% + {\pgf@decorate@inputsegmentobject@endofinputsegments} + }% + \expandafter\let\pgf@decorate@path@storein=\pgf@decorate@inputsegments@@temp% + \let\pgf@decorate@next=\relax% + \else% + \ifx#1\pgf@decorate@inputsegmentobject@moveto% + \let\pgf@decorate@next=\pgf@decorate@inputsegmentobjects@reverse@moveto% + \else% + \ifx#1\pgf@decorate@inputsegmentobject@lineto% + \let\pgf@decorate@next=\pgf@decorate@inputsegmentobjects@reverse@lineto% + \else% + \ifx#1\pgf@decorate@inputsegmentobject@curveto% + \let\pgf@decorate@next=\pgf@decorate@inputsegmentobjects@reverse@curveto% + \else% + \ifx#1\pgf@decorate@inputsegmentobject@closepath% + \let\pgf@decorate@next=\pgf@decorate@inputsegmentobjects@reverse@closepath% + \else% + \fi% + \fi% + \fi% + \fi% + \fi% + \pgf@decorate@next% +}% + +\newif\ifpgf@decorate@inputsegmentobjects@reverse@moveto@waiting + +\def\pgf@decorate@inputsegmentobjects@reverse@moveto#1{% + \ifpgf@decorate@inputsegmentobjects@reverse@closepath@waiting% + \edef\pgf@decorate@current@inputsegmentobject{% + {\noexpand\pgf@decorate@inputsegmentobject@closepath% + {\pgf@decorate@reverse@closepath@length}% + {\pgf@decorate@last}% + {#1}% + }% + }% + \pgf@decorate@inputsegmentobjects@reverse@closepath@waitingfalse% + \else% + \let\pgf@decorate@current@inputsegmentobject=\pgfutil@empty% + \fi% + \pgf@decorate@inputsegmentobjects@reverse@moveto@waitingtrue% + \pgf@decorate@path@@@reverse@pass@second% +}% + +\newif\ifpgf@decorate@inputsegmentobjects@reverse@closepath@waiting + +\def\pgf@decorate@inputsegmentobjects@reverse@closepath#1#2#3{% + \pgf@decorate@inputsegmentobjects@reverse@closepath@waitingtrue% + \def\pgf@decorate@reverse@closepath@length{#1}% + \def\pgf@decorate@current@inputsegmentobject{}% + \pgf@decorate@inputsegmentobjects@reverse@lineto{#1}{#2}{#3}% +}% + +\def\pgf@decorate@inputsegmentobjects@reverse@lineto#1#2#3#4{% + \ifpgf@decorate@inputsegmentobjects@reverse@moveto@waiting% + \def\pgf@decorate@current@inputsegmentobject{{\pgf@decorate@inputsegmentobject@moveto{#3}}}% + \pgf@decorate@inputsegmentobjects@reverse@moveto@waitingfalse% + \else% + \let\pgf@decorate@current@inputsegmentobject=\pgfutil@empty% + \fi% + \def\pgf@decorate@append@inputsegmentobject{{\pgf@decorate@inputsegmentobject@lineto{#1}{#3}{#2}}}% + \ifx#4\pgf@decorate@inputsegmentobject@moveto% + \ifpgf@decorate@inputsegmentobjects@reverse@closepath@waiting% + \def\pgf@decorate@append@inputsegmentobject{{\pgf@decorate@inputsegmentobject@closepath{#1}{#3}{#2}}}% + \pgf@decorate@inputsegmentobjects@reverse@closepath@waitingfalse% + \else% + \def\pgf@decorate@append@inputsegmentobject{{\pgf@decorate@inputsegmentobject@lineto{#1}{#3}{#2}}}% + \fi% + \fi% + \expandafter\expandafter\expandafter\def\expandafter\expandafter\expandafter% + \pgf@decorate@current@inputsegmentobject\expandafter\expandafter\expandafter% + {\expandafter\pgf@decorate@current@inputsegmentobject\pgf@decorate@append@inputsegmentobject} + \def\pgf@decorate@last{#2}% + \pgf@decorate@path@@@reverse@pass@second#4% +}% + +\def\pgf@decorate@inputsegmentobjects@reverse@curveto#1#2#3#4#5{% + \ifpgf@decorate@inputsegmentobjects@reverse@moveto@waiting% + \def\pgf@decorate@current@inputsegmentobject{% + {\pgf@decorate@inputsegmentobject@moveto{#5}}% + {\pgf@decorate@inputsegmentobject@curveto{#1}{#5}{#4}{#3}{#2}}% + }% + \pgf@decorate@inputsegmentobjects@reverse@moveto@waitingfalse% + \else% + \def\pgf@decorate@current@inputsegmentobject{{\pgf@decorate@inputsegmentobject@curveto{#1}{#5}{#4}{#3}{#2}}}% + \fi% + \def\pgf@decorate@last{#2}% + \pgf@decorate@path@@@reverse@pass@second% +}% + +\def\pgf@decorate@path@@@reverse@pass@second{% + \ifx\pgf@decorate@current@inputsegmentobject\pgfutil@empty% + \else% + \expandafter\expandafter\expandafter\def\expandafter\expandafter\expandafter% + \pgf@decorate@inputsegments@@temp\expandafter\expandafter\expandafter% + {\expandafter\pgf@decorate@inputsegments@@temp\pgf@decorate@current@inputsegmentobject}% + \fi% + \pgf@decorate@path@@reverse@pass@second% +}% + + + + + +\let\startpgfdecoration=\pgfdecoration +\let\stoppgfdecoration=\endpgfdecoration + +\let\startpgfmetadecoration=\pgfmetadecoration +\let\stoppgfmetadecoration=\endpgfmetadecoration + + + + + + +\pgfdeclaredecoration{lineto}{initial}{% + \state{initial}[switch if input segment less than=\pgfdecoratedremainingdistance to input segment, + width=\pgfdecoratedremainingdistance] + { + }% + \state{input segment}[width=\pgfdecoratedinputsegmentlength,next state=initial] + { + \pgfpathlineto{\pgfpointdecoratedinputsegmentlast} + }% + + \state{final}{\pgfpathlineto{\pgfpointdecoratedpathlast}}% +}% + +\pgfdeclaredecoration{moveto}{initial}{% + \state{initial}[width=\pgfdecoratedremainingdistance]{}% + \state{final}{\pgfpathmoveto{\pgfpointdecoratedpathlast}}% +}% + + +\pgfdeclaredecoration{curveto}{initial}{% + \state{initial}[width=\pgfdecoratedinputsegmentlength/100] + { + \pgfpathlineto{\pgfpointorigin} + }% + \state{final}{\pgfpathlineto{\pgfpointdecoratedpathlast}}% +}% + + + + + +\let\pgfsnakeremainingdistance=\pgfdecoratedremainingdistance +\let\pgfsnakecompleteddistance=\pgfdecoratedcompleteddistance +\def\pgfsnakeangle{\pgfdecoratedangle}% +\let\pgfdeclaresnake=\pgfdeclaredecoration +\let\pgfsetsnakesegmenttransformation=\pgfsetdecorationsegmenttransformation +\def\pgfpathsnakesto#1#2{% + \edef\pgf@temp{#1}% + \expandafter\pgfdecoration\expandafter{\pgf@temp}% + \pgfpathlineto{#2}% + \endpgfdecoration% +}% +\def\pgfpathsnaketo#1#2{\pgfpathsnakesto{{#1}{\pgfsnakeremainingdistance}}{#2}}% +\def\pgfpathsnakealongvector#1#2#3{% + \pgf@process{#3}% + \pgf@xa=\pgf@x% + \pgf@ya=\pgf@y% + \pgfmathparse{#2}% + \pgf@xa=\pgfmathresult\pgf@xa% + \pgf@ya=\pgfmathresult\pgf@ya% + \advance\pgf@xa by\pgf@path@lastx% + \advance\pgf@ya by\pgf@path@lasty% + \edef\pgf@snake@lib@temp{\noexpand\pgfqpoint{\the\pgf@xa}{\the\pgf@ya}}% + \pgfpathsnaketo{#1}{\pgf@snake@lib@temp}% +}% + +\let\pgfsnakesegmentamplitude=\pgfdecorationsegmentamplitude +\let\pgfsnakesegmentlength=\pgfdecorationsegmentlength +\def\pgfsnakesegmentangle{\pgfdecorationsegmentangle}% +\def\pgfsnakesegmentobjectlength{\pgfkeysvalueof{/pgf/decoration/shape start width}}% +\def\pgfsnakesegmentaspect{\pgfdecorationsegmentaspect}% + +\pgfset{% + /pgf/segment amplitude/.style={/pgf/decoration={amplitude={#1},shape height={2*(#1)}}}, + /pgf/segment length/.style={/pgf/decoration={segment length={#1}}}, + /pgf/segment angle/.style={/pgf/decoration={angle={#1}}}, + /pgf/segment aspect/.style={/pgf/decoration={aspect={#1}}}, + /pgf/segment object length/.style={/pgf/decoration={shape width={#1},radius={#1}}}}% + + + +\endinput diff --git a/preprints/A-polynomial-time-algorithm-for-three-machine-unit-job-scheduling-September-24-2026/build/tex/runtime/generic/pgf/modules/pgfmodulematrix.code.tex b/preprints/A-polynomial-time-algorithm-for-three-machine-unit-job-scheduling-September-24-2026/build/tex/runtime/generic/pgf/modules/pgfmodulematrix.code.tex new file mode 100644 index 0000000000000000000000000000000000000000..b1f5aa62f0d2986a17b0d63e858b05012906f101 --- /dev/null +++ b/preprints/A-polynomial-time-algorithm-for-three-machine-unit-job-scheduling-September-24-2026/build/tex/runtime/generic/pgf/modules/pgfmodulematrix.code.tex @@ -0,0 +1,422 @@ + +\ProvidesFileRCS{pgfmodulematrix.code.tex} + +\usepgfmodule{shapes}% + +\newif\ifpgfmatrix +\newif\ifpgf@matrix@correct@call + +\newif\ifpgf@matrix@fixed + +\newcount\pgfmatrixcurrentrow + +\newcount\pgfmatrixcurrentcolumn +\newcount\pgf@matrix@numberofcolumns + +\let\pgf@matrix@cell=\pgfnodeparttextbox % +\let\pgf@matrix@box=\pgf@matrix@cell % + +\long\def\pgfmatrixbeforeassemblenode#1{% + \long\def\pgf@matrix@before@assemnble@node{#1}} +\pgfmatrixbeforeassemblenode{} + + +\def\pgfsetmatrixrowsep#1{\def\pgfmatrixrowsep{#1}}% +\def\pgfsetmatrixcolumnsep#1{\def\pgfmatrixcolumnsep{#1}}% + +\def\pgfmatrixrowsep{0pt}% +\def\pgfmatrixcolumnsep{0pt}% + + + + +\def\pgfmatrix#1#2#3#4#5#6{% + \ifpgfmatrix% + \pgferror{You cannot nest pgfmatrix environments, yet} + \fi% + \bgroup% + \everycr{}% + #6% + \def\pgf@matrix@par@shape{#1}% + \def\pgf@matrix@par@anchor{#2}% + \def\pgf@matrix@par@name{#3}% + \def\pgf@matrix@par@usage{#4}% + \def\pgf@matrix@par@shift{#5}% + \ifx\pgf@matrix@par@name\pgfutil@empty% + \def\pgf@matrix@par@name{pgf@matrix@internal}% + \fi% + \pgfmatrixcurrentrow=0\relax% + \pgfmatrixcurrentcolumn=0\relax% + \pgf@matrix@numberofcolumns=0\relax% + \let\\=\pgfmatrixendrow% + \tabskip=0pt% + \offinterlineskip% + \pgfmatrixtrue% + \let\pgf@matrix@node@list=\pgfutil@empty% + \expandafter\let\csname pgf@matrix@column@finish@1\endcsname=\pgfutil@empty% + \let\pgf@matrix@row@finish=\pgfutil@empty% + \afterassignment\pgf@matrix@cont\let\pgf@next=% +}% +\def\pgf@matrix@cont{% + \setbox\pgf@matrix@box=\hbox\bgroup\vbox\bgroup% + \halign\bgroup% + \pgf@matrix@init@row% + \pgf@matrix@step@column% + {% + \pgf@matrix@startcell% + ##% + \pgf@matrix@endcell% + }% + &% + ##\pgf@matrix@padding&&% + \pgf@matrix@step@column% + {% + \pgf@matrix@startcell% + ##% + \pgf@matrix@endcell% + }&% + ##\pgf@matrix@padding% + \cr% +}% + +\def\pgf@end@matrix{% + \egroup\egroup% + \egroup% + \pgf@matrix@before@assemnble@node + \ifx\pgf@matrix@node@list\pgfutil@empty% + \else% + \pgf@matrix@compute@origin% + \pgf@matrix@shift@nodes@initial% + \fi% + \pgftransformresetnontranslations% + \pgftransformshift{\pgf@process{\pgftransformreset\pgf@matrix@par@shift\pgf@x=-\pgf@x\pgf@y=-\pgf@y}}% + \pgfmultipartnode{\pgf@matrix@par@shape}{\pgf@matrix@par@anchor}{\pgf@matrix@par@name}{\pgf@matrix@par@usage}% + \ifx\pgf@matrix@node@list\pgfutil@empty% + \else% + {% + \pgftransformreset% + \pgf@process{\pgfpointanchor{\pgf@matrix@par@name}{text}}% + \xdef\pgf@matrix@offset{\noexpand\pgfqpoint{\the\pgf@x}{\the\pgf@y}}% + \pgf@matrix@shift@nodes@secondary{\pgf@matrix@offset}% + }% + \fi% +}% + + + + + +\def\pgfmatrixnextcell{% + \let\pgf@matrix@signal@cell@end=\pgf@matrix@signal@cell@end% + \pgfutil@ifnextchar[{\pgf@matrix@cell@cont}{\pgf@matrix@cell@cont[0pt]}}% + +\def\pgf@matrix@cell@cont[#1]{% + \xdef\pgf@matrix@last@nextcell@options{#1}% + &\pgf@matrix@correct@calltrue&% +}% + + +\def\pgf@matrix@fixed@text{between origins}% +\def\pgf@matrix@natural@text{between borders}% + +\def\pgf@matrix@addtolength#1#2{\edef\pgf@temp{#2}\expandafter\pgf@matrix@addtolength@\expandafter#1#2,\pgf@stop}% +\def\pgf@matrix@addtolength@#1#2,#3\pgf@stop{% + \pgfkeys@spdef\pgf@temp{#2}% + \ifx\pgf@temp\pgfutil@empty% + \else% + \ifx\pgf@temp\pgf@matrix@fixed@text% + \global\pgf@matrix@fixedtrue% + \else% + \ifx\pgf@temp\pgf@matrix@natural@text% + \global\pgf@matrix@fixedfalse% + \else% + \pgfmathaddtolength{#1}{#2}% + \fi% + \fi% + \fi% + \def\pgf@temp{#3}% + \ifx\pgf@temp\pgfutil@empty% + \else% + \pgf@matrix@addtolength@#1#3\pgf@stop% + \fi% +}% + + + + +\def\pgf@matrix@init@row{% + \relax% + \global\advance\pgfmatrixcurrentrow by 1\relax% + \global\pgfmatrixcurrentcolumn=0\relax% + \expandafter\gdef\csname pgf@matrix@maxy\the\pgfmatrixcurrentrow\endcsname{0pt}% + \expandafter\gdef\csname pgf@matrix@miny\the\pgfmatrixcurrentrow\endcsname{0pt}% +}% + + + + +\def\pgf@matrix@step@column{% + \relax% + \global\advance\pgfmatrixcurrentcolumn by1\relax% +}% + + + + +\def\pgf@matrix@startcell{% + \let\pgf@nodecallback=\pgf@matrix@nodecallback% + \pgfinterruptboundingbox% + \pgftransformreset% + \setbox\pgf@matrix@cell=\hbox\bgroup\bgroup% + \pgf@relevantforpicturesizetrue + \pgfsys@beginpicture% + \normalbaselines% + \pgfutil@ifnextchar\let% + {% + \afterassignment\pgf@matrix@empty@check\let\pgf@matrix@temp=% + }% + {% + \pgf@matrix@empty@cell@false% + \pgfmatrixbegincode% + }% +}% + +\def\pgf@matrix@empty@check{\futurelet\pgf@matrix@check@token\pgf@matrix@empty@check@cont}% +\def\pgf@matrix@empty@check@cont{% + \ifx\pgf@matrix@check@token\pgf@matrix@signal@cell@end% + \pgf@matrix@empty@cell@true% + \expandafter% + \pgfmatrixemptycode% + \expandafter% + \let% + \else% + \pgf@matrix@empty@cell@false% + \expandafter% + \pgfmatrixbegincode% + \expandafter + \let% + \fi% +}% + +\newif\ifpgf@matrix@empty@cell@ + + + +\def\pgf@matrix@endcell{% + \ifpgf@matrix@empty@cell@% + \else% + \expandafter\expandafter\expandafter\pgfmatrixendcode% + \expandafter\ifpgf@matrix@last@cell@in@row\expandafter\pgf@matrix@last@cell@in@rowtrue\expandafter\fi% + \fi% + \pgfsys@endpicture% + \ifnum\pgfmatrixcurrentcolumn<\pgf@matrix@numberofcolumns% + \else% + \ifpgf@matrix@last@cell@in@row\else% + {% + \global\pgf@matrix@fixedfalse% + \pgf@x=0pt% + \pgf@matrix@addtolength{\pgf@x}{\pgfmatrixcolumnsep}% + \pgf@matrix@addtolength{\pgf@x}{\pgf@matrix@last@nextcell@options}% + \ifpgf@matrix@fixed% + \expandafter\pgfutil@g@addto@macro\csname pgf@matrix@column@finish@\the\pgfmatrixcurrentcolumn\endcsname% + {\global\pgf@picmaxx=0pt}% + \fi% + \advance\pgfmatrixcurrentcolumn by1\relax % + \expandafter\xdef\csname pgf@matrix@column@sep@\the\pgfmatrixcurrentcolumn\endcsname{\the\pgf@x}% + \ifpgf@matrix@fixed% + \expandafter\gdef\csname pgf@matrix@column@finish@\the\pgfmatrixcurrentcolumn\endcsname{\global\pgf@picminx=0pt}% + \else% + \expandafter\global\expandafter\let\csname pgf@matrix@column@finish@\the\pgfmatrixcurrentcolumn\endcsname=\pgfutil@empty% + \fi% + }% + \fi\fi% + \ifdim\pgf@picmaxx=-16000pt\relax% + \global\pgf@picmaxx=0pt\relax% + \global\pgf@picminx=0pt\relax% + \global\pgf@picmaxy=0pt\relax% + \global\pgf@picminy=0pt\relax% + \fi% + \csname pgf@matrix@column@finish@\the\pgfmatrixcurrentcolumn\endcsname% + \pgf@matrix@row@finish% + \egroup\egroup% + \ifnum\pgfmatrixcurrentcolumn>1\relax% + \pgf@y=\csname pgf@matrix@column@sep@\the\pgfmatrixcurrentcolumn\endcsname\relax% + \advance\pgf@picminx by-\pgf@y% + \fi% + \ifdim\pgf@picmaxy>\csname pgf@matrix@maxy\the\pgfmatrixcurrentrow\endcsname% + \expandafter\xdef\csname pgf@matrix@maxy\the\pgfmatrixcurrentrow\endcsname{\the\pgf@picmaxy}% + \fi% + \ifdim\pgf@picminy<\csname pgf@matrix@miny\the\pgfmatrixcurrentrow\endcsname% + \expandafter\xdef\csname pgf@matrix@miny\the\pgfmatrixcurrentrow\endcsname{\the\pgf@picminy}% + \fi% + \ht\pgf@matrix@cell=\pgf@picmaxy% + \dp\pgf@matrix@cell=-\pgf@picminy% + \ifnum\pgfmatrixcurrentcolumn>\pgf@matrix@numberofcolumns\relax% + \expandafter\xdef\csname pgf@matrix@maxx\the\pgfmatrixcurrentcolumn\endcsname{\the\pgf@picmaxx}% + \expandafter\xdef\csname pgf@matrix@minx\the\pgfmatrixcurrentcolumn\endcsname{\the\pgf@picminx}% + \global\pgf@matrix@numberofcolumns=\pgfmatrixcurrentcolumn\relax% + \else% + \ifdim\pgf@picmaxx>\csname pgf@matrix@maxx\the\pgfmatrixcurrentcolumn\endcsname% + \expandafter\xdef\csname pgf@matrix@maxx\the\pgfmatrixcurrentcolumn\endcsname{\the\pgf@picmaxx}% + \fi% + \ifdim\pgf@picminx<\csname pgf@matrix@minx\the\pgfmatrixcurrentcolumn\endcsname% + \expandafter\xdef\csname pgf@matrix@minx\the\pgfmatrixcurrentcolumn\endcsname{\the\pgf@picminx}% + \fi% + \fi% + \xdef\pgf@matrix@paddingskip{\the\pgf@picmaxx}% + \hfil\hskip-\pgf@picminx% + \wd\pgf@matrix@cell=0pt% + \box\pgf@matrix@cell% + \endpgfinterruptboundingbox% +}% + +\let\pgfmatrixbegincode=\relax +\let\pgfmatrixendcode=\relax +\let\pgfmatrixemptycode=\relax + + + +\def\pgf@matrix@nodecallback#1{% + \def\pgf@temp{#1}% + \ifx\pgf@temp\pgfutil@empty% + \else% + \xdef\pgf@matrix@node@list{\pgf@matrix@node@list,{#1}}% + \expandafter\xdef\csname pgf@matrix@node@location@#1\endcsname{{\the\pgfmatrixcurrentrow}{\the\pgfmatrixcurrentcolumn}}% + \fi% +}% + + + + +\def\pgf@matrix@padding{% + \ifpgf@matrix@correct@call% + \else% + \pgferror{Single ampersand used with wrong catcode}% + \fi% + \hskip\pgf@matrix@paddingskip\hfil% +}% + + + + +\def\pgf@matrix@compute@origin{% + {% + \ifnum\pgfmatrixcurrentrow>0\relax% + \expandafter\gdef\csname pgf@matrix@miny\the\pgfmatrixcurrentrow\endcsname{0pt}% + \fi + \c@pgf@counta=\pgfmatrixcurrentrow\relax% + \loop% + \ifnum\c@pgf@counta>1\relax% + \pgf@y=\csname pgf@matrix@miny\the\c@pgf@counta\endcsname\relax% + \advance\pgf@y by\csname pgf@matrix@maxy\the\c@pgf@counta\endcsname\relax + \advance\c@pgf@counta by-1\relax% + \pgf@ya=\csname pgf@matrix@miny\the\c@pgf@counta\endcsname\relax% + \advance\pgf@y by-\pgf@ya\relax% + \expandafter\xdef\csname pgf@matrix@miny\the\c@pgf@counta\endcsname{\the\pgf@y}% + \repeat% + }% + {% + \ifnum\pgf@matrix@numberofcolumns>0\relax% + \pgf@x=\csname pgf@matrix@minx1\endcsname\relax% + \pgf@x=-\pgf@x% + \expandafter\xdef\csname pgf@matrix@minx1\endcsname{\the\pgf@x}% + \fi + \c@pgf@counta=1\relax% + \loop% + \ifnum\c@pgf@counta<\pgf@matrix@numberofcolumns\relax% + \pgf@x=\csname pgf@matrix@minx\the\c@pgf@counta\endcsname\relax% + \advance\pgf@x by\csname pgf@matrix@maxx\the\c@pgf@counta\endcsname\relax% + \advance\c@pgf@counta by1\relax% + \pgf@xa=\csname pgf@matrix@minx\the\c@pgf@counta\endcsname\relax% + \advance\pgf@x by-\pgf@xa\relax% + \expandafter\xdef\csname pgf@matrix@minx\the\c@pgf@counta\endcsname{\the\pgf@x}% + \repeat% + }% +}% + + + + +\def\pgf@matrix@shift@nodes@initial{% + {% + \pgfutil@for\pgf@matrix@node@name:=\pgf@matrix@node@list\do{% + \ifx\pgf@matrix@node@name\pgfutil@empty% + \else% + \expandafter\ifx\csname pgf@matrix@node@visited@\pgf@matrix@node@name\endcsname\relax% + \pgf@shift@node{\pgf@matrix@node@name}{% + \pgf@x=\csname pgf@matrix@minx% + \expandafter\expandafter\expandafter\pgfutil@secondoftwo\csname pgf@matrix@node@location@\pgf@matrix@node@name\endcsname\endcsname% + \pgf@y=\csname pgf@matrix@miny% + \expandafter\expandafter\expandafter\pgfutil@firstoftwo\csname pgf@matrix@node@location@\pgf@matrix@node@name\endcsname\endcsname% + }% + \expandafter\let\csname pgf@matrix@node@visited@\pgf@matrix@node@name\endcsname=\pgfutil@empty% + \fi% + \fi% + }% + }% +}% + + + +\def\pgf@matrix@shift@nodes@secondary#1{% + \pgfutil@for\pgf@matrix@node@name:=\pgf@matrix@node@list\do{% + \ifx\pgf@matrix@node@name\pgfutil@empty% + \else% + \expandafter\ifx\csname pgf@matrix@node@visited@\pgf@matrix@node@name\endcsname\relax% + \pgf@shift@node{\pgf@matrix@node@name}{#1}% + \expandafter\let\csname pgf@matrix@node@visited@\pgf@matrix@node@name\endcsname=\pgfutil@empty% + \fi% + \fi% + }% +}% + + +\def\pgfmatrixendrow{% + \let\pgf@matrix@signal@cell@end=\pgf@matrix@signal@cell@end + \pgf@matrix@last@cell@in@rowtrue% + \xdef\pgf@matrix@rowsep{\pgfmatrixrowsep}% + &\pgf@matrix@correct@calltrue% + \global\pgf@matrix@fixedfalse% + \pgf@y=0pt% + \pgf@matrix@addtolength\pgf@y{\pgf@matrix@rowsep}% + \pgfutil@ifnextchar[{\pgfmatrixendrow@skip}{\pgf@matrix@finish@line}% +}% +\newif\ifpgf@matrix@last@cell@in@row + +\def\pgf@matrix@signal@cell@end{\pgf@matrix@signal@cell@end}% + +\def\pgfmatrixendrow@skip[#1]{% + \pgf@matrix@addtolength\pgf@y{#1}% + \pgf@matrix@finish@line% +}% + +\def\pgf@matrix@finish@line{% + \global\pgf@y=\pgf@y% + \pgf@ya=-\pgf@y% + \global\advance\pgf@ya by\csname pgf@matrix@miny\the\pgfmatrixcurrentrow\endcsname% + \expandafter\xdef\csname pgf@matrix@miny\the\pgfmatrixcurrentrow\endcsname{\the\pgf@ya}% + \pgfutil@ifnextchar\egroup{\cr\pgf@end@matrix}{\pgf@matrix@no@eom@found}% +}% + +\def\pgf@matrix@eom@found{% + \cr% + \pgf@end@matrix% +}% + +\def\pgf@matrix@no@eom@found{% + \cr% + \noalign{\vskip\pgf@y% + \ifpgf@matrix@fixed% + \vskip\csname pgf@matrix@miny\the\pgfmatrixcurrentrow\endcsname% + \vskip\pgf@y% + \pgf@y=-\pgf@y% + \expandafter\xdef\csname pgf@matrix@miny\the\pgfmatrixcurrentrow\endcsname{\the\pgf@y}% + \gdef\pgf@matrix@row@finish{\global\pgf@picmaxy=0pt}% + \else% + \global\let\pgf@matrix@row@finish=\pgfutil@empty% + \fi% + }% +}% + + +\endinput diff --git a/preprints/A-polynomial-time-algorithm-for-three-machine-unit-job-scheduling-September-24-2026/build/tex/runtime/generic/pgf/modules/pgfmoduleplot.code.tex b/preprints/A-polynomial-time-algorithm-for-three-machine-unit-job-scheduling-September-24-2026/build/tex/runtime/generic/pgf/modules/pgfmoduleplot.code.tex new file mode 100644 index 0000000000000000000000000000000000000000..e4fc5c36163036d0a57231d2ebc4d08f5d314d6d --- /dev/null +++ b/preprints/A-polynomial-time-algorithm-for-three-machine-unit-job-scheduling-September-24-2026/build/tex/runtime/generic/pgf/modules/pgfmoduleplot.code.tex @@ -0,0 +1,338 @@ + +\ProvidesFileRCS{pgfmoduleplot.code.tex} + + + + +\def\pgfplotstreamstart{\pgf@plotstreamstart}% +\def\pgfplotstreampoint#1{\gdef\pgfplotlastpoint{#1}\pgf@plotstreampoint{#1}}% +\def\pgfplotstreampointoutlier#1{\pgfplot@outliers{}{\pgf@plotstreamjump}{\pgfplotstreampoint{#1}}}% +\def\pgfplotstreampointundefined{\pgfplot@outliers{}{\pgf@plotstreamjump}{}}% +\def\pgfplotstreamnewdataset{\pgfplot@outliers{}{\pgf@plotstreamjump}{}}% +\def\pgfplotstreamspecial{\pgf@plotstreamspecial}% +\def\pgfplotstreamend{\pgf@plotstreamend}% + +\def\pgfplot@ignorer#1#2#3{#1}% +\def\pgfplot@jumper#1#2#3{#2}% +\def\pgfplot@plotter#1#2#3{#3}% + +\let\pgfplot@outliers\pgfplot@jumper +\let\pgfplot@undefined\pgfplot@jumper +\let\pgfplot@newdata\pgfplot@jumper + +\pgfset{ + handle outlier points in plots/.is choice, + handle outlier points in plots/ignore/.code=\let\pgfplot@outliers\pgfplot@ignorer, + handle outlier points in plots/jump/.code=\let\pgfplot@outliers\pgfplot@jumper, + handle outlier points in plots/plot/.code=\let\pgfplot@outliers\pgfplot@plotter, + handle undefined points in plots/.is choice, + handle undefined points in plots/ignore/.code=\let\pgfplot@undefined\pgfplot@ignorer, + handle undefined points in plots/jump/.code=\let\pgfplot@undefined\pgfplot@jumper, + handle new data sets in plots/.is choice, + handle new data sets in plots/ignore/.code=\let\pgfplot@newdata\pgfplot@ignorer, + handle new data sets in plots/jump/.code=\let\pgfplot@newdata\pgfplot@jumper, +}% + + + +\def\pgfdeclareplothandler#1#2#3{% + \def#1#2{% + \pgfkeys{% + /pgf/plots/@handler options/.cd, + start=\relax, + end macro=\relax, + point macro=\pgfutil@gobble, + jump macro=\relax, + special macro=\pgfutil@gobble,% + #3% + }% + }% +}% + +\pgfkeys{% + /pgf/plots/@handler options/.cd, + start/.code=% + \gdef\pgf@plotstreamstart{% + \global\pgf@plot@startedfalse% + \global\let\pgf@plotstreamend\pgf@plotstreamend@init% + \global\let\pgf@plotstreampoint\pgf@plotstreampoint@init% + \global\let\pgf@plotstreamjump\pgf@plotstreamjump@init% + \global\let\pgf@plotstreamspecial\pgf@plotstreamspecial@init% + #1% + },% + start macro/.code=% + \gdef\pgf@plotstreamstart{% + \global\pgf@plot@startedfalse% + \global\let\pgf@plotstreamend\pgf@plotstreamend@init% + \global\let\pgf@plotstreampoint\pgf@plotstreampoint@init% + \global\let\pgf@plotstreamjump\pgf@plotstreamjump@init% + \global\let\pgf@plotstreamspecial\pgf@plotstreamspecial@init% + #1% + },% + end/.code=\gdef\pgf@plotstreamend@init{#1}, + end macro/.code=\global\let\pgf@plotstreamend@init#1, + point/.code=\gdef\pgf@plotstreampoint@init##1{#1}, + point macro/.code=\global\let\pgf@plotstreampoint@init#1, + jump/.code=\gdef\pgf@plotstreamjump@init{#1}, + jump macro/.code=\global\let\pgf@plotstreamjump@init#1, + special/.code=\gdef\pgf@plotstreamspecial@init##1{#1}, + special macro/.code=\global\let\pgf@plotstreamspecial@init#1, +}% + +\newif\ifpgf@plot@started + + + + + +\def\pgfsetlinetofirstplotpoint{\let\pgf@plot@first@action=\pgfpathlineto}% + + + +\def\pgfsetmovetofirstplotpoint{\let\pgf@plot@first@action=\pgfpathmoveto}% + +\let\pgf@plot@first@action=\pgfpathmoveto + + + + + + + +\pgfdeclareplothandler{\pgfplothandlerlineto}{}{ + point macro=\pgf@plot@line@handler, + jump=\global\let\pgf@plotstreampoint\pgf@plot@line@handler@move% +}% + +\def\pgf@plot@line@handler#1{% + \pgf@plot@first@action{#1}% + \global\let\pgf@plotstreampoint=\pgfpathlineto% +}% + +\def\pgf@plot@line@handler@move#1{% + \pgfpathmoveto{#1}% + \global\let\pgf@plotstreampoint=\pgfpathlineto% +}% + + + + +\pgfdeclareplothandler{\pgfplothandlerpolygon}{}{% + point macro=\pgf@plot@line@handler@close, + jump macro=\pgf@plot@next@close@and@moveto, + end macro=\pgf@plot@polygon@stop +}% + +\def\pgf@plot@line@handler@close#1{% + \pgfpathmoveto{#1}% + \global\pgf@plot@startedtrue% + \global\let\pgf@plotstreampoint=\pgfpathlineto% +}% + +\def\pgf@plot@polygon@stop{% + \ifpgf@plot@started% + \pgfpathclose% + \fi% + \global\pgf@plot@startedfalse% +}% + +\def\pgf@plot@next@close@and@moveto{% + \ifpgf@plot@started% + \pgfpathclose% + \fi% + \global\let\pgf@plotstreampoint\pgf@plot@line@handler@close% +}% + + + + + + + + +\pgfdeclareplothandler{\pgfplothandlerdiscard}{}{}% + + + + +\pgfdeclareplothandler{\pgfplothandlerrecord}{#1}{% + start=\gdef#1{\pgfplotstreamstart}, + point=\expandafter\gdef\expandafter#1\expandafter{#1\pgfplotstreampoint{##1}}, + jump=\expandafter\gdef\expandafter#1\expandafter{#1\pgf@plotstreamjump}, + special=\expandafter\gdef\expandafter#1\expandafter{#1\pgfplotstreamspecial{##1}}, + end=\expandafter\gdef\expandafter#1\expandafter{#1\pgfplotstreamend}, +}% + + + + +\def\pgfplotxyfile#1{% + \begingroup% + \def\b@pgfplotsxyfile@scanning@for@first{1}% + \pgfplotstreamstart% + \openin\r@pgf@reada=#1 + \ifeof\r@pgf@reada + \pgfwarning{Plot data file `#1' not found.} + \else + \catcode`\#=14 + \catcode`\^^M=5 + \pgf@readxyfile% + \fi + \pgfplotstreamend% + \endgroup% +}% + +\let\pgf@savedpar=\par +\def\pgf@partext{\par}% +\def\pgf@readxyfile{% + \pgfutil@read\r@pgf@reada to \pgf@temp% + \let\par=\pgf@savedpar% + \edef\pgf@temp{\pgf@temp}% + \ifx\pgf@temp\pgfutil@empty% + \if1\b@pgfplotsxyfile@scanning@for@first + \else + \ifeof\r@pgf@reada\else\pgfplotstreamnewdataset\fi% + \fi + \else\ifx\pgf@temp\pgf@partext% + \if1\b@pgfplotsxyfile@scanning@for@first + \else + \ifeof\r@pgf@reada\else\pgfplotstreamnewdataset\fi% + \fi + \else% + \expandafter\pgf@parsexyline\pgf@temp\pgf@stop% + \fi\fi% + \ifeof\r@pgf@reada\else\expandafter\pgf@readxyfile\fi% +}% + +\def\pgf@parsexyline#1 #2 #3\pgf@stop{% + \def\b@pgfplotsxyfile@scanning@for@first{0}% + \edef\pgf@xyline@flag@val{#3}% + \ifx\pgf@xyline@flag@val\pgf@xyline@flag@undef% + \pgfplotstreampointundefined% + \else\ifx\pgf@xyline@flag@val\pgf@xyline@flag@out% + \pgfplotstreampointoutlier{\pgfpointxy{#1}{#2}}% + \else% + \pgfplotstreampoint{\pgfpointxy{#1}{#2}}% + \fi\fi% +}% + +\edef\pgf@xyline@flag@out{o\space}% +\edef\pgf@xyline@flag@undef{u\space}% + + + + + +\def\pgfplotxyzfile#1{% + \begingroup% + \pgfplotstreamstart% + \openin\r@pgf@reada=#1 + \ifeof\r@pgf@reada + \pgfwarning{Plot data file `#1' not found.} + \else + \catcode`\#=14 + \catcode`\^^M=5 + \pgf@readxyzfile% + \fi + \pgfplotstreamend% + \endgroup% +}% + +\def\pgf@readxyzfile{% + \pgfutil@read\r@pgf@reada to \pgf@temp% + \ifx\pgf@temp\pgfutil@empty% + \if1\b@pgfplotsxyfile@scanning@for@first + \else + \ifeof\r@pgf@reada\else\pgfplotstreamnewdataset\fi% + \fi + \else\ifx\pgf@temp\pgf@partext% + \if1\b@pgfplotsxyfile@scanning@for@first + \else + \ifeof\r@pgf@reada\else\pgfplotstreamnewdataset\fi% + \fi + \else% + \expandafter\pgf@parsexyzline\pgf@temp\pgf@stop% + \fi\fi% + \ifeof\r@pgf@reada\else\expandafter\pgf@readxyzfile\fi% +}% + +\def\pgf@parsexyzline#1 #2 #3 #4\pgf@stop{% + \def\b@pgfplotsxyfile@scanning@for@first{0}% + \edef\pgf@xyline@flag@val{#4}% + \ifx\pgf@xyline@flag@val\pgf@xyline@flag@undef% + \pgfplotstreampointundefined% + \else\ifx\pgf@xyline@flag@val\pgf@xyline@flag@out% + \pgfplotstreampointoutlier{\pgfpointxyz{#1}{#2}{#3}}% + \else% + \pgfplotstreampoint{\pgfpointxyz{#1}{#2}{#3}}% + \fi\fi% +}% + + + + + +{% + \catcode`\%=12 + \catcode`\"=12 + \xdef\pgf@gnuplot@head{set table \noexpand\pgf@plottablefile@quoted; set format "%.5f"} +}% + +\let\pgf@plotwrite=\w@pgf@writea +\newif\ifpgf@resample@plot + +\pgfkeys{% + /pgf/plot/gnuplot call/.initial={gnuplot}}% + +\def\pgfplotgnuplot{\pgfutil@ifnextchar[{\pgf@plotgnuplot}{\pgf@plotgnuplot[\jobname]}}% +\def\pgf@plotgnuplot[#1]#2{% + \pgf@resample@plottrue% + \pgfutilpreparefilename{#1.gnuplot}% + \let\pgf@plotgnuplotfile=\pgfretval + \pgfutilpreparefilename{#1.table}% + \let\pgf@plottablefile=\pgfretval + \let\pgf@plottablefile@quoted=\pgfretvalquoted + \openin\pgfutil@inputcheck=\pgf@plotgnuplotfile\relax + \ifeof\pgfutil@inputcheck% + \else% + \pgfutil@read\pgfutil@inputcheck to\pgf@temp% + \pgfutil@read\pgfutil@inputcheck to\pgf@plot@line% + \closein\pgfutil@inputcheck + \edef\pgf@plot@code{#2\space}% + \ifx\pgf@plot@code\pgf@plot@line% + \openin\pgfutil@inputcheck=\pgfretval\relax + \ifeof\pgfutil@inputcheck% + \else% + \closein\pgfutil@inputcheck + \pgf@resample@plotfalse% + \fi% + \fi% + \fi + \ifpgf@resample@plot% + \immediate\openout\pgf@plotwrite=\pgf@plotgnuplotfile\relax + \immediate\pgfutil@write\pgf@plotwrite{\pgf@gnuplot@head}% + \immediate\pgfutil@write\pgf@plotwrite{#2}% + \immediate\closeout\pgf@plotwrite% + \pgfutil@shellescape{% + \pgfkeysvalueof{/pgf/plot/gnuplot call} \pgf@plotgnuplotfile}% + \fi% + \pgfplotxyfile{\pgf@plottablefile}% +}% + + + +\def\pgfplotfunction#1#2#3{% + \pgfplotstreamstart% + \foreach#1in{#2}% + {% + \pgf@process{#3}% + \edef\pgf@marshal{\noexpand\pgfplotstreampoint{\noexpand\pgfqpoint{\the\pgf@x}{\the\pgf@y}}}% + \pgf@marshal% + } + \pgfplotstreamend% +}% + + + + +\endinput diff --git a/preprints/A-polynomial-time-algorithm-for-three-machine-unit-job-scheduling-September-24-2026/build/tex/runtime/generic/pgf/modules/pgfmoduleshapes.code.tex b/preprints/A-polynomial-time-algorithm-for-three-machine-unit-job-scheduling-September-24-2026/build/tex/runtime/generic/pgf/modules/pgfmoduleshapes.code.tex new file mode 100644 index 0000000000000000000000000000000000000000..a176f5a88a8a4dbaad1c0d200a1314b2a7e48f72 --- /dev/null +++ b/preprints/A-polynomial-time-algorithm-for-three-machine-unit-job-scheduling-September-24-2026/build/tex/runtime/generic/pgf/modules/pgfmoduleshapes.code.tex @@ -0,0 +1,925 @@ + +\ProvidesFileRCS{pgfmoduleshapes.code.tex} + + + +\newbox\pgfnodeparttextbox + + + +\def\pgfmultipartnode#1#2#3#4{% + \pgfutil@ifundefined{pgf@sh@s@#1}% + {\pgferror{Unknown shape ``#1''}}% + {% + {% + \ifpgflatenodepositioning% + \pgfsys@beginscope% + \fi% + \pgf@outer@adjust@hook% + \let\pgf@sh@savedmacros=\pgfutil@empty% + \let\pgf@sh@savedpoints=\pgfutil@empty% + \def\pgf@sm@shape@name{#1}% + \csname pgf@sh@s@#1\endcsname% + \pgf@sh@savedpoints% + \pgf@sh@savedmacros% + \pgftransformshift{% + \pgf@sh@reanchor{#1}{#2}% + \pgf@x=-\pgf@x% + \pgf@y=-\pgf@y% + }% + \expandafter\pgfsavepgf@process\csname pgf@sh@sa@#3\endcsname{% + \pgf@sh@reanchor{#1}{#2}% + }% + \edef\pgf@node@name{#3}% + \ifpgflatenodepositioning% + \pgf@shapes@late@pos@begin% + \fi% + \ifx\pgf@node@name\pgfutil@empty% + \else% + \expandafter\xdef\csname pgf@sh@ns@\pgf@node@name\endcsname{#1}% + \edef\pgf@sh@@temp{\noexpand\gdef\expandafter\noexpand\csname pgf@sh@np@\pgf@node@name\endcsname}% + \expandafter\pgf@sh@@temp\expandafter{\pgf@sh@savedpoints}% + \edef\pgf@sh@@temp{\noexpand\gdef\expandafter\noexpand\csname pgf@sh@ma@\pgf@node@name\endcsname}% + \expandafter\pgf@sh@@temp\expandafter{\pgf@sh@savedmacros}% + \pgfgettransform\pgf@temp% + \expandafter\xdef\csname pgf@sh@nt@\pgf@node@name\endcsname{\pgf@temp}% + \expandafter\xdef\csname pgf@sh@pi@\pgf@node@name\endcsname{\pgfpictureid}% + \fi% + \pgfutil@ifundefined{pgf@sh@bbg@#1}{} + {{\pgfusetype{.behind background}\pgfidscope\pgfscope\csname pgf@sh@bbg@#1\endcsname\endpgfscope\endpgfidscope}}% + \pgfutil@ifundefined{pgf@sh@bg@#1}{% + \global\let\pgfpositionnodelaterpath\pgfutil@empty% + }% + {\pgfpushtype% + \pgfusetype{.background}\csname pgf@sh@bg@#1\endcsname% + \ifpgflatenodepositioning% + \pgfsyssoftpath@getcurrentpath\pgfpositionnodelaterpath% + \pgfprocessround{\pgfpositionnodelaterpath}{\pgfpositionnodelaterpath}% + \global\let\pgfpositionnodelaterpath\pgfpositionnodelaterpath% + \fi% + #4\pgfpoptype}% + \pgfutil@ifundefined{pgf@sh@fbg@#1}{} + {{\pgfusetype{.before background}\pgfidscope\pgfscope\csname pgf@sh@fbg@#1\endcsname\endpgfscope\endpgfidscope}}% + {% + \expandafter\pgfutil@for\expandafter\pgf@shape@com\expandafter:\expandafter=\csname pgf@sh@boxes@#1\endcsname\do{% + {% + \pgfusetype{.\pgf@shape@com}% + \pgftransformshift{\pgf@sh@reanchor{#1}{\pgf@shape@com}}% + \pgfapproximatenonlineartransformation% + \expandafter\pgfqboxsynced\expandafter{\csname pgfnodepart\pgf@shape@com box\endcsname}% + }% + }% + }% + \pgfutil@ifundefined{pgf@sh@bfg@#1}{}% + {{\pgfusetype{.behind foreground}\pgfidscope\pgfscope\csname pgf@sh@bfg@#1\endcsname\endpgfscope\endpgfidscope}}% + \pgfutil@ifundefined{pgf@sh@fg@#1}{}% + {\pgfpushtype\pgfusetype{.foreground}\csname pgf@sh@fg@#1\endcsname#4\pgfpoptype}% + \pgfutil@ifundefined{pgf@sh@ffg@#1}{}% + {{\pgfusetype{.before foreground}\pgfidscope\pgfscope\csname pgf@sh@ffg@#1\endcsname\endpgfscope\endpgfidscope}}% + \ifpgflatenodepositioning% + \pgf@shapes@late@pos@end% + \pgfsys@endscope% + \else% + \expandafter\pgf@nodecallback\expandafter{\pgf@node@name}% + \fi% + }% + }% +}% +\let\pgf@outer@adjust@hook\relax + +\def\pgf@shapes@late@pos@begin{% + \edef\pgf@node@name{not yet positionedPGFINTERNAL\pgf@node@name}% + \pgfinterruptboundingbox% + \pgf@relevantforpicturesizetrue% + \setbox\pgfpositionnodelaterbox=\hbox% + \bgroup% + \pgfsys@beginscope% +}% + +\def\pgf@shapes@late@pos@end{% + \pgfsys@endscope% + \egroup% + \ifdim\pgf@picminx>\pgf@picmaxx\relax% + \def\pgfpositionnodelaterminx{0.0pt}% + \let\pgfpositionnodelaterminy\pgfpositionnodelaterminx% + \let\pgfpositionnodelatermaxx\pgfpositionnodelaterminx% + \let\pgfpositionnodelatermaxy\pgfpositionnodelaterminx% + \else% + \edef\pgfpositionnodelaterminx{\the\pgf@picminx}% + \edef\pgfpositionnodelaterminy{\the\pgf@picminy}% + \edef\pgfpositionnodelatermaxx{\the\pgf@picmaxx}% + \edef\pgfpositionnodelatermaxy{\the\pgf@picmaxy}% + \fi% + \let\pgfpositionnodelatername=\pgf@node@name% + \pgf@positionnodelater@macro% + \endpgfinterruptboundingbox% +}% + +\def\pgf@shift@node#1#2{% + {% + \pgfsettransform{\csname pgf@sh@nt@#1\endcsname}% + \pgf@process{#2}% + \advance\pgf@pt@x by\pgf@x% + \advance\pgf@pt@y by\pgf@y% + \pgfgettransform{\pgf@temp}% + \expandafter\xdef\csname pgf@sh@nt@#1\endcsname{\pgf@temp}% + }% +}% + + + + +\def\pgfnode#1#2#3#4#5{% + {% + \setbox\pgfnodeparttextbox=\hbox% + {% + \pgfinterruptpicture% + {#3}% + \endpgfinterruptpicture% + }% + \pgfmultipartnode{#1}{#2}{#4}{#5}% + } +}% + + + + +\def\pgfpositionnodelater#1{% + \let\pgf@positionnodelater@macro=#1% + \ifx\pgf@positionnodelater@macro\relax% + \pgflatenodepositioningfalse% + \else% + \pgflatenodepositioningtrue% + \fi% +}% +\newif\ifpgflatenodepositioning +\pgfpositionnodelater{\relax}% +\def\pgfpositionnodelaterbox{0}% + + + + + +\def\pgfpositionnodenow#1{% + \pgfinterruptpath% + {% + \pgfpointtransformed{#1}% + \edef\pgf@temp@shift{\noexpand\pgfqpoint{\the\pgf@x}{\the\pgf@y}} + \pgftransformreset% + \pgftransformshift{\pgf@temp@shift}% + \pgfsys@pictureboxsynced\pgfpositionnodelaterbox% + \pgf@shift@node{\pgfpositionnodelatername}{\pgf@temp@shift}% + \pgfpointtransformed{\pgfqpoint{\pgfpositionnodelaterminx}{\pgfpositionnodelaterminy}}% + \pgf@protocolsizes{\pgf@x}{\pgf@y} + \pgfpointtransformed{\pgfqpoint{\pgfpositionnodelatermaxx}{\pgfpositionnodelatermaxy}}% + \pgf@protocolsizes{\pgf@x}{\pgf@y} + \expandafter\pgfpositionnodenow@rename\pgfpositionnodelatername\relax% + }% + {% + \csname pgf@lns@\pgfpositionnodelatername\endcsname% + \expandafter\global\expandafter\let\csname pgf@lns@\pgfpositionnodelatername\endcsname\relax% + }% + \endpgfinterruptpath% +}% + +\def\pgfpositionnodenow@rename not yet positionedPGFINTERNAL#1\relax{% + \pgfnoderename{#1}{not yet positionedPGFINTERNAL#1} +}% + + + +\def\pgffakenode#1{% + \edef\pgf@temp{\ifpgflatenodepositioning not yet positionedPGFINTERNAL\fi#1}% + \expandafter\gdef\csname pgf@sh@ns@\pgf@temp\endcsname{coordinate}% + \expandafter\gdef\csname pgf@sh@np@\pgf@temp\endcsname{\def\centerpoint{\pgfpointorigin}}% + \expandafter\gdef\csname pgf@sh@nt@\pgf@temp\endcsname{{1}{0}{0}{1}{0pt}{0pt}}% + \expandafter\global\expandafter\let\csname pgf@sh@ma@\pgf@temp\endcsname\pgfutil@empty% + \expandafter\xdef\csname pgf@sh@pi@\pgf@temp\endcsname{\pgfpictureid}% +}% + + + + +\def\pgfnodepostsetupcode#1#2{% + \ifpgflatenodepositioning% + \expandafter\ifx\csname pgf@lns@not yet positionedPGFINTERNAL#1\endcsname\relax% + \expandafter\gdef\csname pgf@lns@not yet positionedPGFINTERNAL#1\endcsname{#2}% + \else% + \expandafter\pgfutil@g@addto@macro\csname pgf@lns@not yet positionedPGFINTERNAL#1\endcsname{#2}% + \fi% + \fi% +}% + + + + +\def\pgfnodealias#1#2{% + \pgf@node@gnamelet{pgf@sh@ns@#1}{pgf@sh@ns@#2}% + \pgf@node@gnamelet{pgf@sh@np@#1}{pgf@sh@np@#2}% + \pgf@node@gnamelet{pgf@sh@nt@#1}{pgf@sh@nt@#2}% + \pgf@node@gnamelet{pgf@sh@pi@#1}{pgf@sh@pi@#2}% + \pgf@node@gnamelet{pgf@sh@ma@#1}{pgf@sh@ma@#2}% + \pgf@nodecallback{#1}% +}% +\def\pgf@node@gnamelet#1#2{% + \expandafter\let\expandafter\pgf@temp\csname #2\endcsname% + \expandafter\global\expandafter\let\csname #1\endcsname=\pgf@temp% +}% + + +\def\pgfnoderename#1#2{% + \pgfnodealias{#1}{#2}% + \expandafter\global\expandafter\let\csname pgf@sh@ns@#2\endcsname\relax + \expandafter\global\expandafter\let\csname pgf@sh@np@#2\endcsname\relax + \expandafter\global\expandafter\let\csname pgf@sh@nt@#2\endcsname\relax + \expandafter\global\expandafter\let\csname pgf@sh@pi@#2\endcsname\relax + \expandafter\global\expandafter\let\csname pgf@sh@ma@#2\endcsname\relax +}% + + + + + +\def\pgfcoordinate#1#2{% + \edef\pgf@temp{#1}% + \ifx\pgf@temp\pgfutil@empty% + \else% + \ifpgflatenodepositioning% + {\pgftransformshift{#2}\pgfnode{coordinate}{center}{}{#1}{}}% + \else% + \pgf@process{\pgfpointtransformed{#2}}% + \expandafter\gdef\csname pgf@sh@ns@#1\endcsname{coordinate}% + \expandafter\xdef\csname pgf@sh@np@#1\endcsname{% + \noexpand\def\noexpand\centerpoint{\noexpand\pgfqpoint{\the\pgf@x}{\the\pgf@y}}% + } + \expandafter\gdef\csname pgf@sh@nt@#1\endcsname{{1}{0}{0}{1}{0pt}{0pt}}% + \expandafter\global\expandafter\let\csname pgf@sh@ma@#1\endcsname\pgfutil@empty% + \expandafter\xdef\csname pgf@sh@pi@#1\endcsname{\pgfpictureid}% + \fi% + \fi% +}% + +\def\pgf@sh@redeferredanchor#1#2{% + \expandafter\gdef\csname pgf@anchor@\pgf@sm@shape@name @#1\endcsname{#2}}% + + + + + + +\def\pgf@sh@resavedanchor#1#2{% + \pgf@process{#2}% + \edef\pgf@sh@marshal{% + \noexpand\pgfutil@g@addto@macro\noexpand\pgf@sh@savedpoints{% + \noexpand\def\noexpand#1{\noexpand\pgfqpoint{\the\pgf@x}{\the\pgf@y}}% + }}% + \pgf@sh@marshal% +}% + + +\def\pgf@sh@resaveddimen#1#2{% + {#2\global\pgf@x=\pgf@x}% + \edef\pgf@sh@marshal{% + \noexpand\pgfutil@g@addto@macro\noexpand\pgf@sh@savedpoints{% + \noexpand\def\noexpand#1{\the\pgf@x}% + }}% + \pgf@sh@marshal% +}% + +\def\pgf@sh@resavedmacro#1#2{% + \let#1\pgfutil@empty% + \def\addtosavedmacro##1{% + \expandafter\def\expandafter\pgf@sh@addtomacro@temp\expandafter{#1\noexpand\def\noexpand##1{##1}}% + {\expandafter\pgfutil@toks@\expandafter{\pgf@sh@addtomacro@temp}\expandafter}% + \expandafter\def\expandafter#1\expandafter{\the\pgfutil@toks@}% + }% + #2\relax% + \edef\pgf@sh@marshal{% + \noexpand\pgfutil@g@addto@macro\noexpand\pgf@sh@savedmacros{% + \noexpand\def\noexpand#1{#1}% + }}% + \pgf@sh@marshal% +}% + +\def\pgf@sh@reanchor#1#2{% + \pgfutil@ifundefined{pgf@anchor@#1@#2}% + {% + \pgfutil@ifundefined{pgf@anchor@generic@#2}{% + \pgfmathsetcounter{pgf@counta}{#2}% + \csname pgf@anchor@#1@border\endcsname{\pgfqpointpolar{\the\c@pgf@counta}{1pt}}% + }{% + \csname pgf@anchor@generic@#2\endcsname{#1}% + }% + }% + {\csname pgf@anchor@#1@#2\endcsname}% +}% + +\def\pgfdeclaregenericanchor#1#2{% + \expandafter\def\csname pgf@anchor@generic@#1\endcsname##1{#2}% +}% + + + +\def\pgfpointanchor#1#2{% + \pgfutil@ifundefined{pgf@sh@ns@#1} + {\pgfutil@ifundefined{pgf@sh@ns@not yet positionedPGFINTERNAL#1}{% + \pgferror{No shape named `#1' is known}}{ + }% + \pgfpointorigin}% + {% + \pgf@process{% + \edef\pgfreferencednodename{#1}% + \csname pgf@sh@ma@#1\endcsname% + \csname pgf@sh@np@#1\endcsname + \pgf@sh@reanchor{\csname pgf@sh@ns@#1\endcsname}{#2}% + {% + \pgfsettransform{\csname pgf@sh@nt@#1\endcsname}% + \pgf@pos@transform@glob% + }% + \pgf@shape@interpictureshift{#1}% + \pgftransforminvert% + \pgf@pos@transform@glob% + }% + }% +}% + +\def\pgf@shape@interpictureshift#1{% + \expandafter\ifx\csname pgf@sh@pi@#1\endcsname\pgfpictureid% + \else% + \pgfsys@getposition{\csname pgf@sh@pi@#1\endcsname}\pgf@shape@original@pos% + \pgfsys@getposition{\pgfpictureid}\pgf@shape@current@pos% + \pgf@xa=\pgf@x% + \pgf@ya=\pgf@y% + \pgf@process{\pgfpointorigin\pgf@shape@current@pos}% + \advance\pgf@xa by-\pgf@x% + \advance\pgf@ya by-\pgf@y% + \pgf@process{\pgfpointorigin\pgf@shape@original@pos}% + \advance\pgf@xa by\pgf@x% + \advance\pgf@ya by\pgf@y% + \pgf@x=\pgf@xa% + \pgf@y=\pgf@ya% + \fi% +}% + + + + +\def\pgfpointshapeborder#1#2{% + \pgfutil@ifundefined{pgf@sh@ns@#1} + {\pgferror{No shape named `#1' is known}% + \pgfpointorigin}% + {% + \pgf@process{% + \edef\pgfreferencednodename{#1}% + \csname pgf@sh@ma@#1\endcsname% + \csname pgf@sh@np@#1\endcsname% + {% + \pgf@process{\pgfpointtransformed{#2}}% + \pgfsettransform{\csname pgf@sh@nt@#1\endcsname}% + \pgftransforminvert% + \pgf@pos@transform@glob% + \pgf@xa=\pgf@x% + \pgf@ya=\pgf@y% + \pgf@process{\csname pgf@anchor@\csname pgf@sh@ns@#1\endcsname @center\endcsname}% + \pgf@process{\pgf@shape@interpictureshift{#1}}% + \advance\pgf@xa by-\pgf@x% + \advance\pgf@ya by-\pgf@y% + \pgf@xb=10pt + \ifdim\pgf@xa<1pt\ifdim\pgf@xa>-1pt + \pgf@xb=10\pgf@xa + \fi\fi + \pgf@yb=10pt + \ifdim\pgf@ya<1pt\ifdim\pgf@ya>-1pt + \pgf@yb=10\pgf@ya + \fi\fi + \ifdim\dimexpr\expandafter\Pgf@geT\the\pgf@xb\pgf@xb + +\expandafter\Pgf@geT\the\pgf@yb\pgf@yb\relax<0.04pt + \expandafter\pgfutil@firstoftwo + \else + \expandafter\pgfutil@secondoftwo + \fi + {% + \pgfwarning + {Returning node center instead of a point on node border. Did you + specify a point identical to the center of node + ``\pgfreferencednodename''?}% + \pgf@sh@reanchor{\csname pgf@sh@ns@#1\endcsname}{center}% + } + {% + \csname pgf@anchor@\csname pgf@sh@ns@#1\endcsname @border\endcsname + {\pgfqpoint{\pgf@xa}{\pgf@ya}}% + }% + \pgfsettransform{\csname pgf@sh@nt@#1\endcsname}% + \pgf@pos@transform@glob% + \global\pgf@x=\pgf@x% + \global\pgf@y=\pgf@y% + }% + \pgf@shape@interpictureshift{#1}% + \pgftransforminvert% + \pgf@pos@transform@glob% + }% + }% +}% + + + +\def\pgfgetnodeparts#1#2{% + \expandafter\let\expandafter#1\csname pgf@sh@boxes@#2\endcsname% +}% + + + +\long\def\pgfdeclareshape#1#2{% + {% + \def\pgf@sm@shape@name{#1}% + \let\savedanchor=\pgf@sh@savedanchor + \let\saveddimen=\pgf@sh@saveddimen + \let\savedmacro=\pgf@sh@savedmacro% + \let\deferredanchor=\pgf@sh@deferredanchor% + \let\anchor=\pgf@sh@anchor + \let\anchorborder=\pgf@sh@anchorborder + \let\behindbackgroundpath=\pgf@sh@behindbgpath + \let\backgroundpath=\pgf@sh@bgpath + \let\beforebackgroundpath=\pgf@sh@beforebgpath + \let\behindforegroundpath=\pgf@sh@behindfgpath + \let\foregroundpath=\pgf@sh@fgpath + \let\beforeforegroundpath=\pgf@sh@beforefgpath + \let\nodeparts=\pgf@sh@boxes + \let\inheritsavedanchors=\pgf@sh@inheritsavedanchors + \let\inheritanchor=\pgf@sh@inheritanchor + \let\inheritanchorborder=\pgf@sh@inheritanchorborder + \let\inheritbehindbackgroundpath=\pgf@sh@inheritbehindbgpath + \let\inheritbackgroundpath=\pgf@sh@inheritbgpath + \let\inheritbeforebackgroundpath=\pgf@sh@inheritbeforebgpath + \let\inheritbehindforegroundpath=\pgf@sh@inheritbehindfgpath + \let\inheritforegroundpath=\pgf@sh@inheritfgpath + \let\inheritbeforeforegroundpath=\pgf@sh@inheritbeforefgpath + \let\inheritnodeparts=\pgf@sh@inheritboxes + \anchorborder{\csname pgf@anchor@#1@center\endcsname}% + \anchor{text}{\pgfpointorigin}% + \nodeparts{text}% + \expandafter\global\expandafter\let\csname pgf@sh@s@\pgf@sm@shape@name\endcsname=\pgfutil@empty% + #2% + }% +}% + +\def\pgf@sh@savedanchor#1#2{% + \expandafter\pgfutil@g@addto@macro\csname pgf@sh@s@\pgf@sm@shape@name\endcsname{\pgf@sh@resavedanchor{#1}{#2}}}% +\def\pgf@sh@saveddimen#1#2{% + \expandafter\pgfutil@g@addto@macro\csname pgf@sh@s@\pgf@sm@shape@name\endcsname{\pgf@sh@resaveddimen{#1}{#2}}}% + \def\pgf@sh@savedmacro#1#2{% + \expandafter\pgfutil@g@addto@macro\csname pgf@sh@s@\pgf@sm@shape@name\endcsname{\pgf@sh@resavedmacro{#1}{#2}}}% +\def\pgf@sh@anchor#1#2{\expandafter\gdef\csname pgf@anchor@\pgf@sm@shape@name @#1\endcsname{#2}}% +\def\pgf@sh@deferredanchor#1#2{% + \expandafter\pgfutil@g@addto@macro + \csname pgf@sh@s@\pgf@sm@shape@name\endcsname{\pgf@sh@redeferredanchor{#1}{#2}}}% +\def\pgf@sh@anchorborder#1{\expandafter\gdef\csname pgf@anchor@\pgf@sm@shape@name @border\endcsname##1{\pgf@process{##1}#1}}% +\long\def\pgf@sh@behindbgpath#1{\expandafter\gdef\csname pgf@sh@bbg@\pgf@sm@shape@name\endcsname{#1}}% +\long\def\pgf@sh@bgpath#1{\expandafter\gdef\csname pgf@sh@bg@\pgf@sm@shape@name\endcsname{#1}}% +\long\def\pgf@sh@beforebgpath#1{\expandafter\gdef\csname pgf@sh@fbg@\pgf@sm@shape@name\endcsname{#1}}% +\long\def\pgf@sh@behindfgpath#1{\expandafter\gdef\csname pgf@sh@bfg@\pgf@sm@shape@name\endcsname{#1}}% +\long\def\pgf@sh@fgpath#1{\expandafter\gdef\csname pgf@sh@fg@\pgf@sm@shape@name\endcsname{#1}}% +\long\def\pgf@sh@beforefgpath#1{\expandafter\gdef\csname pgf@sh@ffg@\pgf@sm@shape@name\endcsname{#1}}% +\def\pgf@sh@boxes#1{\expandafter\gdef\csname pgf@sh@boxes@\pgf@sm@shape@name\endcsname{#1}}% + +\def\pgf@sh@inheritsavedanchors[from=#1]{% + \expandafter\pgfutil@g@addto@macro\csname pgf@sh@s@\pgf@sm@shape@name\endcsname{\csname pgf@sh@s@#1\endcsname}}% + +\def\pgf@sh@inheritanchor[from=#1]#2{% + \edef\pgf@marshal{\global\let\expandafter\noexpand\csname + pgf@anchor@\pgf@sm@shape@name @#2\endcsname=\expandafter\noexpand\csname + pgf@anchor@#1@#2\endcsname}% + \pgf@marshal% +}% +\def\pgf@sh@inheritanchorborder[from=#1]{% + \edef\pgf@marshal{\global\let\expandafter\noexpand\csname + pgf@anchor@\pgf@sm@shape@name @border\endcsname=\expandafter\noexpand\csname + pgf@anchor@#1@border\endcsname}% + \pgf@marshal% +}% +\def\pgf@sh@inheritor#1#2{% + \edef\pgf@marshal{\global\let\expandafter\noexpand\csname + pgf@sh@#1@\pgf@sm@shape@name\endcsname=\expandafter\noexpand\csname + pgf@sh@#1@#2\endcsname}% + \pgf@marshal% +}% +\def\pgf@sh@inheritbehindbgpath[from=#1]{\pgf@sh@inheritor{bbg}{#1}}% +\def\pgf@sh@inheritbgpath[from=#1]{\pgf@sh@inheritor{bg}{#1}}% +\def\pgf@sh@inheritbeforebgpath[from=#1]{\pgf@sh@inheritor{fbg}{#1}}% +\def\pgf@sh@inheritbehindfgpath[from=#1]{\pgf@sh@inheritor{bfg}{#1}}% +\def\pgf@sh@inheritfgpath[from=#1]{\pgf@sh@inheritor{fg}{#1}}% +\def\pgf@sh@inheritbeforefgpath[from=#1]{\pgf@sh@inheritor{ffg}{#1}}% +\def\pgf@sh@inheritboxes[from=#1]{\pgf@sh@inheritor{boxes}{#1}}% + + + + +\pgfdeclareshape{coordinate} +{% + \savedanchor\centerpoint{% + \pgf@x=.5\wd\pgfnodeparttextbox% + \pgf@y=.5\ht\pgfnodeparttextbox% + \advance\pgf@y by -.5\dp\pgfnodeparttextbox% + }% + + \anchor{center}{\centerpoint}% + \anchor{north}{\centerpoint}% + \anchor{north west}{\centerpoint}% + \anchor{north east}{\centerpoint}% + \anchor{west}{\centerpoint}% + \anchor{east}{\centerpoint}% + \anchor{mid}{\centerpoint}% + \anchor{mid west}{\centerpoint}% + \anchor{mid east}{\centerpoint}% + \anchor{base}{\centerpoint}% + \anchor{base west}{\centerpoint}% + \anchor{base east}{\centerpoint}% + \anchor{south}{\centerpoint}% + \anchor{south west}{\centerpoint}% + \anchor{south east}{\centerpoint}% + \anchorborder{\centerpoint}% + + \nodeparts{}% +}% + + + +\pgfset{ + inner xsep/.initial =.3333em, + inner ysep/.initial =.3333em, + inner sep/.style ={/pgf/inner xsep={#1},/pgf/inner ysep={#1}}, + outer xsep/.initial =.5\pgflinewidth, + outer ysep/.initial =.5\pgflinewidth, + outer sep/.code =\pgf@handle@outer@sep{#1}, + minimum width/.initial =1pt, + minimum height/.initial =1pt, + minimum size/.style ={/pgf/minimum width={#1},/pgf/minimum height={#1}}, +}% + + + +\def\pgf@handle@outer@sep#1{% + \def\pgf@temp{#1}% + \ifx\pgf@temp\pgf@auto@text% + \def\pgf@outer@adjust@hook{% + \pgftransformationadjustments% + \pgfkeyssetvalue{/pgf/outer xsep}{.5\pgflinewidth*\pgfhorizontaltransformationadjustment}% + \pgfkeyssetvalue{/pgf/outer ysep}{.5\pgflinewidth*\pgfverticaltransformationadjustment}% + \pgf@outer@auto@adjust@hook% + }% + \else% + \pgfkeyssetvalue{/pgf/outer xsep}{#1}% + \pgfkeyssetvalue{/pgf/outer ysep}{#1}% + \fi% +}% +\def\pgf@auto@text{auto}% + +\let\pgf@outer@auto@adjust@hook\relax + + + +\newif\ifpgfshapeborderusesincircle +\pgfkeys{/pgf/shape border uses incircle/.is if=pgfshapeborderusesincircle}% +\pgfkeys{/pgf/shape border rotate/.initial=0}% + + +\pgfdeclareshape{rectangle} +{% + \savedanchor\northeast{% + \pgf@x=\the\wd\pgfnodeparttextbox% + \pgfmathsetlength\pgf@xc{\pgfkeysvalueof{/pgf/inner xsep}}% + \advance\pgf@x by 2\pgf@xc% + \pgfmathsetlength\pgf@xb{\pgfkeysvalueof{/pgf/minimum width}}% + \ifdim\pgf@x<\pgf@xb% + \pgf@x=\pgf@xb% + \fi% + \pgf@x=.5\pgf@x% + \advance\pgf@x by.5\wd\pgfnodeparttextbox% + \pgfmathsetlength\pgf@xa{\pgfkeysvalueof{/pgf/outer xsep}}% + \advance\pgf@x by\pgf@xa% + \pgf@y=\ht\pgfnodeparttextbox% + \advance\pgf@y by\dp\pgfnodeparttextbox% + \pgfmathsetlength\pgf@yc{\pgfkeysvalueof{/pgf/inner ysep}}% + \advance\pgf@y by 2\pgf@yc% + \pgfmathsetlength\pgf@yb{\pgfkeysvalueof{/pgf/minimum height}}% + \ifdim\pgf@y<\pgf@yb% + \pgf@y=\pgf@yb% + \fi% + \pgf@y=.5\pgf@y% + \advance\pgf@y by-.5\dp\pgfnodeparttextbox% + \advance\pgf@y by.5\ht\pgfnodeparttextbox% + \pgfmathsetlength\pgf@ya{\pgfkeysvalueof{/pgf/outer ysep}}% + \advance\pgf@y by\pgf@ya% + }% + + \savedanchor\southwest{% + \pgf@x=\wd\pgfnodeparttextbox% + \pgfmathsetlength\pgf@xc{\pgfkeysvalueof{/pgf/inner xsep}}% + \advance\pgf@x by 2\pgf@xc% + \pgfmathsetlength\pgf@xb{\pgfkeysvalueof{/pgf/minimum width}}% + \ifdim\pgf@x<\pgf@xb% + \pgf@x=\pgf@xb% + \fi% + \pgf@x=-.5\pgf@x% + \advance\pgf@x by.5\wd\pgfnodeparttextbox% + \pgfmathsetlength\pgf@xa{\pgfkeysvalueof{/pgf/outer xsep}}% + \advance\pgf@x by-\pgf@xa% + \pgf@y=\ht\pgfnodeparttextbox% + \advance\pgf@y by\dp\pgfnodeparttextbox% + \pgfmathsetlength\pgf@yc{\pgfkeysvalueof{/pgf/inner ysep}}% + \advance\pgf@y by 2\pgf@yc% + \pgfmathsetlength\pgf@yb{\pgfkeysvalueof{/pgf/minimum height}}% + \ifdim\pgf@y<\pgf@yb% + \pgf@y=\pgf@yb% + \fi% + \pgf@y=-.5\pgf@y% + \advance\pgf@y by-.5\dp\pgfnodeparttextbox% + \advance\pgf@y by.5\ht\pgfnodeparttextbox% + \pgfmathsetlength\pgf@ya{\pgfkeysvalueof{/pgf/outer ysep}}% + \advance\pgf@y by-\pgf@ya% + }% + + \anchor{center}{ + \pgf@process{\northeast}% + \pgf@xa=.5\pgf@x% + \pgf@ya=.5\pgf@y% + \pgf@process{\southwest}% + \pgf@x=.5\pgf@x% + \pgf@y=.5\pgf@y% + \advance\pgf@x by \pgf@xa% + \advance\pgf@y by \pgf@ya% + }% + \anchor{mid}{\pgf@anchor@rectangle@center\pgfmathsetlength\pgf@y{.5ex}}% + \anchor{base}{\pgf@anchor@rectangle@center\pgf@y=0pt}% + \anchor{north}{ + \pgf@process{\southwest}% + \pgf@xa=.5\pgf@x% + \pgf@process{\northeast}% + \pgf@x=.5\pgf@x% + \advance\pgf@x by \pgf@xa% + }% + \anchor{south}{ + \pgf@process{\northeast}% + \pgf@xa=.5\pgf@x% + \pgf@process{\southwest}% + \pgf@x=.5\pgf@x% + \advance\pgf@x by \pgf@xa% + }% + \anchor{west}{ + \pgf@process{\northeast}% + \pgf@ya=.5\pgf@y% + \pgf@process{\southwest}% + \pgf@y=.5\pgf@y% + \advance\pgf@y by \pgf@ya% + }% + \anchor{mid west}{\southwest\pgfmathsetlength\pgf@y{.5ex}}% + \anchor{base west}{\southwest\pgf@y=0pt}% + \anchor{north west}{ + \southwest + \pgf@xa=\pgf@x + \northeast + \pgf@x=\pgf@xa}% + \anchor{south west}{\southwest}% + \anchor{east}{% + \pgf@process{\southwest}% + \pgf@ya=.5\pgf@y% + \pgf@process{\northeast}% + \pgf@y=.5\pgf@y% + \advance\pgf@y by \pgf@ya% + }% + \anchor{mid east}{\northeast\pgfmathsetlength\pgf@y{.5ex}}% + \anchor{base east}{\northeast\pgf@y=0pt}% + \anchor{north east}{\northeast}% + \anchor{south east}{ + \northeast + \pgf@xa=\pgf@x + \southwest + \pgf@x=\pgf@xa + }% + \anchorborder{% + \pgf@xb=\pgf@x% + \pgf@yb=\pgf@y% + \southwest% + \pgf@xa=\pgf@x% + \pgf@ya=\pgf@y% + \northeast% + \advance\pgf@x by-\pgf@xa% + \advance\pgf@y by-\pgf@ya% + \pgf@xc=.5\pgf@x% + \pgf@yc=.5\pgf@y% + \advance\pgf@xa by\pgf@xc% + \advance\pgf@ya by\pgf@yc% + \edef\pgf@marshal{% + \noexpand\pgfpointborderrectangle + {\noexpand\pgfqpoint{\the\pgf@xb}{\the\pgf@yb}} + {\noexpand\pgfqpoint{\the\pgf@xc}{\the\pgf@yc}}% + }% + \pgf@process{\pgf@marshal}% + \advance\pgf@x by\pgf@xa% + \advance\pgf@y by\pgf@ya% + }% + + \backgroundpath{ + \pgfpathrectanglecorners + {\pgfpointadd{\southwest}{\pgfpoint{\pgfkeysvalueof{/pgf/outer xsep}}{\pgfkeysvalueof{/pgf/outer ysep}}}} + {\pgfpointadd{\northeast}{\pgfpointscale{-1}{\pgfpoint{\pgfkeysvalueof{/pgf/outer xsep}}{\pgfkeysvalueof{/pgf/outer ysep}}}}} + }% +}% + + +\expandafter\def\csname pgf@sh@ns@current subpath start\endcsname{coordinate}% +\expandafter\def\csname pgf@sh@np@current subpath start\endcsname{% + \def\centerpoint{\expandafter\pgfqpoint\pgfsyssoftpath@lastmoveto}% +}% +\expandafter\def\csname pgf@sh@nt@current subpath start\endcsname{{1}{0}{0}{1}{0pt}{0pt}}% +\expandafter\def\csname pgf@sh@pi@current subpath start\endcsname{\pgfpictureid}% + + + +\expandafter\def\csname pgf@sh@ns@current bounding box\endcsname{rectangle}% +\expandafter\def\csname pgf@sh@np@current bounding box\endcsname{% + \def\southwest{\pgfqpoint{\pgf@picminx}{\pgf@picminy}}% + \def\northeast{\pgfqpoint{\pgf@picmaxx}{\pgf@picmaxy}}% +}% +\expandafter\def\csname pgf@sh@nt@current bounding box\endcsname{{1}{0}{0}{1}{0pt}{0pt}}% +\expandafter\def\csname pgf@sh@pi@current bounding box\endcsname{\pgfpictureid}% + + +\expandafter\def\csname pgf@sh@ns@current path bounding box\endcsname{rectangle}% +\expandafter\def\csname pgf@sh@np@current path bounding box\endcsname{% + \def\southwest{\pgfqpoint{\pgf@pathminx}{\pgf@pathminy}}% + \def\northeast{\pgfqpoint{\pgf@pathmaxx}{\pgf@pathmaxy}}% +}% +\expandafter\def\csname pgf@sh@nt@current path bounding box\endcsname{{1}{0}{0}{1}{0pt}{0pt}}% +\expandafter\def\csname pgf@sh@pi@current path bounding box\endcsname{\pgfpictureid}% + + +\expandafter\def\csname pgf@sh@ns@current page\endcsname{rectangle}% +\expandafter\def\csname pgf@sh@np@current page\endcsname{% + \def\southwest{\pgfpointorigin}% + \def\northeast{\pgfpoint{\pgfsys@thepagewidth}{\pgfsys@thepageheight}}% +}% +\expandafter\def\csname pgf@sh@nt@current page\endcsname{{1}{0}{0}{1}{0pt}{0pt}}% +\expandafter\def\csname pgf@sh@pi@current page\endcsname{pgfpageorigin}% + + +\pgfkeys{/pgf/local bounding box/.code={% + \expandafter\gdef\csname pgf@sh@ns@#1\endcsname{rectangle} + \expandafter\gdef\csname pgf@sh@np@#1\endcsname{% + \def\southwest{\pgfqpoint{\csname pgf@lbb@minx@#1\endcsname}{\csname pgf@lbb@miny@#1\endcsname}}% + \def\northeast{\pgfqpoint{\csname pgf@lbb@maxx@#1\endcsname}{\csname pgf@lbb@maxy@#1\endcsname}}% + } + \expandafter\gdef\csname pgf@sh@nt@#1\endcsname{{1}{0}{0}{1}{0pt}{0pt}} + \expandafter\gdef\csname pgf@sh@pi@#1\endcsname{\pgfpictureid} + \expandafter\gdef\csname pgf@lbb@maxx@#1\endcsname{-16000pt}% + \expandafter\gdef\csname pgf@lbb@minx@#1\endcsname{16000pt}% + \expandafter\gdef\csname pgf@lbb@maxy@#1\endcsname{-16000pt}% + \expandafter\gdef\csname pgf@lbb@miny@#1\endcsname{16000pt}% + \pgf@size@hookedtrue% + \expandafter\def\expandafter\pgf@path@size@hook\expandafter{\pgf@path@size@hook\pgf@lbb@do{#1}} + }, + /pgf/freeze local bounding box/.code={% + {% + \csname pgf@sh@np@#1\endcsname% + \southwest% + \pgf@xa=\pgf@x% + \pgf@ya=\pgf@y% + \northeast% + \expandafter\xdef\csname pgf@sh@np@#1\endcsname{% + \noexpand\def\noexpand\southwest{\noexpand\pgfqpoint{\the\pgf@xa}{\the\pgf@ya}}% + \noexpand\def\noexpand\northeast{\noexpand\pgfqpoint{\the\pgf@x}{\the\pgf@y}}% + }% + }% + }, +}% +\def\pgf@lbb@do#1{% + \ifdim\pgf@size@hook@x<\csname pgf@lbb@minx@#1\endcsname\expandafter\xdef\csname pgf@lbb@minx@#1\endcsname{\the\pgf@size@hook@x}\fi% + \ifdim\pgf@size@hook@x>\csname pgf@lbb@maxx@#1\endcsname\expandafter\xdef\csname pgf@lbb@maxx@#1\endcsname{\the\pgf@size@hook@x}\fi% + \ifdim\pgf@size@hook@y<\csname pgf@lbb@miny@#1\endcsname\expandafter\xdef\csname pgf@lbb@miny@#1\endcsname{\the\pgf@size@hook@y}\fi% + \ifdim\pgf@size@hook@y>\csname pgf@lbb@maxy@#1\endcsname\expandafter\xdef\csname pgf@lbb@maxy@#1\endcsname{\the\pgf@size@hook@y}\fi% +}% + + +\pgfdeclareshape{circle} +{% + \savedanchor\centerpoint{% + \pgf@x=.5\wd\pgfnodeparttextbox% + \pgf@y=.5\ht\pgfnodeparttextbox% + \advance\pgf@y by-.5\dp\pgfnodeparttextbox% + }% + + \saveddimen\radius{% + \pgf@ya=.5\ht\pgfnodeparttextbox% + \advance\pgf@ya by.5\dp\pgfnodeparttextbox% + \pgfmathsetlength\pgf@yb{\pgfkeysvalueof{/pgf/inner ysep}}% + \advance\pgf@ya by\pgf@yb% + \pgf@xa=.5\wd\pgfnodeparttextbox% + \pgfmathsetlength\pgf@xb{\pgfkeysvalueof{/pgf/inner xsep}}% + \advance\pgf@xa by\pgf@xb% + \pgf@process{\pgfpointnormalised{\pgfqpoint{\pgf@xa}{\pgf@ya}}}% + \ifdim\pgf@x>\pgf@y% + \c@pgf@counta=\pgf@x% + \ifnum\c@pgf@counta=0\relax% + \else% + \divide\c@pgf@counta by 255\relax% + \pgf@xa=16\pgf@xa\relax% + \divide\pgf@xa by\c@pgf@counta% + \pgf@xa=16\pgf@xa\relax% + \fi% + \else% + \c@pgf@counta=\pgf@y% + \ifnum\c@pgf@counta=0\relax% + \else% + \divide\c@pgf@counta by 255\relax% + \pgf@ya=16\pgf@ya\relax% + \divide\pgf@ya by\c@pgf@counta% + \pgf@xa=16\pgf@ya\relax% + \fi% + \fi% + \pgf@x=\pgf@xa% + \pgfmathsetlength{\pgf@xb}{\pgfkeysvalueof{/pgf/minimum width}}% + \pgfmathsetlength{\pgf@yb}{\pgfkeysvalueof{/pgf/minimum height}}% + \ifdim\pgf@x<.5\pgf@xb% + \pgf@x=.5\pgf@xb% + \fi% + \ifdim\pgf@x<.5\pgf@yb% + \pgf@x=.5\pgf@yb% + \fi% + \pgfmathsetlength{\pgf@xb}{\pgfkeysvalueof{/pgf/outer xsep}}% + \pgfmathsetlength{\pgf@yb}{\pgfkeysvalueof{/pgf/outer ysep}}% + \ifdim\pgf@xb<\pgf@yb% + \advance\pgf@x by\pgf@yb% + \else% + \advance\pgf@x by\pgf@xb% + \fi% + }% + + \anchor{center}{\centerpoint}% + \anchor{mid}{\centerpoint\pgfmathsetlength\pgf@y{.5ex}}% + \anchor{base}{\centerpoint\pgf@y=0pt}% + \anchor{north}{\centerpoint\advance\pgf@y by\radius}% + \anchor{south}{\centerpoint\advance\pgf@y by-\radius}% + \anchor{west}{\centerpoint\advance\pgf@x by-\radius}% + \anchor{east}{\centerpoint\advance\pgf@x by\radius}% + \anchor{mid west}{\centerpoint\advance\pgf@x by-\radius\pgfmathsetlength\pgf@y{.5ex}}% + \anchor{mid east}{\centerpoint\advance\pgf@x by\radius\pgfmathsetlength\pgf@y{.5ex}}% + \anchor{base west}{\centerpoint\advance\pgf@x by-\radius\pgf@y=0pt}% + \anchor{base east}{\centerpoint\advance\pgf@x by\radius\pgf@y=0pt}% + \anchor{north west}{ + \centerpoint + \pgf@xa=\radius + \advance\pgf@x by-0.707107\pgf@xa + \advance\pgf@y by0.707107\pgf@xa + }% + \anchor{south west}{ + \centerpoint + \pgf@xa=\radius + \advance\pgf@x by-0.707107\pgf@xa + \advance\pgf@y by-0.707107\pgf@xa + }% + \anchor{north east}{ + \centerpoint + \pgf@xa=\radius + \advance\pgf@x by0.707107\pgf@xa + \advance\pgf@y by0.707107\pgf@xa + }% + \anchor{south east}{ + \centerpoint + \pgf@xa=\radius + \advance\pgf@x by0.707107\pgf@xa + \advance\pgf@y by-0.707107\pgf@xa + }% + \anchorborder{% + \pgf@xa=\pgf@x% + \pgf@ya=\pgf@y% + \edef\pgf@marshal{% + \noexpand\pgfpointborderellipse + {\noexpand\pgfqpoint{\the\pgf@xa}{\the\pgf@ya}} + {\noexpand\pgfqpoint{\radius}{\radius}}% + }% + \pgf@marshal% + \pgf@xa=\pgf@x% + \pgf@ya=\pgf@y% + \centerpoint% + \advance\pgf@x by\pgf@xa% + \advance\pgf@y by\pgf@ya% + }% + + \backgroundpath{% + \pgfutil@tempdima=\radius% + \pgfmathsetlength{\pgf@xb}{\pgfkeysvalueof{/pgf/outer xsep}}% + \pgfmathsetlength{\pgf@yb}{\pgfkeysvalueof{/pgf/outer ysep}}% + \ifdim\pgf@xb<\pgf@yb% + \advance\pgfutil@tempdima by-\pgf@yb% + \else% + \advance\pgfutil@tempdima by-\pgf@xb% + \fi% + \pgfpathcircle{\centerpoint}{\pgfutil@tempdima}% + }% +}% + + + + + +\endinput diff --git a/preprints/A-polynomial-time-algorithm-for-three-machine-unit-job-scheduling-September-24-2026/build/tex/runtime/generic/pgf/pgf.revision.tex b/preprints/A-polynomial-time-algorithm-for-three-machine-unit-job-scheduling-September-24-2026/build/tex/runtime/generic/pgf/pgf.revision.tex new file mode 100644 index 0000000000000000000000000000000000000000..12b28e20457e0223f434063a15b5180396cdc03e --- /dev/null +++ b/preprints/A-polynomial-time-algorithm-for-three-machine-unit-job-scheduling-September-24-2026/build/tex/runtime/generic/pgf/pgf.revision.tex @@ -0,0 +1,4 @@ +\def\pgfrevision{3.1.10} +\def\pgfversion{3.1.10} +\def\pgfrevisiondate{2023-01-15} +\def\pgfversiondate{2023-01-15} diff --git a/preprints/A-polynomial-time-algorithm-for-three-machine-unit-job-scheduling-September-24-2026/build/tex/runtime/generic/pgf/systemlayer/pgfsys.code.tex b/preprints/A-polynomial-time-algorithm-for-three-machine-unit-job-scheduling-September-24-2026/build/tex/runtime/generic/pgf/systemlayer/pgfsys.code.tex new file mode 100644 index 0000000000000000000000000000000000000000..d3c11e9b7eae00f623ba92184fc78bb6dcdd9968 --- /dev/null +++ b/preprints/A-polynomial-time-algorithm-for-three-machine-unit-job-scheduling-September-24-2026/build/tex/runtime/generic/pgf/systemlayer/pgfsys.code.tex @@ -0,0 +1,805 @@ + +\ProvidesPackageRCS{pgfsys.code.tex} + + +\ifdefined\pgfkeysloaded\else + \input pgfkeys.code.tex +\fi + +\pgfkeys{/pgf/.is family} +\def\pgfset{\pgfqkeys{/pgf}} + +\newif\ifpgfpicture + + +{\catcode`\p=12\catcode`\t=12\gdef\Pgf@geT#1pt{#1}} + +\def\pgf@sys@tonumber#1{\expandafter\Pgf@geT\the#1} + +\def\pgf@sys@bp@correct#1{#1=0.99627#1} + +\def\pgf@sys@bp#1{% + {% + \pgf@x=#1\relax% + \pgf@x=0.99627\pgf@x% + \edef\temp{\expandafter\Pgf@geT\the\pgf@x\space}% + \pgfutil@toks@\expandafter\expandafter\expandafter{\expandafter\pgfsysprotocol@currentprotocol\temp}% + \xdef\pgfsysprotocol@currentprotocol{\the\pgfutil@toks@}% + }% +} +\def\pgf@sys@pt#1{% + {% + \pgf@x=#1\relax% + \edef\temp{\expandafter\Pgf@geT\the\pgf@x\space}% + \pgfutil@toks@\expandafter\expandafter\expandafter{\expandafter\pgfsysprotocol@currentprotocol\temp}% + \xdef\pgfsysprotocol@currentprotocol{\the\pgfutil@toks@}% + }% +} + + +\newdimen\pgf@x +\newdimen\pgf@y +\newdimen\pgf@xa +\newdimen\pgf@ya +\newdimen\pgf@xb +\newdimen\pgf@yb +\newdimen\pgf@xc +\newdimen\pgf@yc +\newdimen\pgf@xd +\newdimen\pgf@yd + +\newwrite\w@pgf@writea +\newread\r@pgf@reada +\let\pgfutil@inputcheck=\r@pgf@reada + +\newcount\c@pgf@counta +\newcount\c@pgf@countb +\newcount\c@pgf@countc +\newcount\c@pgf@countd + +\newtoks\t@pgf@toka +\newtoks\t@pgf@tokb +\newtoks\t@pgf@tokc + +\let\pgfmath@x\pgf@x +\let\pgfmath@xa\pgf@xa +\let\pgfmath@xb\pgf@xb +\let\pgfmath@xc\pgf@xc + +\let\pgfmath@y\pgf@y +\let\pgfmath@ya\pgf@ya +\let\pgfmath@yb\pgf@yb +\let\pgfmath@yc\pgf@yc + +\let\c@pgfmath@counta\c@pgf@counta +\let\c@pgfmath@countb\c@pgf@countb +\let\c@pgfmath@countc\c@pgf@countc +\let\c@pgfmath@countd\c@pgf@countd + + + + + + +\def\pgf@sys@fail#1{% + \expandafter\ifx\csname pgf@sys@fail@#1\endcsname\pgfutil@empty% + \else% + \expandafter\global\expandafter\let\csname pgf@sys@fail@#1\endcsname=\pgfutil@empty% + \pgfwarning{Your graphic driver \pgfsysdriver\space does not support #1. This warning is given only once}% + \fi% +} + + + + +\def\pgfsys@invoke{\pgf@sys@fail{invoking specials}} + +\def\pgfsys@outerinvoke{\pgfsys@invoke} + + + + +\def\pgfsys@moveto#1#2{\pgf@sys@fail{path constructions}} + +\def\pgfsys@lineto#1#2{\pgf@sys@fail{path constructions}} + +\def\pgfsys@curveto#1#2#3#4#5#6{\pgf@sys@fail{path constructions}} + +\def\pgfsys@rect#1#2#3#4{\pgf@sys@fail{path constructions}} + +\def\pgfsys@closepath{\pgf@sys@fail{path constructions}} + + + + + + + +\def\pgfsys@stroke{\pgf@sys@fail{path usage}} + +\def\pgfsys@fill{\pgf@sys@fail{path usage}} + +\def\pgfsys@fillstroke{\pgf@sys@fail{path usage}} + +\def\pgfsys@clipnext{\pgf@sys@fail{path usage}} + +\def\pgfsys@discardpath{\pgf@sys@fail{path usage}} + +\def\pgfsys@closestroke{\pgfsys@closepath\pgfsys@stroke} + +\newif\ifpgfsys@eorule + + + + +\def\pgfsys@setlinewidth#1{\pgf@sys@fail{setting the line width}} + +\def\pgfsys@buttcap{\pgf@sys@fail{setting the line cap}} + +\def\pgfsys@roundcap{\pgf@sys@fail{setting the line cap}} + +\def\pgfsys@rectcap{\pgf@sys@fail{setting the line cap}} + +\def\pgfsys@miterjoin{\pgf@sys@fail{setting the line join}} + +\def\pgfsys@setmiterlimit#1{\pgf@sys@fail{setting the line cap}} + +\def\pgfsys@roundjoin{\pgf@sys@fail{setting the line cap}} + +\def\pgfsys@beveljoin{\pgf@sys@fail{setting the line cap}} + +\def\pgfsys@setdash#1#2{\pgf@sys@fail{setting the dashing pattern}} + + + + + + +\def\pgfsys@color@rgb@stroke#1#2#3{\pgf@sys@fail{color}} + +\def\pgfsys@color@rgb@fill#1#2#3{\pgf@sys@fail{color}} + +\def\pgfsys@color@rgb#1#2#3{\pgfsys@color@rgb@stroke{#1}{#2}{#3}\pgfsys@color@rgb@fill{#1}{#2}{#3}} + +\def\pgfsys@color@cmyk@stroke#1#2#3#4{\pgf@sys@fail{color}} + +\def\pgfsys@color@cmyk@fill#1#2#3#4{\pgf@sys@fail{color}} + +\def\pgfsys@color@cmyk#1#2#3#4{\pgfsys@color@cmyk@stroke{#1}{#2}{#3}{#4}\pgfsys@color@cmyk@fill{#1}{#2}{#3}{#4}} + +\def\pgfsys@color@cmy@stroke#1#2#3{\pgf@sys@fail{color}} + +\def\pgfsys@color@cmy@fill#1#2#3{\pgf@sys@fail{color}} + +\def\pgfsys@color@cmy#1#2#3{\pgfsys@color@cmy@stroke{#1}{#2}{#3}\pgfsys@color@cmy@fill{#1}{#2}{#3}} + +\def\pgfsys@color@gray@stroke#1{\pgf@sys@fail{color}} + +\def\pgfsys@color@gray@fill#1{\pgf@sys@fail{color}} + +\def\pgfsys@color@gray#1{\pgfsys@color@gray@stroke{#1}\pgfsys@color@gray@fill{#1}} + +\newif\ifpgfsys@color@reset@inorder +\pgfsys@color@reset@inordertrue + + +\def\pgfsys@color@reset{} + +\def\pgfsys@color@unstacked#1{} + +\def\pgfsys@text@to@black@hook{} + + + + +\def\pgfsys@transformcm#1#2#3#4#5#6{\pgf@sys@fail{transformations}} + +\def\pgfsys@transformshift#1#2{\pgfsys@transformcm{1}{0}{0}{1}{#1}{#2}} + +\def\pgfsys@transformxyscale#1#2{\pgfsys@transformcm{#1}{0}{0}{#2}{0bp}{0bp}} + +\def\pgfsys@viewboxmeet#1#2#3#4#5#6#7#8{\pgfsys@beginscope\pgf@sys@default@viewbox@impl{#1}{#2}{#3}{#4}{#5}{#6}{#7}{#8}{<}} + +\def\pgfsys@viewboxslice#1#2#3#4#5#6#7#8{\pgfsys@beginscope\pgf@sys@default@viewbox@impl{#1}{#2}{#3}{#4}{#5}{#6}{#7}{#8}{>}} + +\def\pgfsys@endviewbox{\pgfsys@endscope} + + +\def\pgf@sys@default@viewbox@impl#1#2#3#4#5#6#7#8#9{{% + \pgf@x#1% + \pgf@y#2% + \pgf@xa#3% + \pgf@ya#4% + \advance\pgf@xa by-\pgf@x% + \advance\pgf@ya by-\pgf@y% + \pgfmathdivide@{\pgf@sys@tonumber\pgf@xa}{\pgf@sys@tonumber\pgf@ya}% + \let\aspectr\pgfmathresult% + \pgf@xb#5% + \pgf@yb#6% + \pgf@xc#7% + \pgf@yc#8% + \advance\pgf@xc by-\pgf@xb% + \advance\pgf@yc by-\pgf@yb% + \pgfmathdivide@{\pgf@sys@tonumber\pgf@xc}{\pgf@sys@tonumber\pgf@yc}% + \let\aspects\pgfmathresult% + \ifdim\aspectr pt#9\aspects pt% + \pgfmathdivide@{\pgf@sys@tonumber\pgf@xa}{\pgf@sys@tonumber\pgf@xc}% + \else% + \pgfmathdivide@{\pgf@sys@tonumber\pgf@ya}{\pgf@sys@tonumber\pgf@yc}% + \fi% + \advance\pgf@x by.5\pgf@xa% + \advance\pgf@xb by.5\pgf@xc% + \advance\pgf@y by.5\pgf@ya% + \advance\pgf@yb by.5\pgf@yc% + \pgfsys@transformcm{\pgfmathresult}{0}{0}{\pgfmathresult}{\pgf@x}{\pgf@y}% + \pgfsys@transformcm{1}{0}{0}{1}{-\pgf@xb}{-\pgf@yb}% + }% +} + + + + + +\def\pgfsys@beginscope{\pgf@sys@fail{scoping}} + +\def\pgfsys@endscope{\pgf@sys@fail{scoping}} + +\def\pgfsys@begin@idscope{% + \begingroup% + \edef\pgf@sys@cacheref{\pgfsys@id@refcurrent}% + \expandafter\let\expandafter\pgfsys@beg@save\csname pgf@sys@att@beg@\pgf@sys@cacheref\endcsname% + \expandafter\let\expandafter\pgfsys@end@save\csname pgf@sys@att@end@\pgf@sys@cacheref\endcsname% + \ifx\pgfsys@beg@save\relax% + \ifx\pgfsys@end@save\relax% + \else% + \pgfsys@call@save% + \fi% + \else% + \pgfsys@call@save% + \fi% + \pgfsys@invalidate@currentid% + \begingroup% +} + +\def\pgfsys@call@save{% + \pgfsys@beginscope% + \pgfsys@beg@save% + \expandafter\global\expandafter\let\csname pgf@sys@att@beg@\pgfsys@id@refcurrent\endcsname\relax% + \expandafter\global\expandafter\let\csname pgf@sys@att@end@\pgfsys@id@refcurrent\endcsname\relax% +} + +\def\pgfsys@end@idscope{ + \endgroup% + \ifx\pgfsys@beg@save\relax% + \ifx\pgfsys@end@save\relax% + \else% + \pgfsys@call@end% + \fi% + \else% + \pgfsys@call@end% + \fi% + \endgroup +} + +\def\pgfsys@call@end{% + \pgfsys@end@save% + \pgfsys@endscope% +} + + +\def\pgfsys@begin@text{} + +\def\pgfsys@end@text{} + + + + +\newcount\pgf@sys@id@count + +\def\pgfsys@new@id#1{% + \edef#1{pgf\the\pgf@sys@id@count}% + \global\advance\pgf@sys@id@count by1\relax% +} + +\def\pgfsys@use@id#1{% + \edef\pgf@sys@id@current@id{#1}% + \let\pgfsys@current@type\pgfutil@empty% +} +\let\pgf@sys@id@current@id\pgfutil@empty + +\def\pgfsys@clear@id{% + \let\pgf@sys@id@current@id\pgfutil@empty% +} + +\def\pgfsys@use@type#1{% + \edef\pgfsys@current@type{#1}% + \pgfsys@register@type\pgfsys@current@type% +} +\let\pgfsys@current@type\pgfutil@empty + +\def\pgfsys@append@type#1{% + \ifx\pgfsys@current@type\pgfutil@empty% + \pgfsys@use@type{#1}% + \else% + \pgfsys@use@type{\pgfsys@current@type.#1}% + \fi% +} + +\def\pgfsys@register@type#1{% + \expandafter\let\expandafter\pgf@sys@temp\csname pgf@sys@reg@type@#1\endcsname% + \ifx\pgf@sys@temp\relax% + {% + \c@pgf@counta\pgf@sys@type@count\relax% + \global\advance\c@pgf@counta by1\relax% + \edef\pgf@sys@type@count{\the\c@pgf@counta}% + \expandafter\xdef\csname pgf@sys@reg@type@#1\endcsname{y\the\c@pgf@counta}% + }% + \fi% +} +\def\pgf@sys@reg@type@{} +\def\pgf@sys@reg@type@background{b} +\def\pgf@sys@reg@type@path{p} +\def\pgf@sys@reg@type@text{t} +\expandafter\def\csname pgf@sys@reg@type@background.path\endcsname{bp} +\def\pgf@sys@type@count{0} + +\def\pgfsys@push@type{% + \expandafter\expandafter\expandafter\def\expandafter\expandafter\expandafter\pgf@sys@typestack% + \expandafter\expandafter\expandafter{\expandafter\expandafter\expandafter\def\expandafter\expandafter\expandafter\pgfsys@current@type% + \expandafter\expandafter\expandafter{\expandafter\pgfsys@current@type\expandafter}% + \expandafter\def\expandafter\pgf@sys@typestack\expandafter{\pgf@sys@typestack}}% +} +\let\pgf@sys@typestack\pgfutil@empty + +\def\pgfsys@pop@type{\pgf@sys@typestack} + +\def\pgfsys@id@ref#1#2{#1\csname pgf@sys@reg@type@#2\endcsname} + +\def\pgfsys@id@refcurrent{\pgfsys@id@ref{\pgf@sys@id@current@id}{\pgfsys@current@type}} + +\def\pgfsys@invalidate@currentid{% + \expandafter\global\expandafter\let\csname pgf@sys@id@keylist@\pgfsys@id@refcurrent\endcsname\pgfutil@empty% +} + +\def\pgfsys@attach@to@id#1#2#3#4{% + \pgfsys@register@type{#2}% + \expandafter\def\expandafter\pgf@sys@tempbeg\expandafter{\csname pgf@sys@att@beg@\pgfsys@id@ref{#1}{#2}\endcsname}% + \expandafter\def\expandafter\pgf@sys@tempend\expandafter{\csname pgf@sys@att@end@\pgfsys@id@ref{#1}{#2}\endcsname}% + \expandafter\ifx\pgf@sys@tempbeg\relax% + \expandafter\let\pgf@sys@tempbeg\pgfutil@empty% + \fi% + \expandafter\ifx\pgf@sys@tempend\relax% + \expandafter\let\pgf@sys@tempend\pgfutil@empty% + \fi% + \expandafter\let\expandafter\pgf@sys@tempbeg@cont\pgf@sys@tempbeg% + \expandafter\let\expandafter\pgf@sys@tempend@cont\pgf@sys@tempend% + \expandafter\expandafter\expandafter\gdef\expandafter\pgf@sys@tempbeg\expandafter{\pgf@sys@tempbeg@cont#3}% + \def\pgf@sys@temp{#4}% + \expandafter\expandafter\expandafter\expandafter\expandafter\expandafter\expandafter\gdef\expandafter\expandafter\expandafter\pgf@sys@tempend\expandafter\expandafter\expandafter{\expandafter\pgf@sys@temp\pgf@sys@tempend@cont}% +} + +\def\pgfsys@if@fresh@id#1#2#3#4{% + \edef\pgf@sys@temp{#1}% + \ifx\pgf@sys@temp\pgfutil@empty% + #4% + \else% + \pgfsys@register@type{#2}% + \expandafter\ifx\csname pgf@sys@id@keylist@\pgfsys@id@ref{#1}{#2}\endcsname\pgfutil@empty% + #4% + \else% + #3% + \fi% + \fi% +} + +\def\pgfsys@if@fresh@currentid#1#2{% + \ifx\pgf@sys@id@current@id\pgfutil@empty% + #2% + \else% + \expandafter\ifx\csname pgf@sys@id@keylist@\pgfsys@id@refcurrent\endcsname\pgfutil@empty% + #2% + \else% + #1% + \fi% + \fi% +} + + + + + +\def\pgfsys@imagesuffixlist{} + + + + +\def\pgfsys@opacity#1{\pgfsys@stroke@opacity{#1}\pgfsys@fill@opacity{#1}} + +\def\pgfsys@stroke@opacity#1{\pgf@sys@fail{opacity}} + +\def\pgfsys@fill@opacity#1{\pgf@sys@fail{opacity}} + +\def\pgfsys@transparencygroupfrombox#1{\pgf@sys@fail{transparency groups}} + +\newif\ifpgfsys@transparency@group@isolated +\newif\ifpgfsys@transparency@group@knockout + + +\def\pgfsys@blend@mode#1{\pgf@sys@fail{blend mode}} + + +\def\pgfsys@definemask{\pgf@sys@fail{image masks}} + +\def\pgfsys@fadingfrombox#1#2{\pgf@sys@fail{fadings}} + +\def\pgfsys@usefading#1#2#3#4#5#6#7{\pgf@sys@fail{fadings}} + +\def\pgfsys@clipfading{} + + + +\def\pgfsys@horishading#1#2#3{\pgf@sys@fail{shadings}\pgf@sys@noshading{#1}} + +\def\pgfsys@vertshading#1#2#3{\pgf@sys@fail{shadings}\pgf@sys@noshading{#1}} + +\def\pgfsys@radialshading#1#2#3{\pgf@sys@fail{shadings}\pgf@sys@noshading{#1}} + +\def\pgfsys@functionalshading#1#2#3#4{\pgf@sys@fail{shadings (functional)}\pgf@sys@noshading{#1}} + +\def\pgf@sys@noshading#1{\expandafter\global\expandafter\let\csname @pgfshading#1!\endcsname=\pgfutil@empty} + + + + + +\def\pgfsys@declarepattern#1#2#3#4#5#6#7#8#9{\pgf@sys@fail{patterns}} + +\def\pgfsys@setpatternuncolored#1#2#3#4{\pgf@sys@fail{patterns}} + +\def\pgfsys@setpatterncolored#1{\pgf@sys@fail{patterns}} + + + + + + + +\newif\ifpgfsysanimationsupported + + +\def\pgfsys@animate#1{\pgf@sys@fail{animations}} + + + + + +\def\pgfsys@animation@restart@always{} + +\def\pgfsys@animation@restart@never{} + +\def\pgfsys@animation@restart@whennotactive{} + +\def\pgfsys@animation@repeat#1{} + +\def\pgfsys@animation@repeat@indefinite{} + +\def\pgfsys@animation@repeat@dur#1{} + +\def\pgfsys@animation@freezeatend{} + +\def\pgfsys@animation@removeatend{} + +\def\pgfsys@animation@time#1#2#3#4#5{} + +\def\pgfsys@stay@text{{stay}{0}} +\def\pgfsys@jump@text{{jump}{1}} + +\def\pgfsys@animation@base{} + +\def\pgfsys@animation@offset#1#2{} + +\def\pgfsys@animation@syncbegin#1#2#3#4{} + +\def\pgfsys@animation@syncend#1#2#3#4{} + +\def\pgfsys@animation@event#1#2#3#4#5{} + +\def\pgfsys@animation@repeat@event#1#2#3#4#5{} + +\def\pgfsys@animation@accesskey#1#2#3{} + +\def\pgf@sys@begin@text{begin} + + + +\def\pgfsys@animation@accumulate{} + +\def\pgfsys@animation@noaccumulate{} + + + +\def\pgfsys@animation@whom#1#2{} + + + +\def\pgfsys@animation@rotatealong{} + +\def\pgfsys@animation@norotatealong{} + +\def\pgfsys@animation@movealong#1{} + +\def\pgfsys@animation@tip@markers#1#2{} + +\def\pgfsys@animation@canvas@transform#1#2{} + + + + +\def\pgfsys@animation@val@current{} + +\def\pgfsys@animation@val@text#1{} + +\def\pgfsys@animation@val@scalar#1{} + +\def\pgfsys@animation@val@dimension#1{} + +\def\pgfsys@animation@val@color@rgb#1#2#3{} + +\def\pgfsys@animation@val@color@cmyk#1#2#3#4{} + +\def\pgfsys@animation@val@color@cmy#1#2#3{} + +\def\pgfsys@animation@val@color@gray#1{} + +\def\pgfsys@animation@val@path#1{} + +\def\pgfsys@animation@val@translate#1#2{} + +\def\pgfsys@animation@val@scale#1#2{} + +\def\pgfsys@animation@val@viewbox#1#2#3#4{} + +\def\pgfsys@animation@val@dash#1#2{} + + + + +\def\pgfsys@marker@declare#1#2{% + {% + \pgfpicturetrue% + \pgfsysprotocol@getcurrentprotocol\pgfsys@marker@temp% + \pgfsysprotocol@setcurrentprotocol\pgfutil@empty% + \pgfsysprotocol@bufferedtrue% + \pgfsys@beginscope% + #2% + \pgfsys@endscope% + \xdef#1{\the\pgf@sys@id@count}% + \expandafter\expandafter\expandafter\global\expandafter\pgfsysprotocol@getcurrentprotocol\csname pgf@sys@marker@prot@#1\endcsname% + \pgfsys@marker@declare@% + \pgfsysprotocol@setcurrentprotocol\pgfsys@marker@temp% + \global\advance\pgf@sys@id@count by1\relax% + }% +} +\let\pgfsys@marker@declare@\relax + +\def\pgfsys@marker@use#1{% + \pgfsysprotocol@literal{\csname pgf@sys@marker@prot@#1\endcsname}% +} + + + + + + + +\def\pgfsys@rdf@about#1{\pgf@sys@fail{rdf}} + +\def\pgfsys@rdf@content#1{\pgf@sys@fail{rdf}} + +\def\pgfsys@rdf@datatype#1{\pgf@sys@fail{rdf}} + +\def\pgfsys@rdf@href#1{\pgf@sys@fail{rdf}} + +\def\pgfsys@rdf@inlist{\pgf@sys@fail{rdf}} + +\def\pgfsys@rdf@prefix#1{\pgf@sys@fail{rdf}} + +\def\pgfsys@rdf@property#1{\pgf@sys@fail{rdf}} + +\def\pgfsys@rdf@rel#1{\pgf@sys@fail{rdf}} + +\def\pgfsys@rdf@resource#1{\pgf@sys@fail{rdf}} + +\def\pgfsys@rdf@rev#1{\pgf@sys@fail{rdf}} + +\def\pgfsys@rdf@src#1{\pgf@sys@fail{rdf}} + +\def\pgfsys@rdf@typeof#1{\pgf@sys@fail{rdf}} + +\def\pgfsys@rdf@vocab#1{\pgf@sys@fail{rdf}} + + + + +\def\pgfsys@papersize#1#2{\pgf@sys@fail{papersize}} + +\def\pgfsys@global@papersize#1#2{\pgfsys@papersize{#1}{#2}} + +\def\pgfsys@prepare@papersize#1#2{\pgfsys@papersize{#1}{#2}} + +\ifx\paperheight\@undefined + \def\pgfsys@thepageheight{\the\vsize} % + \def\pgfsys@thepagewidth{\the\hsize} +\else + \def\pgfsys@thepageheight{\paperheight} + \def\pgfsys@thepagewidth{\paperwidth} +\fi + + + +\def\pgfsys@markposition#1{\pgf@sys@fail{marking the current position}} + + +\def\pgfsys@getposition#1#2{\let#2=\relax} + + + + + +\def\pgfsys@shadingoutsidepgfpicture#1{% + #1% +} + + +\def\pgfsys@shadinginsidepgfpicture#1{% + \pgf@makehbox{#1}% + \setbox\pgf@hbox=\hbox{\hskip-.5\wd\pgf@hbox\lower.5\ht\pgf@hbox\hbox{\box\pgf@hbox}}% + \pgfqbox{\pgf@hbox}% +} + + +\def\pgfsys@hbox#1{% + \pgfsys@begin@idscope% + \pgfsys@beginscope% + \setbox#1=\hbox{\box#1}% + \wd#1=0pt% + \ht#1=0pt% + \dp#1=0pt% + \box#1% + \pgfsys@endscope% + \pgfsys@end@idscope% +} + +\def\pgfsys@hboxsynced#1{% + \pgfsys@beginscope\pgflowlevelsynccm\pgfsys@hbox#1\pgfsys@endscope% +}% + +\def\pgfsys@pictureboxsynced#1{% + {% + \setbox0=\hbox{\pgfsys@beginpicture\box#1\pgfsys@endpicture}% + \pgfsys@hboxsynced0% + }% +} + +\def\pgfsys@beginpicture{} + +\def\pgfsys@endpicture{} + +\def\pgfsys@typesetpicturebox#1{% + \pgf@ya=\pgf@shift@baseline\relax% + \advance\pgf@ya by-\pgf@picminy\relax% + \advance\pgf@picmaxy by-\pgf@picminy\relax% + \advance\pgf@picmaxx by-\pgf@picminx\relax% + \setbox#1=\hbox{\hskip-\pgf@picminx\lower\pgf@picminy\box#1}% + \ht#1=\pgf@picmaxy% + \wd#1=\pgf@picmaxx% + \dp#1=0pt% + \leavevmode% + \pgf@xa=\pgf@trimleft@final\relax \ifdim\pgf@xa=0pt \else\kern\pgf@xa\fi + \raise-\pgf@ya\box#1% + \pgf@xa=\pgf@trimright@final\relax \ifdim\pgf@xa=0pt \else\kern\pgf@xa\fi +} + +\def\pgfsys@beginpurepicture{\pgfsys@beginpicture} + +\def\pgfsys@endpurepicture{\pgfsys@endpicture} + +\def\pgfsys@defineimage{% + \ifx\pgf@imagewidth\pgfutil@empty\else\edef\pgf@imagewidth{width=\pgf@imagewidth,}\fi% + \ifx\pgf@imageheight\pgfutil@empty\else\edef\pgf@imageheight{height=\pgf@imageheight,}\fi% + \ifx\pgf@imagepage\pgfutil@empty\else\edef\pgf@imagepage{page=\pgf@imagepage,}\fi% + \edef\pgf@image{\noexpand\includegraphics[\pgf@imageheight\pgf@imagewidth]{\pgf@filename}}% +} + + +\def\pgfsys@defobject#1#2#3#4{% + \pgfsysprotocol@getcurrentprotocol\pgfsys@temp% + {% + \pgfpicturetrue% + \pgfsysprotocol@setcurrentprotocol\pgfutil@empty% + \pgfsysprotocol@bufferedtrue% + \pgfsys@beginscope% + #4% + \pgfsys@endscope% + \pgfsysprotocol@getcurrentprotocol\pgfsys@@temp% + \expandafter\global\expandafter\let\csname #1\endcsname=\pgfsys@@temp% + }% + \pgfsysprotocol@setcurrentprotocol\pgfsys@temp% +} + + +\def\pgfsys@useobject#1#2{% + \pgfsysprotocol@getcurrentprotocol\pgfsys@temp% + {% + \pgfsysprotocol@setcurrentprotocol\pgfutil@empty% + \pgfsysprotocol@bufferedfalse% + #2% + \expandafter\pgfsysprotocol@setcurrentprotocol\csname #1\endcsname% + \pgfsysprotocol@invokecurrentprotocol% + }% + \pgfsysprotocol@setcurrentprotocol\pgfsys@temp% +} + + +\def\pgfsys@begininvisible{\pgfsys@transformcm{1}{0}{0}{1}{2000bp}{2000bp}} + +\def\pgfsys@endinvisible{\pgfsys@transformcm{1}{0}{0}{1}{-2000bp}{-2000bp}} + +\def\pgfsys@begininvisiblescope{\pgfsys@beginscope\pgfsys@begininvisible} + +\def\pgfsys@endinvisiblescope{\pgfsys@endinvisible\pgfsys@endscope} + + +\def\pgfsys@atbegindocument{} + + +\ifdefined\pdfstrcmp + \let\pgfsys@strcmp\pdfstrcmp +\else\ifdefined\strcmp + \let\pgfsys@strcmp\strcmp +\else\ifdefined\directlua + \directlua{ + local lft = lua.get_functions_table() + lft[\string#lft+1] = function() + local lhs = token.scan_string() + local rhs = token.scan_string() + if lhs < rhs then + tex.sprint(-2, "-1") + elseif lhs == rhs then + tex.sprint(-2, "0") + else + tex.sprint(-2, "1") + end + end + token.set_lua("pgfsys@strcmp", \string#lft, "global") + } +\else + \def\pgfsys@strcmp#1#2{\pgf@sys@fail{string comparison}}% +\fi\fi\fi + + + +\ifx\pgfsysdriver\@undefined + \pgfutil@guessdriver +\fi + +\input pgf.cfg + +\immediate\write-1{Driver file for pgf: \pgfsysdriver} +\pgfutil@InputIfFileExists{\pgfsysdriver}{}{% + \pgferror{Driver file ``\pgfsysdriver'' not found.}} + + + +\AtBeginDocument{ + \pgfsys@atbegindocument + \let\pgfsys@atbegindocument=\pgfutil@empty +} + + + + +\pgfutil@aux@read@hook + + +\endinput diff --git a/preprints/A-polynomial-time-algorithm-for-three-machine-unit-job-scheduling-September-24-2026/build/tex/runtime/generic/pgf/systemlayer/pgfsysprotocol.code.tex b/preprints/A-polynomial-time-algorithm-for-three-machine-unit-job-scheduling-September-24-2026/build/tex/runtime/generic/pgf/systemlayer/pgfsysprotocol.code.tex new file mode 100644 index 0000000000000000000000000000000000000000..0bd2f66f377fe54fdbe02002c7d38f34cae78bf4 --- /dev/null +++ b/preprints/A-polynomial-time-algorithm-for-three-machine-unit-job-scheduling-September-24-2026/build/tex/runtime/generic/pgf/systemlayer/pgfsysprotocol.code.tex @@ -0,0 +1,48 @@ + +\ProvidesFileRCS{pgfsysprotocol.code.tex} + + + + + + +\let\pgfsysprotocol@currentprotocol=\pgfutil@empty +\newif\ifpgfsysprotocol@buffered + + +\def\pgfsysprotocol@literalbuffered#1{% + \edef\pgfsysprotocol@temp{{#1\space}}% + \expandafter\pgfutil@g@addto@macro\expandafter\pgfsysprotocol@currentprotocol\pgfsysprotocol@temp% +} + +\def\pgfsysprotocol@literal#1{% + \pgfsysprotocol@literalbuffered{#1}% + \ifpgfsysprotocol@buffered% + \else% + \pgfsysprotocol@flushcurrentprotocol% + \fi% +} + +\def\pgfsysprotocol@getcurrentprotocol#1{% + \let#1=\pgfsysprotocol@currentprotocol% +} + +\def\pgfsysprotocol@setcurrentprotocol#1{% + \global\let\pgfsysprotocol@currentprotocol=#1% +} + + +\def\pgfsysprotocol@invokecurrentprotocol{% + \ifx\pgfsysprotocol@currentprotocol\pgfutil@empty% + \else% + \expandafter\pgfsys@invoke\expandafter{\pgfsysprotocol@currentprotocol}% + \fi% +} + +\def\pgfsysprotocol@flushcurrentprotocol{% + \pgfsysprotocol@invokecurrentprotocol% + \pgfsysprotocol@setcurrentprotocol\pgfutil@empty% +} + + +\endinput diff --git a/preprints/A-polynomial-time-algorithm-for-three-machine-unit-job-scheduling-September-24-2026/build/tex/runtime/generic/pgf/systemlayer/pgfsyssoftpath.code.tex b/preprints/A-polynomial-time-algorithm-for-three-machine-unit-job-scheduling-September-24-2026/build/tex/runtime/generic/pgf/systemlayer/pgfsyssoftpath.code.tex new file mode 100644 index 0000000000000000000000000000000000000000..8fadadb709e253162747d9b83c04a3380c7beb2d --- /dev/null +++ b/preprints/A-polynomial-time-algorithm-for-three-machine-unit-job-scheduling-September-24-2026/build/tex/runtime/generic/pgf/systemlayer/pgfsyssoftpath.code.tex @@ -0,0 +1,141 @@ + +\ProvidesFileRCS{pgfsyssoftpath.code.tex} + + + + + + + +\let\pgfsyssoftpath@thepath=\pgfutil@empty +\let\pgfsyssoftpath@bigbuffer=\pgfutil@empty +\let\pgfsyssoftpath@smallbuffer=\pgfutil@empty +\newcount\pgfsyssoftpath@smallbuffer@items +\newcount\pgfsyssoftpath@bigbuffer@items + +\def\pgfsyssoftpath@flushbuffers{% + \expandafter\expandafter\expandafter\expandafter\expandafter\expandafter\expandafter\gdef% + \expandafter\expandafter\expandafter\expandafter\expandafter\expandafter\expandafter\pgfsyssoftpath@thepath% + \expandafter\expandafter\expandafter\expandafter\expandafter\expandafter\expandafter{% + \expandafter\expandafter\expandafter\pgfsyssoftpath@thepath% + \expandafter\pgfsyssoftpath@bigbuffer\pgfsyssoftpath@smallbuffer}% + \global\let\pgfsyssoftpath@smallbuffer=\pgfutil@empty + \global\let\pgfsyssoftpath@bigbuffer=\pgfutil@empty + \global\pgfsyssoftpath@bigbuffer@items0\relax% + \global\pgfsyssoftpath@smallbuffer@items0\relax% +} + +\def\pgfsyssoftpath@getcurrentpath#1{% + \pgfsyssoftpath@flushbuffers% + \let#1=\pgfsyssoftpath@thepath% +} + +\def\pgfsyssoftpath@setcurrentpath#1{% + \global\let\pgfsyssoftpath@thepath=#1% + \global\let\pgfsyssoftpath@smallbuffer=\pgfutil@empty + \global\let\pgfsyssoftpath@bigbuffer=\pgfutil@empty + \global\pgfsyssoftpath@bigbuffer@items0\relax% + \global\pgfsyssoftpath@smallbuffer@items0\relax% +} + + +\def\pgfsyssoftpath@invokecurrentpath{% + \pgfsyssoftpath@thepath% + \pgfsyssoftpath@bigbuffer% + \pgfsyssoftpath@smallbuffer% +} + +\def\pgfsyssoftpath@flushcurrentpath{% + \pgfsyssoftpath@invokecurrentpath% + \pgfsyssoftpath@setcurrentpath\pgfutil@empty% +} + +\def\pgfsyssoftpath@addtocurrentpath#1{% + \global\advance\pgfsyssoftpath@smallbuffer@items by1\relax% + \ifnum\pgfsyssoftpath@smallbuffer@items<40\relax% + \expandafter\gdef\expandafter\pgfsyssoftpath@smallbuffer\expandafter{\pgfsyssoftpath@smallbuffer#1}% + \else% + \pgfsyssoftpath@smalloverflow{#1}% + \fi% +} +\def\pgfsyssoftpath@smalloverflow#1{% + \global\advance\pgfsyssoftpath@bigbuffer@items by1\relax% + \ifnum\pgfsyssoftpath@bigbuffer@items<30\relax% + \expandafter\expandafter\expandafter\gdef% + \expandafter\expandafter\expandafter\pgfsyssoftpath@bigbuffer% + \expandafter\expandafter\expandafter{\expandafter\pgfsyssoftpath@bigbuffer\pgfsyssoftpath@smallbuffer#1}% + \global\let\pgfsyssoftpath@smallbuffer=\pgfutil@empty% + \global\pgfsyssoftpath@smallbuffer@items0\relax% + \else% + \expandafter\expandafter\expandafter\expandafter\expandafter\expandafter\expandafter\gdef% + \expandafter\expandafter\expandafter\expandafter\expandafter\expandafter\expandafter\pgfsyssoftpath@thepath% + \expandafter\expandafter\expandafter\expandafter\expandafter\expandafter\expandafter{% + \expandafter\expandafter\expandafter\pgfsyssoftpath@thepath + \expandafter\pgfsyssoftpath@bigbuffer\pgfsyssoftpath@smallbuffer#1}% + \global\let\pgfsyssoftpath@smallbuffer=\pgfutil@empty + \global\let\pgfsyssoftpath@bigbuffer=\pgfutil@empty + \global\pgfsyssoftpath@bigbuffer@items0\relax% + \global\pgfsyssoftpath@smallbuffer@items0\relax% + \fi% +} + + + +\def\pgfsyssoftpath@lastmoveto{{0pt}{0pt}} + +\newif\ifpgfsyssoftpathmovetorelevant +\pgfsyssoftpathmovetorelevanttrue + +\def\pgfsyssoftpath@moveto#1#2{% + \edef\pgfsyssoftpath@coord{{#1}{#2}}% + \expandafter\pgfsyssoftpath@addtocurrentpath\expandafter{\expandafter\pgfsyssoftpath@movetotoken\pgfsyssoftpath@coord}% + \ifpgfsyssoftpathmovetorelevant% + \global\let\pgfsyssoftpath@lastmoveto\pgfsyssoftpath@coord% + \fi% +} + +\def\pgfsyssoftpath@lineto#1#2{% + \edef\pgfsyssoftpath@coord{{#1}{#2}}% + \expandafter\pgfsyssoftpath@addtocurrentpath\expandafter{\expandafter\pgfsyssoftpath@linetotoken\pgfsyssoftpath@coord}% +} + + +\def\pgfsyssoftpath@curveto#1#2#3#4#5#6{% + \edef\pgfsyssoftpath@temp{{% + \noexpand\pgfsyssoftpath@curvetosupportatoken{#1}{#2}% + \noexpand\pgfsyssoftpath@curvetosupportbtoken{#3}{#4}% + \noexpand\pgfsyssoftpath@curvetotoken{#5}{#6}% + }}% + \expandafter\pgfsyssoftpath@addtocurrentpath\pgfsyssoftpath@temp% +} + + +\def\pgfsyssoftpath@rect#1#2#3#4{% + \edef\pgfsyssoftpath@temp{{% + \noexpand\pgfsyssoftpath@rectcornertoken{#1}{#2}% + \noexpand\pgfsyssoftpath@rectsizetoken{#3}{#4}% + }}% + \expandafter\pgfsyssoftpath@addtocurrentpath\pgfsyssoftpath@temp% +} + +\def\pgfsyssoftpath@closepath{% + \expandafter\pgfsyssoftpath@addtocurrentpath\expandafter{\expandafter\pgfsyssoftpath@closepathtoken\pgfsyssoftpath@lastmoveto}% +} + +\def\pgfsyssoftpath@specialround#1#2{% + \edef\pgfsyssoftpath@temp{{#1}{#2}}% + \expandafter\pgfsyssoftpath@addtocurrentpath\expandafter{\expandafter\pgfsyssoftpath@specialroundtoken\pgfsyssoftpath@temp}% +} + +\def\pgfsyssoftpath@movetotoken#1#2{\pgfsys@moveto{#1}{#2}} +\def\pgfsyssoftpath@linetotoken#1#2{\pgfsys@lineto{#1}{#2}} +\def\pgfsyssoftpath@rectcornertoken#1#2#3#4#5{\pgfsys@rect{#1}{#2}{#4}{#5}} % +\def\pgfsyssoftpath@curvetosupportatoken#1#2#3#4#5#6#7#8{\pgfsys@curveto{#1}{#2}{#4}{#5}{#7}{#8}} +\def\pgfsyssoftpath@closepathtoken#1#2{\pgfsys@closepath} +\let\pgfsyssoftpath@specialroundtoken=\pgfutil@gobbletwo +\def\pgfsyssoftpath@curvetosupportbtoken#1#2{curvetotokenb} % +\def\pgfsyssoftpath@curvetotoken#1#2{curvetotoken}% + + +\endinput + diff --git a/preprints/A-polynomial-time-algorithm-for-three-machine-unit-job-scheduling-September-24-2026/build/tex/runtime/generic/pgf/utilities/pgffor.code.tex b/preprints/A-polynomial-time-algorithm-for-three-machine-unit-job-scheduling-September-24-2026/build/tex/runtime/generic/pgf/utilities/pgffor.code.tex new file mode 100644 index 0000000000000000000000000000000000000000..fceb8347975ef7797956c8ff4c19121413631596 --- /dev/null +++ b/preprints/A-polynomial-time-algorithm-for-three-machine-unit-job-scheduling-September-24-2026/build/tex/runtime/generic/pgf/utilities/pgffor.code.tex @@ -0,0 +1,720 @@ + +\ProvidesPackageRCS{pgffor.code.tex} + +\ifdefined\pgfmathloaded\else + \input pgfmath.code.tex +\fi + +\newdimen\pgffor@iter +\newdimen\pgffor@skip +\newif\ifpgffor@continue + + +\def\pgffor@reset@hooks{% + \let\pgffor@beginhook=\relax% + \let\pgffor@endhook=\relax% + \let\pgffor@afterhook=\relax% +} +\pgffor@reset@hooks + + +\newtoks\pgffor@stack +\pgffor@stack={{}{}{}} + +\def\pgffor@stackpush#1{% + \def\pgffor@stacktemp{#1}% + \expandafter\expandafter\expandafter\pgffor@stack\expandafter\expandafter\expandafter=% + \expandafter\expandafter\expandafter{\expandafter\expandafter\expandafter% + {\expandafter\pgffor@stacktemp\expandafter}\the\pgffor@stack}% +} + +\def\pgffor@stackpop{\expandafter\pgffor@@stackpop\the\pgffor@stack\pgffor@stackstop} +\def\pgffor@@stackpop#1#2\pgffor@stackstop{\pgffor@stack{#2}#1} + +\def\pgffor@emptyvalues{, \pgffor@stop,}% + +\def\pgffor@foreach{% + \pgffor@atbeginforeach% + \let\pgffor@assign@before@code=\pgfutil@empty% + \let\pgffor@assign@after@code=\pgfutil@empty% + \let\pgffor@assign@once@code=\pgfutil@empty% + \let\pgffor@remember@code=\pgfutil@empty% + \let\pgffor@remember@once@code=\pgfutil@empty% + \pgffor@alphabeticsequencefalse% + \pgffor@contextfalse% + \let\pgffor@var=\pgfutil@empty + \pgffor@vars% +} + +\let\foreach=\pgffor@foreach + +\def\pgffor@vars{% + \pgfutil@ifnextchar i{\pgffor@@vars@end}{% + \pgfutil@ifnextchar[{\pgffor@@vars@opt}{% + \pgfutil@ifnextchar/{\pgffor@@vars@slash@gobble}{% + \pgffor@@vars}}}}% + +\def\pgffor@@vars@end in{\pgfutil@ifnextchar\bgroup{\pgffor@normal@list}{\pgffor@macro@list}} +\def\pgffor@@vars@opt[#1]{\pgfkeys{/pgf/foreach/.cd,#1}\pgffor@vars} +\def\pgffor@@vars#1{\pgffor@var@add#1\pgffor@stop\pgffor@vars} +\def\pgffor@@vars@slash@gobble/{\pgffor@@vars} + +\def\pgffor@var@add#1#2\pgffor@stop{% + \ifx\pgffor@var\pgfutil@empty% + \def\pgffor@var{#1}% + \def\pgffor@mainvar{#1}% + \else% + \expandafter\def\expandafter\pgffor@var\expandafter{\pgffor@var/#1}% + \fi% +} + +\def\pgffor@expand@list@true{\let\pgffor@expand@list\edef} +\def\pgffor@expand@list@false{\let\pgffor@expand@list\def} +\def\pgffor@macro@list#1{% + \expandafter\pgffor@normal@list\expandafter{#1}} +\def\pgffor@normal@list#1{% + \pgffor@expand@list\pgffor@values{#1}% + \expandafter\def\expandafter\pgffor@values\expandafter{\pgffor@values, \pgffor@stop,}% + \ifx\pgffor@values\pgffor@emptyvalues + \def\pgffor@values{\pgffor@stop,}% + \fi% + \let\pgffor@body=\pgfutil@empty% + \global\pgffor@continuetrue% + \pgffor@collectbody} +\def\pgffor@collectbody{% + \pgfutil@ifnextchar\foreach{\pgffor@collectforeach}{% + \pgfutil@ifnextchar\bgroup{\pgffor@collectargument}{\pgffor@collectsemicolon}}% +} + +\def\pgffor@collectforeach\foreach#1in{% + \pgfutil@ifnextchar\bgroup{\pgffor@collectforeach@normal{#1}}{\pgffor@collectforeach@macro{#1}}} +\def\pgffor@collectforeach@macro#1#2{% + \expandafter\long\expandafter\def\expandafter\pgffor@body\expandafter{\pgffor@body\foreach#1in#2}% + \pgffor@collectbody% +} +\def\pgffor@collectforeach@normal#1#2{% + \expandafter\long\expandafter\def\expandafter\pgffor@body\expandafter{\pgffor@body\foreach#1in{#2}}% + \pgffor@collectbody% +} +\long\def\pgffor@collectargument#1{% + \expandafter\pgfutil@in@\expandafter\foreach\expandafter{\pgffor@body}% + \ifpgfutil@in@% + \expandafter\long\expandafter\def\expandafter\pgffor@body\expandafter% + {\pgffor@body{#1}}% + \else% + \expandafter\long\expandafter\def\expandafter\pgffor@body\expandafter% + {\pgffor@body#1}% + \fi% + \pgffor@iterate% +} + +\def\pgffor@collectsemicolon{% + \let\pgffor@next=\pgffor@collectnormalsemicolon% + \ifnum\the\catcode`\;=\active\relax% + \let\pgffor@next=\pgffor@collectactivesemicolon% + \fi% + \pgffor@next% +} + +\def\pgffor@collectnormalsemicolon#1;{% + \expandafter\long\expandafter\def\expandafter\pgffor@body\expandafter{\pgffor@body#1;}% + \pgffor@iterate% +} + +{ + \catcode`\;=\active + + \gdef\pgffor@collectactivesemicolon#1;{% + \expandafter\long\expandafter\def\expandafter\pgffor@body\expandafter{\pgffor@body#1;}% + \pgffor@iterate% + } +} + +\def\pgffor@iterate{% + \let\pgffor@last=\pgfutil@empty% + \let\pgffor@prevlast=\pgfutil@empty% + \let\pgffor@dotsend=\pgfutil@empty% + \let\pgffor@dots@pre=\pgfutil@empty% + \let\pgffor@dots@post=\pgfutil@empty% + \expandafter\pgffor@scan\pgffor@values} + +\def\pgffor@stop{\pgffor@stop}% +\def\pgffor@dots{...}% + +\def\pgffor@scan{\pgfutil@ifnextchar({\pgffor@scanround}{\pgffor@scanone}} +\def\pgffor@scanround(#1)#2,{\def\pgffor@value{(#1)#2}\pgffor@scanned} +\def\pgffor@scanone#1,{\def\pgffor@value{#1}\pgffor@scanned} + + +\newif\ifpgffor@dots@in@ +\def\pgffor@dots@in@#1...#2#3\pgffor@stop{% + \ifx\pgffor@dots@@#2% + \pgffor@dots@in@false% + \else% + \pgffor@dots@in@true% + \fi% +} + +\def\pgffor@dots@@{\pgffor@dots@@} + +\def\pgffor@scanned{% + \ifx\pgffor@value\pgffor@stop% + \let\pgffor@next=\pgffor@after% + \else% + \expandafter\pgffor@dots@in@\pgffor@value\pgffor@dots@...\pgffor@dots@@\pgffor@stop% + \ifpgffor@dots@in@% + \let\pgffor@next=\pgffor@handledots% + \else% + \let\pgffor@next=\pgffor@handlevalue% + \fi% + \ifpgffor@continue% + \else% + \let\pgffor@next=\pgffor@scan% + \fi% + \fi% + \pgffor@next} + + +\def\pgffor@after{% + \global\pgffor@continuetrue% + \pgffor@atendforeach% + \pgffor@afterhook} + +\def\pgffor@handlevalue{% + \let\pgffor@prevlast=\pgffor@last% + \let\pgffor@last=\pgffor@value% + \pgffor@invokebody% + \pgffor@scan% +} + + +\def\pgffor@invokebody{% + \pgffor@begingroup% + \expandafter\pgfutil@in@\expandafter/\expandafter{\pgffor@var}% + \ifpgfutil@in@% + \expandafter\def\expandafter\pgffor@valuerest\expandafter{\pgffor@value//\relax}% + \expandafter\pgffor@multiassign\pgffor@var/\pgffor@stop/\pgffor@stop/\relax% + \else% + \expandafter\expandafter\expandafter\def\expandafter\pgffor@var\expandafter{\pgffor@value}% + \fi% + \ifx\pgffor@assign@once@code\pgfutil@empty% + \else% + \pgffor@assign@once@code% + \fi% + \ifx\pgffor@assign@before@code\pgfutil@empty% + \else% + \pgffor@assign@before@code% + \fi% + \expandafter\expandafter\expandafter\pgffor@reset@hooks\expandafter\pgffor@beginhook\expandafter\pgffor@body\pgffor@endhook% + \ifx\pgffor@assign@after@code\pgfutil@empty% + \else% + \pgffor@assign@after@code% + \fi% + \pgffor@endgroup% +} + + +\def\pgffor@multiassign#1/#2/\relax{% + \def\pgffor@currentvar{#1}% + \def\pgffor@rest{#2}% + \ifx\pgffor@currentvar\pgffor@stop% + \let\pgffor@next=\relax% + \else% + \let\pgffor@next=\pgffor@multiassignrest% + \fi% + \pgffor@next% +} + +\def\pgffor@multiassignrest{\expandafter\pgffor@multiassignfinal\pgffor@valuerest} +\def\pgffor@multiassignfinal#1/#2/\relax{% + \def\pgffor@temp{#1}% + \ifx\pgffor@currentvar\pgffor@mainvar% + \ifpgffor@alphabeticsequence% + \pgffor@makealphabetic\pgffor@temp% + \fi% + \fi% + \def\pgffor@test{#2}% + \ifx\pgffor@test\pgfutil@empty% + \expandafter\def\expandafter\pgffor@valuerest\expandafter{\pgffor@temp//\relax}% + \else% + \def\pgffor@valuerest{#2/\relax}% + \fi% + \expandafter\expandafter\expandafter\def\expandafter\pgffor@currentvar\expandafter{\pgffor@temp}% + \expandafter\pgffor@multiassign\pgffor@rest/\relax% +} + +\def\pgffor@gobblespaces#1{\pgfutil@ifnextchar x{#1}{#1}} + +\def\pgffor@handledots{% + \pgffor@gobblespaces{\expandafter\pgffor@@handledots\pgffor@value\pgffor@@stop}% +} + +\newif\ifpgffor@context + +\def\pgffor@@handledots#1...#2\pgffor@@stop{% + \def\pgffor@dots@pre{#1}% + \def\pgffor@dots@post{#2}% + \def\pgffor@dots@stripcontext#1##1#2\pgffor@@stop{\def\pgffor@dotsvalue{##1}}% + \pgffor@contexttrue% + \ifx\pgffor@dots@pre\pgfutil@empty% + \ifx\pgffor@dots@post\pgfutil@empty% + \pgffor@contextfalse% + \def\pgffor@dots@stripcontext##1\pgffor@@stop{\def\pgffor@dotsvalue{##1}}% + \fi% + \fi% + \pgffor@gobblespaces{\pgffor@dotsscanend}% +} + +\def\pgffor@dots@value@process#1{% + \expandafter\pgffor@dots@stripcontext#1\pgffor@@stop% + \expandafter\pgffor@dots@charcheck\pgffor@dotsvalue\pgffor@@stop% + \ifpgffor@alphabeticsequence + \else + \ifpgffor@assign@parse + \begingroup + \pgfkeys{/pgf/fpu/false/.try}% + \pgfmathparse{\pgffor@dotsvalue}% + \pgfmath@smuggleone\pgfmathresult + \endgroup + \let\pgffor@dotsvalue=\pgfmathresult + \fi + \fi + \let#1=\pgffor@dotsvalue% +} + +\def\pgffor@dotsscanend#1,{% + \pgffor@alphabeticsequencefalse% + \def\pgffor@dotsend{#1}% + \pgffor@dots@value@process{\pgffor@dotsend}% + \pgffor@dots@value@process{\pgffor@last}% + \ifx\pgffor@prevlast\pgfutil@empty% + \ifdim\pgffor@dotsend pt>\pgffor@last pt% + \pgffor@skip=1pt% + \else% + \pgffor@skip=-1pt% + \fi% + \else% + \pgffor@dots@value@process{\pgffor@prevlast}% + \pgffor@skip=\pgffor@last pt% + \pgffor@iter=\pgffor@prevlast pt% + \advance\pgffor@skip by-\pgffor@iter% + \fi% + \pgffor@iter=\pgffor@last pt% + \pgffor@loop% +} + +\def\pgffor@loop{% + \advance\pgffor@iter by\pgffor@skip% + \let\pgffor@next=\pgffor@doloop% + \ifdim\pgffor@skip<0pt% + \ifdim\pgffor@iter<\pgffor@dotsend pt% + \let\pgffor@next=\pgffor@endloop% + \fi% + \else% + \ifdim\pgffor@iter>\pgffor@dotsend pt% + \let\pgffor@next=\pgffor@endloop% + \fi% + \fi% + \ifpgffor@continue% + \else% + \let\pgffor@next=\pgffor@endloop% + \fi% + \pgffor@next% +} + +\def\pgffor@endloop{% + \pgffor@alphabeticsequencefalse% + \pgffor@scan% +} + +{\catcode`\p=12\catcode`\t=12\gdef\Pgffor@geT#1pt{#1}} + +\def\pgffor@doloop{% + \pgffor@begingroup + \edef\pgffor@temp{\expandafter\Pgffor@geT\the\pgffor@iter}% + \edef\pgffor@incheck{{.0/}{\pgffor@temp/}}% + \expandafter\pgfutil@in@\pgffor@incheck% + \ifpgfutil@in@% + \expandafter\pgffor@strip\pgffor@temp% + \fi% + \expandafter\pgfutil@in@\expandafter/\expandafter{\pgffor@var}% + \ifpgfutil@in@% + \expandafter\def\expandafter\pgffor@valuerest\expandafter{\pgffor@temp//\relax}% + \expandafter\pgffor@multiassign\pgffor@var/\pgffor@stop/\pgffor@stop/\relax% + \else% + \ifpgffor@alphabeticsequence% + \pgffor@makealphabetic\pgffor@temp% + \expandafter\let\pgffor@var=\pgffor@temp% + \else% + \expandafter\expandafter\expandafter\def\expandafter\pgffor@var\expandafter{\pgffor@temp}% + \fi% + \fi% + \ifpgffor@context% + \let\pgffor@temp=\pgffor@dots@pre% + \expandafter\pgfutil@append@macrotomacro\expandafter% + {\expandafter\pgffor@temp\expandafter}\expandafter{\pgffor@var}% + \expandafter\pgfutil@append@macrotomacro\expandafter% + {\expandafter\pgffor@temp\expandafter}\expandafter{\pgffor@dots@post}% + \expandafter\let\pgffor@var=\pgffor@temp% + \fi% + \ifx\pgffor@assign@before@code\pgfutil@empty% + \else% + \pgffor@assign@before@code% + \fi% + \expandafter\expandafter\expandafter\pgffor@reset@hooks\expandafter\pgffor@beginhook\expandafter\pgffor@body\pgffor@endhook% + \ifx\pgffor@assign@after@code\pgfutil@empty% + \else% + \pgffor@assign@after@code% + \fi% + \pgffor@endgroup% + \pgffor@loop% +} + +\def\pgffor@strip#1.0{\def\pgffor@temp{#1}} + +\def\breakforeach{\global\pgffor@continuefalse} + +\def\pgffor@gobbletil@pgffor@@stop#1\pgffor@@stop{} + + +\newif\ifpgffor@registeriscount + +\def\pgffor@ifcsregister#1{% + \expandafter\pgffor@@ifcsregister\meaning#1\pgffor@stop} + +\def\pgffor@@ifcsregister#1#2#3#4#5\pgffor@stop{% + \if#1m% + \pgffor@registeriscountfalse% + \let\pgffor@csnext=\pgfutil@secondoftwo% + \else% + \if#1u% + \let\pgffor@csnext=\pgfutil@secondoftwo% + \pgffor@registeriscountfalse% + \else% + \let\pgffor@csnext=\pgfutil@firstoftwo% + \if#4u% + \pgffor@registeriscounttrue% + \else% + \pgffor@registeriscountfalse% + \let\pgffor@csnext=\pgfutil@firstoftwo% + \fi% + \fi% + \fi% + \pgffor@csnext} + +\newif\ifpgffor@alphabeticsequence + +\def\pgffor@dots@charcheck#1\pgffor@@stop{% + \edef\pgffor@dots@charcheck@temp{#1}% + \expandafter\expandafter\expandafter\pgffor@@dotscharcheck\expandafter\meaning\pgffor@dots@charcheck@temp\pgffor@@stop% +} +\def\pgffor@@dotscharcheck#1#2\pgffor@@stop{% + \if#1t% + \afterassignment\pgffor@gobbletil@pgffor@@stop% + \expandafter\chardef\expandafter\pgffor@char\expandafter=\expandafter`\pgffor@dots@charcheck@temp\relax\pgffor@@stop% + \edef\pgffor@char{\the\pgffor@char}% + \ifnum\pgffor@char>64\relax% + \ifnum\pgffor@char<91\relax% + \let\pgffor@dotsvalue=\pgffor@char% + \pgffor@alphabeticsequencetrue% + \else% + \ifnum\pgffor@char>96\relax% + \ifnum\pgffor@char<123\relax% + \let\pgffor@dotsvalue=\pgffor@char% + \pgffor@alphabeticsequencetrue% + \fi% + \fi% + \fi% + \fi% + \fi% +} + +\def\pgffor@makealphabetic#1{% + \pgfutil@tempcnta=#1\relax% + \ifnum\pgfutil@tempcnta>95\relax% + \advance\pgfutil@tempcnta by-96\relax% + \edef#1{\pgffor@alpha\pgfutil@tempcnta}% + \else% + \advance\pgfutil@tempcnta by-64 % + \edef#1{\pgffor@Alpha\pgfutil@tempcnta}% + \fi% +} + +\def\pgffor@Alpha#1{% + \ifcase#1\relax\or A\or B\or C\or D\or E\or F\or G\or H\or I\or J\or K\or L\or M% + \or N\or O\or P\or Q\or R\or S\or T\or U\or V\or W\or X\or Y\or Z\else?\fi% +} + +\def\pgffor@alpha#1{% + \ifcase#1\relax\or a\or b\or c\or d\or e\or f\or g\or h\or i\or j\or k\or l\or m% + \or n\or o\or p\or q\or r\or s\or t\or u\or v\or w\or x\or y\or z\else?\fi% +} + +\newtoks\pgffor@toks + +\def\pgffor@addcstotoks#1{% + \expandafter\def\expandafter\pgffor@tokstemp\expandafter{\expandafter\def\expandafter#1\expandafter{#1}}% + \pgffor@@addcstotoks} + +\def\pgffor@addregtotoks#1{% + \expandafter\def\expandafter\pgffor@tokstemp\expandafter{\expandafter#1\expandafter=\the#1}% + \pgffor@@addcstotoks} + +\def\pgffor@addiftotoks#1{% + \begingroup% + \escapechar=-1\relax% + \expandafter\pgffor@@addiftotoks\string#1\pgffor@stop} +\def\pgffor@@addiftotoks#1#2#3#4\pgffor@stop{% + \endgroup + \csname if#3#4\endcsname% + \expandafter\def\expandafter\pgffor@tokstemp\expandafter{\csname#3#4true\endcsname}% + \else% + \expandafter\def\expandafter\pgffor@tokstemp\expandafter{\csname#3#4false\endcsname}% + \fi% + \pgffor@@addcstotoks% +} + +\def\pgffor@@addcstotoks{% + \expandafter\expandafter\expandafter\pgffor@toks\expandafter\expandafter\expandafter=% + \expandafter\expandafter\expandafter{\expandafter\pgffor@tokstemp\the\pgffor@toks}% +} + + +\def\pgffor@atbeginforeach{% + \begingroup% +} + +\def\pgffor@atendforeach{% + \global\edef\pgffor@remember@expanded{\pgffor@remember@code}% + \ifx\pgffor@remember@expanded\pgfutil@empty% + \else% + \pgffor@remember@expanded% + \global\let\pgffor@remember@expanded=\pgfutil@empty% + \fi% + \endgroup% +} +\def\pgffor@default@begingroup{% + \begingroup% +} + +\let\pgffor@remember@once@expanded=\pgfutil@empty +\let\pgffor@remember@expanded=\pgfutil@empty + +\def\pgffor@default@endgroup{% + \ifx\pgffor@remember@once@code\pgfutil@empty + \else + \xdef\pgffor@remember@once@expanded{\pgffor@remember@once@code}% + \fi + \ifx\pgffor@remember@code\pgfutil@empty + \else + \xdef\pgffor@remember@expanded{\pgffor@remember@code}% + \fi + \endgroup + \ifx\pgffor@remember@once@expanded\pgfutil@empty + \else + \pgffor@remember@once@expanded% + \global\let\pgffor@remember@once@expanded=\pgfutil@empty + \fi + \ifx\pgffor@assign@once@code\pgfutil@empty + \else + \global\let\pgffor@assign@once@code=\pgfutil@empty + \fi + \ifx\pgffor@remember@expanded\pgfutil@empty + \else + \pgffor@remember@expanded% + \global\let\pgffor@remember@expanded=\pgfutil@empty + \fi +} + +\let\pgffor@begingroup=\pgffor@default@begingroup +\let\pgffor@endgroup=\pgffor@default@endgroup + + +\def\pgffor@stack@begingroup{% + \pgffor@toks={}% + \pgffor@addcstotoks{\pgffor@mainvar}% + \pgffor@addcstotoks{\pgffor@var}% + \pgffor@addcstotoks{\pgffor@body}% + \pgffor@addcstotoks{\pgffor@last}% + \pgffor@addcstotoks{\pgffor@prevlast}% + \pgffor@addcstotoks{\pgffor@dotsend}% + \pgffor@addcstotoks{\pgffor@assign@before@code}% + \pgffor@addcstotoks{\pgffor@assign@after@code}% + \pgffor@addcstotoks{\pgffor@remember@code}% + \pgffor@addcstotoks{\pgffor@remember@once@code}% + \pgffor@addcstotoks{\pgffor@dots@pre}% + \pgffor@addcstotoks{\pgffor@dots@post}% + \pgffor@addregtotoks{\pgffor@iter}% + \pgffor@addregtotoks{\pgffor@skip}% + \pgffor@addiftotoks{\ifpgffor@alphabeticsequence}% + \pgffor@addiftotoks{\ifpgffor@context}% + \expandafter\pgffor@stackpush\expandafter{\the\pgffor@toks}% +} + +\def\pgffor@stack@endgroup{\pgffor@stackpop} + + +\newif\ifpgffor@assign@evaluate +\newif\ifpgffor@assign@once +\newif\ifpgffor@assign@parse + +\pgfkeys{/pgf/foreach/.cd, + var/.code=\pgffor@var@add#1\pgffor@stop, + scope iterations/.code={ + \csname if#1\endcsname% + \let\pgffor@begingroup=\pgffor@default@begingroup% + \let\pgffor@endgroup=\pgffor@default@endgroup% + \else% + \let\pgffor@begingroup=\pgffor@stack@begingroup% + \let\pgffor@endgroup=\pgffor@stack@endgroup% + \fi% + }, + evaluate/.code=\pgffor@assign@evaluatetrue\pgffor@assign@oncefalse\pgffor@assign@parse{#1},% + assign/.code=\pgffor@assign@evaluatefalse\pgffor@assign@oncefalse\pgffor@assign@parse{#1},% + evaluate once/.code=\pgffor@assign@evaluatetrue\pgffor@assign@oncetrue\pgffor@assign@parse{#1},% + assign once/.code=\pgffor@assign@evaluatefalse\pgffor@assign@oncetrue\pgffor@assign@parse{#1},% + remember/.code=\pgffor@remember@parse{#1},% + count/.code=\pgffor@count@parse#1\pgffor@stop, + parse/.is if=pgffor@assign@parse, + parse/.default=false, + expand list/.is if=pgffor@expand@list@, + expand list/.default=true, + expand list=false, +} + +\def\pgffor@assign@parse#1{% + \pgfutil@in@;{#1}% + \ifpgfutil@in@% + \else% + \pgfutil@in@={#1}% + \fi% + \ifpgfutil@in@% + \pgffor@assign@@parse#1;\pgffor@stop;% + \else% + \pgffor@assign@parse@old#1\pgffor@stop% + \fi% +} + +\def\pgffor@stop{\pgffor@stop} +\def\pgffor@assign@@parse#1;{% + \def\pgffor@test{#1}% + \ifx\pgffor@test\pgffor@stop% + \let\pgffor@next=\relax% + \else% + \let\pgffor@next=\pgffor@assign@@parse% + \ifx\pgffor@test\pgfutil@empty% + \else% + \pgfutil@in@={#1}% + \ifpgfutil@in@% + \pgffor@assign@@@parse#1\pgffor@stop% + \else% + \pgffor@assign@@@parse#1=#1\pgffor@stop% + \fi% + \fi% + \fi% + \pgffor@next} + +\def\pgffor@assign@@@parse#1=#2\pgffor@stop{% + \ifpgffor@assign@evaluate% + \ifpgffor@assign@once% + \pgfutil@append@tomacro{\pgffor@assign@once@code}{\pgfmathparse{#2}\let#1=\pgfmathresult}% + \pgfutil@append@tomacro{\pgffor@remember@once@code}{\noexpand\def\noexpand#1{#2}}% + \else + \pgfutil@append@tomacro{\pgffor@assign@before@code}{\pgfmathparse{#2}\let#1=\pgfmathresult}% + \fi% + \else% + \ifpgffor@assign@once% + \pgfutil@append@tomacro{\pgffor@assign@once@code}{\def#1{#2}}% + \pgfutil@append@tomacro{\pgffor@remember@once@code}{\noexpand\def\noexpand#1{#2}}% + \else + \pgfutil@append@tomacro{\pgffor@assign@before@code}{\def#1{#2}}% + \fi% + \fi% +} + +\def\pgffor@assign@parse@old#1#2\pgffor@stop{% + \pgffor@assign@@parse@old#2\pgffor@stop as#1using #1\pgffor@@stop} + +\def\pgffor@assign@@parse@old#1as#2#3\pgffor@@stop{% + \pgffor@assign@@@parse@old{#2}#1#3\pgffor@stop\pgffor@@stop} + +\def\pgffor@assign@@@parse@old#1#2using #3\pgffor@stop#4\pgffor@@stop{% + \pgffor@assign@@@parse#1=#3\pgffor@stop} + + + +\def\pgffor@remember@parse#1{% + \pgfutil@in@ a{#1}% + \ifpgfutil@in@% + \pgffor@remember@parse@old#1\pgffor@stop% + \else% + \pgffor@remember@@parse#1;\pgffor@stop;% + \fi% +} + +\def\pgffor@remember@@parse#1;{% + \def\pgffor@test{#1}% + \ifx\pgffor@test\pgffor@stop% + \let\pgffor@next=\relax% + \else% + \let\pgffor@next=\pgffor@remember@@parse% + \ifx\pgffor@test\pgfutil@empty% + \else% + \pgfutil@in@={#1}% + \ifpgfutil@in@% + \pgffor@remember@@@parse#1\pgffor@stop% + \else% + \pgffor@ifcsregister{#1}{% + \pgfutil@append@tomacro{\pgffor@remember@code}{\noexpand#1=\the#1\noexpand\relax}% + }% + {% + \pgfutil@append@tomacro{\pgffor@remember@code}{\noexpand\def\noexpand#1{#1}}% + }% + \fi% + \fi% + \fi% + \pgffor@next% +} + +\def\pgffor@remember@@@parse#1=#2\pgffor@stop{% + \pgffor@ifcsregister{#2}{% + \pgfutil@append@tomacro{\pgffor@assign@after@code}{\expandafter\def\expandafter#1\expandafter{\the#2}}% + \pgfutil@append@tomacro{\pgffor@remember@code}{\noexpand\def\noexpand#1{#1}}% + }% + {% + \pgfutil@append@tomacro{\pgffor@assign@after@code}{\expandafter\def\expandafter#1\expandafter{#2}}% + \pgfutil@append@tomacro{\pgffor@remember@code}{\noexpand\def\noexpand#1{#1}}% + }% +} + +\def\pgffor@remember@parse@old#1#2\pgffor@stop{% + \pgffor@remember@@parse@old#1#2\pgffor@stop as#1(initially 0)\pgffor@@stop} + +\def\pgffor@remember@@parse@old#1#2as#3#4\pgffor@@stop{% + \pgffor@remember@@@parse@old{#1}{#3}#2#4\pgffor@stop\pgffor@@stop} + +\def\pgffor@remember@@@parse@old#1#2#3(initially #4)#5\pgffor@stop#6\pgffor@@stop{% + \pgfutil@append@tomacro{\pgffor@assign@after@code}{\edef#2{#1}}% + \pgfutil@append@tomacro{\pgffor@remember@code}{\noexpand\def\noexpand#2{#2}}% + \pgfutil@append@tomacro{\pgffor@assign@once@code}{\def#2{#4}}% +} + +\def\pgffor@count@parse#1#2\pgffor@stop{% + \pgffor@count@@parse#1#2\pgffor@stop from 1\pgffor@stop\pgffor@@stop} + +\def\pgffor@count@@parse#1#2from#3\pgffor@stop#4\pgffor@@stop{% + \pgfutil@append@tomacro\pgffor@remember@code{\noexpand\def\noexpand#1{#1}}% + \pgfutil@append@tomacro\pgffor@assign@before@code{\pgfmathparse{int(#1+1)}\let#1=\pgfmathresult}% + \pgfmathparse{int(#3-1)}\let#1=\pgfmathresult% +} + + +\def\pgfutil@append@macrotomacro#1#2{% + \expandafter\expandafter\expandafter\def\expandafter\expandafter\expandafter% + #1\expandafter\expandafter\expandafter{\expandafter#1#2}} + +\def\pgfutil@append@tomacro#1#2{% + \expandafter\def\expandafter#1\expandafter{#1#2}} + + +\endinput diff --git a/preprints/A-polynomial-time-algorithm-for-three-machine-unit-job-scheduling-September-24-2026/build/tex/runtime/generic/pgf/utilities/pgfkeys.code.tex b/preprints/A-polynomial-time-algorithm-for-three-machine-unit-job-scheduling-September-24-2026/build/tex/runtime/generic/pgf/utilities/pgfkeys.code.tex new file mode 100644 index 0000000000000000000000000000000000000000..fd8d0de64e777f559b1911a1208f4efe604ecdf3 --- /dev/null +++ b/preprints/A-polynomial-time-algorithm-for-three-machine-unit-job-scheduling-September-24-2026/build/tex/runtime/generic/pgf/utilities/pgfkeys.code.tex @@ -0,0 +1,873 @@ + + + + +\ifx\pgfkeysloaded\undefined + \let\pgfkeysloaded=\relax +\else + \expandafter\endinput +\fi + + +\def\pgfkeys@empty{} +\long\def\pgfkeys@firstoftwo#1#2{#1} +\long\def\pgfkeys@secondoftwo#1#2{#2} +\long\def\pgfkeys@ifcsname#1{% + \ifcsname#1\endcsname + \expandafter\pgfkeys@firstoftwo + \else + \expandafter\pgfkeys@secondoftwo + \fi +} + + +\ifx\PackageError\undefined + \def\pgfkeys@error#1{\errmessage{Package pgfkeys Error: #1}}% +\else + \def\pgfkeys@error#1{\PackageError{pgfkeys}{#1}{}}% +\fi + +\ifx\eTeXrevision\undefined% + \pgfkeys@error{PGF requires etex in extended mode}% + \csname @@end\expandafter\endcsname\expandafter\end% +\fi + +\ifnum\eTeXversion<2 + \pgfkeys@error{PGF requires etex version 2}% + \csname @@end\expandafter\endcsname\expandafter\end% +\fi + + +\ifx\contextversion\undefined + \let\pgfkeys@expanded \expanded + \let\pgfkeys@unexpanded \unexpanded +\else + \let\pgfkeys@expanded \normalexpanded + \let\pgfkeys@unexpanded \normalunexpanded +\fi + +\begingroup + \edef\pgfkeys@tmpa{\string\expanded} + \edef\pgfkeys@tmpb{\meaning\pgfkeys@expanded} + \expandafter +\endgroup +\ifx\pgfkeys@tmpa\pgfkeys@tmpb +\else + \pgfkeys@error{PGF requires the \noexpand\expanded primitive} + \csname @@end\expandafter\endcsname\expandafter\end% +\fi + + + +\long\def\pgfkeyssetvalue#1#2{% + \pgfkeys@temptoks{#2}\expandafter\edef\csname pgfk@#1\endcsname{\the\pgfkeys@temptoks}% +} + +\long\def\pgfkeyssetevalue#1#2{% + \expandafter\edef\csname pgfk@#1\endcsname + {\pgfkeys@unexpanded\expandafter{\pgfkeys@expanded{#2}}}% +} + + +\long\def\pgfkeysaddvalue#1#2#3{% + {% + \toks0{#2}% + \pgfkeysifdefined{#1} + {\pgfkeys@temptoks\expandafter\expandafter\expandafter{\csname pgfk@#1\endcsname}}% + {\pgfkeys@temptoks{}}% + \toks1{#3}% + \xdef\pgfkeys@global@temp{\the\toks0 \the\pgfkeys@temptoks \the\toks1}% + }% + \pgfkeyslet{#1}\pgfkeys@global@temp% +} + + + + +\long\def\pgfkeyslet#1#2{% + \expandafter\let\csname pgfk@#1\endcsname#2% +} + + + +\long\def\pgfkeysgetvalue#1#2{% + \pgfkeys@ifcsname{pgfk@#1}% + {\expandafter\let\expandafter#2\csname pgfk@#1\endcsname}% + {\let#2\relax}} + + + + +\let\pgfkeys@relax\relax +\long\def\pgfkeysvalueof#1{\csname\pgfkeys@ifcsname{pgfk@#1}{pgfk@#1}{pgfkeys@relax}\endcsname} + + + + +\long\def\pgfkeysifdefined#1{% + \ifcsname pgfk@#1\endcsname + \expandafter\pgfkeys@firstoftwo + \else + \expandafter\pgfkeys@secondoftwo + \fi +} + +\long\def\pgfkeysifassignable#1#2#3{% + \pgfkeysifdefined{#1}% + {#2} + {\pgfkeysifdefined{#1/.@cmd}% + {#2}% + {#3}}% +}% + + + + +\newtoks\pgfkeys@pathtoks +\def\pgfkeyscurrentpath{\the\pgfkeys@pathtoks} +\newtoks\pgfkeys@temptoks + +\def\pgfkeys@root{/} +\let\pgfkeysdefaultpath\pgfkeys@root + +\def\pgfkeys{\expandafter\pgfkeys@@set\expandafter{\pgfkeysdefaultpath}}% +\long\def\pgfkeys@@set#1#2{% + \let\pgfkeysdefaultpath\pgfkeys@root% + \pgfkeys@parse#2,\pgfkeys@mainstop% + \def\pgfkeysdefaultpath{#1}} + +\def\pgfkeys@parse{\futurelet\pgfkeys@possiblerelax\pgfkeys@parse@main} +\def\pgfkeys@parse@main{% + \ifx\pgfkeys@possiblerelax\pgfkeys@mainstop% + \expandafter\pgfkeys@cleanup% + \else% + \expandafter\pgfkeys@normal% + \fi% +} +\newif\ifpgfkeys@syntax@handlers +\def\pgfkeys@normal{% + \ifpgfkeys@syntax@handlers% + \expandafter\pgfkeys@syntax@handlers% + \else% + \expandafter\pgfkeys@@normal% + \fi% +} +\def\pgfkeys@syntax@handlers{\pgf@keys@utilifnextchar\relax\pgfkeys@syntax@@handlers\pgfkeys@syntax@@handlers}% +\def\pgfkeys@syntax@@handlers{\futurelet\pgfkeys@first@char\pgfkeys@syntax@handlers@test} +\def\pgfkeys@syntax@handlers@test{% + \pgfkeysgetvalue{/handlers/first char syntax/\meaning\pgfkeys@first@char}\pgfkeys@the@handler% + \ifx\pgfkeys@the@handler\relax% + \expandafter\pgfkeys@@normal% + \else% + \expandafter\pgfkeys@use@handler% + \fi% +} +\long\def\pgfkeys@use@handler#1,{% + \pgfkeys@the@handler{#1}% + \pgfkeys@parse% +} + +\long\def\pgfkeys@@normal#1,{% + \pgfkeys@unpack#1=\pgfkeysnovalue=\pgfkeys@stop% + \pgfkeys@parse% +} +\def\pgfkeys@cleanup\pgfkeys@mainstop{} + +\def\pgfkeys@mainstop{\pgfkeys@mainstop} % +\def\pgfkeys@novalue{} +\def\pgfkeysnovalue{\pgfkeys@novalue} % +\def\pgfkeysnovalue@text{\pgfkeysnovalue} +\def\pgfkeysvaluerequired{\pgfkeysvaluerequired} % + +\long\def\pgfkeys@unpack#1=#2=#3\pgfkeys@stop{% + \pgfkeys@spdef\pgfkeyscurrentkey{#1}% + \edef\pgfkeyscurrentkey{\pgfkeyscurrentkey}% + \ifx\pgfkeyscurrentkey\pgfkeys@empty% + \else% + \pgfkeys@add@path@as@needed% + \pgfkeys@spdef\pgfkeyscurrentvalue{#2}% + \ifx\pgfkeyscurrentvalue\pgfkeysnovalue@text% + \pgfkeysifdefined{\pgfkeyscurrentkey/.@def}% + {\pgfkeysgetvalue{\pgfkeyscurrentkey/.@def}{\pgfkeyscurrentvalue}} + {}% + \fi% + \ifx\pgfkeyscurrentvalue\pgfkeysvaluerequired% + \def\pgf@marshal{\pgfkeysvalueof{/errors/value required/.@cmd}}% + \expandafter\pgf@marshal\expandafter{\pgfkeyscurrentkey}{}\pgfeov% + \else% + \pgfkeys@case@one% + \fi% + \fi} + +\def\pgfkeys@case@one{% + \pgfkeysifdefined{\pgfkeyscurrentkey/.@cmd}% + {\pgfkeysgetvalue{\pgfkeyscurrentkey/.@cmd}{\pgfkeys@code}% + \ifx\pgfkeys@code\relax\expandafter\pgfkeys@firstoftwo\else\expandafter\pgfkeys@secondoftwo\fi + {\pgfkeys@unknown}% + {\expandafter\pgfkeys@code\pgfkeyscurrentvalue\pgfeov}} + {\pgfkeys@case@two}% +} + +\def\pgfkeys@case@two{% + \pgfkeysifdefined{\pgfkeyscurrentkey}% + {\pgfkeys@case@two@extern}% + {\pgfkeys@case@three}% +} + +\def\pgfkeys@case@two@extern{% + \ifx\pgfkeyscurrentvalue\pgfkeysnovalue@text% + \pgfkeysvalueof{\pgfkeyscurrentkey}% + \else% + \pgfkeyslet{\pgfkeyscurrentkey}\pgfkeyscurrentvalue% + \fi% +} + + +\def\pgfkeys@case@three{% + \pgfkeys@split@path% + \pgfkeysifdefined{/handlers/\pgfkeyscurrentname/.@cmd}% + {\pgfkeysgetvalue{/handlers/\pgfkeyscurrentname/.@cmd}{\pgfkeys@code}% + \expandafter\pgfkeys@code\pgfkeyscurrentvalue\pgfeov} + {\pgfkeys@unknown}% +} +\let\pgfkeys@case@three@handleall=\pgfkeys@case@three +\def\pgfkeys@case@three@handle@restricted{% + \pgfkeys@split@path% + \pgfkeysifdefined{/handlers/\pgfkeyscurrentname/.@cmd}{% + \pgfkeys@ifexecutehandler{% + \pgfkeysgetvalue{/handlers/\pgfkeyscurrentname/.@cmd}{\pgfkeys@code}% + \expandafter\pgfkeys@code\pgfkeyscurrentvalue\pgfeov + }{% + \let\pgfkeys@tempa=\pgfkeyscurrentkey + \let\pgfkeys@tempb=\pgfkeyscurrentname + \edef\pgfkeyscurrentkey{\pgfkeyscurrentpath}% + \pgfkeys@split@path% + \let\pgfkeyscurrentkey=\pgfkeys@tempa + \edef\pgfkeyscurrentname{\pgfkeyscurrentname/\pgfkeys@tempb}% + \pgfkeys@unknown + }% + }{% + \pgfkeys@unknown + }% +} + +\def\pgfkeys@ifexecutehandler#1#2{#1}% +\let\pgfkeys@ifexecutehandler@handleall=\pgfkeys@ifexecutehandler +\def\pgfkeys@ifexecutehandler@handleonlyexisting#1#2{% + \pgfkeys@ifcsname{pgfk@excpt@\pgfkeyscurrentname}{% + #1% + }{% + \pgfkeysifdefined{\pgfkeyscurrentpath}{#1}{% + \pgfkeysifdefined{\pgfkeyscurrentpath/.@cmd}{#1}{#2}% + }{}% + }% +}% +\def\pgfkeys@ifexecutehandler@handlefullorexisting#1#2{% + \ifpgfkeysaddeddefaultpath + \pgfkeys@ifcsname{pgfk@excpt@\pgfkeyscurrentname}{% + #1% + }{% + \pgfkeysifdefined{\pgfkeyscurrentpath}{% + #1% + }{% + \pgfkeysifdefined{\pgfkeyscurrentpath/.@cmd}{% + #1% + }{% + #2% + }% + }% + }% + \else + #1% + \fi +}% +\def\pgfkeysaddhandleonlyexistingexception#1{\expandafter\def\csname pgfk@excpt@#1\endcsname{1}}% + +\def\pgfkeys@unknown{% + \pgfkeysifdefined{\pgfkeyscurrentpath/.unknown/.@cmd}% + {% + \pgfkeysgetvalue{\pgfkeyscurrentpath/.unknown/.@cmd}{\pgfkeys@code}% + \expandafter\pgfkeys@code\pgfkeyscurrentvalue\pgfeov} + {% + \pgfkeysgetvalue{/handlers/.unknown/.@cmd}{\pgfkeys@code}% + \expandafter\pgfkeys@code\pgfkeyscurrentvalue\pgfeov% + }% +} + + +\long\def\pgfkey@argumentisspace#1{% + \long\def\pgfkeys@spdef##1##2{% + \futurelet\pgfkeys@possiblespace\pgfkeys@sp@a##2\pgfkeys@stop\pgfkeys@stop#1\pgfkeys@stop\relax##1}% + \def\pgfkeys@sp@a{% + \ifx\pgfkeys@possiblespace\pgfkeys@sptoken% + \expandafter\pgfkeys@sp@b% + \else% + \expandafter\pgfkeys@sp@b\expandafter#1% + \fi}% + \long\def\pgfkeys@sp@b#1##1 \pgfkeys@stop{\pgfkeys@sp@c##1}% +} +\pgfkey@argumentisspace{ } +\long\def\pgfkeys@sp@c#1\pgfkeys@stop#2\relax#3{\pgfkeys@temptoks{#1}\edef#3{\the\pgfkeys@temptoks}} +{\def\:{\global\let\pgfkeys@sptoken= } \: } + + + +\def\pgfkeys@add@path@as@needed{% + \expandafter\futurelet\expandafter\pgfkeys@possibleslash\expandafter\pgfkeys@check@slash\pgfkeyscurrentkey\relax% +} +\newif\ifpgfkeysaddeddefaultpath +\def\pgfkeys@check@slash{% + \ifx\pgfkeys@possibleslash/% + \expandafter\pgfkeys@nevermind% + \else% + \expandafter\pgfkeys@addpath% + \fi% +} + +\def\pgfkeys@nevermind#1\relax{% + \pgfkeysaddeddefaultpathfalse + \let\pgfkeyscurrentkeyRAW\pgfkeyscurrentkey +} +\def\pgfkeys@addpath#1\relax{% + \pgfkeysaddeddefaultpathtrue + \def\pgfkeyscurrentkeyRAW{#1}% + \edef\pgfkeyscurrentkey{\pgfkeysdefaultpath#1}% +} + + +\def\pgfkeys@split@path{% + \pgfkeys@pathtoks{}% + \expandafter\pgfkeys@splitter\pgfkeyscurrentkey//% +} +\def\pgfkeys@splitter#1/#2/{% + \def\pgfkeys@temp{#2}% + \ifx\pgfkeys@temp\pgfkeys@empty% + \def\pgfkeyscurrentname{#1}% + \expandafter\pgfkeys@gobbletoslash% + \else% + \expandafter\pgfkeys@pathtoks\expandafter{\the\pgfkeys@pathtoks#1/}% + \fi% + \pgfkeys@splitter#2/% +} +\def\pgfkeys@gobbletoslash\pgfkeys@splitter/{% + \if\relax\detokenize\expandafter{\the\pgfkeys@pathtoks}\relax\else + \expandafter\pgfkeys@remove@slash\the\pgfkeys@pathtoks\relax + \fi +}% +\def\pgfkeys@remove@slash#1/\relax{\pgfkeys@pathtoks{#1}} + + + + +\def\pgfqkeys{\expandafter\pgfkeys@@qset\expandafter{\pgfkeysdefaultpath}}% +\long\def\pgfkeys@@qset#1#2#3{\def\pgfkeysdefaultpath{#2/}\pgfkeys@parse#3,\pgfkeys@mainstop\def\pgfkeysdefaultpath{#1}} + + + +\long\def\pgfkeysalso#1{\pgfkeys@parse#1,\pgfkeys@mainstop} + + + + +\long\def\pgfqkeysalso#1#2{\def\pgfkeysdefaultpath{#1/}\pgfkeys@parse#2,\pgfkeys@mainstop} + + + + + + +\long\def\pgfkeysdef#1#2{% + \long\def\pgfkeys@temp##1\pgfeov{#2}% + \pgfkeyslet{#1/.@cmd}{\pgfkeys@temp}% + \pgfkeyssetvalue{#1/.@body}{#2}% +} +\long\def\pgfkeysedef#1#2{% + \long\edef\pgfkeys@temp##1\pgfeov{#2}% + \pgfkeyslet{#1/.@cmd}{\pgfkeys@temp}% + \pgfkeyssetevalue{#1/.@body}{#2}% +} + + + +\long\def\pgfkeysdefargs#1#2#3{% + \long\def\pgfkeys@temp#2\pgfeov{#3}% + \pgfkeyslet{#1/.@cmd}{\pgfkeys@temp}% + \pgfkeyssetvalue{#1/.@args}{#2\pgfeov}% + \pgfkeyssetvalue{#1/.@body}{#3}% +} + +\long\def\pgfkeysedefargs#1#2#3{% + \long\edef\pgfkeys@temp#2\pgfeov{#3}% + \pgfkeyslet{#1/.@cmd}{\pgfkeys@temp}% + \pgfkeyssetvalue{#1/.@args}{#2\pgfeov}% + \pgfkeyssetevalue{#1/.@body}{#3}% +} + +\long\def\pgfkeysdefnargs#1#2#3{\pgfkeysdefnargs@{#1}{#2}{#3}{\def}{\pgfkeyssetvalue}}% +\long\def\pgfkeysedefnargs#1#2#3{\pgfkeysdefnargs@{#1}{#2}{#3}{\edef}{\pgfkeyssetevalue}}% +\long\def\pgfkeysdefnargs@#1#2#3#4#5{% + \ifcase#2\relax + \pgfkeyssetvalue{#1/.@args}{}% + \or + \pgfkeyssetvalue{#1/.@args}{##1}% + \or + \pgfkeyssetvalue{#1/.@args}{##1##2}% + \or + \pgfkeyssetvalue{#1/.@args}{##1##2##3}% + \or + \pgfkeyssetvalue{#1/.@args}{##1##2##3##4}% + \or + \pgfkeyssetvalue{#1/.@args}{##1##2##3##4##5}% + \or + \pgfkeyssetvalue{#1/.@args}{##1##2##3##4##5##6}% + \or + \pgfkeyssetvalue{#1/.@args}{##1##2##3##4##5##6##7}% + \or + \pgfkeyssetvalue{#1/.@args}{##1##2##3##4##5##6##7##8}% + \or + \pgfkeyssetvalue{#1/.@args}{##1##2##3##4##5##6##7##8##9}% + \else + \pgfkeys@error{\string\pgfkeysdefnargs: expected <= 9 arguments, got #2}% + \fi + \pgfkeysgetvalue{#1/.@args}\pgfkeys@tempargs + \def\pgfkeys@temp{\expandafter\long\expandafter#4\csname pgfk@#1/.@@body\endcsname}% + \expandafter\pgfkeys@temp\pgfkeys@tempargs{#3}% + \edef\pgfkeys@tempargs{\noexpand\pgfkeysvalueof{#1/.@@body}}% + \def\pgfkeys@temp{\pgfkeysdef{#1}}% + \ifnum#2=1\relax + \expandafter\pgfkeys@temp\expandafter{\pgfkeys@tempargs{##1}}% + \else + \expandafter\pgfkeys@temp\expandafter{\pgfkeys@tempargs##1}% + \fi + #5{#1/.@body}{#3}% +} + + +\pgfkeysdef{/handlers/.code}{\pgfkeysdef{\pgfkeyscurrentpath}{#1}} +\pgfkeysdef{/handlers/.code 2 args}{\pgfkeysdefargs{\pgfkeyscurrentpath}{##1##2}{#1}} +\pgfkeysdef{/handlers/.ecode}{\pgfkeysedef{\pgfkeyscurrentpath}{#1}} +\pgfkeysdef{/handlers/.ecode 2 args}{\pgfkeysedefargs{\pgfkeyscurrentpath}{##1##2}{#1}} +\pgfkeysdefnargs{/handlers/.code args}{2}{\pgfkeysdefargs{\pgfkeyscurrentpath}{#1}{#2}} +\pgfkeysdefnargs{/handlers/.ecode args}{2}{\pgfkeysedefargs{\pgfkeyscurrentpath}{#1}{#2}} +\pgfkeysdefnargs{/handlers/.code n args}{2}{\pgfkeysdefnargs{\pgfkeyscurrentpath}{#1}{#2}} +\pgfkeysdefnargs{/handlers/.ecode n args}{2}{\pgfkeysedefnargs{\pgfkeyscurrentpath}{#1}{#2}} + + +\pgfkeys{/handlers/.add code/.code 2 args=% + \pgfkeysifdefined{\pgfkeyscurrentpath/.@args}% + {% + \pgfkeysaddvalue{\pgfkeyscurrentpath/.@body}{#1}{#2}% + {% + \pgfkeysgetvalue{\pgfkeyscurrentpath/.@args}{\pgfkeys@tempargs}% + \pgfkeysgetvalue{\pgfkeyscurrentpath/.@body}{\pgfkeys@tempbody}% + \def\pgfkeys@marshal{\expandafter\long\expandafter\gdef\expandafter\pgfkeys@global@temp\pgfkeys@tempargs}% + \expandafter\pgfkeys@marshal\expandafter{\pgfkeys@tempbody}% + }% + \pgfkeysifdefined{\pgfkeyscurrentpath/.@@body}{% + \pgfkeyslet{\pgfkeyscurrentpath/.@@body}{\pgfkeys@global@temp}% + }{% + \pgfkeyslet{\pgfkeyscurrentpath/.@cmd}{\pgfkeys@global@temp}% + }% + }% + {% + \edef\pgf@expanded@path{\pgfkeyscurrentpath}% + {% + \toks0{#1}% + \pgfkeysifdefined{\pgf@expanded@path/.@cmd}% + {\expandafter\expandafter\expandafter\pgfkeys@temptoks\expandafter\expandafter\expandafter{\csname pgfk@\pgf@expanded@path/.@body\endcsname}}% + {\expandafter\pgfkeys@temptoks\expandafter{\expandafter\pgfkeyssetvalue\expandafter{\pgf@expanded@path}{##1}}}% + \toks1{#2}% + \xdef\pgfkeys@global@temp{\the\toks0 \the\pgfkeys@temptoks \the\toks1 }% + }% + \def\pgf@temp{\pgfkeyssetvalue{\pgf@expanded@path/.@body}}% + \expandafter\pgf@temp\expandafter{\pgfkeys@global@temp}% + \expandafter\long\expandafter\def\expandafter\pgfkeys@temp\expandafter##\expandafter1\expandafter\pgfeov\expandafter{\pgfkeys@global@temp}% + \pgfkeyslet{\pgf@expanded@path/.@cmd}\pgfkeys@temp% + }% +} +\pgfkeys{/handlers/.prefix code/.code=\pgfkeys{\pgfkeyscurrentpath/.add code={#1}{}}}% +\pgfkeys{/handlers/.append code/.code=\pgfkeys{\pgfkeyscurrentpath/.add code={}{#1}}}% + + + +\pgfkeys{/handlers/.style/.code=\pgfkeys{\pgfkeyscurrentpath/.code=\pgfkeysalso{#1}}} +\pgfkeys{/handlers/.estyle/.code=\pgfkeys{\pgfkeyscurrentpath/.ecode=\noexpand\pgfkeysalso{#1}}} +\pgfkeys{/handlers/.style args/.code 2 args=\pgfkeys{\pgfkeyscurrentpath/.code args={#1}{\pgfkeysalso{#2}}}} +\pgfkeys{/handlers/.estyle args/.code 2 args=\pgfkeys{\pgfkeyscurrentpath/.ecode args={#1}{\noexpand\pgfkeysalso{#2}}}} +\pgfkeys{/handlers/.style 2 args/.code=\pgfkeys{\pgfkeyscurrentpath/.code 2 args=\pgfkeysalso{#1}}} +\pgfkeys{/handlers/.style n args/.code 2 args=\pgfkeys{\pgfkeyscurrentpath/.code n args={#1}{\pgfkeysalso{#2}}}} + + +\pgfkeys{/handlers/.add style/.code 2 args=\pgfkeys{\pgfkeyscurrentpath/.add code={\pgfkeysalso{#1}}{\pgfkeysalso{#2}}}}% +\pgfkeys{/handlers/.prefix style/.code=\pgfkeys{\pgfkeyscurrentpath/.add code={\pgfkeysalso{#1}}{}}}% +\pgfkeys{/handlers/.append style/.code=\pgfkeys{\pgfkeyscurrentpath/.add code={}{\pgfkeysalso{#1}}}}% + + + +\pgfkeys{/handlers/.initial/.code=\pgfkeyssetvalue{\pgfkeyscurrentpath}{#1}} +\pgfkeys{/handlers/.add/.code 2 args=\pgfkeysaddvalue{\pgfkeyscurrentpath}{#1}{#2}} +\pgfkeys{/handlers/.prefix/.code=\pgfkeysaddvalue{\pgfkeyscurrentpath}{#1}{}} +\pgfkeys{/handlers/.append/.code=\pgfkeysaddvalue{\pgfkeyscurrentpath}{}{#1}} +\pgfkeys{/handlers/.get/.code=\pgfkeysgetvalue{\pgfkeyscurrentpath}{#1}} +\pgfkeys{/handlers/.link/.code=\pgfkeyssetvalue{\pgfkeyscurrentpath}{\pgfkeysvalueof{#1}}} + + + +\pgfkeys{/handlers/.default/.code=\pgfkeyssetvalue{\pgfkeyscurrentpath/.@def}{#1}} +\pgfkeys{/handlers/.value required/.code=\pgfkeyssetvalue{\pgfkeyscurrentpath/.@def}{\pgfkeysvaluerequired}} +\pgfkeys{/handlers/.value forbidden/.code=\pgfkeys{\pgfkeyscurrentpath/.add code=% +{% + \ifx\pgfkeyscurrentvalue\pgfkeysnovalue@text% + \else% + \def\pgf@marshal{\pgfkeysvalueof{/errors/value forbidden/.@cmd}}% + \expandafter\expandafter\expandafter\pgf@marshal\expandafter\expandafter\expandafter{\expandafter\pgfkeyscurrentkey\expandafter}\expandafter{\pgfkeyscurrentvalue}\pgfeov% + \fi% +}{}}} + + + +\pgfkeys{/handlers/.store in/.code=\pgfkeysalso{\pgfkeyscurrentpath/.code=\def#1{##1}}} +\pgfkeys{/handlers/.estore in/.code=\pgfkeysalso{\pgfkeyscurrentpath/.code=\edef#1{##1}}} + +\pgfkeys{/handlers/.is if/.code=\pgfkeysalso{% + \pgfkeyscurrentpath/.code=\pgfkeys@handle@boolean{#1}{##1}, + \pgfkeyscurrentpath/.default=true% + }% +} +\def\pgfkeys@handle@boolean#1#2{% + \pgfkeys@ifcsname{#1#2}{% + \csname#1#2\endcsname% + }{% + \def\pgf@marshal{\pgfkeysvalueof{/errors/boolean expected/.@cmd}}% + \expandafter\pgf@marshal\expandafter{\pgfkeyscurrentkey}{#2}\pgfeov% + }% +} + +\pgfkeys{/handlers/.is choice/.code=% + \pgfkeys{% + \pgfkeyscurrentpath/.cd,% + .code=\def\pgfkeys@was@choice{##1}\expandafter\pgfkeysalso\expandafter{\pgfkeyscurrentkey/##1}, + .unknown/.code={% + \def\pgf@marshal{\pgfkeysvalueof{/errors/unknown choice value/.@cmd}}% + {\expandafter\expandafter\expandafter\pgf@marshal\expandafter\expandafter\expandafter{\expandafter\the\expandafter\pgfkeys@pathtoks\expandafter}\expandafter{\pgfkeys@was@choice}\pgfeov}% + }% + }% +} + + + +\pgfkeys{/handlers/.list/.code=% + {% + \ifx\foreach\@undefined% + \pgfkeys@error{You need to load the pgffor package to use the .list key syntax.}% + \fi% + \def\pgf@keys@temp{}% + \foreach \pgf@keys@key in{#1}% + {\expandafter\expandafter\expandafter\gdef% + \expandafter\expandafter\expandafter\pgf@keys@temp% + \expandafter\expandafter\expandafter{\expandafter\pgf@keys@temp\expandafter{\pgf@keys@key}}}% + \edef\pgf@keys@list@path{\pgfkeyscurrentpath}% + \expandafter\expandafter\expandafter\pgf@keys@do@list% + \expandafter\expandafter\expandafter{\expandafter\pgf@keys@list@path\expandafter}\pgf@keys@temp\pgf@stop% + }% +} +\def\pgf@keys@do@list#1{\pgf@keys@utilifnextchar\bgroup{\pgf@keys@do@list@item{#1}}\pgf@keys@gobble} +\def\pgf@keys@do@list@item#1#2{\pgfkeysalso{#1={#2}}\pgf@keys@do@list{#1}} + + + +\pgfkeys{/handlers/.forward to/.code=% + \pgfkeys{\pgfkeyscurrentpath/.add code={}{\pgfkeys{#1={##1}}}} +} + + + +\pgfkeys{/handlers/.show value/.code=\pgfkeysgetvalue{\pgfkeyscurrentpath}{\pgfkeysshower}\show\pgfkeysshower} % +\pgfkeys{/handlers/.show code/.code=\pgfkeysgetvalue{\pgfkeyscurrentpath/.@cmd}{\pgfkeysshower}\show\pgfkeysshower} % + + +\pgfkeys{/handlers/first char syntax/.is if=pgfkeys@syntax@handlers} + + +\def\pgfkeys@searchalso@prepare@unknown@handler#1{% + \global\def\pgfkeys@global@temp##1\pgfeov{}% + \pgfkeys@searchalso@parse#1,\pgfkeys@mainstop + {% + \toks0=\expandafter{\pgfkeys@global@temp##1\pgfeov}% + \toks1={% + \pgfkeysgetvalue{/handlers/.unknown/.@cmd}{\pgfkeys@code}% + \expandafter\pgfkeys@code\pgfkeys@searchalso@temp@value\pgfeov + }% + \xdef\pgfkeys@global@temp{% + \noexpand\def\noexpand\pgfkeys@searchalso@temp@value{####1}% + \noexpand\ifpgfkeysaddeddefaultpath + \noexpand\expandafter\noexpand\pgfkeys@firstoftwo + \noexpand\else + \noexpand\expandafter\noexpand\pgfkeys@secondoftwo + \noexpand\fi{% + \noexpand\pgfkeyssuccessfalse + \noexpand\let\noexpand\pgfkeys@searchalso@name=\noexpand\pgfkeyscurrentkeyRAW + \the\toks0 % + }{% + \the\toks1 % + }% + }% + \expandafter\gdef\expandafter\pgfkeys@global@temp\expandafter##\expandafter1\expandafter\pgfeov\expandafter{\pgfkeys@global@temp}% + }% +}% + +\def\pgfkeys@searchalso@parse{\futurelet\pgfkeys@possiblerelax\pgfkeys@searchalso@parse@main} +\def\pgfkeys@searchalso@parse@main{% + \ifx\pgfkeys@possiblerelax\pgfkeys@mainstop% + \expandafter\pgfkeys@cleanup% + \else% + \expandafter\pgfkeys@searchalso@appendentry% + \fi% +} +\def\pgfkeys@searchalso@appendentry#1,#2{% + \def\pgfkeys@searchalso@nexttok{#2}% + \pgfkeys@spdef\pgfkeys@temp{#1}% + {% + \toks0=\expandafter{\pgfkeys@global@temp##1\pgfeov}% + \toks1=\expandafter{\pgfkeys@temp}% + \xdef\pgfkeys@global@temp{% + \the\toks0 % + \noexpand\ifpgfkeyssuccess\noexpand\else + \noexpand\pgfqkeys{\the\toks1 }{\noexpand\pgfkeys@searchalso@name + \ifx\pgfkeys@searchalso@nexttok\pgfkeys@mainstop\else/.try\fi /.expand once=\noexpand\pgfkeys@searchalso@temp@value}% + \noexpand\fi}% + \expandafter\gdef\expandafter\pgfkeys@global@temp\expandafter##\expandafter1\expandafter\pgfeov\expandafter{\pgfkeys@global@temp}% + }% + \pgfkeys@searchalso@parse#2% +} +\pgfkeys{% + /handlers/.is family/.code=\pgfkeys{\pgfkeyscurrentpath/.ecode=\edef\noexpand\pgfkeysdefaultpath{\pgfkeyscurrentpath/}},% + /handlers/.cd/.code=\edef\pgfkeysdefaultpath{\pgfkeyscurrentpath/},% + /handlers/.search also/.code={% + \pgfkeys@searchalso@prepare@unknown@handler{#1}% + \pgfkeyslet{\pgfkeyscurrentpath/.unknown/.@cmd}{\pgfkeys@global@temp}% + } +}% + + + +\pgfkeys{/handlers/.expand once/.code=\expandafter\pgfkeys@exp@call\expandafter{#1}} +\pgfkeys{/handlers/.expand twice/.code=\expandafter\expandafter\expandafter\pgfkeys@exp@call\expandafter\expandafter\expandafter{#1}} +\pgfkeys{/handlers/.expanded/.code=\edef\pgfkeys@temp{#1}\expandafter\pgfkeys@exp@call\expandafter{\pgfkeys@temp}} + +\long\def\pgfkeys@exp@call#1{\pgfkeysalso{\pgfkeyscurrentpath={#1}}} + +\def\pgfkeys@mathparse{% + \pgfkeys@ifcsname{pgfmathparse}{% + \pgfmathparse + }{% + \pgfkeys@error{You have to load `pgfmath' to use \string\pgfmathparse}% + \def\pgfmathresult + }% +} +\pgfkeys{/handlers/.evaluated/.code=\pgfkeys@mathparse{#1}\expandafter\pgfkeys@exp@call\expandafter{\pgfmathresult}} + + +\newif\ifpgfkeyssuccess +\pgfkeys{/handlers/.try/.code=\pgfkeys@try} +\pgfkeys{/handlers/.retry/.code=\ifpgfkeyssuccess\else\pgfkeys@try\fi} +\def\pgfkeys@try{% + \edef\pgfkeyscurrentkey{\pgfkeyscurrentpath}% + \ifx\pgfkeyscurrentvalue\pgfkeysnovalue@text% + \pgfkeysifdefined{\pgfkeyscurrentpath/.@def}% + {\pgfkeysgetvalue{\pgfkeyscurrentpath/.@def}{\pgfkeyscurrentvalue}} + {}% + \fi% + \pgfkeysifdefined{\pgfkeyscurrentpath/.@cmd}% + {% + \pgfkeysgetvalue{\pgfkeyscurrentpath/.@cmd}{\pgfkeys@code}% + \expandafter\pgfkeys@code\pgfkeyscurrentvalue\pgfeov% + \pgfkeyssuccesstrue% + }% + {% + \pgfkeysifdefined{\pgfkeyscurrentpath}% + {% + \ifx\pgfkeyscurrentvalue\pgfkeysnovalue@text% + \pgfkeysvalueof{\pgfkeyscurrentpath}% + \else% + \pgfkeyslet{\pgfkeyscurrentpath}\pgfkeyscurrentvalue% + \fi% + \pgfkeyssuccesstrue% + }% + {% + \pgfkeys@split@path% + \pgfkeysifdefined{/handlers/\pgfkeyscurrentname/.@cmd}{% + \pgfkeys@ifexecutehandler{% + \pgfkeysgetvalue{/handlers/\pgfkeyscurrentname/.@cmd}{\pgfkeys@code}% + \expandafter\pgfkeys@code\pgfkeyscurrentvalue\pgfeov + \pgfkeyssuccesstrue% + }{% + \pgfkeyssuccessfalse + }% + }{% + \pgfkeyssuccessfalse + }% + }% + }% +} + + +\pgfkeys{/utils/exec/.code={#1}} % + + + +\pgfkeys{ + /errors/boolean expected/.code 2 args=% + {% + \toks1={#1}% + \toks2={#2}% + \pgfkeys@error{% + Boolean parameter of key '\the\toks1' must be 'true' or 'false', not + '\the\toks2'. I am going to ignore it% + }% + }, + /errors/value required/.code 2 args=% + {% + \toks1={#1}% + \pgfkeys@error{% + The key '\the\toks1' requires a value. I am going to ignore this + key% + }% + }, + /errors/value forbidden/.code 2 args=% + {% + \toks1={#1}% + \toks2={#2}% + \pgfkeys@error{% + You may not specify a value for the key '\the\toks1'. I am going to ignore + the value '\the\toks2' that you provided% + }% + }, + /errors/unknown choice value/.code 2 args=% + {% + \toks1={#1}% + \toks2={#2}% + \pgfkeys@error{% + Choice '\the\toks2' unknown in choice key '\the\toks1'. I am + going to ignore this key% + }% + }, + /errors/unknown key/.code 2 args=% + {% + \toks1={#1}% + \toks2={#2}% + \def\pgf@temp{#2}% + \pgfkeys@error{% + I do not know the key '\the\toks1'\ifx\pgf@temp\pgfkeysnovalue@text\space\else, to which you passed + '\the\toks2', \fi and I am going to ignore it. Perhaps you + misspelled it% + }% + } +} + +\pgfkeys{/handlers/.unknown/.code=% + {% + \def\pgf@marshal{\pgfkeysvalueof{/errors/unknown key/.@cmd}}% + {\expandafter\expandafter\expandafter\pgf@marshal\expandafter\expandafter\expandafter{\expandafter\pgfkeyscurrentkey\expandafter}\expandafter{\pgfkeyscurrentvalue}\pgfeov}% + }% +} + +\pgfkeys{ + /handler config/.is choice, + /handler config/all/.code={% + \let\pgfkeys@case@three=\pgfkeys@case@three@handleall + \let\pgfkeys@ifexecutehandler=\pgfkeys@ifexecutehandler@handleall + }, + /handler config/only existing/.code={% + \let\pgfkeys@case@three=\pgfkeys@case@three@handle@restricted + \let\pgfkeys@ifexecutehandler=\pgfkeys@ifexecutehandler@handleonlyexisting + }, + /handler config/full or existing/.code={% + \let\pgfkeys@case@three=\pgfkeys@case@three@handle@restricted + \let\pgfkeys@ifexecutehandler=\pgfkeys@ifexecutehandler@handlefullorexisting + }, + /handler config/only existing/add exception/.code={\pgfkeysaddhandleonlyexistingexception{#1}}, +}% +\pgfkeysaddhandleonlyexistingexception{.cd}% +\pgfkeysaddhandleonlyexistingexception{.try}% +\pgfkeysaddhandleonlyexistingexception{.retry}% +\pgfkeysaddhandleonlyexistingexception{.lastretry}% +\pgfkeysaddhandleonlyexistingexception{.unknown}% +\pgfkeysaddhandleonlyexistingexception{.expand once}% +\pgfkeysaddhandleonlyexistingexception{.expand twice}% +\pgfkeysaddhandleonlyexistingexception{.expanded}% + + +\def\pgf@keys@gobble#1{} +\long\def\pgf@keys@utilifnextchar#1#2#3{% + \let\pgf@keys@utilreserved@d=#1% + \def\pgf@keys@utilreserved@a{#2}% + \def\pgf@keys@utilreserved@b{#3}% + \futurelet\pgf@keys@utillet@token\pgf@keys@utilifnch} +\def\pgf@keys@utilifnch{% + \ifx\pgf@keys@utillet@token\pgf@keys@utilsptoken + \let\pgf@keys@utilreserved@c\pgf@keys@utilxifnch + \else + \ifx\pgf@keys@utillet@token\pgf@keys@utilreserved@d + \let\pgf@keys@utilreserved@c\pgf@keys@utilreserved@a + \else + \let\pgf@keys@utilreserved@c\pgf@keys@utilreserved@b + \fi + \fi + \pgf@keys@utilreserved@c} +{% + \def\:{\global\let\pgf@keys@utilsptoken= } \: + \def\:{\pgf@keys@utilxifnch} \expandafter\gdef\: {\futurelet\pgf@keys@utillet@token\pgf@keys@utilifnch} +} + +\chardef\pgfkeys@inputcheck0 +\def\pgfkeys@IfFileExists#1{% + \openin\pgfkeys@inputcheck=#1 + \ifeof\pgfkeys@inputcheck + \expandafter\closein\expandafter\pgfkeys@inputcheck + \expandafter\pgfkeys@secondoftwo + \else + \expandafter\closein\expandafter\pgfkeys@inputcheck + \expandafter\pgfkeys@firstoftwo + \fi +} + +\def\usepgfkeyslibrary{\pgf@keys@utilifnextchar[{\pgfkeys@@uselibrary}{\pgfkeys@uselibrary}} +\def\pgfkeys@@uselibrary[#1]{\pgfkeys@uselibrary{#1}} +\def\pgfkeys@uselibrary#1{\pgfkeys@uselibrary@parse#1,\pgfkeys@mainstop} +\def\pgfkeys@uselibrary@parse{% + \pgf@keys@utilifnextchar\pgfkeys@mainstop{\pgfkeys@cleanup}{\pgfkeys@uselibrary@normal}% +}% +\def\pgfkeys@uselibrary@normal#1,{% + \pgfkeys@uselibrary@load{#1}% + \pgfkeys@uselibrary@parse +} +\def\pgfkeys@uselibrary@load#1{% + \expandafter\pgfkeys@spdef\expandafter\pgf@temp\expandafter{#1}% + \ifx\pgf@temp\pgfkeys@empty\expandafter\pgfkeys@firstoftwo\else\expandafter\pgfkeys@secondoftwo\fi{}{% + \pgfkeys@ifcsname{pgfkeys@library@\pgf@temp @loaded}{}{% + \expandafter\let\csname pgfkeys@library@\pgf@temp @loaded\endcsname=\pgfkeys@empty + \expandafter\edef\csname pgfkeys@library@#1@atcode\endcsname{\the\catcode`\@}% + \expandafter\edef\csname pgfkeys@library@#1@barcode\endcsname{\the\catcode`\|}% + \expandafter\edef\csname pgfkeys@library@#1@dollarcode\endcsname{\the\catcode`\$}% + \catcode`\@=11 + \catcode`\|=12 + \catcode`\$=3 + \pgfkeys@IfFileExists{pgfkeyslibrary\pgf@temp.code.tex}{% + \input pgfkeyslibrary\pgf@temp.code.tex + }{% + \pgfkeys@error{I did not find the pgfkeys library '\pgf@temp'. I looked for the + file named pgfkeyslibrary\pgf@temp.code.tex, but could not find it in the + current texmf trees.}% + }% + \catcode`\@=\csname pgfkeys@library@#1@atcode\endcsname + \catcode`\|=\csname pgfkeys@library@#1@barcode\endcsname + \catcode`\$=\csname pgfkeys@library@#1@dollarcode\endcsname + }% + }% +} + +\let\pgfkeys@library@filtered@loaded\pgfkeys@empty +\input pgfkeyslibraryfiltered.code.tex + +\endinput diff --git a/preprints/A-polynomial-time-algorithm-for-three-machine-unit-job-scheduling-September-24-2026/build/tex/runtime/generic/pgf/utilities/pgfkeyslibraryfiltered.code.tex b/preprints/A-polynomial-time-algorithm-for-three-machine-unit-job-scheduling-September-24-2026/build/tex/runtime/generic/pgf/utilities/pgfkeyslibraryfiltered.code.tex new file mode 100644 index 0000000000000000000000000000000000000000..e6c357b885922534418600735ae7030100671e81 --- /dev/null +++ b/preprints/A-polynomial-time-algorithm-for-three-machine-unit-job-scheduling-September-24-2026/build/tex/runtime/generic/pgf/utilities/pgfkeyslibraryfiltered.code.tex @@ -0,0 +1,571 @@ + + + +\let\pgfkeys@orig@case@one=\pgfkeys@case@one +\let\pgfkeys@orig@@set=\pgfkeys@@set +\let\pgfkeys@orig@@qset=\pgfkeys@@qset +\let\pgfkeys@orig@try=\pgfkeys@try +\let\pgfkeys@orig@unknown=\pgfkeys@unknown + +\newif\ifpgfkeysfilteringisactive +\newif\ifpgfkeysfiltercontinue +\let\pgfkeys@key@predicate=\pgfkeys@empty +\let\pgfkeys@filtered@handler=\pgfkeys@empty +\newtoks\pgfkeys@tmptoks + +\def\pgfkeysfiltered{% + \expandafter\pgfkeysfiltered@@install\expandafter{\pgfkeysdefaultpath}% +} + +\long\def\pgfkeysfiltered@@install#1#2{% + \pgfkeys@install@filter@and@invoke{% + \let\pgfkeysdefaultpath\pgfkeys@root% + \pgfkeys@parse#2,\pgfkeys@mainstop% + \def\pgfkeysdefaultpath{#1}% + }% +} + +\long\def\pgfkeysalsofrom#1{% + \expandafter\pgfkeysalso\expandafter{#1}% +} + +\long\def\pgfkeysalsofilteredfrom#1{% + \expandafter\pgfkeysalsofiltered\expandafter{#1}% +} + +\long\def\pgfkeysalsofiltered#1{% + \pgfkeys@install@filter@and@invoke{\pgfkeysalso{#1}}% +}% + +\long\def\pgfqkeysfiltered#1{% + \expandafter\pgfqkeysfiltered@@install\expandafter{\pgfkeysdefaultpath}{#1}% +} + +\long\def\pgfqkeysfiltered@@install#1#2#3{% + \pgfkeys@install@filter@and@invoke{% + \def\pgfkeysdefaultpath{#2/}\pgfkeys@parse#3,\pgfkeys@mainstop\def\pgfkeysdefaultpath{#1}% + }% +} + + + +\def\pgfkeysactivatefamily#1{% + \pgfkeysiffamilydefined + {#1}% + {\csname pgfk@#1/familyactivetrue\endcsname}% + {\pgfkeysvalueof{/errors/family unknown/.@cmd}{#1}\pgfeov}% +} + +\def\pgfkeysdeactivatefamily#1{% + \pgfkeysiffamilydefined + {#1}% + {\csname pgfk@#1/familyactivefalse\endcsname}% + {\pgfkeysvalueof{/errors/family unknown/.@cmd}{#1}\pgfeov}% +} + +\def\pgfkeysactivatefamilies#1#2{% + \pgfkeyssavekeyfilterstateto\pgfkeys@cur@state + \expandafter\pgfkeysactivatefamilies@impl\expandafter{\pgfkeys@cur@state}{#1}{#2}% +} +\def\pgfkeysactivatefamilies@impl#1#2#3{% + \pgfkeysinstallkeyfilter{/pgf/key filters/false}{}% + \let#3=\pgfkeys@empty% + \def\pgfkeys@filtered@handler{\pgfkeys@family@activate@handler{#3}}% + \pgfkeysalsofiltered{#2}% + #1% +} + +\def\pgfkeys@family@activate@handler#1{% + \pgfkeysactivatefamily{\pgfkeyscurrentkey}% + \pgfkeys@tmptoks=\expandafter\expandafter\expandafter{\expandafter#1\expandafter\pgfkeysdeactivatefamily\expandafter{\pgfkeyscurrentkey}}% + \edef#1{\the\pgfkeys@tmptoks}% +} + +\long\def\pgfkeysiffamilydefined#1#2#3{\ifcsname ifpgfk@#1/familyactive\endcsname#2\else#3\fi} + +\def\pgfkeysisfamilyactive#1{% + \pgfkeysiffamilydefined{#1}{% + \expandafter\let\expandafter\ifpgfkeysfiltercontinue\csname ifpgfk@#1/familyactive\endcsname + }{% + \pgfkeysvalueof{/errors/family unknown/.@cmd}{#1}\pgfeov% + \expandafter\expandafter\expandafter\let\csname ifpgfkeysfiltercontinue\endcsname\csname iffalse\endcsname + }% +}% + +\def\pgfkeysgetfamily#1#2{% + \pgfkeysifdefined{#1/family}{\pgfkeysgetvalue{#1/family}{#2}\pgfkeyssuccesstrue}{\pgfkeyssuccessfalse}% +} + +\def\pgfkeyssetfamily#1#2{% + \pgfkeysiffamilydefined{#2}{% + \pgfkeyssetvalue{#1/family}{#2}% + }{% + \pgfkeysalso{/errors/family unknown={#2}}% + }% +}% + + +\long\def\pgfkeys@cur@is@descendant@of@errors{% + \expandafter\pgfkeys@cur@is@descendant@of@errors@impl\pgfkeyscurrentkey/errors\pgf@@eov +}% +\long\def\pgfkeys@cur@is@descendant@of@errors@impl#1/errors#2\pgf@@eov{% + \def\pgfkeyspred@TMP{#1}% + \ifx\pgfkeyspred@TMP\pgfkeys@empty + \pgfkeysfiltercontinuetrue + \else + \pgfkeysfiltercontinuefalse + \fi +}% + +\def\pgfkeysinterruptkeyfilter{% + \ifpgfkeysfilteringisactive + \let\pgfkeys@case@one=\pgfkeys@orig@case@one + \let\pgfkeys@try=\pgfkeys@orig@try + \let\pgfkeys@unknown=\pgfkeys@orig@unknown + \fi +} + +\def\endpgfkeysinterruptkeyfilter{% + \ifpgfkeysfilteringisactive + \let\pgfkeys@case@one=\pgfkeys@case@one@filtered + \let\pgfkeys@try=\pgfkeys@try@filtered + \let\pgfkeys@unknown=\pgfkeys@unknown@filtered + \fi +} + +\def\pgfkeysactivatefamiliesandfilteroptions#1#2{% + \pgfkeysactivatefamilies{#1}{\pgfkeys@family@deactivation}% + \pgfkeysfiltered{#2}% + \pgfkeys@family@deactivation +} + +\def\pgfqkeysactivatefamiliesandfilteroptions#1#2#3{% + \pgfkeysactivatefamilies{#1}{\pgfkeys@family@deactivation}% + \pgfqkeysfiltered{#2}{#3}% + \pgfkeys@family@deactivation +} + +\def\pgfkeyssplitpath{\pgfkeys@split@path}% + + +\def\pgfkeysactivatesinglefamilyandfilteroptions#1#2{% + \pgfkeysactivatefamily{#1}% + \pgfkeysfiltered{#2}% + \pgfkeysdeactivatefamily{#1}% +} + +\def\pgfqkeysactivatesinglefamilyandfilteroptions#1#2#3{% + \pgfkeysactivatefamily{#1}% + \pgfqkeysfiltered{#2}{#3}% + \pgfkeysdeactivatefamily{#1}% +} + +\def\pgfkeysinstallkeyfilter#1#2{% + \pgfkeysifdefined{#1/.@cmd}{% + \edef\pgfkeyscurrentkeyfilter{#1}% + \def\pgfkeyscurrentkeyfilterargs{#2}% + \pgfkeysgetvalue{#1/.@cmd}{\pgfkeys@key@predicate@}% + \def\pgfkeys@key@predicate{\pgfkeys@key@predicate@#2\pgfeov}% + }{% + \pgfkeysvalueof{/errors/no such key filter/.@cmd}{#1}{#2}\pgfeov% + }% +} + +\def\pgfkeysinstallkeyfilterhandler#1#2{% + \pgfkeysifdefined{#1/.@cmd}{% + \edef\pgfkeyscurrentkeyfilterhandler{#1}% + \def\pgfkeyscurrentkeyfilterhandlerargs{#2}% + \pgfkeysgetvalue{#1/.@cmd}{\pgfkeys@filtered@handler@}% + \def\pgfkeys@filtered@handler{\pgfkeys@filtered@handler@#2\pgfeov}% + }{% + \pgfkeysvalueof{/errors/no such key filter handler/.@cmd}{#1}{#2}\pgfeov% + }% +} + +\def\pgfkeyssavekeyfilterstateto#1{% + \pgfkeys@tmptoks={\pgfkeysinstallkeyfilter}% + \pgfkeys@tmptoks=\expandafter\expandafter\expandafter{\expandafter\the\expandafter\pgfkeys@tmptoks\expandafter{\pgfkeyscurrentkeyfilter}}% + \pgfkeys@tmptoks=\expandafter\expandafter\expandafter{\expandafter\the\expandafter\pgfkeys@tmptoks\expandafter{\pgfkeyscurrentkeyfilterargs}\pgfkeysinstallkeyfilterhandler}% + \pgfkeys@tmptoks=\expandafter\expandafter\expandafter{\expandafter\the\expandafter\pgfkeys@tmptoks\expandafter{\pgfkeyscurrentkeyfilterhandler}}% + \pgfkeys@tmptoks=\expandafter\expandafter\expandafter{\expandafter\the\expandafter\pgfkeys@tmptoks\expandafter{\pgfkeyscurrentkeyfilterhandlerargs}}% + \edef#1{% + \the\pgfkeys@tmptoks + }% +} + + +\pgfkeys{% + /errors/family unknown/.code=\pgfkeys@error{% + Sorry, I do not know family '#1' and can't work with any assoicated family handling. Perhaps you misspelled it?}, + /errors/no such key filter/.code 2 args=\pgfkeys@error{Sorry, there is no such key filter '#1'.}, + /errors/no such key filter handler/.code 2 args=\pgfkeys@error{Sorry, there is no such key filter handler '#1'.}, + /handlers/.is family/.append code={% + \edef\pgfkeyspred@TMP{pgfk@\pgfkeyscurrentpath/familyactive}% + \expandafter\pgfkeys@non@outer@newif\expandafter{\pgfkeyspred@TMP}% + \edef\pgfkeyspred@TMP{\pgfkeyscurrentpath/.belongs to family=\pgfkeyscurrentpath}% + \expandafter\pgfkeysalso\expandafter{\pgfkeyspred@TMP}% + },% + /handlers/.activate family/.code=\pgfkeysactivatefamily{\pgfkeyscurrentpath}, + /handlers/.deactivate family/.code=\pgfkeysdeactivatefamily{\pgfkeyscurrentpath}, + /handlers/.belongs to family/.code={\pgfkeyssetfamily{\pgfkeyscurrentpath}{#1}},% + /handlers/.lastretry/.code={% + \ifpgfkeyssuccess\else + \pgfkeys@try + \ifpgfkeyssuccess\else + \pgfkeys@split@path% + \pgfkeys@unknown + \fi + \fi + }, + /handlers/.install key filter/.code={% + \pgfkeysinstallkeyfilter{\pgfkeyscurrentpath}{#1}% + },% + /handlers/.install key filter handler/.code={% + \pgfkeysinstallkeyfilterhandler{\pgfkeyscurrentpath}{#1}% + },% + /pgf/key filter handlers/append filtered to/.code={% + \pgfkeys@tmptoks=\expandafter\expandafter\expandafter{\expandafter\pgfkeyscurrentkeyRAW\expandafter=\expandafter{\pgfkeyscurrentvalue}}% + \ifx#1\pgfkeys@empty + \else + \pgfkeys@tmptoks=\expandafter\expandafter\expandafter{\expandafter#1\expandafter,\the\pgfkeys@tmptoks}% + \fi + \edef#1{\the\pgfkeys@tmptoks}% + },% + /pgf/key filter handlers/ignore/.code={}, + /pgf/key filter handlers/ignore/.install key filter handler, + /pgf/key filter handlers/log/.code={% + \pgf@typeout{LOG: the option '\pgfkeyscurrentkey' (was originally '\pgfkeyscurrentkeyRAW') (case \pgfkeyscasenumber) has not been processed due to pgfkeysfiltered.}% + }, + /pgf/key filters/active families/.code={% + \if\pgfkeyscasenumber0% + \pgfkeysfiltercontinuetrue + \else + \if\pgfkeyscasenumber3% + \pgfkeysgetfamily\pgfkeyscurrentpath\pgfkeyspred@TMP + \else + \pgfkeysgetfamily\pgfkeyscurrentkey\pgfkeyspred@TMP + \fi + \ifpgfkeyssuccess + \pgfkeysisfamilyactive{\pgfkeyspred@TMP}% + \else% + \pgfkeysfiltercontinuefalse + \fi + \fi + },% + /pgf/key filters/active families or no family/.code 2 args={% + \if\pgfkeyscasenumber0% + \pgfkeysevalkeyfilterwith{#2}% + \else + \if\pgfkeyscasenumber3% + \pgfkeysgetfamily\pgfkeyscurrentpath\pgfkeyspred@TMP + \else + \pgfkeysgetfamily\pgfkeyscurrentkey\pgfkeyspred@TMP + \fi + \ifpgfkeyssuccess + \pgfkeysisfamilyactive{\pgfkeyspred@TMP}% + \else% + \pgfkeysevalkeyfilterwith{#1}% + \fi + \fi + }, + /pgf/key filters/active families or no family DEBUG/.code 2 args={% + \if\pgfkeyscasenumber0% + \pgf@typeout{[pgfkeyshasactivefamilyornofamily(\pgfkeyscurrentkey, \pgfkeyscasenumber) invoking unknown handler '#2']}% + \pgfkeysevalkeyfilterwith{#2}% + \else + \if\pgfkeyscasenumber3% + \pgfkeysgetfamily\pgfkeyscurrentpath\pgfkeyspred@TMP + \else + \pgfkeysgetfamily\pgfkeyscurrentkey\pgfkeyspred@TMP + \fi + \ifpgfkeyssuccess + \pgfkeysisfamilyactive{\pgfkeyspred@TMP}% + \ifpgfkeysfiltercontinue + \pgf@typeout{[pgfkeyshasactivefamilyornofamily(\pgfkeyscurrentkey, \pgfkeyscasenumber) family is ACTIVE]}% + \else + \pgf@typeout{[pgfkeyshasactivefamilyornofamily(\pgfkeyscurrentkey, \pgfkeyscasenumber) family is NOT active.]}% + \fi + \else% + \pgf@typeout{[pgfkeyshasactivefamilyornofamily(\pgfkeyscurrentkey, \pgfkeyscasenumber) invoking has-no-family-handler '#1']}% + \pgfkeysevalkeyfilterwith{#1}% + \fi + \fi + }, + /pgf/key filters/active families and known/.code={% + \if\pgfkeyscasenumber0% + \pgfkeysfiltercontinuefalse + \else + \if\pgfkeyscasenumber3% + \pgfkeysgetfamily\pgfkeyscurrentpath\pgfkeyspred@TMP + \else + \pgfkeysgetfamily\pgfkeyscurrentkey\pgfkeyspred@TMP + \fi + \ifpgfkeyssuccess + \pgfkeysisfamilyactive{\pgfkeyspred@TMP}% + \else% + \pgfkeysfiltercontinuefalse + \fi + \fi + }, + /pgf/key filters/active families or descendants of/.code={% + \if\pgfkeyscasenumber0% + \pgfkeysfiltercontinuefalse + \else + \if\pgfkeyscasenumber3% + \pgfkeysgetfamily\pgfkeyscurrentpath\pgfkeyspred@TMP + \else + \pgfkeysgetfamily\pgfkeyscurrentkey\pgfkeyspred@TMP + \fi + \ifpgfkeyssuccess + \pgfkeysisfamilyactive{\pgfkeyspred@TMP}% + \else% + \def\pgfkeysisdescendantof@impl##1#1##2\pgf@@eov{% + \def\pgfkeyspred@TMP{##1}% + \ifx\pgfkeyspred@TMP\pgfkeys@empty + \pgfkeysfiltercontinuetrue + \else + \pgfkeysfiltercontinuefalse + \fi + }% + \expandafter\pgfkeysisdescendantof@impl\pgfkeyscurrentkey#1\pgf@@eov + \fi + \fi + }, + /pgf/key filters/is descendant of/.code={% + \if\pgfkeyscasenumber0% + \pgfkeysfiltercontinuetrue + \else + \def\pgfkeysisdescendantof@impl##1#1##2\pgf@@eov{% + \def\pgfkeyspred@TMP{##1}% + \ifx\pgfkeyspred@TMP\pgfkeys@empty + \pgfkeysfiltercontinuetrue + \else + \pgfkeysfiltercontinuefalse + \fi + }% + \expandafter\pgfkeysisdescendantof@impl\pgfkeyscurrentkey#1\pgf@@eov + \fi + },% + /pgf/key filters/equals/.code={% + \if\pgfkeyscasenumber0% + \pgfkeysfiltercontinuetrue + \else + \def\pgfkeyspred@TMP{#1}% + \ifx\pgfkeyscurrentkey\pgfkeyspred@TMP + \pgfkeysfiltercontinuetrue + \else + \pgfkeysfiltercontinuefalse + \fi + \fi + },% + /pgf/key filters/not/.code={% + \pgfkeysevalkeyfilterwith{#1}% + \ifpgfkeysfiltercontinue + \pgfkeysfiltercontinuefalse + \else + \pgfkeysfiltercontinuetrue + \fi + },% + /pgf/key filters/and/.code 2 args={% + \pgfkeysevalkeyfilterwith{#1}% + \ifpgfkeysfiltercontinue + \pgfkeysevalkeyfilterwith{#2}% + \fi + },% + /pgf/key filters/or/.code 2 args={% + \pgfkeysevalkeyfilterwith{#1}% + \ifpgfkeysfiltercontinue + \else + \pgfkeysevalkeyfilterwith{#2}% + \fi + },% + /pgf/key filters/true/.code={\pgfkeysfiltercontinuetrue},% + /pgf/key filters/true/.install key filter, + /pgf/key filters/false/.code={% + \pgfkeysfiltercontinuefalse + },% + /pgf/key filters/defined/.code={% + \if\pgfkeyscasenumber0% + \pgfkeysfiltercontinuefalse + \else + \pgfkeysfiltercontinuetrue + \fi + }, +}% + + + +\def\pgfkeys@case@one@filtered{% + \pgfkeys@cur@is@descendant@of@errors + \ifpgfkeysfiltercontinue + \pgfkeys@orig@case@one + \else + \pgfkeysfiltercontinuetrue + \pgfkeysifdefined{\pgfkeyscurrentkey/.@cmd}{% + \def\pgfkeyscasenumber{1}% + \pgfkeys@key@predicate% + \ifpgfkeysfiltercontinue + \pgfkeysgetvalue{\pgfkeyscurrentkey/.@cmd}{\pgfkeys@code}% + \expandafter\pgfkeys@code\pgfkeyscurrentvalue\pgfeov + \else + \pgfkeys@filtered@handler% + \fi + }{% + \pgfkeysifdefined{\pgfkeyscurrentkey}{% + \def\pgfkeyscasenumber{2}% + \pgfkeys@key@predicate% + \ifpgfkeysfiltercontinue + \pgfkeys@case@two@extern + \else + \pgfkeys@filtered@handler% + \fi + }{% + \pgfkeys@split@path + \pgfkeysifdefined{/handlers/\pgfkeyscurrentname/.@cmd}{% + \pgfkeys@ifexecutehandler{% + \def\pgfkeyscasenumber{3}% + \pgfkeys@key@predicate% + \ifpgfkeysfiltercontinue + \pgfkeysgetvalue{/handlers/\pgfkeyscurrentname/.@cmd}{\pgfkeys@code}% + \expandafter\pgfkeys@code\pgfkeyscurrentvalue\pgfeov + \else + \pgfkeys@filtered@handler% + \fi + }{% + \pgfkeys@unknown + }% + }{% + \pgfkeys@unknown + }% + }% + }% + \fi +}% + +\def\pgfkeys@unknown@filtered{% + \def\pgfkeyscasenumber{0}% + \pgfkeys@key@predicate% + \ifpgfkeysfiltercontinue + \pgfkeys@orig@unknown + \else + \pgfkeys@filtered@handler% + \fi +} + +\def\pgfkeys@try@filtered{% + \ifpgfkeysfiltercontinue + \pgfkeys@orig@try + \else + \pgfkeysfiltercontinuetrue + \edef\pgfkeyscurrentkey{\pgfkeyscurrentpath}% + \ifx\pgfkeyscurrentvalue\pgfkeysnovalue@text% + \pgfkeysifdefined{\pgfkeyscurrentpath/.@def}% + {\pgfkeysgetvalue{\pgfkeyscurrentpath/.@def}{\pgfkeyscurrentvalue}} + {}% + \fi% + \pgfkeysifdefined{\pgfkeyscurrentpath/.@cmd}% + {% + \def\pgfkeyscasenumber{1}% + \pgfkeys@key@predicate% + \ifpgfkeysfiltercontinue + \pgfkeysgetvalue{\pgfkeyscurrentkey/.@cmd}{\pgfkeys@code}% + \expandafter\pgfkeys@code\pgfkeyscurrentvalue\pgfeov% + \else + \pgfkeys@filtered@handler% + \fi + \pgfkeyssuccesstrue% + }% + {% + \pgfkeysifdefined{\pgfkeyscurrentpath}% + {% + \def\pgfkeyscasenumber{2}% + \pgfkeys@key@predicate% + \ifpgfkeysfiltercontinue + \ifx\pgfkeyscurrentvalue\pgfkeysnovalue@text% + \pgfkeysvalueof{\pgfkeyscurrentpath}% + \else% + \pgfkeyslet{\pgfkeyscurrentpath}\pgfkeyscurrentvalue% + \fi% + \else + \pgfkeys@filtered@handler% + \fi + \pgfkeyssuccesstrue% + }% + {% + \pgfkeys@split@path% + \pgfkeysifdefined{/handlers/\pgfkeyscurrentname/.@cmd}{% + \pgfkeys@ifexecutehandler{% + \def\pgfkeyscasenumber{3}% + \pgfkeys@key@predicate% + \ifpgfkeysfiltercontinue + \pgfkeysgetvalue{/handlers/\pgfkeyscurrentname/.@cmd}{\pgfkeys@code}% + \expandafter\pgfkeys@code\pgfkeyscurrentvalue\pgfeov + \else + \pgfkeys@filtered@handler% + \fi + \pgfkeyssuccesstrue% + }{% + \pgfkeyssuccessfalse + }% + }{% + \pgfkeyssuccessfalse + }% + }% + }% + \fi +} + + +\long\def\pgfkeys@install@filter@and@invoke#1{% + \ifpgfkeysfilteringisactive + \pgfkeys@error{Sorry, nested calls to key filtering routines are not allowed. (reason: It is not possible to properly restore the previous filtering state after returning from the nested call)}% + \fi + \pgfkeysfilteringisactivetrue + \let\pgfkeys@case@one=\pgfkeys@case@one@filtered + \let\pgfkeys@try=\pgfkeys@try@filtered + \let\pgfkeys@unknown=\pgfkeys@unknown@filtered + #1% + \let\pgfkeys@case@one=\pgfkeys@orig@case@one + \let\pgfkeys@try=\pgfkeys@orig@try + \let\pgfkeys@unknown=\pgfkeys@orig@unknown + \pgfkeysfilteringisactivefalse +} + + + + +\long\def\pgfkeysevalkeyfilterwith#1{% + \pgfkeys@eval@key@filter@subroutine@unpack#1=\pgfkeysnovalue=\pgfkeys@stop +}% +\long\def\pgfkeys@eval@key@filter@subroutine@unpack#1=#2=#3\pgfkeys@stop{% + \pgfkeys@spdef\pgfkeyspred@TMP{#1}% + \edef\pgfkeyspred@TMP{\pgfkeyspred@TMP}% + \pgfkeys@spdef\pgfkeyspred@TMPB{#2}% + \ifx\pgfkeyspred@TMPB\pgfkeysnovalue@text% + \pgfkeysifdefined{\pgfkeyspred@TMP/.@def}% + {\pgfkeysgetvalue{\pgfkeyspred@TMP/.@def}{\pgfkeyspred@TMPB}} + {}% + \fi% + \ifx\pgfkeyspred@TMPB\pgfkeysvaluerequired% + \pgfkeysvalueof{/errors/value required/.@cmd}\pgfkeyspred@TMP\pgfkeyspred@TMPB\pgfeov% + \else% + \pgfkeysifdefined{\pgfkeyspred@TMP/.@cmd}{% + \pgfkeysgetvalue{\pgfkeyspred@TMP/.@cmd}{\pgfkeys@code}% + \expandafter\pgfkeys@code\pgfkeyspred@TMPB\pgfeov + }{% + \pgfkeysvalueof{/errors/no such key filter/.@cmd}\pgfkeyspred@TMP\pgfkeyspred@TMPB\pgfeov% + }% + \fi% +} + + +\def\pgfkeys@non@outer@newif@#1#2{% + \expandafter\edef\csname #2true\endcsname{\noexpand\let\noexpand#1=\noexpand\iftrue}% + \expandafter\edef\csname #2false\endcsname{\noexpand\let\noexpand#1=\noexpand\iffalse}% + \csname #2false\endcsname +}% + +\def\pgfkeys@non@outer@newif#1{% + \expandafter\pgfkeys@non@outer@newif@\csname if#1\endcsname{#1}% +} +\endinput diff --git a/preprints/A-polynomial-time-algorithm-for-three-machine-unit-job-scheduling-September-24-2026/build/tex/runtime/generic/pgf/utilities/pgfrcs.code.tex b/preprints/A-polynomial-time-algorithm-for-three-machine-unit-job-scheduling-September-24-2026/build/tex/runtime/generic/pgf/utilities/pgfrcs.code.tex new file mode 100644 index 0000000000000000000000000000000000000000..f4a390b828c2f79eecd0b9ce6f5a0636fe4f3700 --- /dev/null +++ b/preprints/A-polynomial-time-algorithm-for-three-machine-unit-job-scheduling-September-24-2026/build/tex/runtime/generic/pgf/utilities/pgfrcs.code.tex @@ -0,0 +1,121 @@ + +\ifx\eTeXrevision\undefined% + \pgfutil@packageerror{PGF}{PGF requires etex in extended mode}{}% + \csname @@end\expandafter\endcsname\expandafter\end% +\fi + +\ifnum\eTeXversion<2 + \pgfutil@packageerror{PGF}{PGF requires etex version 2}{}% + \csname @@end\expandafter\endcsname\expandafter\end% +\fi + + +\pgfutil@IfFileExists{pgf.revision.tex}{\input pgf.revision.tex } {% + \def\pgfrevision{0.0}% + \def\pgfversion{0.0}% + \def\pgfversiondate{2014-07-01}% + \def\pgfrevisiondate{2014-07-01}% +} + +\begingroup +\catcode`\"=12 +\pgfutil@IfUndefined{directlua}{}{% + \directlua{pgf = {}; pgf.pgfversion = "\pgfversion"}% +}% +\endgroup + +\let\pgftypesetversion=\pgfrevision + + +\ifx\pgfrcsloaded\undefined +\def\pgfrcsloaded{} + +\edef\pgfrcsatcode{\the\catcode`\@} +\catcode`\@=11 + +\def\ProvidesFileRCS{% + \let\pgfrcs@final=\ProvidesFileRCS@ + \pgfrcs@parseinput +} +\def\ProvidesFileRCS@#1{% + \ProvidesFile{#1}[\pgfversiondate\space v\pgfversion\space(\pgfrevision)]% +} + +\def\pgf@remove@ext#1.#2\relax{% + \def\pgfretval{#1}% +}% + +\def\ProvidesPackageRCS{% + \let\pgfrcs@final=\ProvidesPackageRCS@ + \pgfrcs@parseinput +} +\def\ProvidesPackageRCS@#1{% + \pgf@remove@ext#1.\relax + \edef\pgfrcs@marshal{{\pgfretval}[\pgfversiondate\space v\pgfversion\space(\pgfrevision)]}% + \expandafter\ProvidesPackage\pgfrcs@marshal +} + +\def\ProvidesClassRCS{% + \let\pgfrcs@final=\ProvidesClassRCS@ + \pgfrcs@parseinput +} +\def\ProvidesClassRCS@#1{% + \pgf@remove@ext#1.\relax + \edef\pgfrcs@marshal{{\pgfretval}[\pgfversiondate\space v\pgfversion\space(\pgfrevision)]}% + \expandafter\ProvidesClass\pgfrcs@marshal +} + +\def\pgfrcs@parseinput@bracket{[}% +\def\pgfrcs@parseinput@dollarsign{$}% + +\def\pgfrcs@parseinput#1{% + \def\pgfrcs@parseinput@{#1}% + \ifx\pgfrcs@parseinput@\pgfrcs@parseinput@bracket% + \let\next=\pgfrcs@parseinput@opt + \else + \ifx\pgfrcs@parseinput@\pgfrcs@parseinput@dollarsign + \let\next\pgfrcs@parseinput@dollar + \else + \def\next{\pgfrcs@final{#1}}% + \fi + \fi + \next +} +\def\pgfrcs@parseinput@opt#1] ${% + \pgfrcs@parseinput@dollar +}% +\def\pgfrcs@parseinput@dollar#1${% + \pgf@parseid $#1$% + \edef\pgfrcs@parseinput@{\pgf@rcsfile.\pgf@rcssuffix}% + \expandafter\pgfrcs@final\expandafter{\pgfrcs@parseinput@}% +}% +\def\pgf@parseid $#1: #2.#3,v #4 #5/#6/#7 #8${% + \pgf@parsercsfile$#2/$ + \def\pgf@rcssuffix{#3} + \def\pgf@rcsrevision{#4} + \def\pgf@rcsdate{#5/#6/#7} +} +\def\pgf@parsercsfile$#1/#2${ + \def\pgf@temp{#2} + \ifx\pgf@temp\pgfutil@empty + \def\pgf@rcsfile{#1} + \else + \pgf@parsercsfile$#2$ + \fi} + + +\ifx\ProvidesPackage\@undefined + \def\ProvidesPackage#1[#2]{\wlog{Loading package #1 version #2.}} +\fi + +\ifx\ProvidesFile\@undefined + \def\ProvidesFile#1[#2]{\wlog{Loading file #1 version #2.}} +\fi + +\ProvidesPackageRCS{pgfrcs.code.tex} + +\catcode`\@=\pgfrcsatcode + +\fi + + diff --git a/preprints/A-polynomial-time-algorithm-for-three-machine-unit-job-scheduling-September-24-2026/build/tex/runtime/generic/pgf/utilities/pgfutil-common.tex b/preprints/A-polynomial-time-algorithm-for-three-machine-unit-job-scheduling-September-24-2026/build/tex/runtime/generic/pgf/utilities/pgfutil-common.tex new file mode 100644 index 0000000000000000000000000000000000000000..4ed7e7b5016d6446913e213b2cc86b64c0510a0d --- /dev/null +++ b/preprints/A-polynomial-time-algorithm-for-three-machine-unit-job-scheduling-September-24-2026/build/tex/runtime/generic/pgf/utilities/pgfutil-common.tex @@ -0,0 +1,623 @@ + + + + + +\catcode`\@=11\relax + + +\newif\ifpgfutil@format@is@latex +\newif\ifpgfutil@format@is@plain +\newif\ifpgfutil@format@is@context + + +\let\pgfutil@exp\romannumeral +\chardef\pgfutil@exp@end=0 + + +\catcode`\Q=3 +\def\pgfutil@trimspaces#1{% + \romannumeral-`0\pgfutil@trimspaces@\noexpand#1Q Q} +\def\pgfutil@trimspaces@#1 Q{\pgfutil@trimspaces@@#1Q} +\def\pgfutil@trimspaces@@#1Q#2{#1} +\catcode`\Q=11 + +\long\def\pgfutil@ifx#1#2{% + \ifx#1#2% + \expandafter\pgfutil@firstoftwo + \else + \expandafter\pgfutil@secondoftwo + \fi} + +\long\def\pgfutil@ifstrequal#1#2{% + \ifnum\pgfsys@strcmp{\pgfutil@unexpanded{#1}}{\pgfutil@unexpanded{#2}}=0 + \expandafter\pgfutil@firstoftwo + \else + \expandafter\pgfutil@secondoftwo + \fi +} + +\def\pgfutil@ifxempty{\pgfutil@ifx\pgfutil@empty} + +\long\def\pgfutil@ifempty#1{% + \if\relax\detokenize{#1}\relax + \expandafter\pgfutil@firstoftwo + \else + \expandafter\pgfutil@secondoftwo + \fi} + +\def\pgfutil@ifundefined#1{% + \expandafter\pgfutil@ifx\csname#1\endcsname\relax} + +\def\pgfutil@IfUndefined#1{% + \ifcsname#1\endcsname + \expandafter\pgfutil@secondoftwo + \else + \expandafter\pgfutil@firstoftwo + \fi +} + +\long\def\pgfutil@firstofone#1{#1} +\long\def\pgfutil@firstoftwo#1#2{#1} +\long\def\pgfutil@secondoftwo#1#2{#2} +\def\pgfutil@empty{} +\long\def\pgfutil@gobble@until@relax#1\relax{} +\long\def\pgfutil@gobble#1{} +\long\def\pgfutil@gobbletwo#1#2{} +\def\pgfutil@namedef#1{\expandafter\def\csname #1\endcsname} +\def\pgfutil@namelet#1{\expandafter\pgfutil@@namelet\csname#1\endcsname} +\def\pgfutil@@namelet#1#2{\expandafter\let\expandafter#1\csname#2\endcsname} +\long\def\pgfutil@g@addto@macro#1#2{% + \begingroup + \pgfutil@toks@\expandafter{#1#2}% + \xdef#1{\the\pgfutil@toks@}% + \endgroup} +\newif\ifpgfutil@tempswa +\newif\ifpgfutil@tempswb + + +\let\pgfutil@pushmacro@string\pgfutil@empty + +\def\pgfutil@pushmacro#1{% + \xdef\pgfutil@pushmacro@string{\string#1}% + \ifcsname pgfutil@pushedmacro@\pgfutil@pushmacro@string\endcsname\else + \csname newcount\expandafter\endcsname\csname pgfutil@pushedmacro@\pgfutil@pushmacro@string\endcsname + \fi + \global\advance\csname pgfutil@pushedmacro@\pgfutil@pushmacro@string\endcsname 1\relax + \global\expandafter\let\csname\the\csname pgfutil@pushedmacro@\pgfutil@pushmacro@string\endcsname\pgfutil@pushmacro@string\endcsname#1% +} + +\def\pgfutil@popmacro#1{% + \xdef\pgfutil@pushmacro@string{\string#1}% + \expandafter\let\expandafter#1\csname\the\csname pgfutil@pushedmacro@\pgfutil@pushmacro@string\endcsname\pgfutil@pushmacro@string\endcsname + \global\advance\csname pgfutil@pushedmacro@\pgfutil@pushmacro@string\endcsname -1\relax +} + + +\long\def\pgfutil@ifnextchar#1#2#3{% + \let\pgfutil@reserved@d=#1% + \def\pgfutil@reserved@a{#2}% + \def\pgfutil@reserved@b{#3}% + \futurelet\pgfutil@let@token\pgfutil@ifnch} +\def\pgfutil@ifnch{% + \ifx\pgfutil@let@token\pgfutil@sptoken + \let\pgfutil@reserved@c\pgfutil@xifnch + \else + \ifx\pgfutil@let@token\pgfutil@reserved@d + \let\pgfutil@reserved@c\pgfutil@reserved@a + \else + \let\pgfutil@reserved@c\pgfutil@reserved@b + \fi + \fi + \pgfutil@reserved@c} +{% + \def\:{\global\let\pgfutil@sptoken= } \: + \def\:{\pgfutil@xifnch} \expandafter\gdef\: {\futurelet\pgfutil@let@token\pgfutil@ifnch} +} + + +\def\pgfutil@ignorespaces + {\ifx\pgfutil@let@token\pgfutil@sptoken + \expandafter\pgfutil@ignorespaces@helper + \else + \expandafter\pgfutil@next + \fi} + +{ + \def\:{\pgfutil@ignorespaces@helper} + \expandafter\gdef\: {\futurelet\pgfutil@let@token\pgfutil@ignorespaces} +} + + +\newif\ifpgfutil@in@ + +\def\pgfutil@in@#1#2{% + \def\pgfutil@in@@##1#1##2##3\pgfutil@in@@{% + \ifx\pgfutil@in@##2\pgfutil@in@false\else\pgfutil@in@true\fi}% + \pgfutil@in@@#2#1\pgfutil@in@\pgfutil@in@@} + + + +\def\pgfutil@nnil{\pgfutil@nil} +\def\pgfutil@fornoop#1\@@#2#3{} +\long\def\pgfutil@for#1:=#2\do#3{% + \expandafter\def\expandafter\pgfutil@fortmp\expandafter{#2}% + \ifx\pgfutil@fortmp\pgfutil@empty \else + \expandafter\pgfutil@forloop#2,\pgfutil@nil,\pgfutil@nil\@@#1{#3}\fi} +\long\def\pgfutil@forloop#1,#2,#3\@@#4#5{\def#4{#1}\ifx #4\pgfutil@nnil \else + #5\def#4{#2}\ifx #4\pgfutil@nnil \else#5\pgfutil@iforloop #3\@@#4{#5}\fi\fi} +\long\def\pgfutil@iforloop#1,#2\@@#3#4{\def#3{#1}\ifx #3\pgfutil@nnil + \expandafter\pgfutil@fornoop \else + #4\relax\expandafter\pgfutil@iforloop\fi#2\@@#3{#4}} +\def\pgfutil@tfor#1:={\pgfutil@tf@r#1 } +\long\def\pgfutil@tf@r#1#2\do#3{\def\pgfutil@fortmp{#2}\ifx\pgfutil@fortmp\pgfutil@space\else + \pgfutil@tforloop#2\pgfutil@nil\pgfutil@nil\@@#1{#3}\fi} +\long\def\pgfutil@tforloop#1#2\@@#3#4{\def#3{#1}\ifx #3\pgfutil@nnil + \expandafter\pgfutil@fornoop \else + #4\relax\expandafter\pgfutil@tforloop\fi#2\@@#3{#4}} + +\def\pgfutil@space{ } + + +\chardef\pgfutil@inputcheck0 +\def\pgfutil@IfFileExists#1#2#3{% + \openin\pgfutil@inputcheck=#1 % + \ifeof\pgfutil@inputcheck + #3\relax + \else + #2\relax + \fi + \closein\pgfutil@inputcheck} + +\def\pgfutil@InputIfFileExists#1#2#3{\pgfutil@IfFileExists{#1}{\input #1\relax#2}{#3}}% + + + +\def\pgfutil@loop#1\pgfutil@repeat{\def\pgfutil@body{#1}\pgfutil@iterate} +\def\pgfutil@iterate{\pgfutil@body \let\pgfutil@next\pgfutil@iterate \else\let\pgfutil@next\relax\fi \pgfutil@next} +\let\pgfutil@repeat=\fi % + + +\let\pgfutil@scan@mark\relax +\let\pgfutil@scan@stop\relax + +\long\def\pgfutil@switch#1#2#3#4#5{% + \pgfutil@exp + \pgfutil@switch@@{#1}% + {#2}#3% + {#2}{}% + \pgfutil@scan@mark{#4}% + \pgfutil@scan@mark{#5}% + \pgfutil@scan@stop +} +\long\def\pgfutil@switch@@#1#2#3#4{% + #1{#2}{#3}% + {\pgfutil@switch@end{#4}}% + {\pgfutil@switch@@{#1}{#2}}% +} +\long\def\pgfutil@switch@end#1#2#3\pgfutil@scan@mark#4#5\pgfutil@scan@stop{\pgfutil@exp@end#1#4} + + +\let\pgfutil@aux@read@hook=\relax + + + +\newtoks\pgfutil@everybye + + + + +\def\pgfutil@addpdfresource@extgs#1{\pgf@sys@addpdfresource@extgs@plain{#1}} +\def\pgfutil@addpdfresource@colorspaces#1{\pgf@sys@addpdfresource@colorspaces@plain{#1}} +\def\pgfutil@addpdfresource@patterns#1{\pgf@sys@addpdfresource@patterns@plain{#1}} +\def\pgfutil@setuppdfresources{\pgf@sys@setuppdfresources@plain} + + + +\let\pgfutil@insertatbegincurrentpage=\relax + + + +\def\pgfutil@raggedright{\rightskip\z@ plus2em \spaceskip.3333em \xspaceskip.5em\relax} +\def\pgfutil@raggedleft{\leftskip\z@ plus2em \rightskip\z@ \spaceskip.3333em \xspaceskip.5em\parfillskip0pt\relax} + + +\def\pgfutil@packageerror#1#2#3{\errhelp{#3}\errmessage{Package #1 Error: #2}} +\def\pgfutil@packagewarning#1#2{\immediate\write17{Package #1: Warning! #2.}} +\def\pgferror#1{\pgfutil@packageerror{pgf}{#1}{}} +\def\pgfwarning#1{\pgfutil@packagewarning{pgf}{#1}} + + + + + +\def\usepgflibrary{\pgfutil@ifnextchar[{\use@pgflibrary}{\use@@pgflibrary}}% +\def\use@pgflibrary[#1]{\use@@pgflibrary{#1}} +\def\use@@pgflibrary#1{% + \edef\pgf@list{#1}% + \pgfutil@for\pgf@temp:=\pgf@list\do{% + \expandafter\pgfkeys@spdef\expandafter\pgf@temp\expandafter{\pgf@temp}% + \ifx\pgf@temp\pgfutil@empty + \else + \expandafter\ifx\csname pgf@library@\pgf@temp @loaded\endcsname\relax% + \expandafter\let\csname pgf@library@\pgf@temp @loaded\endcsname=\pgfutil@empty% + \expandafter\edef\csname pgf@library@#1@atcode\endcsname{\the\catcode`\@} + \expandafter\edef\csname pgf@library@#1@barcode\endcsname{\the\catcode`\|} + \expandafter\edef\csname pgf@library@#1@dollarcode\endcsname{\the\catcode`\$} + \catcode`\@=11 + \catcode`\|=12 + \catcode`\$=3 + \pgfutil@InputIfFileExists{pgflibrary\pgf@temp.code.tex}{}{% + \pgferror{I did not find the pgf library + '\pgf@temp'. I looked for the file named + pgflibrary\pgf@temp.code.tex, but could not find it in + the current texmf trees.} + }% + \catcode`\@=\csname pgf@library@#1@atcode\endcsname + \catcode`\|=\csname pgf@library@#1@barcode\endcsname + \catcode`\$=\csname pgf@library@#1@dollarcode\endcsname + \fi% + \fi + }% +} + + + +\def\usepgfmodule{\pgfutil@ifnextchar[{\use@pgfmodule}{\use@@pgfmodule}}% +\def\use@pgfmodule[#1]{\use@@pgfmodule{#1}} +\def\use@@pgfmodule#1{% + \edef\pgf@list{#1}% + \pgfutil@for\pgf@temp:=\pgf@list\do{% + \expandafter\ifx\csname pgf@module@\pgf@temp @loaded\endcsname\relax% + \expandafter\let\csname pgf@module@\pgf@temp @loaded\endcsname=\pgfutil@empty% + \expandafter\edef\csname pgf@module@#1@atcode\endcsname{\the\catcode`\@} + \expandafter\edef\csname pgf@module@#1@barcode\endcsname{\the\catcode`\|} + \expandafter\edef\csname pgf@module@#1@dollarcode\endcsname{\the\catcode`\$} + \catcode`\@=11 + \catcode`\|=12 + \catcode`\$=3 + \input pgfmodule\pgf@temp.code.tex + \catcode`\@=\csname pgf@module@#1@atcode\endcsname + \catcode`\|=\csname pgf@module@#1@barcode\endcsname + \catcode`\$=\csname pgf@module@#1@dollarcode\endcsname + \fi% + }% +} + +\def\pgfutilensuremath#1{% + \ifmmode#1\else$#1$\fi +} + + + +\begingroup + \catcode`\"=12 + \edef\pgf@loc@TMPa{"}% + \gdef\pgfutilpreparefilename#1{% + \begingroup + \ifnum\the\catcode`\"=13 + \pgfutilconvertdcolon + \fi + \xdef\pgf@temp{#1}% + \endgroup + \expandafter\pgfutil@in@\expandafter"\expandafter{\pgf@temp}% + \ifpgfutil@in@ + \def\pgf@loc@TMPa{\pgfutilstrreplace{"}{}}% + \expandafter\pgf@loc@TMPa\expandafter{\pgf@temp}% + \edef\pgfretval{"\pgfretval"}% + \let\pgfretvalquoted=\pgfretval + \else + \let\pgfretval=\pgf@temp + \edef\pgfretvalquoted{"\pgfretval"}% + \fi + }% + \catcode`\"=13 + \xdef\pgfutilconvertdcolon{% + \noexpand\def\noexpand"{\pgf@loc@TMPa}% + }% +\endgroup + +\def\pgfutil@command@to@string#1#2{% + \expandafter\pgfutil@command@to@string@@\meaning#1\pgfutil@EOI{#2}% +}% +\xdef\pgfutil@glob@TMPa{\meaning\pgfutil@empty}% +\expandafter\def\expandafter\pgfutil@command@to@string@@\pgfutil@glob@TMPa#1\pgfutil@EOI#2{% + \def#2{#1}% +}% +\begingroup +\catcode`\|=0 +\catcode`\\=12 +|gdef|pgfutil@backslash@as@other{\}% +|endgroup + +\def\pgfutilifcontainsmacro#1#2#3{% + \def\pgf@marshal{#1}% + \pgfutil@command@to@string\pgf@marshal\pgf@marshal + \edef\pgf@marshal{\noexpand\pgfutil@in@{\pgfutil@backslash@as@other}{\pgf@marshal}}% + \pgf@marshal + \ifpgfutil@in@ + \def\pgf@marshal{#2}% + \else + \def\pgf@marshal{#3}% + \fi + \pgf@marshal +}% + +\def\pgfutilifstartswith#1#2#3#4{% + \def\pgfutilifstartswith@ ##1#1##2\pgf@EOI{% + \def\pgfutil@tmp{##1}% + \ifx\pgfutil@tmp\pgfutil@empty + \def\pgfutil@tmp#1####1\pgf@EOI{% + \def\pgfretval{####1}% + }% + \pgfutil@tmp#2\pgf@EOI% + #3\relax% + \else + #4\relax% + \fi + }% + \pgfutilifstartswith@#2--#1\pgf@EOI% +}% + +\long\def\pgfutilstrreplace#1#2#3{% + \def\pgfretval{}% + \long\def\pgfutil@search@and@replace@@##1#1##2\pgf@EOI{% + \expandafter\def\expandafter\pgfretval\expandafter{\pgfretval ##1#2}% + \pgfutil@search@and@replace@loop{#1}{##2}% + }% + \pgfutil@search@and@replace@loop{#1}{#3}% +} +\long\def\pgfutil@search@and@replace@loop#1#2{% + \pgfutil@in@{#1}{#2}% + \ifpgfutil@in@ + \def\pgf@loc@TMPa{\pgfutil@search@and@replace@@ #2\pgf@EOI}% + \else + \expandafter\def\expandafter\pgfretval\expandafter{\pgfretval #2}% + \let\pgf@loc@TMPa=\relax + \fi + \pgf@loc@TMPa +}% + +\def\pgfutilsolvetwotwoleq#1#2{% + \begingroup + \dimendef\aa=0 + \dimendef\ab=1 + \dimendef\ba=2 + \dimendef\bb=3 + \dimendef\ra=4 + \dimendef\rb=5 + \dimendef\tmpa=6 + \dimendef\tmpb=7 + \edef\pgf@temp{#1}% + \expandafter\pgfutilsolvetwotwoleq@A\pgf@temp + \edef\pgf@temp{#2}% + \expandafter\pgfutilsolvetwotwoleq@r\pgf@temp + \pgfutilsolvetwotwoleq@ifislarger\aa\ba{% + \def\Pa{a}% + \def\Pb{b}% + }{% + \def\Pa{b}% + \def\Pb{a}% + }% + \tmpa=\csname m\Pa a\endcsname pt % + \ifdim\tmpa<0pt \tmpa=-\tmpa\fi + \ifdim\tmpa<0.0001pt + \let\pgfmathresult=\pgfutil@empty + \else + \pgfmathreciprocal@ + {\csname m\Pa a\endcsname}% + \let\pivot=\pgfmathresult + \csname \Pb a\endcsname=\pivot\csname \Pb a\endcsname + \edef\factor{\expandafter\pgf@sys@tonumber\csname \Pb a\endcsname}% + \tmpa=-\factor\csname \Pa b\endcsname + \advance\csname \Pb b\endcsname by\tmpa + \tmpa=-\factor\csname r\Pa\endcsname + \advance\csname r\Pb\endcsname by\tmpa + \tmpa=\csname \Pb b\endcsname% + \ifdim\tmpa<0pt \tmpa=-\tmpa\fi + \ifdim\tmpa<0.0001pt + \let\pgfmathresult=\pgfutil@empty + \else + \pgfmathdivide@ + {\expandafter\pgf@sys@tonumber\csname r\Pb\endcsname} + {\expandafter\pgf@sys@tonumber\csname \Pb b\endcsname}% + \expandafter\let\csname pgfmathresult\Pb\endcsname=\pgfmathresult + \tmpa=\csname pgfmathresult\Pb\endcsname\csname \Pa b\endcsname + \advance\csname r\Pa\endcsname by-\tmpa + \tmpa=\pivot\csname r\Pa\endcsname + \expandafter\edef\csname pgfmathresult\Pa\endcsname{\pgf@sys@tonumber\tmpa}% + \edef\pgfmathresult{% + {\csname pgfmathresult\Pa\endcsname}% + {\csname pgfmathresult\Pb\endcsname}% + }% + \fi + \fi + \pgfmath@smuggleone\pgfmathresult + \endgroup +}% + +\def\pgfutilsolvetwotwoleq@ifislarger#1#2#3#4{% + \tmpa=#1 + \ifdim\tmpa<0pt + \multiply\tmpa by-1 + \fi + \tmpb=#2 + \ifdim\tmpb<0pt + \multiply\tmpb by-1 + \fi + \ifdim\tmpa>\tmpb + #3% + \else + #4% + \fi +}% + +\def\pgfutilsolvetwotwoleqfloat@ifislarger#1#2#3#4{% + \pgfmathfloatabs@{#1}\let\tmpa=\pgfmathresult + \pgfmathfloatabs@{#2}\let\tmpb=\pgfmathresult + \pgfmathfloatlessthan@{\tmpb}{\tmpa}% + \ifpgfmathfloatcomparison + #3% + \else + #4% + \fi +}% +\def\pgfutilsolvetwotwoleq@A#1#2#3#4{% + \def\maa{#1}\def\mab{#2}% + \def\mba{#3}\def\mbb{#3}% + \aa=#1pt \ab=#2pt + \ba=#3pt \bb=#4pt +} +\def\pgfutilsolvetwotwoleq@r#1#2{% + \ra=#1pt \rb=#2pt +}% +\def\pgfutilsolvetwotwoleqfloat@A#1#2#3#4{% + \pgfmathfloatparsenumber{#1}\let\maa=\pgfmathresult + \pgfmathfloatparsenumber{#2}\let\mab=\pgfmathresult + \pgfmathfloatparsenumber{#3}\let\mba=\pgfmathresult + \pgfmathfloatparsenumber{#4}\let\mbb=\pgfmathresult +} +\def\pgfutilsolvetwotwoleqfloat@r#1#2{% + \pgfmathfloatparsenumber{#1}\let\ra=\pgfmathresult + \pgfmathfloatparsenumber{#2}\let\rb=\pgfmathresult +}% + +\def\pgfutilsolvetwotwoleqfloat#1#2{% + \begingroup + \pgfmathfloatcreate{1}{1.0}{-4}% + \let\thresh=\pgfmathresult + \edef\pgf@temp{#1}% + \expandafter\pgfutilsolvetwotwoleqfloat@A\pgf@temp + \edef\pgf@temp{#2}% + \expandafter\pgfutilsolvetwotwoleqfloat@r\pgf@temp + \pgfutilsolvetwotwoleqfloat@ifislarger\maa\mba{% + \def\Pa{a}% + \def\Pb{b}% + }{% + \def\Pa{b}% + \def\Pb{a}% + }% + \expandafter\pgfmathfloatabs@\expandafter{\csname m\Pa a\endcsname}% + \let\tmpa=\pgfmathresult + \pgfmathfloatlessthan@{\tmpa}{\thresh}% + \ifpgfmathfloatcomparison + \let\pgfmathresult=\pgfutil@empty + \else + \expandafter\pgfmathfloatreciprocal@\expandafter{\csname m\Pa a\endcsname}% + \let\pivot=\pgfmathresult + \expandafter\pgfmathfloatmultiply@\expandafter{\csname m\Pb a\endcsname}{\pivot}% + \let\factor=\pgfmathresult + \expandafter\let\csname m\Pb a\endcsname=\factor + \expandafter\pgfmathfloatmultiply@\expandafter{\csname m\Pa b\endcsname}{\factor}% + \let\tmpa=\pgfmathresult + \expandafter\pgfmathfloatsubtract@\expandafter{\csname m\Pb b\endcsname}{\tmpa}% + \expandafter\let\csname m\Pb b\endcsname=\pgfmathresult + \expandafter\pgfmathfloatmultiply@\expandafter{\csname r\Pa\endcsname}{\factor}% + \let\tmpa=\pgfmathresult + \expandafter\pgfmathfloatsubtract@\expandafter{\csname r\Pb\endcsname}{\tmpa}% + \expandafter\let\csname r\Pb\endcsname=\pgfmathresult + \expandafter\pgfmathfloatabs@\expandafter{\csname m\Pb b\endcsname}% + \let\tmpa=\pgfmathresult + \pgfmathfloatlessthan@{\tmpa}{\thresh}% + \ifpgfmathfloatcomparison + \let\pgfmathresult=\pgfutil@empty + \else + \edef\pgf@marshal{% + \noexpand\pgfmathfloatdivide@ + {\csname r\Pb\endcsname} + {\csname m\Pb b\endcsname}% + }% + \pgf@marshal + \expandafter\let\csname pgfmathresult\Pb\endcsname=\pgfmathresult + \let\tmpa=\pgfmathresult + \expandafter\pgfmathfloatmultiply@\expandafter{\csname m\Pa b\endcsname}{\tmpa}% + \let\tmpa=\pgfmathresult + \expandafter\pgfmathfloatsubtract@\expandafter{\csname r\Pa\endcsname}{\tmpa}% + \expandafter\let\csname r\Pa\endcsname=\pgfmathresult + \expandafter\pgfmathfloatmultiply@\expandafter{\csname r\Pa\endcsname}{\pivot}% + \expandafter\let\csname pgfmathresult\Pa\endcsname=\pgfmathresult + \edef\pgfmathresult{% + {\csname pgfmathresult\Pa\endcsname}% + {\csname pgfmathresult\Pb\endcsname}% + }% + \expandafter\pgfutilsolvetwotwoleqfloat@to@pgf@range\pgfmathresult + \fi + \fi + \pgfmath@smuggleone\pgfmathresult + \endgroup +}% + + +\def\pgfutilsolvetwotwoleqfloat@to@pgf@range#1#2{% + \pgfmathfloatcreate{1}{1.6}{4}\let\pgfutilsolvetwotwoleqfloat@to@pgf@range@max=\pgfmathresult + \pgfmathfloatabs@{#1}% + \expandafter\pgfmathfloatlessthan@\expandafter{\pgfmathresult}{\pgfutilsolvetwotwoleqfloat@to@pgf@range@max}% + \ifpgfmathfloatcomparison + \pgfmathfloatabs@{#2}% + \expandafter\pgfmathfloatlessthan@\expandafter{\pgfmathresult}{\pgfutilsolvetwotwoleqfloat@to@pgf@range@max}% + \ifpgfmathfloatcomparison + \pgfmathfloattofixed{#1}\let\tmpa=\pgfmathresult + \pgfmathfloattofixed{#2}% + \edef\pgfmathresult{{\tmpa}{\pgfmathresult}}% + \else + \let\pgfmathresult=\pgfutil@empty + \fi + \else + \let\pgfmathresult=\pgfutil@empty + \fi +}% + +\let\pgfutil@write=\write +\let\pgfutil@read=\read + +\let\pgfutil@protect\relax + +\def\pgfutil@check@rerun#1#2{} + +\newdimen\pgfutil@tempdima +\newdimen\pgfutil@tempdimb + + +\let\pgfutil@ifluatex\iffalse +\begingroup\expandafter\expandafter\expandafter\endgroup +\expandafter\ifx\csname directlua\endcsname\relax +\else + \expandafter\let\csname pgfutil@ifluatex\expandafter\endcsname + \csname iftrue\endcsname +\fi + +\pgfutil@ifluatex + \let\pgfutil@directlua\directlua + \pgfutil@directlua{% + tex.enableprimitives('pgfutil@',{'luaescapestring'})} +\else + \def\pgfutil@directlua#1{} + \def\pgfutil@luaescapestring#1{} +\fi + + +\def\pgfutil@shellescape@lua@eightseven#1{% + \pgfutil@directlua{os.execute("\pgfutil@luaescapestring{#1}")}% +}% +\def\pgfutil@shellescape#1{% + \immediate\write18{#1}% +}% +\pgfutil@IfUndefined{directlua}{% +}{% + \pgfutil@IfUndefined{lastsavedimageresourcepages}{% + }{% + \let\pgfutil@shellescape=\pgfutil@shellescape@lua@eightseven + }% +}% + + +\def\pgfutil@advancestringcounter#1{% + \begingroup + \c@pgf@counta=#1\relax + \advance\c@pgf@counta by1 + \edef#1{\the\c@pgf@counta}% + \pgfmath@smuggleone#1% + \endgroup +}% + +\endinput diff --git a/preprints/A-polynomial-time-algorithm-for-three-machine-unit-job-scheduling-September-24-2026/paper.pdf b/preprints/A-polynomial-time-algorithm-for-three-machine-unit-job-scheduling-September-24-2026/paper.pdf new file mode 100644 index 0000000000000000000000000000000000000000..db1c9e9064b992ecdb7880ef94d2d11a342d2a9a --- /dev/null +++ b/preprints/A-polynomial-time-algorithm-for-three-machine-unit-job-scheduling-September-24-2026/paper.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:2080c4ad5953f225c1db7fe803c6d8602ae499510719cfe0cb4f22f99bccff1e +size 246589 diff --git a/preprints/A-polynomial-time-construction-of-strong-thin-trees-September-23-2026/README.md b/preprints/A-polynomial-time-construction-of-strong-thin-trees-September-23-2026/README.md new file mode 100644 index 0000000000000000000000000000000000000000..b32069bbf8a671a001d56d36ea42f9821acfd989 --- /dev/null +++ b/preprints/A-polynomial-time-construction-of-strong-thin-trees-September-23-2026/README.md @@ -0,0 +1,18 @@ +# [A polynomial-time construction of strong thin trees](paper.pdf) + +**Author:** OpenAI + +**Date:** September 23, 2026 + +## Citation + +```bibtex +@misc{OAI:A-polynomial-time-construction-of-strong-thin-trees-September-23-2026, + author = {{OpenAI}}, + title = {{A polynomial-time construction of strong thin trees}}, + howpublished = {OpenAI Math Release preprint + \href{https://github.com/openai/math/blob/main/preprints/A-polynomial-time-construction-of-strong-thin-trees-September-23-2026/paper.pdf}{OAI:A-polynomial-time-construction-of-strong-thin-trees-September-23-2026}}, + year = {2026} +} +``` + diff --git a/preprints/A-polynomial-time-construction-of-strong-thin-trees-September-23-2026/build/source/figures/metric_geometry.tex b/preprints/A-polynomial-time-construction-of-strong-thin-trees-September-23-2026/build/source/figures/metric_geometry.tex new file mode 100644 index 0000000000000000000000000000000000000000..86d89ead46c269367a6672260ca52dc6e2216b20 --- /dev/null +++ b/preprints/A-polynomial-time-construction-of-strong-thin-trees-September-23-2026/build/source/figures/metric_geometry.tex @@ -0,0 +1,43 @@ +\begin{figure}[tb] +\centering +\begin{tikzpicture}[font=\small,>=Stealth] + \node[font=\bfseries] at (2.7,2.6) {Two realizations of the vertex labels}; + \node[draw,rounded corners,align=center,minimum width=4.5cm, + fill=MidnightBlue!5] (euclidean) at (2.7,1.55) + {$p_v$ and $Y_v=Up_v$ in Euclidean space\\[3pt] + $\norm{Y_u-Y_v}_2^2\le d(u,v)$}; + \node[draw,rounded corners,align=center,minimum width=4.5cm, + fill=ForestGreen!5] (profiles) at (2.7,-.2) + {$c_v$ in the profile polytope $Q$\\[3pt] + $\norm{c_u-c_v}_\infty=d(u,v)$}; + \draw[->,thick] (euclidean) -- node[right,align=left,font=\footnotesize] + {selected vertex labels} (profiles); + \node[font=\bfseries] at (10.2,2.6) {Exact radial motion in $Q$}; + \coordinate (center) at (7.6,1.3); + \coordinate (point) at (10.2,-.2); + \coordinate (opposite) at (12.8,1.3); + \draw[->,thick,MidnightBlue] (point) -- (center); + \draw[->,thick,ForestGreen] (point) -- (opposite); + \fill (center) circle (2pt); + \fill (point) circle (2pt); + \fill (opposite) circle (2pt); + \node[anchor=south,align=center,font=\footnotesize] + at (7.6,1.48) {$c_x$\\[2pt]radius $0$}; + \node[anchor=south,align=center,font=\footnotesize] + at (12.8,1.48) {$c_x^-$\\[2pt]radius $R$}; + \node[anchor=north,align=center,font=\footnotesize, + text=MidnightBlue] at (8.55,.03) + {move inward\\[2pt]length $\rho$}; + \node[anchor=north,align=center,font=\footnotesize, + text=ForestGreen] at (11.85,.03) + {move outward\\[2pt]length $R-\rho$}; + \node[anchor=north,align=center,font=\footnotesize] + at (10.2,-.4) {$q$\\[2pt]radius $\rho$}; +\end{tikzpicture} +\caption{Spectral packing uses Euclidean points; balls and shells use +the distance-profile realization with the same vertex labels. +The radial paths are schematic: in the ambient norm, distance +traveled along either indicated segment equals change of radius. +The two segments need not form one straight segment in $Q$.} +\label{tg:geometry-figure} +\end{figure} diff --git a/preprints/A-polynomial-time-construction-of-strong-thin-trees-September-23-2026/build/source/macros.tex b/preprints/A-polynomial-time-construction-of-strong-thin-trees-September-23-2026/build/source/macros.tex new file mode 100644 index 0000000000000000000000000000000000000000..9ae8c02aa4edf01f537aa156cb16f8c4156ed8e4 --- /dev/null +++ b/preprints/A-polynomial-time-construction-of-strong-thin-trees-September-23-2026/build/source/macros.tex @@ -0,0 +1,48 @@ +\usepackage[T1]{fontenc} +\usepackage[utf8]{inputenc} +\usepackage{lmodern} +\usepackage[margin=1in]{geometry} +\usepackage{microtype} +\usepackage{amsmath,amssymb,amsthm,mathtools} +\usepackage{enumitem,booktabs,array} +\usepackage{needspace} +\usepackage[dvipsnames]{xcolor} +\usepackage{tikz} +\usetikzlibrary{arrows.meta,positioning,calc,fit,backgrounds} +\usepackage[colorlinks=true,linkcolor=MidnightBlue,citecolor=MidnightBlue,urlcolor=MidnightBlue]{hyperref} +\usepackage[nameinlink,noabbrev,capitalise]{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} +\newtheorem{algorithm}[theorem]{Algorithm} +\theoremstyle{remark} +\newtheorem{remark}[theorem]{Remark} +\newcommand{\R}{\mathbb R} +\newcommand{\Q}{\mathbb Q} +\newcommand{\C}{\mathbb C} +\newcommand{\N}{\mathbb N} +\newcommand{\E}{\mathbb E} +\newcommand{\Prob}{\mathbb P} +\newcommand{\eps}{\varepsilon} +\newcommand{\Id}{I} +\newcommand{\one}{\mathbf 1} +\DeclareMathOperator{\tr}{tr} +\DeclareMathOperator{\rank}{rank} +\DeclareMathOperator{\diag}{diag} +\DeclareMathOperator{\conv}{conv} +\DeclareMathOperator{\Span}{span} +\DeclareMathOperator{\Var}{Var} +\DeclareMathOperator{\Vol}{vol} +\DeclareMathOperator{\supp}{supp} +\DeclarePairedDelimiter{\norm}{\lVert}{\rVert} +\DeclarePairedDelimiter{\abs}{\lvert}{\rvert} +\DeclarePairedDelimiterX{\ip}[2]{\langle}{\rangle}{#1,#2} +\newcommand{\poly}{\operatorname{poly}} +\newcommand{\op}{\mathrm{op}} +\newcommand{\TV}{\mathrm{TV}} +\setlist{itemsep=3pt,topsep=5pt} +\setlength{\emergencystretch}{2em} diff --git a/preprints/A-polynomial-time-construction-of-strong-thin-trees-September-23-2026/build/source/main.tex b/preprints/A-polynomial-time-construction-of-strong-thin-trees-September-23-2026/build/source/main.tex new file mode 100644 index 0000000000000000000000000000000000000000..038b1ec49020c40a352b4dede449ee12f64321d1 --- /dev/null +++ b/preprints/A-polynomial-time-construction-of-strong-thin-trees-September-23-2026/build/source/main.tex @@ -0,0 +1,33 @@ +\documentclass[11pt]{article} +\input{macros} +\pdftrailerid{} +\pdfinfoomitdate=1 +\pdfsuppressptexinfo=15 +\title{A polynomial-time construction of strong thin trees} +\author{OpenAI} +\date{September 23, 2026} +\hypersetup{pdftitle={A polynomial-time construction of strong thin trees}, + pdfauthor={OpenAI},pdfsubject={Thin spanning trees and polynomial bit complexity}} +\begin{document} +\maketitle +\begin{abstract} +We give a deterministic polynomial-time construction of strong thin trees. +Given a finite $k$-edge-connected loopless multigraph on at least one vertex, +the algorithm constructs a spanning tree meeting every cut in at most a +$C/k$ fraction of its edges, for a universal constant $C$. The running time +is polynomial in the binary input length, including when parallel-edge +multiplicities are encoded in binary. +\end{abstract} +\tableofcontents +\input{sections/introduction} +\input{sections/tree_geometry} +\input{sections/extraction} +\input{sections/normalization} +\input{sections/signing} +\input{sections/iteration} +\input{sections/fractional_rounding} +\appendix +\input{sections/signing_interface} +\bibliographystyle{plain} +\bibliography{refs/sources} +\end{document} diff --git a/preprints/A-polynomial-time-construction-of-strong-thin-trees-September-23-2026/build/source/refs/sources.bib b/preprints/A-polynomial-time-construction-of-strong-thin-trees-September-23-2026/build/source/refs/sources.bib new file mode 100644 index 0000000000000000000000000000000000000000..f4859899255b32269cdd85266ed46af8859e8e30 --- /dev/null +++ b/preprints/A-polynomial-time-construction-of-strong-thin-trees-September-23-2026/build/source/refs/sources.bib @@ -0,0 +1,329 @@ +@book{GLS1988, + author = {Gr{\"o}tschel, Martin and Lov{\'a}sz, L{\'a}szl{\'o} and Schrijver, Alexander}, + title = {\href{https://doi.org/10.1007/978-3-642-97881-4}{Geometric Algorithms and Combinatorial Optimization}}, + series = {Algorithms and Combinatorics}, + volume = {2}, + publisher = {Springer}, + address = {Berlin}, + year = {1988}, + doi = {10.1007/978-3-642-97881-4}, + url = {https://www.zib.de/userpage/groetschel/pubnew/paper/groetschellovaszschrijver1988.pdf} +} + +@article{Kaiser2012, + author = {Kaiser, Tom{\'a}{\v s}}, + title = {\href{https://arxiv.org/abs/0911.2809v3}{A short proof of the tree-packing theorem}}, + journal = {Discrete Mathematics}, + volume = {312}, + number = {10}, + pages = {1689--1691}, + year = {2012}, + doi = {10.1016/j.disc.2012.01.020}, + eprint = {0911.2809v3}, + archivePrefix = {arXiv}, + url = {https://arxiv.org/abs/0911.2809v3} +} + +@inproceedings{Terao2023, + author = {Terao, Tatsuya}, + title = {\href{https://doi.org/10.4230/LIPIcs.ICALP.2023.104}{Faster Matroid Partition Algorithms}}, + booktitle = {50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)}, + series = {Leibniz International Proceedings in Informatics}, + volume = {261}, + pages = {104:1--104:20}, + publisher = {Schloss Dagstuhl--Leibniz-Zentrum f{\"u}r Informatik}, + year = {2023}, + doi = {10.4230/LIPIcs.ICALP.2023.104}, + url = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2023.104} +} + +@misc{AnariOveisGharan2015, + author = {Anari, Nima and Oveis Gharan, Shayan}, + title = {\href{https://arxiv.org/abs/1411.4613v4}{Effective-Resistance-Reducing Flows, Spectrally Thin Trees, and Asymmetric {TSP}}}, + year = {2015}, + howpublished = {arXiv:1411.4613v4}, + eprint = {1411.4613v4}, + archivePrefix = {arXiv}, + primaryClass = {cs.DS}, + note = {Full version, revised 1 September 2015}, + url = {https://arxiv.org/abs/1411.4613v4} +} + +@article{MarcusSpielmanSrivastava2015, + author = {Marcus, Adam W. and Spielman, Daniel A. and Srivastava, Nikhil}, + title = {\href{https://doi.org/10.4007/annals.2015.182.1.8}{Interlacing Families {II}: Mixed Characteristic Polynomials and the {Kadison--Singer} Problem}}, + journal = {Annals of Mathematics}, + volume = {182}, + number = {1}, + pages = {327--350}, + year = {2015}, + doi = {10.4007/annals.2015.182.1.8}, + url = {https://annals.math.princeton.edu/2015/182-1/p08} +} + +@misc{Goddyn2004, + author = {Goddyn, Luis A.}, + title = {Some Open Problems {I} Like}, + year = {2004}, + note = {Problem 4; webpage modified March 17, 2004}, + url = {https://web.archive.org/web/20211025204852/https://www.sfu.ca/~goddyn/Problems/problems.html} +} + +@article{AsadpourEtAl2017, + author = {Asadpour, Arash and Goemans, Michel X. and M{\k{a}}dry, Aleksander and {Oveis Gharan}, Shayan and Saberi, Amin}, + title = {An {$O(\log n/\log\log n)$}-Approximation Algorithm for the Asymmetric Traveling Salesman Problem}, + journal = {Operations Research}, + volume = {65}, + number = {4}, + pages = {1043--1061}, + year = {2017}, + doi = {10.1287/opre.2017.1603}, + note = {Preliminary version in SODA 2010. 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J.}, + title = {Remarks to {Maurice Fr{\'e}chet}'s Article + ``Sur la d{\'e}finition axiomatique d'une classe d'espace + distanci{\'e}s vectoriellement applicable sur l'espace de {Hilbert}''}, + journal = {Annals of Mathematics}, + series = {Second Series}, + volume = {36}, + number = {3}, + pages = {724--732}, + year = {1935}, + doi = {10.2307/1968654}, + url = {https://www.convexoptimization.com/TOOLS/Schoenberg2.pdf} +} + +@article{AroraRaoVazirani2009, + author = {Arora, Sanjeev and Rao, Satish and Vazirani, Umesh}, + title = {Expander Flows, Geometric Embeddings and Graph Partitioning}, + journal = {Journal of the ACM}, + volume = {56}, + number = {2}, + pages = {5:1--5:37}, + year = {2009}, + doi = {10.1145/1502793.1502794}, + url = {https://www.cs.princeton.edu/~arora/pubs/arvfull.pdf} +} + +@misc{ThinTreesCompanion, + author = {{OpenAI}}, + title = {{The strong thin tree conjecture}}, + howpublished = {OpenAI Math Release preprint + \href{https://github.com/openai/math/blob/main/preprints/The-strong-thin-tree-conjecture-September-23-2026/paper.pdf}{OAI:The-strong-thin-tree-conjecture-September-23-2026}}, + year = {2026} +} +@misc{EzeunalaJiang2026, + author = {Ezeunala, Ekene and Jiang, Haotian}, + title = {Rank-One Matrix Discrepancy and Algorithmic {Kadison--Singer}}, + year = {2026}, + howpublished = {arXiv:2609.17266v1}, + eprint = {2609.17266v1}, + archivePrefix = {arXiv}, + url = {https://arxiv.org/abs/2609.17266v1} +} + +@article{Bansal2024, + author = {Nikhil Bansal}, + title = {On a Generalization of Iterated and Randomized Rounding}, + journal = {Theory of Computing}, + volume = {20}, + number = {6}, + year = {2024}, + pages = {1--23}, + doi = {10.4086/toc.2024.v020a006}, + url = {https://theoryofcomputing.org/articles/v020a006/} +} + +@inproceedings{BansalGarg2017, + author = {Nikhil Bansal and Shashwat Garg}, + title = {Algorithmic Discrepancy Beyond Partial Coloring}, + booktitle = {Proceedings of the 49th Annual ACM SIGACT Symposium + on Theory of Computing}, + year = {2017}, + pages = {914--926}, + doi = {10.1145/3055399.3055490}, + eprint = {1611.01805}, + archivePrefix = {arXiv}, + note = {Full version: arXiv:1611.01805v2, Theorem 6} +} + +@inproceedings{HarveyOlver2014, + author = {Nicholas J. A. Harvey and Neil Olver}, + title = {Pipage Rounding, Pessimistic Estimators and Matrix Concentration}, + booktitle = {Proceedings of the Twenty-Fifth Annual ACM-SIAM Symposium + on Discrete Algorithms}, + editor = {Chandra Chekuri}, + publisher = {Society for Industrial and Applied Mathematics}, + year = {2014}, + pages = {926--945}, + doi = {10.1137/1.9781611973402.69}, + eprint = {1307.2274}, + archivePrefix = {arXiv}, + note = {Full version: arXiv:1307.2274v1, Appendix F} +} + diff --git a/preprints/A-polynomial-time-construction-of-strong-thin-trees-September-23-2026/build/source/sections/extraction.tex b/preprints/A-polynomial-time-construction-of-strong-thin-trees-September-23-2026/build/source/sections/extraction.tex new file mode 100644 index 0000000000000000000000000000000000000000..c509b2f646a851c63c8a63b2d353445acbf9e6da --- /dev/null +++ b/preprints/A-polynomial-time-construction-of-strong-thin-trees-September-23-2026/build/source/sections/extraction.tex @@ -0,0 +1,510 @@ +\section{Constructive shortcut extraction} +\label{sec:ta-algorithm} + +We now turn the simultaneous response of \Cref{prop:tg-response} into an +algorithm. Throughout this section, an explicit multigraph has distinct +labels for its parallel edges, and loops are discarded. Write +$E_H(\mathcal P)$ for the edges crossing a partition $\mathcal P$ of the +vertex set. Matrices act on $\one^\perp$, with its inherited Euclidean +inner product. We retain the cone $\mathcal N$ and compact convex set +$\mathcal K_H$ from the preceding section: +\begin{equation} + \mathcal K_H=\{X\succeq L_H: + X\mathbin\bullet Z\le3L_H\mathbin\bullet Z\quad(Z\in\mathcal N)\}. + \label{eq:ta-K} +\end{equation} +Here $A\mathbin\bullet Z=\tr(AZ)$ always means the inherited trace, even +when rational, nonorthonormal coordinates are used for computation. + + + +The construction follows the packing-preserving sparsification strategy +of \cite{ThinTreesCompanion}. Convex continuation produces a rational +shortcut together with a large integer packing supported on its +low-resistance edges. Nonnegative capacities permit partition uncrossing; +the nearly tight partitions then have highly connected parts, which is +precisely the input required by the simultaneous geometric response. + +\subsection{A ramp of packing capacities} + +We use two standard algorithmic inputs in precise forms. The +Nash--Williams--Tutte tree-packing theorem +\cite{NashWilliams1961,Tutte1961} says that a multigraph contains $q$ edge-disjoint +spanning trees if and only if +\begin{equation} + |E_H(\mathcal P)|\ge q(|\mathcal P|-1) + \quad\hbox{for every partition }\mathcal P; + \label{eq:ta-integer-packing} +\end{equation} +see \cite[Theorem~1]{Kaiser2012}. A maximum union of $q$ graphic matroids +can be constructed by matroid partition, originating with Edmonds +\cite{Edmonds1968}; we use the polynomial-time independence-oracle +algorithm of \cite[Theorem~14]{Terao2023}. The algorithm +returns disjoint forests of maximum total cardinality. Their total size +is $q(n-1)$ exactly when all $q$ forests are spanning trees. Acyclicity +is a polynomial-time independence test, including for parallel edges. +We reject $q(n-1)>m$ before invoking this procedure; thus the number of +matroid copies is polynomial for an explicit graph. + +\begin{proposition}[Constructive shortcut extraction] +\label{prop:ta-extraction} +There are absolute constants $A$ and $r_0$ such that the following holds. +If an explicit multigraph $H$ on at least two vertices contains $r\ge r_0$ +edge-disjoint spanning trees, one can deterministically compute a rational +$X\in\mathcal K_H$ and at least +\[ + (1-Ar^{-1/16})r +\] +edge-disjoint spanning trees, every edge $e$ of which satisfies +$b_e^TX^{-1}b_e\le r^{-1/4}$. The running time and output encoding length +are polynomial in $|V(H)|+|E(H)|$. +\end{proposition} + +Fix such an $H$, put $m=|E(H)|$, and set +\begin{equation} + \kappa=\frac1{m+1},\qquad t=\frac\kappa{32},\qquad + r^{-1/8}\le g\le2r^{-1/8}, + \label{eq:ta-parameters} +\end{equation} +where $g$ is dyadic. Comparing eighth powers computes $g$ exactly with +$O(\log r)$ bits. At a parameter $s>0$ and matrix $X\in\mathcal K_H$, define +\begin{equation} + R_e(X)=b_e^TX^{-1}b_e,\qquad + p_e=\min\{2,(sR_e(X))^{-1}\},\qquad + u_e=\max\{0,p_e-1\}. + \label{eq:ta-ramp} +\end{equation} +The maintained invariant is +\begin{equation} + \sum_{e\in E_H(\mathcal P)}(u_e+\kappa) + \ge Q(|\mathcal P|-1)\qquad\hbox{for every }\mathcal P. + \label{eq:ta-invariant} +\end{equation} +Initially take $s=1/2$, $X=L_H$, and $Q=r$. Since +$L_H\succeq b_eb_e^T$, each $R_e(L_H)\le1$, so $u_e=1$ and the initial +invariant follows from the given packing. + +One step replaces +\begin{equation} + s^+=(1+t)s,\qquad X^+=(1-t)X+tY,\qquad Q^+=Q-2grt, + \label{eq:ta-step} +\end{equation} +where $Y\in\mathcal K_H$ is to be found. Steps are performed while +$s^40$. Choose variables +$0\le y_e\le1$ for $e\in E(G)$ and impose +\begin{equation} + Y\succeq6s y_e b_eb_e^T. + \label{eq:ta-response-LMI} +\end{equation} +For these variables define concave lower capacities on all edges of $H$: +\begin{equation} + a_e(y)= + \begin{cases} + \kappa+\min\{1,u_e+t(5y_e-4)\},&e\in E(G),\\ + \kappa,&e\notin E(G). + \end{cases} + \label{eq:ta-lower-capacity} +\end{equation} +They are nonnegative, in fact $a_e\ge\kappa-4t=7\kappa/8$. +This small baseline will also make the uncrossing argument valid. + +To check the lower bound, recall that $X\succeq\alpha b b^T$ is +equivalent to $\alpha\le(b^TX^{-1}b)^{-1}$ for $X\succ0$. +Thus $X\succeq sp_e b_eb_e^T$, and \Cref{eq:ta-response-LMI} gives +\[ + \frac1{s^+R_e(X^+)}\ge + \frac{(1-t)p_e+6ty_e}{1+t} + \ge p_e+t(5y_e-4). +\] +Indeed the numerator of the difference in the last inequality, divided +by $t$, is $(1-5t)y_e+4+4t-2p_e\ge0$, since $p_e\le2$ and $t<1/5$. +Taking the ramp at $s^+,X^+$, including both its upper cap and its lower +truncation, proves +\begin{equation} + u_e^++\kappa\ge a_e(y). + \label{eq:ta-ramp-lower} +\end{equation} +Off $G$ this follows just from $u_e^+\ge0$. + +\begin{lemma}[A feasible continuation response] +\label{lem:ta-continuation} +Under \Cref{eq:ta-invariant,eq:ta-target-bound}, with +$s\le r^{1/4}$, there exist $Y\in\mathcal K_H$ and $y$ satisfying +\Cref{eq:ta-response-LMI} and +\begin{equation} + \sum_{e\in E_H(\mathcal P)}a_e(y) + \ge (Q-grt)(|\mathcal P|-1) + \qquad\hbox{for every }\mathcal P. + \label{eq:ta-strong-target} +\end{equation} +\end{lemma} + +\begin{proof} +Let $\mathcal C$ be the compact convex set of admissible $(Y,y)$ before +the partition inequalities are imposed. It is nonempty: take $Y=L_H$ +and $y=0$. Put $q=Q-grt>0$ and let +\[ + S_{\mathcal P}(y)=\sum_{e\in E_H(\mathcal P)}a_e(y) + -q(|\mathcal P|-1). +\] +Each slack is continuous and concave. Its vector image need not be +convex, but the downward set +\[ + \mathcal D=\{v:\ v_{\mathcal P}\le S_{\mathcal P}(y) + \text{ for some }(Y,y)\in\mathcal C\} +\] +is convex and closed. Convexity follows coordinatewise from concavity; +closedness follows by taking a convergent subsequence of witnesses in +$\mathcal C$. The set $\mathcal D$ is unbounded downward. If it does not +meet the nonnegative orthant, its distance from that orthant is positive: +the norm of the negative part of $S(y)$ has a positive minimum on the +compact set $\mathcal C$, and passing downward cannot reduce that norm. +Separation therefore supplies nonnegative weights $w_{\mathcal P}$, +normalized to sum to one, for which +\begin{equation} + F(w):=\max_{(Y,y)\in\mathcal C} + \sum_{\mathcal P}w_{\mathcal P}S_{\mathcal P}(y)<0. + \label{eq:ta-separated} +\end{equation} + +There are finitely many partitions. Hence $F$ is continuous on their +probability simplex and has a minimizer. Among minimizers choose one +maximizing $\sum_{\mathcal P}w_{\mathcal P}|\mathcal P|^2$. +Its support is a chain under refinement. To prove this, suppose +incomparable $\mathcal P,\mathcal Q$ have positive mass. Move equal mass +from them to their common refinement $\mathcal P\wedge\mathcal Q$ and +finest common coarsening $\mathcal P\vee\mathcal Q$. For each edge, +\[ + \one_{E_H(\mathcal P\wedge\mathcal Q)}+ + \one_{E_H(\mathcal P\vee\mathcal Q)} + \le \one_{E_H(\mathcal P)}+\one_{E_H(\mathcal Q)}. +\] +Moreover, +\[ + |\mathcal P\wedge\mathcal Q|+|\mathcal P\vee\mathcal Q| + \ge |\mathcal P|+|\mathcal Q|. +\] +For the second inequality, form the bipartite graph whose vertices are +parts of $\mathcal P$ and $\mathcal Q$ and whose edges are their nonempty +intersections. Its edges count refinement parts, and its connected +components count coarsening parts; edges plus components are at least +vertices. Since $a_e\ge0$ and $q>0$, uncrossing decreases the weighted +slack for every $(Y,y)$, hence cannot increase $F$. +Writing $a=|\mathcal P|$, $b=|\mathcal Q|$, $c=|\mathcal P\wedge\mathcal Q|$ +and $d=|\mathcal P\vee\mathcal Q|$, incomparability gives +$c>\max(a,b)$ and $d<\min(a,b)$. Thus +\[ + c^2+d^2-a^2-b^2 + =(c-a)(c-b)+(d-a)(d-b)+(a+b)(c+d-a-b)>0. +\] +This contradicts the tie-breaking choice. The one-part partition has +zero slack and is comparable with every partition, so it creates no +exception. + +It remains to satisfy \Cref{eq:ta-strong-target} simultaneously on any +chain. Let the old slack in \Cref{eq:ta-invariant} be +$\sigma_{\mathcal P}\ge0$. If $\sigma_{\mathcal P}>1$, the worst possible +loss $4tm<1$ already leaves the desired inequality. Retain only partitions +with $\sigma_{\mathcal P}\le1$, omitting the one-part partition. Every +part of every retained partition induces an +$h$-edge-connected graph in $G$, where $h=\lfloor r/2\rfloor$. +Indeed, split one such part by any nontrivial internal cut. Subtracting +the old partition capacity from the invariant for the refined partition +shows that the internal cut has capacity at least $Q-1$. Removing all +baselines loses less than $\kappa m<1$, and $u_e\le1$ then shows that +the active internal cut has more than $Q-2\ge h$ edges. The same argument +using a two-part partition at the root gives more than $Q-1$ active +edges across every root cut. Singleton parts require no check. + +Apply \Cref{prop:tg-response} to this chain, with its root, and set $y_e=1$ +on edges satisfying $Y\succeq6s b_eb_e^T$, and $y_e=0$ otherwise. +These choices satisfy \Cref{eq:ta-response-LMI}. Fix a retained partition, +write $d=|\mathcal P|\ge2$ and $M=|E_G(\mathcal P)|$, and call its at most +$r^{-1/3}M$ exceptional crossing edges bad. Good edges never lose +capacity; bad edges lose at most $4t$ each. If $M\le8r(d-1)$, the total +loss is at most +\[ + 4t r^{-1/3}M\le32tr^{2/3}(d-1)\le grt(d-1) +\] +for sufficiently large $r$, as required. + +If $M>8r(d-1)$, near-tightness gives +$\sum_{E_G(\mathcal P)}u_e\le Q(d-1)+1\le r(d-1)+1$. +Fewer than half of these $M$ edges can have $u_e>1/2$: their number is +at most $2r(d-1)+2\mu$. +The other two LMIs have margins at least $\mu/2$. +The $\lambda$ and lower $y$ faces have margin $P^{-8}$, the upper $y$ +face has margin at least $1/2$, and the $z$ lower and upper faces have +margins at least $\kappa/2$ and $1$, respectively. Finally +\[ + \kappa+u_e+t(5y_{e,0}-4)-z_{e,0} + \ge\kappa/2-4t=3\kappa/8. +\] +All scalar normal norms and affine matrix Lipschitz constants are at +most $20P^3$ in these coordinates. It follows that the entire ball of +radius +\begin{equation} + \rho_0=P^{-20} + \label{eq:ta-aux-radius} +\end{equation} +about this auxiliary point satisfies all non-packing constraints. +In particular these estimates are uniform even when an active $u_e$ +is extremely small. + +By \Cref{lem:ta-continuation}, choose a stronger feasible point $x_*$, +using $z_e=a_e(y)$ and any representation +\Cref{eq:ta-conic-representation}. Let $\beta=grt$. Every nontrivial +partition has slack at least $\beta(|\mathcal P|-1)\ge\beta$ at $x_*$ +relative to \Cref{eq:ta-computational-packing}. Moreover, +\[ + \beta\ge\frac1{32(m+1)}\ge\frac1{32P}. +\] +At the auxiliary point every such slack is at least $-P^2$, because +all capacities are nonnegative and $Q^+\le r\le m$. +Mix a fraction $\theta=P^{-8}$ of the auxiliary point into $x_*$. +Since $P\ge5$, the new partition slacks are at least +$(1-\theta)\beta-\theta P^2\ge\beta/2$. +Convexity gives a ball of radius $\theta\rho_0=P^{-28}$ satisfying the +non-packing constraints about the mixture. A partition normal has at +most $m$ unit coefficients in the $z$ coordinates, so its norm is at +most $\sqrt m$. Thus the smaller ball of radius +\begin{equation} + \rho=P^{-30} + \label{eq:ta-inradius} +\end{equation} +satisfies every partition constraint as well. Their exponential number +does not change this uniform estimate. + +There is also a uniform outer ball. Testing \Cref{eq:ta-K} against the +centered identity gives $\tr Y\le6m$. Taking traces in the second LMI +of \Cref{eq:ta-conic-representation} gives +$\sum\lambda_{ijk}\le3m$. Together with $0\le y_e\le1$ and +$0\le z_e\le\kappa+1\le2$, these bounds place all free coordinates in +the ball of radius $P^3$. We may impose the larger radius $2P^3$ with +slack. This ball also has an exact rational separator: for a rational +query $a$ with $\|a\|>R$, the inequality +$ a\cdot x\le(R^2+\|a\|^2)/2$ contains the radius-$R$ ball +and strictly excludes $a$, by Cauchy--Schwarz and the arithmetic--geometric +mean inequality. All ball constants have logarithmic encoding length in $P$. +The rational central-cut ellipsoid procedure, using +\Cref{lem:ta-separation}, now finds an exactly feasible point in +polynomial work \cite[Theorem~3.2.1 and its proof]{GLS1988}. +Indeed, use our exact membership test at every oracle query and choose +the enclosing-volume cutoff below $(2\rho/D)^D$, the volume of a +coordinate cube contained in the proved radius-$\rho$ inner ball. +The small-volume outcome is impossible, so the algorithm must terminate +at an exactly accepted rational point. This argument uses the displayed +positive inradius; it does not assume exact solvability of an arbitrary +semidefinite feasibility problem. + +To bound the returned denominators independently of those in $u$, put +$L=\lceil\log_2P\rceil$, $\eta=2^{-32L}$ and $\gamma=2^{-36L}$, +and instead find $x$ such that +\begin{equation} + x\pm\eta e_j\in\mathcal F + \qquad(1\le j\le D), + \label{eq:ta-translated} +\end{equation} +where $\mathcal F$ is the full step feasible set, including the outer +bound. The intersection of these $2D$ translates still contains a ball +of radius at least $\rho-\eta\ge3\rho/4$, and has a separation oracle +by translating the queries. Use the corresponding volume cutoff with +$3\rho/4$ in place of $\rho$. Convexity implies that the cross-polytope +$\{x+v:\|v\|_1\le\eta\}$ lies in $\mathcal F$. +Round each coordinate of $x$ to the fixed dyadic mesh $\gamma$. +Its total rounding error is at most +$D\gamma/2\le\eta/(2P)<\eta$, so the rounded point remains feasible. +The outer bound gives at most $39L+O(1)$ bits per coordinate. +This argument does not round a +boundary solution without a margin. + +Finally $s$ and $Q$ have polynomial encoding length throughout the +prescribed schedule. At a rational $X\succeq L_H$, its centered inverse +and all old $u_e$ are computed exactly with polynomial bit operations. +Every new $Y$ lies on the fixed grid just described. Forming +$X^+=(1-t)X+tY$ increases encoding length by only a polynomial additive +amount per step. Over the polynomial number of steps neither +denominators nor oracle work can cascade beyond polynomial size. +\end{proof} + +\begin{proof}[Proof of \Cref{prop:ta-extraction}] +Apply \Cref{lem:ta-continuation,lem:ta-bits} at every step. +The computed capacities satisfy \Cref{eq:ta-computational-packing}; +\Cref{eq:ta-ramp-lower} therefore preserves +\Cref{eq:ta-invariant} with target $Q^+$. At termination $s\ge r^{1/4}$, +and every active edge has $R_e(X)<1/s\le r^{-1/4}$. +Dropping the baselines loses less than one across any partition, and +$u_e\le1$ gives +\[ + |E_G(\mathcal P)|\ge Q(|\mathcal P|-1)-1 + \ge (Q-1)(|\mathcal P|-1) + \quad(|\mathcal P|\ge2). +\] +Hence \Cref{eq:ta-integer-packing} supplies +$\lfloor Q-1\rfloor$ disjoint spanning trees. Construct them by the +graphic-matroid union procedure stated above. By +\Cref{eq:ta-target-bound}, their number is +$r(1-O(r^{-1/8}(1+\log r)))-O(1)$, which is at least +$(1-Ar^{-1/16})r$ after enlarging $A$ and $r_0$. All steps have the +claimed polynomial bit complexity. +\end{proof} diff --git a/preprints/A-polynomial-time-construction-of-strong-thin-trees-September-23-2026/build/source/sections/fractional_rounding.tex b/preprints/A-polynomial-time-construction-of-strong-thin-trees-September-23-2026/build/source/sections/fractional_rounding.tex new file mode 100644 index 0000000000000000000000000000000000000000..8a4a22cd58e9d8e72ecd913af1ce9aec32136fcb --- /dev/null +++ b/preprints/A-polynomial-time-construction-of-strong-thin-trees-September-23-2026/build/source/sections/fractional_rounding.tex @@ -0,0 +1,175 @@ +\section{Simultaneous cost-and-cut rounding} +\label{sec:fractional-rounding} + +The all-cut theorem also permits control of one nonnegative cost +functional. The reduction combines the truncation and rounding in +\eqref{eq:ta-binary-reduction} with repeated deletion and selection of +the least-cost tree, as in Oveis Gharan and Saberi +\cite[full preprint, Proposition~5.1]{OveisGharanSaberi2011}. +The rational compression below keeps the auxiliary graph polynomial +in the input bit length. + +\begin{corollary}[Thin and cheap rounding of fractional cuts] +\label{cor:ta-fractional-rounding} +Fix once and for all an integer $B\ge\max\{1,C\}$, where $C$ is the +constant in Theorem~\ref{thm:tt-algorithm}. Let $G=(V,E)$ be a finite undirected +graph on $n\ge2$ vertices, given by an explicit list of edge records; +in this corollary, loops and parallel records are allowed. Given +nonnegative rational numbers $x_e,c_e$ in binary, suppose that +\[ + x(\delta_G(S)):=\sum_{e\in\delta_G(S)}x_e\ge1 + \qquad(\varnothing\ne S\subsetneq V). +\] +There is a deterministic algorithm, polynomial in the total binary +input length, that returns a spanning tree +$T\subseteq\{e\in E:x_e>0\}$ satisfying simultaneously +\begin{align} + |\delta_T(S)|&\le24B\,x(\delta_G(S)) + &&(\varnothing\ne S\subsetneq V), + \label{eq:ta-fractional-cuts}\\ + c(T):=\sum_{e\in T}c_e&\le24B\sum_{e\in E}c_ex_e. + \label{eq:ta-fractional-cost} +\end{align} +The cut inequalities are a promise on the input; no metric assumption +on the costs is required. +\end{corollary} + +\begin{proof} +Discard loops, which affect no cut and can only reduce the nonnegative +cost sum. For each unordered nonloop pair $p$, let $E_p$ be its set of +edge records and put $X_p=\sum_{e\in E_p}x_e$. Discard pairs with $X_p=0$; +otherwise choose an original edge $e_p$ with $x_{e_p}>0$ of minimum +cost among such edges, and put $d_p=c_{e_p}$. This preserves every +cut mass and gives +\[ + \sum_p d_pX_p\le\sum_{e\in E}c_ex_e. +\] +Set $K=n^2$, $y_p=\min\{X_p,1\}$, and +$a_p=\lfloor2K y_p\rfloor$. Form an explicit loopless multigraph $A$ +with $a_p$ labelled copies of pair $p$, each of cost $d_p$. +Every cut has $y$-mass at least one: if a crossing coordinate was +truncated, that coordinate alone contributes one, and otherwise the +cut mass is unchanged. A cut crosses at most $n^2/4$ unordered pairs, +so rounding down loses at most $n^2/4$ from its scaled mass. Hence +\begin{align} + |\delta_A(S)|&\ge2K-n^2/4\ge K, + \label{eq:ta-fractional-lower}\\ + |\delta_A(S)|&\le2K\,x(\delta_G(S)), + \qquad + c(A):=\sum_p a_pd_p\le2K\sum_{e\in E}c_ex_e. + \label{eq:ta-fractional-upper} +\end{align} +Each $a_p\le2n^2$, and fewer than $n^2/2$ pairs occur, so +$|E(A)|0$. More generally, +every rational point in a spanning-tree polytope satisfies the cut +promise, since each constituent spanning tree meets every nontrivial +cut. The bit guarantee here is not a claim for arbitrary real-oracle +inputs. + +The earlier SODA version instead fixes an optimum Held--Karp solution +and defines its cost bound as $c(T)\le s\,OPT_{\rm HK}$ in +Definition~5.1; its Theorem~6.1 is the tour-augmentation theorem, not +the full-manuscript conjecture +\cite[Definition~5.1 and Theorem~6.1]{AsadpourEtAl2010}. +Constant-factor approximation for ATSP was already established +independently by Svensson, Tarnawski, and V\'egh and improved by Traub +and Vygen \cite{SvenssonTarnawskiVegh2020,TraubVygen2022}. +The consequence proved here concerns the underlying simultaneous +rounding formulation; no new approximation-order claim is made. diff --git a/preprints/A-polynomial-time-construction-of-strong-thin-trees-September-23-2026/build/source/sections/introduction.tex b/preprints/A-polynomial-time-construction-of-strong-thin-trees-September-23-2026/build/source/sections/introduction.tex new file mode 100644 index 0000000000000000000000000000000000000000..a175bba682e85710f8920552e31ca1f17572ae69 --- /dev/null +++ b/preprints/A-polynomial-time-construction-of-strong-thin-trees-September-23-2026/build/source/sections/introduction.tex @@ -0,0 +1,164 @@ +\section{Introduction} +\label{sec:introduction} + +A spanning tree retains connectivity with only $n-1$ edges. A thin +spanning tree must also use a small fraction of every cut, with the +same tree serving all cuts simultaneously. We give a deterministic polynomial-bit +construction at the inverse-connectivity scale. + +Throughout, graphs are finite undirected loopless multigraphs, and +every parallel copy counts separately. For $S\subseteq V(G)$, let +$\delta_G(S)$ be the multiset of edges with one endpoint in $S$ and +the other in its complement. The graph is $k$-edge-connected if +$|\delta_G(S)|\ge k$ for every nonempty proper $S$. A spanning tree +$T$ is \emph{$\alpha$-thin} in $G$ if +\[ + |\delta_T(S)|\le\alpha|\delta_G(S)| + \qquad(\varnothing\ne S\subsetneq V(G)). +\] +We consider two input representations: an explicit list of labelled +edge copies, or a list of endpoint pairs with nonnegative integer +multiplicities written in binary. In the second representation, the +number of edge copies may be exponential in the input length. + +\input{sections/main-theorem} + +The universal $C/k$ bound is the strong thin-tree conjecture. The +companion article \emph{The strong thin tree conjecture} +\cite{ThinTreesCompanion} proves the existence statement and supplies +the complete hierarchy geometry used here. Theorem~\ref{thm:tt-algorithm} +gives the positive algorithmic resolution in both finite input +models. Its order in $k$ is necessary: for two vertices joined by +$k$ parallel edges, the unique cut contains exactly one tree edge. + +The construction also gives simultaneous cost-and-cut rounding. Given +nonnegative rational edge weights whose mass on every cut is at least one, and any +nonnegative rational edge costs, we find a tree in the positive support +with every cut load and its total cost bounded by constant multiples of +the corresponding fractional quantities +(Corollary~\ref{cor:ta-fractional-rounding}). No metric assumption on +the costs is needed. + +\subsection{Context and prior work} + +Goddyn's qualitative conjecture asks whether every positive thinness +bound follows from sufficiently large edge connectivity, independently +of graph order \cite[Problem~4]{Goddyn2004}. The strong form specifies +the rate $C/k$; the distinction is made explicitly in +\cite[Conjecture~1.3 and the following discussion]{AnariOveisGharan2015}. +One source of algorithmic interest is the asymmetric traveling salesman +problem, which asks for a minimum-cost tour through the vertices of a +directed metric. Asadpour, Goemans, M\k{a}dry, Oveis Gharan, and Saberi +used maximum-entropy spanning-tree distributions to obtain an +$O(\log n/\log\log n)$ approximation +\cite{AsadpourEtAl2017}. Their rounding method also gives the +constructive $O(\log n/(k\log\log n))$ thin-tree scale; see +\cite[Section~1]{AnariOveisGharan2015}. + +For planar graphs, and more generally on each fixed orientable +surface, Oveis Gharan and Saberi obtained constructive inverse-connectivity +bounds \cite[Theorem~5.1]{OveisGharanSaberi2011}. +For general multigraphs, Anari and Oveis Gharan introduced +effective-resistance-reducing flows and locally connected hierarchies, +and proved the existence of +$\operatorname{poly}(\log\log n)/k$-thin trees +\cite[Corollary~1.8]{AnariOveisGharan2015}. Their geometric method is +the principal foundation of the hierarchy argument in the companion +and its extension below. The existence theorem does not itself give +a polynomial-time construction of those trees. + +Recent work has obtained inverse-connectivity bounds for structured +families of cuts. Klein and Olver treat every prescribed laminar +family \cite{KleinOlver2023}. Klein, Olver, and Yeoh construct a tree +crossing each cut of size less than $(1+1/40)k$ at most $88$ times +\cite[Theorem~1]{KleinOlverYeoh2026}. Our conclusion controls all cuts +of an arbitrary multigraph. Thin trees remain an independent +structural problem: constant-factor approximation for asymmetric +traveling salesman was already established by Svensson, Tarnawski, +and V\'egh and subsequently improved by Traub and Vygen +\cite{SvenssonTarnawskiVegh2020,TraubVygen2022}. + +\subsection{From the geometric existence proof to an algorithm} + +High edge connectivity supplies many edge-disjoint spanning trees +by the Nash--Williams--Tutte theorem +\cite{NashWilliams1961,Tutte1961}. A packing bounds the average +tree load on each fixed cut, but does not identify one tree that is +small on every cut. The companion retains a large packing through +successive sparsifications, reducing cut sizes and packing size by +almost the same factor. Two vector representations allow the same +edge subset to preserve cut control and enough rank information to +certify all partition inequalities for tree packing. Iterated matrix +halving with a bounded product of losses was used by Harvey and Olver +to obtain spectrally thin trees under an effective-resistance hypothesis +\cite[full version, Appendix~F]{HarveyOlver2014}. Here the additional +packing representation permits iteration from edge connectivity. + +A shortcut matrix is a positive quadratic form that bounds cut sizes; +its inverse assigns each edge a resistance, and edges of low resistance +are the candidates retained for sparsification. Three steps of the +existence argument require new computational work. The shortcut matrix +must be consistent with partitions determined by that same matrix; its +construction by compactness and a fixed point does not provide a uniform +finite search bound. Real whitening may introduce irrational entries, +so a polynomial-bit algorithm must specify its numerical data and control +every precision loss. Finally, the existential matrix partition theorem +of Marcus, Spielman, and Srivastava must be replaced by an algorithm +that finds the signs. We use the public constructive method of Ezeunala +and Jiang, with a quantitative implementation proved below, while +retaining the packing argument. + +\paragraph{A larger cone of geometric tests.} +The companion bounds a shortcut matrix on cut vectors. For optimization +we use centered positive semidefinite matrices whose squared distances +satisfy triangle inequalities. This cone has polynomially many displayed +constraints. Enlarging the cone strengthens the required shortcut +inequality, so the companion's cut-only statement cannot simply be +applied. We extend its proof by constructing a doubled distance-profile +polytope in which radii about vertex images are affine and radial motion +has exactly its metric length. The underlying distance-profile isometry +is classical \cite{Frechet1910,Kuratowski1935}; the additional radial +identities make the bag and charging arguments work in the new geometry. + +\paragraph{Continuous capacities and finite optimization.} +Instead of selecting edges abruptly at a resistance threshold, we assign +capacities through a continuous ramp. A convex response lowers the +resistance cutoff while maintaining the partition inequalities with a +controlled loss in packing size. +Uncrossing reduces a putative obstruction to a chain of partitions, +where the geometric response applies simultaneously. Exact rational +separation and a quantitative interior ball turn each response into a +finite computation using the ellipsoid method \cite{GLS1988}. +Constructive graphic matroid partition then recovers an integer packing +\cite{Edmonds1968,Terao2023}. + +\paragraph{Exact data and certified sparsification.} +A log-determinant normalization, following the variational method behind +radial isotropic position \cite[Section~2.2]{Barthe1998}, gives the second +vector representation. We approximate the stacked vectors rationally, +complete their covariance exactly to the identity, and apply constructive +matrix signing. This step builds on the Kadison--Singer partition theorem +of Marcus, Spielman, and Srivastava +\cite{MarcusSpielmanSrivastava2015} and the public rank-one discrepancy +method of Ezeunala and Jiang \cite{EzeunalaJiang2026}; the exact interface +and the constant used are proved in Appendix~\ref{app:signing}. Rational matrix +certificates imply every cut and packing estimate needed for the next +step. The product of the multiplicative losses remains bounded through +the iteration. A separate truncation and rounding reduction handles +binary multiplicities before any edge expansion: it produces a graph +with polynomially many explicit copies, independently of the numerical +values of the multiplicities and $k$. The final tree lifts to original +edge copies. + +\paragraph{Organization.} +Section~\ref{tg:section} proves the metric-cone hierarchy estimate and +its simultaneous response to a partition chain. Section~\ref{sec:ta-algorithm} +constructs a rational shortcut and a large packing of low-resistance edges. +Section~\ref{sec:normalization} builds the rank-preserving frame. +Section~\ref{sec:signing} provides exact rational padding and signing +certificates. Section~\ref{sec:iteration} combines the two representations, +iterates the sparsification, and proves the binary-multiplicity reduction. +Section~\ref{sec:fractional-rounding} records a simultaneous cost-and-cut +rounding consequence for rational fractional inputs. +Appendix~\ref{app:signing} gives the quantitative rational implementation +of the public rank-one signing method. diff --git a/preprints/A-polynomial-time-construction-of-strong-thin-trees-September-23-2026/build/source/sections/iteration.tex b/preprints/A-polynomial-time-construction-of-strong-thin-trees-September-23-2026/build/source/sections/iteration.tex new file mode 100644 index 0000000000000000000000000000000000000000..283a8db7e5cf3281b71ebfce32098933aa933216 --- /dev/null +++ b/preprints/A-polynomial-time-construction-of-strong-thin-trees-September-23-2026/build/source/sections/iteration.tex @@ -0,0 +1,150 @@ +\section{Packing-preserving sparsification and the final tree} +\label{sec:iteration} + + +\begin{lemma}[One sparsification step] +\label{lem:ta-sparsification} +For all sufficiently large integers $r$, from an explicit graph $H$ +containing $r$ disjoint spanning trees one can construct a subgraph +$H'=(V(H),A)$ and an integer $r'$ such that +\begin{align} + &r/4\le r'\le r/2,\qquad H'\text{ contains }r'\text{ disjoint spanning trees}, + \label{eq:ta-packing-halving}\\ + &|\delta_{H'}(S)|\le a_r|\delta_H(S)|\quad\text{for every cut}, + \qquad a_r=\tfrac12+3C_s r^{-1/8}, + \label{eq:ta-cut-halving}\\ + &\frac{a_r}{r'/r}\le1+C_1r^{-1/16} + \label{eq:ta-ratio-loss} +\end{align} +for an absolute $C_1$. The construction is deterministic and takes +polynomial bit time, with the certificate in \Cref{lem:ta-signing}. +\end{lemma} + +\begin{proof} +Use \Cref{prop:ta-extraction} and discard excess trees to obtain a union +$B$ of $q$ trees with $(1-Ar^{-1/16})r\le q\le r$. +Increase the cutoff so $q\ge r/2$, and apply +\Cref{lem:ta-frame,lem:ta-signing}. Put $\epsilon_r=C_s r^{-1/8}$. +The two principal blocks of \Cref{eq:ta-half-frame} give +\begin{equation} + L_A\preceq\tfrac12L_B+\epsilon_rX + \preceq\tfrac12L_H+\epsilon_rX, + \qquad \sum_{e\in A}z_ez_e^T\succeq(\tfrac12-\epsilon_r)I. + \label{eq:ta-two-blocks} +\end{equation} +A centered cut indicator $x$ has $xx^T\in\mathcal N$. +By \Cref{eq:ta-K}, $x^TXx\le3x^TL_Hx$; evaluating the first inequality +on $x$ proves \Cref{eq:ta-cut-halving}. + +For packing, fix a partition into $d$ parts. Let $W$ be the span of all +$z_e$ with endpoints in a common part, and let $\Pi$ be the orthogonal +projection onto $W^\perp$. Rank preservation from \Cref{lem:ta-frame} +gives $\dim W\le\sum_{P\in\mathcal P}(|P|-1)=n-d$, hence +$\rank\Pi\ge d-1$. Taking the trace after projection in the second +inequality of \Cref{eq:ta-two-blocks}, and then using +\Cref{eq:ta-frame}, gives +\[ + (\tfrac12-\epsilon_r)(d-1) + \le\sum_{e\in E_A(\mathcal P)}\|\Pi z_e\|^2 + \le |E_A(\mathcal P)|\frac{1+4/r}{q}. +\] +The integer packing criterion thus supplies at least +\begin{equation} + \left\lfloor + \frac{q(1/2-\epsilon_r)}{1+4/r} + \right\rfloor + \label{eq:ta-projected-packing} +\end{equation} +spanning trees in $H'$. +Compute its maximum packing number, capped at $\lfloor r/2\rfloor$, +by binary search with the constructive graphic-matroid union algorithm, +and denote the result by $r'$. The lower bound in +\Cref{eq:ta-projected-packing} is already at most $\lfloor r/2\rfloor$, +so capping does not reduce that guarantee. Consequently +\[ + \frac{r'}r\ge\frac12(1-C_2r^{-1/16}) +\] +for an absolute $C_2$, absorbing the floor loss $1/r$, the extraction +loss, and $\epsilon_r$. Taking a sufficiently large absolute cutoff +gives \Cref{eq:ta-packing-halving}. Dividing +$a_r=\tfrac12(1+6\epsilon_r)$ by this lower bound and enlarging the +absolute constant proves \Cref{eq:ta-ratio-loss}. +\end{proof} + +\begin{proof}[Proof of \Cref{thm:tt-algorithm} for explicit graphs] +First suppose $n\ge2$. Put +$r_{\mathrm{init}}=\max\{1,\lfloor k/2\rfloor\}$. +For a partition into $d\ge2$ parts, summing the $k$-edge-connectivity +bounds on its parts gives $|E_G(\mathcal P)|\ge kd/2$. +Thus \Cref{eq:ta-integer-packing} supplies +$\lfloor k/2\rfloor$ trees, and connectedness supplies one when $k=1$. +Construct this initial packing. Notice that +$r_{\mathrm{init}}\ge k/3$. + +Starting with $H_0=G$ and $r_0=r_{\mathrm{init}}$, apply +\Cref{lem:ta-sparsification} whenever the current parameter is at least +the fixed absolute cutoff $R_0$. Write $H_i,r_i$ for the resulting +sequence. +At termination $1\le r_Jc_{uv}$, a tree uses at +most one edge of any parallel pair. Every selected pair has +$c_{uv}>0$, so its tree edge maps to a genuine original edge. +All arithmetic in \Cref{eq:ta-binary-reduction} is polynomial in +$\log k$, the multiplicity encoding lengths, and $n$; the main routine +uses the new parameter $n^2$. Enlarging the universal constant by two +completes \Cref{thm:tt-algorithm} in the binary model. diff --git a/preprints/A-polynomial-time-construction-of-strong-thin-trees-September-23-2026/build/source/sections/main-theorem.tex b/preprints/A-polynomial-time-construction-of-strong-thin-trees-September-23-2026/build/source/sections/main-theorem.tex new file mode 100644 index 0000000000000000000000000000000000000000..75961993960cf5224369be0c45025d4fb21dc306 --- /dev/null +++ b/preprints/A-polynomial-time-construction-of-strong-thin-trees-September-23-2026/build/source/sections/main-theorem.tex @@ -0,0 +1,15 @@ +\begin{theorem}[Algorithmic strong thin trees] +\label{thm:tt-algorithm} +There is an absolute constant $C$ with the following property. Given a +finite $k$-edge-connected multigraph $G$ on at least one vertex and +an integer $k\ge1$, +a deterministic algorithm returns a spanning tree $T$ satisfying +\begin{equation} + |\delta_T(S)|\le \frac Ck\,|\delta_G(S)| + \qquad(\varnothing\ne S\subsetneq V(G)) + \label{eq:ta-thin} +\end{equation} +in time polynomial in the total binary input length, whether edges +are listed explicitly or multiplicities are encoded in binary. +For a one-vertex graph the answer is the empty tree. +\end{theorem} diff --git a/preprints/A-polynomial-time-construction-of-strong-thin-trees-September-23-2026/build/source/sections/normalization.tex b/preprints/A-polynomial-time-construction-of-strong-thin-trees-September-23-2026/build/source/sections/normalization.tex new file mode 100644 index 0000000000000000000000000000000000000000..de1f5543c2735d9ce4cf463956fb1e335dc5fe48 --- /dev/null +++ b/preprints/A-polynomial-time-construction-of-strong-thin-trees-September-23-2026/build/source/sections/normalization.tex @@ -0,0 +1,130 @@ +\section{A rank-preserving normalized frame} +\label{sec:normalization} + + +The shortcut controls effective resistances. A second vector system +encodes the spanning-tree packing, so that signing can preserve both +properties simultaneously. The normalization uses the log-determinant +variational method for radial isotropic position +\cite[Section~2.2]{Barthe1998}. We prove the approximate positive-weight +form and its bit bounds directly, so no attainment assumption is needed. + +\Needspace{4\baselineskip} +\begin{lemma}[A computable normalized frame] +\label{lem:ta-frame} +Suppose $B$ is the union of $q$ edge-disjoint spanning trees, where +$r/2\le q\le r$ and $r$ exceeds an absolute constant. There are vectors +$z_e\in\R^{n-1}$, obtained from faithful reduced incidence columns by one +common invertible linear map and positive individual scalings, such that +\begin{equation} + \sum_{e\in B}z_ez_e^T=I,\qquad + \|z_e\|^2\le\frac{1+4/r}{q}. + \label{eq:ta-frame} +\end{equation} +They admit rational approximations to Euclidean error $2^{-b}$ in time +polynomial in $n+|B|+r+b$. An exact specification uses polynomial-bit +positive rational edge weights followed by whitening; in particular, +every subset has exactly its original incidence rank. +\end{lemma} + +\begin{proof} +Delete one row of the incidence matrix and write the resulting columns +as $c_e$. Put $M=|B|=q(n-1)$ and consider +\begin{equation} + N(d)=\sum_{e\in B}\exp(d_e)c_ec_e^T,\qquad + f(d)=\log\det N(d)-\frac1q\sum_{e\in B}d_e. + \label{eq:ta-logdet} +\end{equation} +By Cauchy--Binet, +$\det N(d)=\sum_T\exp(\sum_{e\in T}d_e)$, where $T$ ranges over the +spanning trees of $B$: reduced incidence determinants of trees have +absolute value one, and other maximal minors vanish. Keeping the $q$ +specified disjoint trees and applying the arithmetic--geometric mean +inequality gives +\[ + \det N(d)\ge q\exp\left(\frac1q\sum_{e\in B}d_e\right), + \qquad f(d)\ge\log q. +\] +Also $f(0)\le(n-1)\log(2M)$, so the initial gap above the infimum is +bounded by $n\log(2M)$. Differentiation gives +\begin{equation} + \partial_e f(d)=\exp(d_e)c_e^TN(d)^{-1}c_e-\frac1q. + \label{eq:ta-gradient} +\end{equation} +The Hessian is the covariance matrix of the tree indicator under the +probability distribution proportional to +$\exp(\sum_{e\in T}d_e)$. Hence it is PSD with operator norm at most +$M$, since these indicators have squared norm $n-1\le M$. +Each leverage score in \Cref{eq:ta-gradient} lies in $[0,1]$; consequently +$\|\nabla f\|\le\sqrt M$. + +Start at $d=0$. At each step compute a rational gradient estimate $v$ +with $\|v-\nabla f(d)\|\le1/(10r^2)$. If $\|v\|\le2/r^2$, stop; +otherwise update +$d\leftarrow d-v/(2(M+1))$. The norm comparison can be made by squaring +rational quantities. The Hessian bound yields +\[ + f(d-\alpha v)\le f(d)-\alpha\nabla f(d)\cdot v + +\tfrac12M\alpha^2\|v\|^2, + \qquad \alpha=\frac1{2(M+1)}. +\] +At a nonstopping step the gradient error is at most $\|v\|/20$. +Thus the decrease is at least +$\alpha\|v\|^2/2\ge2\alpha/r^4$. +The gap bound permits at most +$O((M+1)r^4 n\log(2M))$ such steps. At termination +\begin{equation} + \|\nabla f(d)\|\le3/r^2. + \label{eq:ta-gradient-stop} +\end{equation} + +Here is an explicit precision justification for this descent. Its +iteration bound is prescribed in advance, and each coordinate changes +by at most two in one step. Thus $|d_e|\le B_0$ for a known polynomial +$B_0$ in $n+M+r$. On that entire box, +\[ + e^{-B_0}N(0)\preceq N(d)\preceq e^{B_0}N(0). +\] +The matrix $N(0)$ is positive definite, has norm at most $2M$, and has +positive integral determinant. Its smallest eigenvalue is therefore +at least $(2M)^{-(n-2)}$. The upper norm and inverse-norm bounds for +$N(d)$ have polynomial logarithms. Computing exponentials to +polynomially many bits, forming rational approximations to $N(d)$, +and solving rational linear systems therefore evaluates +\Cref{eq:ta-gradient} to the requested error in polynomial bit work. +For example, the identity +$\widetilde N^{-1}-N^{-1}=\widetilde N^{-1}(N-\widetilde N)N^{-1}$ +controls the inverse error once +$\|\widetilde N-N\|\le\lambda_{\min}(N)/2$. +Gradient estimates may be rounded on one fixed dyadic grid $2^{-b_0}$ +sufficiently fine for the stated error. Starting at zero, every iterate +then lies on the grid $[2(M+1)2^{b_0}]^{-1}\mathbb Z^M$. +The prescribed iteration and magnitude bounds therefore keep every +$d_e$ a polynomial-bit rational. + +At the terminal $d$, choose positive rational weights $a_e$ satisfying +$|a_e/\exp(d_e)-1|\le1/(100r^2)$ and put +$N_a=\sum_e a_ec_ec_e^T$. +Compute its rational diagonal elimination $N_a=L D_aL^T$ with positive +diagonal $D_a$, set $F=D_a^{-1/2}L^{-1}$, and define +$z_e=\sqrt{a_e}Fc_e$. Then $FN_aF^T=I$. The relative matrix bounds on $N_a$ and +\Cref{eq:ta-gradient-stop} imply +\[ + \|z_e\|^2 + \le\frac{1+1/(100r^2)}{1-1/(100r^2)} + \left(\frac1q+\frac3{r^2}\right) + \le\frac1q+\frac4{r^2}\le\frac{1+4/r}{q}. +\] +Their outer products sum exactly to $I$. All $a_e$ are strictly positive +and $F$ is invertible, proving rank preservation. + +For the promised approximations, perform exact rational diagonal +elimination on the positive definite $N_a$ and then approximate the +positive square roots of its pivots. Every pivot and inverse pivot +has polynomial logarithmic bounds, either by the preceding spectral +bounds or by determinant bounds for rational principal submatrices. +Triangular solves and scalar square roots therefore approximate $F$ +and all $z_e$ to $2^{-b}$ with polynomially many guard bits and +polynomial work. These approximations refer to the single exact frame +just defined; rounding does not redefine its subset ranks. +\end{proof} diff --git a/preprints/A-polynomial-time-construction-of-strong-thin-trees-September-23-2026/build/source/sections/signing.tex b/preprints/A-polynomial-time-construction-of-strong-thin-trees-September-23-2026/build/source/sections/signing.tex new file mode 100644 index 0000000000000000000000000000000000000000..29a5568fe5debb006b53233be3f8495b9b8d654a --- /dev/null +++ b/preprints/A-polynomial-time-construction-of-strong-thin-trees-September-23-2026/build/source/sections/signing.tex @@ -0,0 +1,206 @@ +\section{Exact rational padding and certified signing} +\label{sec:signing} + + +The deterministic signing theorem proved in Appendix~\ref{app:signing} +has an exact input promise. For explicit +rational columns $v_1,\ldots,v_M\in(\Q+i\Q)^d$ satisfying +$\sum_i v_iv_i^*=I$, it returns signs $\sigma_i\in\{-1,1\}$ with +\begin{equation} + \left\|\sum_i\sigma_i v_iv_i^*\right\|_{\op} + \le40\sqrt{\Delta},\qquad + \Delta=\max_i\|v_i\|^2, + \label{eq:ta-signing-input} +\end{equation} +in time polynomial in the full binary input length. This is +\Cref{thm:rational-signing}, a quantitative form of the method of +Ezeunala and Jiang~\cite{EzeunalaJiang2026}. Real rational columns are a +special case. We first establish an elementary exact factorization, then prove the +transfer required here, including identity completion and a rational +success certificate. + +\begin{lemma}[Rational positive semidefinite decomposition] +\label{lem:rational-psd} +Every rational Hermitian positive semidefinite matrix $A$ can be written +exactly as +\[ + A=\sum_{j=1}^M v_jv_j^*, + \qquad v_j\in\Q(\mathrm i)^n, +\] +in polynomial bit time, with $M$ and the total encoding length of the +columns polynomial in the encoding length of $A$. If $A$ is real, +the columns may be taken real. For $A=0$, the sum may be empty. +\end{lemma} + +\begin{proof} +If a positive semidefinite matrix has all diagonal entries zero, it is +zero, since its two-by-two principal minors give +$|a_{ij}|^2\le a_{ii}a_{jj}$. Otherwise permute a positive diagonal +entry into the first position and write +\[ + A=\begin{pmatrix}d&b^*\\ b&C\end{pmatrix} + =d\begin{pmatrix}1\\b/d\end{pmatrix} + \begin{pmatrix}1\\b/d\end{pmatrix}^{\!*} + +\begin{pmatrix}0&0\\0&C-bb^*/d\end{pmatrix}. +\] +Here $d$ is positive rational and the Schur complement is positive +semidefinite: evaluate the quadratic form of $A$ at +$(-b^*x/d,x)$. Repetition gives a rational pivoted $LDL^*$ +factorization, or equivalently $A=\sum_j d_j u_ju_j^*$ with positive +rational $d_j$ and rational complex $u_j$. + +All elimination entries have polynomial encoding length. To see this +directly, clear the input denominators by one positive integer $q$. +The resulting Gaussian-integer matrix has polynomial-bit entries, +and every minor has polynomial bit length by the determinant +expansion. At any pivot order, the pivots and elimination coefficients +are ratios of these minors, with an additional factor $q$ for the +pivots. There are at most $n$ pivots, so rational elimination takes +polynomial bit time. + +For a pivot $d=a/b>0$ with positive integers $a,b$, expand +$ab=\sum_k c_k4^k$, where $c_k\in\{0,1,2,3\}$. Then +\[ + d=\sum_k c_k\left(\frac{2^k}{b}\right)^2. +\] +Replace $d u u^*$ by $c_k$ copies of the outer product of +$(2^k/b)u$ for each $k$. The number of copies and their bit lengths +are polynomial in those of $a,b,u$. This proves every assertion, +including the real case. +\end{proof} + + +\begin{lemma}[Checked signing of the stacked frame] +\label{lem:ta-signing} +Let $X$ and the union $B$ of $q$ trees be as in +\Cref{prop:ta-extraction}, retaining $r/2\le q\le r$ trees, and let +$z_e$ be the frame of \Cref{lem:ta-frame}. Put +\begin{equation} + w_e=(X^{-1/2}b_e,z_e)\qquad(e\in B). + \label{eq:ta-stacked} +\end{equation} +For an absolute constant $C_s$, a deterministic polynomial-time procedure +finds $A\subseteq B$ satisfying +\begin{equation} + \left\|\sum_{e\in A}w_ew_e^T + -\frac12\sum_{e\in B}w_ew_e^T\right\|_{\op} + \le C_s r^{-1/8}. + \label{eq:ta-half-frame} +\end{equation} +An exact polynomial-time rational PSD test certifies the output. +\end{lemma} + +\begin{proof} +The first block of \Cref{eq:ta-stacked} is expressed in orthonormal +coordinates on $\one^\perp$; its particular orthogonal basis is +irrelevant. Since $L_B\preceq L_H\preceq X$, its outer-product sum is +at most $I$, while the second block sums to $I$. Block +Cauchy--Schwarz therefore gives +\begin{equation} + \sum_{e\in B}w_ew_e^T\preceq2I. + \label{eq:ta-subisotropic} +\end{equation} +Individual squared norms are at most +$r^{-1/4}+(1+4/r)/q\le5r^{-1/4}$ for the chosen cutoff. + +Write $n=|V(H)|$, $m=|E(H)|$, and $P=n+m+2$, as in +\Cref{lem:ta-bits}. The centered positive definite $X$ has polynomial-bit rational entries, +with smallest eigenvalue at least $P^{-3}$ and norm at most $6m$. +As in the proof of \Cref{lem:ta-frame}, elimination, square-root +approximation, and solves compute a whitening of this block to any +requested polynomial number of bits. More explicitly, in the rational +coordinates used above write $X=CA_XC^T$ and $b_e=Cc_e$. Then +$c_e^TA_X^{-1}c_f=b_e^TX^{-1}b_f$, so whitening $A_X$ gives the same +Gram matrix as the displayed block and differs from it only by an +orthogonal map. The smallest eigenvalue of $A_X$ is at least +$P^{-3}/n$, because the centered projection of $(v,0)$ has squared norm +at least $\|v\|^2/n$. Thus this computation has the same polynomial +precision bounds. Combining it with \Cref{lem:ta-frame}, compute +rational columns $a_e$ approximating $w_e/2$ to Euclidean error at most +$1/(1000|B|r)$. Because \Cref{eq:ta-subisotropic} implies +$\|w_e/2\|\le1$, the outer-product identity +\[ + aa^T-vv^T=(a-v)a^T+v(a-v)^T +\] +shows that +\begin{equation} + \sum_{e\in B}\|a_ea_e^T-w_ew_e^T/4\|_{\op} + \le\frac1{100r},\qquad + \Sigma:=\sum_{e\in B}a_ea_e^T\preceq\frac34I. + \label{eq:ta-approximate-data} +\end{equation} +The same estimate controls every signed sum, without knowing the signs. +Also $\|a_e\|^2\le3r^{-1/4}$. + +Apply \Cref{lem:rational-psd} to the rational PSD residual +$I-\Sigma$, obtaining polynomially many polynomial-bit real rational +columns $v_1,\ldots,v_J$ with +\begin{equation} + VV^T=\sum_{j=1}^Jv_jv_j^T=I-\Sigma, + \qquad V=[v_1\ \cdots\ v_J]. + \label{eq:ta-padding-factor} +\end{equation} +In particular $\|V\|_{\op}\le1$ and each $\|v_j\|\le1$. +Extend the data columns by zeros in $J$ additional coordinates. Set +$N=2r$. For each $j$, add $N^2$ columns +\begin{equation} + p_{j,k}=\frac1N(v_j,\tau_{j,k}f_j), + \qquad 1\le k\le N^2, + \label{eq:ta-padding-columns} +\end{equation} +where $f_j$ is the $j$th extra coordinate vector and exactly half the +$\tau_{j,k}$ equal $1$, half $-1$. The parity requirement holds because +$N$ is even. Their unsigned sum is +\[ + \sum_{j,k}p_{j,k}p_{j,k}^T + =\begin{pmatrix}I-\Sigma&0\\0&I_J\end{pmatrix}. +\] +Together with the extended data this is exactly the identity. +Every padding squared norm is at most $2/N^2$, so the maximum squared +norm $\Delta$ of the entire rational system satisfies +$\Delta\le3r^{-1/4}$. The number $|B|+JN^2$ of explicit columns and +their binary lengths are polynomial, since $r\le m$. + +Apply \Cref{eq:ta-signing-input}. Write $S$ for the augmented signed +matrix, and distinguish the output signs $\sigma_{j,k}$ from the fixed +construction signs $\tau_{j,k}$. Its lower principal block is +\[ + D_0=\diag\left(N^{-2}\sum_k\sigma_{1,k},\ldots, + N^{-2}\sum_k\sigma_{J,k}\right). +\] +The padding contribution to its upper principal block is exactly +\begin{equation} + VD_0V^T. + \label{eq:ta-padding-removal} +\end{equation} +If $\|S\|\le T$, then $\|D_0\|\le T$ and +$\|VD_0V^T\|\le T$. Subtracting the latter from the full upper block +gives +\begin{equation} + \left\|\sum_{e\in B}\sigma_ea_ea_e^T\right\| + \le2T. + \label{eq:ta-data-signing} +\end{equation} +No assertion about the signed off-diagonal block is needed. + +For a fully rational certificate, choose a dyadic $d_0$ with +$\sqrt\Delta\le d_0\le2\sqrt\Delta$ by squared comparisons and set +$T=2^{15}d_0$. Check $TI-S\succeq0$ and $TI+S\succeq0$ exactly by +rational elimination. Since $40<2^{15}$, the signing theorem guarantees +that both tests pass. The certified output satisfies +\Cref{eq:ta-data-signing} and $T\le2^{16}\sqrt\Delta$. + +Return the positive data signs $A=\{e:\sigma_e=1\}$. +By \Cref{eq:ta-approximate-data,eq:ta-data-signing}, its centered half +sum has norm at most +\[ + \frac12\left\|\sum_e\sigma_e w_ew_e^T\right\| + \le4T+\frac2{100r} + \le2^{20}r^{-1/8}. +\] +Thus one may take $C_s=2^{20}$. The rational PSD tests have polynomial +cost and certify all the required matrix estimates simultaneously. +The signing algorithm and all numerical routines above have prescribed +polynomial precision and iteration bounds. This proves deterministic +polynomial bit complexity. +\end{proof} diff --git a/preprints/A-polynomial-time-construction-of-strong-thin-trees-September-23-2026/build/source/sections/signing_interface.tex b/preprints/A-polynomial-time-construction-of-strong-thin-trees-September-23-2026/build/source/sections/signing_interface.tex new file mode 100644 index 0000000000000000000000000000000000000000..36cb304f079802129d7911e149490e66d6ee6b46 --- /dev/null +++ b/preprints/A-polynomial-time-construction-of-strong-thin-trees-September-23-2026/build/source/sections/signing_interface.tex @@ -0,0 +1,581 @@ +\section{A finite-bit signing interface} +\label{app:signing} + +We record a conservative quantitative form of the rank-one signing +method of Ezeunala and Jiang~\cite{EzeunalaJiang2026}. The constants +below suffice for the exact padding argument. We include their +accounting and the finite-precision implementation to make the input +interface explicit. + +\begin{theorem}[Rational rank-one signing] +\label{thm:rational-signing} +Given rational Hermitian matrices $H_1,\ldots,H_N$ of rank at most one, +a deterministic algorithm returns signs $\sigma_i\in\{-1,1\}$ such that +\[ + \left\|\sum_i\sigma_iH_i\right\| + \le40\left\|\sum_iH_i^2\right\|^{1/2}. +\] +Its running time is polynomial in the total binary encoding length. +Consequently, for rational columns $v_i$ with $\sum_i v_iv_i^*=I$, +the same algorithm gives discrepancy at most +$40\sqrt{\max_i\|v_i\|^2}$. +\end{theorem} + +The last assertion follows from +$(v_iv_i^*)^2=\|v_i\|^2v_iv_i^*$ and positivity. Forming the outer +products has polynomial bit cost. We prove the matrix assertion next. + +\subsection{The potential and its quantitative estimates} + +In dimension zero, any signs suffice. Otherwise discard zero matrices; +if none remain, any signs again suffice. A nonzero +rank-one Hermitian matrix has one nonzero eigenvalue, equal to its +trace. With $D=\sum_i|\operatorname{Tr}H_i|$, put +\[ + A_i=D^{-1}\operatorname{sign}(\operatorname{Tr}H_i)H_i, + \qquad \nu_0=\left\|\sum_iA_i^2\right\|. +\] +These are rational positive semidefinite matrices and +$\sum_i\operatorname{Tr}A_i=1$. Matrix Cauchy--Schwarz gives +$1/(Nd^2)\le\nu_0\le1$, where $d$ is the matrix order. Rational +positive-semidefiniteness tests and bisection compute rational parameters +\[ + \nu_0\le\nu\le\min\{1,1.001\nu_0\},\qquad + \varepsilon=\nu/d,\qquad + \frac{\sqrt\nu}{200N}\le\lambda\le\frac{\sqrt\nu}{100N}. +\] +Set $c=384$, $K=100$, and $\sigma=\lambda^2/\nu$. Thus +$\varepsilon\ge1/(Nd^3)$ and +$1/(40000N^2)\le\sigma\le1/(10000N^2)$. + +For $x\in[-1,1]^N$, let +\[ + S(x)=\sum_i x_iA_i,\quad + \psi_i=(1-x_i^2)^{1/3},\quad + \eta_x(Z)=c\sum_i\psi_iA_iZA_i. +\] +Define $R(x)$ as the minimum of +$t+\varepsilon\operatorname{Tr}(X+Y)$ over $X,Y\succ0$ subject to +\begin{equation} + X^{-1}+S(x)+\eta_x(Y)\preceq tI, + \qquad Y^{-1}-S(x)+\eta_x(X)\preceq tI, + \label{eq:sign-potential} +\end{equation} +and put $\Psi(x)=R(x)+\lambda\sum_i\psi_i$. + +The variational argument of Ezeunala and Jiang +\cite[Section~4.1]{EzeunalaJiang2026} uses only $c>0$ and gives +\begin{equation} + \|S(x)\|\le R(x)\le\|S(x)\|+2\sqrt{(c+2)\nu}, + \qquad \Psi(0)\le(2\sqrt{386}+0.01)\sqrt\nu. + \label{eq:sign-potential-bound} +\end{equation} +For clarity, its quantitative ingredients are as follows. The minimizer +is unique; with its positive definite dual multipliers $P,Q$ it satisfies +\begin{gather} + (2K)^{-1}I\preceq X,Y\preceq(K/\varepsilon)I, + \quad P,Q\preceq I,\quad \operatorname{Tr}(P+Q)=1,\nonumber\\ + P=X\eta_x(Q)X+\varepsilon X^2, + \qquad Q=Y\eta_x(P)Y+\varepsilon Y^2, + \label{eq:sign-dual} +\end{gather} +and both inequalities in \eqref{eq:sign-potential} are tight. Indeed, +the scalar choice $X=Y=aI$ proves the upper bound in +\eqref{eq:sign-potential-bound}; its value is below $420$ such that the right side is at most $(1-\delta)(P,Q)$ +in componentwise Loewner order. Thus $\mathcal T$ is a strict +contraction in the norm +\[ + \|(U,V)\|_{P,Q} + =\inf\{s\ge0:-s(P,Q)\preceq(U,V)\preceq s(P,Q)\}. +\] +The convergent series +$\mathcal L^{-1}=\sum_{j\ge0}\mathcal T^j\mathcal C^{-1}$ is positive. +Its value on $(I,I)$ is $(P,Q)/\varepsilon\preceq(I,I)/\varepsilon$. +Consequently its induced norm for +$\|(U,V)\|_{\max}=\max\{\|U\|_{\op},\|V\|_{\op}\}$ is at most +$\varepsilon^{-1}$. Finally, $\mathcal L$ and its inverse are +self-adjoint for the Frobenius pairing. The Frobenius operator norm +of $\mathcal L^{-1}$ therefore equals its spectral radius, which is +bounded by every induced norm. This proves \eqref{eq:sign-L}. + +The Hessian $\mathcal H$ of the Lagrangian in $(X,Y)$ +satisfies +\begin{equation} + \frac{2\varepsilon^2}{K}I\preceq\mathcal H, + \qquad \|\mathcal H\|\le16K^3. + \label{eq:sign-H} +\end{equation} +The lower bound follows by substituting $P\succeq\varepsilon X^2$ in +$2\operatorname{Tr}(PX^{-1}UX^{-1}UX^{-1})$ and using +$X^{-1}\succeq(\varepsilon/K)I$; the other block is identical. +The upper bound uses three inverse factors of norm at most $2K$ in +each of the two terms of a Hessian block. + +\subsection{A quantified direction of decrease} + +A coordinate is active when $|x_i|<1$; let $n$ be their number. +In the following differential calculations, vectors and derivatives +are restricted to these coordinates, with the other coordinates fixed +on the current open face. +Write $A_i=u_iu_i^*$ in this analysis and set +\[ + a_i=u_i^*Xu_i,\quad b_i=u_i^*Yu_i,\quad + p_i=u_i^*Pu_i,\quad q_i=u_i^*Qu_i. +\] +The factors $u_i$ need not be rational: the algorithm below uses only +the rational matrices $A_i$. If $c\psi_i b_i\ge1-x_i$, moving $x_i$ +to $1$ preserves feasibility in \eqref{eq:sign-potential} with the +same $(t,X,Y)$ and decreases $\Psi$ by at least $\lambda\psi_i$. +The symmetric condition $c\psi_i a_i\ge1+x_i$ permits the move to $-1$. +We may therefore assume +\begin{equation} + c\psi_i a_i<1+x_i,\qquad c\psi_i b_i<1-x_i + \quad\text{for all active }i. + \label{eq:sign-light} +\end{equation} + +Put $\alpha_i=1+c\psi_i'b_i$, $\beta_i=1-c\psi_i'a_i$, and +\[ + T_i^2=\alpha_i\beta_i/\psi_i,\quad + \ell_i=\alpha_i/T_i,\quad m_i=\beta_i/T_i,\qquad + \widetilde a_i=\ell_i a_i,\quad \widetilde b_i=m_i b_i,\quad + z_i=\ell_i p_i,\quad w_i=m_i q_i. +\] +For $k_i=-\psi_i''$, elementary substitution in +\eqref{eq:sign-light} gives +\begin{equation} + \frac{k_i}{\alpha_i\beta_i}\ge\frac13, + \quad\frac{k_i}{T_i^2}\ge\frac13, + \quad T_i^{-2}\le\frac32, + \quad\frac{|\psi_i'|}{\psi_i k_i}\le1, + \quad\widetilde a_i\widetilde b_i\le c^{-2}, + \quad\widetilde a_i^2,\widetilde b_i^2\le3c^{-2}. + \label{eq:sign-scalar} +\end{equation} +Here is a direct check. By symmetry take $x_i\ge0$ and put +$r=1-x_i^2$. Then +$2/3\le\alpha_i\beta_i\le2/r$ and +$k_i=(2/3)r^{-5/3}(1+x_i^2/3)$. +Also $\widetilde a_i\widetilde b_i=\psi_i a_i b_i +\le r^{2/3}/c^2$. For +$u=c\psi_i a_i/(1+x_i)$ and $v=c\psi_i b_i/(1-x_i)$ in $(0,1)$, +\[ + c^2\widetilde a_i^2 + =r^{2/3}u^2\frac{1+x_i-2x_iv/3}{1-x_i+2x_iu/3},\qquad + c^2\widetilde b_i^2 + =r^{2/3}v^2\frac{1-x_i+2x_iu/3}{1+x_i-2x_iv/3}. +\] +Maximizing the first in $u$ and minimizing $v$, and maximizing the +second in $u,v$, bounds them by $3$ and $1$. +For negative $x_i$ the roles interchange. These formulas also prove +the remaining assertions of \eqref{eq:sign-scalar}. + +We quantify the covariance construction in +\cite[Lemma~4.5]{EzeunalaJiang2026}. Define the real symmetric, +entrywise nonnegative positive semidefinite matrices +\[ + \mathsf A_{ij}=c\ell_i\ell_j|u_i^*Xu_j|^2, + \qquad \mathsf B_{ij}=cm_im_j|u_i^*Yu_j|^2, +\] +and $D_F=\operatorname{diag}(\widetilde a_i z_i)$, +$D_G=\operatorname{diag}(\widetilde b_i w_i)$. +The dual identities give $\mathsf A w\le z$ and +$\mathsf Bz\le w$ coordinatewise. Moreover, +\[ + \|D_G^{1/2}\mathsf A D_F^{-1/2}\|_F^2\le n/c, + \qquad + \|D_F^{1/2}\mathsf B D_G^{-1/2}\|_F^2\le n/c. +\] +For example, $\mathsf A_{ij}^2\le +c\widetilde a_i\widetilde a_j\mathsf A_{ij}$ and +$\widetilde a_i\widetilde b_i\le c^{-2}$ bound the first sum by +$c^{-1}\sum_j(\mathsf A w)_j/z_j$. + +Restrict $D_F^{1/2}F$ and $D_G^{1/2}G$ to right singular directions +with singular value at most $1/2$ for these two maps. Each restriction +has codimension at most $4n/c$. Impose also +$(I-\mathsf A)F+(I-\mathsf B)G=0$. The resulting subspace +$\mathcal G\subseteq\mathbb R^{2n}$ has dimension at least $n-8n/c$. +The covariance theorem of Bansal and Garg~\cite[Theorem~6]{BansalGarg2017}, +in the form stated by Bansal~\cite[Theorem~2.5]{Bansal2024} and recalled +in \cite[Lemma~2.1]{EzeunalaJiang2026}, supplies a +positive semidefinite $\Gamma$ with +\[ + \operatorname{range}\Gamma\subseteq\mathcal G,\quad + \Gamma\preceq4\operatorname{diag}\Gamma,\quad + \Gamma_{jj}\le1,\quad + \operatorname{Tr}\Gamma\ge(1/2-8/c)n=23n/48. +\] +Take a centered finitely supported $(F,G)$ with covariance $\Gamma$ +and set $H=F-\mathsf BG=\mathsf AF-G$, $h_i=H_i/T_i$, and +$\omega_i=\mathbb E H_i^2$. +The two singular-value restrictions give +\[ + \|F\|_{D_F}^2+\|G\|_{D_G}^2 + \le4\sum_i(\widetilde a_i z_i+\widetilde b_i w_i)H_i^2. +\] +The response matrices are +$\dot X=X(\sum_i\ell_iF_iA_i)X$ and +$\dot Y=Y(\sum_i m_iG_iA_i)Y$. +The diagonal of their inversion-energy quadratic form +\[ + E_h=\operatorname{Tr}(PX^{-1}\dot X X^{-1}\dot X X^{-1}) + +\operatorname{Tr}(QY^{-1}\dot Y Y^{-1}\dot Y Y^{-1}) +\] +is $(D_F,D_G)$. Hence +\begin{equation} + \mathbb E E_h\le16\sum_i + (\widetilde a_i z_i+\widetilde b_i w_i)\omega_i. + \label{eq:sign-energy} +\end{equation} + +We expand the first and second variation calculation of +\cite[Lemma~4.3 and its proof]{EzeunalaJiang2026} to obtain +\[ + \partial_iR=T_i(z_i-w_i),\qquad + h^T\nabla^2Rh\le3E_h-\frac c2\sum_i k_i h_i^2(b_ip_i+a_iq_i). +\] +The first identity is the envelope formula +$\partial_iR=\alpha_i p_i-\beta_i q_i$. +For the second, keep $t$ fixed and replace $x$ by $x+\tau h$. +Invertibility of $\mathcal L$ and the implicit function theorem +give a local curve $(X(\tau),Y(\tau))$ satisfying both tight +constraints. Its first derivatives are the response matrices above: +the identities $H=F-\mathsf BG=\mathsf AF-G$ verify the linearized +constraints. This curve stays positive definite for small $\tau$, +and its objective bounds $R(x+\tau h)$ from above with equality at +$\tau=0$. Differentiating twice and pairing with $P,Q$, using +$\mathcal L(P,Q)=\varepsilon(I,I)$, therefore gives +\begin{align*} + h^T\nabla^2Rh + &\le2E_h-c\sum_i k_i h_i^2(b_ip_i+a_iq_i)\\ + &\quad+2c\sum_i\psi_i'h_i + \bigl(q_i u_i^*\dot X u_i+p_i u_i^*\dot Y u_i\bigr). +\end{align*} +Write $E_X,E_Y$ for the two summands of $E_h$. Since +$P\succeq X\eta_x(Q)X$, +\[ + E_X\ge\operatorname{Tr}(\eta_x(Q)\dot X X^{-1}\dot X) + =c\sum_i\psi_iq_i u_i^*\dot X X^{-1}\dot X u_i. +\] +The inequality +$|u_i^*\dot X u_i|^2\le a_i u_i^*\dot X X^{-1}\dot X u_i$ +and weighted Cauchy--Schwarz bound the $X$ mixed sum by +\[ + \left|c\sum_i\psi_i'h_iq_i u_i^*\dot X u_i\right| + \le\sqrt{E_X B_X},\qquad + B_X=c\sum_i(\psi_i')^2\psi_i^{-1}h_i^2a_iq_i. +\] +The corresponding $Y$ bound uses +$B_Y=c\sum_i(\psi_i')^2\psi_i^{-1}h_i^2b_ip_i$. +Thus the full mixed term is at most $2\sqrt{E_hB_h}$, where +$B_h=B_X+B_Y$. Finally, +$(\psi_i')^2/(\psi_i k_i)\le1/2$ gives +$B_h\le(c/2)\sum_i k_i h_i^2(b_ip_i+a_iq_i)$, and +$2\sqrt{E_hB_h}\le E_h+B_h$ proves the asserted Hessian bound. + +Let $\Sigma=\mathbb E hh^T$ and choose +$\mu_i=-24T_i^{-1}(\widetilde a_i-\widetilde b_i)\omega_i$. +Using \eqref{eq:sign-scalar} and \eqref{eq:sign-energy}, the operator +$\mathcal Df=\nabla f\cdot\mu+\tfrac12\operatorname{Tr}(\nabla^2f\Sigma)$ +satisfies +\begin{align*} + \mathcal DR + &\le(24-c/12)\sum_i + (\widetilde a_i w_i+\widetilde b_i z_i)\omega_i\le0,\\ + \mathcal D\Bigl(\sum_i\psi_i\Bigr) + &\le-\left(\frac12-\frac{48}{c}\right) + \sum_i\frac{k_i\omega_i}{T_i^2} + \le-\frac18\sum_i\omega_i. +\end{align*} +The first line uses cancellation of +$24\sum_i(\widetilde a_i z_i+\widetilde b_i w_i)\omega_i$ by the drift. +For the second, the identity $T_i(\widetilde a_i+\widetilde b_i)=a_i+b_i$ +and \eqref{eq:sign-light} give +$|\psi_i'\mu_i|\le(48/c)k_i\omega_i/T_i^2$. + +All movement bounds are polynomial and uniform in the state. +Indeed \eqref{eq:sign-dual} gives +\[ + \mathsf Aw\le(1-s_0)z,\quad + \mathsf Bz\le(1-s_0)w,\quad + s_0=\varepsilon/(4K^2),\quad + \|\mathsf A\|,\|\mathsf B\|\le3N. +\] +The spectral radius of $\mathsf A\mathsf B$ is at most $(1-s_0)^2$. +Using its positive semidefinite conjugate yields +\[ + \|(I-\mathsf A\mathsf B)^{-1}\|, + \|(I-\mathsf B\mathsf A)^{-1}\| + \le J:=1+36K^2N^2/\varepsilon. +\] +Solving the two response equations gives +$\|F\|^2+\|G\|^2\le E_0\|H\|^2$, where +$E_0=2J^2(1+3N)^2$. Consequently, with +\[ + C_1=\max\{1,48E_0/23,4(1+9N^2)\},\qquad C_0=8C_1, +\] +we have $\sum_i\omega_i\ge n/C_1$, $\omega_i\le C_1$ and +\begin{equation} + \|\mu\|_\infty\le C_0,\quad + \operatorname{Tr}\Sigma\le C_0n,\quad + \mathcal D\Psi\le-\lambda n/C_0. + \label{eq:sign-progress} +\end{equation} +This proves a quantitative descent alternative for $c=384$. +The covariance and drift are used only to prove that a direction of +decrease exists. The algorithm below computes neither of them nor +the possibly irrational spectral subspaces used in their construction. + +Here is the deterministic alternative that the algorithm will use. +Suppose at least one coordinate is active, every active coordinate is +farther than $\sigma$ from its endpoints, and put +$a=\lambda/(8C_0^2)$. Either an endpoint test above +succeeds, giving a decrease of at least $\lambda\sigma$, or +\begin{equation} + \|\nabla\Psi(x)\|_\infty\ge a + \quad\text{or}\quad + \lambda_{\min}(\nabla^2\Psi(x))\le-8a. + \label{eq:sign-deterministic-alternative} +\end{equation} +The endpoint estimate uses $\psi_i\ge\sigma$. If all endpoint tests +fail, \eqref{eq:sign-light} holds and the preceding calculation +applies. If also $\|\nabla\Psi\|_\infty0$ is computable in +polynomial time in the input length and $\log\eta^{-1}$. +Approximate every $\psi_i$ by a nonnegative rational in $[0,1]$ to +error $\delta$. This changes the optimal value by at most +$(cKd+\lambda N)\delta$: in either direction, increase $t$ by +$cKd\delta$ to absorb the perturbation of the matrix constraints. +Indeed, $\sum_iA_i^2\preceq\nu I$ and +$\|Y\|\le K/\varepsilon$ bound the change in $\eta_x(Y)$ by +$c\delta\nu\|Y\|\le c\delta\nu K/\varepsilon=cKd\delta$; +the other block is identical. +The resulting rational program is a semidefinite program by the +Schur-complement formula. Impose the rational bounds +\[ + I/400\preceq X,Y\preceq(200/\varepsilon)I, + \qquad 0\le t\le100. +\] +The scalar upper bound and sublevel estimates hold for every coefficient +vector in $[0,1]^N$, so these boxes contain the minimizers both before +and after approximation. The preceding comparison of unboxed optimal +values therefore applies to the boxed programs as well. +The point $X=Y=I/20$, $t=50$ has +constraint slack at least $50-20-2-384/20=8.8$ and lies strictly +inside the bounds. It supplies an inverse-polynomial interior +radius: each Schur block dominates +$\left(\begin{smallmatrix}28.8I&I\\I&0.05I\end{smallmatrix}\right) +\succeq I/100$, since subtracting $I/100$ leaves the scalar +determinant $28.79\cdot0.04-1=0.1516>0$. +In upper-triangular real and imaginary coordinates, a ball of radius +$10^{-6}$ preserves this margin and all the boxes: a matrix-pair +perturbation changes either Schur block by at most +$|\Delta t|+384\|\Delta Y\|_F+\|\Delta X\|_F$ or its symmetric +counterpart. An outer radius $400(d+1)/\varepsilon$ suffices. +Rational weak optimization~\cite{GLS1988} therefore evaluates the +rational program to error $\eta/2$ in polynomial work. Choosing +$\delta\le\eta/[2(cKd+\lambda N)]$ gives the claimed oracle. +Rational bisection computes the cube roots. This value oracle also +works on lower-dimensional faces and at vertices. + +\subsection{Rational descent and termination} + +At the start of an iteration, move every active coordinate at distance +$z\le\sigma$ from its nearest endpoint to that endpoint. This never +increases the exact potential, since +\[ + \Delta\Psi\le z\sqrt\nu-\lambda[z(2-z)]^{1/3} + \le z\sqrt\nu-\lambda\sqrt z\le0. +\] +There are at most $N$ such permanent freezes. Every remaining active +coordinate is more than $\sigma$ from either endpoint. If none remain, +return the resulting vertex; all derivative and direction computations +below are performed only when $n\ge1$. + +With $a=\lambda/(8C_0^2)$ as above, choose positive powers of two $s,h$, each within +a factor two below its indicated bound, with +\[ + s\le\min\{\sigma/4,a/(4M),1/4\},\qquad + h\le\min\{\sigma/8,1,a/(64MN)\},\qquad + \Delta=as^2. +\] +Central first differences, central diagonal second differences, and +four-corner mixed second differences of the value oracle, each query +to error at most $ah^2/(128N)$, give rational approximations +$\widehat g,\widehat H$ to $\nabla\Psi,\nabla^2\Psi$ satisfying +\[ + \|\widehat g-\nabla\Psi\|_\infty2\Delta$. +Evaluate all final candidates to error $\Delta/4$ and choose the +smallest reported value. Selection loses at most $\Delta/2$ against +the best true candidate. Every accepted step thus decreases the +exact potential by at least $3\Delta/2$. + +Starting from $x=0$, every live coordinate stays on the fixed mesh +$\rho\mathbb Z$; frozen coordinates equal $\pm1$. All coordinates, +parameters, and query points have polynomial encoding length. +There are at most $\Psi(0)/\Delta$ main iterations, each with +polynomially many rational semidefinite value queries and one exact +rational positive-semidefiniteness test. The normalized lower bounds +on $\nu$, $\varepsilon$, $\lambda$, and $\sigma$ imply that $C_0$ and +$M$ are bounded by fixed polynomials in $N+d$. In turn, $a$ and the +chosen $s,h$ are bounded below by fixed inverse polynomials, as is +$\Delta=as^2$. Since $\Psi(0)<40$, the iteration count is polynomial +as well. Together with the encoding and oracle bounds, this proves +deterministic polynomial bit complexity on every execution. + +At the final vertex $x^f$, the coordinate term vanishes and +\eqref{eq:sign-potential-bound} gives +\[ + \left\|\sum_i x_i^f A_i\right\| + \le(2\sqrt{386}+0.01)\sqrt\nu + \le(2\sqrt{386}+0.01)\sqrt{1.001\nu_0} + <40\sqrt{\nu_0}. +\] +Undoing the rational normalization and folding the signs of the +original traces into $x_i^f$ proves +Theorem~\ref{thm:rational-signing}. diff --git a/preprints/A-polynomial-time-construction-of-strong-thin-trees-September-23-2026/build/source/sections/tree_geometry.tex b/preprints/A-polynomial-time-construction-of-strong-thin-trees-September-23-2026/build/source/sections/tree_geometry.tex new file mode 100644 index 0000000000000000000000000000000000000000..be0982cf643821b6c0074e6e4b675dfd82ee1ad9 --- /dev/null +++ b/preprints/A-polynomial-time-construction-of-strong-thin-trees-September-23-2026/build/source/sections/tree_geometry.tex @@ -0,0 +1,885 @@ +\section{Metric-cone shortcuts for thin trees}\label{tg:section} + +We next develop the geometric input to the thin-tree algorithm. The +shortcut theorem of the companion \cite{ThinTreesCompanion} tests a matrix on cuts. Convex +optimization in the next section will use a larger, finitely described +cone of tests. We establish the corresponding hierarchy estimate by +extending its complete geometric arguments, which follow the hierarchy +and effective-resistance framework of Anari and Oveis Gharan +\cite[Sections~5--7]{AnariOveisGharan2015}. + +Throughout this section graphs are finite loopless multigraphs on a +vertex set $V$, with $n=|V|\ge2$; edge sets retain multiplicities. +All symmetric matrices act on the Euclidean space +$\mathsf E=\one^\perp\subset\R^V$. Equivalently, they are centered full +matrices, with their zero action on $\Span\{\one\}$ understood. +Write $\Id_{\mathsf E}$ for the identity on this space, +$b_{uv}=e_u-e_v$, and $L_G=\sum_{e\in E(G)}b_eb_e^T$, using arbitrary +edge orientations. Matrix pairings are $\ip{A}{Z}=\tr(AZ)$. +Define the closed convex cone +\begin{equation}\label{tg:test-cone} + \mathcal N=\left\{Z\succeq0: + b_{uw}^TZb_{uw}\le b_{uv}^TZb_{uv}+b_{vw}^TZb_{vw} + \quad(u,v,w\in V)\right\}. +\end{equation} +Its trace-dual cone is +$\mathcal N^*=\{A:\ip{A}{Z}\ge0\text{ for all }Z\in\mathcal N\}$. +The Gram-matrix representation of squared Euclidean distances is +classical \cite{Schoenberg1935}; the additional triangle inequalities +give the standard $\ell_2^2$ metric relaxation +\cite[Section~2]{AroraRaoVazirani2009}. +For $Z\in\mathcal N$, the numbers $d(u,v)=b_{uv}^TZb_{uv}$ form +a squared Euclidean pseudometric. The centered identity belongs to +$\mathcal N$. So does $c_Sc_S^T$, where +$c_S=\one_S-(|S|/n)\one$: its squared distances are precisely the +$0$--$1$ cut metric of $S$. + +A \emph{hierarchy} is a rooted tree of distinct nonempty subsets of +$V$, with root $V$, singleton leaves, and children partitioning their +parent into at least two sets. For a nonroot node $A$, with parent +$A^*$, set +\[ + P(A)=\delta_G(A),\qquad O(A)=E_G(A,A^*\setminus A). +\] +The hierarchy is \emph{locally $h$-connected} if every $G[A]$ is +$h$-edge-connected, with the singleton condition vacuous. +Each $|O(A)|\ge h$. Each graph edge belongs to exactly two parent +boundaries: those of the two children of its endpoints' least common +ancestor. In particular its multiplicity in any selected collection +of parent boundaries is at most two. + +\Needspace{4\baselineskip} +\begin{theorem}[Hierarchy bound for the metric test cone] +\label{thm:tg-hierarchy} +There is an absolute integer $h_0$ such that, for every integer +$h\ge h_0$, every locally $h$-connected hierarchy of $G$, and every +collection $\mathcal M$ of nonroot nodes satisfying +$|O(A)|\ge |P(A)|/h$, there is a matrix $D\succ0$ on $\mathsf E$ with +\begin{align} + \ip{D}{Z}&\le\ip{L_G}{Z} &&(Z\in\mathcal N),\label{tg:shortcut}\\ + \frac1{|O(A)|}\sum_{e\in O(A)}b_e^TD^{-1}b_e + &\le h^{-31/40} &&(A\in\mathcal M).\label{tg:hierarchy-resistance} +\end{align} +The cutoff is independent of the graph and the hierarchy depth. +\end{theorem} + +The proof first expresses the shortcut objective as a ratio whose +denominator is total edge length. We then represent its numerator by +collections of disjoint balls and charge their radii to edge lengths; +the resulting estimate will give simultaneous responses to partition +chains. + +\subsection{The conic dual} + +\begin{lemma}[A geometric upper bound for the shortcut program] +\label{tg:dual} +Let $G$ be connected and $\mathcal M$ a nonempty finite collection +of nonroot nodes with nonempty parent boundaries. Let $p_*$ be the infimum, over +$D\succ0$ satisfying \Cref{tg:shortcut}, of the largest marked +average resistance. Then +\begin{equation}\label{tg:dual-ratio} + p_*\le \sup_{p,U} + \frac{\displaystyle\sum_{A\in\mathcal M}\frac1{|O(A)|} + \left(\sum_{e=uv\in O(A)}a_e\right)^2} + {\displaystyle\sum_{e=uv\in E(G)}d(u,v)}, + \qquad + a_e=|\ip{u_e}{p_u-p_v}|. +\end{equation} +Here $p_v$ are Euclidean points whose squared distances +$d(u,v)=\norm{p_u-p_v}_2^2$ satisfy the triangle inequalities, the +denominator is positive, and $U$ has orthonormal rows $u_e^T$. +The Euclidean dimension may be enlarged. +\end{lemma} + +\begin{proof} +Put $A_A=|O(A)|^{-1}\sum_{e\in O(A)}b_eb_e^T$ and +$f_A(D)=\tr(A_AD^{-1})$. These functions are convex, since +\[ + b^TD^{-1}b=\sup_x(2b^Tx-x^TDx). +\] +The conic constraint is $L_G-D\in\mathcal N^*$. +Choose $D_0=\delta\Id_{\mathsf E}$ with +$0<\delta<\lambda_{\min}(L_G|_{\mathsf E})$. +Then $L_G-D_0\succ0$ lies in the interior of $\mathcal N^*$, +because the positive semidefinite cone is contained in +$\mathcal N^*$. Taking an epigraph height above all $f_A(D_0)$ +makes the other inequalities strict as well. + +For completeness, finite-dimensional separation gives the needed +strong duality even if the infimum is not attained. Separate +$(0,0,p_*)$ from the open convex set of $(a,M,z)$ for which some +$D\succ0$ and $t\in\R$ satisfy +\[ + f_A(D)-t0$. Writing +$N_0=\tr((Z^{1/2}A(\lambda)Z^{1/2})^{1/2})$, optimization over the +scale $\tau Z$ gives +\[ + \sup_{\tau\ge0}(2\sqrt\tau N_0-\tau\mathcal E) + =N_0^2/\mathcal E. +\] +The zero multiplier contributes zero. Represent $Z$ as the centered +Gram matrix of points $p_v$, and put +\[ + w_e=\sum_{A:e\in O(A)}\frac{\lambda_A}{|O(A)|}. +\] +Then $N_0$ is the nuclear norm of the matrix with columns +$\sqrt{w_e}(p_u-p_v)$. The singular value decomposition, after +adding zero coordinates if needed, gives +\[ + N_0=\max_{UU^T=I}\sum_e\sqrt{w_e}\ip{u_e}{p_u-p_v}. +\] +Changing individual row signs preserves row orthonormality, so take +the displayed contributions nonnegative. For such a maximizing $U$, +\begin{align*} + N_0 + &\le\sum_A\sqrt{\frac{\lambda_A}{|O(A)|}} + \sum_{e\in O(A)}a_e,\\ + N_0^2 + &\le\sum_A\frac1{|O(A)|} + \left(\sum_{e\in O(A)}a_e\right)^2. +\end{align*} +The first line uses $\sqrt{\sum_jx_j}\le\sum_j\sqrt{x_j}$ and +the second uses $\sum_A\lambda_A=1$. Overlapping boundaries are +allowed. Substitution into \Cref{tg:lagrange} proves the claim. +\end{proof} + +\subsection{A convex realization with exact radial motion} + +Starting from the classical distance-profile isometry into +$\ell_\infty$ \cite{Frechet1910,Kuratowski1935}, we construct a +continuous geometry for an arbitrary finite metric. It will be used for balls and shells; the Euclidean +points $p_v$ remain the inputs to the spectral argument. +\Cref{tg:geometry-figure} contrasts the Euclidean spectral points with +the distance-profile realization of the same vertex labels and +illustrates the exact radial motion of \Cref{tg:profiles}. + +\begin{lemma}[Doubled distance profiles]\label{tg:profiles} +Every finite metric $(S,d)$ has an isometric realization +$x\mapsto c_x$ in a convex polytope $Q$ in a finite-dimensional +normed space with the following properties. Distance +$\rho_x(q)=\norm{q-c_x}$ is affine on $Q$. There is a common $R>0$ +and a point $c_x^-\in Q$ such that +\begin{equation}\label{tg:opposite-radii} + \norm{q-c_x}+\norm{q-c_x^-}=R\qquad(q\in Q). +\end{equation} +From any $q\in Q$ one can reach every radial value in $[0,R]$ by +moving toward $c_x$ or $c_x^-$, and the change in radius equals +the distance moved. Every straight segment in $Q$ has its norm +length, and clamped radial functions are piecewise affine on it. +\end{lemma} + +\begin{proof} +Take $R>2\max_{x,y}d(x,y)$ and two copies $x^+,x^-$ of each point. +On the doubled set define +\[ + \widehat d(x^+,y^+)=\widehat d(x^-,y^-)=d(x,y),\qquad + \widehat d(x^+,y^-)=R-d(x,y). +\] +This is a metric. A triangle with all signs equal is an original +triangle. A triangle with two equal signs has two cross-copy sides; +an inequality with a cross-copy side on the left reduces to an +original triangle inequality. The remaining inequality follows +from $d(x,y)+d(x,z)+d(z,y)\le3\operatorname{diam}(d)<2R$. + +For a doubled point $a$, let +$c_a=(\widehat d(a,z))_z$ be its distance profile in $\ell_\infty$, +and set $Q=\conv\{c_a\}$. The triangle inequality and coordinate +$a$ show $\norm{c_a-c_b}_\infty=\widehat d(a,b)$. +More generally, for $q=\sum_b\theta_bc_b$, +\begin{equation}\label{tg:affine-radius} + \norm{q-c_a}_\infty + =\sum_b\theta_b\widehat d(a,b)=q_a, + \qquad \theta_b\ge0,\quad\sum_b\theta_b=1. +\end{equation} +Indeed, convexity of the norm gives the upper bound, and coordinate +$a$ gives the lower bound. Thus the radius is the restriction of +a linear coordinate function, independently of the representation +of $q$. Set $c_x=c_{x^+}$ and $c_x^-=c_{x^-}$. +The identity +$\widehat d(x^+,b)+\widehat d(x^-,b)=R$ for every doubled point $b$ +proves \Cref{tg:opposite-radii}. + +If $\rho=\rho_x(q)$, the segment $q_t=(1-t)q+tc_x$ has +$\norm{q_t-q}=t\rho$ and $\rho_x(q_t)=(1-t)\rho$. +The segment $q_t^-=(1-t)q+tc_x^-$ has +$\norm{q_t^--q}=t(R-\rho)$ and +$\rho_x(q_t^-)=\rho+t(R-\rho)$. +These formulas prove the exact-motion assertion, including the +endpoint cases. Convexity of $Q$ keeps all segments in $Q$; +the final assertion follows from \Cref{tg:affine-radius} and clamping. +\end{proof} + +For a pseudometric, first identify zero-distance points to apply +\Cref{tg:profiles}. This identification concerns only the geometric +point set. Every graph vertex, edge copy, hierarchy node, and +center-vertex label is retained. In particular coincident vertex +images do not contract the graph. + +\input{figures/metric_geometry} + +\subsection{Transferring the bag and charging estimates} + +The next lemma extends the bag-selection and shell-charging arguments of +\cite[Propositions B.7, B.8, and B.10]{ThinTreesCompanion} to $Q$. +We retain those constructions and their scalar accounting, but verify +the geometric steps below: their stated cube hypotheses do not cover +all tests in $\mathcal N$. The two charging alternatives explain the +division into compact and assigned bags. Compact bags charge disjoint +geometric objects. Assigned bags may overlap geometrically, provided +their hierarchy labels separate the graph edges used for charging. + +Use relative open balls $B(c,\delta)=\{q\in Q:\norm{q-c}<\delta\}$ +and strict shells $B(c;a,b)=\{q\in Q:a<\norm{q-c}0$ and +$N_A\ge\alpha\sum_{e\in O(A)}d_e$, where $0<\alpha\le1$. +Define +\begin{align*} + J&=1+\lceil\log_2(256/\alpha^2)\rceil,& + C_2&=128J^4,&\sigma&=\alpha/C_2,\\ + L&=2+\lceil\log_2(1/\sigma)\rceil,& + H_0&=1+\lceil\log_2(24/(\rho\sigma^2))\rceil,& + C_1&=1024\epsilon^{-2}. +\end{align*} +If $\beta>1$, $0<\epsilon<1/3$, $0<\rho<1$, and +$\sigma^\epsilon\le(48\beta C_1)^{-1}$, there is a +$\rho$-geometric sequence consisting either of $\beta$-compact +families or of $\sigma^{1+2\epsilon}$-assigned families, with +\begin{equation}\label{tg:captured-mass} + \mathsf M\ge + \frac{\sigma^\epsilon}{384\beta C_1C_2LH_0} + \sum_{A\in T}N_A. +\end{equation} +\item\label{tg:transfer-compact} +For $G$ $k$-edge-connected, a $\rho$-geometric sequence of $\beta$-compact families +with $\rho\le1/12$ and $\beta\ge36$ satisfies +$\mathcal E\ge(k/4)\mathsf M$. +\item\label{tg:transfer-assigned} +For a locally $k$-connected hierarchy, selected nodes satisfying +$|O(A)|\ge k\rho|P(A)|$, and a $\rho$-geometric sequence of +$(24C_3/k)$-assigned families, one has +\begin{equation}\label{tg:assigned-charge} + \mathcal E\ge\frac{kC_4}{96C_3}\mathsf M +\end{equation} +provided +$C_4\ge3$, $\rho\le(6C_4)^{-1}$, and +$C_3\ge2((C_4+1)+4(C_4+2)^2)$. +\end{enumerate} +\end{lemma} + +\begin{proof} +We follow the constructions in Sections B.2--B.4 of +\cite{ThinTreesCompanion}, keeping their scale choices and scalar +potentials. All balls, shells, and intersections now lie in the +continuous polytope $Q$. New shells inherit vertex centers, so +\Cref{tg:affine-radius} remains available. We first obtain the bags, +then explain how each of the two charging constructions operates in +this realization. + +\emph{Extraction and the spectral step.} +The two truncations and homogeneity arguments in +\cite[Section B.2.2]{ThinTreesCompanion} use only +$a_e^2\le d_e$, nonnegative boundary sums, and multiplicity at most +two. They select nonempty subsets $O'(A)\subseteq O(A)$ retaining +a fixed fraction of each squared boundary sum, and group the nodes +so that both positive quantities $a_e^2$ and $d_e$ vary by a factor +less than four within $F=\bigcup_AO'(A)$. For one such group put +$N=\sum_A|O(A)|$, $N'=|F|$, +$c_1=\min_{e\in F}a_e^2$, and $c_2=\max_{e\in F}d_e$. +The truncation and grouping estimates give $c_1N'\ge\sigma Nc_2$. +For the Euclidean points $Y_v=Up_v$ and +$\widetilde\alpha=\sigma N/N'$ this implies +\[ + (\E_{e\in F}a_e)^2\ge c_1 + \ge\widetilde\alpha c_2 + \ge\widetilde\alpha\E_{e\in F}\norm{Y_u-Y_v}_2^2. +\] +This is the exact hypothesis of the Euclidean spectral-packing +Lemma B.3 of \cite{ThinTreesCompanion}; that lemma places no binary +restriction on its points. Its selected centers have squared +Euclidean separation at least $4r$. Since +\begin{equation}\label{tg:spectral-to-metric} + \norm{Y_u-Y_v}_2^2\le d(u,v)=\norm{c_u-c_v}_\infty, +\end{equation} +the corresponding $Q$-balls of radius $r$ are disjoint. If $b_0$ +is their number, the spectral lemma gives +$b_0r\ge\widetilde\alpha^\epsilon c_1N'/C_1$. Shrink their common +radius to $\delta$ defined by +$b_0\delta=\widetilde\alpha^\epsilon c_1N'/C_1$. +Only the selected vertex labels pass from the Euclidean geometry +to $Q$; a map on intermediate points is unnecessary. + +The rest of the one-family construction uses a maximal disjoint +collection of balls of radius $D=\max\{\delta,2c_2\}$, centered at +images $c_u$ for which $u\in A$ and $uv\in O'(A)$ for some $A$. +Maximality puts every candidate center within $2D$ of a selected +center: intersection of two relative balls implies this distance +bound by the triangle inequality. Every endpoint center is within +$c_2$ of an assigned endpoint, hence within $3D$ of a selected center. +Grouping by those centers gives diameter at most $6D$ in the compact +case. In the assigned case the witnessing edge has length +at most $c_23(b-a)$, retain intervals +$(r_1,a-2\delta)$ and $(b+2\delta,r_2)$ and insert the $m$ balls. +The expanded radial interval and Lipschitzness give every required +separation. For the sparse case $m\delta\le3(b-a)$, retain +$(r_1,a)$ and $(b,r_2)$ and create the central shells +$H_3=B(c;a+\delta,b-\delta)$ and $H_4=B(c;a,b)$. +Give $H_3$ owner $A$, no avoided label, and the old excluded nodes +strictly below $A$. Give $H_4$ the old owner and excluded set, +and avoided label $A$. Their domains are respectively +$C(H)\cap\mathcal T[A]$ and $C(H)\setminus\mathcal T[A]$. +The old owner is a strict ancestor of $A$. An original ball with +owner $A$ cannot come from an earlier scale, since assigned nodes +are distinct across families; at the current scale it is disjoint +from every unprocessed candidate. A synthetic shell with owner +$A$ can only have been created while processing this same bag. +It has already treated every then-interior candidate in its expanded +interval; later restrictions only reduce its support, and border +candidates never become interior. Hence neither an original ball +nor a synthetic shell with owner $A$ can be the chosen record. +Thus both new labels have the required form. +For the changed-label path witnesses, extremal neighbors +$v_1,v_2$ in the selected piece and their adjacent candidates +$u_1,u_2\in A$ satisfy +\[ + \varrho(v_1)\le a,\quad \varrho(v_2)\ge b,\qquad + \varrho(u_1)b-\delta. +\] +Connectivity of $G[A]$ supplies $k$ paths for the inner central +shell; connectivity of the selected piece supplies at least +$k/4$ paths for the avoiding shell. Their initial vertices +avoid excluded subtrees: membership in one would make the +short edge $u_1v_1$ an excluded boundary edge with an endpoint +inside the old outer radius, or make that excluded subtree an +ancestor of $A$, contradicting $A$'s conflict membership. +The sparse-case width bound gives $a+\delta0$, and set +\[ + \alpha=h^{-4/5},\qquad \epsilon=1/10,\qquad + \beta=36,\qquad \rho=(2h^2)^{-1},\qquad + C_3=10^4,\qquad C_4=3. +\] +Let $T$ consist of the marked nodes with positive $N_A$ and +$N_A\ge\alpha\sum_{e\in O(A)}d_e$. +Boundary multiplicity at most two gives +\begin{equation}\label{tg:good-nodes} + \sum_{A\in\mathcal M\setminus T}N_A\le2\alpha\mathcal E. +\end{equation} +If $T$ is nonempty, apply \Cref{tg:transfer}(i). +Its parameters obey +\[ + C_2=O((1+\log h)^4),\quad L,H_0=O(1+\log h),\quad + \sigma=h^{-4/5}/C_2. +\] +The smallness condition holds for all sufficiently large absolute +$h$. So does the assignment requirement, because +\[ + h\sigma^{1+2\epsilon} + =h^{1/25}/C_2^{6/5}\longrightarrow\infty, + \qquad \sigma^{1+2\epsilon}\ge24C_3/h. +\] +The marked-node condition implies +$|O(A)|\ge|P(A)|/h\ge h\rho|P(A)|$. +For large $h$, $\rho\le1/18$; also +$10^4\ge2(4+4\cdot25)=208$. +Both charging conclusions of \Cref{tg:transfer} therefore apply, +according to the family type, and give the common bound +\[ + \mathcal E\ge c_*h\mathsf M, + \qquad c_*=\frac{C_4}{96C_3}=\frac{1}{320000}. +\] +Together with \Cref{tg:captured-mass}, this yields +\[ + \frac{\sum_{A\in T}N_A}{\mathcal E} + \le \frac{384\beta C_1C_2LH_0}{c_*h\sigma^\epsilon} + =O\bigl(h^{-23/25}(1+\log h)^7\bigr). +\] +Indeed $\sigma^\epsilon=h^{-2/25}C_2^{-1/10}$, so the exact +logarithmic exponent before rounding upward is $32/5<7$. +Including \Cref{tg:good-nodes}, or using it alone when $T$ is empty, +every ratio in \Cref{tg:dual-ratio} is at most +\begin{equation}\label{tg:final-ratio} + 2h^{-4/5}+O\bigl(h^{-23/25}(1+\log h)^7\bigr) + =o(h^{-31/40}). +\end{equation} +The exponent margins are $1/40$ and $29/200$, respectively. +Increase $h_0$ so the bound is at most $h^{-31/40}/2$. +Then $p_*\le h^{-31/40}/2$, and the defining property of the +infimum supplies a feasible positive definite $D$ with marked +averages less than $h^{-31/40}$. No attainment assumption is used. +\end{proof} + +For a fixed connected ambient graph $H$, define +\begin{equation}\label{tg:compact-domain} + \mathcal K_H=\{X\succeq L_H: + \ip{X}{Z}\le3\ip{L_H}{Z}\quad(Z\in\mathcal N)\}. +\end{equation} +This set is nonempty, convex, and compact. It contains $L_H$; +closedness is immediate, and the test $\Id_{\mathsf E}$ bounds +$\tr X$ by $3\tr L_H$. For $X\succ0$ write +$R_X(e)=b_e^TX^{-1}b_e$. Two elementary identities used below are +\begin{equation}\label{tg:resistance-order} + X\succeq D\succ0\ \Longrightarrow\ R_X(e)\le R_D(e), + \qquad + X\succeq t b_eb_e^T\ \Longleftrightarrow\ tR_X(e)\le1 + \quad(t>0). +\end{equation} +The first follows by inverse order, and the second by congruence +with $X^{-1/2}$ and the single nonzero eigenvalue of a rank-one matrix. + +\begin{proposition}[Simultaneous response]\label{prop:tg-response} +Let $H$ be connected and let $G\subseteq H$ be a spanning subgraph. +Let $r$ be a sufficiently large integer, put $h=\lfloor r/2\rfloor$, +and suppose $G$ and the subgraphs induced by all parts of a nested +sequence of partitions are $h$-edge-connected, with the singleton +condition vacuous. If $01/(6s)\}| + \le6s\,h^{-31/40}|O(A)|. +\] +Thus at most $2(h^{-1}+6s h^{-31/40})M_j$ disappearing edges +fail \Cref{tg:response-edge}. Double counting only increases +this upper bound. + +There is at most one representative per dyadic bin, and its bin +index strictly decreases. Hence +$\sum_{j=i}^{\ell-1}M_j<4M_i$. Each edge of $B_i$ disappears +at exactly one subsequent transition, so its bad-edge count is +at most $8(h^{-1}+6s h^{-31/40})M_i$. +For a skipped partition with positive crossing count $M$, its +representative is finer and has $M_i<2M$. Consequently its +bad-edge count is at most +\begin{equation}\label{tg:response-fraction} + 16(h^{-1}+6s h^{-31/40})M + =O(r^{-1}+r^{-21/40})M\le r^{-1/3}M +\end{equation} +for a sufficiently large absolute cutoff. The final inequality +uses $21/40-1/3=23/120>0$. Partitions with no crossing edges +need no estimate. One matrix $Y$ therefore works for the whole +nested sequence. +\end{proof} diff --git a/preprints/A-polynomial-time-construction-of-strong-thin-trees-September-23-2026/paper.pdf b/preprints/A-polynomial-time-construction-of-strong-thin-trees-September-23-2026/paper.pdf new file mode 100644 index 0000000000000000000000000000000000000000..c1d30fd66841f4dac3ee52b864865c25cce11bc4 --- /dev/null +++ b/preprints/A-polynomial-time-construction-of-strong-thin-trees-September-23-2026/paper.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:da0e75ba53ff393fd57e9028170fce25aed47ae02954131c9eb04513bdffa658 +size 615951 diff --git a/preprints/A-positive-solution-to-Tingleys-problem-September-23-2026/README.md b/preprints/A-positive-solution-to-Tingleys-problem-September-23-2026/README.md new file mode 100644 index 0000000000000000000000000000000000000000..f2911edb8a7a0f0c83f04c874b9cdc0cf3584128 --- /dev/null +++ b/preprints/A-positive-solution-to-Tingleys-problem-September-23-2026/README.md @@ -0,0 +1,18 @@ +# [A positive solution to Tingley's problem](paper.pdf) + +**Author:** OpenAI + +**Date:** September 23, 2026 + +## Citation + +```bibtex +@misc{OAI:A-positive-solution-to-Tingleys-problem-September-23-2026, + author = {{OpenAI}}, + title = {{A positive solution to Tingley's problem}}, + howpublished = {OpenAI Math Release preprint + \href{https://github.com/openai/math/blob/main/preprints/A-positive-solution-to-Tingleys-problem-September-23-2026/paper.pdf}{OAI:A-positive-solution-to-Tingleys-problem-September-23-2026}}, + year = {2026} +} +``` + diff --git a/preprints/A-positive-solution-to-Tingleys-problem-September-23-2026/build/main.tex b/preprints/A-positive-solution-to-Tingleys-problem-September-23-2026/build/main.tex new file mode 100644 index 0000000000000000000000000000000000000000..ab9bd82b388ec4e55a333d05c61c853b7941798b --- /dev/null +++ b/preprints/A-positive-solution-to-Tingleys-problem-September-23-2026/build/main.tex @@ -0,0 +1,544 @@ +\documentclass[11pt]{article} +\usepackage[T1]{fontenc} +\usepackage{mathpazo,microtype} +\usepackage[margin=1in]{geometry} +\usepackage{amsmath,amssymb,amsthm,mathtools} +\usepackage[hidelinks]{hyperref} +\urlstyle{same} +\hypersetup{pdftitle={A positive solution to Tingley's problem},pdfauthor={OpenAI}, + pdfsubject={Sphere isometries of real Banach spaces},pdfkeywords={Tingley's problem, sphere isometry, Banach space}} +\pdfinfoomitdate=1 +\pdfsuppressptexinfo=15 +\newtheorem{theorem}{Theorem}[section] +\newtheorem{proposition}[theorem]{Proposition} +\newtheorem{lemma}[theorem]{Lemma} +\newtheorem{corollary}[theorem]{Corollary} +\theoremstyle{remark} +\newtheorem{remark}[theorem]{Remark} +\newcommand{\R}{\mathbb R} +\newcommand{\norm}[1]{\left\lVert#1\right\rVert} +\newcommand{\abs}[1]{\left|#1\right|} +\newcommand{\co}{\operatorname{conv}} +\newcommand{\clco}{\overline{\operatorname{conv}}} +\newcommand{\diam}{\operatorname{diam}} +\newcommand{\dist}{\operatorname{dist}} +\newcommand{\supp}{\operatorname{supp}} +\makeatletter +\renewcommand{\@maketitle}{% + \newpage\null\vskip 1em + \begin{center}{\LARGE\@title\par}\vskip .6em{\large\@author\par}\vskip .4em{\normalsize\@date\par}\end{center} + \vskip .5em} +\newenvironment{chordfigure}{% + \par\medskip\noindent\begin{minipage}{\linewidth} + \def\@captype{figure}\centering +}{\end{minipage}\par\medskip} +\makeatother +\newcommand{\needroom}[1]{% + \par\ifdim\dimexpr\pagegoal-\pagetotal\relax<#1\relax\newpage\fi} +\title{A positive solution to Tingley's problem} +\author{OpenAI} +\date{September 23, 2026} +\begin{document} +\maketitle +\begin{abstract} +Every surjective isometry between the unit spheres of real Banach spaces extends uniquely to a surjective real-linear isometry, giving an affirmative solution to Tingley's problem. If distances between different radii are not preserved, we realize the positive maximal defect in a possibly enlarged pair of Banach spaces. We then align extremal chords using common supports and Darbo's fixed-point theorem and obtain a contradiction from support and convexity estimates. +\end{abstract} + +\section{Introduction}\label{sec:intro} + +For a real normed space $X$, write $S_X=\{x\in X:\norm{x}=1\}$, with the distance inherited from $X$. Tingley's problem asks whether every surjective isometry $f:S_X\to S_Y$ extends to a real-linear isometry of the spaces \cite{Tingley1987}. Any such extension must be the radial map below. This candidate preserves norms and distances at each fixed radius; the difficulty is to recover distances between points at different radii from the metric on a single sphere. We prove the full assertion. + +\begin{theorem}\label{thm:main} +Let $X,Y$ be nonzero real Banach spaces and let $f:S_X\to S_Y$ be a surjective isometry. Then +\[ +T(0)=0,\qquad T(x)=\norm{x}f\left(\frac{x}{\norm{x}}\right)\quad(x\ne0) +\] +is a surjective real-linear isometry. It is the unique linear extension of $f$. +\end{theorem} + +The spaces may have arbitrary dimension and need not be separable, reflexive, smooth, or strictly convex. Applied to complex Banach spaces regarded as real spaces, the conclusion is real linearity; complex linearity is not asserted. + +The problem belongs to the extension theory of metric isometries. +Mazur and Ulam proved that every surjective isometry between real normed +spaces is affine \cite{MazurUlam1932}; Mankiewicz obtained an extension +theorem for surjective isometries between convex sets with nonempty interiors +\cite{Mankiewicz1972}. A unit sphere has empty ambient interior, so this +does not directly provide the desired extension. Tingley's original +paper established preservation of antipodal points for surjective +isometries between finite-dimensional Banach-space spheres +\cite{Tingley1987}. + +Early positive results used the structure of concrete spaces. Wang +treated surjective isometries between spheres of complex $C_0$-spaces +\cite{Wang1994}, and Ding obtained extension results for +$\ell^p(\Gamma)$-type sequence spaces with $p>1$, $p\ne2$ +\cite{Ding2003}. +It is useful to distinguish results for specified source and target classes from the +\emph{Mazur--Ulam property}: every surjective sphere isometry from the +given space to an arbitrary real Banach target extends linearly. +Cheng and Dong formulated this property explicitly +\cite{ChengDong2011}. Tan proved it for real $L^p(\mu)$ on +$\sigma$-finite measure spaces when $10$, there are nonzero real Banach spaces, a surjective sphere isometry, still denoted by $f:S_X\to S_Y$, and $x_0,y\in S_X$, $00. +\end{aligned} +\end{equation} +\end{lemma} +\begin{proof} +Choose triples approaching the absolute supremum. Pass to a subsequence of one sign and, if necessary, replace the map by its inverse. After a further subsequence the radius parameters converge to $t\in[0,1]$. The uniform Lipschitz bound allows us to replace them all by $t$, retaining convergence of the signed defects to $M$. By \eqref{eq:defectbound}, $M/2\le t\le1-M/2$. + +We describe the limit construction to justify attainment. Choose a free ultrafilter $\mathcal U$ on $\mathbb N$: a maximal proper filter containing all cofinite sets. It exists by the maximal principle, since a union of a chain of such filters is again a proper filter. We use the usual set-theoretic framework with choice here. For any subset, either it or its complement belongs to $\mathcal U$. Every bounded real sequence has a unique limit along $\mathcal U$. Indeed, in a compact interval containing the sequence, the closures of its values on sets in $\mathcal U$ have the finite intersection property; their intersection is nonempty and the ultrafilter property makes it a singleton. These limits preserve sums, inequalities, and continuous real functions. + +On bounded sequences in $X$ use the seminorm +\[ +\norm{(a_n)}_{\mathcal U}=\lim_{\mathcal U}\norm{a_n}. +\] +Let $Q_X$ be the quotient by its kernel, with the induced norm, and define $Q_Y$ in the same way. Constant sequences embed $X,Y$ isometrically, so these quotients are nonzero. Every unit vector in either quotient has a representative of unit vectors: normalize each nonzero term and replace zero terms by a fixed unit vector. The change has norm $|\norm{a_n}-1|$, whose ultralimit is zero. + +Applying $f$ termwise to unit representatives defines $F:S_{Q_X}\to S_{Q_Y}$. The identity +\[ +\lim_{\mathcal U}\norm{f(a_n)-f(b_n)} +=\lim_{\mathcal U}\norm{a_n-b_n} +\] +proves both independence of representative and distance preservation. Applying $f^{-1}$ termwise gives the inverse of $F$, so it is surjective. For any fixed $q\in[0,1]$, +\[ +\norm{F([a_n])-qF([b_n])}-\norm{[a_n]-q[b_n]} +=\lim_{\mathcal U}D_q(a_n,b_n)\in[-M,M]. +\] +Since the pair and $q$ were arbitrary, the bound holds for every pair and every radius. The chosen maximizing sequences give equality at $t$. + +Complete both quotient spaces. Their unit spheres are the metric completions of their previous spheres, by density and normalization. Extending the induced isometry and its inverse gives inverse maps by continuity and density, hence a surjective isometry of these completed spheres. Continuity preserves the defect bound and the selected equality. These spaces and this map have the asserted properties. +\end{proof} + +It therefore suffices to rule out the attained configuration \eqref{eq:maximal}, which we retain throughout the next two sections. + +\section{Aligning extremal chords}\label{sec:alignment} + +For a bounded subset $H$ of a normed space, its Kuratowski measure of noncompactness \cite{Kuratowski1930} is +\[ +\chi(H)=\inf\{\delta>0:H\text{ is covered by finitely many sets of diameter at most }\delta\}. +\] +We use the uniform strict-contraction form of Darbo's fixed-point theorem \cite{Darbo1955}, proved in Appendix~\ref{sec:appendix}. + +\begin{theorem}\label{thm:condensing} +Let $C$ be a nonempty closed bounded convex subset of a Banach space. If $g:C\to C$ is continuous and, for some $0\le\gamma<1$, +\[ +\chi(g(H))\le\gamma\chi(H)\qquad(H\subset C), +\] +then $g$ has a fixed point. +\end{theorem} + +\begin{samepage} +\begin{proposition}\label{prop:alignment} +Under \eqref{eq:maximal}, put $b=f(y)$ and keep $y,t$ fixed. There are endpoints $x,z\in S_X$, directions $v\in S_X$, $w\in S_Y$, and positive numbers $p,r,s,u$ such that +\begin{equation}\label{eq:chords} +\begin{aligned} +x&=ty+pv\in S_X,& z&=ty-rv\in S_X,\\ +f(x)&=tb+sw\in S_Y,& f(z)&=tb-uw\in S_Y, +\end{aligned} +\end{equation} +and +\begin{equation}\label{eq:radii} +s-p=r-u=M,\qquad d:=p+r=s+u=\norm{x-z}\le2. +\end{equation} +All four radii belong to $[1-t,1+t]$. +\end{proposition} +\end{samepage} +\begin{proof} +Starting from a direction at $ty$, we map its sphere contact to $S_Y$ and take the opposite sphere contact on the line through $tb$. Pull this opposite endpoint back to $S_X$, and reverse its direction from $ty$. A fixed direction for this return map will align the two chords. + +For $v\in S_X$, use Lemma~\ref{lem:radial} to define $p=p(v)>0$ by $x(v)=ty+pv\in S_X$. Define successively +\begin{equation}\label{eq:return} +f(x(v))=tb+sw,\qquad f(z)=tb-uw\in S_Y,\qquad +r=\norm{z-ty},\quad g(v)=\frac{ty-z}{r}, +\end{equation} +where $w\in S_Y$ and $u>0$ is determined by radial contact in direction $-w$. These quantities are continuous in $v$, and their positive radii lie in $[1-t,1+t]$. + +Let $E=\{v\in S_X:s(v)-p(v)=M\}$. It is closed in $X$, since $S_X$ is closed and $s-p$ is continuous on $S_X$. It is nonempty: $v_0=(x_0-ty)/\norm{x_0-ty}$ satisfies $x(v_0)=x_0$ by uniqueness of radial contact, and belongs to $E$ by \eqref{eq:maximal}. For $v\in E$, distance preservation and the global defect bound give +\[ +s+u=\norm{x(v)-z}\le p+r,\qquad r-u\le M. +\] +Since $s=p+M$, both inequalities are equalities. Thus +\begin{equation}\label{eq:equality} +r=u+M,\qquad \norm{pv+rg(v)}=p+r. +\end{equation} +In particular $v$ and $g(v)$ share a support. Moreover, every unit direction $l$ sharing a support with $g(v)$ belongs to $E$: for the fixed $z,r,u$ associated with $v$, +\[ +p(l)+r=\norm{x(l)-z} +=\norm{f(x(l))-f(z)}\le s(l)+u, +\] +so $s(l)-p(l)\ge r-u=M$. The reverse inequality is the global defect bound. In particular $g(v)\in E$. + +\smallskip\noindent\emph{An invariant convex set.} +Start with $v_0\in E$. Suppose $\co\{v_0,\ldots,v_n\}\subset E$. Choose a strictly positive convex combination $a_n$ of these vectors, and set $v_{n+1}=g(a_n)$. A common support $\varphi$ for $a_n,v_{n+1}$, extended to their finite span with the $v_i$, takes value one at each $v_i$ by strict positivity. For every $l$ in the enlarged convex hull, +\[ +1=\varphi(l)\le\norm{l}\le1. +\] +Thus $l$ is a unit vector sharing $\varphi$ with $v_{n+1}=g(a_n)$, and belongs to $E$ by the preceding propagation rule. + +Choose the coefficients so that the $a_n$ are dense in +$C=\clco\{v_0,v_1,\ldots\}$. +Explicitly, enumerate all finite rational probability vectors on initial lists of indices, repeat each arbitrarily late, and at stage $n$ use one whose list is contained in $\{0,\ldots,n\}$. Mix its zero-padded coefficients with the uniform positive vector on that list of $n+1$ indices, using a weight $0<\lambda_n<1$ tending to zero. This changes the prescribed combination by at most $2\lambda_n$, because every $v_i$ is unit. Each prescribed rational combination is therefore a limit of $a_n$'s. Such combinations are dense in $C$, proving the assertion. The set $C$ is nonempty, closed, bounded, convex, and contained in $E\subset S_X$. Since $g(a_n)=v_{n+1}\in C$, continuity gives $g(C)\subset C$. Only this constructed set is separable; the ambient spaces are unrestricted. + +\smallskip\noindent\emph{A strict decrease of noncompactness.} +Compare $v,v'\in E$, marking their associated variables by primes. From \eqref{eq:return}, +\begin{align*} +r\norm{g(v)-g(v')} +&\le |r-r'|+\norm{z-z'}\\ +&\le |r-r'|+|u-u'|+u\norm{w-w'},\\ +s\norm{w-w'} +&\le |s-s'|+\norm{f(x(v))-f(x(v'))}\\ +&\le |s-s'|+|p-p'|+p\norm{v-v'}. +\end{align*} +Using $s=p+M$ and $r=u+M$ yields +\begin{equation}\label{eq:contraction} +\norm{g(v)-g(v')} +\le\frac{up}{rs}\norm{v-v'} + +\frac{2|u-u'|}{r}+\frac{2u|p-p'|}{rs}. +\end{equation} +Set $a=1-t>0$ and $U=1+t$. All radii lie in $[a,U]$, and $r=u+M$, $s=p+M$ give $00$, intersect its parts with $H$ and subdivide them according to finite partitions of the ranges of $p$ and $u$ into intervals of length at most $\eta$. Since $r,s\ge a$ and $u/r<1$, the last two terms in \eqref{eq:contraction} sum to at most $4\eta/a$ on each resulting part. The images of these parts cover $g(H)$ with diameters at most $\gamma\delta+4\eta/a$. Letting $\eta\downarrow0$ for this fixed cover and then taking the infimum over diameter covers gives $\chi(g(H))\le\gamma\chi(H)$. + +Theorem~\ref{thm:condensing} now gives $v\in C$ with $g(v)=v$. Set $x=x(v)$; this endpoint may differ from $x_0$. Substitution in \eqref{eq:return} gives \eqref{eq:chords}. Equation \eqref{eq:equality} and distance preservation give \eqref{eq:radii}. +\end{proof} + +\begin{chordfigure} +\setlength{\unitlength}{1pt} +\begin{picture}(340,148) +\linethickness{0.7pt} +\put(10,116){\line(1,0){270}} +\put(10,36){\line(1,0){270}} +\put(10,116){\circle*{3}}\put(190,116){\circle*{3}}\put(280,116){\circle*{3}} +\put(10,36){\circle*{3}}\put(135,36){\circle*{3}}\put(280,36){\circle*{3}} +\put(10,130){\makebox(0,0){$z$}}\put(190,130){\makebox(0,0){$ty$}}\put(280,130){\makebox(0,0){$x$}} +\put(10,20){\makebox(0,0){$f(z)$}}\put(135,20){\makebox(0,0){$tb$}}\put(280,20){\makebox(0,0){$f(x)$}} +\linethickness{0.35pt} +\put(100,100){\vector(-1,0){90}}\put(100,100){\vector(1,0){90}} +\put(235,100){\vector(-1,0){45}}\put(235,100){\vector(1,0){45}} +\put(100,87){\makebox(0,0){$r$}}\put(235,87){\makebox(0,0){$p$}} +\put(72.5,52){\vector(-1,0){62.5}}\put(72.5,52){\vector(1,0){62.5}} +\put(207.5,52){\vector(-1,0){72.5}}\put(207.5,52){\vector(1,0){72.5}} +\put(72.5,65){\makebox(0,0){$u$}}\put(207.5,65){\makebox(0,0){$s$}} +\put(310,116){\makebox(0,0){$X$}}\put(310,36){\makebox(0,0){$Y$}} +\end{picture} +\caption{The aligned chords of Proposition~\ref{prop:alignment}. Each segment is drawn along its own unit direction. The inner points are $ty$ and $tb$; the endpoints lie on the respective unit spheres. The diagram records lengths only, with $p+r=s+u$ and $r-u=s-p=M$.} +\label{fig:chords} +\end{chordfigure} + +\section{The chord contradiction}\label{sec:comparison} + +Fix the configuration supplied by Proposition~\ref{prop:alignment}, illustrated in Figure~\ref{fig:chords}, and set +\begin{equation}\label{eq:AB} +A=\frac p d,\qquad B=\frac u d,\qquad \epsilon=1-t. +\end{equation} +Then $A,B>0$ and $A+B=1-M/d<1$. + +We first derive a secant estimate from the defect bound. If either $A$ or $B$ exceeds $\epsilon$, this estimate leads to a radial perturbation that increases the maximal defect. In the remaining case, convexity gives incompatible bounds for the preserved distance from $y$ to $z$. + +\needroom{11\baselineskip} +\begin{lemma}\label{lem:secant} +Put +\[ +h=\min\left\{\frac t s,\frac\epsilon u\right\},\qquad +\theta=\frac{hs}{p},\qquad +K=\frac{\norm{(r/u)v+hy}-\norm{(p/s)v+hy}}{r/u-p/s}. +\] +For every support $\varphi$ at $v+\theta y$, +\begin{equation}\label{eq:K} +\frac\epsilon p\le\varphi(v)\le K\le +\begin{cases} +1-B,& B\le\epsilon,\\ +B,& B>\epsilon. +\end{cases} +\end{equation} +\end{lemma} +\begin{proof} +At radius $t-sh$, the signed defect for $x,y$ is +\[ +\Delta_x:=D_{t-sh}(x,y)=s\bigl(\norm{w+hb}-\norm{(p/s)v+hy}\bigr). +\] +At radius $t+uh$, the negative of the signed defect for $z,y$ is +\[ +\Delta_z:=-D_{t+uh}(z,y)=u\bigl(\norm{(r/u)v+hy}-\norm{w+hb}\bigr). +\] +Both radii lie in $[0,1]$, so both defects are at most $M$. The choice of $h$ makes at least one radius an endpoint, where its defect is zero. The weighted sum $u\Delta_x+s\Delta_z$ cancels the common target norm; since $sr-up=Md$, +\[ +\frac{u\Delta_x+s\Delta_z}{d}=MK. +\] +The inequality $t/s\le\epsilon/u$ is equivalent to $B\le\epsilon$. In that case $\Delta_x=0$ and $K\le s/d=1-B$; otherwise $\Delta_z=0$ and $K\le u/d=B$. At $B=\epsilon$ both defects vanish, so $K=0$ and the stated upper bound remains valid. + +The supported vector is nonzero: if $v+\theta y=0$, then $\theta=1$ and $v=-y$, whereas $\norm{ty+pv}=1$ forces $p=1+t$ and hence $\theta\le t/p<1$. + +Since $r/u>p/s>0$, a support at $v+\theta y$, hence also at $(p/s)v+hy$, bounds the secant from below by $\varphi(v)$. It remains to prove $\varphi(v)\ge\epsilon/p$. Write $\beta=t/p$, so $0<\theta\le\beta$. If $\theta=\beta$, then +\[ +\varphi(v)=\frac{1-t\varphi(y)}p\ge\frac\epsilon p +\] +because $\norm{pv+ty}=1$. If $\theta<\beta$, take a support $\psi$ at $v+\beta y$. The two support inequalities give +\[ +\theta\bigl(\psi(y)-\varphi(y)\bigr) +\le\varphi(v)-\psi(v) +\le\beta\bigl(\psi(y)-\varphi(y)\bigr). +\] +Thus $\psi(y)\ge\varphi(y)$ and $\varphi(v)\ge\psi(v)\ge\epsilon/p$. +\end{proof} + +\begin{lemma}\label{lem:large} +Neither $A>\epsilon$ nor $B>\epsilon$ is possible. +\end{lemma} +\begin{proof} +First suppose $B>\epsilon$. Then $h=\epsilon/u$ and Lemma~\ref{lem:secant} gives +\begin{equation}\label{eq:product} +\frac{pu}{d\epsilon}\ge1. +\end{equation} +Let $L$ be any support at $v$, and write $j=L(y)$. For a support $\varphi$ at $v+\theta y$, Lemma~\ref{lem:secant} gives +\begin{equation}\label{eq:strictchain} +B\ge\varphi(v)=\norm{v+\theta y}-\theta\varphi(y) +\ge1+\theta j-\theta. +\end{equation} +In fact $B>1+\theta j-\theta$. If equality held throughout \eqref{eq:strictchain}, then $\norm{v+\theta y}=L(v+\theta y)$, so $L$ would also support at $v+\theta y$. But Lemma~\ref{lem:secant} applies to every support there and would give $1=L(v)\le B<1$. Therefore +\[ +1-j>\frac{1-B}{\theta}=\frac{pu}{d\epsilon}\ge1. +\] +Every support at $v$ is strictly negative on $y$. Applying one to $rv-ty=-z$ gives $r-tj\le1$, so $u\epsilon$. + +We record the inverse configuration explicitly. For $f^{-1}$ use $f(z)$ as the first endpoint, $b$ as the base direction, and $f(x)$ as the opposite endpoint. The new parameters are +\begin{equation}\label{eq:inverse} +(p',s',r',u')=(u,r,s,p),\quad +(v',w',y',b')=(-w,-v,b,y). +\end{equation} +By \eqref{eq:inverse-defect}, the inverse retains the bound in \eqref{eq:maximal}, and its defect at $(f(z),b,t)$ is $s'-p'=r-u=M$. The new configuration therefore has the same hypotheses, with $(A',B')=(B,A)$. The preceding argument thus says that $A>\epsilon$ implies $B>\epsilon$ and that every support at $-w$ is strictly negative on $b$. Equivalently, every support at $w$ is strictly positive on $b$. + +Consequently, if either strict inequality holds, both support-sign conclusions hold. For $0<\etas$. Also $\norm{pv+\eta y}1. +\] +By \eqref{eq:ratios} and monotonicity of secant slopes, +\[ +F(r/\epsilon)\ge1+\frac{r/\epsilon}{p/t}(1/t-1)>1/t. +\] +On the other hand, $u/\epsilon\in[0,s/t]$, so convexity gives $G(u/\epsilon)\le1/t$. But sphere distance preservation gives +\[ +\epsilon F(r/\epsilon)=\norm{y-z} +=\norm{b-f(z)}=\epsilon G(u/\epsilon), +\] +a contradiction. +\end{proof} + +\section{The linear extension}\label{sec:extension} + +\begin{proof}[Proof of Theorem~\ref{thm:main}] +If the defect of the original sphere map were positive, Lemma~\ref{lem:attainment} and Proposition~\ref{prop:no-defect} would contradict one another. Thus $M=0$. For $a=\alpha x$ and $c=\beta y$, where $x,y$ are unit and $\alpha\ge\beta>0$, put $q=\beta/\alpha$. The radial map satisfies +\[ +\norm{T(a)-T(c)} +=\alpha\norm{f(x)-qf(y)} +=\alpha\norm{x-qy}=\norm{a-c}. +\] +Exchanging the points handles the other ordering; distances involving zero follow from norm preservation. For $b\in Y\setminus\{0\}$, choose $y\in S_X$ with $f(y)=b/\norm{b}$. Then $T(\norm{b}y)=b$. Hence $T$ is a surjective metric isometry fixing zero. + +For completeness we prove the affine step of Mazur--Ulam \cite{MazurUlam1932} in the short reflection presentation of Nica \cite{Nica2012}, who explicitly credits V\"ais\"al\"a and Vogt \cite{Vaisala2003,Vogt1973}. If $F:E\to F_0$ is an onto isometry of real normed spaces and $a,c\in E$, let +\[ +m=(a+c)/2,\qquad m'=(F(a)+F(c))/2, +\qquad \delta=\norm{F(m)-m'}. +\] +The triangle inequality gives $\delta\le\norm{a-c}/2$. Reflection $R(z)=2m'-z$ is an onto isometry of $F_0$. Its conjugate $H=F^{-1}RF$ interchanges $a,c$ and has midpoint defect +\[ +\norm{H(m)-m}=\norm{RF(m)-F(m)}=2\delta. +\] +Apply the same construction repeatedly to the resulting onto self-isometries of $E$. Their midpoint defects double but obey the same bound $\norm{a-c}/2$, so $\delta=0$. Thus every onto isometry preserves midpoints. + +In particular $T((a+c)/2)=(T(a)+T(c))/2$. Since $T(0)=0$, first $T(2a)=2T(a)$ and then $T(a+c)=T(a)+T(c)$. Rational homogeneity follows from additivity, and real homogeneity from continuity. Finally, a linear extension is determined on every nonzero vector by its norm and its unit direction, proving uniqueness. +\end{proof} + +\appendix +\section{The fixed-point ingredients}\label{sec:appendix} + +We prove Theorem~\ref{thm:condensing}, including the compact convex fixed-point theorem it uses. The measure-of-noncompactness argument is the classical mechanism of Darbo \cite{Darbo1955}; the compact case is the Schauder fixed-point theorem \cite{Schauder1930}, obtained below from the simplex argument using Sperner's labeling lemma \cite{Sperner1928}. + +\begin{lemma}\label{lem:chi} +For every bounded nonempty set $H$ in a normed space, +\[ +\chi(\overline H)=\chi(H),\qquad +\chi(\clco H)=\chi(H). +\] +If $C_n$ are nonempty closed bounded subsets of a Banach space with $C_{n+1}\subset C_n$ and $\chi(C_n)\to0$, then their intersection is nonempty and compact. +\end{lemma} +\begin{proof} +Closing each member of a finite diameter cover preserves its diameter and covers the closure, proving the first identity. To prove the second, cover $H$ by finitely many nonempty subsets $H_1,\ldots,H_N$ of diameter at most $\delta$, and let $R$ bound the norms of their points. Each $\co H_i$ has diameter at most $\delta$: for two convex combinations, expand their difference using products of their coefficients and apply the triangle inequality. Every point of $\co H$ has the form +\[ +\sum_{i=1}^N\lambda_i z_i,\qquad z_i\in\co H_i, +\quad\lambda_i\ge0,\quad\sum_i\lambda_i=1, +\] +choosing arbitrary $z_i$ when $\lambda_i=0$. Partition the coefficient simplex into finitely many sets of $\ell_1$-diameter at most $\eta$. For each part, take all sums admitting a representation with coefficients in that part. These sets cover $\co H$. Two sums in one such set differ in norm by at most $\delta+R\eta$, by adding and subtracting the combination with one coefficient vector and the other set of $z_i$'s. Thus $\chi(\co H)\le\delta+R\eta$. Let $\eta\downarrow0$, take the infimum in $\delta$, and use the closure identity and monotonicity for the reverse inequality. + +For the intersection assertion, choose $x_n\in C_n$. For each positive integer $k$, a sufficiently late $C_n$ has a finite cover of diameter less than $2^{-k}$. Successive infinite subsequences, followed by a diagonal choice, produce a Cauchy subsequence of the $x_n$ with indices tending to infinity. Its limit belongs to every $C_n$, by completeness, nesting, and closedness. Their intersection is closed and has measure of noncompactness zero. Finite covers of arbitrarily small diameter give total boundedness, so the intersection is compact. +\end{proof} + +\needroom{6\baselineskip} +\begin{lemma}\label{lem:simplex} +Every continuous self-map of a finite-dimensional probability simplex has a fixed point. +\end{lemma} +\begin{proof} +The zero-dimensional case is immediate. Let $P$ be a continuous self-map of the simplex +$\Delta_m=\{a\in\R^{m+1}:a_i\ge0,\ \sum_i a_i=1\}$. +Subdivide it into simplices, and label each subdivision vertex $a$ by an index $i$ such that $a_i>0$ and $a_i\ge P(a)_i$. Such an index exists: otherwise summing over the positive coordinates of $a$ would give $1<\sum_i P(a)_i=1$. The label of a vertex lies among the vertices of every original face containing it. + +Here is the required combinatorial fact, including its parity proof. In a finite face-compatible triangulation of an $m$-simplex with labels $0,\ldots,m$ satisfying that boundary condition, the number of small simplices carrying all labels is odd. Count incidences of small $m$-simplices with facets labeled $0,\ldots,m-1$, modulo two. Internal facets count twice. Such a boundary facet can lie only in the original face missing label $m$; by induction its number is odd, starting with the single vertex in dimension zero. A small $m$-simplex contributes one incidence if it carries all labels, two if its vertex labels include $0,\ldots,m-1$ with one of these repeated, and none otherwise. This proves the claim. + +There are finite compatible subdivisions of arbitrarily small mesh: subdivide faces first and cone their subdivisions to the barycenter. The vertices of each resulting simplex are barycenters of a nested chain of faces. If two nested faces have $k$ and $l$ vertices, $k\le l$, their barycenters differ by at most $(1-k/l)$ times the original diameter. Thus a subdivision in dimension $m\ge1$ reduces each diameter by a factor at most $m/(m+1)$. Repeat. + +In each subdivision choose a fully labeled small simplex. Compactness and the shrinking mesh give a subsequence whose vertices converge to one point $a$. For every coordinate $i$, the vertex with label $i$ satisfies the corresponding inequality; continuity yields $a_i\ge P(a)_i$. Summing gives equality in every coordinate, so $a=P(a)$. +\end{proof} + +\begin{lemma}\label{lem:compactfixed} +Every continuous self-map $g$ of a nonempty compact convex subset $K$ of a normed space has a fixed point. +\end{lemma} +\begin{proof} +For $\delta>0$, choose a finite strict $\delta$-net $P_1,\ldots,P_N\in K$. For $x\in K$ set +\[ +\lambda_i(x)= +\frac{\max\{\delta-\norm{P_i-g(x)},0\}} + {\sum_j\max\{\delta-\norm{P_j-g(x)},0\}}. +\] +The denominator is positive, and these are continuous probability weights. Map the probability simplex to itself by sending coefficients $(a_i)$ to $(\lambda_i(\sum_j a_jP_j))$. By Lemma~\ref{lem:simplex} there is a fixed coefficient vector. Its convex combination $x\in K$ satisfies +\[ +\norm{x-g(x)} +\le\sum_i\lambda_i(x)\norm{P_i-g(x)}<\delta, +\] +since a positive weight implies the corresponding distance is less than $\delta$. Letting $\delta\downarrow0$ and taking a convergent subsequence in $K$ gives a fixed point. +\end{proof} + +\begin{proof}[Proof of Theorem~\ref{thm:condensing}] +Set $C_0=C$ and $C_{n+1}=\clco(g(C_n))$. These are nonempty closed bounded convex subsets of $C$ satisfying $C_{n+1}\subset C_n$. By Lemma~\ref{lem:chi}, +\[ +\chi(C_{n+1})=\chi(g(C_n))\le\gamma\chi(C_n), +\] +so their intersection $K$ is nonempty, compact, and convex. If $x\in K$, then $g(x)\in g(C_n)\subset C_{n+1}$ for every $n$, so $g(K)\subset K$. Apply Lemma~\ref{lem:compactfixed}. +\end{proof} + +\input{references.tex} +\end{document} diff --git a/preprints/A-positive-solution-to-Tingleys-problem-September-23-2026/build/references.tex b/preprints/A-positive-solution-to-Tingleys-problem-September-23-2026/build/references.tex new file mode 100644 index 0000000000000000000000000000000000000000..796debae8df956385c1aee3398c3d578fda1cdf5 --- /dev/null +++ b/preprints/A-positive-solution-to-Tingleys-problem-September-23-2026/build/references.tex @@ -0,0 +1,151 @@ +\begin{thebibliography}{99} + +\bibitem{Banach1929} +S.~Banach, +\emph{Sur les fonctionnelles lin\'eaires}, +Studia Mathematica \textbf{1} (1929), 211--216. +\url{https://doi.org/10.4064/sm-1-1-211-216}. + +\bibitem{Banakh2022} +T.~Banakh, +\emph{Every 2-dimensional Banach space has the Mazur--Ulam property}, +Linear Algebra and its Applications \textbf{632} (2022), 268--280. +\url{https://doi.org/10.1016/j.laa.2021.09.020}. + +\bibitem{ChengDong2011} +L.~Cheng and Y.~Dong, +\emph{On a generalized Mazur--Ulam question: Extension of isometries between unit spheres of Banach spaces}, +Journal of Mathematical Analysis and Applications \textbf{377} (2011), no.~2, 464--470. +\url{https://doi.org/10.1016/j.jmaa.2010.11.025}. + +\bibitem{DacunhaCastelle1972} +D.~Dacunha-Castelle, +\emph{Ultraproduits d'espaces de Banach}, +S\'eminaire Goulaouic--Schwartz (1971--1972), expos\'e no.~9, 11 pp. +\url{https://www.numdam.org/item/SEDP_1971-1972____A9_0/}. + +\bibitem{Darbo1955} +G.~Darbo, +\emph{Punti uniti in trasformazioni a codominio non compatto}, +Rendiconti del Seminario Matematico della Universit\`a di Padova +\textbf{24} (1955), 84--92. +\url{https://www.numdam.org/item/RSMUP_1955__24__84_0/}. + +\bibitem{Ding2003} +G.~G.~Ding, +\emph{The isometric extension problem in the unit spheres of $\ell^p(\Gamma)$ $(p>1)$ type spaces}, +Science in China Series A: Mathematics \textbf{46} (2003), 333--338. +\url{https://doi.org/10.1360/03ys9035}. + +\bibitem{Fakhoury2025} +M.~Fakhoury, +\emph{Tingley's problem for Schreier spaces and their $p$-convexifications}, +Journal of Functional Analysis \textbf{289} (2025), no.~10, 111122. +\url{https://doi.org/10.1016/j.jfa.2025.111122}. + +\bibitem{FernandezPoloPeralta2018VN} +F.~J.~Fern\'andez-Polo and A.~M.~Peralta, +\emph{On the extension of isometries between the unit spheres of von Neumann algebras}, +Journal of Mathematical Analysis and Applications \textbf{466} (2018), no.~1, 127--143. +\url{https://doi.org/10.1016/j.jmaa.2018.05.062}. + +\bibitem{HuangZhu2026Haagerup} +J.~Huang and Y.~Zhu, +\emph{Tingley's problem for Haagerup noncommutative $L_p$-spaces, $10$ such that, for every sufficiently +large integer $n$, one can choose $n$ points in the unit square so that every +triangle they determine has area at least $c_1n^{-2+\eta}$. +Thus the almost $n^{-2}$ upper-bound formulation of Heilbronn's triangle +problem is false. The exponent $\eta$ is fixed but extremely small. +\end{abstract} +\tableofcontents +\clearpage +\input{sections/01-introduction} +\input{sections/02-lattices} +\input{sections/03-norm-digits} +\input{sections/04-orbits} +\input{sections/05-auxiliary-cap} +\input{sections/06-sampling} +\input{sections/07-zero-determinants} +\input{sections/08-alteration} +\appendix +\input{sections/09-prime-interval} +\begingroup +\raggedright +\bibliographystyle{plain} +\bibliography{references} +\endgroup +\end{document} diff --git a/preprints/A-power-improvement-in-the-Heilbronn-triangle-lower-bound-September-25-2026/build/references.bib b/preprints/A-power-improvement-in-the-Heilbronn-triangle-lower-bound-September-25-2026/build/references.bib new file mode 100644 index 0000000000000000000000000000000000000000..2625647629cff023bd587fd456d9dce8b3ad9984 --- /dev/null +++ b/preprints/A-power-improvement-in-the-Heilbronn-triangle-lower-bound-September-25-2026/build/references.bib @@ -0,0 +1,192 @@ +@article{Roth1951, + author = {Roth, K. F.}, + title = {On a Problem of {Heilbronn}}, + journal = {Journal of the London Mathematical Society}, + volume = {26}, + number = {3}, + pages = {198--204}, + year = {1951}, + doi = {10.1112/jlms/s1-26.3.198} +} + +@article{KPS1982, + author = {Koml{\'o}s, J{\'a}nos and Pintz, J{\'a}nos and Szemer{\'e}di, Endre}, + title = {A Lower Bound for {Heilbronn}'s Problem}, + journal = {Journal of the London Mathematical Society (2)}, + volume = {25}, + number = {1}, + pages = {13--24}, + year = {1982}, + doi = {10.1112/jlms/s2-25.1.13} +} + +@article{Schmidt1972, + author = {Schmidt, Wolfgang M.}, + title = {On a Problem of {Heilbronn}}, + journal = {Journal of the London Mathematical Society (2)}, + volume = {4}, + number = {3}, + pages = {545--550}, + year = {1972}, + doi = {10.1112/jlms/s2-4.3.545} +} + +@article{Roth1972a, + author = {Roth, K. F.}, + title = {On a Problem of {Heilbronn}, {II}}, + journal = {Proceedings of the London Mathematical Society (3)}, + volume = {25}, + number = {2}, + pages = {193--212}, + year = {1972}, + doi = {10.1112/plms/s3-25.2.193} +} + +@article{Roth1972b, + author = {Roth, K. F.}, + title = {On a Problem of {Heilbronn}, {III}}, + journal = {Proceedings of the London Mathematical Society (3)}, + volume = {25}, + number = {3}, + pages = {543--549}, + year = {1972}, + doi = {10.1112/plms/s3-25.3.543} +} + +@article{KPS1981, + author = {Koml{\'o}s, J{\'a}nos and Pintz, J{\'a}nos and Szemer{\'e}di, Endre}, + title = {On {Heilbronn}'s Triangle Problem}, + journal = {Journal of the London Mathematical Society (2)}, + volume = {24}, + number = {3}, + pages = {385--396}, + year = {1981}, + doi = {10.1112/jlms/s2-24.3.385} +} + +@misc{CPZ2023, + author = {Cohen, Alex and Pohoata, Cosmin and Zakharov, Dmitrii}, + title = {A New Upper Bound for the {Heilbronn} Triangle Problem}, + year = {2023}, + eprint = {2305.18253}, + archivePrefix = {arXiv}, + note = {\href{https://arxiv.org/abs/2305.18253v1}{arXiv:2305.18253v1}, 29 May 2023}, + url = {https://arxiv.org/abs/2305.18253v1} +} + +@article{CPZ2025, + author = {Cohen, Alex and Pohoata, Cosmin and Zakharov, Dmitrii}, + title = {Lower Bounds for Incidences}, + journal = {Inventiones mathematicae}, + volume = {240}, + pages = {1045--1118}, + year = {2025}, + doi = {10.1007/s00222-025-01331-2}, + eprint = {2409.07658}, + archivePrefix = {arXiv}, + url = {https://arxiv.org/abs/2409.07658v2} +} + +@article{Zakharov2026, + author = {Zakharov, Dmitrii}, + title = {Small Triangles}, + journal = {Journal of the London Mathematical Society}, + year = {2026}, + volume = {113}, + number = {3}, + pages = {e70447}, + doi = {10.1112/jlms.70447}, + url = {https://londmathsoc.onlinelibrary.wiley.com/doi/10.1112/jlms.70447} +} + +@article{Erdos1932, + author = {Erd{\H o}s, Paul}, + title = {Beweis eines {Satzes} von {Tschebyschef}}, + journal = {Acta Litt. Sci. Szeged, Sect. Sci. Math.}, + volume = {5}, + pages = {194--198}, + year = {1932}, + url = {https://www.renyi.hu/~p_erdos/1932-01.pdf} +} + +@book{LidlNiederreiter1994, + author = {Lidl, Rudolf and Niederreiter, Harald}, + title = {Introduction to Finite Fields and their Applications}, + publisher = {Cambridge University Press}, + edition = {Second}, + year = {1994}, + doi = {10.1017/CBO9781139172769} +} + +@article{Smith1861, + author = {Smith, Henry John Stephen}, + title = {On Systems of Linear Indeterminate Equations and Congruences}, + journal = {Philosophical Transactions of the Royal Society of London}, + volume = {151}, + pages = {293--326}, + year = {1861}, + doi = {10.1098/rstl.1861.0016} +} + +@article{SalemSpencer1942, + author = {Salem, R. and Spencer, D. C.}, + title = {On Sets of Integers Which Contain No Three Terms in Arithmetical Progression}, + journal = {Proceedings of the National Academy of Sciences of the United States of America}, + volume = {28}, + number = {12}, + pages = {561--563}, + year = {1942}, + doi = {10.1073/pnas.28.12.561} +} + +@article{Behrend1946, + author = {Behrend, F. A.}, + title = {On Sets of Integers Which Contain No Three Terms in Arithmetical Progression}, + journal = {Proceedings of the National Academy of Sciences of the United States of America}, + volume = {32}, + number = {12}, + pages = {331--332}, + year = {1946}, + doi = {10.1073/pnas.32.12.331} +} + +@article{Barlotti1956, + author = {Barlotti, Adriano}, + title = {Un'osservazione sulle $k$-calotte degli spazi lineari finiti di dimensione tre}, + journal = {Bollettino dell'Unione Matematica Italiana (3)}, + volume = {11}, + number = {2}, + pages = {248--252}, + year = {1956}, + url = {https://www.bdim.eu/item?id=BUMI_1956_3_11_2_248_0} +} + +@misc{Ellmann2025, + author = {Ellmann, Gabor}, + title = {A Lower Bound for {Heilbronn}'s Triangle-Problem}, + year = {2025}, + eprint = {1703.03297}, + archivePrefix = {arXiv}, + note = {\href{https://arxiv.org/abs/1703.03297v12}{arXiv:1703.03297v12}, 12 November 2025}, + url = {https://arxiv.org/abs/1703.03297v12} +} + +@misc{Agama2026, + author = {Agama, Theophilus}, + title = {New Bounds for the {Heilbronn} Triangle Problem}, + year = {2026}, + eprint = {2006.05269}, + archivePrefix = {arXiv}, + note = {\href{https://arxiv.org/abs/2006.05269v13}{arXiv:2006.05269v13}, 6 May 2026}, + url = {https://arxiv.org/abs/2006.05269v13} +} + +@misc{Henk2002, + author = {Henk, Martin}, + title = {Successive Minima and Lattice Points}, + year = {2002}, + eprint = {math/0204158}, + archivePrefix = {arXiv}, + url = {https://arxiv.org/abs/math/0204158v1}, + note = {Theorem 1.3 (Minkowski's second theorem)} +} diff --git a/preprints/A-power-improvement-in-the-Heilbronn-triangle-lower-bound-September-25-2026/build/sections/01-introduction.tex b/preprints/A-power-improvement-in-the-Heilbronn-triangle-lower-bound-September-25-2026/build/sections/01-introduction.tex new file mode 100644 index 0000000000000000000000000000000000000000..38bf195a8b87615c4fe85818ddf5cbc62bed4bcc --- /dev/null +++ b/preprints/A-power-improvement-in-the-Heilbronn-triangle-lower-bound-September-25-2026/build/sections/01-introduction.tex @@ -0,0 +1,159 @@ +\section{Introduction}\label{intro:section} + +For three points $p,q,r\in\mathbb R^2$, write +\[ + \operatorname{Area}(pqr)=\frac12|\det(q-p,r-p)|. +\] +If $P\subset[0,1]^2$ is finite and $|P|\ge3$, define +\[ + \Delta(P)=\min_{\{p,q,r\}\in\binom P3}\operatorname{Area}(pqr), + \qquad + \Delta(n)=\max_{\substack{P\subset[0,1]^2\\|P|=n}}\Delta(P). +\] +The minimum is over all unordered triples of distinct points. In particular, +a collinear triple makes $\Delta(P)=0$. +The maximum defining $\Delta(n)$ exists: the minimum area is continuous on +the compact space of ordered $n$-tuples, it is positive for suitable points +on a parabola, and a tuple with a repeated point has minimum area zero. + +The original Heilbronn conjecture asked whether $\Delta(n)=O(n^{-2})$; +see Roth~\cite[p.~198]{Roth1951}. Two elementary constructions attain this +scale. Erd\H{o}s's finite-field parabola, recorded in +Roth~\cite[Appendix]{Roth1951}, gives a grid configuration whose triangle +determinants are nonzero integers before scaling. Random sampling followed +by deleting one point from each small triangle also gives +$\Delta(n)\gg n^{-2}$. Koml\'os, Pintz and Szemer\'edi improved this to +$\Delta(n)\gg(\log n)/n^2$, disproving the original +conjecture~\cite{KPS1982}. Their argument represents the small-area triples +in a random point set as edges of a hypergraph, removes short cycles, +and uses a stronger independence bound to select a large subset containing +none of those triples~\cite[Section~2]{KPS1982}. + +The almost $n^{-2}$ formulation, discussed in +Zakharov's survey~\cite[Section~3]{Zakharov2026}, asks whether, for every +$\varepsilon>0$, there are constants $C_\varepsilon$ and +$n_0(\varepsilon)$ such that +\begin{equation}\label{intro:almost} + \Delta(n)\le C_\varepsilon n^{-2+\varepsilon} + \qquad(n\ge n_0(\varepsilon)). +\end{equation} +We disprove this formulation by a power improvement in the lower bound. + +\begin{theorem}\label{intro:main} +There are absolute constants $\eta,c_1>0$ and an integer $n_0\ge3$ such that +\[ + \Delta(n)\ge c_1n^{-2+\eta} + \qquad\text{for every integer }n\ge n_0. +\] +\end{theorem} + +In particular, $\Delta(n)\ge n^{-2+\eta/2}$ for every sufficiently large +$n$. The ratio of the theorem's lower bound to $n^{-2+\eta/2}$ is +$c_1n^{\eta/2}\to\infty$, which contradicts \eqref{intro:almost} at +$\varepsilon=\eta/2$. Section~\ref{alt:section} gives an explicit, though +extremely small, admissible exponent. + +On the upper-bound side, Roth~\cite{Roth1951} proved the first +$o(n^{-1})$ estimate by a density-increment argument, followed by refinements +of Schmidt~\cite{Schmidt1972} and Roth~\cite{Roth1972a,Roth1972b}. +Koml\'os, Pintz and Szemer\'edi~\cite{KPS1981} refined Roth's analytic method +to obtain $\Delta(n)\ll_\varepsilon n^{-8/7+\varepsilon}$. +Cohen, Pohoata and Zakharov developed new incidence-geometric approaches, +first obtaining $\Delta(n)\le n^{-8/7-1/2000}$ for sufficiently large +$n$~\cite[Theorem~1.1]{CPZ2023}, and subsequently proving +$\Delta(n)\ll_\varepsilon n^{-7/6+\varepsilon}$ for every +$\varepsilon>0$~\cite[Theorem~1.8]{CPZ2025}. + +Earlier preprints claim stronger power lower bounds than ours, with +unresolved issues in the following versions. Ellmann's version~12 +acknowledges an unproved local uniformity assumption and describes the +argument as heuristic~\cite[p.~3]{Ellmann2025}; the deletion estimate uses +this assumption~\cite[Theorem~4, pp.~5--6]{Ellmann2025}. +In Agama's version~13, Theorem~4.1~\cite[pp.~12--13]{Agama2026}, +counting the center with the boundary points introduces a collinear triple +consisting of the two endpoints of a diameter and their midpoint. +Omitting the center leaves the every-triple estimate unproved, since the +argument estimates triangles formed by the center and adjacent boundary +points. These objections concern the cited arguments, not the impossibility +of their asserted bounds. + +\subsection{The two congruence conditions} + +The proof works first with integer columns $u=(u_1,u_2,u_3)^{\mathsf T}$ +in a box of the form +\[ + 0\le u_1,u_20$, then +\[ + \det(\Lambda\cap y^\perp)=\frac{I|y|}{g}. +\] +\end{lemma} + +\begin{proof} +Choose $u\in\Lambda$ with $y\cdot u=g$. The map +$v\mapsto y\cdot v/g$ maps $\Lambda$ onto $\mathbb Z$, has kernel +$\Lambda\cap y^\perp$, and splits by $1\mapsto u$. Hence a basis of +the kernel together with $u$ is a basis of $\Lambda$. +The height of $u$ above $y^\perp$ is $g/|y|$, which proves the formula. +For $\Lambda=\mathbb Z^3$, primitivity gives $g=1$. +\end{proof} + +We next count matrices with unequal column bounds. The difficult case +occurs when the third-column radius reaches only one direction of its +integral plane; we count those planes through their shortest vectors. + +\begin{lemma}[Unequal column bounds]\label{lat:anisotropic} +Let $R_1\ge R_2\ge R_3\ge1$ and let $t\in\mathbb Z\setminus\{0\}$. +The number of triples $u_1,u_2,u_3\in\mathbb Z^3$ satisfying +$|u_j|\le R_j$ and $\det(u_1,u_2,u_3)=t$ is at most +\[ + C(R_1R_2R_3)^2\bigl(\log(2R_1)\bigr)^2. +\] +The constant is independent of $t$ and of the three radii. +\end{lemma} + +\begin{proof} +The columns $u_2,u_3$ are independent. Choose a primitive normal $y$ +to their plane, fixing its sign by requiring its first nonzero coordinate +to be positive. Then $u_2\times u_3=a y$ for some nonzero integer $a$, +so $|y|\le R_2R_3$. If $\mu_1,\mu_2$ are the successive lengths of +$L_y$, the columns imply $\mu_1\le R_3$ and $\mu_2\le R_2$. + +For fixed $u_2,u_3$, the determinant equation is $u_1\cdot y=t/a$. +Its integral solutions are either absent or form a translate of $L_y$. +By Lemmas~\ref{lat:plane-count} and~\ref{lat:plane-covolume}, at most +$C R_1^2/|y|$ satisfy the first-column bound; the required hypothesis is +$\mu_2\le R_2\le R_1$. For a fixed normal $y$, there are likewise at +most $C R_2^2/|y|$ choices for $u_2$. + +First suppose $\mu_2\le R_3$. There are at most +$C R_3^2/|y|$ choices for $u_3$. The resulting bound, summed over +normals, is +\[ + C(R_1R_2R_3)^2 + \sum_{\substack{y\in\mathbb Z^3\\0<|y|\le R_2R_3}}|y|^{-3} + \ll (R_1R_2R_3)^2\log(2R_2R_3). +\] +The last estimate follows by splitting the integer vectors into dyadic +shells; a shell of radius $Y\ge1$ has $O(Y^3)$ vectors. + +It remains to treat $\mu_2>R_3$. Group the normals by powers of two +$U,Z\ge1$ with +\[ + U\le\mu_1<2U,\qquad Z\le\mu_2<2Z. +\] +Lemma~\ref{lat:successive} gives $|y|\asymp UZ$. +There are at most $C U^4Z^2$ normals in such a group. Indeed, choose +a shortest nonzero vector $z\in L_y$. It is primitive in $\mathbb Z^3$, +because division by a common divisor would produce a shorter vector +in $L_y$. Its length is in $[U,2U)$, giving $O(U^3)$ choices. +For each such $z$, the possible normals lie in $L_z$ and have length +less than $4UZ$. If $\nu_1,\nu_2$ are the successive lengths of +$L_z$, integrality gives $\nu_1\ge1$, and the preceding lemmas give +\[ + \nu_2\le\nu_1\nu_2\le2|z|<4U\le4UZ. +\] +Consequently Lemma~\ref{lat:plane-count}, applied at radius $4UZ$, +bounds their number by +$C(UZ)^2/|z|\le C UZ^2$. This proves the claimed normal count. + +For each of these normals, all eligible third columns lie on the line +of a shortest vector. Lemma~\ref{lat:plane-count} gives at most +$1+2R_3/\mu_1\ll R_3/U$ choices, since $U\le\mu_1\le R_3$. +Thus the contribution of one dyadic group is at most +\[ + C U^4Z^2\, + \frac{R_1^2}{UZ}\, + \frac{R_2^2}{UZ}\, + \frac{R_3}{U} + =C R_1^2R_2^2R_3U + \le C(R_1R_2R_3)^2. +\] +There are $O(\log(2R_3)\log(2R_2))$ groups, since +$1\le U\le R_3$ and $1\le Z\le R_2$. Combining both cases proves +the assertion. +\end{proof} + +\begin{proposition}[A fixed nonzero determinant]\label{lat:fixed-det} +Let $\Lambda\subseteq\mathbb Z^3$ have index $I$, let $X\ge2$, and +suppose its third successive length satisfies $\lambda_3\le X$. +For every $t\in\mathbb Z\setminus\{0\}$, the number of $3$ by $3$ +matrices with rows in $\Lambda$, row lengths at most $X$, and +determinant $t$ is at most +\[ + C\bigl(\log(2X)\bigr)^2\frac{X^6}{I^2}. +\] +The constant is absolute. +\end{proposition} + +\begin{proof} +Let $V$ be the matrix whose rows are the successive shortest vectors +$v_1,v_2,v_3$, and let $\Lambda_0$ be their integer span. By +Lemma~\ref{lat:successive}, +\[ + m=[\Lambda:\Lambda_0]=\frac{|\det V|}{I}\le3^{3/2}. +\] +Since the finite group $\Lambda/\Lambda_0$ has order $m$, each +$u\in\Lambda$ has coordinates in $m^{-1}\mathbb Z$ relative to this +basis. If $u=\sum_j\alpha_jv_j$ and $|u|\le X$, Cramer's rule gives +\[ + |\alpha_j| + \le\frac{X\prod_{i\ne j}\lambda_i}{|\det V|} + \le 3^{3/2}\frac{X}{\lambda_j}. +\] +Thus the map $A\mapsto Q=mAV^{-1}$ is injective and takes the matrices +under consideration to integral matrices. Their common determinant is +$m^3t/\det V\ne0$. If this number is not an integer, there are no +such matrices; otherwise Lemma~\ref{lat:anisotropic} applies. + +The $j$th column of $Q$ has length at most +$R_j=C_0X/\lambda_j$, where $C_0$ is an absolute constant chosen +sufficiently large. These radii satisfy $R_1\ge R_2\ge R_3\ge1$. +Moreover, integrality of $\Lambda$ gives $\lambda_j\ge1$, and +Lemma~\ref{lat:successive} gives +\[ + R_j\le C_0X,\qquad + R_1R_2R_3=\frac{C_0^3X^3}{\lambda_1\lambda_2\lambda_3} + \le\frac{C_0^3X^3}{I}. +\] +The bound in Lemma~\ref{lat:anisotropic} is therefore the claimed one. +\end{proof} diff --git a/preprints/A-power-improvement-in-the-Heilbronn-triangle-lower-bound-September-25-2026/build/sections/03-norm-digits.tex b/preprints/A-power-improvement-in-the-Heilbronn-triangle-lower-bound-September-25-2026/build/sections/03-norm-digits.tex new file mode 100644 index 0000000000000000000000000000000000000000..8b6dadaeb9400ce8030a9bb0a5743f12d78f42eb --- /dev/null +++ b/preprints/A-power-improvement-in-the-Heilbronn-triangle-lower-bound-September-25-2026/build/sections/03-norm-digits.tex @@ -0,0 +1,230 @@ +\section{A determinant obstruction from field norms} +\label{norm:section} + +We construct random columns modulo a prime power, each carrying a label in +a finite field. Three distinct labels will force the determinant residue +to have no small integer representative. The construction first expresses +a field norm as a sum of determinants, then places these summands at one +specified coefficient in a base expansion. + +\subsection{The norm polynomial} + +Fix +\begin{equation}\label{norm:fixed-parameters} + d=41,\qquad M=\binom{4d-1}{d},\qquad + T=\binom{M}{3},\qquad k=T^2+1. +\end{equation} +Let $r$ be an odd prime. We recall the elementary finite-field facts +needed here; see \cite[Chapter~2]{LidlNiederreiter1994} for background. +In a splitting field over $\mathbb F_r$, the roots of +$X^{r^d}-X$ form a field: the Frobenius identity shows closure under +addition, subtraction and multiplication, and nonzero roots are closed +under inversion. The derivative is $-1$, so there are exactly $r^d$ +distinct roots. This field, denoted $K=\mathbb F_{r^d}$, consequently +has dimension $d$ over $\mathbb F_r$. Its automorphism $t\mapsto t^r$ +has $d$th power equal to the identity, and its fixed elements are exactly +the $r$ roots of $X^r-X$, namely $\mathbb F_r$. +For $t\in K$, the product $\prod_{j=0}^{d-1}t^{r^j}$ is therefore fixed +by Frobenius and belongs to $\mathbb F_r$; it is nonzero if $t\ne0$. +Choose an $\mathbb F_r$-basis $\beta_1,\ldots,\beta_d$ of $K$. +For a group $X=(X_{i\nu})_{1\le i\le3,\,1\le\nu\le d}$ of $3d$ +variables, define +\[ + Z(X)=\left(\sum_{\nu=1}^d\beta_\nu X_{1\nu}, + \sum_{\nu=1}^d\beta_\nu X_{2\nu}, + \sum_{\nu=1}^d\beta_\nu X_{3\nu}\right)^{\mathsf T}. +\] +For three disjoint variable groups $X^{(1)},X^{(2)},X^{(3)}$, put +\[ + \mathcal D=\det\bigl(Z(X^{(1)}),Z(X^{(2)}),Z(X^{(3)})\bigr). +\] +Let $\sigma$ act on this polynomial ring by applying $c\mapsto c^r$ +to coefficients in $K$ and fixing every variable. Define +\begin{equation}\label{norm:polynomial} + \mathcal N(X^{(1)},X^{(2)},X^{(3)}) + =\prod_{j=0}^{d-1}\sigma^j(\mathcal D). +\end{equation} +The action of $\sigma$ permutes the factors, so $\mathcal N$ has +coefficients in $\mathbb F_r$. It is homogeneous of degree $d$ in each +group. At arguments in $\mathbb F_r^{3d}$, it evaluates to the field norm +of the corresponding determinant: +\begin{align*} + \mathcal N(U^{(1)},U^{(2)},U^{(3)}) + &=\operatorname{Nm}_{K/\mathbb F_r} + \det\bigl(Z(U^{(1)}),Z(U^{(2)}),Z(U^{(3)})\bigr),\\ + \operatorname{Nm}_{K/\mathbb F_r}(t)&=\prod_{j=0}^{d-1}t^{r^j}. +\end{align*} +Indeed, coefficient conjugation commutes with evaluation at such arguments. +Transposing two variable groups changes the sign of each factor in +\eqref{norm:polynomial}. Since $d$ is odd, $\mathcal N$ is alternating. + +\begin{lemma}\label{norm:decomposition} +There exist homogeneous degree-$d$ polynomials $f_{ia}\in\mathbb F_r[X]$ +(allowing the zero polynomial), for +$1\le i\le3$ and $0\le a\tau. +\] +By \eqref{norm:centered}, every other representative $S_C+mh$, +$m\in\mathbb Z\setminus\{0\}$, has absolute value greater than +$h/2>\tau$. This proves the assertion. +\end{proof} + +In particular, if $G\in\operatorname{SL}_3(R_h)$ and an integer matrix +$A$ satisfies $A\bmod h=GC$ and $|\det A|\le\tau$, then at least two +labels of $C$ coincide: multiplication by $G$ preserves its determinant +residue. For three independently sampled labels, the probability of +such a coincidence is at most $3r^{-d}$ by the union bound. diff --git a/preprints/A-power-improvement-in-the-Heilbronn-triangle-lower-bound-September-25-2026/build/sections/04-orbits.tex b/preprints/A-power-improvement-in-the-Heilbronn-triangle-lower-bound-September-25-2026/build/sections/04-orbits.tex new file mode 100644 index 0000000000000000000000000000000000000000..4a4f89254c1c9b4b6426e34473f1ffcc2da02550 --- /dev/null +++ b/preprints/A-power-improvement-in-the-Heilbronn-triangle-lower-bound-September-25-2026/build/sections/04-orbits.tex @@ -0,0 +1,228 @@ +\section{Residue orbits and a conditional divisor estimate} +\label{orb:section} + +We now describe the lattice of possible rows after a change of coordinates +modulo a prime power. The size of the corresponding special-linear orbit +will control the probability of a prescribed lifted matrix. Two divisors +measure the singularity of the residue matrix; the smaller one has bounded +expectation even after all three labels have been fixed. + +Throughout this section, let $B$ be a prime, let $k\geq 1$ be an +integer, and put $h=B^k$ and $R_h=\mathbb Z/h\mathbb Z$. We interpret +$B^k$ as zero when it occurs as an entry of a matrix over $R_h$, but as the +positive integer $h$ in counting formulas. The row span of a matrix over +$R_h$ is the $R_h$-submodule generated by its rows. + +The diagonal reduction below is the prime-power-ring form of Smith's +normal form; see Smith~\cite[Articles~12--16]{Smith1861} for the classical +integer reduction and determinantal divisors. We include the elementary +proof needed here, including singular cases. + +\begin{lemma}[Diagonal form and the row lattice] +\label{orb:diagonal} +Let $C\in M_3(R_h)$ have at least one unit entry. There are invertible +matrices $P,Q\in\operatorname{GL}_3(R_h)$ and integers +$0\leq b\leq e\leq k$ such that +\[ + C=P\,\operatorname{diag}(1,B^b,B^e)\,Q. +\] +Set $D=B^b$, $E=B^e$, and $I=DE$, all as positive integers. The subgroup +\[ + \Lambda_C=\{v\in\mathbb Z^3: + v\bmod h\text{ belongs to the row span of }C\} +\] +is a full lattice satisfying +\begin{equation} + [\mathbb Z^3:\Lambda_C]=I, + \qquad E\mathbb Z^3\subseteq\Lambda_C. + \label{orb:lattice} +\end{equation} +Moreover, $b\geq j$, for $1\leq j\leq k$, if and only if every +$2\times2$ minor of $C$ vanishes modulo $B^j$. +\end{lemma} + +\begin{proof} +Move a unit entry into the first position, scale it to $1$, and clear the +rest of its row and column by invertible operations. In the remaining +$2\times2$ block choose a nonzero entry of least $B$-adic valuation $b$, +move it to the first position of that block, and multiply by a unit to +make it $B^b$. Every other entry of the block is divisible by $B^b$ in +$R_h$, so row and column subtraction clear its row and column. The final +entry is still divisible by $B^b$; scaling it by a unit makes it $B^e$ +with $b\leq e\leq k$. If the block is zero, take $b=e=k$. +This proves the stated form, including all zero-pivot cases. + +Invertible row and column operations preserve the ideal generated by +the $2\times2$ minors: each new minor is a linear combination of old +minors, and the inverse operations give the reverse inclusion. For the +diagonal matrix this ideal is $B^bR_h$. Its image modulo $B^j$ is zero +exactly when $b\geq j$, proving the last assertion and, in particular, +showing that $b$ is independent of the diagonalization. + +Write $\mathscr M$ for the row span of $C$. Left multiplication by $P$ does not +change the row span, while multiplication on the right by $Q$ is an +automorphism of $R_h^3$. Hence +\[ + |\mathscr M|=|R_h\times DR_h\times ER_h| + =h\frac hD\frac hE=\frac{h^3}{I}. +\] +The reduction map identifies $\mathbb Z^3/\Lambda_C$ with $R_h^3/\mathscr M$, +which proves the index formula. Since $D$ divides $E$, the diagonal row +span contains $ER_h^3$. Right multiplication by $Q$ preserves +$ER_h^3$, so $\mathscr M$ also contains $ER_h^3$. Taking inverse images proves +the containment in \eqref{orb:lattice}. +\end{proof} + +The preceding description permits singular matrices: either or both of +the last two diagonal entries may be zero. The orbit estimate must +retain these cases, because zero determinants are included in the +triangle problem. + +\begin{lemma}[A special-linear orbit bound] +\label{orb:orbit} +Under the hypotheses and notation of Lemma~\ref{orb:diagonal}, define +\[ + \mathcal O_C=\{GC:G\in\operatorname{SL}_3(R_h)\}. +\] +There is an absolute constant $c_0>0$ such that +\begin{equation} + |\mathcal O_C|\geq c_0\frac{h^8}{D^3E^2}. + \label{orb:orbit-bound} +\end{equation} +For a uniformly chosen $G\in\operatorname{SL}_3(R_h)$, the matrix $GC$ +is uniform on $\mathcal O_C$. Every matrix in this orbit has the same +row span and determinant as $C$. +\end{lemma} + +\begin{proof} +Let $\mathcal G=\operatorname{GL}_3(R_h)$, and let $\mathcal H$ be the +stabilizer of $C$ under left multiplication. Conjugation by $P$ identifies +$\mathcal H$ with the stabilizer of $\operatorname{diag}(1,D,E)$ and +preserves determinants. An element of this latter stabilizer has first +column exactly $(1,0,0)^{\mathsf T}$. Its second column differs from +$(0,1,0)^{\mathsf T}$ by three entries annihilated by $D$, and its third +column differs from $(0,0,1)^{\mathsf T}$ by three entries annihilated by +$E$. The respective annihilators have $D$ and $E$ elements. Discarding +the invertibility condition gives +\begin{equation} + |\mathcal H|\leq D^3E^3. + \label{orb:stabilizer} +\end{equation} + +For every unit $u\in R_h^\times$ with +$u\equiv1\pmod{h/E}$, the matrix $\operatorname{diag}(1,1,u)$ +stabilizes $\operatorname{diag}(1,D,E)$. If $e3Hw\text{ for every }1\le m\le3H\}. +\end{equation} +Choose $a$ uniformly from $\mathcal A$, and independently choose +$G_q$ uniformly from $\operatorname{GL}_3(\mathbb F_q)$. Our auxiliary +set is +\begin{equation}\label{aux:set-definition} + V=G_q(a+S). +\end{equation} +Here and below a triple $u^{(1)},u^{(2)},u^{(3)}$ is +\emph{affinely independent} if +$u^{(2)}-u^{(1)}$ and $u^{(3)}-u^{(1)}$ are linearly independent. +This allows the three columns to have linear rank two. + +\begin{lemma}\label{aux:random-set} +Let $H\ge1$ be an integer, let $q\ge2000H^2$ be prime, and put +$w=\lfloor q/(1000H^2)\rfloor$. Suppose +$S\subset\{0,\ldots,w-1\}^3$ contains no three distinct collinear +points. The set $\mathcal A$ in \eqref{aux:allowed-shifts} has at least +$q^3/2$ elements. For every $a\in\mathcal A$ and every invertible +$G_q$, the set $V$ in \eqref{aux:set-definition} has $|S|$ elements, +does not contain zero, and contains no three distinct collinear points. + +Moreover, let $x=(x_1,x_2,x_3)\in\mathbb Z^3\setminus\{0\}$ satisfy +$|x|\le H$, where $|x|$ is the Euclidean norm. For any +$u^{(1)},u^{(2)},u^{(3)}\in V$, the relation +\begin{equation}\label{aux:short-relation} + x_1u^{(1)}+x_2u^{(2)}+x_3u^{(3)}=0 + \quad\text{in }\mathbb F_q^3 +\end{equation} +is impossible if $x_1+x_2+x_3\ne0$. If $x_1+x_2+x_3=0$, it is +impossible whenever the three columns are affinely independent. +Consequently, \eqref{aux:short-relation} is impossible for three +distinct elements of $V$. +\end{lemma} + +\begin{proof} +For each integer $1\le m\le3H0$, no column of $A\bmod q$ is zero, and its columns are +either affinely independent or include two equal columns. For an +affinely independent triple with $W(A)>0$, there is no nonzero +$x\in\mathbb Z^3$ with $|x|\le H$ and $Ax=0\bmod q$. +\end{proposition} + +\begin{proof} +First replace the allowed shift by an unconditional uniform +$a\in\mathbb F_q^3$. Under this replacement the map +$v\mapsto G_qv+G_qa$ is uniform in the affine group: conditional on +$G_q$, its translation $G_qa$ is uniform. Conditioning back on +$a\in\mathcal A$ increases the probability of any event by at most +two, by Lemma~\ref{aux:random-set}. + +The affine group acts transitively on the ordered affinely independent +triples in $\mathbb F_q^3$, whose number is +\[ + q^3(q^3-1)(q^3-q). +\] +There are $s(s-1)(s-2)$ such triples in $S$. Therefore, for an +affinely independent triple of columns, +\[ + W(A)\le + 2\frac{q^9}{s^3} + \frac{s(s-1)(s-2)}{q^3(q^3-1)(q^3-q)}\ll1. +\] + +For the other two bounds, fix any allowed $a$ and put $S_a=a+S$. +If two specified target columns are linearly independent, their +inverse images under a uniform $G_q$ are uniform among the +$(q^3-1)(q^3-q)$ ordered independent pairs. At most $s^2$ of these +pairs belong to $S_a^2$. Requiring all three columns to belong to +$G_qS_a$ can only reduce this probability. Consequently, +\[ + W(A)\le\frac{q^9}{s^3}\frac{s^2}{(q^3-1)(q^3-q)} + \ll\frac{q^3}{s} +\] +whenever the target matrix has rank at least two. This estimate +holds for each allowed shift, so also after averaging over $a$. + +If any target column is zero, its inclusion probability is zero. +Otherwise fix one target column. Its inverse image under $G_q$ is +uniform among the $q^3-1$ nonzero vectors, and $S_a$ consists of $s$ +nonzero vectors. Thus +\[ + W(A)\le\frac{q^9}{s^3}\frac{s}{q^3-1} + \ll\left(\frac{q^3}{s}\right)^2. +\] +This also covers all patterns of equal columns. The remaining +assertions follow from the line property and short-relation +conclusion in Lemma~\ref{aux:random-set}. +\end{proof} + +In the sampling construction, choose $(a,G_q)$ independently of all +labels, all digit choices, and the random change of coordinates at the main modulus. +Given $V$, choose the auxiliary residues of the sampled columns +independently and uniformly from $V$. Since $|V|=s$ for every +outcome, the probability of any prescribed ordered triple of +auxiliary residues is +\begin{equation}\label{aux:normalized-probability} + \frac{\Pr(\text{all three residues belong to }V)}{s^3} + =\frac{W(A)}{q^9}, +\end{equation} +where $A$ is any integer lift of that triple. This identity includes +repeated residues and is unchanged on conditioning on any main-modulus +data. It is the normalization used when the two residue conditions +are combined by the Chinese remainder theorem. diff --git a/preprints/A-power-improvement-in-the-Heilbronn-triangle-lower-bound-September-25-2026/build/sections/06-sampling.tex b/preprints/A-power-improvement-in-the-Heilbronn-triangle-lower-bound-September-25-2026/build/sections/06-sampling.tex new file mode 100644 index 0000000000000000000000000000000000000000..6542ada7630b4fdfaef1430b3c8c5ed6c70e5610 --- /dev/null +++ b/preprints/A-power-improvement-in-the-Heilbronn-triangle-lower-bound-September-25-2026/build/sections/06-sampling.tex @@ -0,0 +1,172 @@ +\section{Sampling integral columns}\label{sam:section} + +We now combine the two residue constructions and lift them to integral +columns. The purpose of this section is to express the probability of a +small determinant as a weighted lattice count. The main and auxiliary +changes of coordinates are shared by all samples; fresh labels, digits, +auxiliary columns, and lifts are chosen for each sample. + +Set +\begin{equation}\label{sam:box} + N=(hq)^{10},\qquad + \mathcal B=\bigl([0,N)^2\times[N,2N)\bigr)\cap\mathbb Z^3. +\end{equation} +For $u\in\mathcal B$, define its projection by +\[ + \pi(u)=(u_1/u_3,u_2/u_3)\in[0,1)^2. +\] +Choose $G_h$ uniformly in $\operatorname{SL}_3(\mathbb Z/h\mathbb Z)$, +independently of the auxiliary set $V$. Conditional on $(G_h,V)$, sample +columns $u\in\mathcal B$ independently as follows: +\begin{enumerate} + \item Draw a label uniformly from $K$ and its digit column $c$ by the + main-modulus construction, using fresh independent digit choices. + Prescribe $u\bmod h=G_hc$. + \item Independently choose $u\bmod q$ uniformly from $V$. + \item Among columns with those residues, choose $u$ uniformly in + $\mathcal B$. +\end{enumerate} +Since $h$ and $q$ are coprime and $hq$ divides $N$, each pair of prescribed +residues has exactly $(N/(hq))^3$ lifts. This also shows that the sampling +rule is defined for every outcome of the residue choices. The samples +need not be independent after averaging over $(G_h,V)$; all estimates below +retain the shared choices. + +\begin{lemma}[Proportional pairs]\label{sam:proportional} +For two sampled columns $u,v$, one has +\begin{equation}\label{sam:pair-bound} + \Pr\bigl(\pi(u)=\pi(v)\bigr)\ll (hq)^6N^{-3}. +\end{equation} +\end{lemma} +\begin{proof} +Equality of the projections is equivalent to proportionality of $u$ and +$v$, because their third coordinates are positive. Every column in +$\mathcal B$ is a positive integer multiple of a unique primitive integral +vector $z$ with $z_3>0$. The number of its multiples in $\mathcal B$ is at +most $\sqrt6N/|z|$. Therefore the total number of ordered proportional +pairs in $\mathcal B^2$, including equal columns, is at most +\[ + 6N^2\sum_{0<|z|\le\sqrt6N}|z|^{-2}\ll N^3. +\] +For the last estimate, dyadic shells of radius $R$ contain $O(R^3)$ +integral vectors and contribute $O(R)$ to the sum. + +Condition on both columns' residues at both moduli and on the shared +random choices. Their lifts are independent, each uniform on exactly +$(N/(hq))^3$ columns. The preceding count bounds the number of +proportional pairs within any two such sets of lifts. Dividing by +$(N/(hq))^6$ and averaging proves the claim. +\end{proof} + +For three samples, write $A=(u^{(1)},u^{(2)},u^{(3)})$ for the integral +matrix of columns, and $C=(c^{(1)},c^{(2)},c^{(3)})$ for their digit matrix +before multiplication by $G_h$. With $C$ fixed, use +Lemma~\ref{orb:diagonal} to define $D,E,I=DE$ and the row lattice +$\Lambda=\Lambda_C$, and write $\mathcal O=\mathcal O_C$ for the orbit in +Lemma~\ref{orb:orbit}. Thus $\Lambda$ is the inverse image in $\mathbb Z^3$ of the row +span of $C$ modulo $h$, and +$\mathcal O=\{GC:G\in\operatorname{SL}_3(\mathbb Z/h\mathbb Z)\}$. +Recall also the auxiliary weight from \eqref{aux:weight}: +\[ + W(A)=\left(\frac{q^3}{s}\right)^3 + \Pr\bigl(u^{(1)},u^{(2)},u^{(3)}\bmod q\in V\bigr). +\] +Here the probability defining $W$ averages only over the auxiliary choices. + +\begin{lemma}[Exact lifting identity]\label{sam:lifting} +Condition on all three labels and digit columns, hence on $C$. For any +set $\mathcal E\subseteq\mathcal B^3$ of integral matrices, +\begin{equation}\label{sam:crt-identity} + \Pr(A\in\mathcal E\mid C,\text{labels}) + =\frac{h^9}{N^9|\mathcal O|} + \sum_{\substack{A\in\mathcal E\\ A\bmod h\in\mathcal O}}W(A). +\end{equation} +\end{lemma} +\begin{proof} +The matrix $G_hC$ is uniform on $\mathcal O$, since every point of an orbit +has the same number of group elements mapping $C$ to it. Given $V$, the +three auxiliary residues are independent and uniform on its $s$ elements. +Consequently, for each specified ordered triple of auxiliary residues, +its probability is its inclusion probability in $V$, divided by $s^3$. +This remains true when some residues coincide. Independence of the main +and auxiliary choices and the exact number of lifts give, for each fixed +integral $A$ with $A\bmod h\in\mathcal O$, the conditional probability +\[ + \frac{1}{|\mathcal O|}\, + \frac{\Pr(A\bmod q\text{ has all columns in }V)}{s^3} + \left(\frac{hq}{N}\right)^9 + =\frac{h^9}{N^9|\mathcal O|}W(A). +\] +For other matrices the probability is zero. Summation proves the identity. +\end{proof} + +We call a sampled triple \emph{bad} if its projected points are pairwise +distinct and $|\det A|\le\tau$. Every matrix in $\mathcal O$ has determinant +$\det C$ modulo $h$. Since $2\tauh^{100}$, the box +$\mathcal B=[0,N)^2\times[N,2N)\cap\mathbb Z^3$, and the weight +$W(A)$ from \eqref{aux:weight}. For a column $u\in\mathcal B$, its +projection means $(u_1/u_3,u_2/u_3)$. + +\begin{proposition}\label{zero:weighted-zero} +For the lattice $\Lambda$ and weight $W$ above, let $\mathcal Z$ be +the set of integer $3\times3$ matrices whose columns belong to +$\mathcal B$, whose rows belong to $\Lambda$, whose determinant is +zero, and whose three projected columns are pairwise distinct. Then +\begin{equation}\label{zero:target} + \sum_{A\in\mathcal Z}W(A) + \ll \log(2N)\frac{N^6}{I^2}. +\end{equation} +The implied constant is independent of the fixed digit matrix $C$. +\end{proposition} + +The orbit restriction in Proposition~\ref{sam:weighted-count} only +reduces this sum. We first record the lattice estimates needed for +the null vectors. + +\begin{lemma}\label{zero:plane} +For a primitive vector $x\in\mathbb Z^3$, put +\[ + \Lambda_x=\Lambda\cap x^\perp,\qquad + x\cdot\Lambda=g(x)\mathbb Z,\qquad + J_x=\det(\Lambda_x), +\] +where $g(x)>0$. Then +\begin{equation}\label{zero:plane-data} + g(x)\mid E,\qquad J_x=\frac{I|x|}{g(x)},\qquad + \frac{g(x)^3}{I}\le h^2. +\end{equation} +The reduction of $\Lambda_x$ modulo $q$ is the entire plane +\[ + \Pi_x=\{v\in\mathbb F_q^3:v\cdot\bar x=0\}, + \qquad \bar x=x\bmod q\ne0. +\] +If $\Gamma\subseteq\Lambda_x$ is a sublattice of index $m$, the +number of ordered triples of vectors in $\Gamma$, all of length at +most $4N$ and spanning a plane over $\mathbb Q$, is at most +\begin{equation}\label{zero:row-count} + \ll \frac{N^6}{(mJ_x)^3}. +\end{equation} +\end{lemma} + +\begin{proof} +Primitivity gives $x\cdot\mathbb Z^3=\mathbb Z$. Since +$E\mathbb Z^3\subseteq\Lambda$, we have +$E\mathbb Z\subseteq g(x)\mathbb Z$, so $g(x)\mid E$. +Lemma~\ref{lat:plane-covolume} gives the formula for $J_x$, and +$g(x)^3/I\le E^2/D\le h^2$. + +Primitivity also implies $\bar x\ne0$, with no restriction on the +size of $x$. To prove surjectivity, choose $z\in\mathbb Z^3$ with +$x\cdot z=1$. If an integral lift $v$ of a point of $\Pi_x$ has +$x\cdot v=qa$, then $v-qaz\in\mathbb Z^3\cap x^\perp$ is another +lift of the same point. Thus $\mathbb Z^3\cap x^\perp$ reduces +onto $\Pi_x$. Its multiple by $E$ lies in $\Lambda_x$ and still +reduces onto $\Pi_x$, because $q\nmid E$. + +Finally, a triple counted in \eqref{zero:row-count} contains two +independent vectors of length at most $4N$. Therefore the second +successive minimum of $\Gamma$ is at most $4N$ whenever the count +is nonzero. Lemma~\ref{lat:plane-count} then bounds the choices for +each vector by $O(N^2/\det\Gamma)$, and +$\det\Gamma=mJ_x$. +\end{proof} + +\begin{lemma}\label{zero:moment} +For $R\ge h$, +\begin{equation}\label{zero:moment-bound} + \sum_{\substack{x\in\mathbb Z^3\ \mathrm{primitive}\\ + R\le |x|<2R}}g(x)^3 + \ll R^3 I. +\end{equation} +In particular, this estimate holds when $R>H/2$. +\end{lemma} + +\begin{proof} +Since $g(x)\mid B^e$, write $g(x)=B^{a(x)}$ with +$0\le a(x)\le e$. For $0\le j\le e$, the condition +$B^j\mid g(x)$ is equivalent to +$Cx=0\pmod{B^j}$. Indeed, if $\widetilde C$ is any integral lift +of $C$, then $\Lambda$ is the sum of the integral row span of +$\widetilde C$ and $h\mathbb Z^3$, and $B^j\mid h$. +The diagonal form of $C$ shows that the fraction of residue +classes modulo $B^j$ satisfying this congruence is +\[ + B^{-j-\max(j-b,0)}. +\] +Indeed, the first diagonal coordinate must vanish, the second +must be divisible by $B^{\max(j-b,0)}$, and the third is +unrestricted since $j\le e$. + +Because $R\ge h\ge B^j$, each residue class contains +$O((R/B^j)^3)$ integral vectors of length less than $2R$. +We may discard primitivity for this upper bound. Consequently, +\begin{align*} + \sum_{\substack{x\ \mathrm{primitive}\\R\le |x|<2R}}g(x)^3 + &\le \sum_{j=0}^e B^{3j} + \#\{x\in\mathbb Z^3:|x|<2R,\ Cx=0\pmod{B^j}\}\\ + &\ll R^3\sum_{j=0}^e B^{2j-\max(j-b,0)} + \ll R^3 B^{b+e}=R^3I. +\end{align*} +The exponents in the last sum strictly increase to $b+e$, so its +bound is uniform in $B$. Finally, $h\ge2$ implies +$H/2=h^2/2\ge h$. +\end{proof} + +\begin{proof}[Proof of Proposition~\ref{zero:weighted-zero}] +Every $A\in\mathcal Z$ has rational rank two. Indeed, its columns +are nonzero, and no two are proportional because their projections +are distinct. Its right nullspace therefore has a primitive +integral generator $x$, unique up to sign. Every coordinate of +$x$ is nonzero: otherwise the relation would make two columns +proportional. Taking the cross product of two independent rows +and dividing by the gcd of its coordinates gives +\begin{equation}\label{zero:null-range} + Ax=0,\qquad x_1x_2x_3\ne0,\qquad |x|\ll N^2. +\end{equation} +Every row of $A$ lies in $\Lambda_x$ and has length at most $4N$. +Thus $x$ records a relation among the columns, while the rows are lattice +vectors perpendicular to $x$. We may sum over both choices of sign of +$x$ for an upper bound. + +Partition the possible vectors into dyadic shells +$R\le |x|<2R$, $R=1,2,4,\ldots$. By +\eqref{zero:null-range}, there are $O(\log(2N))$ shells. We will +bound the weighted contribution of each shell by $O(N^6/I^2)$. +For a fixed primitive $x$ in such a shell, a condition confining every +row to an index-$m$ sublattice of $\Lambda_x$ gives at most +\begin{equation}\label{zero:fixed-null-count} + \ll \frac{N^6g(x)^3}{m^3I^3R^3} +\end{equation} +matrices of rational rank two, by \eqref{zero:row-count} and +$J_x=I|x|/g(x)$. In the affine case we will sum $g(x)^3$ using +Lemma~\ref{zero:moment}. In the equal-pair cases we instead use +$g(x)^3\le Ih^2$, together with an extra row congruence or a smaller +number of possible null vectors. +The support statement and the three weight bounds in +Proposition~\ref{aux:weights} give the following exhaustive cases. + +\paragraph{Affinely independent residues.} +If $W(A)>0$ and the columns of $A\bmod q$ are affinely +independent, Lemma~\ref{aux:random-set} excludes the +relation $Ax=0$ when $|x|\le H$. To see both parts of this +exclusion, if $x_1+x_2+x_3\ne0$ use the restricted shift; if the +sum is zero, use affine independence and $\bar x\ne0$. +Hence a contributing shell satisfies $R>H/2$. +Using $W(A)\ll1$, \eqref{zero:fixed-null-count} with $m=1$, and +Lemma~\ref{zero:moment}, its contribution is at most +\begin{equation}\label{zero:affine-count} + \ll \frac{N^6}{I^3R^3} + \sum_{\substack{x\ \mathrm{primitive}\\R\le |x|<2R}}g(x)^3 + \ll \frac{N^6}{I^2}. +\end{equation} + +\paragraph{An equal pair with an additional equation.} +Every other matrix of positive weight has two equal columns +modulo $q$. Fix one pair $ih^{100}$. +This argument includes matrices of every rank modulo $q$. + +\paragraph{An equal pair without an additional equation.} +It remains to consider $\bar x\in\mathbb F_q\delta$. +The coordinate of $x$ outside the pair $i,j$ is divisible by +$q$ and, by \eqref{zero:null-range}, is a nonzero integer. +Thus $|x|\ge q$, and a contributing shell has $R>q/2$. +The nonzero multiples of $\delta$ give $q-1$ residue classes +for $x$ modulo $q$. Each class contains +$O((1+R/q)^3)$ vectors of length less than $2R$, so the number +of possible $x$ in this shell is +\begin{equation}\label{zero:special-normals} + \ll R^3/q^2. +\end{equation} +\smallskip +\noindent\textbf{Rank at least two modulo $q$.} +The weight bound +$W(A)\ll q^3/s$ now gives +\begin{equation}\label{zero:special-rank-two} + \ll \frac{N^6}{I^2}\,h^2q^{-2}(q^3/s) + \ll \frac{N^6}{I^2}\,\frac{h^{14}}q + \ll \frac{N^6}{I^2} +\end{equation} +for the contribution of the shell. + +\smallskip +\noindent\textbf{Rank at most one modulo $q$.} +All rows reduce into +one of the $q+1$ one-dimensional linear subspaces of $\Pi_x$. +This also covers the zero row space. For each such line +$\ell\subseteq\Pi_x$, surjectivity in Lemma~\ref{zero:plane} +shows that +\[ + \{v\in\Lambda_x:v\bmod q\in\ell\} +\] +has index $q$ in $\Lambda_x$. Formula~\eqref{zero:row-count} +thus bounds the rank-two rational row triples, summed over +all these lines, by +\[ + \ll(q+1)\frac{N^6}{(qJ_x)^3} + \ll\frac{N^6}{q^2J_x^3}. +\] +Using \eqref{zero:special-normals} and the general weight bound, +the weighted contribution is at most +\begin{equation}\label{zero:special-rank-one} + \ll \frac{N^6}{I^2}\,h^2q^{-4}(q^3/s)^2 + \ll \frac{N^6}{I^2}\,\frac{h^{26}}{q^2} + \ll \frac{N^6}{I^2}. +\end{equation} + +The affine and equal-pair cases cover the support of $W$. +Their bounds are uniform over the dyadic shells, so summing +over the $O(\log(2N))$ shells proves \eqref{zero:target}. +\end{proof} diff --git a/preprints/A-power-improvement-in-the-Heilbronn-triangle-lower-bound-September-25-2026/build/sections/08-alteration.tex b/preprints/A-power-improvement-in-the-Heilbronn-triangle-lower-bound-September-25-2026/build/sections/08-alteration.tex new file mode 100644 index 0000000000000000000000000000000000000000..b5e502e1cbb3c302526469dcc5da4b1dc01b8002 --- /dev/null +++ b/preprints/A-power-improvement-in-the-Heilbronn-triangle-lower-bound-September-25-2026/build/sections/08-alteration.tex @@ -0,0 +1,137 @@ +\section{Deletion and projective normalization}\label{alt:section} + +The preceding estimates allow us to remove every repeated projected point +and every small triangle while retaining many columns. We first construct +a set for each sufficiently large prime $r$, compare its area bound with +its cardinality, and then pass to every sufficiently large cardinality. +All constants below may depend on the fixed integer $k$; none depends on +the growing prime $r$. + +This final step is an elementary alteration argument. The earlier +probabilistic lower bound of Koml\'os, Pintz and +Szemer\'edi~\cite{KPS1982} uses a stronger hypergraph independence lemma; +no such lemma is needed below. + +\begin{proof}[Proof of Theorem~\ref{intro:main}] +Keep the parameters constructed above, with $d=41$, and set +\begin{equation}\label{alt:sample-size} + a_r=r\sqrt{N^3/\tau},\qquad n_r=\lfloor a_r\rfloor. +\end{equation} +Here $k\ge2$ and $B$ is an odd prime, so $\tau\ge1$. Take $2n_r$ samples +with the shared random choices and independent conditional draws specified +in Section~\ref{sam:section}. Let $Z$ count the unordered pairs of sample +indices with equal projected points, plus the unordered triples whose +projected points are pairwise distinct and whose determinant has absolute +value at most $\tau$. Linearity of expectation, Lemma~\ref{sam:proportional}, +and Corollary~\ref{sam:bad-triple} give +\begin{equation}\label{alt:expected-events} + \frac{\mathbb EZ}{n_r} + \ll n_r(hq)^6N^{-3} + +n_r^2(\log(2N))^2\frac{h}{N^3r^d}. +\end{equation} +No independence between the counted events is needed. + +Both terms tend to zero. Indeed, $N=(hq)^{10}$ and $\tau\ge1$ give +\[ + n_r(hq)^6N^{-3}\le r\tau^{-1/2}(hq)^{-9}=o(1). +\] +For the other term, the definition of $n_r$ gives +\[ + n_r^2(\log(2N))^2\frac{h}{N^3r^d} + \le (\log(2N))^2\frac{h}{\tau}r^{2-d}. +\] +Since $B^{k-1}\ge3$, the definition of $\tau$ implies +$\tau\ge B^{k-1}/3$, and hence $h/\tau\le3B=O_k(r^{30})$. +The bounds $B=O_k(r^{30})$, $q\le2h^{100}$, and $N=(hq)^{10}$ +also give $\log(2N)=O_k(\log r)$. Thus the second term in +\eqref{alt:expected-events} is +$O_k((\log r)^2r^{-9})=o(1)$. +Thus $\mathbb EZ=o(n_r)$. + +For sufficiently large $r$, choose an outcome with $Z\tau$. In particular, the construction excludes every +zero-area triple as well. + +For a retained triple, divide the $j$th column of $A$ by its positive +third coordinate $A_{3j}$. The resulting columns have the form +$(p_{j1},p_{j2},1)^{\mathsf T}$, so the determinant formula for triangle +area gives the exact identity +\begin{equation}\label{alt:area-identity} + \operatorname{Area}(p_1p_2p_3) + =\frac{|\det A|}{2A_{31}A_{32}A_{33}}. +\end{equation} +Since $N\le A_{3j}<2N$ for each $j$, every retained triangle has area at +least $\tau/(16N^3)$. Consequently +\begin{equation}\label{alt:area-bound} + \Delta(P_r)\ge\frac{\tau}{16N^3}. +\end{equation} +Also $\taurh^{29/2}\to\infty$. Hence $n_r\to\infty$ and eventually +$n_r\ge a_r/2$. Combining this with \eqref{alt:area-bound} yields +\begin{equation}\label{alt:scaled-area} + n_r^2\Delta(P_r)\ge\frac{r^2}{64}. +\end{equation} + +We now express both the cardinality and the area bound in terms of $r$. +The parameter choices give +\[ + \begin{gathered} + 100k^2r^{30}0$ such that +\begin{equation}\label{alt:cardinality-growth} + A_kr^\alpha\le n_r\le D_kr^\alpha, + \qquad \alpha=1+30\beta=45435k+16, +\end{equation} +for every sufficiently large prime $r$. In particular, put +\begin{equation}\label{alt:exponent} + \eta=\frac2\alpha=\frac{2}{45435k+16}. +\end{equation} +The upper bound in \eqref{alt:cardinality-growth} gives +$r^2\ge D_k^{-\eta}n_r^\eta$. Combining this with +\eqref{alt:scaled-area}, we obtain +\begin{equation}\label{alt:prime-area} + \Delta(P_r)\ge c_*n_r^{-2+\eta}, + \qquad c_*=\frac1{64D_k^\eta}>0. +\end{equation} +Thus the same cardinality estimate that will permit interpolation also +converts the factor $r^2$ into a fixed power of the number of points. + +For a sufficiently large integer $n$, put +$m=\lceil(n/A_k)^{1/\alpha}\rceil$. +Lemma~\ref{norm:prime-interval} supplies a prime $r$ with $m2$ is composite, so $P(x)=P(x-1)$. For $x=2m+1$, every prime in +$(m+1,2m+1]$ divides $\binom{2m+1}{m}$. The two central binomial +coefficients are equal and their sum is strictly less than +$2^{2m+1}$, whence $\binom{2m+1}{m}<4^m$. Thus +\[ + P(2m+1)\le P(m+1)\binom{2m+1}{m}<4^{2m+1}. +\] + +Suppose now that there is no prime in $(n,2n]$. The largest binomial +coefficient in $(1+1)^{2n}$ gives +$\binom{2n}{n}\ge4^n/(2n+1)$. +For any prime $p$, its exponent in this coefficient is +\[ + \sum_{j\ge1}\left(\left\lfloor\frac{2n}{p^j}\right\rfloor + -2\left\lfloor\frac{n}{p^j}\right\rfloor\right). +\] +Each summand is zero or one, so the entire prime power contributed +by $p$ is at most $2n$. The primes at most $\sqrt{2n}$ therefore +contribute at most $(2n)^{\sqrt{2n}}$. Larger primes have exponent +at most one. Those in $(2n/3,n]$ have exponent zero, and by assumption +there are none in $(n,2n]$. For sufficiently large $n$ it follows that +\[ + \frac{4^n}{2n+1}\le\binom{2n}{n} + \le(2n)^{\sqrt{2n}}P(2n/3) + <(2n)^{\sqrt{2n}}4^{2n/3}. +\] +Taking logarithms contradicts +$(n/3)\log4>\sqrt{2n}\log(2n)+\log(2n+1)$ for sufficiently large $n$. +\end{proof} diff --git a/preprints/A-power-improvement-in-the-Heilbronn-triangle-lower-bound-September-25-2026/main.pdf b/preprints/A-power-improvement-in-the-Heilbronn-triangle-lower-bound-September-25-2026/main.pdf new file mode 100644 index 0000000000000000000000000000000000000000..7c9f43e52300cd1441aedadbf395fc9665cf1d2d --- /dev/null +++ b/preprints/A-power-improvement-in-the-Heilbronn-triangle-lower-bound-September-25-2026/main.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:e43687a84b82e23fd73f8025d8d8c91e7fee91c8ceead9bc41502b0595f09f0a +size 443755 diff --git a/preprints/A-power-saving-for-intersective-polynomial-differences-with-an-exponent-depending-only-on-the-degree-October-5-2026/README.md b/preprints/A-power-saving-for-intersective-polynomial-differences-with-an-exponent-depending-only-on-the-degree-October-5-2026/README.md new file mode 100644 index 0000000000000000000000000000000000000000..eae247b2437c5c7023df5ca996c1228da1358bae --- /dev/null +++ b/preprints/A-power-saving-for-intersective-polynomial-differences-with-an-exponent-depending-only-on-the-degree-October-5-2026/README.md @@ -0,0 +1,18 @@ +# [A power saving for intersective polynomial differences with an exponent depending only on the degree](power-saving-intersective-polynomial-differences.pdf) + +**Author:** OpenAI + +**Date:** October 5, 2026 + +## Citation + +```bibtex +@misc{OAI:A-power-saving-for-intersective-polynomial-differences-with-an-exponent-depending-only-on-the-degree-October-5-2026, + author = {{OpenAI}}, + title = {{A power saving for intersective polynomial differences with an exponent depending only on the degree}}, + howpublished = {OpenAI Math Release preprint + \href{https://github.com/openai/math/blob/main/preprints/A-power-saving-for-intersective-polynomial-differences-with-an-exponent-depending-only-on-the-degree-October-5-2026/power-saving-intersective-polynomial-differences.pdf}{OAI:A-power-saving-for-intersective-polynomial-differences-with-an-exponent-depending-only-on-the-degree-October-5-2026}}, + year = {2026} +} +``` + diff --git a/preprints/A-power-saving-for-intersective-polynomial-differences-with-an-exponent-depending-only-on-the-degree-October-5-2026/build/manuscript/figures/list-regrouping.tex b/preprints/A-power-saving-for-intersective-polynomial-differences-with-an-exponent-depending-only-on-the-degree-October-5-2026/build/manuscript/figures/list-regrouping.tex new file mode 100644 index 0000000000000000000000000000000000000000..f7e971ce1906da1ae2e560b834b98d2bcf5339c5 --- /dev/null +++ b/preprints/A-power-saving-for-intersective-polynomial-differences-with-an-exponent-depending-only-on-the-degree-October-5-2026/build/manuscript/figures/list-regrouping.tex @@ -0,0 +1,54 @@ +\begin{figure}[htbp] +\centering +\begin{tikzpicture}[ + x=1cm,y=1cm, + font=\small, + list/.style={draw=black!65,rounded corners=1pt, + minimum width=3.1cm,minimum height=.56cm}, + selected/.style={fill=blue!9,draw=blue!60!black}, + every path/.style={line width=.55pt} +] + \node[align=center] at (1.55,.85) + {$r_*$ old plus lists\\addresses $(u,q)$, $q\in\Pi_+$}; + \node[align=center] at (8.65,.85) + {one new list at $u$\\length $\ell_{s+1}=r_*\ell_s$}; + + \node[list] at (1.55,0) {$1\quad\cdots\quad\ell_s$}; + \node[anchor=east] at (-.15,0) {$q_1$}; + \node at (1.55,-.62) {$\vdots$}; + \node[list,selected] at (1.55,-1.25) + {$1\quad\cdots\quad\ell_s$}; + \node[anchor=east] at (-.15,-1.25) {$q=v_b$}; + \node at (1.55,-1.88) {$\vdots$}; + \node[list] at (1.55,-2.5) {$1\quad\cdots\quad\ell_s$}; + \node[anchor=east] at (-.15,-2.5) {$q_{r_*}$}; + + \draw[<->,>=Stealth] (3.38,-1.25)--(6.82,-1.25); + \node[above,align=center] at (5.1,-1.13) {bijective relabeling}; + \node[below,align=center] at (5.1,-1.37) {no restriction}; + + \draw[black!65,rounded corners=1pt] (7.05,.32) rectangle (10.25,-2.82); + \node at (8.65,0) {$\{q_1\}\times[\ell_s]$}; + \node at (8.65,-.62) {$\vdots$}; + \filldraw[selected] (7.05,-.97) rectangle (10.25,-1.53); + \node at (8.65,-1.25) {$\{v_b\}\times[\ell_s]$}; + \node at (8.65,-1.88) {$\vdots$}; + \node at (8.65,-2.5) {$\{q_{r_*}\}\times[\ell_s]$}; + + \draw[->,>=Stealth] (8.65,-2.92)--(8.65,-3.72); + \node[anchor=east,align=right] at (8.45,-3.32) + {restrict the index\\to $q=v_b$}; + \node[anchor=west] at (8.85,-3.32) {probability $1/r_*$}; + \node[list,selected] at (8.65,-4.1) {$j'_v=(v_b,i)$}; + \node[below] at (8.65,-4.42) {$i$ uniform on $[\ell_s]$}; +\end{tikzpicture} +\caption{Regrouping at a fixed new address +$u=u_{S\cup\{b\}}(v)$ and restricting the index of one slot $v\in V_+$. +The horizontal arrow only relabels the same independent entries; it +incurs no probability factor. The highlighted sublist is prescribed +by $v_b$. Restricting all $n_*$ independent slot indices to their +respective prescribed sublists has probability $r_*^{-n_*}$. +The ellipses denote the other words in $\Pi_+$; boxes describe index +positions, and their field values may coincide.} +\label{tuple:fig-regrouping} +\end{figure} diff --git a/preprints/A-power-saving-for-intersective-polynomial-differences-with-an-exponent-depending-only-on-the-degree-October-5-2026/build/manuscript/main.tex b/preprints/A-power-saving-for-intersective-polynomial-differences-with-an-exponent-depending-only-on-the-degree-October-5-2026/build/manuscript/main.tex new file mode 100644 index 0000000000000000000000000000000000000000..5633fe7625cdf492190c6c806d6a4bdf99dcc8ca --- /dev/null +++ b/preprints/A-power-saving-for-intersective-polynomial-differences-with-an-exponent-depending-only-on-the-degree-October-5-2026/build/manuscript/main.tex @@ -0,0 +1,75 @@ +\documentclass[11pt]{article} +\usepackage[T1]{fontenc} +\usepackage{lmodern} +\input{glyphtounicode-cmex} +\usepackage{amsmath,amssymb,amsthm,mathtools} +\usepackage[margin=1in]{geometry} +\usepackage{microtype} +\usepackage{booktabs} +\usepackage{tikz} +\usetikzlibrary{arrows.meta,positioning,calc,fit} +\usepackage{xcolor} +\usepackage{url} +\usepackage[colorlinks=true,linkcolor=blue!45!black,citecolor=blue!45!black, + urlcolor=blue!45!black,pdfauthor={OpenAI}, + pdftitle={A power saving for intersective polynomial differences with an exponent depending only on the degree}]{hyperref} + +\newtheorem{theorem}{Theorem}[section] +\newtheorem{proposition}[theorem]{Proposition} +\newtheorem{lemma}[theorem]{Lemma} +\newtheorem{corollary}[theorem]{Corollary} +\theoremstyle{definition} +\newtheorem{definition}[theorem]{Definition} +\theoremstyle{remark} +\newtheorem{remark}[theorem]{Remark} +\numberwithin{equation}{section} +\newcommand{\N}{\mathbb N} +\newcommand{\Z}{\mathbb Z} +\newcommand{\Q}{\mathbb Q} +\newcommand{\R}{\mathbb R} +\newcommand{\C}{\mathbb C} +\newcommand{\F}{\mathbb F} +\newcommand{\E}{\mathbb E} +\newcommand{\1}{\mathbf 1} +\newcommand{\eps}{\varepsilon} +\newcommand{\abs}[1]{\lvert#1\rvert} +\newcommand{\norm}[1]{\lVert#1\rVert} +\DeclareMathOperator{\supp}{supp} +\DeclareMathOperator{\rad}{rad} +\DeclareMathOperator{\id}{id} +\newcommand{\articleid}[1]{\nolinkurl{#1}} +\allowdisplaybreaks[1] + +\title{A power saving for intersective polynomial differences\\ +with an exponent depending only on the degree} +\author{OpenAI} +\date{October 5, 2026} + +\begin{document} +\maketitle +\begin{abstract} +An integer polynomial is intersective if it has a root modulo every positive +integer. For each degree $k\ge2$, we prove that there is an exponent $c_k>0$ +such that every set $A\subseteq\{1,\ldots,N\}$ whose differences avoid all nonzero values $h(1),h(2),\ldots$, +where $h$ is an intersective polynomial of degree $k$ with positive leading +coefficient, satisfies +$|A|=O_h(N^{1-c_k})$. The implied constant may depend on $h$, but the +power-saving exponent depends only on its degree. +\end{abstract} + +\tableofcontents + +\input{sections/introduction} +\input{sections/local} +\input{sections/tuples} +\input{sections/lifts} +\input{sections/kernel} +\input{sections/transfer} +\input{sections/induction} + +\bibliographystyle{alpha} +\clearpage +\phantomsection +\addcontentsline{toc}{section}{References} +\bibliography{references/references} +\end{document} diff --git a/preprints/A-power-saving-for-intersective-polynomial-differences-with-an-exponent-depending-only-on-the-degree-October-5-2026/build/manuscript/references/references.bib b/preprints/A-power-saving-for-intersective-polynomial-differences-with-an-exponent-depending-only-on-the-degree-October-5-2026/build/manuscript/references/references.bib new file mode 100644 index 0000000000000000000000000000000000000000..98a04c63f7d702ea8299782cf8330cd2e4db4207 --- /dev/null +++ b/preprints/A-power-saving-for-intersective-polynomial-differences-with-an-exponent-depending-only-on-the-degree-October-5-2026/build/manuscript/references/references.bib @@ -0,0 +1,194 @@ +@article{Furstenberg1977, + author = {Furstenberg, H.}, + title = {Ergodic behavior of diagonal measures and a theorem of {Szemer{\'e}di} on arithmetic progressions}, + journal = {Journal d'Analyse Math{\'e}matique}, + volume = {31}, + pages = {204--256}, + year = {1977}, + doi = {10.1007/BF02813304}, + url = {https://doi.org/10.1007/BF02813304}, + note = {\href{https://doi.org/10.1007/BF02813304}{doi:10.1007/\allowbreak{}BF02813304}} +} + +@article{Sarkozy1978, + author = {S{\'a}rk{\"o}zy, A.}, + title = {On difference sets of sequences of integers. {I}}, + journal = {Acta Mathematica Academiae Scientiarum Hungaricae}, + volume = {31}, + number = {1--2}, + pages = {125--149}, + year = {1978}, + doi = {10.1007/BF01896079}, + url = {https://doi.org/10.1007/BF01896079}, + note = {\href{https://doi.org/10.1007/BF01896079}{doi:10.1007/\allowbreak{}BF01896079}} +} + +@article{KamaeMendesFrance1978, + author = {Kamae, T. and Mend{\`e}s France, M.}, + title = {Van der {Corput}'s difference theorem}, + journal = {Israel Journal of Mathematics}, + volume = {31}, + number = {3--4}, + pages = {335--342}, + year = {1978}, + doi = {10.1007/BF02761498}, + url = {https://doi.org/10.1007/BF02761498}, + note = {\href{https://doi.org/10.1007/BF02761498}{doi:10.1007/\allowbreak{}BF02761498}} +} + +@article{PintzSteigerSzemeredi1988, + author = {Pintz, J. and Steiger, W. L. and Szemer{\'e}di, E.}, + title = {On sets of natural numbers whose difference set contains no squares}, + journal = {Journal of the London Mathematical Society}, + series = {2}, + volume = {37}, + number = {2}, + pages = {219--231}, + year = {1988}, + doi = {10.1112/jlms/s2-37.2.219}, + url = {https://doi.org/10.1112/jlms/s2-37.2.219}, + note = {\href{https://doi.org/10.1112/jlms/s2-37.2.219}{doi:10.1112/\allowbreak{}jlms/s2-37.2.219}} +} + +@article{Lucier2006, + author = {Lucier, Jason}, + title = {Intersective sets given by a polynomial}, + journal = {Acta Arithmetica}, + volume = {123}, + number = {1}, + pages = {57--95}, + year = {2006}, + doi = {10.4064/aa123-1-4}, + url = {https://doi.org/10.4064/aa123-1-4}, + note = {\href{https://doi.org/10.4064/aa123-1-4}{doi:10.4064/\allowbreak{}aa123-1-4}} +} + +@article{Rice2019, + author = {Rice, Alex}, + title = {A maximal extension of the best-known bounds for the {Furstenberg--S{\'a}rk{\"o}zy} theorem}, + journal = {Acta Arithmetica}, + volume = {187}, + number = {1}, + pages = {1--41}, + year = {2019}, + doi = {10.4064/aa170828-26-8}, + url = {https://doi.org/10.4064/aa170828-26-8}, + eprint = {1612.01760}, + archivePrefix = {arXiv}, + primaryClass = {math.NT}, + note = {\href{https://doi.org/10.4064/aa170828-26-8}{doi:10.4064/\allowbreak{}aa170828-26-8}. + Revised version: \href{https://arxiv.org/abs/1612.01760v5}{arXiv:\allowbreak1612.01760v5}, June 8, 2020} +} + +@article{BloomMaynard2022, + author = {Bloom, Thomas F. and Maynard, James}, + title = {A new upper bound for sets with no square differences}, + journal = {Compositio Mathematica}, + volume = {158}, + number = {8}, + pages = {1777--1798}, + year = {2022}, + doi = {10.1112/S0010437X22007679}, + url = {https://doi.org/10.1112/S0010437X22007679}, + eprint = {2011.13266}, + archivePrefix = {arXiv}, + primaryClass = {math.NT}, + note = {\href{https://doi.org/10.1112/S0010437X22007679}{doi:10.1112/\allowbreak{}S0010437X22007679}; + \href{https://arxiv.org/abs/2011.13266}{arXiv:\allowbreak2011.13266}} +} + +@article{Arala2024, + author = {Arala, Nuno}, + title = {A maximal extension of the {Bloom--Maynard} bound for sets without square differences}, + journal = {Functiones et Approximatio Commentarii Mathematici}, + volume = {71}, + number = {2}, + pages = {271--296}, + year = {2024}, + doi = {10.7169/facm/2158}, + url = {https://doi.org/10.7169/facm/2158}, + eprint = {2303.03345}, + archivePrefix = {arXiv}, + primaryClass = {math.NT}, + note = {\href{https://doi.org/10.7169/facm/2158}{doi:10.7169/\allowbreak{}facm/2158}. + Preprint (title uses ``with no square differences''): + \href{https://arxiv.org/abs/2303.03345v1}{arXiv:\allowbreak2303.03345v1}, March 6, 2023} +} + +@misc{GreenSawhney2025, + author = {Green, Ben and Sawhney, Mehtaab}, + title = {New bounds for the {Furstenberg--S{\'a}rk{\"o}zy} theorem}, + year = {2025}, + eprint = {2411.17448}, + archivePrefix = {arXiv}, + primaryClass = {math.NT}, + doi = {10.48550/arXiv.2411.17448}, + url = {https://arxiv.org/abs/2411.17448v2}, + howpublished = {\href{https://arxiv.org/abs/2411.17448v2}{arXiv:\allowbreak2411.17448v2}}, + note = {Version 2, January 10, 2025} +} + +@misc{AdajarEtAl2026, + author = {Adajar, Carlo Francisco E. and Agrawal, Rishika and Choudhuri, Mukul Rai and Chuah, Chian Yeong and Fan, Steve and Hegde, Swaroop and Lott, Andrew and Nandakumar, Krishnamohan and Ponagandla, Nagendar Reddy}, + title = {Extensions of the {Furstenberg--S{\'a}rk{\"o}zy} theorem via the arithmetic level-$d$ inequality}, + year = {2026}, + eprint = {2605.16216}, + archivePrefix = {arXiv}, + primaryClass = {math.NT}, + doi = {10.48550/arXiv.2605.16216}, + url = {https://arxiv.org/abs/2605.16216v2}, + howpublished = {\href{https://arxiv.org/abs/2605.16216v2}{arXiv:\allowbreak2605.16216v2}}, + note = {Version 2, August 28, 2026} +} + +@article{FILS1978, + author = {Fr{\"o}hlich, J{\"u}rg and Israel, Robert B. and Lieb, Elliott H. and Simon, Barry}, + title = {Phase transitions and reflection positivity. {I}. General theory and long range lattice models}, + journal = {Communications in Mathematical Physics}, + volume = {62}, + number = {1}, + pages = {1--34}, + year = {1978}, + doi = {10.1007/BF01940327}, + url = {https://doi.org/10.1007/BF01940327}, + note = {\href{https://doi.org/10.1007/BF01940327}{doi:10.1007/\allowbreak{}BF01940327}} +} + +@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://doi.org/10.1007/s11856-010-0005-1}, + eprint = {0806.0047}, + archivePrefix = {arXiv}, + note = {\href{https://doi.org/10.1007/s11856-010-0005-1}{doi:10.1007/\allowbreak{}s11856-010-0005-1}; + \href{https://arxiv.org/abs/0806.0047}{arXiv:\allowbreak0806.0047}} +} + +@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}, + eprint = {1611.05784}, + archivePrefix = {arXiv}, + primaryClass = {math.CO}, + note = {\href{https://doi.org/10.1016/j.aim.2017.05.009}{doi:10.1016/\allowbreak{}j.aim.2017.05.009}; + \href{https://arxiv.org/abs/1611.05784}{arXiv:\allowbreak1611.05784}} +} + +@misc{squarepower, + author = {{OpenAI}}, + title = {{A power saving for square-difference-free sets}}, + howpublished = {OpenAI Math Release preprint + \href{https://github.com/openai/math/blob/main/preprints/A-power-saving-for-square-difference-free-sets-September-24-2026/paper.pdf}{OAI:A-power-saving-for-square-difference-free-sets-September-24-2026}}, + year = {2026} +} diff --git a/preprints/A-power-saving-for-intersective-polynomial-differences-with-an-exponent-depending-only-on-the-degree-October-5-2026/build/manuscript/sections/induction.tex b/preprints/A-power-saving-for-intersective-polynomial-differences-with-an-exponent-depending-only-on-the-degree-October-5-2026/build/manuscript/sections/induction.tex new file mode 100644 index 0000000000000000000000000000000000000000..a4a6e0fe28d973165c245854113d9af1a3d1afb4 --- /dev/null +++ b/preprints/A-power-saving-for-intersective-polynomial-differences-with-an-exponent-depending-only-on-the-degree-October-5-2026/build/manuscript/sections/induction.tex @@ -0,0 +1,363 @@ +\section{Exceptional terms and the power-saving induction} +\label{sec:induction} + +We now combine the shorter-scale comparison with cancellation on good +residue assignments to obtain a recurrence for $\mathcal Y_l(N)$. +The exceptional parts of the prime laws contribute a summable family of +additional shorter-scale terms. The expansion and contraction argument +adapt \cite[Proposition~9.2 and Theorem~9.3]{squarepower}. + +Retain all parameters from \eqref{eq:14}. For $B\subseteq\mathcal P_N$, put +\[ + D_B=M_0q_B,\qquad + n_B=\left\lceil\frac{N}{\lambda(D_B)}\right\rceil, + \qquad + \eta_{l,B}=\prod_{p\in B}\eta_{l,p}. +\] +Thus $q_\varnothing=\eta_{l,\varnothing}=1$ and +$D_\varnothing=M_0$. Recall that $\rho_l$ is the proportion of good +residue assignments for $h_l$, and that +$\rho_l\ge\rho_*=M_0^{1-L}>0$ by +Lemma~\ref{transfer:good-assignments}. + +\begin{proposition}[Recurrence for the maximal functional] +\label{induction:recurrence} +There are constants $C_1\ge1$ and $N_1\ge3$, depending only on the fixed +polynomial $h$, its chosen roots, and the fixed structural choices, such that +for every integer $N\ge N_1$ and every positive integer $l\le N^\zeta$, +\begin{equation} + \begin{split} + \mathcal Y_l(N)\le C_1N^{-\sigma} + +(1+u_N)\Bigg(& + (1-\rho_l)\mathcal Y_{lD_\varnothing}(n_\varnothing)\\ + &+\rho_l + \sum_{\substack{\varnothing\ne B\subseteq\mathcal P_N\\q_B\le N^\tau}} + \eta_{l,B}\mathcal Y_{lD_B}(n_B)\Bigg), + \end{split} + \label{eq:31} +\end{equation} +where +\[ + u_N=\bigl(1+M_0^kN^{k\tau-1}\bigr)^d-1\longrightarrow0. +\] +Every scale $n_B$ appearing on the right belongs to $[1,N-1]$. +\end{proposition} + +\begin{proof} +Fix $A\subseteq[N]$ satisfying \eqref{eq:24}. Use the functions $f_s$ and +their lifts $G_s$ from Section~\ref{sec:transfer}, and, for a residue +assignment $\mathbf s=(s_v)_{v\in V}$, write +\[ + \Phi_{\mathbf s}((z_v)_{v\in V})=\prod_{v\in V}G_{s_v}(z_v). +\] +Since $\E_s f_s=\1_A$, linearity of the lift and multilinearity of the +integral give +\begin{equation} + Y_l(N,A)=\E_{\mathbf s} + \int\Phi_{\mathbf s}\,d\bigotimes_{p\in\mathcal P_N}\mu_{l,p}, + \label{induction:residue-average} +\end{equation} +where the residues $s_v$ are independent and uniform modulo $M_0$. + +For each $B\subseteq\mathcal P_N$, define the nonnegative measure +\[ + \kappa_{l,B}= + \bigotimes_{p\in B}\eps_{l,p} + \otimes\bigotimes_{p\in\mathcal P_N\setminus B}\mu_{l,p}. +\] +Its mass is $\eta_{l,B}$. Expanding the finite product gives the identity +of signed measures +\begin{equation} + \bigotimes_{p\in\mathcal P_N}\mu_{l,p} + -\bigotimes_{p\in\mathcal P_N}(\mu_{l,p}-\eps_{l,p}) + =\sum_{\varnothing\ne B\subseteq\mathcal P_N} + (-1)^{|B|+1}\kappa_{l,B}. + \label{induction:exceptional-expansion} +\end{equation} +The complement of $B$ therefore carries the full laws $\mu_{l,p}$, to +which Proposition~\ref{transfer:shorter-scale} applies. The functions +$\Phi_{\mathbf s}$ can have either sign; we will take the absolute value +of each integral on the right of +\eqref{induction:exceptional-expansion}. + +The exceptional masses have a uniform bound with an extra factor $q_B$: +\begin{equation} + \sum_{\varnothing\ne B\subseteq\mathcal P_N}q_B\eta_{l,B} + =\prod_{p\in\mathcal P_N}(1+p\eta_{l,p})-1 + \le\prod_{p\ge P}(1+p^{-3})-1\le\frac14. + \label{eq:32} +\end{equation} +Here we used $\eta_{l,p}\le p^{-4}$ and the choice of $P$ in +\eqref{eq:9}. This bound first allows us to discard the terms with +$q_B>N^\tau$. + +To see this, suppose $\eta_{l,B}>0$. Simultaneous translation invariance +of $\eps_{l,p}$ implies that each single-label marginal of +$\eps_{l,p}/\eta_{l,p}$ is uniform on $\F_p$. The same holds for the full +laws. Thus each single-slot marginal of $\kappa_{l,B}/\eta_{l,B}$ is +uniform on $X_{\mathcal P_N}$. Ordinary H\"older's inequality and the +moment estimate \eqref{eq:12}, at the fixed even exponent $d$, give +\begin{equation} + \left|\int\Phi_{\mathbf s}\,d\kappa_{l,B}\right| + \le\eta_{l,B}\prod_{v\in V}\|G_{s_v}\|_d + \le C_\epsilon\eta_{l,B}N^\epsilon + \qquad(\epsilon>0). + \label{induction:exceptional-moment} +\end{equation} +The constants are uniform in $l,B,A$ and $\mathbf s$; the bound +$|f_s|\le M_0$ only introduces a fixed factor. Terms of zero mass vanish. +Combining \eqref{induction:exceptional-moment} with \eqref{eq:32} yields +\[ + \sum_{\substack{\varnothing\ne B\subseteq\mathcal P_N\\q_B>N^\tau}} + \left|\int\Phi_{\mathbf s}\,d\kappa_{l,B}\right| + \le \frac{C_\epsilon}{4}N^{-\tau+\epsilon} + =O(N^{-\sigma}), +\] +where, for example, $\epsilon=\tau/2$ suffices because +$\sigma=\gamma\tau/4<\tau/2$. + +For $q_B\le N^\tau$ and positive mass, condition instead on all the +labels at primes in $B$. The laws at the remaining primes are unchanged, +and the function in each slot is the corresponding specialization of +$G_{s_v}$. Proposition~\ref{transfer:shorter-scale}, which allows arbitrary +slotwise specializations, gives +\begin{equation} + \left|\int\Phi_{\mathbf s}\,d\kappa_{l,B}\right| + \le\eta_{l,B}\left( + \Lambda_B^d\mathcal Y_{lD_B}(n_B)+C_2N^{-\sigma}\right), + \qquad + \Lambda_B=\frac{\lambda(D_B)n_B}{N}. + \label{induction:exceptional-transfer} +\end{equation} +The bound is uniform in every conditioned tuple, so integration over +the prescribed labels preserves it. It also holds for zero-mass terms. +By \eqref{eq:27}, $\Lambda_B^d\le1+u_N$; by \eqref{eq:32}, +$\sum_{B\ne\varnothing}\eta_{l,B}\le1/4$. Therefore summing the errors +in \eqref{induction:exceptional-transfer} still costs $O(N^{-\sigma})$. + +For a good assignment $\mathbf s$, Proposition~\ref{transfer:cancellation} +bounds the integral against +$\bigotimes_p(\mu_{l,p}-\eps_{l,p})$ by $O(N^{-\sigma})$ in absolute +value. The expansion \eqref{induction:exceptional-expansion} and the +two exceptional estimates consequently give +\[ + \int\Phi_{\mathbf s}\,d\bigotimes_p\mu_{l,p} + \le O(N^{-\sigma})+(1+u_N) + \sum_{\substack{\varnothing\ne B\subseteq\mathcal P_N\\q_B\le N^\tau}} + \eta_{l,B}\mathcal Y_{lD_B}(n_B). +\] +For every other assignment, apply +Proposition~\ref{transfer:shorter-scale} directly with $B=\varnothing$. +The resulting bound is +$(1+u_N)\mathcal Y_{lM_0}(n_\varnothing)+O(N^{-\sigma})$. +Averaging these estimates in \eqref{induction:residue-average} gives +the coefficients $\rho_l$ and $1-\rho_l$ in \eqref{eq:31}. +The bounds are uniform in $A$, so we may take the maximum defining +$\mathcal Y_l(N)$. + +Finally, $k\tau<1$ by \eqref{eq:14}, so $u_N\to0$, and the bound +$1\le n_B0$ and $C_0\ge1$, depending only on the fixed +polynomial $h$, its chosen roots, and the fixed structural choices, such that +for all positive integers $l,N$, +\begin{equation} + \mathcal Y_l(N)\le C_0l^{b_0}N^{-a_0}. + \label{eq:33} +\end{equation} +The exponents $a_0,b_0$ can be chosen as functions only of the structural +parameters $k,\zeta,\sigma,M_0,\rho_*$. If the chosen roots of $h$ +satisfy Condition~\eqref{local:unit-condition} and the choices depending +only on $k$ from Sections~\ref{sec:local} and~\ref{sec:tuples} are used, then +$a_0,b_0,d$ depend only on $k$, while $C_0$ may still depend on the fixed data. +\end{proposition} + +\begin{proof} +To select the exponents, suppose that at a scale where the recurrence applies +the trial bound +$\mathcal Y_{lD_B}(n_B)\le C(lD_B)^b n_B^{-a}$ is available at every +successor scale, with $C\ge1$ and $a,b>0$. Put $s=ka+b$. Since +$n_B\ge N/\lambda(D_B)$ and $\lambda(D_B)\le D_B^k$, each trial bound gives +\[ + C(lD_B)^b n_B^{-a} + \le C l^bN^{-a}\lambda(D_B)^aD_B^b + \le C l^bN^{-a}D_B^s. +\] +If $s\le1$, then $q_B^s\le q_B$, so Equation~\eqref{eq:32} bounds the +coefficient of $C l^bN^{-a}$ inside the +parentheses of \eqref{eq:31} by +\[ + M_0^s\left(1-\rho_l+ + \rho_l\sum_{\substack{\varnothing\ne B\subseteq\mathcal P_N\\q_B\le N^\tau}} + \eta_{l,B}q_B^s\right) + \le M_0^s(1-3\rho_*/4). +\] +This upper bound tends to $1-3\rho_*/4<1$ as $s$ tends to zero. Two +further restrictions handle the other parts of the argument. +For any proposed $a>0$, Corollary~\ref{lift:density} gives a bound +$\mathcal Y_l(N)\le C_aN^a$ for all positive $l,N$. Taking +$b=2a/\zeta$ makes $l^bN^{-a}>N^a$ whenever $l>N^\zeta$, so this +subpower bound covers the range outside the recurrence after increasing +$C$. Finally, $a<\sigma$ makes $C_1N^{-\sigma}=o(N^{-a})$, allowing +the additive error to be absorbed at sufficiently large scales. + +Choose explicitly +\[ + a_0=\frac12\min\left\{ + \sigma,\quad \frac1{k+2/\zeta},\quad + \frac{-\log(1-3\rho_*/4)}{(k+2/\zeta)\log M_0} + \right\}, + \qquad b_0=\frac{2a_0}{\zeta}. +\] +These numbers are positive because $M_0\ge2$ and $0<\rho_*\le1$. +They satisfy +\begin{equation} + a_0<\sigma,\qquad + s_0:=ka_0+b_0\le1,\qquad + \theta:=M_0^{s_0}(1-3\rho_*/4)<1. + \label{induction:exponent-choice} +\end{equation} +Indeed, $s_0=(k+2/\zeta)a_0\le1/2$, and the third term in the minimum gives +\[ + s_0\log M_0\le-\tfrac12\log(1-3\rho_*/4), + \qquad + \theta\le(1-3\rho_*/4)^{1/2}<1. +\] +This displays the asserted structural dependence. Under +Condition~\eqref{local:unit-condition}, Sections~\ref{sec:local} +and~\ref{sec:tuples} permit $L,P,M_0,d$ to be fixed in terms of $k$ only. +Then $\rho_*=M_0^{1-L}$ and all parameters in \eqref{eq:14} depend only +on $k$, so the displayed choices of $a_0,b_0$ do as well. The constants +in the estimates below, and consequently $C_0$, may depend on the fixed data. +All quantities in \eqref{induction:exponent-choice} are fixed from now on. + +Corollary~\ref{lift:density} supplies a constant +$C_*\ge1$ such that +\begin{equation} + \mathcal Y_l(N)\le C_*N^{a_0} + \qquad(l,N\ge1). + \label{induction:subpower-bound} +\end{equation} +If $l>N^\zeta$, then +\[ + l^{b_0}N^{-a_0}>N^{\zeta b_0-a_0}=N^{a_0}, +\] +so \eqref{eq:33} follows from \eqref{induction:subpower-bound} for any +$C_0\ge C_*$. This establishes the estimate outside the range of +the recurrence at every scale. + +We choose the initial range before fixing $C_0$. Set +$\kappa=(1-\theta)/2>0$, and take an integer $N_*\ge N_1$ so large that +for every $N\ge N_*$, +\begin{equation} + \theta u_N\le\kappa,\qquad + C_1N^{a_0-\sigma}\le\kappa. + \label{induction:threshold} +\end{equation} +This is possible because $u_N\to0$ and $a_0<\sigma$; the choice is +independent of $C_0$. Now choose +$C_0\ge\max\{1,C_*N_*^{2a_0}\}$. For all $l\ge1$ and $NN^\zeta$ was established above, so assume +$l\le N^\zeta$ and apply Proposition~\ref{induction:recurrence}. +Its successor scales satisfy $1\le n_B0$, it gives +\[ + |A|=\sum_{j=1}^M|A_j| + \le M C_0^{1/d}\left\lceil\frac NM\right\rceil^{1-c_k} + \le 2^{1-c_k}M^{c_k}C_0^{1/d}N^{1-c_k}. +\] +Taking $C_h=2^{1-c_k}M^{c_k}C_0^{1/d}\ge1$ and $N_h=M$ proves the theorem. +The constants $C_h,N_h$ may depend on every coefficient of $h$, while +$c_k$ depends only on $k$. +\end{proof} diff --git a/preprints/A-power-saving-for-intersective-polynomial-differences-with-an-exponent-depending-only-on-the-degree-October-5-2026/build/manuscript/sections/introduction.tex b/preprints/A-power-saving-for-intersective-polynomial-differences-with-an-exponent-depending-only-on-the-degree-October-5-2026/build/manuscript/sections/introduction.tex new file mode 100644 index 0000000000000000000000000000000000000000..e5db249b0a4de89590e048685326ea3ffe5af9ad --- /dev/null +++ b/preprints/A-power-saving-for-intersective-polynomial-differences-with-an-exponent-depending-only-on-the-degree-October-5-2026/build/manuscript/sections/introduction.tex @@ -0,0 +1,228 @@ +\section{Introduction}\label{sec:introduction} + +For a polynomial $h\in\Z[x]$, consider how large a set of integers can be +if no difference between two of its elements is a nonzero value of $h$. +Write $\N_+=\{1,2,\ldots\}$, $[N]=\{1,\ldots,N\}$, and +$A-A=\{a-b:a,b\in A\}$. We call $h$ \emph{intersective} if, for every +integer $q\ge1$, some integer $t$ satisfies $h(t)\equiv0\pmod q$. +This is the necessary local condition for polynomial differences in every +set of positive density: if $h$ has no root modulo $q$, the multiples of +$q$ contain no difference in $h(\N_+)$. The condition allows the roots at +different moduli to arise from different factors of $h$; a rational root is +not required. + +\begin{theorem}\label{thm:main} +For every integer $k\ge2$, there is a constant $c_k\in(0,1)$ with the +following property. For every intersective polynomial $h\in\Z[x]$ of degree +$k$ with positive leading coefficient, there are a constant $C_h\ge1$ and an +integer $N_h\ge1$ such that, whenever $N\ge N_h$ and $A\subseteq[N]$ satisfy +\[ + (A-A)\cap h(\N_+)\subseteq\{0\}, +\] +one has +\[ + |A|\le C_h N^{1-c_k}. +\] +\end{theorem} + +The exponent is common to all polynomials of a fixed degree. The constants +$C_h,N_h$ may depend on every coefficient of $h$, and $h$ remains fixed as +$N$ grows. The theorem concerns ordinary integer differences; zero values of +$h$ impose no restriction on $A$. + +\subsection{History and relation to earlier work} + +Furstenberg~\cite{Furstenberg1977} and S\'ark\"ozy~\cite{Sarkozy1978} +proved that every set of positive upper density contains two elements +whose difference is a nonzero square. Their proofs introduced distinct +ergodic and Fourier-analytic approaches to this recurrence problem. +Kamae and Mend\`es France~\cite{KamaeMendesFrance1978} extended the +positive-density conclusion to nonconstant intersective polynomials. Together with +the elementary congruence obstruction above, this identifies the exact +class of nonconstant polynomials for which the density of an avoiding set must tend +to zero. + +The quantitative problem has driven successive refinements of Fourier +density-increment methods. For square differences, Pintz, Steiger, and +Szemer\'edi~\cite{PintzSteigerSzemeredi1988} introduced a substantially +stronger iteration. Bloom and Maynard~\cite{BloomMaynard2022} later +obtained the bound +\[ + |A|\ll N(\log N)^{-c\log\log\log N} +\] +using estimates for the additive energy of rational numbers. +For general intersective polynomials, Lucier~\cite{Lucier2006} established +quantitative bounds using auxiliary polynomials adapted to progressions, +and Rice~\cite{Rice2019} developed stronger estimates by combining +iteration with sieved polynomial sums. +Arala~\cite{Arala2024} extended the Bloom--Maynard bound to every +intersective polynomial of degree at least two, with a constant depending +only on the degree in the exponent of $\log N$. + +Green and Sawhney~\cite[Theorem~1.1]{GreenSawhney2025} obtained the +square-difference bound +\[ + |A|\ll N\exp(-c\sqrt{\log N}). +\] +Their arithmetic +level-$d$ inequality and random sparsification also underpin the recent +general-polynomial result of Adajar et al.~\cite[Theorem~1.1]{AdajarEtAl2026}: +for every fixed intersective $h$ of degree at least two and every fixed +$0<\mu<1/2$, +\[ + |A|\le N\exp\bigl(-c_{h,\mu}(\log N)^\mu\bigr) +\] +for sufficiently large $N$. Theorem~\ref{thm:main} gives a fixed power +of $N$ as the saving, with an exponent common to all polynomials of each +fixed degree. + +The immediate predecessor for our method is the power-saving theorem +for square differences in~\cite[Theorem~1.1]{squarepower}. Its proof +uses finite reflection-positive tuple laws and a signed multilinear +functional. Hatami~\cite{Hatami2010} gave a multilinear H\"older criterion +for graph norms. Reflection and Cauchy--Schwarz arguments are central to +the reflection-group constructions of Conlon and Lee~\cite{ConlonLee2017}. +Here we adapt the specific laws and functional of~\cite{squarepower}, +with all required proofs included, and supply the polynomial and +change-of-scale arguments described next. + +\subsection{The argument and its principal constructions} + +The square-difference method constructs a multilinear form +which dominates a fixed power of density, and proves that this form contracts +on passing to shorter intervals. We retain this organization, but polynomial +values require two additional compatibilities: the polynomial changes when +one passes to a progression, and the local distribution supplied by the +multilinear construction need not be the natural distribution of polynomial +values. + +\paragraph{Normalizing the local coefficients.} +We first pass from the polynomial in Theorem~\ref{thm:main} to a polynomial +$g(x)=h(r+D_0x)/M$ for which the first nonzero Taylor coefficient at one +chosen root in each $\Z_p$ is a unit. Only finitely many primes require the +extra division by $M$. The multiplicity of each chosen root is retained, and +the representative $r$ is chosen so that positive inputs for $g$ give positive +inputs for $h$. The purpose is to make the local numerical choices that +determine the final exponent depend only on the degree. + +\paragraph{A family closed under passage to progressions.} +Section~\ref{sec:local} retains the auxiliary construction for an arbitrary +fixed intersective polynomial $h$. It gives integer polynomials $h_l$, indexed by +$l\in\N_+$, with $h_1=h$, and a fixed completely multiplicative function +$\lambda:\N_+\to\N_+$ such that, for all positive integers $l,D$, +\[ + h_l(Dx-i)=\lambda(D)h_{lD}(x) + \quad\text{for some }0\le ik$, its reduced derivative is nonzero, and the complete-sum estimates hold +beyond a threshold depending only on $k$. At the finitely many smaller primes below that +threshold, interpolation bounds the derivative valuation at some argument in +terms of $k$ and $p$. These bounds make the lifting moduli and cycle length, +and hence the number of tuple slots below, depend only on $k$. + +\paragraph{Weighted forms that follow the family.} +For each sufficiently large prime $p$, Section~\ref{sec:tuples} constructs a +probability distribution on tuples $(z_v)_{v\in V}\in\F_p^V$, where $V$ is +one fixed finite set of even cardinality $d$. A distinguished cycle of slots +has increments among the values of $h_l$ at arguments where its derivative is +nonzero modulo $p$, apart from a small exceptional part of the distribution. +The constraint on an edge is weighted by the number of such arguments giving +its increment. This weight has a Fourier bound uniform in the auxiliary +polynomial and transforms exactly under the progression identity above. + +The laws are reflection positive: the quadratic form obtained by splitting +the slots across any of a specified family of reflections is positive +semidefinite. They also have uniform individual marginals and a conditional +mixing estimate. Section~\ref{sec:lifts} derives the resulting signed +H\"older inequality and estimates for Fourier lifts. At scale $N$, the lift +retains the normalized Fourier coefficients of $1_A$ at rational frequencies +whose denominators are at most a small fixed power of $N$. Integrating its +product over the $d$ slots gives a nonnegative quantity $Y_l(N,A)$ satisfying +\[ + Y_l(N,A)\ge (|A|/N)^d. +\] +The associated seminorm permits comparison along progressions with a factor +tending to one, rather than a fixed loss at every scale. + +\paragraph{A kernel realizing the resulting increment distribution.} +The increment distribution of two slots is determined by the whole tuple +construction. It is supported on regular polynomial values, but there is no +reason for it to equal the distribution obtained by evaluating $h_l$ at a +uniform argument. Section~\ref{sec:kernel} constructs a signed kernel on +actual positive values of $h_l$ whose Fourier transform approximates the +desired local multipliers. At each sufficiently large prime, a density on regular arguments +realizes the prescribed increment distribution. A truncated divisor +expansion combines these densities, while uniform lifting handles +characters of higher prime-power conductor. Minor arcs are controlled by +elementary Weyl differencing. + +Only support is needed to use this kernel: its bilinear form vanishes on two +ordered subsets of an avoiding set, regardless of the signs of the kernel +weights. Its Fourier approximation therefore gives cancellation for the +chosen pair of slots. This separates the construction of a useful tuple law +from the problem of realizing its pair distribution on integer polynomial +values. + +\paragraph{Contraction while the auxiliary parameter grows.} +Section~\ref{sec:transfer} combines the kernel estimate with the seminorm +comparison. A fixed positive proportion of assignments of residue classes +at the finitely many small primes gives cancellation. The other assignments, +together with the summable exceptional parts of the tuple laws, are bounded +using shorter intervals and polynomials farther along the family. +Section~\ref{sec:induction} closes the resulting recurrence with a bound +that holds for some fixed $a,b>0$, simultaneously for every $l$ and every set $A$ +avoiding nonzero $h_l$-differences: +\[ + Y_l(N,A)\le C l^bN^{-a}. +\] +The factor $l^b$ is necessary to accommodate the +growth of the auxiliary polynomial: when $l$ is large relative to $N$, a +uniform subpower bound suffices, while the kernel supplies contraction in +the remaining range. For the normalized polynomial $g$, the parameters can +be chosen so that $a/d$ depends only on $k$. Taking $l=1$ and using density +domination bounds sets avoiding nonzero $g$-differences. Partitioning $A$ +into residue classes modulo $M$, and using $Mg(n)=h(r+D_0n)$ with +$r+D_0n\ge1$, then transfers the bound to $h$ and proves +Theorem~\ref{thm:main}. + +The polynomial argument thus adds exact covariance of the weighted tuple laws +across the auxiliary family and a signed polynomial kernel realizing their +pair multipliers. The initial normalization controls the structural parameters +that determine the exponent. All the estimates needed from the +reflection-positive method are proved below. + +\subsection{Notation and uniformity} + +We write $e(x)=\exp(2\pi i x)$ and use the negative sign in Fourier +coefficients. For $\xi\in\Q/\Z$, let $q(\xi)$ be its reduced positive +denominator, with $q(0)=1$. Averages, measures, and norms on finite sets use +uniform probability measure unless a different measure is specified. +For a positive integer $n$, write $\rad(n)=\prod_{p\mid n}p$, with +$\rad(1)=1$. +Empty products equal $1$, and a product space with no coordinates is a +singleton. Conditional expectations are with respect to the law specified +in the surrounding statement. + +The auxiliary-family development applies to any fixed intersective polynomial +$h$ with chosen roots, and we write $k=\deg h$. Constants in $O(\cdot)$ and +$\ll$ may depend on this fixed datum and on parameters already fixed from it; +they are independent of the varying integers, sets, and auxiliary index $l$, +unless a restriction is explicitly stated. Likewise, sufficiently-large +thresholds are uniform in those varying data. We explicitly identify the +structural choices that depend only on $k$ when the first nonzero coefficient +at every chosen root is a unit. Bounds denoted $O_k(\cdot)$ are uniform over +all fixed data of degree $k$ satisfying those choices. A bound $N^{o(1)}$ +means $C_\epsilon N^\epsilon$ for every $\epsilon>0$ under the same +uniformity convention. All structural parameters are fixed before the +asymptotic scale $N$ varies. diff --git a/preprints/A-power-saving-for-intersective-polynomial-differences-with-an-exponent-depending-only-on-the-degree-October-5-2026/build/manuscript/sections/kernel.tex b/preprints/A-power-saving-for-intersective-polynomial-differences-with-an-exponent-depending-only-on-the-degree-October-5-2026/build/manuscript/sections/kernel.tex new file mode 100644 index 0000000000000000000000000000000000000000..8d042410141a51214d15beb81127783e7270ff6e --- /dev/null +++ b/preprints/A-power-saving-for-intersective-polynomial-differences-with-an-exponent-depending-only-on-the-degree-October-5-2026/build/manuscript/sections/kernel.tex @@ -0,0 +1,539 @@ +\section{A signed kernel for polynomial differences} +\label{sec:kernel} + +The tuple laws constrain differences modulo primes. We now connect their +pair distributions to actual values of $h_l$ at positive integers. The +connection is a uniform Fourier approximation: a kernel supported on those +polynomial values will have the same small-denominator multipliers as the +pair law, up to a power error. We adapt the signed square-kernel construction +of~\cite[Section~6, especially Proposition~6.1]{squarepower}. For a general +polynomial, we realize the pair distribution by weighting regular argument +classes; its explicit probability formula is not needed. + +\subsection{Pair multipliers and the ordered estimate} + +Fix an integer $l\ge1$, put $F=h_l$, and choose a directed edge +$v_j\to v_{j+1}$ of the designated cycle. For $p\ge P$, let $\pi_p$ +be the distribution of $z_{v_{j+1}}-z_{v_j}$ under $\mu_{l,p}^{\circ}$. +Write $\pi_p(y)$ for its point probabilities and define +\[ + \Gamma_p(a'/p)=\sum_{y\in\F_p}\pi_p(y)e(a'y/p). +\] +The distribution is supported on values of $F$ at regular arguments. +Conditioning the character of the second endpoint on the other slots, +then pairing with the opposite character of the first endpoint, gives +\[ + |\Gamma_p(a'/p)|\le p^{-\gamma}\qquad(p\nmid a') +\] +by~\eqref{eq:8} and Cauchy--Schwarz. For a rational $\xi$ with squarefree +denominator supported on primes at least $P$, decompose $\xi$ additively +into its prime-denominator parts and define $\Gamma(\xi)$ as the product +of the corresponding $\Gamma_p$. In particular, +\begin{equation} + \Gamma(0)=1,\qquad |\Gamma(\xi)|\le q(\xi)^{-\gamma}. + \label{eq:15} +\end{equation} +All estimates below are uniform in the selected cycle edge. + +\begin{proposition}[Ordered polynomial-kernel estimate] +\label{kernel:estimate} +Use the fixed parameters in~\eqref{eq:14}, and let $N$ be a sufficiently +large integer. Suppose $1\le l\le N^\zeta$ and +\[ + D=M_0\prod_{p\in B}p\le N^\beta, +\] +where $B$ is a finite set of primes at least $P$. Let +$J_1,J_2:[N]\to\C$ satisfy $|J_1|,|J_2|\le1$ and be supported in +residue classes $a_1,a_2$ modulo $D$, respectively. Suppose that +$r=a_2-a_1$ is admissible for $l$ modulo $D$, that $x0$, one regular +argument $x_y$ satisfying $F(x_y)=y$ in $\F_p$. Define +\[ + \psi_p(x_y)=p\pi_p(y),\qquad \psi_p(x)=0 + \quad\text{at the other arguments}. +\] +Then $\psi_p$ is nonnegative, has uniform mean one, and pushes forward +under $F$ to $\pi_p$. In particular $0\le\psi_p\le p$. For positive +squarefree integers $u$ coprime to $D$, define periodic functions +\[ + B_p(m)=\psi_p(m)-1,\qquad + B_u(m)=\prod_{p\mid u}B_p(m),\qquad B_1=1. +\] +Each $B_p$ has mean zero, and $|B_u|\le u$. We will use the truncated +sum of these products to select the prescribed local distributions at +small denominators. Indeed, for positive squarefree $R$ coprime to $D$, +\[ + \sum_{u\mid R}B_u(m)=\prod_{p\mid R}(1+B_p(m)) + =\prod_{p\mid R}\psi_p(m). +\] +The cutoff $u\le H$ preserves this identity when $R\le H$, and we +will use local exponential-sum cancellation for larger denominators. +The truncated sum may have either sign. + +All coefficients of $F$ are $O_h(a_l)$. Hence one integer $x_*\ge1$, +depending only on $h$, can be chosen for the entire auxiliary family, +independently of $l$, so that for $x\ge x_*$, +\[ + F(x)\asymp a_lx^k,\qquad F'(x)\asymp a_lx^{k-1},\qquad + |F''(x)|\ll a_lx^{k-2}, +\] +with $F'(x)>0$. The lower-degree terms have total absolute value +$O_h(a_lx^{k-1})$ for $x\ge1$. Enlarging this same $x_*$ therefore also +arranges +\[ + F(x)\ge \frac{a_lx^k}{2}>0\qquad(x\ge x_*), +\] +uniformly in $l$. Put $U=(N/a_l)^{1/k}$. Since $k\zeta=\eta/20$, +the hypothesis $l\le N^\zeta$ gives +\[ + k!a_l\le k!a l^k\le k!a N^{\eta/20}. +\] +The exponent gap $\eta/8-\eta/20=3\eta/40$ is positive. Thus one +sufficiently large threshold for $N$, depending on $h$ and the fixed +parameters but independent of $l$, ensures $k!a\le N^{3\eta/40}$ and hence, +simultaneously for all $l\le N^\zeta$, +\[ + k!a_l\le N^{\eta/8},\qquad + N^{1-\eta/8}\le U^k\le N. +\] +The last inequality uses that $a_l$ is a positive integer. The lower bound +for $U$ tends to infinity uniformly in this range of $l$, so increasing the +same threshold ensures that there is a unique real $X>x_*$ with $F(X)=N$, +and $X\asymp U$. The lower bound on $F(X)$ gives +\[ + X\le (2N/a_l)^{1/k}=2^{1/k}U<2U. +\] +With $m$ ranging over integers and $u$ over positive integers, define +\begin{equation} + k_1(n)=\sum_{x_*N^\eta$, and in particular +\[ + N^{\eta/2}\le q'\le N^{1-\eta}, +\] +using $A_l\le N^{\eta/8}$. + +Choose $M=\lceil2U\rceil$. Since $X<2U$, this integer is larger than all +the interval lengths under consideration. Also $U\ge1$, so the absolute +comparison $2U\le M\le2U+1\le3U$ holds. Repeated differencing gives +\begin{equation} + \begin{split} + \left|\sum_{m\in I}e\bigl(\alpha F(m)+y'm\bigr)\right|^{2^{k-1}} + &\le C_k M^{2^{k-1}-k}\\ + &\quad\times\sum_{|b_1|,\ldots,|b_{k-1}|0$ their multiplicity is +$O_{\epsilon,k}(m^\epsilon)$. Indeed, at a prime power $p^a$ the +number of exponent choices is a binomial coefficient of fixed degree +in $a$; it is at most $p^{a\epsilon}$ for all sufficiently large $p$. +The finitely many smaller primes contribute a bounded extra factor, +since a fixed polynomial in $a$ divided by $p^{a\epsilon}$ is bounded. + +Partition the positive product indices into blocks of +$\lfloor q'/2\rfloor$ consecutive integers. If $0E-s$. For $j>E$, uniform +arguments in the restricted class modulo $p^j$ give uniform $z$ +modulo $p^{j-E+s}$, hence uniform reduction modulo $p^{j-E}$. +The $F$-values are consequently uniform among the lifts of $r$ +modulo $p^j$, and a primitive character has mean zero. +The joint factor from primes dividing $D$ is therefore +\[ + \chi_D(\alpha_0)= + \begin{cases} + e(\alpha_Dr),&q(\alpha_D)\mid D,\\ + 0,&q(\alpha_D)\nmid D. + \end{cases} +\] + +Put $q_1=q(\alpha_1)$. If $u$ has a prime not dividing $q_1$, +its factor $B_p$ has mean zero independently of all the other +factors, so $A_u=0$. Thus only divisors of +$\operatorname{rad}(q_1)=\prod_{p\mid q_1}p$ contribute. +For $p^j\Vert q_1$, let $\alpha_p$ be the primitive $p^j$-denominator +part of $\alpha_1$ and set +\[ + a_{p,j}=\E_{m\bmod p^j}e(\alpha_pF(m)),\qquad + b_{p,j}=\E_{m\bmod p^j}B_p(m)e(\alpha_pF(m)). +\] +The Chinese remainder theorem now expresses the truncated coefficient as +\begin{equation} + c(\alpha_0)=\chi_D(\alpha_0) + \sum_{\substack{u\mid\operatorname{rad}(q_1)\\u\le H}} + \left(\prod_{\substack{p^j\Vert q_1\\p\mid u}}b_{p,j}\right) + \left(\prod_{\substack{p^j\Vert q_1\\p\nmid u}}a_{p,j}\right). + \label{kernel:truncated-coefficient} +\end{equation} +The mean against $1+B_p=\psi_p$ is +\[ + a_{p,j}+b_{p,j}= + \begin{cases} + \Gamma_p(\alpha_p),&j=1,\\ + 0,&j\ge2. + \end{cases} +\] +For $j=1$ this is the defining pushforward of $\psi_p$. For +$j\ge2$, condition on a regular argument modulo $p$. Its lifts +are mapped uniformly to the lifts of its value, so their primitive +character mean vanishes. This holds on each argument class separately, +regardless of the size of $\psi_p$ there. + +If $q_1\le H$, all divisors of $\operatorname{rad}(q_1)$ occur. +Thus~\eqref{kernel:truncated-coefficient} factors as +\[ + c(\alpha_0)=\chi_D(\alpha_0) + \prod_{p^j\Vert q_1}(a_{p,j}+b_{p,j}). +\] +For squarefree $q_1$ this is +$\chi_D(\alpha_0)\Gamma(\alpha_1)$; if any exponent is at least +two, it is zero. These are precisely the cases in~\eqref{eq:20}. +The empty product, when $q_1=1$, also gives the required coefficient. + +Suppose instead that $q_1>H$. The model coefficient is then zero. +By~\eqref{eq:2}, $|a_{p,j}|\le p^{-8\delta j}$. For $j=1$ the +preceding identity and~\eqref{eq:15} give +$|b_{p,1}|\le2p^{-\gamma}$; for $j\ge2$ the identity gives +$b_{p,j}=-a_{p,j}$, hence again +$|b_{p,j}|\le2p^{-\gamma j}$. Taking absolute values +in~\eqref{kernel:truncated-coefficient} and dropping the divisor cutoff yields +\begin{align*} + |c(\alpha_0)| + &\le\prod_{p^j\Vert q_1}(|a_{p,j}|+|b_{p,j}|)\\ + &\le3^{\omega(q_1)}q_1^{-\gamma} + \le q_1^{-\gamma/2}0\), take a maximizing assignment. Each of its +folded integrals is nonnegative and at most \(M\), while their product +is at least \(M^2\) by~\eqref{lift:fold-cs}. Both therefore equal +\(M\). Hence either fold preserves maximality. Apply the pullbacks +in reverse order, first \(\phi_m\), then \(\phi_{m-1}\), and so on. +The final assignment is +\(a\circ\phi_m\circ\cdots\circ\phi_1\), which is constant. It +follows that \(M\) is a diagonal value. Conversely, constant +assignments are included in the maximum. We have proved that the +maximum absolute mixed integral equals the largest diagonal value in +the list; this is also true when \(M=0\). + +For \(\epsilon>0\), apply this conclusion to the list +\[ + \frac{g_v}{(Z_{l,\mathcal P}(g_v)+\epsilon)^{1/d}}, + \qquad v\in V. +\] +Each diagonal value in this list is at most one. Multilinearity gives +\( |\mathcal Z_{l,\mathcal P}((g_v)_v)| + \le\prod_v(Z_{l,\mathcal P}(g_v)+\epsilon)^{1/d}\). +Let \(\epsilon\downarrow0\). This proves the second assertion, +including cases where some diagonal values vanish. +\end{proof} + +Define +\[ + \|g\|_{l,\mathcal P}=Z_{l,\mathcal P}(g)^{1/d} + \quad\text{for real }g. +\] +For \(B\subseteq\mathcal P\), let \(\E_B g\) mean uniform averaging +over the coordinates in \(B\), with all other coordinates held fixed. +The resulting function may be viewed on either \(X_{\mathcal P}\) or +\(X_{\mathcal P\setminus B}\). + +\begin{lemma}[Seminorm and changes of coordinates] +\label{lift:seminorm} +The map \(g\mapsto\|g\|_{l,\mathcal P}\) is a seminorm on the real +functions on \(X_{\mathcal P}\). It satisfies +\[ + |\E g|\le\|g\|_{l,\mathcal P}, + \qquad + \|\E_Bg\|_{l,\mathcal P\setminus B} + =\|\E_Bg\|_{l,\mathcal P} + \le\|g\|_{l,\mathcal P}. +\] +Translations of the input are isometries. Moreover, if \(D\) is a +positive integer coprime to every prime of \(\mathcal P\), then for +any \(a'\in X_{\mathcal P}\), +\begin{equation} + \left\|g\bigl(\lambda(D)^{-1}(\,\cdot-a')\bigr) + \right\|_{l,\mathcal P} + =\|g\|_{lD,\mathcal P}. + \label{lift:transport} +\end{equation} +Here the affine change is interpreted separately at each prime. +\end{lemma} + +\begin{proof} +Nonnegativity follows from Proposition~\ref{lift:signed-holder}. +Since \(d\) is even, homogeneity of \(Z\) gives absolute homogeneity +of its \(d\)-th root. Expanding \(Z(g+h)\) by its slots and applying +\eqref{eq:10} termwise gives +\[ + Z_{l,\mathcal P}(g+h) + \le\sum_{S\subseteq V}\|g\|_{l,\mathcal P}^{|S|} + \|h\|_{l,\mathcal P}^{d-|S|} + =\bigl(\|g\|_{l,\mathcal P}+\|h\|_{l,\mathcal P}\bigr)^d. +\] +Taking roots proves the triangle inequality. +Putting \(g\) in one slot and 1 in all others in~\eqref{eq:10} +proves mean domination, since the single-slot marginal is uniform and +\(Z(1)=1\). + +The simultaneous translation invariance of the tuple laws proves that +translations are isometries. Let \(G_B\subseteq X_{\mathcal P}\) +be the additive subgroup of vectors supported on \(B\). Then +\[ + (\E_Bg)(z)=\frac1{|G_B|}\sum_{a\in G_B}g(z+a). +\] +The triangle inequality gives the asserted contraction. Since +\(\E_Bg\) is independent of \(B\), the probability laws at those +primes integrate out, proving the equality of the two seminorms. +Finally, \(\lambda(D)\) is a unit at each remaining prime, and +\eqref{eq:7} pushes the product law at \(l\) exactly to that at +\(lD\). This proves~\eqref{lift:transport}. +\end{proof} + +\subsection{Decay at large Fourier denominators} + +Conditional mixing gives a different estimate, valid also after +removing the exceptional part of each prime law. Its proof requires +orthogonality between different exact supports; applying the triangle +inequality to all Fourier coefficients would lose this decay. + +\begin{lemma}[Large-denominator estimate] +\label{lift:large-denominator} +Fix \(l\ge1\) and a finite prime set \(\mathcal P\) as above. At +each prime choose either \(\rho_p=\mu_{l,p}\) or +\(\rho_p=\mu_{l,p}^{\circ}\). Let \(g_v:X_{\mathcal P}\to\C\) +be arbitrary functions. If, in one distinguished slot \(v\), every +Fourier coefficient at a denominator at most \(R\) vanishes, where +\(R\ge1\), then +\begin{equation} + \left|\int\prod_{u\in V}g_u(z_u)\, + d\bigotimes_{p\in\mathcal P}\rho_p\right| + \le R^{-\gamma}\|g_v\|_2 + \prod_{u\ne v}\|g_u\|_{2(d-1)}. + \label{eq:11} +\end{equation} +\end{lemma} + +\begin{proof} +At one prime, let \(\mathfrak m_p\) denote the marginal law of the labels in all +slots other than \(v\), and let +\[ + T_p f=\E_{\rho_p}\bigl(f(z_v)\mid(z_u)_{u\ne v}\bigr). +\] +This is an operator from uniform \(L^2(\F_p)\) to \(L^2(\mathfrak m_p)\). +It preserves constants. It sends uniform-mean-zero inputs to +\(\mathfrak m_p\)-mean-zero outputs, since \(\E_{\mathfrak m_p}T_pf=\E f\). +By~\eqref{eq:8}, its norm on the mean-zero subspace is at most +\(p^{-\gamma}\). + +The product law makes the conditional operator for all primes equal +to \(T=\bigotimes_pT_p\): this identity holds first for products of +one-prime functions, by independence across primes, and then for every +function by linearity. The exact-support \(B\) Fourier space has a +mean-zero factor at primes in \(B\) and a constant factor elsewhere. +On this space the operator norm is at most \(q_B^{-\gamma}\). +Furthermore, the images of two distinct exact-support spaces are +orthogonal in \(L^2(\bigotimes_p\mathfrak m_p)\). At a prime in the symmetric +difference of the supports, one image factor is constant and the other +has mean zero. Their inner product is zero, and tensoring preserves +this orthogonality. + +Decompose \(g_v=\sum_{q_B>R}g_{v,B}\) by exact supports. Both the +domain summands and their images are orthogonal, so +\[ + \|Tg_v\|_2^2 + =\sum_{q_B>R}\|Tg_{v,B}\|_2^2 + \le\sum_{q_B>R}q_B^{-2\gamma}\|g_{v,B}\|_2^2 + \le R^{-2\gamma}\|g_v\|_2^2. +\] +Condition the original integral on the other slots and apply +Cauchy--Schwarz. The remaining factor is bounded by +\[ + \left\|\prod_{u\ne v}g_u(z_u)\right\|_2 + \le\prod_{u\ne v} + \left(\E|g_u(z_u)|^{2(d-1)}\right)^{1/(2(d-1))} + =\prod_{u\ne v}\|g_u\|_{2(d-1)}, +\] +using ordinary H\"older and the uniform single-slot marginals. +This proves~\eqref{eq:11}. If \(\mathcal P\) is empty, the frequency +hypothesis forces \(g_v=0\), and the conclusion is immediate. +\end{proof} + +\subsection{Fourier lifts and uniform moments} + +A bounded function on an integer interval gives Fourier coefficients at +rational frequencies. We transfer a finite set of these coefficients +to \(X_{\mathcal P}\). Such a transfer need not preserve pointwise +bounds, so we prove bounds for every fixed moment, following the +Fourier-lift argument of~\cite[Lemmas~5.1--5.2]{squarepower}. We must also allow +arbitrary subfamilies of complete exact supports, because fixing some +residue coordinates changes which denominators remain. + +Let \(I\) be a nonempty consecutive interval of integers and let +\(f:I\to\C\). Put +\[ + \widehat f_I(\xi)=\frac1{|I|}\sum_{x\in I}f(x)e(-x\xi). +\] +For \(B\subseteq\mathcal P\), define its exact-support contribution +\[ + f_B(z)=\sum_{q(\xi)=q_B}\widehat f_I(\xi)e(z\xi), +\] +where the frequencies are characters of \(X_{\mathcal P}\). For +\(Q\ge1\), the Fourier lift is +\[ + \mathcal L_{I,\mathcal P,Q}f(z) + =\sum_{\substack{B\subseteq\mathcal P\\q_B\le Q}}f_B(z) + =\sum_{\substack{\xi\text{ on }X_{\mathcal P}\\q(\xi)\le Q}} + \widehat f_I(\xi)e(z\xi). +\] +Each \(f_B\) depends only on the coordinates in \(B\), and has mean +zero in each of those coordinates. If \(f\) is real, all these +functions are real because each summation set is closed under negation. +Retaining a support \(B\) always means retaining all its frequencies. + +\begin{proposition}[Uniform lift moments] +\label{lift:moments} +Let \(\mathcal P\) be any finite set of primes, \(Q\ge1\), and let +\(I\) be a consecutive integer interval with \(|I|\ge10Q^6\). +For every integer \(r\ge1\), every \(\epsilon>0\), every function +\(f:I\to\C\), and every subfamily +\(\mathcal F\subseteq\{B\subseteq\mathcal P:q_B\le Q\}\), +\begin{equation} + \left\|\sum_{B\in\mathcal F}f_B\right\|_{2r} + \le C_{r,\epsilon}\|f\|_\infty Q^\epsilon. + \label{eq:12} +\end{equation} +The constant is independent of the prime set, the interval and its +location, the function, and the subfamily. +\end{proposition} + +\begin{proof} +The zero function is immediate; otherwise scale so that +\(\|f\|_\infty\le1\). Write \(N_I=|I|\), and let \(u\) be the +largest cardinality of a support whose prime product is at most \(Q\). +We first control conditional sums of squares of the \(f_B\), then their +square-function moments. Symmetrization will recover the moment of +their sum. + +\smallskip +\noindent\emph{Conditional energy.} +For \(J\subseteq\mathcal P\) with \(q_J\le Q\), and any fixed +coordinate vector \(z_J\), we claim that +\begin{equation} + \sum_{\substack{B\in\mathcal F\\B\supseteq J}} + \E\bigl(|f_B|^2\mid z_J\bigr)\ll4^{|J|}. + \label{eq:13} +\end{equation} +The conditional expectation here averages uniformly over all coordinates +outside \(J\). If \(B\supseteq J\), the coefficient of an outside +character \(\eta\) of exact support \(B\setminus J\), after fixing +\(z_J\), is +\[ + c_J(\eta;z_J) + =\frac1{N_I}\sum_{x\in I}f(x)e(-x\eta) + \prod_{p\in J}\bigl(p\1_{\{x\equiv z_p\pmod p\}}-1\bigr). +\] +Indeed, the sum over the nonzero frequencies at \(p\) is +\(\sum_{a=1}^{p-1}e(a(z_p-x)/p) +=p\1_{\{x\equiv z_p\pmod p\}}-1\). +Different supports \(B\supseteq J\) have disjoint sets of outside +characters. Conditional Parseval therefore identifies the left side of +\eqref{eq:13} with the squared Euclidean norm of this coefficient vector +on the union of those character sets. + +Expand the product in \(c_J\) over subsets \(J'\subseteq J\). +Apart from a sign, the corresponding vector has coordinates +\[ + \frac{q_{J'}}{N_I} + \sum_{\substack{x\in I\\x\equiv z_{J'}\pmod{q_{J'}}}} + f(x)e(-x\eta). +\] +The congruence denotes the class determined by the fixed coordinates in +\(J'\); for the empty subset it imposes no restriction. Write its +progression as \(x=a+q_{J'}n\), with \(n\) in a consecutive interval +of length \(N'\). Then +\[ + N'\ge0.9N_I/q_{J'}, + \qquad N'q_{J'}/N_I\le1.1. +\] +In these coordinates the frequencies are \(q_{J'}\eta\), up to +constant phases. They remain distinct because their denominators are +coprime to \(q_J\). Their denominators are at most \(Q\), so they +number at most \(Q^2\) and have pairwise circle distance at least +\(Q^{-2}\). + +The Gram matrix of these characters in normalized \(L^2\) on the +progression has diagonal 1. By geometric summation, every off-diagonal +entry has modulus at most \(Q^2/(2N')\). Its operator norm is +therefore at most its maximum absolute row sum, which is bounded by +\[ + 1+\frac{Q^4}{2N'} + \le1+\frac{Q^5}{1.8N_I} + \le1+\frac1{18Q}. +\] +The associated Fourier analysis map has norm squared equal to this +Gram-matrix norm. Since \(\|f\|_\infty\le1\) and +\(N'q_{J'}/N_I\le1.1\), the displayed coefficient vector has +Euclidean norm bounded by an absolute constant. The triangle inequality +over the \(2^{|J|}\) subsets \(J'\), followed by squaring, proves +\eqref{eq:13}. + +\smallskip +\noindent\emph{Square-function moments.} +Set +\[ + S(z)=\left(\sum_{B\in\mathcal F}|f_B(z)|^2\right)^{1/2}. +\] +Expand \(\E S^{2r}\) and fix the supports of its first \(r-1\) +factors. Their union \(W'\) has size at most \((r-1)u\), and their +product depends only on \(z_{W'}\). For the final support \(B\), +integration outside \(W'\) leaves exactly +\(\E(|f_B|^2\mid z_J)\), where \(J=B\cap W'\). +Every intersection that occurs satisfies \(q_J\le q_B\le Q\). +For each fixed intersection \(J\), the sum over supports with +\(B\cap W'=J\) is bounded by the larger sum over all \(B\supseteq J\) +in~\eqref{eq:13}. There are at most \(2^{(r-1)u}\) possible +intersections and each has at most \(u\) elements. Thus +\[ + \E S^{2r}\le C\,2^{(r-1)u}4^u\E S^{2r-2}. +\] +The same argument with \(r=1\) uses \(J=W'=\varnothing\). +Iteration gives +\begin{equation} + \|S\|_{2r}\le C_r^{1+u}. + \label{lift:square-function} +\end{equation} +The overlaps among supports have now been controlled. It remains to +bound their sum by this square function with a loss exponential only in +\(u\). + +\smallskip +\noindent\emph{Recovering the sum.} +Fix a support size \(j\ge1\) and color every prime coordinate with one +of \(j\) colors. Call a support \emph{transversal} when it has exactly +one prime of each color, and let \(\mathcal T\) be the transversal +supports in \(\mathcal F\) for this coloring. Take independent uniform +pairs \(X_p^0,X_p^1\) at every prime. The operator +\(\operatorname{swap}_p\) interchanges these two copies. Define +\[ + D_B=\prod_{p\in B}(1-\operatorname{swap}_p)f_B(X^0), + \qquad B\in\mathcal T. +\] +Averaging over all second copies leaves \(f_B(X^0)\), since every +other term in this expansion has mean zero in at least one copied +coordinate. Jensen's inequality gives +\[ + \left\|\sum_{B\in\mathcal T}f_B\right\|_{2r} + \le\left\|\sum_{B\in\mathcal T}D_B\right\|_{L^{2r}(X^0,X^1)}. +\] +Independent swaps preserve the joint law of the copies and multiply +\(D_B\) by the product of the corresponding signs. Thus for independent +uniform signs \(\epsilon_p\), indexed by primes, +\[ + \left\|\sum_{B\in\mathcal T}D_B\right\|_{L^{2r}(X^0,X^1)} + =\left\|\sum_{B\in\mathcal T}D_B\prod_{p\in B}\epsilon_p + \right\|_{L^{2r}(X^0,X^1,\epsilon)}. +\] + +For completeness, the scalar sign estimate +\[ + \left\|\sum_i a_i\epsilon_i\right\|_{2r} + \le C_r\left(\sum_i|a_i|^2\right)^{1/2} +\] +follows by expanding the even moment. Surviving terms have even +multiplicity at every index. For nonnegative coefficients their sum is +at most +\[ + \frac{(2r)!}{2^r r!}\left(\sum_i a_i^2\right)^r, +\] +because \((2m)!\ge2^m m!\). For complex coefficients, take absolute +values of the surviving terms in +\((\sum_i a_i\epsilon_i)^r(\sum_i\overline{a_i}\epsilon_i)^r\) +and use the nonnegative case. Iterating this estimate over the color +classes gives, for fixed copies, +\[ + \left\|\sum_{B\in\mathcal T}D_B\prod_{p\in B}\epsilon_p + \right\|_{L^{2r}(\epsilon)} + \le C_r^j\left(\sum_{B\in\mathcal T}|D_B|^2\right)^{1/2}. +\] +At each step Minkowski is used in the form +\(\| (\sum_i|B_i|^2)^{1/2}\|_{2r} +\le(\sum_i\|B_i\|_{2r}^2)^{1/2}\). +The signs are indexed by individual primes; transversality ensures that +each monomial uses exactly one sign from each color class. + +In expanding the swaps in \(D_B\), index its \(2^j\) terms by a choice +of copy for each color. For any fixed such choice, made consistently +across all supports, the selected prime coordinates have the original +product law. The triangle inequality in +\(L^{2r}(\ell^2(\mathcal T))\) therefore gives +\[ + \left\|\left(\sum_{B\in\mathcal T}|D_B|^2\right)^{1/2}\right\|_{2r} + \le2^j\left\|\left(\sum_{B\in\mathcal T}|f_B|^2\right)^{1/2} + \right\|_{2r} + \le2^j\|S\|_{2r}. +\] +Consequently the transversal sum has norm at most +\((2C_r)^j\|S\|_{2r}\). + +Now color independently and uniformly. A fixed size-\(j\) support is +transversal with probability \(j!/j^j\ge e^{-j}\), where here \(e\) +is Euler's number; the inequality follows from +\(\log(j!)\ge\int_1^j\log x\,dx\). +Thus the full size-\(j\) sum is exactly the coloring average of its +transversal sum divided by this probability. Minkowski yields +\[ + \left\|\sum_{\substack{B\in\mathcal F\\|B|=j}}f_B\right\|_{2r} + \le(2eC_r)^j\|S\|_{2r}. +\] +Include the constant term of size zero and sum over \(j\le u\). +Together with~\eqref{lift:square-function}, this bounds the norm of +the full sum by \((C'_r)^{1+u}\). +Finally, the product of \(u\) distinct primes is at least \((u+1)!\), +so \((u+1)!\le Q\). Hence \(u=o(\log Q)\) as \(Q\to\infty\), +uniformly in the prime set. For every fixed \(r\) and \(\epsilon>0\), +\((C'_r)^{1+u}\le C_{r,\epsilon}Q^\epsilon\). +Restoring \(\|f\|_\infty\) proves~\eqref{eq:12} with all the stated +uniformities. +\end{proof} + +\begin{lemma}[Specialization of residue coordinates] +\label{lift:specialization} +Let \(B\subseteq\mathcal P\), fix \(z_B\in X_B\), and let +\(G:X_{\mathcal P}\to\C\). For \(1\le s<\infty\), +\[ + \|G(z_B,\cdot)\|_{L^s(X_{\mathcal P\setminus B})} + \le q_B^{1/s}\|G\|_{L^s(X_{\mathcal P})}. +\] +In particular, a Fourier lift or a subfamily of its complete supports +satisfies~\eqref{eq:12} after specialization with the additional factor +\(q_B^{1/(2r)}\). The same bound holds after retaining either the +remaining denominators at most a given \(H\ge1\), or those greater +than \(H\). +\end{lemma} + +\begin{proof} +The specified fiber has uniform probability \(1/q_B\). Its contribution +to \(\E|G|^s\) is +\(q_B^{-1}\E|G(z_B,\cdot)|^s\). Dropping the other nonnegative +contributions proves the first assertion. For the last assertion, +a character with original exact support \(C\) retains exact support +\(C\setminus B\) after specialization. The two cuts therefore arise, +before specialization, from the complete-support subfamilies +\[ + \{C\in\mathcal F:q_{C\setminus B}\le H\}, + \qquad + \{C\in\mathcal F:q_{C\setminus B}>H\}. +\] +Apply Proposition~\ref{lift:moments} to these subfamilies and then use +the fiber bound. Combining coefficients can cancel frequencies but +cannot create any outside the indicated ranges. +\end{proof} + +In particular, applying~\eqref{eq:11} with the same set of coordinates +fixed in every slot incurs total specialization factor +\[ + q_B^{1/2}\left(q_B^{1/(2(d-1))}\right)^{d-1}=q_B. +\] +Two specialized \(L^2\) norms incur the same factor. All moment orders +needed here are fixed after \(h\) is fixed. Finite products of their +subpower bounds remain subpower; none of these constants depends on +\(l\), the specialized residues, or the retained supports. + +\subsection{Parameters and the density quantity} + +We now choose the analytic parameters used in the kernel estimate and +the scale comparison. The constants \(k,d,\gamma\) have already been +fixed. Set +\begin{equation} + \begin{aligned} + \eta&=\frac1{100k},& + \zeta&=\frac{\eta}{20k},& + \nu&=\frac{\eta}{10\,2^k},\\ + \beta&=\frac{\nu}{100},& + \tau&=\frac{\beta\gamma}{1000k(d+1)},& + \sigma&=\frac{\gamma\tau}{4}. + \end{aligned} + \label{eq:14} +\end{equation} +The displayed parameters are functions only of the already fixed numbers +\(k,d,\gamma\). When Condition~\eqref{local:unit-condition} holds and the +designated choices of Sections~\ref{sec:local} and~\ref{sec:tuples} are made, +these numbers, and hence every parameter in~\eqref{eq:14}, depend only on +\(k\). +Here \(\eta\) is distinct from the exceptional masses \(\eta_{l,p}\). +At an integer scale \(N\ge1\), the kernel proof will use \(\eta\) to +set the denominators and widths of neighborhoods of rational frequencies, +\(\nu\) for the exponential-sum saving \(N^{-\nu}\) outside them, +and \(\zeta\) for the allowed range \(l\le N^\zeta\). +The exponent \(\beta\) fixes the Fourier denominator cutoff \(N^\beta\); +\(\tau\) sets the subdivision into about \(N^\tau\) intervals and +the bound \(q_B\le N^\tau\) in the shorter-scale comparison, while +\(\sigma\) is the common error exponent in \(N^{-\sigma}\). +Put +\[ + Q=N^\beta,\qquad H=N^{\beta/2},\qquad + \mathcal P_N=\{p\text{ prime}:P\le p\le Q\}. +\] +For \(A\subseteq[N]\), define +\[ + Y_l(N,A)=Z_{l,\mathcal P_N} + \bigl(\mathcal L_{[N],\mathcal P_N,Q}\1_A\bigr). +\] +This is the quantity that will be compared between scales. + +\begin{corollary}[Density domination and a uniform upper bound] +\label{lift:density} +For all positive integers \(l,N\) and all \(A\subseteq[N]\), +\[ + Y_l(N,A)\ge\left(\frac{|A|}{N}\right)^d\ge0. +\] +For every \(\epsilon>0\), there is a constant \(C_\epsilon\), +independent of \(l,N,A\), such that +\[ + Y_l(N,A)\le C_\epsilon N^\epsilon. +\] +\end{corollary} + +\begin{proof} +The lift is real and has uniform mean \(|A|/N\). The lower bound +follows from Lemma~\ref{lift:seminorm}. Ordinary H\"older and the +uniform single-slot marginals give +\[ + Y_l(N,A)\le + \left\|\mathcal L_{[N],\mathcal P_N,Q}\1_A\right\|_d^d. +\] +The integer \(d\) is even. Also \(6\beta<1\), so for all sufficiently +large \(N\) one has \(N\ge10Q^6\). Apply~\eqref{eq:12} with moment +order \(d\), choosing its exponent small enough that its \(d\)-th +power is at most \(C_\epsilon N^\epsilon\). In the bounded remaining +range of \(N\), the lift has a bounded number of coefficients, each of +modulus at most one. The probability normalization then supplies a +uniform bound independent of \(l\), and enlarging \(C_\epsilon\) +finishes the proof. +\end{proof} diff --git a/preprints/A-power-saving-for-intersective-polynomial-differences-with-an-exponent-depending-only-on-the-degree-October-5-2026/build/manuscript/sections/local.tex b/preprints/A-power-saving-for-intersective-polynomial-differences-with-an-exponent-depending-only-on-the-degree-October-5-2026/build/manuscript/sections/local.tex new file mode 100644 index 0000000000000000000000000000000000000000..c96f97eef37a252cb3655ebd8c02796b25992ca9 --- /dev/null +++ b/preprints/A-power-saving-for-intersective-polynomial-differences-with-an-exponent-depending-only-on-the-degree-October-5-2026/build/manuscript/sections/local.tex @@ -0,0 +1,493 @@ +\section{Auxiliary polynomials and local estimates}\label{sec:local} + +Passing to an arithmetic progression changes the polynomial whose values are +forbidden differences. We first show that one such change can make the first +nonzero Taylor coefficient at each chosen local root a unit. We then organize +all subsequent changes into a coherent family \(h_l\), indexed by positive +integers, and show that this local property is preserved. The family and its +local estimates remain available for an arbitrary fixed intersective +polynomial; the unit condition will let us choose their structural parameters +using only the degree. The estimates give decay of complete exponential sums +at large primes, together with residue classes on which polynomial values +lift uniformly and support cycles of one fixed length. + +Throughout this section, \(h\in\Z[x]\) is the fixed intersective polynomial of +degree \(k\ge2\), with positive leading coefficient \(a\). + +\subsection{A family compatible with progressions} + +We use Lucier's auxiliary-polynomial construction +\cite[Section~8]{Lucier2006}, in which a compatible root at each prime +determines the normalization by its multiplicity. We give the construction +and its inheritance rule explicitly, before proving the uniform local +estimates needed here. + +Here \(\Z_p\) denotes the ring of compatible residues modulo the powers of +a prime \(p\), and \(v_p\) its valuation, with \(v_p(0)=\infty\). +Every nonzero element is a power of \(p\) times a unit, so \(\Z_p\) is an +integral domain. Intersectivity supplies a root of \(h\) modulo every power +of \(p\). These roots can be chosen compatibly: at each stage, retain a +residue class having root extensions to arbitrarily high powers, and then +choose one of its finitely many refinements with the same property. Fix the +resulting root \(\mathfrak z_p\in\Z_p\) for each prime \(p\), and write +\[ + h(\mathfrak z_p+X)=\sum_{j=m_p}^k b_{j,p}X^j, + \qquad 1\le m_p\le k, + \qquad b_{m_p,p}\ne0. +\] +Thus \(m_p\) is the multiplicity of the chosen root. + +The coefficient \(b_{m_p,p}\) is a unit for all sufficiently large primes, +with a threshold depending only on \(h\). To see this, factor +\(h=c\prod_i g_i^{d_i}\), where \(c\) is a nonzero integer and the \(g_i\) +are pairwise nonassociate primitive irreducible polynomials in \(\Z[x]\). Cleared +B\'ezout identities for \((g_i,g_j)\), \(i\ne j\), and for \((g_i,g_i')\) +have nonzero integer right-hand sides. Exactly one factor \(g_i\) vanishes +at \(\mathfrak z_p\), and \(g_i'(\mathfrak z_p)\ne0\). Outside the finite +set of primes dividing these right-hand sides or \(c\), the derivative and +all other factor values at \(\mathfrak z_p\) are units. Hence \(m_p=d_i\) +and +\[ + b_{m_p,p} + =c\,g_i'(\mathfrak z_p)^{d_i} + \prod_{j\ne i}g_j(\mathfrak z_p)^{d_j} +\] +is a unit. This argument allows repeated factors of \(h\). + +We will sometimes impose the following additional condition on the chosen +roots: +\begin{equation*}\tag{U}\label{local:unit-condition} + b_{m_p,p}\in\Z_p^\times\qquad\text{for every prime }p. +\end{equation*} +The construction below works without this condition. Its role is to make the +local parameter choices depend only on \(k\). Every intersective polynomial +can be reduced to this case by one progression change. + +\begin{lemma}[Normalization at the chosen roots]\label{local:normalization} +There are positive integers \(D_0,M\), an integer \(-D_00\}\) is finite. For \(p\in S_0\), take +\(u_p=u'_p+1\), and set +\[ + D_0=\prod_{p\in S_0}p^{u_p}, + \qquad M=\prod_{p\in S_0}p^{u_pm_p+u'_p}. +\] +Choose the representative \(-D_0m_p\), its valuation is at least +\[ + u_p(j-m_p)-u'_p\ge1. +\] +If \(p\notin S_0\), both \(D_0\) and \(M\) are \(p\)-adic units, so every +coefficient is integral and the coefficient at \(j=m_p\) is again a unit. +Thus the centered expansion belongs to \(\Z_p[X]\) for every prime \(p\). +Translation by \(-w_p\in\Z_p\) shows that \(g\) itself belongs to +\(\Z_p[x]\). Every rational coefficient of \(g\) is therefore integral at +every prime and hence is an integer. + +The elements \(w_p\) are roots of \(g\); reducing them at the prime-power +factors of any modulus and applying the Chinese remainder theorem proves +intersectivity. The degree and sign of the leading coefficient are preserved +because \(D_0,M>0\). Finally, the defining identity for \(g\) gives +\(Mg(n)=h(r+D_0n)\), and \(r>-D_0\) gives \(r+D_0n\ge1\) for \(n\ge1\). +\end{proof} + +In the coordinates of any progression of step \(M\), a nonzero difference +\(g(n)\), \(n\ge1\), therefore gives the nonzero original difference +\(h(r+D_0n)\) at a positive argument. This will transfer the final bound for +\(g\) back to the original polynomial. We continue below with the arbitrary +fixed polynomial \(h\) and its chosen roots, assuming +Condition~\eqref{local:unit-condition} only when explicitly stated. + +Define a completely multiplicative function \(\lambda:\N_+\to\N_+\) by +\(\lambda(p)=p^{m_p}\). For each \(l\ge1\), let \(r_l\) be the unique +integer in \((-l,0]\) satisfying +\[ + r_l\equiv\mathfrak z_p\pmod{p^{v_p(l)}} + \qquad(p\mid l), +\] +and set +\[ + h_l(x)=\frac{h(r_l+lx)}{\lambda(l)}, + \qquad a_l=\frac{a l^k}{\lambda(l)}. +\] +For \(l=1\), this convention gives \(r_1=0\) and \(h_1=h\). + +\begin{lemma}[The auxiliary family]\label{local:family} +For every \(l\ge1\), the polynomial \(h_l\) belongs to \(\Z[x]\), is +intersective, and has degree \(k\) and positive leading coefficient \(a_l\). +Every coefficient of \(h_l\) is \(O_h(a_l)\). For every prime \(p\), the +element \(\mathfrak z_{p,l}=(\mathfrak z_p-r_l)/l\) belongs to \(\Z_p\), and +\begin{equation}\label{local:centered-expansion} + \begin{aligned} + h_l(\mathfrak z_{p,l}+X) + &=\sum_{j=m_p}^k b_{j,p}\frac{l^j}{\lambda(l)}X^j,\\ + v_p\!\left(b_{m_p,p}\frac{l^{m_p}}{\lambda(l)}\right) + &=v_p(b_{m_p,p}). + \end{aligned} +\end{equation} +In particular, the root multiplicity remains \(m_p\), and +Condition~\eqref{local:unit-condition}, when it holds for \(h\), holds for +every \(h_l\) at these roots. For all positive integers \(l,D\), there is an +integer \(i\), \(0\le i0\), and use the +affine coordinate +\[ + Y=\frac{r_l-\mathfrak z_p}{p^u}+\frac{l}{p^u}x + \quad\text{over }\Z_p. +\] +Its slope is a unit, and expansion at \(\mathfrak z_p\) gives +\begin{equation}\label{local:normalized-expansion} + h_l(x)= + \left(\frac{\lambda(l)}{p^{u m_p}}\right)^{-1} + \sum_{j=m_p}^k b_{j,p}p^{u(j-m_p)}Y^j. +\end{equation} +The prefactor is a \(p\)-adic unit, and the displayed polynomial is integral +over \(\Z_p\). This proves \(h_l\in\Z[x]\). + +For every prime \(p\), the element +\((\mathfrak z_p-r_l)/l\) belongs to \(\Z_p\): when \(p\mid l\), this +uses the defining congruence for \(r_l\), and otherwise \(l\) is a unit. +It is a root of \(h_l\). Reducing these roots at the prime-power factors of +an arbitrary modulus and applying the Chinese remainder theorem proves +intersectivity. + +Substitution of \(\mathfrak z_{p,l}+X\) into the definition of \(h_l\) +gives the centered expansion in \eqref{local:centered-expansion}. Complete +multiplicativity gives \(v_p(\lambda(l))=m_pv_p(l)\), so +\(l^{m_p}/\lambda(l)\) is a unit in \(\Z_p\). This proves the exact valuation +there, including the preservation of the multiplicity and of the unit +condition. The assertion is local at \(p\); the displayed quotient need not +be an integer. + +The degree and leading coefficient follow from the definition. Expanding +\(h(r_l+lx)\) and using \(|r_l|k\), the reduced degree \(k'\) +lies in \(\{1,\ldots,k\}\), so \(h_l'\bmod p\) is not the zero polynomial. +The reduced degree may be smaller than the degree over the integers. + +Set +\[ + \delta=\frac{1}{100k2^k},\qquad \gamma=\frac{\delta}{10}. +\] +An argument \(x\in\F_p\) is called \emph{regular for \(h_l\)} if +\(h_l'(x)\ne0\) in \(\F_p\). + +\begin{lemma}[Complete polynomial sums]\label{local:sums} +There is a threshold \(P_1>\max(k,2)\), depending only on \(h\), such that +for every prime \(p\ge P_1\) and every \(l\ge1\), all but at most \(k-1\) +arguments modulo \(p\) are regular for \(F=h_l\), and at least half are +regular. If Condition~\eqref{local:unit-condition} holds, one may choose +\(P_1=P_1(k)\), uniformly over all such polynomials and chosen roots of +degree \(k\). For every integer \(u\) with \(p\nmid u\), +\begin{equation}\label{eq:2} + \left|\E_{x\bmod p^j}e\!\left(\frac{uF(x)}{p^j}\right)\right| + \le p^{-8\delta j}\quad(j\ge1), + \qquad + \left|\E_{\substack{x\bmod p\\x\text{ regular}}} + e\!\left(\frac{uF(x)}p\right)\right| + \le p^{-4\delta}. +\end{equation} +The second average is normalized by the number of regular arguments. +\end{lemma} + +\begin{proof} +Choose an initial threshold so that the preceding nonconstancy holds and +\(p>k\). Under Condition~\eqref{local:unit-condition}, this initial threshold +can be chosen using only \(k\). The nonzero polynomial \(F'\bmod p\) has +degree at most \(k-1\), which gives the count of regular arguments; taking +\(p\ge2(k-1)\) also makes at least half of the arguments regular. + +First consider the sum modulo \(p\), and let \(k'\) be the degree of +\(F\bmod p\). If \(k'=1\), the sum is zero. If \(k'\ge2\), repeated +differencing gives +\[ + \left|\E_{x\bmod p}e(uF(x)/p)\right|^{2^{k'-1}} + \le + \E_{b_1,\ldots,b_{k'-1}\bmod p} + \left|\E_{x\bmod p} + e\!\left(\frac{u\,\Delta_{b_1}\cdots\Delta_{b_{k'-1}}F(x)}p\right) + \right|, +\] +where \(\Delta_bF(x)=F(x+b)-F(x)\). To obtain the inequality, square the +mean, express it as the mean of multiplicative differences, and iterate the +triangle inequality and Cauchy--Schwarz. The reduction modulo \(p\) of +\(u\Delta_{b_1}\cdots\Delta_{b_{k'-1}}F\) has degree at most one, with +coefficient of \(x\) equal to \(u k'!\) times the leading coefficient +of \(F\bmod p\) times \(b_1\cdots b_{k'-1}\). Its character mean vanishes +unless one increment is zero. The right-hand side is therefore at most +\((k'-1)/p\). Since \(8\delta<2^{1-k}\le2^{1-k'}\) and +\(1\le k'\le k\), increasing the threshold by an amount depending only on +\(k\) gives the first bound in \eqref{eq:2} for \(j=1\). Removing at most +\(k-1\) arguments gives +\[ + \left|\E_{\substack{x\bmod p\\x\text{ regular}}}e(uF(x)/p)\right| + \le\frac{p}{p-k+1} + \left(\left|\E_{x\bmod p}e(uF(x)/p)\right|+\frac{k-1}{p}\right). +\] +The first bound and another increase depending only on \(k\) give the +second bound in \eqref{eq:2}. + +For \(j\ge2\), put \(n=\lfloor j/2\rfloor\). Write +\(x=x_0+p^{j-n}y\), where \(x_0\) runs modulo \(p^{j-n}\) and \(y\) +runs modulo \(p^n\). Taylor expansion modulo \(p^j\) leaves only the +linear variation, because \(2(j-n)\ge j\). The average over \(y\) vanishes +unless \(F'(x_0)\equiv0\pmod{p^n}\). We claim that these arguments occupy +at most \(k-1\) classes modulo \(p^v\), where +\(v=\lceil n/(k-1)\rceil\). + +Otherwise choose \(k\) integer representatives \(x_1,\ldots,x_k\) in +distinct such classes with \(p^n\mid F'(x_i)\). Interpolation of the +degree-at-most-\(k-1\) polynomial \(F'\) gives +\[ + F'(X)=\sum_{i=1}^k F'(x_i) + \frac{\prod_{j\ne i}(X-x_j)}{\prod_{j\ne i}(x_i-x_j)}. +\] +Each denominator has valuation at most \((k-1)(v-1)8\delta\), one fixed increase of +the prime threshold absorbs the factor \(k-1\) for every \(j\ge2\). +All choices of threshold are uniform in \(l\) and \(u\); fix \(P_1\) +large enough to satisfy them all. After the initial nonconstancy condition, +every increase used only \(k\). This proves the stated degree-only choice +under Condition~\eqref{local:unit-condition}. +\end{proof} + +\subsection{Admissible residues and uniform lifting} + +At large primes we have many regular arguments. +At the finitely many smaller primes, the derivative may be divisible by +\(p\) at every argument. We therefore need, uniformly in \(l\), an argument +at which \(v_p(h_l'(x))\) is bounded above. +Restrictions based on derivative nonvanishing and lifting at prime powers +also appear in Rice's treatment +\cite[Sections~2.4--2.5 and~4 of the revised version]{Rice2019}. + +\begin{lemma}[A derivative bound at small primes]\label{local:small-primes} +Let +\[ + V_k=(i^j)_{0\le i,j\le k-1},\qquad + T_{k,p}=\max_{1\le j\le k}v_p(j). +\] +For every prime \(pE\), a uniform argument in +that class modulo \(p^j\) gives a uniform value among the residues modulo +\(p^j\) congruent to \(F(x_0)\bmod p^E\). +\end{lemma} + +\begin{proof} +Taylor expansion over the integers gives +\[ + F(x_0+p^{E-s}y)=F(x_0)+p^E\bigl(c_1y+pC(y)\bigr), + \qquad p\nmid c_1,\quad C\in\Z[y]. +\] +Indeed the linear coefficient is \(p^EF'(x_0)/p^s\), and every term of +degree at least two is divisible by \(p^{E+1}\), since \(E-2s\ge1\). +The polynomial \(T(y)=c_1y+pC(y)\) induces a bijection modulo every power +of \(p\): its difference at two arguments is their difference times a +\(p\)-adic unit, so the induced map on each finite residue ring is +injective and hence bijective. + +For a uniform argument in the stated class modulo \(p^j\), the variable +\(y\) is uniform modulo \(p^{j-E+s}\). Its reduction modulo +\(p^{j-E}\) is still uniform, with each residue occurring \(p^s\) times. +The bijectivity of \(T\) at this latter modulus proves the asserted +distribution of \(F\). +\end{proof} + +For an admissible residue of \(F=h_l\), take its defining argument \(x_0\) +and put \(s=v_p(F'(x_0))\); at a large prime any integer representative of +the regular argument has \(s=0\). The chosen exponent always satisfies +\(E_p\ge2s+1\). Lemma~\ref{local:lifting} therefore supplies an argument +class of modulus \(p^{E_p-s}\), which divides \(p^{E_p}\), on which +\(F\) has precisely the required uniform lifting property. + +\subsection{Admissible cycles of a fixed length} + +We finally choose one cycle length that works simultaneously at every prime +and for every member of the auxiliary family. Fix an integer \(L>2\) such +that +\[ + p^{E_p}\mid L\quad(p2. +\] +There are only finitely many divisibility conditions, so such a choice exists. +For example, take the least multiple of \(\prod_{pP_1\) while retaining the exponents +\(E_p\) just fixed. In particular, \(E_p=1\) still holds on +\(P_1\le p0, +\] +the claimed error is $O(N^{-\sigma})$. The denominator cuts are +symmetric under negation, so the functions used here are real. + +We next express $g_{\le H}$ as an average of shorter lifts. Let +$a'\pmod D$ be the residue in Lemma~\ref{transfer:specialization}. +For each representative $j\in[L_D]$ with $j\equiv a'\pmod D$, set +\[ + c_j=j-L_D,\qquad + A'_j=\{x'\in[n_B]:L_Dx'+c_j\in A\}. +\] +These progressions partition the integers of $[N]$ in the class +$a'\pmod D$: the first positive term is $j$, and $n_B$ terms suffice +for every such integer. Terms above $N$ contribute no points to $A'_j$. +Moreover, $A'_j$ satisfies \eqref{eq:24} with $lD$ in place of $l$. +For if a nonzero difference in $A'_j-A'_j$ were $h_{lD}(m)$ with +$m\ge1$, the corresponding difference in $A-A$ would be +\[ + L_Dh_{lD}(m)=h_l(Dm-i),\qquad 0\le i0. +\] +\end{lemma} + +\begin{proof} +At every prime-power factor of $M_0$, the construction in +Section~\ref{sec:local} supplies a length-$L$ sequence of admissible +residues whose sum is zero. The Chinese remainder theorem combines them +into such a sequence modulo $M_0$. +Fix one sequence. There are $M_0$ choices of the initial cycle residue; +the other cycle residues are then determined, and the zero-sum condition +closes the cycle. Its $L$ vertices are distinct. The remaining $d-L$ +residues can be arbitrary, giving at least $M_0^{d-L+1}$ good assignments +out of $M_0^d$. +\end{proof} + +The proportion $\rho_l$ may vary with $h$ and $l$. When +Condition~\eqref{local:unit-condition} holds and the designated choices of +Sections~\ref{sec:local} and~\ref{sec:tuples} are made, both $M_0$ and $L$, +and therefore the lower bound $\rho_*=M_0^{1-L}$, depend only on $k$. + +\begin{proposition}[Cancellation on good assignments]\label{transfer:cancellation} +Let $N$ be sufficiently large, let $l\le N^\zeta$, and let +$A\subseteq[N]$ satisfy \eqref{eq:24}. If $(s_v)_{v\in V}$ is good +for $l$, then +\begin{equation}\label{eq:29} + \left| + \int\prod_{v\in V}G_{s_v}(z_v)\, + d\bigotimes_{p\in\mathcal P_N} + (\mu_{l,p}-\eps_{l,p}) + \right|\ll N^{-\sigma}. +\end{equation} +The bound is uniform in $l,A$ and the good assignment. +\end{proposition} + +\begin{proof} +We may replace each $\mu_{l,p}-\eps_{l,p}$ by its normalized law +$\mu_{l,p}^{\circ}$: passing back multiplies the integral by +$\prod_{p\in\mathcal P_N}(1-\eta_{l,p})\le1$. +We first locate an ordered pair of intervals along the cycle. +We then truncate the other inputs and condition on their prime supports; +the remaining pair integral will have exactly the multipliers appearing +in Proposition~\ref{kernel:estimate}. + +Partition $[N]$ into $\lfloor N^\tau\rfloor$ consecutive intervals +whose lengths differ by at most one. Each $G_s$ is the average of the +interval lifts $\mathcal L_{I,\mathcal P_N,Q}f_s$ with weights $|I|/N$. +Thus we may average independently over an interval assignment to every +slot. These intervals have length $\gg N^{1-\tau}$, so the moment +estimate \eqref{eq:12} applies: indeed +$6\beta+\tau<7\beta<1$ by \eqref{eq:14}. +For any interval assignment, ordinary H\"older, uniform single-slot +marginals and \eqref{eq:12} bound the integral in absolute value by +$N^{o(1)}$. +The probability that all $L$ cycle slots choose the same interval is +\[ + \sum_I\left(\frac{|I|}{N}\right)^L + \le\left(\max_I\frac{|I|}{N}\right)^{L-1} + \ll N^{-(L-1)\tau}. +\] +Their contribution is $O(N^{-\sigma})$, since +$(L-1)\tau>\sigma=\gamma\tau/4$. + +In every other assignment, some directed cycle edge has its first +interval $I_1$ strictly before its second interval $I_2$. +Otherwise the interval indices would be nonincreasing all the way +around the cycle, and therefore constant. Fix such an edge for the +current assignment. The constants below are uniform in this choice. + +Set $T=N^\tau$ and truncate every interval lift except the two selected +ones to denominators at most $T$. +Telescoping the other slots, \eqref{eq:11} and \eqref{eq:12} give total +error $O(T^{-\gamma}N^{o(1)})=O(N^{-\sigma})$, because +$\gamma\tau>\sigma$. +Expand the truncated inputs into characters. Each coefficient has +modulus at most $M_0$, and there are at most $T^2$ rational frequencies +of denominator at most $T$. Thus the sum of the absolute coefficients +in the resulting expansion is $O(T^{2(d-2)})$. + +Fix one term, let $B$ be the union of the supports of its $d-2$ +characters, and put +\[ + R=q_B\le T^{d-2},\qquad + \mathcal S=\mathcal P_N\setminus B. +\] +Condition on all tuple labels at every prime in $B$. +The expanded characters become constants of modulus one. +By independence between primes, the remaining tuple labels still have +law $\bigotimes_{p\in\mathcal S}\mu_{l,p}^{\circ}$. +Let $g_1,g_2$ denote the two selected interval lifts after specialization +at this conditioning. They are functions on $X_{\mathcal S}$. +For the chosen directed edge, let $\Gamma$ be the increment multiplier +defined in Section~\ref{sec:kernel}. The remaining pair integral is +exactly +\begin{equation}\label{eq:30} + \sum_{\xi\text{ on }X_{\mathcal S}} + \widehat g_1(-\xi)\widehat g_2(\xi)\Gamma(\xi). +\end{equation} +Indeed, simultaneous translation of the two labels kills all pairs of +characters whose frequencies do not sum to zero. The surviving pair +with frequencies $-\xi,\xi$ has value +$e(\xi(z_2-z_1))$, whose expectation is $\Gamma(\xi)$. + +For the terms with $q(\xi)>H$, \eqref{eq:15} and Parseval give +\[ + \left|\sum_{q(\xi)>H} + \widehat g_1(-\xi)\widehat g_2(\xi)\Gamma(\xi)\right| + \le H^{-\gamma}\|g_1\|_2\|g_2\|_2 + \ll H^{-\gamma}R N^{o(1)}. +\] +The last inequality uses the interval moment bound and the cost +$R^{1/2}$ of each specialization in $L^2$. + +We apply the kernel estimate to the remaining terms. +Put $D=M_0R$. The parameter choice gives +\[ + (d-2)\tau<\frac\beta2, + \qquad HR\le Q, + \qquad D\le N^\beta +\] +for large $N$; the first inequality follows directly from +$\tau=\beta\gamma/[1000k(d+1)]$. +By Lemma~\ref{transfer:specialization}, their Fourier coefficient +formulas for $q(\xi)\le H$ restrict the underlying integers to classes +$a'_1,a'_2\pmod D$. +The difference $a'_2-a'_1$ is admissible modulo $M_0$ by goodness. +It is admissible at each prime in $B$ because every conditioning of +positive probability is in the cycle support of the nonexceptional law. +It is therefore admissible modulo $D$. + +The remaining frequencies of denominator at most $H$ are precisely the +set $\Xi$ in Proposition~\ref{kernel:estimate}. +Indeed, their denominators are squarefree and coprime to $D$. +Conversely, a squarefree denominator at most $H$ and coprime to $D$ +has no prime below $P$, since all those primes divide $M_0$, and no +prime in $B$; every prime divisor is at most $H\le Q$. + +Define functions on $[N]$ by +\[ + J_i(x)=\1_A(x)\1_{I_i}(x) + \1_{\{x\equiv a'_i\pmod D\}},\qquad i=1,2. +\] +They are bounded by one and have ordered supports. +Every difference from the first support to the second is positive, +so a zero polynomial value cannot be such a difference; +\eqref{eq:24} excludes all nonzero values of $h_l(\N_+)$. +Thus all hypotheses of Proposition~\ref{kernel:estimate} hold at scale +$N$. The Fourier normalizations in \eqref{eq:25} are +\[ + \widehat g_i(\xi) + =\frac{DN}{|I_i|}\widehat{J_i}_{[N]}(\xi) + \qquad(q(\xi)\le H). +\] +Since \eqref{eq:16} has a factor $D$ outside its absolute value, +the low-denominator part of \eqref{eq:30} is at most +\[ + C H^{-\gamma/2}D\frac{N^2}{|I_1||I_2|} + \ll H^{-\gamma/2}D N^{2\tau}. +\] + +Both parts of \eqref{eq:30} are consequently bounded, uniformly in the +conditioning and in the term of the character expansion, by +\[ + O\!\left(N^{-\beta\gamma/4+(d+2)\tau+o(1)}\right). +\] +Averaging the conditioning costs no factor, and summing the character +expansion costs at most $O(N^{2d\tau})$. +The final exponent satisfies +\[ + \frac{\beta\gamma}{4}-(3d+2)\tau-\sigma + =\tau\left(250k(d+1)-(3d+2)-\frac\gamma4\right)>0. +\] +Thus the resulting error is $O(N^{-\sigma})$. +The same-interval contribution and the truncation error already satisfy +this bound. Finally, the interval assignments were averaged with +probabilities summing to one, so the uniform estimate proves +\eqref{eq:29}. +\end{proof} diff --git a/preprints/A-power-saving-for-intersective-polynomial-differences-with-an-exponent-depending-only-on-the-degree-October-5-2026/build/manuscript/sections/tuples.tex b/preprints/A-power-saving-for-intersective-polynomial-differences-with-an-exponent-depending-only-on-the-degree-October-5-2026/build/manuscript/sections/tuples.tex new file mode 100644 index 0000000000000000000000000000000000000000..00f7385ae31eeafbd4fd7adbd49e65975c444a6c --- /dev/null +++ b/preprints/A-power-saving-for-intersective-polynomial-differences-with-an-exponent-depending-only-on-the-degree-October-5-2026/build/manuscript/sections/tuples.tex @@ -0,0 +1,776 @@ +\section{Weighted reflection-positive tuple laws} +\label{sec:tuples} + +We now construct the finite-field probability laws used to average products +of Fourier lifts. The two requirements that must be reconciled are +positivity across certain reflections of the slots and support on +polynomial differences around a prescribed cycle. A measure obtained +simply by imposing the cycle constraints need not have the required +positivity. We instead construct several measures, indexed by sets of +\emph{marked blocks}. Marking a block removes some constraints and creates +a positive contribution at reflections in that block. A suitably weighted +mixture lets this contribution dominate the possible negative contribution +from the corresponding unmarked measure. + +The graph estimates and marked-block construction adapt +\cite[Lemmas~2.1--2.2 and Section~3]{squarepower}. Here the edge constraints +are weighted by regular fibers of $h_l$. Their directions are retained +throughout: neither the polynomial value set nor its weight is assumed to +be invariant under negation. We prove all the properties needed below, +including the exact covariance under changes of the auxiliary parameter. + +\subsection{Slots, reflections, and mixing of edge weights} + +Retain the constants $\delta,\gamma$ and the integer $L>2$ from +Section~\ref{sec:local}. Choose the least integer $t\ge1$ such that +$(t+1)\delta/2>5$, and set +\[ + w=2t+1,\qquad K=L!,\qquad r_*=K/2,\qquad + V=\Pi^w,\qquad d=K^w,\qquad n_*=d/2, +\] +where $\Pi$ is the set of bijections from the positions $[L]$ to the +symbols $[L]$. Thus an element of $\Pi$ is a permutation word, and a slot +$v\in V$ consists of $w$ such words, which we call its blocks. Fix an +$L$-cycle $c$ acting on positions. Right composition $q\mapsto qc$ +shifts a word, whereas left composition permutes its symbols. The +designated cycle of slots is +\[ + v_j=(c^j,\ldots,c^j)\quad(0\le jw-s$ unmarked blocks. +The resulting graph on list entries satisfies +Lemma~\ref{tuple:list-mixing}. The same threshold also preserves the +designated cycle: if $s\le t$, at least $t+1$ blocks remain unmarked, +and each consecutive pair of cycle vertices shifts in every one of +them. At $s=t+1$ only $t$ blocks remain, and we set $I_S=1$; +we call these sets $S$ terminal. + +Each slot will choose an entry of its address list. To obtain strict +positivity at a marked cut, we also weight the choices themselves. +Set $\Delta=1/(10L^2)$. For an unordered pair of distinct slots +$x,y\in V$, set $H^S_{xy}=0$ unless $x,y$ agree in every block except +one marked block and their words in that block differ by a +transposition of symbols. If the exchanged positions have gap $g$, +set $H^S_{xy}=\Delta^g$. Choose independent uniform indices +$j_v\in[\ell_s]$ and put +\[ + R_S(j)=\exp\left(\sum_{\{x,y\}\subseteq V} + H^S_{xy}\1_{\{j_x=j_y\}}\right). +\] +Define a finite, not yet normalized, measure $\nu_S=\nu_{l,p,S}$ by +\[ + \int \Phi(z)\,d\nu_S(z) + =\E_{Y,j}\left[ + I_S(Y)R_S(j) + \Phi\bigl((Y_{u_S(v),j_v})_{v\in V}\bigr)\right]. +\] +Thus the output label at $v$ is $z_v=Y_{u_S(v),j_v}$. All underlying +averages in this definition are independent uniform averages. +Since the index set is fixed, +\[ + 0\le I_S\le1,\qquad 1\le R_S\le C, + \qquad \nu_S(1)\le C, +\] +with a constant independent of $l,p,S$. + +Every $\nu_S$ is invariant under the slot involution $\theta$ of any +reflection cut. Indeed, simultaneous symbol relabeling preserves +$H^S$: it conjugates the transposition defining an interaction and +leaves the exchanged positions unchanged. If the cut block is +marked, addresses are unchanged. If it is unmarked, reflection +permutes the lists, preserving each directed edge because left symbol +permutations commute with right composition by $c$. Reindexing the +independent lists and indices therefore preserves the measure. In +particular, every component has a symmetric matrix at every cut. + +\subsection{Positivity across a cut} + +Fix a reflection cut in block $b$. For a real function +$J:\F_p^{V_+}\to\R$, write +\[ + Q_S(J)=\int J((z_v)_{v\in V_+}) + J((z_{\theta v})_{v\in V_+})\,d\nu_S(z). +\] +We compare $Q_S$ with $Q_{S\cup\{b\}}$. When $b$ is marked, +the index interactions give a fixed positive lower bound. When $b$ +is unmarked, list mixing bounds any negative contribution by a power +of $p$. The same half-label square occurs in both bounds after +concatenating the lists. + +\begin{lemma}[Positivity at a marked block] +\label{tuple:marked-positive} +There is a constant $c_*>0$, independent of $l,p,S$ and the cut, such +that if $b\in S$, then +\begin{equation} + Q_S(J)\ge c_*\, + \E_{Y,j_+} I_S(Y) + J\bigl((Y_{u_S(v),j_v})_{v\in V_+}\bigr)^2. + \label{eq:4} +\end{equation} +Here $j_+$ consists of independent uniform indices on $V_+$. +\end{lemma} + +\begin{proof} +The interaction between the two halves is governed by the symmetric +matrix +\[ + \mathsf A_{xy}=H^S_{x,\theta y},\qquad x,y\in V_+. +\] +Fix $x\in V_+$, and suppose that the cut symbols occur at positions +$i0. +\] + +For $j\in[\ell_s]^{V_+}$ define +$u(j)=(\1_{\{j_x=i\}})_{x\in V_+,\,i\in[\ell_s]}$. +After identifying the minus side with the plus side by reflection, +the cross-interaction matrix on half-index assignments is +\[ + C_{j,j'}=\exp\bigl(u(j)^T + (\mathsf A\otimes\operatorname{Id})u(j')\bigr). +\] +Split $\mathsf A=(\mathsf A-\kappa\operatorname{Id})+ +\kappa\operatorname{Id}$. The entrywise exponential associated with +the first summand is positive semidefinite: its power-series +expansion is a sum of Gram matrices of tensor powers. Its diagonal +entries are at least one. The exponential associated with the second +summand is a tensor product, over $x\in V_+$, of +\[ + \mathbf1\mathbf1^T+(\exp(\kappa)-1)\operatorname{Id} +\] +on $[\ell_s]$. It therefore dominates +$a_*\operatorname{Id}$, where +$a_*=(\exp(\kappa)-1)^{n_*}$. The entrywise product of positive +semidefinite matrices is positive semidefinite, as follows directly +from their Gram representations. Multiply the lower bound for the +second exponential entrywise by the first. The resulting diagonal +lower bound is at least $a_*\operatorname{Id}$ because the first +exponential has diagonal entries at least one. Thus +$C\succeq a_*\operatorname{Id}$. + +The interactions within the two halves give the same diagonal factor +on either side of $C$, with all entries at least one. The full index +matrix retains the lower bound $a_*\operatorname{Id}$. For fixed +lists $Y$, reflected vertices have the same address because $b$ is +marked. The half-label evaluations therefore form the same vector +on both sides of this matrix, even when some field labels coincide. +There are $\ell_s^{n_*}$ half-index assignments. Averaging uniformly +on both sides converts the matrix lower bound into +\[ + \E_{j_+,j_-}R_S(j) + J((z_v)_{v\in V_+})J((z_{\theta v})_{v\in V_+}) + \ge a_*\ell_s^{-n_*}\, + \E_{j_+}J((z_v)_{v\in V_+})^2. +\] +Multiplying by $I_S(Y)$ and averaging $Y$ proves \eqref{eq:4} with +$c_*=a_*\ell_{t+1}^{-n_*}$. +\end{proof} + +\begin{lemma}[The contribution at an unmarked block] +\label{tuple:unmarked-defect} +Suppose $b\notin S$. Split the address lists into the two halves of +the cut. Denote the plus lists by $Y_+$, and let $I_+(Y_+)$ be the +product of the weights on edges internal to them. Put +\[ + \mathcal D_S(J)=\E_{Y_+,j_+} I_+(Y_+) + J\bigl((Y_{u_S(v),j_v})_{v\in V_+}\bigr)^2. +\] +If $|S|\le t$, then +\begin{equation} + Q_S(J)\ge-Cp^{-\delta}\mathcal D_S(J). + \label{eq:5} +\end{equation} +If $|S|=t+1$, then $Q_S(J)\ge0$. +\end{lemma} + +\begin{proof} +Reflection identifies the minus lists and their internal edge factors +with the plus lists and their internal factors. No index interaction +crosses this cut: an interaction changes only a marked block, whereas +the cut is determined by the unmarked block $b$. The two within-half +index weights are likewise identified by reflection; write $R_+$ +for the one on the plus half. + +Let $K(Y_+,Y_-)$ be the product of all cross-list edge weights. Then +$Q_S(J)$ is the quadratic form of this kernel on the half function +\[ + g(Y_+)=I_+(Y_+)\E_{j_+} + R_+(j_+)J\bigl((Y_{u_S(v),j_v})_{v\in V_+}\bigr). +\] +Cauchy--Schwarz, the uniform bound on $R_+$, and $I_+^2\le I_+$ give +$\|g\|_2^2\le C\mathcal D_S(J)$. By \eqref{eq:3}, the kernel +$K$ differs in operator norm by $O(p^{-\delta})$ from the constant +$c_p^{e_{\rm cross}}$, where $e_{\rm cross}$ is its number of edges. +The constant contributes $c_p^{e_{\rm cross}}(\E g)^2\ge0$. +This proves \eqref{eq:5}. In the terminal case $I_S=1$, so the +cross kernel is exactly one and the entire form is nonnegative. +\end{proof} + +\begin{lemma}[Domination after marking one more block] +\label{tuple:marking-domination} +If $b\notin S$ and $|S|\le t$, then, with $S'=S\cup\{b\}$, +\begin{equation} + Q_{S'}(J)\ge c_*r_*^{-n_*}\mathcal D_S(J). + \label{eq:6} +\end{equation} +\end{lemma} + +\begin{proof} +Let $\Pi_+$ be the $r_*$ words in block $b$ satisfying the cut order. +For each new address $u\in U_{S'}$, the old plus addresses above it +are $(u,q)$ with $q\in\Pi_+$; here $q$ is inserted in block $b$. +Concatenate their lists by identifying +\[ + [\ell_{s+1}]\ \cong\ \Pi_+\times[\ell_s], + \qquad + Y'_{u,(q,i)}=Y_{(u,q),i}. +\] +This is a bijection of independent uniform list entries, so it changes +neither their joint distribution nor the value of an average. +Figure~\ref{tuple:fig-regrouping} separates this relabeling from the +index restriction used below. + +\input{figures/list-regrouping} + +A new address edge requires $t+1$ shifted blocks outside $S'$. Those blocks +also certify an old edge for every choice of the two words in block +$b$. Hence every factor of $I_{S'}(Y')$ occurs, with its direction +unchanged, among the factors of $I_+(Y_+)$. Since all factors belong +to $[0,1]$, +\[ + I_+(Y_+)\le I_{S'}(Y'). +\] +When $S'$ is terminal this inequality follows from $I_{S'}=1$. + +Apply \eqref{eq:4} to $S'$. In its nonnegative right-hand side, +restrict the new index at each $v\in V_+$ to the sublist +$\{v_b\}\times[\ell_s]$. Each restriction has probability $1/r_*$, +independently across the $n_*$ slots. On this event the labels and +the remaining independent indices are exactly those in +$\mathcal D_S(J)$. Its probability is $r_*^{-n_*}$, and the +pointwise weight comparison above now yields \eqref{eq:6}. +\end{proof} + +\subsection{The mixture, its support, and covariance} + +We can now choose the mixture. The gain in \eqref{eq:6} is a fixed +positive constant, whereas the possible loss in \eqref{eq:5} tends +to zero with $p$. Thus a component with one more marked block may +receive a smaller coefficient and still compensate for that loss. + +\begin{proposition}[Reflection-positive laws with polynomial cycle support] +\label{tuple:laws} +For all sufficiently large primes $p$, uniformly in $l\ge1$, define +\begin{align*} + M_{l,p}&=\sum_{\substack{S\subseteq[w]\\|S|\le t+1}} + p^{-|S|\delta/2}\nu_{l,p,S},\\ + \mu_{l,p}&=\frac{M_{l,p}}{M_{l,p}(1)},\\ + \eps_{l,p}&=\frac{1}{M_{l,p}(1)} + \sum_{|S|=t+1}p^{-(t+1)\delta/2}\nu_{l,p,S}. +\end{align*} +Then $\mu_{l,p}$ is a reflection-positive probability law and +$0\le\eps_{l,p}\le\mu_{l,p}$. Its exceptional mass satisfies +\[ + \eta_{l,p}:=\eps_{l,p}(1)\le p^{-4}. +\] +The probability law +\[ + \mu_{l,p}^{\circ} + =\frac{\mu_{l,p}-\eps_{l,p}}{1-\eta_{l,p}} +\] +has only admissible increments $z_{v_{j+1}}-z_{v_j}$ along the +designated cycle. Every component and both measures $\mu_{l,p}$, +$\eps_{l,p}$ are invariant under simultaneous translation of all +labels. Every positive-mass component or subsum therefore has uniform +single-slot marginals after normalization. In either $\mu_{l,p}$ or +$\mu_{l,p}^{\circ}$, the total probability of components with +$S\ne\varnothing$ is $O(p^{-\delta/2})$. +\end{proposition} + +\begin{proof} +At a cut in block $b$, pair each $S$ with $b\notin S$, $|S|\le t$, +with $S\cup\{b\}$. Each component containing $b$ appears in exactly +one pair. Equations~\eqref{eq:5} and \eqref{eq:6} bound the quadratic +form of the pair below by +\[ + p^{-|S|\delta/2} + \left(p^{-\delta/2}c_*r_*^{-n_*}-Cp^{-\delta}\right) + \mathcal D_S(J). +\] +This is nonnegative once $p$ is sufficiently large. The only unpaired +sets are terminal sets omitting $b$; their forms are nonnegative by +Lemma~\ref{tuple:unmarked-defect}. Thus $M_{l,p}$ is reflection +positive at every cut. + +For $S=\varnothing$, each list has length one and $R_S=1$. +Lemma~\ref{tuple:list-mixing} and $c_p\ge1/(2k)$ show that +$\nu_{\varnothing}(1)$ is bounded below by a fixed positive constant +for all sufficiently large $p$. All component masses are bounded +above, and the number of components is fixed. It follows that +\[ + 05$ gives the stated bound $p^{-4}$ after +increasing the prime threshold. Summing the nonleading coefficients +also gives total probability $O(p^{-\delta/2})$ for $S\ne\varnothing$. +Removing the terminal components retains the positive leading mass, +so the same estimate holds after normalization to $\mu_{l,p}^{\circ}$. + +Simultaneous translation of all list entries preserves their product +law, all differences in $I_S$, and the index weight $R_S$. It +therefore translates every output label and proves the asserted +invariances and marginal uniformity. + +Finally, a nonterminal $S$ leaves at least $t+1$ blocks unmarked. +Consecutive cycle vertices shift by $c$ in all these blocks, including +the edge from $v_{L-1}$ to $v_0$. Their addresses are consequently +joined by a directed edge. Since its weights include every pair of +list entries, any positive-weight label tuple has admissible +increment on this edge, regardless of the chosen indices. This +holds around the entire cycle in every component retained by +$\mu_{l,p}^{\circ}$. +\end{proof} + +The auxiliary parameter changes the edge weight, so we need covariance +between different laws rather than dilation invariance of a single law. + +\begin{lemma}[Covariance under progression changes] +\label{tuple:covariance} +For every $l\ge1$, every sufficiently large prime $p$, and every +integer $D\ge1$ with $p\nmid D$, the following holds. For each +$a'\in\F_p$, the map +\begin{equation} + (z_v)_{v\in V}\longmapsto + \bigl(\lambda(D)^{-1}(z_v-a')\bigr)_{v\in V} + \label{eq:7} +\end{equation} +pushes $\mu_{l,p}$ to $\mu_{lD,p}$ and $\eps_{l,p}$ to +$\eps_{lD,p}$. +\end{lemma} + +\begin{proof} +Equation~\eqref{eq:1} states that +$h_l(Dx-i)=\lambda(D)h_{lD}(x)$. Both $D$ and $\lambda(D)$ are +units modulo $p$. The argument map $x\mapsto Dx-i$ is therefore a +bijection and, on differentiating this identity, preserves regularity. +Counting regular arguments gives +\[ + W_{lD,p}(y)=W_{l,p}(\lambda(D)y). +\] +Apply \eqref{eq:7} to every list entry in a component measure. The +independent uniform list law and all index weights are unchanged; +each old edge weight becomes exactly the corresponding new edge +weight. Thus the map pushes $\nu_{l,p,S}$ to $\nu_{lD,p,S}$ for +each $S$. All component masses, and in particular the normalizing +masses $M_{l,p}(1)$ and $M_{lD,p}(1)$, agree. The two claimed +pushforwards follow with no additional factor. +\end{proof} + +\subsection{Conditional mixing and the prime threshold} + +The mixture has the required positivity and support. To control high +Fourier frequencies later, we also need to know that the other labels +give little information about a mean-zero function of any one label. +The leading component has this property by list mixing. The remaining +components have small total probability, and their uniform marginals +give the bound needed to include them. + +\begin{lemma}[Conditional mixing] +\label{tuple:conditional-mixing} +For all sufficiently large $p$, uniformly in $l$, let $\rho_p$ be +either $\mu_{l,p}$ or $\mu_{l,p}^{\circ}$. For every $v\in V$ and +every function $f:\F_p\to\C$ of uniform mean zero, +\begin{equation} + \left\|\E_{\rho_p}\bigl(f(z_v)\mid + (z_{v'})_{v'\ne v}\bigr)\right\|_2 + \le p^{-\gamma}\|f\|_2. + \label{eq:8} +\end{equation} +The norm on the left is taken in the other-label marginal of +$\rho_p$; the norm on the right is uniform on $\F_p$. +\end{lemma} + +\begin{proof} +First normalize the leading component $\nu_{\varnothing}$, and +write $x=z_v$, $y=(z_{v'})_{v'\ne v}$. Before normalization its +density relative to uniform labels is $K_1(x,y)K_2(y)$, where $K_1$ +is the product of the $m$ weights on edges incident to $v$, and +$K_2$ contains the remaining weights. Put +\[ + A(y)=\E_xK_1(x,y),\qquad + B_f(y)=\E_x f(x)K_1(x,y). +\] +The incident edges form a graph between the singleton list $x$ and +the list $y$. Equation~\eqref{eq:3}, applied to this kernel, gives +\[ + \|A-c_p^m\|_2\ll p^{-\delta},\qquad + \|B_f\|_2\ll p^{-\delta}\|f\|_2. +\] +This includes $m=0$, when $A=1$ and $B_f=0$. + +Let $Z_0=\nu_{\varnothing}(1)$, which is bounded below uniformly. +The other-label marginal has density $Z_0^{-1}K_2(y)A(y)$, and the +conditional expectation equals $B_f(y)/A(y)$ where $A(y)>0$. +Its squared norm is therefore exactly +\[ + \frac1{Z_0}\E_y K_2(y)\frac{|B_f(y)|^2}{A(y)}, +\] +with the ratio taken as zero if $A(y)=0$. On $A\ge c_p^m/2$, the +bound on $B_f$ makes this $O(p^{-2\delta})\|f\|_2^2$. The +complement has uniform probability $O(p^{-2\delta})$, by the bound +on $A-c_p^m$ and the fixed positive lower bound on $c_p^m$. +Everywhere, weighted Cauchy--Schwarz gives +\[ + \frac{|B_f(y)|^2}{A(y)} + \le\E_x |f(x)|^2K_1(x,y)\le\|f\|_2^2. +\] +Since $K_2\le1$, the complement has the same required bound. Thus +the squared conditional norm in the normalized leading component is +$O(p^{-2\delta})\|f\|_2^2$. + +For either mixture, condition additionally on its component index. +The tower property and Jensen's inequality can only increase the +squared conditional norm under this extra conditioning. In each +nonleading component the trivial bound is $\|f\|_2^2$, because its +single-slot marginal is uniform. Proposition~\ref{tuple:laws} bounds +their total probability by $O(p^{-\delta/2})$. The squared norm for +the mixture is consequently $O(p^{-\delta/2})\|f\|_2^2$, and the +norm is $O(p^{-\delta/4})\|f\|_2$. Since $\gamma=\delta/10$, the +fixed implicit constant is absorbed by taking $p$ sufficiently large, +proving \eqref{eq:8}. +\end{proof} + +We require a threshold $P>P_1$ such that all the preceding +assertions hold for every prime $p\ge P$, uniformly in $l$. We +further require +\begin{equation} + 3\le p^{\gamma/2}\quad(p\ge P),\qquad + \prod_{\substack{p\ge P\\p\ \mathrm{prime}}}(1+p^{-3})-1 + \le\frac14. + \label{eq:9} +\end{equation} +The second requirement is possible by convergence of the product. +For such a choice, define the small-prime modulus +\[ + M_0=\prod_{\substack{p5. +\] +The normalizing mass $M_{l,p}(1)$ is at least the leading mass, so +\[ + \eta_{l,p}\le C_{\mathrm{mass}}p^{-\alpha_{\mathrm{exc}}}. +\] +It is at most $p^{-4}$ if +$p^{\alpha_{\mathrm{exc}}-4}\ge C_{\mathrm{mass}}$. In either +$\mu_{l,p}$ or $\mu_{l,p}^{\circ}$ the probability of nonleading +components is at most $C_{\mathrm{mass}}p^{-\delta/2}$. For the latter +law, its unnormalized mass after removing the terminal terms still +contains $\nu_{\varnothing}(1)\ge Z_*$; hence the same denominator +bound applies without an additional normalization factor. Cycle support +and translation invariance in Proposition~\ref{tuple:laws} are exact +properties of these components and impose no further threshold. + +For the leading component in Lemma~\ref{tuple:conditional-mixing}, the +incident graph has $m\le e_*$ edges. Its list operator error is at most +$C_{\mathrm{gr}}p^{-\delta}$. The split into $A\ge c_p^m/2$ and its +complement in that proof therefore bounds the squared conditional norm by +\[ + Z_0^{-1}C_{\mathrm{gr}}^2 + \bigl(2c_p^{-m}+4c_p^{-2m}\bigr)p^{-2\delta}\|f\|_2^2. +\] +Using $Z_0\ge Z_*$ and $c_p^{-m}\le(2k)^{e_*}$ bounds this by +\[ + C_{\mathrm{lead}}p^{-2\delta}\|f\|_2^2, + \qquad + C_{\mathrm{lead}}:=Z_*^{-1}C_{\mathrm{gr}}^2 + \bigl(2(2k)^{e_*}+4(2k)^{2e_*}\bigr). +\] +The case $m=0$ already has zero conditional norm. For either mixture, +conditioning additionally on the component index and applying Jensen's +inequality bounds its squared conditional norm by +\[ + \bigl(C_{\mathrm{lead}}p^{-2\delta} + +C_{\mathrm{mass}}p^{-\delta/2}\bigr)\|f\|_2^2 + \le (C_{\mathrm{lead}}+C_{\mathrm{mass}}) + p^{-\delta/2}\|f\|_2^2. +\] +Here the bound in every other component is $\|f\|_2^2$ because its +single-slot marginal is uniform. As +$\delta/2-2\gamma=3\delta/10>0$, the desired squared bound +$p^{-2\gamma}\|f\|_2^2$ follows once +\[ + p^{\delta/2-2\gamma}\ge C_{\mathrm{lead}}+C_{\mathrm{mass}}. +\] + +Choose an integer $P>P_1$ large enough that the displayed requirements +on $p$ above hold for every $p\ge P$, and that $P^{\gamma/2}\ge3$. +This is possible because $\delta>0$, $\alpha_{\mathrm{exc}}-4>0$, and +$\delta/2-2\gamma>0$, and all the constants depend only on $k,L,t$. +The product condition in \eqref{eq:9} can be +included in the same choice: for integer $P\ge2$, +\[ + \prod_{\substack{p\ge P\\p\ \mathrm{prime}}}(1+p^{-3}) + \le \exp\left(\sum_{n\ge P}n^{-3}\right) + \le \exp\left(\frac{1}{2(P-1)^2}\right), +\] +whose last expression tends to one. The covariance proof uses only the +exact progression identity and bijections when $p\nmid D$, so it imposes +no further restriction on $P$. This proves the stated dependence of $P$. +Under Condition~\eqref{local:unit-condition}, Section~\ref{sec:local} +allows $P_1,L$ to be chosen from $k$ and each $E_p$ from $k,p$. Also, +$t$ is chosen from $\delta$, which depends only on $k$. The product defining $M_0$ +then uses a degree-dependent set of primes and degree-dependent +exponents. For a general fixed polynomial, $M_0$ retains any dependence +of the local exponents $E_p$ as well as that of $P_1$ and $L$. +\end{proof} + +For the rest of the paper, fix a choice of $P$ supplied by +Proposition~\ref{tuple:threshold-dependence} and the corresponding $M_0$. diff --git a/preprints/A-power-saving-for-intersective-polynomial-differences-with-an-exponent-depending-only-on-the-degree-October-5-2026/power-saving-intersective-polynomial-differences.pdf b/preprints/A-power-saving-for-intersective-polynomial-differences-with-an-exponent-depending-only-on-the-degree-October-5-2026/power-saving-intersective-polynomial-differences.pdf new file mode 100644 index 0000000000000000000000000000000000000000..4878b3a1caaa51499eb72d433a4f0ab6e9ac6a39 --- /dev/null +++ b/preprints/A-power-saving-for-intersective-polynomial-differences-with-an-exponent-depending-only-on-the-degree-October-5-2026/power-saving-intersective-polynomial-differences.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:d69e417ac953581d7f373c9457e809e47ad4b534c1f79407ff5c0e13f382b605 +size 644894 diff --git a/preprints/A-power-saving-for-planar-halving-lines-September-25-2026/README.md b/preprints/A-power-saving-for-planar-halving-lines-September-25-2026/README.md new file mode 100644 index 0000000000000000000000000000000000000000..8e7f265ce89a61eb62c037fa87925854c5fdd2e4 --- /dev/null +++ b/preprints/A-power-saving-for-planar-halving-lines-September-25-2026/README.md @@ -0,0 +1,18 @@ +# [A power saving for planar halving lines](main.pdf) + +**Author:** OpenAI + +**Date:** September 25, 2026 + +## Citation + +```bibtex +@misc{OAI:A-power-saving-for-planar-halving-lines-September-25-2026, + author = {{OpenAI}}, + title = {{A power saving for planar halving lines}}, + howpublished = {OpenAI Math Release preprint + \href{https://github.com/openai/math/blob/main/preprints/A-power-saving-for-planar-halving-lines-September-25-2026/main.pdf}{OAI:A-power-saving-for-planar-halving-lines-September-25-2026}}, + year = {2026} +} +``` + diff --git a/preprints/A-power-saving-for-planar-halving-lines-September-25-2026/build/main.tex b/preprints/A-power-saving-for-planar-halving-lines-September-25-2026/build/main.tex new file mode 100644 index 0000000000000000000000000000000000000000..c5705fbed719317880cbb220aeaa2ed4bc7967af --- /dev/null +++ b/preprints/A-power-saving-for-planar-halving-lines-September-25-2026/build/main.tex @@ -0,0 +1,53 @@ +\documentclass[11pt]{article} +\usepackage[T1]{fontenc} +\usepackage[utf8]{inputenc} +\usepackage{lmodern} +\usepackage[margin=1in]{geometry} +\usepackage{amsmath,amssymb,amsthm,mathtools} +\usepackage{microtype,enumitem} +\usepackage{tikz} +\usepackage[numbers,sort&compress]{natbib} +\usepackage[hidelinks,breaklinks]{hyperref} +\pdfinfoomitdate=1 +\pdftrailerid{} +\pdfsuppressptexinfo=15 +\hypersetup{pdftitle={A power saving for planar halving lines},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} +\setlength{\emergencystretch}{2em} +\setlist{itemsep=3pt,topsep=5pt} +\title{A power saving for planar halving lines} +\author{OpenAI} +\date{September 25, 2026} +\begin{document} +\maketitle +\begin{abstract} +There are absolute constants $\varepsilon>0$ and $C$ such that every +sufficiently large even $n$-point set in the plane with no three collinear +has at most $Cn^{4/3-\varepsilon}$ unordered halving pairs. This gives a +power saving over the classical $O(n^{4/3})$ bound for planar halving lines. +The proof is nonquantitative and does not supply explicit constants. +\end{abstract} +\tableofcontents +\clearpage +\input{sections/introduction} +\input{sections/finite} +\input{sections/rank-limits} +\input{sections/differentiability} +\input{sections/zoom} +\input{sections/spatial} +\input{sections/boundary} +\input{sections/recurrence} +\begingroup +\raggedright +\bibliographystyle{plainnat} +\bibliography{references} +\endgroup +\end{document} diff --git a/preprints/A-power-saving-for-planar-halving-lines-September-25-2026/build/references.bib b/preprints/A-power-saving-for-planar-halving-lines-September-25-2026/build/references.bib new file mode 100644 index 0000000000000000000000000000000000000000..ae4a4d4d2bc11b0c6405fac18ce98c76ef24514c --- /dev/null +++ b/preprints/A-power-saving-for-planar-halving-lines-September-25-2026/build/references.bib @@ -0,0 +1,204 @@ +@article{Dey1998, + author={Dey, Tamal K.}, + title={Improved Bounds for Planar {$k$}-Sets and Related Problems}, + journal={Discrete \& Computational Geometry}, + volume={19}, number={3}, pages={373--382}, year={1998}, + doi={10.1007/PL00009354}, url={https://doi.org/10.1007/PL00009354} +} +@article{AgarwalAronovChanSharir1998, + author={Agarwal, Pankaj K. and Aronov, Boris and Chan, Timothy M. and Sharir, Micha}, + title={On Levels in Arrangements of Lines, Segments, Planes, and Triangles}, + journal={Discrete \& Computational Geometry}, + volume={19}, pages={315--331}, year={1998}, + doi={10.1007/PL00009348}, + url={https://tmc.web.engr.illinois.edu/kset.pdf}, + note={Numbered and page locators refer to the author manuscript dated August 30, 1997} +} +@article{Toth2001, + author={T{\'o}th, G{\'e}za}, + title={Point Sets with Many {$k$}-Sets}, + journal={Discrete \& Computational Geometry}, + volume={26}, number={2}, pages={187--194}, year={2001}, + doi={10.1007/s004540010022}, url={https://doi.org/10.1007/s004540010022} +} +@incollection{Nivasch2008, + author={Nivasch, Gabriel}, + title={An Improved, Simple Construction of Many Halving Edges}, + booktitle={Surveys on Discrete and Computational Geometry: Twenty Years Later}, + series={Contemporary Mathematics}, volume={453}, pages={299--305}, + publisher={American Mathematical Society}, year={2008}, + doi={10.1090/conm/453/08804}, url={https://doi.org/10.1090/conm/453/08804} +} +@article{AlonsoLopezRodrigo2024, + author={Alonso, Estrella and L{\'o}pez, Maril{\'o} and Rodrigo, Javier}, + title={An Improvement of the Upper Bound for the Number of Halving Lines of Planar Sets}, + journal={Symmetry}, volume={16}, number={7}, pages={936}, year={2024}, + doi={10.3390/sym16070936}, url={https://doi.org/10.3390/sym16070936} +} +@article{AjtaiEtAl1982, + author={Ajtai, M. and Chv{\'a}tal, V. and Newborn, M. M. and Szemer{\'e}di, E.}, + title={Crossing-Free Subgraphs}, + journal={Annals of Discrete Mathematics}, volume={12}, pages={9--12}, year={1982}, + doi={10.1016/S0304-0208(08)73484-4}, url={https://doi.org/10.1016/S0304-0208(08)73484-4} +} +@book{Leighton1983, + author={Leighton, Frank Thomson}, + title={Complexity Issues in {VLSI}: Optimal Layouts for the Shuffle-Exchange Graph and Other Networks}, + series={Foundations of Computing}, publisher={MIT Press}, year={1983}, + isbn={9780262121040}, url={https://mitpress.mit.edu/9780262121040/complexity-issues-in-vlsi/} +} +@article{ChabrillacCrouzeix1987, + author={Chabrillac, Yves and Crouzeix, Jean-Pierre}, + title={Continuity and Differentiability Properties of Monotone Real Functions of Several Real Variables}, + journal={Mathematical Programming Study}, volume={30}, pages={1--16}, year={1987}, + doi={10.1007/BFb0121151}, url={https://doi.org/10.1007/BFb0121151} +} +@article{BorweinBurkeLewis2004, + author={Borwein, Jonathan M. and Burke, James V. and Lewis, Adrian S.}, + title={Differentiability of Cone-Monotone Functions on Separable {Banach} Space}, + journal={Proceedings of the American Mathematical Society}, + volume={132}, number={4}, pages={1067--1076}, year={2004}, + doi={10.1090/S0002-9939-03-07149-1}, + url={https://sites.math.washington.edu/~burke/papers/reprints/46-cone-monotone.pdf} +} +@misc{VianaRokhlin, + author={Viana, Marcelo}, + title={Disintegration into Conditional Measures: {Rokhlin}'s Theorem}, + year={n.d.}, note={Undated author lecture notes}, + url={https://w3.impa.br/~viana/out/rokhlin.pdf} +} + +@article{Lovasz1971, + author = {Lov{\'a}sz, L{\'a}szl{\'o}}, + title = {On the Number of Halving Lines}, + journal = {Annales Universitatis Scientiarum Budapestinensis de Rolando E{\"o}tv{\"o}s Nominatae, Sectio Mathematica}, + volume = {14}, + pages = {107--108}, + year = {1971} +} + + +@incollection{ErdosLovaszSimmonsStraus1973, + author = {Erd{\H o}s, Paul and Lov{\'a}sz, L{\'a}szl{\'o} and Simmons, A. and Straus, Ernst G.}, + title = {Dissection Graphs of Planar Point Sets}, + booktitle = {A Survey of Combinatorial Theory}, + publisher = {North-Holland}, + address = {Amsterdam}, + pages = {139--149}, + year = {1973}, + url = {https://users.renyi.hu/~p_erdos/1973-07.pdf} +} + + +@article{PachSteigerSzemeredi1992, + author = {Pach, J{\'a}nos and Steiger, William and Szemer{\'e}di, Endre}, + title = {An Upper Bound on the Number of Planar {$k$}-Sets}, + journal = {Discrete \& Computational Geometry}, + volume = {7}, + number = {2}, + pages = {109--123}, + year = {1992}, + doi = {10.1007/BF02187829}, + url = {https://doi.org/10.1007/BF02187829} +} + + +@inproceedings{Pinchasi2009, + author = {Pinchasi, Rom}, + title = {Halving Lines and Measure Concentration in the Plane}, + booktitle = {Proceedings of the 25th Annual Symposium on Computational Geometry}, + series = {SCG '09}, + publisher = {Association for Computing Machinery}, + address = {New York, NY, USA}, + pages = {141--147}, + year = {2009}, + doi = {10.1145/1542362.1542393}, + url = {https://doi.org/10.1145/1542362.1542393} +} + +@article{ArtsteinSlomka2017, + author = {Artstein-Avidan, Shiri and Slomka, Boaz A.}, + title = {The Fundamental Theorems of Affine and Projective Geometry Revisited}, + journal = {Communications in Contemporary Mathematics}, + volume = {19}, + number = {5}, + pages = {1650059}, + year = {2017}, + doi = {10.1142/S0219199716500590}, + eprint = {1604.01762}, + archivePrefix = {arXiv}, + primaryClass = {math.GM}, + url = {https://doi.org/10.1142/S0219199716500590} +} + + +@article{PachSolymosiTardos2010, + author = {Pach, J{\'a}nos and Solymosi, J{\'o}zsef and Tardos, G{\'a}bor}, + title = {Crossing Numbers of Imbalanced Graphs}, + journal = {Journal of Graph Theory}, + volume = {64}, + number = {1}, + pages = {12--21}, + year = {2010}, + doi = {10.1002/jgt.20435}, + url = {https://doi.org/10.1002/jgt.20435} +} + +@article{Rademacher1919, + author = {Rademacher, Hans}, + title = {{\"U}ber partielle und totale Differenzierbarkeit von Funktionen mehrerer Variabeln und {\"u}ber die Transformation der Doppelintegrale}, + journal = {Mathematische Annalen}, + volume = {79}, + number = {4}, + pages = {340--359}, + year = {1919}, + doi = {10.1007/BF01498415}, + url = {https://doi.org/10.1007/BF01498415} +} + + +@misc{Simon2014GMT, + author = {Simon, Leon}, + title = {Introduction to Geometric Measure Theory}, + year = {2014}, + howpublished = {Tsinghua lecture notes}, + note = {2014 draft}, + url = {https://web.stanford.edu/class/math285/ts-gmt.pdf} +} + + +@misc{PayneRedaelli2025preprint, + author = {Payne, Kevin R. and Redaelli, Davide Francesco}, + title = {A Primer on Semiconvex Functions in General Potential Theories}, + year = {2025}, + eprint = {2303.14477}, + archivePrefix = {arXiv}, + primaryClass = {math.AP}, + note = {Version 3, 15 April 2025; Appendix B, Corollary B.4, p. 117}, + url = {https://arxiv.org/abs/2303.14477v3} +} + + +@article{PascolettiZanolin2010, + author = {Pascoletti, Anna and Zanolin, Fabio}, + title = {A Path Crossing Lemma and Applications to Nonlinear Second Order Equations under Slowly Varying Perturbations}, + journal = {Le Matematiche}, + volume = {65}, + number = {2}, + pages = {121--168}, + year = {2010}, + doi = {10.4418/2010.65.2.13}, + url = {https://doi.org/10.4418/2010.65.2.13} +} + +@article{Siebenmann2005, + author = {Siebenmann, L.}, + title = {The {Osgood--Schoenflies} Theorem Revisited}, + journal = {Russian Mathematical Surveys}, + volume = {60}, + number = {4}, + pages = {645--672}, + year = {2005}, + doi = {10.1070/RM2005v060n04ABEH003672}, + url = {https://doi.org/10.1070/RM2005v060n04ABEH003672} +} diff --git a/preprints/A-power-saving-for-planar-halving-lines-September-25-2026/build/sections/boundary.tex b/preprints/A-power-saving-for-planar-halving-lines-September-25-2026/build/sections/boundary.tex new file mode 100644 index 0000000000000000000000000000000000000000..c2c9f8b27b0b4521336a861221001260f8031b9d --- /dev/null +++ b/preprints/A-power-saving-for-planar-halving-lines-September-25-2026/build/sections/boundary.tex @@ -0,0 +1,343 @@ +\section{Boundary activity and the packing contradiction} +\label{sec:boundary} + +We now use the spatial measure constructed above to rule out the assumed +sequence of extremal batches. The essential finite observation is that, +when the directions of the cutting lines vary little, every participating +vertex either lies in one of three narrow strips or is accounted for by +a marked token crossing the boundary of a fixed region. We first prove +this observation without taking limits. + +\begin{lemma}[Three strips] +\label{lem:three-strips} +Let $P$ be a finite subset of $\mathbb R^2$, let $B$ be a bounded open set, +and let $D\geq1$ satisfy $|x|\leq D$ for $x\in B$. Suppose that +$M_0,\ldots,M_m\subset P$ have the same cardinality and are given by +strictly separating cuts +\[ + M_i=\{x\in P:\nu_i\cdot x] (0,0)--(8.1,0); + \draw (1,-.08)--(1,.08); + \draw (4,-.08)--(4,.08); + \draw (7,-.08)--(7,.08); + \node[below] at (1,-.2) {$c_-$}; + \node[below] at (4,-.2) {$c_0$}; + \node[below] at (7,-.2) {$c_+$}; + \node[anchor=east] at (8.1,-.8) {$t=\nu_0\cdot x$}; + \node at (1,1.95) {$S_{c_-}$}; + \node at (4,1.95) {$S_{c_0}$}; + \node at (7,1.95) {$S_{c_+}$}; + \draw[<->] (1.4,.45)--(3.6,.45); + \draw[<->] (4.4,.45)--(6.6,.45); + \node[align=center] at (2.5,1.12) + {Active, initially\\marked\\absent at $i_-$}; + \node[align=center] at (5.5,1.12) + {Active, initially\\unmarked\\present at $i_+$}; +\end{tikzpicture} +\caption{Projection onto $t=\nu_0\cdot x$. The shaded bands have +half-width $3\rho$; their separation is schematic, and in general they +may overlap. An initially marked participant in $B$ outside the bands is lost +by a state $i_-$ attaining $c_-$; this endpoint comparison has no gains +inside $B$. +The symmetric statement holds for initially unmarked participants and +a state $i_+$ attaining $c_+$. Token conservation bounds these two +numbers by exits and entries, respectively. This is an aggregate count; +it does not assign each participant's own token to a boundary crossing.} +\label{fig:three-strip-projection} +\end{figure} + +Consider first an $x\in(A\cap B)\setminus S$ that is initially marked. +Avoidance of $S_{c_0}$ gives $t(x)c_i\geq c_-$ at some $i$. Therefore +$t(x)>c_- -\rho$. Avoidance of $S_{c_-}$ improves this to +$t(x)>c_-+3\rho$. At an index $i_-$ attaining the minimum threshold, +\[ + \nu_{i_-}\cdot x\geq t(x)-\rho>c_-; +\] +thus $x$ is lost between states $0$ and $i_-$. The existence of even one +such $x$ also forces $c_0-c_->6\rho$. Consequently every initially +unmarked $y\in P\cap B$ is still unmarked at state $i_-$, because +\[ + \nu_{i_-}\cdot y\geq t(y)-\rho>c_0-\rho>c_-. +\] +There are therefore no gains anywhere inside $B$ in this comparison, +including among vertices lying in the strips. + +To count the losses, place one distinct token on each point of $M_0$. +At every step retain the tokens on unchanged marked vertices and move +the others along the chosen matching. There is exactly one token on +every marked vertex at each state. It follows that +\[ + \#(M_0\cap B)-\#(M_{i_-}\cap B) + =\text{token exits from }B-\text{token entries into }B +\] +over that prefix of steps. Since all differences between these two states inside $B$ are losses, +their number equals this difference and is at most $C_B$. This bounds +the initially marked participants outside the strips; if there are +none, the bound holds without imposing a threshold gap. + +Now let $x\in(A\cap B)\setminus S$ be initially unmarked. It satisfies +$t(x)>c_0+3\rho$ and is marked at some later state, so +$t(x)6\rho$. Every initially marked $y\in P\cap B$ consequently +remains marked at state $i_+$, since +\[ + \nu_{i_+}\cdot y\leq t(y)+\rho0,\qquad \mu_J\leq\mu,\qquad + \lambda_J(\widetilde Z)=|J|,\\ + \lambda_J(F)^3\leq C s_J^3\mu_J(F). + \end{gathered} + \label{eq:boundary-measure-input} +\end{equation} +for every Borel set $F\subset\widetilde Z$ and every dyadic $J\subset K_0$. +Here $s_J$ is the limiting fraction of distinct participating batch +endpoints. At every dyadic resolution $\delta$, +\begin{equation} + \#\{J\subset K_0:|J|=\delta,\ s_J>u\} + \leq C u^{-4/3}. + \label{eq:boundary-packing-input} +\end{equation} +The spatial projection $\beta$ of $\mu|_G$ is boundedly supported and +satisfies +\begin{equation} + \beta(B(x,r))\leq C_G r^d, + \qquad 0\frac74. + \label{eq:boundary-disk-input} +\end{equation} +These masses are unchanged because the enhancement preserved all the +original measure projections. + +The directions throughout $K_0$ lie between the directions at its two +fixed bracketing times. The common linear normalization sends those +two projective directions to distinct fixed directions. Passing to a +subsequence selects one of the two projective arcs between them and +one of its two oriented lifts. Hence all forward transfer vectors lie +in a fixed closed cone $\mathcal C$ of aperture less than $\pi$. +We may enlarge this cone slightly and discard finitely many indices. Each intermediate +forward transfer vector is a positive linear combination of the two +bracketing forward vectors; this identifies the relevant cone after +normalization, including its orientation. + +Choose linearly independent covectors $\ell_1,\ell_2$ in the interior +of its dual cone. Thus $\ell_j(v)>0$ for every nonzero +$v\in\mathcal C$. Choose a bounded open parallelogram +\[ + B=\{x:a_1<\ell_1(x)1$. Applying this bound to the at most three +limiting strips proves the result, including $\theta_J=0$. +\end{proof} + +It remains to combine this activity bound with the endpoint estimate. +Fix $M\geq1$ and subdivide $K_0$ into dyadic windows of length +$\delta$. By \eqref{eq:boundary-packing-input}, the number with +$s_J>M\delta^{3/4}$ is at most +$C M^{-4/3}\delta^{-1}$. Each has total $\lambda_J$-mass $\delta$, +so these windows contribute at most $C M^{-4/3}$ to +$\lambda_{K_0}(G)$. + +For the other windows, \eqref{eq:boundary-measure-input} gives +\[ + \sum_{s_J\leq M\delta^{3/4}}\lambda_J(G) + \leq C M\delta^{3/4}\sum_J\mu_J(G)^{1/3}. +\] +There are at most $\delta^{-1}$ windows. Set +$\alpha=d-1=15/17\in(0,1)$. H\"older's inequality, +Lemma~\ref{lem:limiting-boundary-activity}, and +\eqref{eq:boundary-budgets} imply +\[ + \sum_J\mu_J(G)^{1/3} + \leq \delta^{-2/3}\Bigl(\sum_J\mu_J(G)\Bigr)^{1/3} + \leq C_G\delta^{-2/3} + \Bigl(1+\sum_J\theta_J^\alpha\Bigr)^{1/3}. +\] +Concavity and the angular budget give +\[ + \sum_J\theta_J^\alpha + \leq \delta^{\alpha-1}\Bigl(\sum_J\theta_J\Bigr)^\alpha + \leq C\delta^{\alpha-1}. +\] +Consequently +\begin{align*} + \lambda_{K_0}(G) + &\leq C M^{-4/3} + +C_G M\delta^{-1/4}(\delta+\delta^{d-1})^{1/3}\\ + &\leq C M^{-4/3} + +C_G M\bigl(\delta^{1/12}+\delta^{3/68}\bigr), +\end{align*} +where +\[ + \frac{d-1}{3}-\frac14 + =\frac{15}{51}-\frac14=\frac{3}{68}>0. +\] +First let $\delta\downarrow0$ with $M$ fixed, and then let +$M\to\infty$. This contradicts $\lambda_{K_0}(G)>0$ and excludes +the proposed sequence of extremal batches. Together with the finite +reductions, it establishes the uniform little-$o$ activity-packing +estimate. The numerical exponent $3/68$ concerns only this limiting +contradiction; the power saving in the final finite theorem is chosen +separately. diff --git a/preprints/A-power-saving-for-planar-halving-lines-September-25-2026/build/sections/differentiability.tex b/preprints/A-power-saving-for-planar-halving-lines-September-25-2026/build/sections/differentiability.tex new file mode 100644 index 0000000000000000000000000000000000000000..42d707c44823ce5fede3b9ebcf2e4ffc1e64a1e8 --- /dev/null +++ b/preprints/A-power-saving-for-planar-halving-lines-September-25-2026/build/sections/differentiability.tex @@ -0,0 +1,305 @@ +\section{Differentiable coordinates and transverse directions} +\label{sec:differentiability} + +We now convert the crossing measures into local geometric information. +The input is the countable family $\mathcal D$ of dyadic time windows, +the edge endpoint marginal $\lambda$ on the rank cube $Z$, its conditional time +probabilities $h_z$, and the crossing measures $\sigma_{CD}$ constructed +above. We use three established facts: $h_z$ is nonatomic for +$\lambda$-almost every $z$; +\begin{equation} + \lambda_J=h_z(J)\lambda, + \qquad + \lambda_J\ll\sum_{\substack{C,D\subset J\\ C\cap D=\varnothing}} + \sigma_{CD}; + \label{eq:analytic-input-ac} +\end{equation} +and the $(C,D)$-coordinate projection of $\sigma_{CD}$ is dominated by +planar Lebesgue measure. The sum in \eqref{eq:analytic-input-ac} specifies +null sets; it may equivalently be replaced by a finite measure formed +from a countable sum with strictly positive weights. Our objective is +to find a positive part of $\lambda$ on which +three different sweep directions have three different tangent directions. + +\subsection{Calculus for the limiting coordinates} + +We first record the two analytic facts needed below. Differentiability +of coordinatewise monotone functions is classical +\cite{ChabrillacCrouzeix1987}; the precise finite-dimensional statement +is also given in \cite[Theorem~1(f)]{BorweinBurkeLewis2004}. +We include the argument needed here. Monotonicity in +each coordinate may have either prescribed sign; reflecting a coordinate +reduces the first lemma to the nondecreasing case. + +\begin{lemma}\label{lem:monotone-differentiability} +A bounded Borel function on an open rectangle in $\mathbb R^2$ that is +monotone in each coordinate is Fr\'echet differentiable almost everywhere +with respect to planar Lebesgue measure. +\end{lemma} + +\begin{proof} +It suffices to work in a compact subrectangle and assume that both +monotonicities are nondecreasing. Slice monotonicity shows that the +distributional derivatives $D_1f,D_2f$ are locally finite positive +measures. Put $\nu=D_1f+D_2f$ and +$Q_r(x)=x+[-r,r]^2$. For a square lying sufficiently far inside the +domain, monotonicity and integration over +$x+[-2r,-r]^2$ give +\begin{equation} + \operatorname{osc}_{Q_r(x)}f + \le f(x+(r,r))-f(x-(r,r)) + \le \frac{\nu(Q_{3r}(x))}{r}. + \label{eq:monotone-oscillation} +\end{equation} +Indeed, compare every point $u$ of that lower square with +$u+(3r,3r)$. The difference of their function values is at least the +middle difference in \eqref{eq:monotone-oscillation}. Integrating the +horizontal increment and then the vertical increment by slices bounds +the resulting integral by $r\nu(Q_{3r}(x))$; the lower square has area +$r^2$. + +The upper area density of $\nu$ is finite at almost every $x$. +Consequently, at almost every $x$ there are $M_x,r_x>0$ such that +$|f(y)-f(x)|\le M_x|y-x|$ whenever $|y-x|0$. For each $x\in S$ choose arbitrarily small radii +$r<1$ such that $B(x,5r)\subset\Omega$ and +\[ + |F(y)-F(x)-DF(x)(y-x)|\le\varepsilon|y-x| + \quad\text{for }y\in B(x,5r). +\] +The elementary $5r$ covering lemma gives disjoint balls $B(x_i,r_i)$ +whose fivefold enlargements cover $S$. Since their centers lie in +$B(0,R)$, we have $\sum_i r_i^2\le (R+1)^2$. +The set $F(B(x_i,5r_i))$ lies in the $5\varepsilon r_i$ neighborhood +of a segment of length at most $10Mr_i$. Its outer area is therefore +at most $C(M\varepsilon+\varepsilon^2)r_i^2$. +Summing these covers and letting $\varepsilon\downarrow0$ proves +that $F(S)$ is null. A countable union over $R,M$ completes the proof. +\end{proof} + +\subsection{A positive part with independent gradients} + +Because $h_z$ is a nonatomic probability, for almost every $z$ it gives +positive mass to two dyadic windows whose closures are disjoint and +contained in $(0,1)$. There are only countably many pairs, so we can +fix such windows $K,K'$ for which +\[ + E=\{z:h_z(K)>0,\ h_z(K')>0\} +\] +has positive $\lambda$-measure. Equation +\eqref{eq:analytic-input-ac} for $K'$ implies that $\lambda|_E$ is +absolutely continuous with respect to the sum of its crossing measures. +We may further restrict $E$ to a positive-measure set on which +\begin{equation} + \lambda|_E\ll\sigma_{AB} + \quad\text{for some disjoint }A,B\subset K'. + \label{eq:one-crossing-part} +\end{equation} +For completeness, form a finite measure $\Sigma$ from the countable sum +by assigning strictly positive weights to its summands. +If $g_{AB}=d\sigma_{AB}/d\Sigma$, the sets $\{g_{AB}>0\}$ cover +$\Sigma$-almost all points. One of them has positive +$\lambda|_E$-measure; restriction to it proves +\eqref{eq:one-crossing-part}. +Thus the projection of $\lambda|_E$ to +\[ + p=(z_A,z_B)\in(0,1)^2 +\] +is absolutely continuous with respect to planar Lebesgue measure. +The boundary of the square has zero mass by the same projection bound. + +For each $C\subset K$, express the finite coordinate $r_C$ as a function +of $(r_A,r_B)$. These functions are bounded and coordinatewise monotone, +with signs determined by the relative order of the windows. Extract a +subsequence on a countable dense grid, simultaneously for all $C$. +The lower monotone extensions of the grid limits give Borel monotone +functions $f_C$. At every continuity point $p$ of $f_C$, squeezing +between grid points below and above $p$ shows convergence even along +moving arguments $p_m\to p$. + +To apply this convergence to the endpoint measure, take +$z\in\operatorname{supp}\lambda$ with pair coordinate $p\in(0,1)^2$. +There are actual finite endpoint labels $z_m\to z$ along a subsequence: +every open neighborhood of $z$ has positive limiting mass and therefore +contains finite endpoint labels at all sufficiently late indices. +Their pair coordinates tend to $p$, so moving-argument convergence gives, +whenever $f_C$ is continuous at $p$, +\begin{equation} + z_C=f_C(p). + \label{eq:coordinate-identities} +\end{equation} +By Lemma~\ref{lem:monotone-differentiability} and the absolute continuity +of the pair projection, we can discard a $\lambda|_E$-null set so that +all these identities hold and all the $f_C$ are differentiable at $p$. + +The fixed pair $A,B$ supplies a coordinate chart in which every $f_C$ +is differentiable at the pair coordinate of each remaining label. +We now use crossing +measures from pairs $C,D$ inside each positively active window +$J\subset K$ to find independent gradients. Those crossing measures +are controlled in their own $(C,D)$ coordinates; the critical-image +lemma connects this control to derivatives in the $(A,B)$ chart. + +\begin{lemma}\label{lem:independent-gradients} +After discarding a further $\lambda|_E$-null set, every label $z\in E$ +has the following property: if $J\subset K$ is dyadic and $h_z(J)>0$, +then some disjoint dyadic $C,D\subset J$ satisfy +\[ + \det\begin{pmatrix}\nabla f_C(p)\\ \nabla f_D(p)\end{pmatrix}\ne0. +\] +\end{lemma} + +\begin{proof} +Fix $J$. Let $B_J\subset E$ be the labels with $h_z(J)>0$ for which +every such determinant vanishes. For each disjoint $C,D\subset J$, +Lemma~\ref{lem:critical-image-null}, applied to $F=(f_C,f_D)$, gives a +Borel planar null set containing +\[ + \{(f_C(p),f_D(p)):p\text{ is a differentiability point of }F, + \ \det DF(p)=0\}. +\] +By \eqref{eq:coordinate-identities}, the $(C,D)$ projection of $B_J$ +lies in that null set. Projection domination therefore gives +$\sigma_{CD}(B_J)=0$. Equation \eqref{eq:analytic-input-ac} implies +$\lambda_J(B_J)=0$. Since +$\lambda_J(B_J)=\int_{B_J}h_z(J)\,d\lambda(z)$ and $h_z(J)>0$ on +$B_J$, we get $\lambda(B_J)=0$. Discard these null sets for all +countably many $J$. +\end{proof} + +\subsection{Allowed tangent directions} + +Fix a remaining label $z$ and its pair coordinate $p$. +At small rank scales $h$, the preceding convergence and differentiability +have the following iterated consequence: for every fixed $R<\infty$, +\begin{equation} + r_C=f_C(p)+h\nabla f_C(p)\cdot w+o(h), + \qquad w=\frac{(r_A,r_B)-p}{h},\quad |w|\le R. + \label{eq:rank-first-order} +\end{equation} +Here first $h\downarrow0$, and for each fixed $h$ the configuration index +is taken sufficiently large. The estimate is uniform on the displayed +ball, simultaneously for any fixed finite collection of $C$. +Indeed, differentiate $f_C$ at $p$, then bracket the ball by a finite +grid of continuity points with mesh $o(h)$ and use monotonicity. +Choosing the indices successively gives the estimate for all $C$ along +any sufficiently fast diagonal. No rate uniform in $p$ is asserted. + +Orient each physical sweep direction by increasing original horizontal +coordinate. For $t\in K^\circ$, forward displacements on a line of +direction $t$ have a common prescribed sign in each of the two rank +coordinates. Let $Q$ be the resulting closed signed quadrant. +For each $t\in K^\circ$, define $D_p(t)\subset Q\cap S^1$ by the +linear inequalities +\begin{align*} + \nabla f_C(p)\cdot d&\ge0 + &&\text{if $C\subset K$ lies strictly before $t$},\\ + \nabla f_C(p)\cdot d&\le0 + &&\text{if $C\subset K$ lies strictly after $t$}. +\end{align*} +``Strictly'' here requires a positive time gap. +It is an intersection of closed halfplanes with the quadrant, restricted +to the unit circle, so it is a closed angular interval when nonempty. + +Any pair of forward-ordered points on a line of direction $t$, with +bounded zoom coordinates tending to distinct limits $w,w'$, satisfies +\begin{equation} + \frac{w'-w}{|w'-w|}\in D_p(t). + \label{eq:all-pairs-direction} +\end{equation} +This follows from monotonicity of $r_C$ along the line and +\eqref{eq:rank-first-order}. The base point and the line may vary with +the configuration. Moreover, $D_p(t)$ is nonempty. In each configuration +follow the forward line through the physical preimage of $p$ until its +rank image first exits the zoom unit ball. This exit exists: the rank +map is a homeomorphism onto the open square, and the preimage of a compact +subset is compact. Along a sufficiently fast diagonal the exit points +have a convergent subsequence, and their unit limiting displacement +belongs to $D_p(t)$. + +\begin{lemma}\label{lem:separated-tangent-directions} +If $s0$, then +$D_p(s)\cap D_p(t)=\varnothing$. There are three times +$\tau_1<\tau_2<\tau_3$ in $K^\circ$ for which these intervals are +singletons with distinct projective directions. +\end{lemma} + +\begin{proof} +Choose a dyadic $J$ whose closure lies in $(s,t)$ and for which +$h_z(J)>0$. Lemma~\ref{lem:independent-gradients} provides disjoint +$C,D\subset J$ with independent gradients. A vector belonging to both +$D_p(s)$ and $D_p(t)$ would have zero scalar product with both gradients, +and hence be zero, contrary to unit length. + +Consider the support of the nonatomic measure $h_z|_{K^\circ}$. +Remove the endpoints of its complementary intervals and the endpoints +of $K$. Only countably many points are removed, and any two distinct +remaining support points have positive measure strictly between them. +Their intervals $D_p(t)$ are consequently pairwise disjoint. At most +countably many disjoint angular intervals can have positive length, since +each contains a different rational angle. Removing those times still +leaves full $h_z|_K$-measure. Three remaining times give the claim. +All directions lie in the same quadrant, so distinct unit directions +are also projectively distinct. +\end{proof} + +The singleton times in this lemma may depend on $z$. We also need one +fixed triple for a common normalization of the configurations. For +every remaining $z$, nonatomicity and $h_z(K)>0$ allow rational times +$t_10$ throughout a positive-measure subset, again denoted $E$. +Thus +\[ + \lambda_{K_0}(E)>0, + \qquad + D_p(t_1),\ D_p(t_2),\ D_p(t_3) + \text{ are pairwise disjoint compact angular intervals.} +\] +All restrictions made here are restrictions of the original spatial +measure $\lambda$. Positive conditional time mass was used only in +the stated measurable events; positive spatial mass follows from the +countable selections. These conclusions supply the directional input +for the local geometric argument. diff --git a/preprints/A-power-saving-for-planar-halving-lines-September-25-2026/build/sections/finite.tex b/preprints/A-power-saving-for-planar-halving-lines-September-25-2026/build/sections/finite.tex new file mode 100644 index 0000000000000000000000000000000000000000..34b5138ddc4f6354ae9b4efa2466da0f7b30050d --- /dev/null +++ b/preprints/A-power-saving-for-planar-halving-lines-September-25-2026/build/sections/finite.tex @@ -0,0 +1,254 @@ +\section{Sweeps, activity, and exchange graphs} +\label{sec:finite} + +The chain and crossing mechanism below is the one underlying Dey's bound~\cite{Dey1998}. We first convert intervals containing many participating points into +straight-line graphs with controlled crossings. This gives the classical +exponent $4/3$ and isolates the packing estimate that must be improved. + +We first justify the genericity convention of the introduction. The +additional conditions on coordinates, slopes, and proper segment +intersections can be achieved by an arbitrarily small perturbation. Indeed, the signs +of the finitely many nonzero oriented areas of triples are unchanged by +a sufficiently small perturbation. Consequently every pair has exactly +the same points on its two sides, and in particular the halving pairs are +preserved. The additional conditions exclude finitely many proper +algebraic sets of perturbations. + +Write $P=\{(x_i,y_i):1\le i\le N\}$ and fix a rank $k$, with +$0\le k\le N$. At a slope $s$ that is not a determined slope, let +$M_k(s)$ be the set of the $k$ smallest values of $y_i-sx_i$. +Choose one representative slope in each interval between consecutive +determined slopes, including the two unbounded intervals. These are +the \emph{states}; adjacent states form a \emph{step}. A step either +leaves $M_k$ unchanged or replaces one point by one other point. +The latter kind of step is a \emph{switch}. An interval of states +includes its endpoints, and its step set consists of the intervening +steps. Its \emph{activity} is the number of distinct points whose +membership in $M_k$ changes at one of those steps. Thus activity counts +points, rather than switches. For two sets of size $k$ put +\[ + d(M,M')=|M\setminus M'|=\tfrac12|M\mathbin{\triangle}M'|. +\] +Activity records membership changes throughout the interval, whereas +$d(M,M')$ records only the difference of its endpoint sets. +This is a metric, and one step changes distance from a fixed state by +at most one. At middle rank, the same endpoint-set distance occurs +in Pinchasi's study of slope-ordered balanced cuts and measure +concentration~\cite{Pinchasi2009}; the activity reduction below also +handles points that return to their initial membership. + +At a switch the gained point has larger first coordinate than the lost +point: the difference of their two projections changes sign as $s$ +increases. The sum of first coordinates in $M_k$ therefore increases +strictly at every switch. In particular an interval of positive +activity has different endpoint sets. + +Our main intermediate result strengthens the classical crossing bound +for collections of disjoint sweep intervals. Its quantifier is uniform +in both the rank and the activity threshold. + +\begin{theorem}[Uniform activity packing]\label{thm:activity-packing} +For every $\xi>0$ there is $Q_\xi<\infty$ such that, for every generic +$N$-point configuration, every rank, and every integer $1\le b\le N$ +with $N/b\ge Q_\xi$, any collection of intervals with disjoint step +sets and activity at least $b$ has at most $\xi(N/b)^{4/3}$ members. +\end{theorem} + +We first establish the finite reductions and the corresponding +$O((N/b)^{4/3})$ bound. Sections~\ref{sec:rank-limits}--\ref{sec:boundary} +prove the improvement to the displayed arbitrarily small constant. + +\begin{lemma}[Activity and endpoint distance]\label{lem:activity-distance} +Fix a generic $N$-point set and a rank $k$. For an integer $b\ge4$, +any collection of intervals of activity at least $b$ can be divided +into two classes. At rank $k+\lfloor b/4\rfloor$ the intervals in +the first class have endpoint distance at least $\lfloor b/4\rfloor$; +at rank $k-\lfloor b/4\rfloor$ the intervals in the second class do +so. Each rank used by a nonempty class is feasible. For $1\le b<4$, +the original rank already gives endpoint distance at least one. +\end{lemma} + +\begin{proof} +Set $c=\lfloor b/4\rfloor$. Classify an interval according to whether +at least $b/2$ of its active points were unmarked initially; put all +remaining intervals in the second class. In the first case let $E$ +be the initially marked set at rank $k+c$. At most $c$ of the initially +unmarked active points lie in $E$, so at least +$\lceil b/2\rceil-c\ge c$ of them lie outside $E$. + +Take such a point $u$. At some state it enters rank $k$, and is then +below at least $c+1$ of the $k+c$ points of $E$ in projection order. +Initially it was above all of $E$. For each of those $c+1$ points the +projection difference is affine in the slope, so the new comparison +persists to the final state. Let $E'$ be the final rank-$(k+c)$ set. +If at least $c$ of the specified points outside $E$ lie in $E'$, then +$|E'\setminus E|\ge c$. Otherwise one of them, say $u$, is outside +$E'$. All $c+1$ points of $E$ that remain above $u$ are then outside +$E'$, so again the endpoint distance is at least $c$. +There were at least $\lceil b/2\rceil$ initially unmarked points, which +also proves $k+c\le N$. + +The other class has at least $b/2$ initially marked active points. +Apply the preceding argument to the complementary marks and the +negated projections; enlarging the complementary rank by $c$ is +the same as reducing the original rank by $c$. Its feasibility follows +in the same way. The case $b<4$ follows from the strict increase of +the sum of first coordinates at switches. +\end{proof} + +\begin{lemma}[Greedy batches]\label{lem:greedy-batches} +Fix a rank and an integer $d\ge1$, and put $a=\lceil d/2\rceil$. +Starting at the first state, end a batch at the first subsequent state +at distance $a$ from its start, and repeat while possible. If $R$ +complete batches are obtained, every collection of intervals with +disjoint step sets and endpoint distance at least $d$ has at most +$R$ members. +\end{lemma} + +\begin{proof} +Distances change by at most one at a step, so every completed batch has +distance exactly $a$. If an interval contains no batch end strictly +after its start, both its endpoints are at distance at most $a-1$ +from the same preceding batch start. Their mutual distance is at most +$2(a-1)0$, $C<\infty$, and $n_0$ +such that, for every even $n\ge n_0$ and every $n$-point set +$P\subset\mathbb R^2$ with no three collinear, +\[ + h(P)\le C n^{4/3-\varepsilon}. +\] +\end{theorem} + +The improvement is in the exponent and is uniform over all configurations. +The proof is nonquantitative: it supplies neither a numerical value of +$\varepsilon$ nor effective values of $C$ and $n_0$. + +\subsection{The halving-line problem and earlier bounds} +A \emph{$k$-set} of a planar point configuration is a $k$-point subset +strictly separable from the remaining points by a line. The $k$-set +problem asks for the largest possible number of these subsets at a +specified size and rank. Halving pairs concern its middle rank and +measure the complexity of balanced line separations. + +The early work of Lov\'asz and of Erd\H{o}s, Lov\'asz, Simmons and +Straus established the $O(n^{3/2})$ scale for halving lines and developed +the rotating-line and dissection-graph viewpoint +\cite{Lovasz1971,ErdosLovaszSimmonsStraus1973}. +Pach, Steiger and Szemer\'edi improved the upper bound by an +iterated-logarithm factor \cite{PachSteigerSzemeredi1992}. +Dey then proved the bound $O(n(k+1)^{1/3})$ for $k$-sets with +$1\le k\le n/2$, giving $O(n^{4/3})$ for halving lines +\cite{Dey1998}. His use of convex chains and graph crossings provides +the finite starting point of the present argument. + +The lower bounds developed on a different scale. +Erd\H{o}s, Lov\'asz, Simmons and Straus constructed configurations with +$\Omega(n\log n)$ halving lines +\cite{ErdosLovaszSimmonsStraus1973}. T{\'o}th obtained +$n\exp(\Omega(\sqrt{\log n}))$ halving pairs, and Nivasch subsequently +simplified the construction and improved its exponential constant +\cite{Toth2001,Nivasch2008}. These bounds remain far below a fixed +power $n^{1+\delta}$. More recently, Alonso, L{\'o}pez and Rodrigo +improved a lower-order term in the upper bound while retaining the +exponent $4/3$ \cite{AlonsoLopezRodrigo2024}. + +A related route through measures was developed by Pinchasi +\cite{Pinchasi2009}. A line avoiding the point set and leaving half +its points on each side gives a balanced cut. Pinchasi studied the +changes between such cuts ordered by slope and related endpoint-change +bounds to planar measure concentration. Our activity parameter also +counts points that change status during an interval and later return; +Section~\ref{sec:finite} supplies the reduction from that parameter +to endpoint changes, uniformly over all ranks. + +\subsection{Rank sweeps and the stronger conclusion} +For the sweep arguments, call a configuration \emph{generic} if no three +points are collinear, its first coordinates are distinct, its determined +slopes are distinct, and no proper intersection of two determined +segments lies on a third determined segment. Section~\ref{sec:finite} +justifies reducing the halving-pair theorem to this case. + +Fix a generic configuration and an integer rank $0\le k\le n$. For a slope $s$ distinct from the determined slopes, mark the +$k$ points with smallest values of +$y-sx$. As $s$ increases, the marked set changes by exchanging one +point for another. We call such a change a \emph{switch}. For an +interval of this sweep, its \emph{activity} is the number of distinct +points whose membership changes. At the middle rank, halving pairs +are exactly the switches. Write $S_k(P)$ for the total number of switches +as the slope ranges from $-\infty$ to $+\infty$. + +\begin{theorem}[Uniform bound over all ranks]\label{thm:all-ranks} +There are absolute constants $\varepsilon>0$ and $C<\infty$ such that +for every integer $N\ge1$, every generic $N$-point planar configuration +$P$, and every integer $0\le k\le N$, +\[ + S_k(P)\le C N^{4/3-\varepsilon}. +\] +\end{theorem} + +The uniformity in $k$ is needed even for the middle-rank conclusion: +restricting a block of switches to its participating points changes the +rank. At ranks $0$ and $N$ the marked set is constant and $S_k(P)=0$. + +The same uniformity gives a bound throughout the planar $k$-set range. +For an arrangement $\mathcal L$ of nonvertical straight lines, let +$\Lambda_k(\mathcal L)$ be the closure of the points on its lines with +exactly $k$ lines strictly below them. Write +$\operatorname{comp}(\Lambda_k(\mathcal L))$ for the total number of +vertices and edge pieces of this polygonal chain. + +\begin{corollary}[Planar $k$-sets and straight-line levels] +\label{cor:k-sensitive} +There are absolute constants $\varepsilon_0>0$ and $C_0<\infty$ such +that, for every integer $n\ge2$ and every integer $1\le k\le n/2$, +the number of $k$-sets of any $n$-point set $Q\subset\mathbb R^2$ with +no three collinear is at most +\begin{equation}\label{eq:k-sensitive} + C_0 n(k+1)^{1/3-\varepsilon_0}. +\end{equation} +The same bound holds for $\operatorname{comp}(\Lambda_k(\mathcal L))$ +for any arrangement $\mathcal L$ of $n$ nonvertical straight lines +with no parallel pair and no three concurrent. +\end{corollary} + +The all-ranks theorem first bounds each level in an arrangement of a given +size. A shallow cutting covers a low level by $O(n/(k+1))$ cells, each +crossed by only $O(k+1)$ lines; within each cell, the level has a fixed local rank. +Applying the uniform bound in every cell gives the dependence on $k$ above. +The full argument, including the passage from levels to $k$-sets, appears +in Section~\ref{sec:k-sensitive-proof} after the main theorem is proved. + +This corollary concerns planar point sets and complete straight lines +only; it asserts no corresponding saving for pseudolines or line segments. + +\subsection{Proof strategy} + +The central estimate concerns disjoint intervals of sweep steps with +activity at least $b$. The usual crossing argument bounds their number +by $O((n/b)^{4/3})$. We prove a uniform little-$o$ improvement of this +packing bound as $n/b$ tends to infinity. Choosing one sufficiently +large fixed scale then gives a contracting recurrence and the power +saving in Theorem~\ref{thm:halving-power}. + +Here is the geometric mechanism behind the packing improvement. +For an interval whose endpoint marked sets differ, let $L$ be the points +marked only at its start and $G$ those marked only at its end. If +$|L|=|G|=a$, joining every point of $L$ to every point of $G$ gives a +complete bipartite exchange graph with $a^2$ edges. Convex and concave chain partitions +control its crossings. If the packing bound were attained to within +a constant factor, we could pass to limiting measures of vertices, +edges, and crossings after assigning points compact rank coordinates. +Each such coordinate is a smoothed fraction of a family of convex +chain graphs lying below the point. +Two separated time windows provide a pair of rank coordinates that +parametrizes the plane. The crossing counts imply that a positive +portion of edge activity has an absolutely continuous projection in +such a coordinate pair. + +Compact rank coordinates alone do not control physical distances: +a small region in the rank square could represent a highly distorted +part of the original plane. We use the order of the sweep directions +to recover Euclidean control on a positive part of the limiting measure. +The coordinates belonging to other time windows become coordinatewise +monotone functions. Their almost-everywhere derivatives force three +transverse families of sweep lines to have three transverse limiting +directions. A local rigidity lemma for these three families prevents +collapse after a suitable linear normalization. Crucially, its limits +are taken first at a fixed small rank scale and then far enough along +the sequence of configurations. We preserve that order when passing +to a Hausdorff limit of coordinate graphs. + +The crossing bound also makes edge endpoint mass absolutely continuous +with respect to vertex mass. Together with the absolutely continuous +edge projection in the rank square, this lets us select a portion with +positive original edge activity whose vertex projection has bounded +density there. The geometric control transfers this +vertex measure to the normalized Euclidean plane, where a disk of +radius $r$ has mass $O(r^{32/17})$, and hence a thin strip of width $r$ +has mass $O(r^{15/17})$. In a fixed bounded region, the active vertices +of a narrow angular window lie either in three such strips or among +vertices accounted for by marked tokens crossing the boundary. A token +follows a marked point through the successive exchanges. Each +token crosses that boundary only a bounded number of times. Summing +these activity bounds contradicts the positive limiting edge mass. +The exponents $32/17$ and $15/17$ are auxiliary spatial exponents; +the final saving $\varepsilon$ comes from the separate fixed-scale +recurrence. + +Section~\ref{sec:finite} proves the finite reductions and states the +uniform packing estimate. Section~\ref{sec:rank-limits} constructs the +rank coordinates and limiting measures. Sections~\ref{sec:differentiability} +and~\ref{sec:zoom} establish the directional and metric control, and +Section~\ref{sec:spatial} constructs the spatial measure. +Section~\ref{sec:boundary} proves the boundary estimate and packing +contradiction. Section~\ref{sec:recurrence} completes the finite induction +and derives the $k$-set and level bounds. + +\paragraph{Background.} +We use weak compactness and regularity of finite Borel measures on +compact metric spaces, the Radon--Nikodym and Lebesgue decomposition +theorems and finite-measure differentiation +\cite[Chapter~1, Section~3]{Simon2014GMT}, almost-everywhere differentiability +of scalar Lipschitz functions, and planar separation by a Jordan curve. +The conditional-measure and Lipschitz-differentiability statements are +specified at their applications. The crossing, monotone-coordinate, +critical-image, three-foliation, and boundary estimates needed for the +new packing argument are proved in the paper. diff --git a/preprints/A-power-saving-for-planar-halving-lines-September-25-2026/build/sections/rank-limits.tex b/preprints/A-power-saving-for-planar-halving-lines-September-25-2026/build/sections/rank-limits.tex new file mode 100644 index 0000000000000000000000000000000000000000..4e3333153bf7ffee24f9b44e753611657e900eea --- /dev/null +++ b/preprints/A-power-saving-for-planar-halving-lines-September-25-2026/build/sections/rank-limits.tex @@ -0,0 +1,345 @@ +\section{Rank coordinates and limiting measures}\label{sec:rank-limits} +We now suppose, for a contradiction, that there is a sequence of batch +systems for which +\[ + N/a\longrightarrow\infty,\qquad R\asymp(N/a)^{4/3}. +\] +Here each of the $R$ batches exchanges $a$ lost points for $a$ gained +points, and contributes the $a^2$ rightward edges between them. Write +\[ + E_*=Ra^2,\qquad M=aN. +\] +Thus $E_*^3\asymp a^2N^4$ and $E_*/M\to\infty$. The constants below may +depend on the two constants in the displayed comparison for $R$. +Our objective in this section is to assign compact labels to points so +that edge mass is controlled by vertex mass, and every positive portion +of edge mass sees crossings between different time windows. + +Give batch $i$ the time $(i-1/2)/R$. A \emph{window} is a dyadic +subinterval of $[0,1]$, with the half-open convention except at the right +endpoint $1$. Its edges are those of its batches. The consecutive +batches in a nonempty window have a slope span, from the state just +before its first batch to the state just after its last batch. Disjoint +windows have slope spans with disjoint interiors. Every edge slope is +strictly inside its batch span. Each fixed window is nonempty eventually. +For a fixed time $t\in(0,1)$, choose batch $i=1+\lfloor Rt\rfloor$ +and take the arithmetic mean of its two endpoint-state slopes. +A \emph{line of direction $t$} means a line parallel to this +configuration-dependent slope. The half-open convention fixes the +choice at cell boundaries. For fixed times and windows separated by +positive gaps, these directions have the same strict order as their +times for all sufficiently late configurations. + +Let $S_J$ denote the number of distinct endpoints of the exchange edges +in $J$. These are exactly the points whose membership changes between +the endpoint states of at least one batch in $J$. A point that changes +membership within a batch and returns before its end contributes to +the sweep activity but need not contribute to $S_J$. Thus $S_J$ is +at most the activity of the full sweep interval covered by $J$. +The finite estimates proved above give +\begin{equation}\label{eq:finite-window-crossings} + \operatorname{cr}(J)\le C a^2S_J^2. +\end{equation} +They also give convex and concave partitions of the full edge set into +at most $M$ chains; the edges of a window have such partitions into at +most $aS_J$ chains. + +\subsection{A pair of rank coordinates parametrizes the plane} +For a nonempty window $J$, extend every chain in its convex partition to +a full convex piecewise linear graph. On the left choose a ray slope +between the left endpoint of the window span and the first edge slope; +on the right choose one between the last edge slope and the right +endpoint of that span. Add affine graphs until there are exactly $M$ +graphs, denoted $g_{J,1},\ldots,g_{J,M}$. All their slopes lie in a +closed interval $[\ell_J,u_J]$ strictly inside the window span. The +finitely many choices of ray slopes and affine translates can avoid +all proper crossings between an edge in $J$ and an edge outside $J$. +At each such crossing exactly one of these $M$ graphs passes through +the crossing point: its actual edge graph. This uses the genericity +assumption excluding a third determined segment through a proper +crossing. For an empty window use arbitrary affine graphs; assertions +about any fixed finite collection of windows are made after those +windows have become nonempty. + +Choose a continuous strictly positive probability density on $\mathbb R$ +with distribution function $\Phi$. For a sufficiently small +$\varepsilon_J>0$ put +\[ + r_J(x,y)=\frac1M\sum_{i=1}^M + \Phi\!\left(\frac{y-g_{J,i}(x)}{\varepsilon_J}\right). +\] +This is the expected fraction of graphs below $(x,y)$ after a random +vertical translation. Choose $\varepsilon_J$ so that, at every proper +crossing under consideration, the contribution of all graphs other +than the one through the point differs from their unsmoothed count by +at most $1$. The choice is possible because there are finitely many +crossings and all these other vertical distances are nonzero. Hence +at a crossing on the $i$th graph in height order, +\begin{equation}\label{eq:rank-slot-error} + |r_J-i/M|\le 2/M. +\end{equation} +The choices can be made separately for all countably many windows. + +For fixed $x$, the function $y\mapsto r_J(x,y)$ is continuous, strictly +increasing, and has limits $0$ and $1$. Every level $r_J=v$, where +$0x$ the inequalities for the slopes of the original graphs give +\[ + \ell_J(x'-x)\le g_{J,v}(x')-g_{J,v}(x) + \le u_J(x'-x). +\] +For example, translating $(x,y)$ by $(h,u_Jh)$ can only increase $r_J$; +the upper inequality follows by applying this at a level point. The +lower inequality is analogous. + +Suppose $A$ precedes the disjoint window $B$. Then $u_A<\ell_B$. +Consequently the difference of any $A$-level graph and any $B$-level +graph is strictly decreasing, with opposite infinite limits at the +two ends of the real line. The graphs meet exactly once. Their +intersection depends continuously on the two levels: bracket its +$x$-coordinate on either side by strict inequalities, which persist +under small changes of the levels, and then use continuity of a level +graph. We have proved that +\begin{equation}\label{eq:rank-homeomorphism} + (r_A,r_B):\mathbb R^2\longrightarrow(0,1)^2 + \quad\hbox{is a homeomorphism}. +\end{equation} +If $C$ is disjoint from both $A$ and $B$, expressing $r_C$ in these +coordinates gives a function monotone in each coordinate. Indeed, +along an $A$-level graph, $r_B$ decreases strictly as $x$ increases; +along a $B$-level graph, $r_A$ increases strictly. Compare the slope +range of $C$ with that of the level graph to determine the sign of +$r_C$. The two signs, in the order $(r_A,r_B)$, are $(+,-)$ when +$C0$ the number of edges whose label +oscillation exceeds $\rho$ is at most $2M/\rho=o(E_*)$. + +\subsection{The limiting measures} +Vertices, edges, and crossings require different normalizations. +We attach the rank label $z(q)$ to each vertex and crossing point; +an edge $e$ receives the label $(z(q_e),t_e)$, where $q_e$ is its left +endpoint and $t_e$ its batch time. The following table defines the +finite measures and names their weak limits: +\begin{center} +\begin{tabular}{lll} +Object & Weight of one finite object & Weak limit\\[3pt] +Vertex & $N^{-1}$ & $\mu$ on $Z$\\ +Participating vertex in $J$ & $N^{-1}$ & $\mu_J$ on $Z$\\ +Edge with its batch time & $E_*^{-1}$ & $\nu$ on $Z\times[0,1]$\\ +Crossing between $I$ and $J$ & $M^{-2}$ & $\sigma_{IJ}$ on $Z$ +\end{tabular} +\end{center} +Here $E_*=Ra^2$ and $M=aN$, and crossing measures are defined for +disjoint windows $I,J$. Each participating vertex is counted once +within a window. Thus, for example, the full vertex and edge measures +are $N^{-1}\sum_{q\in P}\delta_{z(q)}$ and +$E_*^{-1}\sum_e\delta_{(z(q_e),t_e)}$. +Compactness and bounded masses give a common weakly convergent +subsequence for this countable collection of measures. In particular +$\mu$ and $\nu$ are probability measures. Define the endpoint marginals +and the total participating-vertex masses by +\[ + s_J=\mu_J(Z),\qquad + \lambda=\pi_{Z\,*}\nu,\qquad + \lambda_J(F)=\nu(F\times J). +\] +The measures $\lambda$ and $\lambda_J$ count edge endpoints with edge +weights, rather than counting distinct vertices. In particular, +$\lambda_J$ is additive over disjoint time windows, whereas the +participating sets counted by $\mu_J$ may overlap. +We have $\mu_J\le\mu$ and $S_J/N\to s_J$. +The time marginal of $\nu$ is Lebesgue measure, since each batch has +the same number $a^2$ of edges and the batch times form a uniform grid. +In particular every fixed window boundary is $\nu$-null. Restrictions +to windows therefore converge weakly, and $\lambda_J(Z)=|J|$. +Disintegration on the compact metrizable spaces~\cite[Example~2 and Theorem~3]{VianaRokhlin} supplies probability +measures $h_z$ on $[0,1]$, measurable in $z$, such that +\begin{equation}\label{eq:conditional-time} + \lambda_J=h_z(J)\lambda. +\end{equation} +Equivalently, $\lambda_J(F)=\int_F h_z(J)\,d\lambda(z)$ for every +Borel $F\subset Z$. Thus $h_z$ is the conditional distribution of batch +time under edge mass, given the endpoint rank label. +On the $\lambda$-null set where a version has not been specified, choose +any probability measure. Finally, the preceding grid argument gives +\begin{equation}\label{eq:crossing-projection} + (z_I,z_J)_*\sigma_{IJ}\le\mathcal L^2\big|_{[0,1]^2}. +\end{equation} + +The crossing projection bound concerns $\sigma_{IJ}$, whereas our +eventual contradiction must control the edge mass $\lambda$ using +the participating-vertex measures $\mu_J$. We next establish the +relations between these measures. For +any dyadic partition into windows of length $\delta$, the finite +activity-packing bound implies +\begin{equation}\label{eq:activity-tail} + \#\{J: |J|=\delta,\ s_J>u\}\le C u^{-4/3} + \qquad(u>0). +\end{equation} +Indeed a window with $S_J>uN$ has activity exceeding $uN$, and these +windows have disjoint step sets. Apply the uniform finite estimate +and then take the limit for this finite collection of windows. + +\begin{lemma}[Edge mass and vertex mass]\label{lem:edge-vertex-mass} +For every window $J$ and every Borel set $F\subset Z$, +\begin{equation}\label{eq:local-mass} + \lambda_J(F)^3\le C s_J^3\mu_J(F). +\end{equation} +In particular $\lambda\ll\mu$. The same estimate holds if additional +labels in a compact metrizable space are attached to vertices, +inherited by their edge endpoints, and a further weak limit is taken. +\end{lemma} +\begin{proof} +For a set $U$ of labels in a finite configuration, take the edges in +$J$ whose left endpoint belongs to $U$. Write $e(U)$ for their number +and $m(U)$ for their number of distinct left endpoints. The graph has +at most $S_J$ vertices, and all its edges have a designated endpoint +in a set of $m(U)$ vertices. The designated-endpoint crossing +inequality, with its sparse-graph alternative, and +\eqref{eq:finite-window-crossings} give +\[ + e(U)^3\le C m(U)S_J\operatorname{cr}(J)+C S_J^3 + \le C a^2m(U)S_J^3+C S_J^3. +\] +After division by $E_*^3\asymp a^2N^4$, the last term tends to zero, +since it is at most $C/(a^2N)$. Also $m(U)/N$ is at most the +participating-vertex measure of $U$. + +For a compact $F$, use its open $\rho$-neighborhood as $U$ and bound +its vertex measure by that of the closed $2\rho$-neighborhood. +Weak convergence and the open-set and closed-set inequalities give +\[ + \lambda_J(F)^3\le C s_J^3\mu_J(F^{[2\rho]}). +\] +Let $\rho\downarrow0$. Inner regularity of $\lambda_J$ then extends +the estimate from compact to all Borel sets. Take $J=[0,1]$ to obtain +absolute continuity. The finite argument makes no use of the nature +of the labels, proving the last assertion as well. +\end{proof} + +\begin{lemma}[No atoms in conditional time]\label{lem:nonatomic-time} +For $\lambda$-almost every $z$, the probability measure $h_z$ is +nonatomic. +\end{lemma} +\begin{proof} +Fix $\beta>0$ and a dyadic partition of mesh $\delta$. Put +$A_J=\{z:h_z(J)>\beta\}$. At each $z$ at most $1/\beta$ of these +sets occur. Thus $\sum_J\mu_J(A_J)\le1/\beta$. +For $M_0\ge1$, the windows with $s_J>M_0\delta^{3/4}$ contribute at +most $C M_0^{-4/3}$ to $\sum_J\lambda_J(A_J)$, by +\eqref{eq:activity-tail} and $\lambda_J(Z)=\delta$. +The remaining windows contribute, by +\eqref{eq:local-mass} and H\"older's inequality, at most +\[ + C M_0\delta^{3/4}\sum_J\mu_J(A_J)^{1/3} + \le C M_0\beta^{-1/3}\delta^{1/12}. +\] +First let $\delta\downarrow0$, then $M_0\to\infty$. +If $h_z$ has an atom of mass greater than $\beta$, the member of every +partition containing that atom contributes at least its mass to +$\sum_Jh_z(J)\mathbf1_{A_J}(z)$. Fatou's lemma therefore shows that +such $z$ form a $\lambda$-null set. A countable sequence of positive +$\beta$ tending to zero proves the claim. +\end{proof} + +\begin{lemma}[Edge mass sees crossings]\label{lem:edge-crossing-ac} +For every window $K$, +\begin{equation}\label{eq:crossing-ac} + \lambda_K\ll\sum_{I,J\subset K,\ I\cap J=\varnothing}\sigma_{IJ}, +\end{equation} +where the sum specifies null sets. Thus, if a Borel set $F\subset Z$ is null for every +$\sigma_{IJ}$ with $I,J\subset K$ disjoint windows, then +$\lambda_K(F)=0$. +\end{lemma} +\begin{proof} +By inner regularity it suffices to consider a compact set $F$ that is +null for every crossing measure on the right of \eqref{eq:crossing-ac}. +Fix a dyadic resolution $\delta$ subdividing $K$, and discard windows with +$s_J>\delta^{2/3}$. Their number is at most $C\delta^{-8/9}$, so the +limiting edge mass lost is at most $C\delta^{1/9}$. +In the remaining windows keep edges whose left-endpoint label lies in +the open $\rho$-neighborhood of $F$ and whose label oscillation is at +most $\rho$. For fixed $\delta,\rho>0$, the oscillation condition +costs $o(E_*)$ edges. If $e_{\delta,\rho}$ is the total number retained, +weak convergence consequently gives +\[ + \liminf \frac{e_{\delta,\rho}}E_* + \ge \lambda_K(F)-C\delta^{1/9}. +\] +A crossing between retained edges from different windows has label +in the closed $2\rho$-neighborhood of $F$. For this fixed finite +partition, its normalized limsup is therefore bounded by +\[ + \sum_{I\ne J}\sigma_{IJ}(F^{[2\rho]}), +\] +where each unordered pair is counted once. This tends to zero as +$\rho\downarrow0$. Crossings within one retained window are bounded +by \eqref{eq:finite-window-crossings}; after division by $M^2$ their +total limsup is at most +\[ + C\sum_{J:s_J\le\delta^{2/3}}s_J^2 + \le C\delta^{-1}\delta^{4/3}=C\delta^{1/3}. +\] +The ordinary crossing inequality, including its sparse alternative, +reads $e_{\delta,\rho}^3\le C N^2\operatorname{cr}+C N^3$. +Divide by $E_*^3\asymp a^2N^4$, take the configuration limit, and then +let $\rho\downarrow0$. It follows that +\[ + \bigl(\lambda_K(F)-C\delta^{1/9}\bigr)_+^3 + \le C\delta^{1/3}. +\] +Finally let $\delta\downarrow0$. This gives $\lambda_K(F)=0$, as +required. Equivalently, $\lambda_K$ is absolutely continuous with +respect to any countable mixture of the crossing measures in the +statement whose coefficients are all positive and whose total mass is finite. +\end{proof} + +The compact labels now retain three useful features: two separated +rank coordinates parametrize the plane; conditional edge activity +has no time atoms; and every positive portion of activity is detected +by a crossing measure with a bounded planar projection. These are +the inputs to the local geometric argument. diff --git a/preprints/A-power-saving-for-planar-halving-lines-September-25-2026/build/sections/recurrence.tex b/preprints/A-power-saving-for-planar-halving-lines-September-25-2026/build/sections/recurrence.tex new file mode 100644 index 0000000000000000000000000000000000000000..c7883e9b1401d408de2fc6c0fe712bc89122e2a9 --- /dev/null +++ b/preprints/A-power-saving-for-planar-halving-lines-September-25-2026/build/sections/recurrence.tex @@ -0,0 +1,159 @@ +\section{From activity packing to a power saving} +\label{sec:recurrence} + +We now turn the uniform activity-packing estimate into a fixed power +improvement. The scale is chosen once and then used at every size in +an induction. Throughout this section, a sweep is the ordering by +increasing values of $y-sx$ as $s$ runs from $-\infty$ to $+\infty$. +Its rank-$k$ marked set consists of the $k$ smallest values. The +\emph{activity} of an interval of sweep steps is the number of distinct +points whose membership in that set changes during the interval. + +\begin{proposition}\label{prop:packing-to-power} +Suppose that, for every $\eta>0$, there is $R_\eta<\infty$ such that +for every generic planar configuration of $N$ points, every rank, and +every integer $1\le b\le N$ with $N/b\ge R_\eta$, the number of +step-disjoint intervals of activity at least $b$ is at most +$\eta(N/b)^{4/3}$. Then there are absolute constants $\varepsilon>0$ +and $C$ such that every such rank sweep has at most +$CN^{4/3-\varepsilon}$ switches. +\end{proposition} + +\begin{proof} +Write $a=4/3$, set $\eta=(4\cdot2^a)^{-1}$, and choose an integer +$q\ge\max\{4,R_\eta\}$. This integer remains fixed. For $N\ge4q$, +put +\[ + d=\lfloor N/q\rfloor,\qquad b=d-1. +\] +Then $N/(2q)\le b0. +\] +Choose $C$ so that $F(n)\le Cn^\beta$ for $1\le n<4q$. Suppose this +bound holds at every size smaller than $N\ge4q$. If $t_j\le N/q$ +are the active-subset sizes of the blocks, the restriction argument +and the induction hypothesis give +\[ + F(N)\le\sum_j F(t_j) + \le C\sum_j t_j^\beta + \le C\frac{q^a}{2}\left(\frac Nq\right)^\beta + =\frac{C}{\sqrt2}N^\beta + \le CN^\beta. +\] +Here every nonempty subset has size less than $N$, while empty subsets +contribute zero. Thus induction proves the assertion for all sizes +and ranks. The selection of $q$ from $R_\eta$ is nonquantitative; +this argument does not provide a numerical value of $\varepsilon$. +\end{proof} + +Proposition~\ref{prop:packing-to-power} and Theorem~\ref{thm:activity-packing} +prove Theorem~\ref{thm:all-ranks}. It remains to transfer its middle-rank +case to the configurations of Theorem~\ref{thm:halving-power}. + +For a generic configuration of $N=2m$ points, a pair is halving +precisely when its critical slope ties positions $m$ and $m+1$: +there are then $m-1$ points on each side of its line. The two tied +points exchange order, so halving pairs are exactly the middle-rank +switches. Each unordered pair has one critical slope in this sweep. +The perturbation argument in Section~\ref{sec:finite} preserves all +halving pairs of an arbitrary configuration with no three collinear. Consequently +Proposition~\ref{prop:packing-to-power}, applied to Theorem~\ref{thm:activity-packing}, gives the asserted bound for every even +size and every such configuration. + +\subsection{Planar \texorpdfstring{$k$}{k}-sets and straight-line levels} +\label{sec:k-sensitive-proof} + +We finish by deriving the rank-sensitive consequence stated in the +introduction. The local ranks created by a shallow cutting can vary from +cell to cell, which is why the uniform sweep theorem was needed. + +\begin{proof}[Proof of Corollary~\ref{cor:k-sensitive}] +Take $\varepsilon$ from Theorem~\ref{thm:all-ranks} and set +$\varepsilon_0=\min\{\varepsilon,1/6\}$ and +$\beta=4/3-\varepsilon_0>1$. The theorem remains valid with exponent +$\beta$ in place of $4/3-\varepsilon$. + +Dualize a point $(a,b)$ to the line $y=b-as$. For a generic dual point +set $P$, let $V_j$ be the arrangement vertices with exactly $j$ lines +strictly below them. Then $|V_j|=S_{j+1}(P)$: at such a vertex the lines +in positions $j+1$ and $j+2$ exchange order. The vertices of +$\Lambda_j$ are $V_{j-1}\cup V_j$, with out-of-range sets empty +\cite[Section~1, p.~2]{AgarwalAronovChanSharir1998}. Thus +Theorem~\ref{thm:all-ranks} bounds every level of $M$ dual lines by +$O(M^\beta)$ vertices, uniformly in its rank; its number of edge pieces +is one more than its number of vertices. For any line arrangement in +the corollary, the dual point set has no three collinear and distinct +first coordinates. The perturbation in Section~\ref{sec:finite} can +also preserve the order of those coordinates, so its preservation of +all triple-orientation signs preserves every dual vertex's number of +lines below it. The uniform $O(M^\beta)$ level bound therefore holds +for every such general-position line arrangement. + +Apply Matou\v{s}ek's shallow-cutting lemma in the form stated by +Agarwal, Aronov, Chan, and Sharir +\cite[Lemma~3.2 and the following reduction, pp.~10--11]{AgarwalAronovChanSharir1998}. +In dimension two, take $r=n/(k+1)\ge1$ and +$q=kr/n+1<2$. The cutting has $O(rq)=O(n/(k+1))$ cells, each crossed +by at most $n/r=k+1$ lines. Within the interior of a cell, the +noncrossing lines give a fixed rank offset, so the part of the global +level has one fixed local rank in the arrangement of crossing lines. +This local rank need not +equal $k$, which is why the all-ranks bound is used. Boundary pieces +cost $O(k+1)$ per cell: each crossing line meets the simplex boundary +at most twice. An input line not crossing the interior can contribute +only along a side or at a vertex. General position allows at most one +such line per side and at most two through each vertex; the already +counted intersections split their contributions into $O(k+1)$ pieces +in total. Since $\beta>1$, these terms are absorbed by the local +$O((k+1)^\beta)$ bound. The cited shallow-cutting reduction now gives +\[ + \operatorname{comp}\bigl(\Lambda_k(\mathcal L)\bigr) + =O\bigl(r(n/r)^\beta\bigr) + =O\bigl(n(k+1)^{1/3-\varepsilon_0}\bigr). +\] + +For a generic point set $Q$, every $k$-set consists of the $k$ smallest +values of some linear functional. Normal directions with positive +second coordinate give the bottom-$k$ sweep, while those with negative +second coordinate give complements of the bottom-$(n-k)$ sweep. +The horizontal directions are included by limits, since first +coordinates are distinct. Hence the number of $k$-sets is at most +$S_k(Q)+S_{n-k}(Q)+2$. In the dual arrangement, +$S_k(Q)=|V_{k-1}|$; vertical reflection sends the vertices counted by +$S_{n-k}(Q)=|V_{n-k-1}|$ to $V_{k-1}$ of the reflected arrangement. +Each of these two vertex sets +lies on a $k$-th level, so the level bound just proved applies twice. +Finally, for any fixed point set, choose one strict separating line for +each of its finitely many $k$-sets. All these strict separations survive +a sufficiently small generic perturbation. Its original number of +$k$-sets is therefore no larger than the generic count. +\end{proof} diff --git a/preprints/A-power-saving-for-planar-halving-lines-September-25-2026/build/sections/spatial.tex b/preprints/A-power-saving-for-planar-halving-lines-September-25-2026/build/sections/spatial.tex new file mode 100644 index 0000000000000000000000000000000000000000..053c555901d0aee9b85587cf1e3d52047ffb6c62 --- /dev/null +++ b/preprints/A-power-saving-for-planar-halving-lines-September-25-2026/build/sections/spatial.tex @@ -0,0 +1,267 @@ +\section{A spatial measure with dimension greater than seven quarters} +\label{sec:spatial} + +The boundary argument will count participating vertices, so it needs a +strip estimate for vertex mass while retaining a positive amount of edge +mass. We now construct a single Euclidean limit with these two properties. +The metric estimate from the preceding section concerns one fixed rank +point and radius at a time. A Hausdorff limit of graphs will allow us to +use it with configuration thresholds that still depend on the point +and radius. + +Write $D=(0,1)^2$, and let $\pi:Z\to[0,1]^2$ be the projection onto the +rank coordinates $(z_A,z_B)$. Denote the inverse rank maps, in the common +Euclidean normalization, by $L_n:D\to\mathbb R^2$. These are +homeomorphisms. For $\overline B(p,h)\subset D$, put +\[ + U_n(p,h)=\max_{|u-p|=h}|L_n(u)-L_n(p)|, + \qquad + V_n(p,h)=\min_{|u-p|=h}|L_n(u)-L_n(p)|. +\] +The input from the preceding section has the following exact quantifiers. +For each selected $p$, there is $C_p<\infty$ such that, for every integer +$T\ge2$, all sufficiently small fixed $h>0$ satisfy, eventually in $n$, +\begin{equation} + V_n(p,h/T)\ge\frac{U_n(p,h)}{C_pT}, + \qquad + U_n(p,h/T)\le\frac{C_pV_n(p,h)}{T}. + \label{eq:spatial-metric-input} +\end{equation} +The constant $C_p$ is independent of $T$. The allowed radii may depend on +$p,T$, and the eventual index may depend on $p,T,h$. + +We use the positive good part already selected above: a Borel set +$E\subset Z$ with $\lambda_{K_0}(E)>0$, whose restricted endpoint measure +has absolutely continuous pair-coordinate projection, and for which +\eqref{eq:spatial-metric-input} holds at every $p\in\pi(E)$ after deleting +a null set. We also retain $\lambda\ll\mu$. + +The absolutely continuous projection here belongs to edge mass +$\lambda|_E$, not yet to the vertex mass $\mu|_E$ whose strip mass we +must bound. We will discard labels over a null set carrying the singular +part of the vertex projection and restrict its density to a bounded +range. The first deletion loses no selected edge mass, by absolute +continuity of its projection, and some finite density bound still +retains positive $\lambda_{K_0}$-mass. The metric construction will +then transfer this vertex-density bound from +rank coordinates to Euclidean disks and strips. + +\begin{proposition}\label{prop:spatial-limit} +There is a Borel set $G\subset E$ with $\lambda_{K_0}(G)>0$, a further +subsequence, and a translation and uniform rescaling of each normalized +Euclidean plane with the following properties. Over every +$p\in\pi(G)$ the limiting graph of the maps $L_n$ has a unique position +$l(p)$ in the closed unit disk. With $\eta=1/16$, there is $c_G>0$ such +that +\begin{equation} + |l(p)-l(q)|\ge c_G|p-q|^{1+\eta} + \qquad(p,q\in\pi(G)). + \label{eq:inverse-holder} +\end{equation} +Attach Euclidean position as an additional compactified coordinate to the +vertex measures, and take a weak limit $\widetilde\mu$. Its restriction +by the old-label condition $z\in G$ has position marginal $\nu_G$ +satisfying +\begin{equation} + \nu_G(B(x,r))\le C_G r^{d_*}, + \qquad d_*=\frac{2}{1+\eta}=\frac{32}{17}, + \qquad x\in\mathbb R^2,\quad 0\frac74. +\] +This is an auxiliary spatial exponent; it is not the exponent saving in +the final halving-line bound. diff --git a/preprints/A-power-saving-for-planar-halving-lines-September-25-2026/build/sections/zoom.tex b/preprints/A-power-saving-for-planar-halving-lines-September-25-2026/build/sections/zoom.tex new file mode 100644 index 0000000000000000000000000000000000000000..12048343466a3ed40b226bd324a128d930a2b0b9 --- /dev/null +++ b/preprints/A-power-saving-for-planar-halving-lines-September-25-2026/build/sections/zoom.tex @@ -0,0 +1,271 @@ + +\section{Three transverse foliations determine the zoom}\label{sec:zoom} + +Exact preservation of three transverse families of parallel lines is +an affine rigidity phenomenon; see Artstein-Avidan and Slomka +\cite{ArtsteinSlomka2017}. We need a compactness statement that also +rules out collapse before a limit map has been obtained. +The following elementary lemma prevents a change of coordinates from +collapsing at the scale under consideration. Its monotonicity hypothesis +is imposed before any fixed linear change of the target coordinates. +For a vector $v$, write $[v]$ for its unoriented line through the origin. + +\begin{lemma}[Three-foliation compactness]\label{lem:three-foliation} +Let $\Omega_j\subset\mathbb R^2$ be open sets containing every compact +subset of $\mathbb R^2$ for all sufficiently large $j$. Let +$F_j:\mathbb R^2\longrightarrow\Omega_j$ be homeomorphisms. Put +$v_1=(1,0)$, $v_2=(0,1)$ and $v_3=(1,1)$. Suppose the following hold. +\begin{enumerate} +\item There is a fixed invertible matrix $P$ such that each coordinate +of $P F_j$ is monotone on each line parallel to $v_i$. For each $i$, +the two monotonicity signs are independent of the line and of $j$. +\item Along every subsequence, if $x_j,y_j$ lie on one line parallel to $v_i$ and +$F_j(x_j)\to a$, $F_j(y_j)\to b$, then $b-a\in\mathbb R v_i$. +The points and their common line may depend on $j$. +\item $F_j(0)=0$ and $|F_j(v_1)|=1$. +\end{enumerate} +After passage to a subsequence there is $\sigma\in\{-1,1\}$ such that +$F_j\to \sigma\operatorname{id}$ uniformly on compact sets. The inverse +maps converge to $\sigma\operatorname{id}$ uniformly on compact sets as +well. +\end{lemma} + +\begin{proof} +We first control images of entire lines and intersections between +families. Those intersections then determine the map on rational grids; coordinate +monotonicity will extend this control to compact sets. The final step +will control the inverse maps. + +To begin, we show that entire line images, including lines with moving base +points, converge to the prescribed target lines. Fix a family $i$ and +points $x_j$ with $a_j=F_j(x_j)\to a$. Orient the source lines in the +direction $v_i$. Their forward target displacements belong to a fixed +closed pointed cone $Q_i$, the inverse under $P$ of the signed coordinate +quadrant specified by the first hypothesis. Choose a linear functional +$\ell_i$ strictly positive on $Q_i\setminus\{0\}$. There is $M_i<\infty$ +such that +\[ + |q|\le M_i\ell_i(q)\qquad(q\in Q_i). +\] +Consequently $s=\ell_i(F_j(x)-a_j)$ is a strictly increasing continuous +parameter on the image of this oriented source line: injectivity rules +out equality at two points. In this parameter its image $\gamma_j$ is +$M_i$-Lipschitz. + +The parameter intervals exhaust $\mathbb R$. Indeed, for any fixed +$R$, a ball containing all points at distance at most $M_iR+1$ from +$a_j$ lies in $\Omega_j$ for large $j$. The preimage of its closure is +compact, so each source ray eventually leaves that ball. The cone +inequality forces its parameter to exceed $R$ in the positive direction +or to be less than $-R$ in the negative direction. This proves the +claim without any control of the Euclidean locations $x_j$. + +At parameter $s=1$, a convergent subsequence gives a nonzero vector +in $Q_i\cap\mathbb R v_i$: its $\ell_i$ value is one. Thus this +intersection contains a ray, and pointedness makes the ray unique. +Compactness of equi-Lipschitz curves and the second hypothesis now imply +\[ + \gamma_j(s)\longrightarrow a+s\frac{d_i}{\ell_i(d_i)} + \quad\hbox{locally uniformly in }s, +\] +where $d_i$ is the unit vector on $\mathbb R v_i$ belonging to $Q_i$. +There is exactly one such ray; $\ell_i(d_i)>0$. The formula also shows +that the limit is unique, so no additional subsequence is needed for +this assertion. In particular the line image traces the entire +limiting line in both directions. + +It follows that intersections of two different families converge. +More precisely, let two source lines of types $i\ne k$ have base +images tending to $a$ and $b$. The two limiting lines intersect at a +unique point $c$. In a small parallelogram around $c$ with sides +parallel to these limiting directions, the preceding parameterized +convergence produces arcs close to the two transverse diameters and +joining their respective opposite sides. They intersect by planar +separation, in the opposite-sides form of +\cite[Lemma~2.11]{PascolettiZanolin2010}. The source lines have exactly one intersection, and their +images do also because $F_j$ is a homeomorphism. Hence this unique +intersection is inside every sufficiently small such parallelogram +eventually, proving convergence to $c$. The argument allows $a=b$; +transversality concerns the two line types, not the two base points. + +We have controlled intersections even when their base points move. +We next use those intersections to determine the scale at every rational +point, starting from the normalized pair $0,v_1$. + +Pass to a subsequence so $F_j(v_1)$ converges. Its limit is +$\sigma v_1$ with $\sigma=\pm1$ by the second and third hypotheses. +Starting with $0,v_1$, intersections of horizontal, vertical and +slope-one lines generate the integer lattice. For example the +diagonal through $0$ meets the vertical through $v_1$ at $(1,1)$; +its horizontal meets the vertical through $0$ at $(0,1)$. +The diagonal through $v_1$ meets that vertical at $(0,-1)$. +These operations and their reverses give each integer step and then +every lattice intersection; Figure~\ref{fig:three-foliation-grid} shows +the first steps. Intersection convergence proves +$F_j(z)\to\sigma z$ at all integer lattice points. + +\begin{figure}[htbp] +\centering +\begin{tikzpicture}[x=1.5cm,y=1.5cm,font=\small] + \draw[gray!65] (-.3,0)--(1.4,0); + \draw[gray!65] (0,-1.2)--(0,1.3); + \draw[gray!65] (1,-1.2)--(1,1.3); + \draw[gray!65,dashed] (-.25,1)--(1.4,1); + \draw[gray!65,dashed] (-.25,-1)--(1.4,-1); + \draw[thick] (0,0)--(1,1); + \draw[thick] (0,-1)--(1,0); + \fill (0,0) circle (2.1pt); + \fill (1,0) circle (2.1pt); + \draw[fill=white,thick] (1,1) circle (2.1pt); + \draw[fill=white,thick] (0,1) circle (2.1pt); + \draw[fill=white,thick] (0,-1) circle (2.1pt); + \node[below left] at (0,0) {$0$}; + \node[below right] at (1,0) {$v_1$}; + \node[above right] at (1,1) {$v_3=(1,1)$}; + \node[above left] at (0,1) {$v_2=(0,1)$}; + \node[below left] at (0,-1) {$-v_2=(0,-1)$}; + \node[align=left,anchor=west] at (1.9,.55) + {\textbullet\ Initial pair $0,v_1$\\[3pt] + $\circ$\ Points obtained by intersections}; +\end{tikzpicture} +\caption{The source-plane intersection construction using horizontal, +vertical and slope-one lines. The diagonal through $0$ meets the vertical +through $v_1$ at $v_3$; its horizontal gives $v_2$. The diagonal through +$v_1$ meets the vertical through $0$ at $-v_2$. Convergence of the image +intersections is proved in the text.} +\label{fig:three-foliation-grid} +\end{figure} + +Fix $m\ge1$. By the original two coordinate monotonicities, +$P F_j(v_1/m)$ is in the coordinate rectangle with vertices $0$ and +$P F_j(v_1)$, so it is bounded. Every subsequential limit is $b v_1$ +by the second hypothesis. Apply the same finite intersection +construction with seed $v_1/m$ in place of $v_1$. It constructs +$v_1$ after $m$ additions, and intersection convergence shows that +its limiting image is $mbv_1$. Thus $mb=\sigma$. +This argument is valid even if one initially supposes $b=0$, since +every intersection is between two different line types. Therefore +$F_j(v_1/m)\to\sigma v_1/m$ without a noncollapse assumption. +Repeating the grid construction proves convergence on +$(m^{-1}\mathbb Z)^2$, and hence at every rational point. + +Finally squeeze each coordinate of $P F_j$ between the appropriate +corners of a rational grid cell. The signs can differ between the +two coordinates and the two source axes; choose the lower and upper +corners for each coordinate separately. The limiting corner values +are those of $\sigma P x$, whose oscillation on a cell is bounded by +a constant times its mesh. A finite grid covering a compact set +therefore proves uniform convergence there. This uses monotonicity +of $P F_j$, not an unjustified monotonicity of $F_j$. + +For inverse convergence, choose a circle sufficiently large to contain +a given compact target set with positive margin. Uniform convergence +on that circle and the Jordan curve theorem +(see \cite{Siebenmann2005}) show that its image +encloses the target set for large $j$; this image bounds the image of +the source disk. The inverse images of the target set are thus +uniformly bounded. Uniform convergence of $F_j$ on that disk gives +uniform convergence of the inverses. +\end{proof} + +\subsection{Application to the rank coordinates} + +Fix a rank point $p$ where all functions $f_C$ are differentiable and +where the directional conclusion already proved holds. Choose three +times $\tau_1,\tau_2,\tau_3$ whose allowed direction sets are singletons +and are pairwise projectively distinct. Let their unit directions +be $d_1,d_2,d_3$. Fix an invertible matrix $B$ carrying +$[d_1],[d_2],[d_3]$ to $[v_1],[v_2],[v_3]$. + +Consider any radii $h_j\downarrow0$ and indices tending sufficiently +fast that, on every bounded set of $w$ and for every fixed $C$, +\begin{equation}\label{eq:zoom-expansion} + r_{C,j}(p+h_jw)=f_C(p)+h_j\nabla f_C(p)\cdot w+o(h_j). +\end{equation} +Here $r_{C,j}$ denotes the $C$ coordinate as a function of the pair +of rank coordinates, and the error is uniform on the bounded set. +This diagonal condition follows from monotone squeezing at continuity +points and differentiability at $p$. It asserts no uniform rate in +the configuration index. + +Normalize the original Euclidean plane linearly so its three +$\tau_i$ directions become $[v_i]$, translate the preimage of $p$ to +$0$, and scale the whole plane so the image of $v_1$ under the rank +zoom $B(r-p)/h_j$ has norm one. Such a scale exists: on the chosen +horizontal ray the rank zoom is continuous, starts at zero, and exits +every fixed ball contained in its expanding target domain. +Let $F_j$ be this map from the normalized Euclidean plane to rank +zoom coordinates. + +The first hypothesis of Lemma~\ref{lem:three-foliation} holds with +$P=B^{-1}$, by the two original rank-coordinate monotonicities; +passing to a subsequence fixes any orientation signs. Its second +hypothesis follows from \eqref{eq:zoom-expansion}: every bounded limiting displacement on +a $\tau_i$ line satisfies all its directional sign constraints, so +it is parallel to $B d_i$, whether or not its base point moves. +The target domains exhaust the plane since $p$ is interior to the +rank square. The lemma therefore applies. In this normalization +the zoom and its inverse converge locally uniformly to a nonsingular +linear map; with the stated choice they converge to $\sigma I$. + +We must compare this point-dependent normalization with the common +normalization based on three fixed times $t_1K_p$. For each integer $T\ge2$, there exists +$h_{p,T}>0$ such that for every fixed $0=stealth] + \draw[->] (-0.5,0)--(10.7,0) node[right] {$s$}; + \fill[blue!13] (0,0.15) rectangle (3,0.55); + \fill[black!9] (3,0.15) rectangle (6,0.55); + \fill[blue!13] (6,0.15) rectangle (9,0.55); + \draw[blue!65!black,thick] (0,0.35)--(3,0.35); + \draw[blue!65!black,thick] (6,0.35)--(9,0.35); + \node at (1.5,0.95) {$I_-$}; + \node at (4.5,0.95) {different bits}; + \node at (7.5,0.95) {$I_+$}; + \foreach \x/\lab in {0/{b-T},3/{b},6/{b'},9/{b'+T}} { + \draw (\x,-0.08)--(\x,0.1); + \node[below] at (\x,-0.1) {$\lab$}; + } + \draw[orange!75!black,very thick] (1.2,-1.0)--(7.8,-1.0); + \draw[orange!75!black] (1.2,-0.85)--(1.2,-1.15); + \draw[orange!75!black] (7.8,-0.85)--(7.8,-1.15); + \node[below] at (4.5,-1.1) {central window $J$}; + \draw[densely dotted] (3,-0.65)--(3,-1.05); + \draw[densely dotted] (6,-0.65)--(6,-1.05); +\end{tikzpicture} diff --git a/preprints/A-power-saving-for-planar-unit-distances-September-23-2026/build/figures/strict-crosses.tex b/preprints/A-power-saving-for-planar-unit-distances-September-23-2026/build/figures/strict-crosses.tex new file mode 100644 index 0000000000000000000000000000000000000000..4d48a720f184abb09aff8f9fb4f69fa3968149da --- /dev/null +++ b/preprints/A-power-saving-for-planar-unit-distances-September-23-2026/build/figures/strict-crosses.tex @@ -0,0 +1,27 @@ +\begin{figure}[tb] +\centering +\begin{tikzpicture}[x=1.45cm,y=1.15cm] + \draw[gray!45] (-.55,0)--(2.55,0) (-.55,2)--(2.55,2); + \draw[gray!45] (0,-.5)--(0,2.5) (2,-.5)--(2,2.5); + \node[draw,circle,fill=white,inner sep=2.2pt] (y1) at (0,0) {}; + \node[draw,circle,fill=white,inner sep=2.2pt] (y2) at (2,2) {}; + \node[circle,fill=blue!65!black,inner sep=2.5pt] (x1) at (2,0) {}; + \node[circle,fill=blue!65!black,inner sep=2.5pt] (x2) at (0,2) {}; + \node[below left=3pt of y1] {\(y_1=(a,b)\)}; + \node[above right=3pt of y2] {\(y_2=(a',b')\)}; + \node[below right=3pt of x1] {\((a,b')\)}; + \node[above left=3pt of x2] {\((a',b)\)}; + \node[rotate=90,fill=white,inner sep=2pt] at (0,1) {\(Y=0\)}; + \node[rotate=90,fill=white,inner sep=2pt] at (2,1) {\(Y=0\)}; + \node[fill=white,inner sep=2pt] at (1,0) {\(Y=1\)}; + \node[fill=white,inner sep=2pt] at (1,2) {\(Y=1\)}; +\end{tikzpicture} +\caption{Two candidate center states differing in both coordinates admit +exactly two point states in their common strict cross. The filled points are +those opposite corners. Within this four-state configuration, the bit of a +consistent edge and the state at either endpoint determine the state at +the other endpoint. Rows have fixed first label and columns have fixed +second label. The diagram displays coordinate +equalities of states, not geometric positions of the original plane points.} +\label{com:fig-crosses} +\end{figure} diff --git a/preprints/A-power-saving-for-planar-unit-distances-September-23-2026/build/main.tex b/preprints/A-power-saving-for-planar-unit-distances-September-23-2026/build/main.tex new file mode 100644 index 0000000000000000000000000000000000000000..9e3a287bf54d839cbd92f11121241d0491b194ec --- /dev/null +++ b/preprints/A-power-saving-for-planar-unit-distances-September-23-2026/build/main.tex @@ -0,0 +1,62 @@ +\documentclass[11pt]{article} +\usepackage[T1]{fontenc} +\usepackage{lmodern} +\usepackage[margin=1.05in]{geometry} +\usepackage{amsmath,amssymb,amsthm,mathtools} +\usepackage{microtype,booktabs,enumitem} +\usepackage{needspace} +\usepackage{tikz} +\usetikzlibrary{arrows.meta,calc,positioning,patterns} +\usepackage[colorlinks=true,linkcolor=blue!45!black,citecolor=blue!45!black,urlcolor=blue!45!black]{hyperref} +\usepackage[capitalise,noabbrev,nameinlink]{cleveref} +\ifdefined\pdftrailerid\pdftrailerid{}\fi +\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} +\newtheorem{example}[theorem]{Example} +\newcommand{\R}{\mathbb R} +\newcommand{\C}{\mathbb C} +\newcommand{\Q}{\mathbb Q} +\newcommand{\Z}{\mathbb Z} +\newcommand{\N}{\mathbb N} +\newcommand{\E}{\mathbb E} +\newcommand{\Prob}{\mathbb P} +\newcommand{\1}{\mathbf 1} +\newcommand{\dd}{\,\mathrm d} +\newcommand{\abs}[1]{\lvert #1\rvert} +\newcommand{\norm}[1]{\lVert #1\rVert} +\newcommand{\set}[1]{\{#1\}} +\newcommand{\ip}[2]{\langle #1,#2\rangle} +\DeclareMathOperator{\supp}{supp} +\DeclareMathOperator{\Var}{Var} +\DeclareMathOperator{\Cov}{Cov} +\DeclareMathOperator{\KL}{D_{\mathrm{KL}}} +\DeclareMathOperator{\TV}{TV} +\numberwithin{equation}{section} +\setlist[enumerate]{itemsep=3pt,topsep=5pt} +\setlist[itemize]{itemsep=3pt,topsep=5pt} +\title{A power saving for planar unit distances} +\author{OpenAI} +\hypersetup{pdftitle={A power saving for planar unit distances},pdfauthor={OpenAI}} +\date{September 23, 2026} +\AddToHook{cmd/thebibliography/after}{\addcontentsline{toc}{section}{\refname}} +\begin{document} +\maketitle +\begin{abstract} +We prove a power saving for the planar unit-distance problem: for some absolute \(\beta<4/3\), every set of \(n\) points in the Euclidean plane determines \(O(n^\beta)\) unordered pairs at unit distance. +\end{abstract} +\input{sections/00-introduction} +\input{sections/01-incidence} +\input{sections/02-prediction} +\input{sections/03-heights} +\input{sections/04-budgets} +\input{sections/05-common-levels} +\input{sections/06-extraction} +\input{sections/07-algebra} +\bibliographystyle{plain} +\bibliography{references} +\end{document} diff --git a/preprints/A-power-saving-for-planar-unit-distances-September-23-2026/build/references.bib b/preprints/A-power-saving-for-planar-unit-distances-September-23-2026/build/references.bib new file mode 100644 index 0000000000000000000000000000000000000000..bebb33aac3a787bf1f2c842168977350d2c9a932 --- /dev/null +++ b/preprints/A-power-saving-for-planar-unit-distances-September-23-2026/build/references.bib @@ -0,0 +1,272 @@ +@article{erdos1946, + author = {Erd{\H{o}}s, P.}, + title = {On sets of distances of {$n$} points}, + journal = {The American Mathematical Monthly}, + volume = {53}, + number = {5}, + pages = {248--250}, + year = {1946}, + month = may, + doi = {10.1080/00029890.1946.11991674}, + url = {https://users.renyi.hu/~p_erdos/1946-03.pdf} +} + +@incollection{sst1984, + author = {Spencer, J. and Szemer{\'e}di, E. and Trotter, Jr., W. 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M. and Szemer{\'e}di, E.}, + title = {Crossing-free subgraphs}, + booktitle = {Theory and Practice of Combinatorics}, + editor = {Rosa, Alexander and Sabidussi, Gert and Turgeon, Jean}, + series = {North-Holland Mathematics Studies}, volume = {60}, + publisher = {North-Holland}, address = {Amsterdam}, + pages = {9--12}, year = {1982}, + doi = {10.1016/S0304-0208(08)73484-4} +} + +@book{leighton1983, + author = {Leighton, F. Thomson}, + title = {Complexity Issues in {VLSI}: Optimal Layouts for the Shuffle-Exchange Graph and Other Networks}, + series = {Foundations of Computing}, + publisher = {MIT Press}, address = {Cambridge, MA}, year = {1983}, + isbn = {9780262121040}, + url = {https://mitpress.mit.edu/9780262121040/complexity-issues-in-vlsi/} +} + +@article{agostonPalvolgyi2022, + author = {{\'A}goston, P{\'e}ter and P{\'a}lv{\"o}lgyi, D{\"o}m{\"o}t{\"o}r}, + title = {An improved constant factor for the unit distance problem}, + journal = {Studia Scientiarum Mathematicarum Hungarica}, + volume = {59}, number = {1}, pages = {40--57}, year = {2022}, + doi = {10.1556/012.2022.01517}, + eprint = {2006.06285}, archivePrefix = {arXiv}, + url = {https://arxiv.org/abs/2006.06285v3} +} + +@misc{katzSilier2026, + author = {Katz, Nets and Silier, Olivine}, + title = {Structure of cell decompositions in extremal {Szemer{\'e}di--Trotter} examples}, + year = {2026}, eprint = {2303.17186}, archivePrefix = {arXiv}, + url = {https://arxiv.org/abs/2303.17186v2}, + note = {Version 2, 1 June 2026; first submitted 2023}, + howpublished = {\url{https://arxiv.org/abs/2303.17186v2}} +} + +@inproceedings{pachRazSolymosi2026, + author = {Pach, J{\'a}nos and Raz, Orit E. and Solymosi, J{\'o}zsef}, + title = {{Erd{\H o}s}'s unit distance problem and rigidity}, + booktitle = {42nd International Symposium on Computational Geometry (SoCG 2026)}, + editor = {Ahn, Hee-Kap and Hoffmann, Michael and Nayyeri, Amir}, + series = {Leibniz International Proceedings in Informatics}, + volume = {367}, pages = {83:1--83:9}, year = {2026}, + publisher = {Schloss Dagstuhl--Leibniz-Zentrum f{\"u}r Informatik}, + doi = {10.4230/LIPIcs.SoCG.2026.83}, + url = {https://doi.org/10.4230/LIPIcs.SoCG.2026.83} +} + +@article{Tardos2003, + author = {Tardos, G{\'a}bor}, + title = {On distinct sums and distinct distances}, + journal = {Advances in Mathematics}, + volume = {180}, number = {1}, pages = {275--289}, year = {2003}, + doi = {10.1016/S0001-8708(03)00004-5}, + url = {https://www.renyi.hu/~tardos/distinctsums.pdf} +} + +@article{golodShafarevich1964, + author = {Golod, E. S. and Shafarevich, I. R.}, + title = {On the class field tower}, + journal = {Izvestiya Akademii Nauk SSSR. Seriya Matematicheskaya}, + volume = {28}, number = {2}, pages = {261--272}, year = {1964}, + note = {Russian; English translation in American Mathematical Society Translations, + Series 2, 48 (1965), 91--102}, + url = {https://www.mathnet.ru/eng/im2955} +} diff --git a/preprints/A-power-saving-for-planar-unit-distances-September-23-2026/build/sections/00-introduction.tex b/preprints/A-power-saving-for-planar-unit-distances-September-23-2026/build/sections/00-introduction.tex new file mode 100644 index 0000000000000000000000000000000000000000..9aa7ef1e02d2c940061b5a7ad84d3f385ae6d0d9 --- /dev/null +++ b/preprints/A-power-saving-for-planar-unit-distances-September-23-2026/build/sections/00-introduction.tex @@ -0,0 +1,210 @@ +\section{Introduction}\label{sec:introduction} + +For a finite set \(X\subset\R^2\), write +\[ + u(X)=\#\bigl\{\{x,y\}\subset X:\ \norm{x-y}_2=1\bigr\}, + \qquad + u(n)=\max_{\substack{X\subset\R^2\\ |X|=n}}u(X). +\] +Thus pairs are unordered and their endpoints are distinct. We prove a fixed +power saving over the exponent \(4/3\). + +\begin{theorem}\label{thm:main} +There are absolute constants \(00\). +All constants are independent of the configuration. + +\subsection{The unit-distance problem and earlier methods} + +Erd\H{o}s introduced the repeated-distance problem in 1946 +\cite{erdos1946}. Rescaling a configuration turns any specified positive +distance into a unit distance, so the problem concerns the largest possible +multiplicity of one distance. His lattice construction gives the lower bound +\(n^{1+c/\log\log n}\) for a positive constant \(c\) and sufficiently large +\(n\) \cite[Theorem~2]{erdos1946}. +Spencer, Szemer\'edi and Trotter established the upper bound +\(u(n)=O(n^{4/3})\) in 1984 \cite{sst1984}. +Sz\'ekely gave a crossing-number proof and a general incidence theorem +for families of curves \cite[Theorems~4 and~8]{Szekely1997}. +The crossing inequality was developed independently by Ajtai, +Chv\'atal, Newborn and Szemer\'edi and by Leighton +\cite{ajtaiEtAl1982,leighton1983}. +Sz\'ekely's incidence theorem applies to points and unit circles using two elementary +properties: two unit circles meet in at most two points, and two points +belong to at most two unit circles. +Later work improved the leading constant +\cite{agostonPalvolgyi2022} and described the structure of configurations +near the extremal incidence scale \cite{katzSilier2026}. +Pach, Raz and Solymosi connected the problem to rigidity and obtained +a logarithmic improvement conditional on a rigidity conjecture +\cite[Conjecture~7 and Theorem~8]{pachRazSolymosi2026}. + +The historical unit-distance conjecture of Erd\H{o}s proposed the stronger +bound \(u(n)=n^{1+o(1)}\). An OpenAI construction disproved it by giving +at least \(n^{1+\varepsilon}\) unit distances, for some fixed +\(\varepsilon>0\), along an unbounded sequence of sizes +\cite[Theorem~1.1]{openaiUnitLower2026}. Alon et al.\ gave a simplified +account and explained the construction's number-theoretic ancestry +\cite[Sections~1.2--1.3]{alonEtAl2026}. They credit the class-group +ideas to Michel, Soundararajan, Ellenberg and Venkatesh. The construction +combines these ideas with Golod--Shafarevich +number-field towers having prescribed splitting, including the +tower-cutting method of Hajir, Maire and Ramakrishna +\cite{ellenbergVenkatesh2007,golodShafarevich1964,hajirMaireRamakrishna2021}. +Sawin sharpened the construction to \(u(n)\gg n^{1.014114}\) along an +unbounded sequence of sizes \cite[Theorem~1]{sawin2026}. +These are construction lower bounds. Theorem~\ref{thm:main} bounds every +configuration from above, leaving a gap between the known exponents. + +Our proof begins with the same point--circle incidence interpretation. +Random cuttings isolate a piece of the incidence graph with controlled +degrees; their sampled-cell construction belongs to the framework of +Clarkson and Clarkson--Shor \cite{Clarkson1987,ClarksonShor1989}. +The next ingredient is information: two points sampled as neighbors of +one center share little information. Entropy methods already played a +role in the distinct-distance estimates of Tardos and Katz--Tardos +\cite{Tardos2003,KatzTardos}. In particular, the conditional-copy +argument of Katz--Tardos retains the marginal law when resampling +within a class. Here that principle enters a short-history prediction +lemma whose tests may depend on a neighboring center. Its exact +conditional laws and the independence of the fresh test are proved in +\Cref{sec:prediction}. + +The arithmetic input is the normalized product formula +\cite{ProductFormula}. We use the associated absolute height of +algebraic numbers: the weighted sum of the positive parts of their +logarithmic absolute values. The product formula converts +relations between unit-edge jumps into bounds summed over all absolute +values. The proof then combines these bounds with prediction and, at its +last stage, algebraic independence and quadratic extensions. The +intermediate statements connecting probability, height and algebra are +proved in the paper. + +\subsection{The proof in outline} + +Suppose that no fixed power saving exists. Set \(t=n^{1/3}\) along a sequence +of counterexamples, so their ordered unit pairs number \(t^{4-o(1)}\). +The following steps produce a contradiction. + +\paragraph{An incidence graph and its sampling laws.} +Use two copies of the configuration as points and unit-circle centers. +An edge joins a point to a center at unit distance. Algebraic +specialization preserves these equations and distinctness. In +\Cref{sec:incidence} we restrict this graph to a piece with controlled +degrees and an expansion estimate that limits the fraction of edges +captured by small vertex subsets. Throughout, a random edge is uniform +on the retained graph, so its vertex marginals are proportional to +degrees. A \emph{wedge} is sampled by choosing a center with this +marginal and then two of its neighbors independently. Its endpoint +mutual information---the relative entropy of their joint law from the +product of their marginals---is \(o(\log t)\). + +\paragraph{Prediction from a short history.} +A test assigns a finite code to each point. A history records its codes +at a short list of previous tests. Given an original point, sample a +prediction from the point marginal conditional on having the same +history, and sample a neighboring center from the original point. +The prediction keeps the original point marginal. The information +bound allows a history to be fixed for which the prediction's answers +are sufficiently compatible with the sampled center at a fresh test, +independent of this whole experiment. Section~\ref{sec:prediction} +proves this statement for general finite point--center laws and then +adapts it to the coordinate tests used below. + +\paragraph{Arithmetic scale and the bounded case.} +In the complex coordinates \(z=x+iy\) and \(w=x-iy\), a unit edge +has jumps \(d\) and \(d^{-1}\). At an absolute value \(v\) and a real +threshold \(s\), record the bit \(\1_{\{-\log|d|_v>s\}}\). +Let \(f(v,s)\) be the smaller of the fractions of edges with bit zero +and bit one. The scale \(S\) is the integral of \(f\) over all +thresholds, summed with the normalized place weights. Thus \(S\) +measures how much the logarithmic edge-jump profiles vary. + +If \(S\) stays bounded, the product formula transfers this control +to heights of normalized point coordinates. At a fresh complex +embedding, quantization tests produce a predicted point separated +from the original point, but approximately satisfying the unit +equations at three of its neighbors. The three normalized jumps are +also separated in that embedding. A determinant calculation excludes +this configuration. Section~\ref{sec:heights} proves the required +height bounds and this contradiction; the remaining case is +\(S\to\infty\). + +\paragraph{Common and rare levels.} +The same thresholds give each vertex two labels, recording closeness +in its two coordinates; their pair is called its state. +Except for an integrated error of \(O(1)\), +exactly one label agrees across an edge, and the bit tells us which one. +Call levels common when both bits have appreciable frequency, and +rare when one bit has small frequency. The rare levels with large +consistency error have total length \(o(S)\) and can be removed. +On the remaining rare levels one coordinate label occurs at almost +every vertex; its exceptions are measured by their total level length. + +For two edges sharing a point or sharing a center, assign scores that +reward certain matches of the other endpoints and charge bit +disagreements. The product formula bounds their integrals in +\Cref{sec:budgets}. The prediction lemma and the elementary geometry +of two coordinate labels give lower bounds on common levels, while +the rare-level terms cancel when the two orientations are added. +One step requires care: concentration of the predicted states on a +row (a fixed first label), a column (a fixed second label), or at most +two states must be transferred to the actual point, even when that +support is selected using sampled histories. Lemma~\ref{com:transfer} +proves this comparison under the actual edge and history laws. +Section~\ref{sec:common} concludes that common levels contribute +negligibly to center-wedge variation, but a positive amount of +point-exception mass remains on rare levels. + +\paragraph{Height-weighted extraction.} +The remaining mass provides many endpoint pairs \((p,r)\) of wedges, +each with a chosen center \(q\) and ratio +\(\lambda_{pr}=(z_p-z_q)/(z_r-z_q)\). +The logarithms of these ratios are approximately differences of +profiles attached to the two endpoints. Consequently, on a complete +rectangle of retained pairs, the alternating product +\[ + \frac{\lambda_{pr}\lambda_{p'r'}}{\lambda_{pr'}\lambda_{p'r}} +\] +has small height. Section~\ref{sec:extraction} produces a dense graph +where all individual ratios have height at least \(J/3\), with +\(J\to\infty\), and every rectangle product has height \(o(J)\). +The selection first removes errors in their height-weighted measure, +then chooses a common height band: small exceptional probability alone +would not control lost height. Pairs on any unit circle containing +many points are removed as well. + +\paragraph{An algebraic obstruction.} +Section~\ref{sec:algebra} rules out this dense pair graph. In a field +formed from the sequence, elements of height \(o(J)\) form a +relatively algebraically closed subfield. The rectangle products lie +in this subfield, while the individual ratios are transcendental over +it. We select a fixed complete bipartite pattern on plane or curve +loci. Over the field generated by the defining coefficients of these +loci, its positions have no polynomial relations beyond those required +by the loci. Differentiation over the negligible-height subfield +then gives a finite system involving radicals derived from the +pair-distance identity. + +After adjoining one side's positions and derivative data, a point on +the other side still has the independence required by its locus. +For each radicand we construct, over this enlarged +coefficient field, a valuation that is odd on that radicand and +zero on the others. These valuations permit independent sign changes +of the square roots. An unused sign change contradicts the +differentiated equations. The popular-unit-circle exclusion removes +the case in which the required square-class independence fails. + +The proof excludes every sequence whose unit-distance counts approach +the exponent \(4/3\), establishing the existence of a fixed positive +gap without estimating its size. + +Throughout the proof, asymptotic notation refers to the chosen sequence +\(t\to\infty\). Constants denoted by \(c,C\) may change between occurrences. +All logarithms are natural. Every appeal to a slowly varying parameter +will be made after the quantities it must dominate have been specified. diff --git a/preprints/A-power-saving-for-planar-unit-distances-September-23-2026/build/sections/01-incidence.tex b/preprints/A-power-saving-for-planar-unit-distances-September-23-2026/build/sections/01-incidence.tex new file mode 100644 index 0000000000000000000000000000000000000000..278d93e72bee95aaf7ec1a2c92d7329ef45eef87 --- /dev/null +++ b/preprints/A-power-saving-for-planar-unit-distances-September-23-2026/build/sections/01-incidence.tex @@ -0,0 +1,552 @@ +\section{An incidence piece and its probability laws}\label{sec:incidence} + +The input to this section is a sequence of planar configurations with +\(t=n^{1/3}\to\infty\) and at least \(t^{4-o(1)}\) ordered unit pairs. +We construct an unbalanced incidence graph whose degrees are controlled +on both sides. Its edge law has a uniform expansion estimate, including +for arbitrarily small sets. We then prove two consequences of that +estimate and show that the endpoints of a random two-edge path have +mutual information \(o(\log t)\). + +Such an input sequence exists if \Cref{thm:main} fails. Indeed, for each +\(j\ge2\), failure of a bound with exponent \(4/3-1/j\) allows the +constant in that bound to be chosen large enough to force a counterexample +with \(n\ge j\). Its ordered unit-pair count is at least +\(n^{4/3-1/j}=t^{4-3/j}\). Passing to an increasing sequence of sizes +gives the claimed input. We use separate copies of the configuration as +points and as centers of unit circles, so its ordered unit pairs are +exactly the incidences between these two copies. + +\subsection{Preserving incidences under algebraic specialization} + +Later arguments use number-field absolute values. The following +specialization permits this without losing any of the unit pairs. + +\begin{lemma}\label{inc:specialization} +Let \(x_1,\ldots,x_n\) be distinct points of \(\R^2\), and let +\(\mathcal E\) be any collection of pairs \((i,j)\) for which +\(\norm{x_i-x_j}_2=1\). There are distinct points +\(x'_1,\ldots,x'_n\) with real algebraic coordinates, arbitrarily close +to the respective original points, such that +\(\norm{x'_i-x'_j}_2=1\) for every \((i,j)\in\mathcal E\). +\end{lemma} + +\begin{proof} +Fix a neighborhood of each original point and choose a rational open +rectangle containing that point inside its neighborhood. For variables +\(X_i=(X_{i,1},X_{i,2})\), impose membership in these rectangles, +the equations +\[ + (X_{i,1}-X_{j,1})^2+(X_{i,2}-X_{j,2})^2=1 + \qquad ((i,j)\in\mathcal E), +\] +and the strict inequalities +\[ + (X_{i,1}-X_{j,1})^2+(X_{i,2}-X_{j,2})^2>0 + \qquad (i\ne j). +\] +This finite system has rational coefficients and a real solution, namely +the original configuration. The transfer principle for real closed fields +\cite[Theorems~3.1 and~4.1]{RCFTransfer} gives a solution in the field +of real algebraic numbers. The equations preserve the required unit pairs, +the inequalities preserve distinctness, and the rectangles give the +prescribed closeness. +\end{proof} + +We henceforth apply this lemma separately to each configuration. Extra +unit pairs created by the specialization are harmless; all subsequent +graphs may retain only the incidences under consideration. + +\subsection{Cuttings and the incidence bound} + +For finite sets \(\mathcal P\) of points and \(\mathcal Q\) of distinct +unit-circle centers, let \(I(\mathcal P,\mathcal Q)\) denote the number +of incidences. Two distinct unit circles meet in at most two points. +Also, two distinct points lie on at most two unit circles: their possible +centers are intersections of the two unit circles centered at the points. +These two facts provide the pair-counting bounds used below. +The cutting proof uses the random-sampling principle for regions with +bounded defining data developed by Clarkson \cite[Section~4]{Clarkson1987}. +We give the circle boundary conventions explicitly so that every retained +point belongs to its assigned open region. + +\begin{lemma}\label{inc:cutting} +Let \(\mathcal P\) be a finite planar point set and let \(\mathcal Q\) +be a nonempty set of \(l\) distinct unit circles. For every \(R\ge1\), +sample +\(s=\lceil C R\log(2+l)\rceil\) circles independently and uniformly +with replacement, where \(C\) is a sufficiently large absolute constant. +With probability at least \(9/10\), after deleting the points on sampled +circles, all remaining points can be assigned to at most \(C'(1+s^2)\) +open regions, each of which intersects at most \(l/R\) of the original +circles. Each assigned point belongs to its assigned region. +\end{lemma} + +\begin{proof} +Choose an auxiliary horizontal direction before taking the sample. +Require that an input point and an intersection point of two original +circles have different horizontal coordinates unless they are the same +point. Require also that an input point have a horizontal coordinate +different from that of a horizontal extremum of an original circle, +unless it is that extremum. Only finitely many directions are excluded. +For the latter assertion, if \(c\) is a circle center and \(p\) is a +point, an equality with an extremal abscissa has the form +\(e\cdot(p-c)=\pm1\) for a unit horizontal direction \(e\), and has +at most finitely many solutions in \(e\). This auxiliary direction +does not change the coordinates used elsewhere in the proof. + +Ignore repetitions among the sampled circles. Express each sampled +circle as its upper and lower semicircle graphs on their open horizontal +domains. The event points are their horizontal extrema and all their +pairwise intersections, including tangencies. Between consecutive event +abscissas, the graphs have fixed vertical order; consecutive graphs, +with infinite outer boundaries allowed, bound open gaps. + +Sweep these gaps from left to right. Continue a gap across an event +abscissa precisely when its two boundary graphs and their adjacency +continue unchanged and its closed limiting vertical segment contains no +event point. Otherwise start a new piece on the right. Each resulting +piece is the open region between two fixed semicircle graphs, or infinite +boundaries, over an open interval. It contains no part of a sampled +circle. In particular, a continued gap has no start, end, tangency or +crossing inside it at the abscissa across which it continues. An input +point at an event abscissa must, by the choice of direction, itself be an +event point and therefore lies on a sampled circle. Thus every point not +deleted is in one of the open pieces. + +There are \(O(1+s^2)\) pieces, including in the presence of multiple +intersections or several events with the same abscissa. A new right-hand +gap can start only if an event point lies in its closed limiting vertical +segment. If \(c_x\) distinct sampled circles pass through an event point +\(x\), only \(O(1+c_x)\) right-hand gaps can have this property, by +their vertical ordering. Summing over event points costs \(O(s+s^2)\): +extrema contribute \(O(s)\), and at an intersection with \(c_x\ge2\) +one may charge \(c_x\) to \(2\binom{c_x}{2}\), while each pair of +circles has at most two intersections. This also covers extrema that +coincide with intersections. There is only a constant number of initial +unbounded gaps. + +Finally, all possible pieces have a description from the full circle +set using a bounded number of circles. Each of their two horizontal +endpoints is an extremal or intersection abscissa, or infinity; each +boundary graph is specified by a circle and a sign, or is infinite. +Consequently at most \(C_0(1+l)^6\) candidate regions can occur, with +an absolute constant \(C_0\). For any fixed candidate region intersecting +more than \(l/R\) circles, the probability that no sampled circle +intersects it is at most +\[ + (1-1/R)^s\le e^{-s/R}, +\] +with the evident interpretation when \(R=1\). An actual open piece +avoids every sampled circle. A union bound over the candidate regions, +with \(C\) sufficiently large, therefore gives the asserted conflict +bound with failure probability at most \(1/10\). +\end{proof} + +\begin{proposition}\label{inc:incidence} +For \(m\) distinct points and \(l\) distinct unit circles in the plane, +\begin{equation}\label{inc:incidence-bound} + I\le C\log^2(2+m+l) + \bigl(m^{2/3}l^{2/3}+m+l\bigr), +\end{equation} +where \(C\) is an absolute constant. In particular, when \(m,l\) are +bounded by a fixed power of \(t\), the logarithmic factor is +\((\log t)^{O(1)}\). +\end{proposition} + +\begin{proof} +Sz\'ekely's point--curve incidence theorem +\cite[Theorem~8]{Szekely1997} gives the stronger bound +\(I=O(m^{2/3}l^{2/3}+m+l)\) when distinct curves meet in at most two +points and distinct points belong to at most two curves. Distinct unit +circles satisfy both hypotheses. If simple curves are taken to be arcs, +cut each circle at two points outside the finite input point set; the +resulting \(2l\) arcs satisfy the same pair bounds and preserve every +incidence. The logarithmically weakened form +\eqref{inc:incidence-bound} follows, including the empty-set cases. +\end{proof} + +\subsection{Degree control and a maximizing rectangle} + +We now use the cutting to obtain the prescribed imbalance, and then +choose a rectangle of the incidence graph that cannot gain too much +edge density by further restriction. This last maximization supplies +both expansion and lower degree bounds. + +\begin{proposition}\label{inc:piece} +From a sequence of \(n=t^3\) real planar points with at least +\(t^{4-o(1)}\) ordered unit pairs, one can obtain real algebraic point +and center sets \(P,Q\), and a nonempty graph \(E\subseteq P\times Q\) +consisting of unit incidences, with the following properties. There is +a parameter \(a=a(t)>0\) such that \(a\to0\), +\(a\log t\to\infty\), and +\begin{equation}\label{inc:scales} + |P|=t^{1+2a+o(a)},\qquad + |Q|=t^{2+a+o(a)},\qquad + |E|=t^{2+2a+o(a)}. +\end{equation} +Writing \(d(p),d(q)\) for graph degrees, uniformly over all vertices, +\begin{equation}\label{inc:degrees} + d(p)=t^{1+o(a)},\qquad d(q)=t^{a+o(a)}. +\end{equation} +The uniform law on \(E\) has marginals +\(\mu(p)=d(p)/|E|\) and \(\kappa(q)=d(q)/|E|\) satisfying +\begin{equation}\label{inc:marginal-atoms} + \mu(p)\ge\frac{\gamma}{|P|},\qquad + \kappa(q)\ge\frac{\gamma}{|Q|},\qquad + \gamma=\frac{27}{50}=0.54. +\end{equation} +Set \(\eta=\gamma-\tfrac12=1/25\). There is a single sequence +\(\xi_t\to0\) such that, for every \(F\subseteq P\) and +\(G\subseteq Q\), with counting fractions +\(u=|F|/|P|\) and \(v=|G|/|Q|\), +\begin{equation}\label{inc:expansion} + \Prob_E(p\in F,q\in G) + \le \sqrt{uv}\bigl((uv)^\eta+\xi_t\bigr). +\end{equation} +The sequence \(\xi_t\) is independent of the chosen subsets. +\end{proposition} + +\begin{proof} +First use \Cref{inc:specialization}. Choose \(a\to0\) sufficiently +slowly that the initial incidence count is at least \(t^{4-o(a)}\) +and every fixed power of \(\log t\) is \(t^{o(a)}\). In particular +\(a\log t\to\infty\). Apply \Cref{inc:cutting} with +\(R=t^{1-a}\) and \(s=O(t^{1-a}\log t)\). There are \(t^3\) +points and circles. Incidences on the sampled circles number at most +\(s t^3\), and incidences at deleted points with other circles number +at most \(2s t^3\). Thus the deletion costs +\(O(t^{4-a}\log t)=o(t^{4-o(a)})\). + +Each remaining region meets at most \(t^{2+a}\) circles. Further split +its assigned point set into sets of size at most \(2t^{1+2a}\). +The total number of resulting pieces is at most the original region +count plus \(O(t^3/t^{1+2a})\), hence at most +\(t^{2-2a+o(a)}\). Some piece has at least +\(t^{2+2a-o(a)}\) incidences. Denote its two vertex sets temporarily +by \(P_0,Q_0\), and its edge count by \(e_0\). At this stage +\[ + |P_0|\le 2t^{1+2a},\qquad |Q_0|\le t^{2+a},\qquad + t^{2+2a-o(a)}\le e_0\le t^{2+2a+o(a)}, +\] +where the upper bound follows from \eqref{inc:incidence-bound}. + +Choose \(b=o(a)\) so slowly that \(b\log t\to\infty\) and \(b\) +dominates all exponent errors in these estimates and the logarithmic +factor in \eqref{inc:incidence-bound}. Delete point vertices of degree +greater than \(t^{1+b}\), and center vertices of degree greater than +\(t^{a+b}\), with degrees measured before these deletions. The counts +of such vertices are at most +\[ + t^{1+2a-b/2}\quad\hbox{and}\quad t^{2+a-b/2}, +\] +respectively, for all sufficiently large \(t\). Apply +\eqref{inc:incidence-bound} to the excessive vertices on one side and +all vertices on the other. In either case the mixed term is at most +\(t^{2+2a-b/3+o(b)}\); the two linear terms are bounded by +\(t^{1+2a+o(b)}+t^{2+a+o(b)}\). Their sum is \(o(e_0)\). +Both deletions therefore retain \(t^{2+2a+o(a)}\) edges and give +the degree upper bounds \(t^{1+o(a)}\), \(t^{a+o(a)}\). +Dividing the retained edge count by these two upper bounds shows that +both retained vertex counts attain their stated upper scales within +\(t^{o(a)}\). Relabel this trimmed graph as \((P_0,Q_0,E_0)\). + +For subsets of these trimmed vertex sets, let \(u,v\) be their +respective counting fractions and let \(w\) be their captured edge +fraction. The incidence bound and the maximum-degree bounds give, +with one uniform factor \(T_t=t^{o(a)}\), +\begin{align} + w&\le T_t\bigl((uv)^{2/3}+t^{-a}v+t^{-1}u\bigr), + \label{inc:preliminary-expansion}\\ + w&\le T_t\min(u,v).\nonumber +\end{align} +Taking the minimum with the degree bounds separately in the two linear +terms, and using \(\min(x,y)\le\sqrt{xy}\), yields +\begin{equation}\label{inc:preliminary-square-root} + w\le T_t\sqrt{uv} + \bigl((uv)^{1/6}+2t^{-a/2}\bigr), +\end{equation} +after increasing \(T_t\) if necessary. For example, +\(\min(t^{-a}v,u)\le t^{-a/2}\sqrt{uv}\); the other linear term +uses \(t^{-1/2}\le t^{-a/2}\) for large \(t\). + +Put \(\tau=t^{-a/8}\). Among all induced rectangles with counting +fractions \(u,v\ge\tau\) and captured edge fraction \(w\), maximize +\[ + \frac{w}{(uv)^\gamma},\qquad \gamma=\frac{27}{50}. +\] +We choose \(\gamma>1/2\) so that a bound by \((uv)^\gamma\) +improves on \(\sqrt{uv}\) for small sets. The inequality +\(\gamma<2/3\), where \(2/3\) is the incidence exponent in +\eqref{inc:preliminary-expansion}, will ensure that the maximizing +rectangle retains the scales \eqref{inc:scales}. +This is a maximum over a nonempty finite family, and its value is at +least one because the full rectangle is feasible. Write \(U,V\) +for the maximizing counting fractions and \(W\) for its captured +edge fraction. We claim +\begin{equation}\label{inc:winning-fractions} + U,V,W\ge t^{-o(a)}. +\end{equation} +Indeed \eqref{inc:preliminary-square-root} gives +\[ + 1\le T_t\left((UV)^{2/3-\gamma} + +2t^{-a/2}(UV)^{-\eta}\right). +\] +By feasibility, \(UV\ge t^{-a/4}\), so the second term on the right +is at most \(2t^{-49a/100+o(a)}\), which tends to zero. If +\(UV\le t^{-ca}\) along a subsequence for any fixed \(c>0\), the +first term tends to zero as well, since +\(2/3-\gamma=19/150>0\). This is impossible. Thus +\(UV\ge t^{-o(a)}\), which implies the same for each of \(U,V\), +and \(W\ge(UV)^\gamma\) gives the claim for \(W\). + +Let \((P,Q,E)\) be this maximizing rectangle, retaining precisely its +edges from the trimmed graph. The scale estimates +\eqref{inc:scales} and the degree upper bounds survive by +\eqref{inc:winning-fractions}. For further subsets with fractions +\(u,v\) inside \(P,Q\), optimality gives +\[ + \Prob_E(F,G)\le(uv)^\gamma +\] +whenever \(Uu,Vv\ge\tau\). Otherwise +\(uv\le\tau/\min(U,V)=t^{-a/8+o(a)}\). Apply +\eqref{inc:preliminary-square-root} with the old fractions \(Uu,Vv\) +and divide by \(W\). Uniformly in all such infeasible subsets, this +gives +\[ + \Prob_E(F,G) + \le t^{o(a)}\sqrt{uv} + \bigl((uv)^{1/6}+2t^{-a/2}\bigr) + \le\xi_t\sqrt{uv}, +\] +where one may take \(\xi_t=t^{-a/48+o(a)}\to0\), with the exponent +error depending only on the chosen rectangle and the previous uniform +bounds. Combining the feasible and infeasible cases proves +\eqref{inc:expansion}. Empty subsets satisfy it trivially. + +Finally \eqref{inc:winning-fractions} implies that removing any single +vertex of the maximizing rectangle remains feasible for all sufficiently +large \(t\). Optimality with a point \(p\) removed gives +\[ + 1-\mu(p)\le(1-|P|^{-1})^\gamma\le1-\gamma|P|^{-1}, +\] +the last inequality following from concavity. The center comparison is +identical. This proves \eqref{inc:marginal-atoms}, and multiplying by +\(|E|\) supplies the lower degree bounds in \eqref{inc:degrees}. +\end{proof} + +In particular, the minimum degree on either side tends to infinity, +and \(\max_p\mu(p)=t^{-1+o(1)}\). The lower marginal bounds also +convert \eqref{inc:expansion} to the form needed for probability laws. +For \(h=\mu(F)\), \(k=\kappa(G)\), we have +\(|F|/|P|\le h/\gamma\), \(|G|/|Q|\le k/\gamma\), and hence +\begin{equation}\label{inc:marginal-expansion} + \Prob_E(F,G)\le C\sqrt{hk}\bigl((hk)^\eta+\xi_t\bigr), +\end{equation} +with a fixed absolute \(C\). The same \(\xi_t\) works for all subsets. + +\subsection{Matching labels and uniformly rare sets} + +We will repeatedly assign finite labels to the vertices, with no bound +on the number of labels. Expansion says that if almost all edges match, +one label occupies almost all the mass. A second application of +expansion makes the error proportional to the actual mismatch, even +when that mismatch is smaller than \(\xi_t\). + +\begin{lemma}\label{inc:matching} +For the sequence of edge laws in \Cref{inc:piece}, there are absolute +constants \(s_0>0,C<\infty\) and an index after which the following +holds uniformly for every pair of label maps +\(\ell_P:P\to\mathcal A\), \(\ell_Q:Q\to\mathcal A\) into a +finite set. If +\[ + s=\Prob_E\bigl(\ell_P(p)\ne\ell_Q(q)\bigr)\le s_0, +\] +some label \(a_0\in\mathcal A\) satisfies +\[ + \mu\{\ell_P\ne a_0\}+\kappa\{\ell_Q\ne a_0\}\le Cs. +\] +Moreover, for independent \(p,p'\sim\mu\), at every value of \(s\), +\begin{equation}\label{inc:point-label-mismatch} + \Prob\bigl(\ell_P(p)\ne\ell_P(p')\bigr)\le Cs. +\end{equation} +The analogous statement holds for independent centers of law \(\kappa\). +\end{lemma} + +\begin{proof} +Let \(u_i,v_i\) be the counting fractions of the two classes with label +\(i\), including zero fractions. Sum \eqref{inc:expansion} over the +matched rectangles. Cauchy--Schwarz gives +\(\sum_i\sqrt{u_iv_i}\le1\), and therefore +\[ + 1-s\le\max_i(u_iv_i)^\eta+\xi_t. +\] +For sufficiently small fixed \(s_0\) and sufficiently large \(t\), +one label \(a_0\) thus has both counting fractions arbitrarily close +to one. Apply \eqref{inc:expansion} to either complementary class +against the whole opposite side. The marginal exception masses +\[ + h=\mu\{\ell_P\ne a_0\},\qquad + k=\kappa\{\ell_Q\ne a_0\} +\] +are then as small as any prescribed fixed constant, uniformly over +the label maps. This first conclusion need not be proportional to \(s\). + +To obtain proportionality, put +\(J=\Prob_E(\ell_P\ne a_0,\ell_Q\ne a_0)\). +By \eqref{inc:marginal-expansion} and \(\sqrt{hk}\le(h+k)/2\), +we may choose the preceding constants so that +\(J\le(h+k)/4\). The events on which exactly one endpoint has label +\(a_0\) are mismatches, so +\[ + s\ge h+k-2J\ge\tfrac12(h+k). +\] +This proves the first assertion, including when \(s=0\), with no +additive error. Two independent point labels differ with probability +at most \(2h\), giving \eqref{inc:point-label-mismatch} for small +\(s\). For \(s>s_0\), the bound follows by increasing \(C\) so +that \(Cs_0\ge1\). The center assertion is identical. +\end{proof} + +Define the \emph{wedge law} on triples \((p,q,r)\in P\times Q\times P\) +by first choosing \(q\sim\kappa\) and then independently choosing +\(p,r\) according to the conditional edge law given \(q\). Explicitly, +\begin{equation}\label{inc:wedge-law} + \Prob(p,q,r)= + \begin{cases} + (|E|d(q))^{-1},&(p,q),(r,q)\in E,\\ + 0,&\text{otherwise}. + \end{cases} +\end{equation} +Each point marginal is \(\mu\), and each of the two point--center +marginals is the uniform edge law. + +\begin{lemma}\label{inc:rare} +Let \(\delta_t\to0\) be any nonnegative sequence. The estimates below +hold uniformly over all indicated subsets, with error factors depending +only on \(\delta_t,\xi_t\) and the fixed constants in +\eqref{inc:marginal-expansion}: +\begin{enumerate}[label=\textup{(\roman*)}] +\item If \(F\subseteq P\), \(G\subseteq Q\), and +\(h=\mu(F)\le\delta_t\), while \(k=\kappa(G)\) is arbitrary, then +\begin{equation}\label{inc:one-small-set} + \Prob_E(F,G)\le C(\delta_t^\eta+\xi_t)\sqrt{hk} + =o(1)\sqrt{hk}. +\end{equation} +The same assertion holds with the sides interchanged. +\item If also \(k\le\delta_t\), then +\begin{equation}\label{inc:joint-rare-edge} + \Prob_E(F,G)\le\tfrac C2(\delta_t^{2\eta}+\xi_t)(h+k) + =o(1)(h+k). +\end{equation} +\item If \(F\subseteq P\) and \(h=\mu(F)\le\delta_t\), then +under the wedge law, +\begin{equation}\label{inc:joint-rare-wedge} + \Prob(p\in F,r\in F)=o(1)h. +\end{equation} +\end{enumerate} +In particular, the error factors may be taken outside sums or integrals +over any family of sets whose marginal masses obey the same uniform +bound \(\delta_t\). +\end{lemma} + +\begin{proof} +The first two assertions follow directly from +\eqref{inc:marginal-expansion}, using \(hk\le\delta_t\) in the +first case and \(hk\le\delta_t^2\) and +\(\sqrt{hk}\le(h+k)/2\) in the second. + +For the third, let \(g(q)=\Prob(p\in F\mid q)\), so that +\(\E_\kappa g=h\) and the probability in question is +\(\E_\kappa g^2\). There is nothing to prove when \(h=0\). +For any \(d\in(0,1]\), put \(G_d=\{q:g(q)>d\}\). Markov's +inequality gives \(\kappa(G_d)\le h/d\). Splitting at this threshold, +\[ + \E_\kappa g^2 + \le dh+\E_\kappa[g\1_{G_d}] + =dh+\Prob_E(F,G_d) + \le h\left[d+C d^{-1/2} + \bigl((\delta_t^2/d)^\eta+\xi_t\bigr)\right]. +\] +Choose \(d=d_t\to0\) sufficiently slowly that +\(\delta_t^{2\eta}d_t^{-1/2-\eta}\to0\) and +\(\xi_t d_t^{-1/2}\to0\). The bracket then tends to zero, +independently of \(F\). This proves the uniform assertion and hence +its integrated form. +\end{proof} + +\subsection{The information carried by a wedge} + +The small point side and the two-center codegree bound together ensure +that the two point endpoints of a wedge share little information. We +record the atom estimates as well, since they later convert a probability +of wedges into a count of distinct point pairs. + +For probability laws \(a,b\) on the same finite set, define their relative +entropy by +\begin{equation}\label{inc:relative-entropy} + \KL(a\Vert b)=\sum_x a(x)\log\frac{a(x)}{b(x)}, +\end{equation} +with a zero summand when \(a(x)=0\) and value \(+\infty\) if +\(a(x)>0=b(x)\). + +\begin{lemma}\label{inc:entropy} +For the wedge law \eqref{inc:wedge-law}, let +\(d_{\min,Q}=\min_{q\in Q}d(q)\). If \(p\ne r\), then +\begin{equation}\label{inc:wedge-atoms} + \Prob(p,r)\le\frac{2}{|E|d_{\min,Q}} + =t^{-2-3a+o(a)}. +\end{equation} +The diagonal probability is +\begin{equation}\label{inc:wedge-diagonal} + \Prob(p=r)=\frac{|Q|}{|E|}=t^{-a+o(a)}=o(1). +\end{equation} +The mutual information of the point endpoints satisfies +\begin{equation}\label{inc:information-bound} + D:=\KL\bigl(\mathcal L(p,r)\,\Vert\,\mu\otimes\mu\bigr) + =o(\log t). +\end{equation} +\end{lemma} + +\begin{proof} +For fixed \(p,r\), summing \eqref{inc:wedge-law} gives +\[ + \Prob(p,r)=\frac1{|E|} + \sum_{q:(p,q),(r,q)\in E}\frac1{d(q)}. +\] +There are at most two summands when \(p\ne r\), proving the +off-diagonal estimate by \eqref{inc:scales}--\eqref{inc:degrees}. +Summing instead over the diagonal yields +\[ + \Prob(p=r)=\frac1{|E|}\sum_{q\in Q} + \frac{d(q)}{d(q)}=\frac{|Q|}{|E|}. +\] +The asymptotics follow from the same estimates and +\(a\log t\to\infty\). + +For a finite random variable \(X\), write +\(\mathsf H(X)=-\sum_x\Prob(X=x)\log\Prob(X=x)\), with +\(0\log0=0\). Every off-diagonal atom is at most +\(M_t=2/(|E|d_{\min,Q})\). Dropping the nonnegative diagonal +entropy contributions therefore gives +\[ + \mathsf H(p,r) + \ge\bigl(1-\Prob(p=r)\bigr)\log(1/M_t) + \ge(2-o(1))\log t. +\] +On the other hand, +\(\mathsf H(p)=\mathsf H(r)\le\log|P| +=(1+o(1))\log t\). Expanding the definition of relative entropy +gives +\[ + 0\le D=\mathsf H(p)+\mathsf H(r)-\mathsf H(p,r) + =o(\log t), +\] +as required. +\end{proof} + +We retain this graph and these probability laws for the rest of the +proof: the uniform edge law, its point and center marginals, and the +wedge law. The expansion consequences hold for every finite labeling +and every sufficiently small exceptional set, with constants independent +of any tests subsequently assigned to the vertices. diff --git a/preprints/A-power-saving-for-planar-unit-distances-September-23-2026/build/sections/02-prediction.tex b/preprints/A-power-saving-for-planar-unit-distances-September-23-2026/build/sections/02-prediction.tex new file mode 100644 index 0000000000000000000000000000000000000000..ffb998bab6ff778f40fc276c94f81f68b3ac53e8 --- /dev/null +++ b/preprints/A-power-saving-for-planar-unit-distances-September-23-2026/build/sections/02-prediction.tex @@ -0,0 +1,391 @@ +\section{Prediction from short histories}\label{sec:prediction} + +The incidence reduction supplies a probability law on point--center pairs whose +point endpoints in a random wedge have mutual information \(D=o(\log t)\). +We now turn this information bound into predictions from short histories of +point tests. The prediction uses only the history of the point; its accuracy +is measured against a test that may depend on the center. We first prove this +statement for an arbitrary finite probability law and then give the two forms +needed for coordinate quantization and coordinate equality. + +The proof sums the information contributed by successive tests. Since +that sum is at most \(D\), a sufficiently long history leaves little +new dependence on average. Related short conditioning arguments occur in +Ahlswede's +wringing method \cite[Section~4, Lemma~3]{Ahlswede1982}. +Here the proof must additionally compare each endpoint with its own +posterior law and leave the next test independent of the selected history. + +\subsection{The probability laws and the entropy comparison} + +Let \(P,Q\) be finite nonempty sets and let \(\omega(p,q)\) be a probability +mass function on \(P\times Q\), with marginals \(\mu\) and \(\kappa\). +Delete atoms of zero marginal mass. Write +\[ + \omega(p\mid q)=\frac{\omega(p,q)}{\kappa(q)}, + \qquad + \omega(q\mid p)=\frac{\omega(p,q)}{\mu(p)}. +\] +The associated wedge law and its point-pair marginal are +\begin{equation}\label{pred:wedge-law} + \Omega(p,q,r)=\kappa(q)\omega(p\mid q)\omega(r\mid q), + \qquad + \pi(p,r)=\sum_{q\in Q}\Omega(p,q,r). +\end{equation} +Both marginals of \(\pi\) are \(\mu\). Relative entropy is defined in +\eqref{inc:relative-entropy}. For finite probability laws \(a,b\), set +\[ + \TV(a,b)=\frac12\sum_x\abs{a(x)-b(x)}. +\] +The information available to the argument is +\begin{equation}\label{pred:information} + D=\KL(\pi\Vert\mu\otimes\mu). +\end{equation} +It is finite, since \(\pi(p,r)>0\) implies \(\mu(p)\mu(r)>0\). + +We use the standard chain rule for relative entropy +\cite[Theorem~2.5.3]{CoverThomas2006}: +\begin{equation}\label{pred:chain-rule} + \KL(a_{XY}\Vert b_{XY}) + =\KL(a_X\Vert b_X) + +\sum_x a_X(x)\KL(a_{Y\mid x}\Vert b_{Y\mid x}). +\end{equation} +Nonnegativity and the chain rule imply that applying a deterministic map +cannot increase relative entropy: adjoin the map to the outcome and then +discard the nonnegative conditional term. We also use Pinsker's inequality +\cite[Lemma~11.6.1]{CoverThomas2006}, which in our natural-logarithm +convention is +\begin{equation}\label{pred:pinsker} + 2\TV(a,b)^2\le\KL(a\Vert b). +\end{equation} +Finally, total variation controls bounded expectations by +\begin{equation}\label{pred:tv-expectation} + \abs{\E_a F-\E_b F}\le2\norm{F}_{\infty}\TV(a,b). +\end{equation} + +A test \(T\) is a random element of a probability space, independent of +\((p,q,r)\), which specifies a code map \(c_T:P\to[M]\), where +\([M]=\{1,\ldots,M\}\). The test space may be infinite; all maps and +integrands below are assumed measurable. Only finite probability laws of +codes are used. All vector norms and inner products in this section are +Euclidean. For a fixed history map \(H:P\to\mathcal H\), put +\begin{equation}\label{pred:posterior-point} + \mu_h(x)=\frac{\mu(x)\mathbf1_{\{H(x)=h\}}}{\mu(H=h)} + \quad\text{when }\mu(H=h)>0. +\end{equation} +Its values on histories of zero mass are immaterial. + +\begin{lemma}[Prediction device]\label{pred:device} +Let \(\omega,\pi,D\) be as in \eqref{pred:wedge-law}--\eqref{pred:information}, +let \(K,M\ge1\) be integers, and let \(T_1,\ldots,T_K\) be independent tests +with the same law as \(T\). Each test specifies, in addition to \(c_T\), +a feature map \(f_T:[M]\to\R^d\) and a coefficient map +\(g_T:Q\to\R^d\), for an integer \(d\ge1\) and constants +\(B,B'\ge0\), satisfying +\[ + \norm{f_T(c)}\le B,\qquad \norm{g_T(q)}\le B' +\] +for all codes, centers and tests. Choose \(J\) uniformly from \([K]\), +independently of all tests, and retain only the strict prefix +\[ + H(p)=(c_{T_1}(p),\ldots,c_{T_{J-1}}(p)). +\] +Conditioned on \(J,T_1,\ldots,T_{J-1}\), let \(T\) be a fresh test and define +\[ + m_{T,h}=\sum_{x\in P}\mu_h(x)f_T(c_T(x)). +\] +Then +\begin{equation}\label{pred:device-bound} + \E_{J,T_1,\ldots,T_{J-1},T} + \left|\E_{(p,q)\sim\omega} + \ip{f_T(c_T(p))-m_{T,H(p)}}{g_T(q)}\right| + \le4BB'\left(\frac DK\right)^{1/4}. +\end{equation} +Every realized history has at most \(M^{K-1}\) values. In particular, a +history can be fixed before drawing the fresh test so that its average +absolute discrepancy in \eqref{pred:device-bound} is at most the right side. +\end{lemma} + +\begin{proof} +First freeze all \(K\) tests. Apply their complete transcript map to both +coordinates of the pair \((p,r)\). The actual transcript law is the image of +\(\pi\); the reference law is the image of \(\mu\otimes\mu\). +Data processing bounds their relative entropy by \(D\). +For \(j\in[K]\), write \(H_j\) for the transcript strictly before step \(j\). +Conditional on prefixes \((h,h')\), the reference next-code law is exactly +\[ + b_{j,h,h'}= + (c_{T_j})_*\mu_h\ \otimes\ (c_{T_j})_*\mu_{h'}. +\] +Here \(\mu_h\) is formed with \(H_j\). Let \(a_{j,h,h'}\) denote the actual +conditional law of the next two codes. All prefix pairs of positive actual +probability have positive reference probability. Iterating +\eqref{pred:chain-rule} therefore gives +\begin{equation}\label{pred:conditional-kl} + \sum_{j=1}^K + \E_{(p,r)\sim\pi} + \KL\bigl(a_{j,H_j(p),H_j(r)}\Vert + b_{j,H_j(p),H_j(r)}\bigr)\le D. +\end{equation} +Consequently, by \eqref{pred:pinsker} and Cauchy--Schwarz, +\begin{equation}\label{pred:conditional-tv} + \frac1K\sum_{j=1}^K + \E_{(p,r)\sim\pi} + \TV\bigl(a_{j,H_j(p),H_j(r)},b_{j,H_j(p),H_j(r)}\bigr) + \le\sqrt{\frac{D}{2K}}. +\end{equation} + +For this fixed test sequence and step \(j\), subtract the prediction belonging +to each point's own prefix: +\[ + U_j(p)=f_{T_j}(c_{T_j}(p))- + \sum_x\mu_{H_j(p)}(x)f_{T_j}(c_{T_j}(x)). +\] +The norm is at most \(2B\). Under each reference conditional product, both +residuals are centered and independent, so the conditional expectation of +\(\ip{U_j(p)}{U_j(r)}\) is zero. This inner product has absolute value at +most \(4B^2\). Thus \eqref{pred:tv-expectation} and +\eqref{pred:conditional-tv} imply +\begin{equation}\label{pred:residual-variance} + \frac1K\sum_{j=1}^K + \E_{(p,r)\sim\pi}\ip{U_j(p)}{U_j(r)} + \le8B^2\sqrt{\frac{D}{2K}}. +\end{equation} +Each summand is nonnegative, for conditional independence in the wedge law +gives the exact identity +\begin{equation}\label{pred:wedge-square} + \E_{(p,r)\sim\pi}\ip{U_j(p)}{U_j(r)} + =\E_{q\sim\kappa} + \left\|\E_{p\sim\omega(\cdot\mid q)}U_j(p)\right\|^2. +\end{equation} +It follows, by Cauchy--Schwarz first over \(q\) and then over the tests and +step, that the average absolute discrepancy is at most +\[ + B'\left(8B^2\sqrt{\frac{D}{2K}}\right)^{1/2} + \le4BB'(D/K)^{1/4}. +\] +Everything proved so far holds after averaging the initially frozen tests. +The residual at step \(j\) uses only \(T_1,\ldots,T_j\). Since the tests +are independent and identically distributed, its joint law with the strict +prefix is unchanged when \(T_j\) is replaced by a fresh test after that +prefix. This proves \eqref{pred:device-bound}. Finally average over the +strict prefixes and choose one attaining at most the average. There are at +most \(M^{j-1}\le M^{K-1}\) possible histories for a prefix of length \(j-1\). +\end{proof} + +The important comparison in this proof is conditional on two possibly +different histories. The reference law uses the first endpoint's prediction +at the first coordinate and the second endpoint's prediction at the second. +No independence of the actual endpoints after their histories have been +specified is asserted. Their independence is used only conditional on the +center, in \eqref{pred:wedge-square}. + +\subsection{Acceptance tests and the prediction coupling} + +\begin{corollary}[Finite-code acceptance]\label{pred:acceptance} +In the setting of \Cref{pred:device}, suppose each test specifies a set +\(\mathcal A_T(q)\subseteq[M]\) of accepted codes for every \(q\in Q\). +Put +\[ + s_0=\E_T\Prob_{(p,q)\sim\omega} + \{c_T(p)\notin\mathcal A_T(q)\}. +\] +There is a fixed strict-prefix history \(H\), with at most \(M^{K-1}\) +values, such that +\begin{equation}\label{pred:acceptance-bound} + \E_T\sum_{p,q}\omega(p,q) + \sum_x\mu_{H(p)}(x) + \mathbf1_{\{c_T(x)\notin\mathcal A_T(q)\}} + \le s_0+4\sqrt M(D/K)^{1/4}. +\end{equation} +The fresh test \(T\) is independent of the fixed history construction. +\end{corollary} + +\begin{proof} +Use \(f_T(c)=e_c\in\R^M\), of norm one, and use the acceptance indicator +vector as \(g_T(q)\), of norm at most \(\sqrt M\). Actual and predicted +failure probabilities differ by the negative of their acceptance +probabilities. Averaging \eqref{pred:device-bound} and then fixing a strict +prefix proves the assertion. +\end{proof} + +We record the sampling laws used with this result. For any fixed history +\(H\) and integer \(r\ge1\), draw one point, one prediction, and \(r\) +neighbors according to +\begin{equation}\label{pred:shared-prediction-law} + \Prob\{p,\widehat p,q_1,\ldots,q_r\} + =\mu(p)\mu_{H(p)}(\widehat p) + \prod_{j=1}^r\omega(q_j\mid p), +\end{equation} +and draw the fresh test independently of this entire tuple. The two point +marginals are both \(\mu\), each \((p,q_j)\) has law \(\omega\), and each +\(q_j\) has marginal \(\kappa\). In particular each pair +\((p,\widehat p)\) is independent of the fresh test, although its two points +need not be independent. These assertions follow by summing +\eqref{pred:shared-prediction-law}, using +\(\sum_{p:H(p)=h}\mu(p)\mu_h(x)=\mu(x)\mathbf1_{\{H(x)=h\}}\). +For every fixed history, +\begin{equation}\label{pred:point-collision} + \Prob\{p=\widehat p\} + =\sum_h\frac{\sum_{p:H(p)=h}\mu(p)^2}{\mu(H=h)} + \le\abs{H(P)}\max_p\mu(p). +\end{equation} +Given \(H(p)\), the prediction is independent of all neighbors in +\eqref{pred:shared-prediction-law}. Applying a union bound to +\eqref{pred:acceptance-bound} therefore bounds the chance that at least one +of the \(r\) acceptance comparisons rejects this \emph{same} predicted point +at the same fresh test by +\(r(s_0+4\sqrt M(D/K)^{1/4})\). The union bound needs no independence of +the rejection events. + +There is also a useful equivalent way to draw the point pair: draw +\(h\) with probability \(\mu(H=h)\), then draw two points independently +from \(\mu_h\). This again has both marginals \(\mu\), and its collision +probability satisfies \eqref{pred:point-collision}. Once \(H\) is fixed, +this pair law can be used unchanged for every fresh test. Sampling a second +object from the first object's fiber also appears in the entropy argument +of Katz and Tardos \cite[Section~2, proof of Lemma~1]{KatzTardos}; the +identities above specify the marginals and independent test needed here. + +For the coordinate-quantization application, a test will be an independently +chosen field embedding, a code will specify bounded coordinate bins together +with an overflow code, and \(\mathcal A_T(q)\) will consist of codes achieved +by actual neighbors of \(q\). The exact hypotheses needed from that +construction are a fixed finite alphabet of size \(M\) and acceptance of +every actual edge, so \(s_0=0\). If \(K=o(\log t)\), +\(D/K=o(1)\), and \(\max_p\mu(p)\le t^{-1+o(1)}\), then the prediction +failure and the collision probability both tend to zero. Formula +\eqref{pred:shared-prediction-law} with \(r=3\) is the simultaneous +three-neighbor experiment. Geometric estimates valid for all couplings with +these marginals remain valid for each fixed history. More generally, +additional nonnegative error quantities can be averaged together with the +prediction error before choosing a prefix; this always leaves the test fresh. + +\subsection{Equality of one coordinate} + +The next application allows an arbitrarily large set of state labels without +paying for its size in the feature norm. A state is a pair in a product +\(\mathcal A\times\mathcal B\). For a state \(y=(a,b)\), define its +\emph{strict cross} +\begin{equation}\label{pred:strict-cross} + R_y=\{(a',b'): + (a'=a\text{ and }b'\ne b) + \text{ or }(a'\ne a\text{ and }b'=b)\}. +\end{equation} +Thus exactly one coordinate agrees. A fresh test specifies state maps +\(y_T(x)=(A_T(x),B_{1,T}(x))\) for all \(x\in P\cup Q\). +Only the finitely many labels attained on these vertex sets need be retained. +The labels and their equality relations are assumed measurable in \(T\). + +\begin{corollary}[Prediction of strict crosses]\label{pred:cross} +Let \(\omega,\pi,D\) be as above, let \(K\ge1,L\ge2\) be integers, and +let \(T\) be an independent random test giving the state maps just described. +Set +\[ + s_0=\E_T\Prob_{(p,q)\sim\omega} + \{y_T(p)\notin R_{y_T(q)}\}. +\] +There exists a fixed history \(H:P\to\mathcal H\), with +\(\abs{H(P)}\le L^{2K}\), for which the laws of the \emph{true}, unhashed +states +\[ + \nu_{h,T}=(y_T)_*\mu_h +\] +satisfy +\begin{equation}\label{pred:cross-bound} + \E_T\E_{(p,q)\sim\omega} + [1-\nu_{H(p),T}(R_{y_T(q)})] + \le s_0+\frac4L+18(D/K)^{1/4}. +\end{equation} +The history is formed from earlier tests and does not use \(T\). +\end{corollary} + +\begin{proof} +Augment each test by two fresh independent random maps from its attained +coordinate labels into \([L]\), assigning independent uniform values to +distinct labels. Numbering attained labels by their first occurrence in a +fixed vertex order makes this construction measurable. Use the same map +on both vertex sets for each coordinate, +and use independent maps for the two coordinates and for different tests. +The point code is its two hashed coordinates, so the alphabet has size +\(M=L^2\). For a code \((a,b)\in[L]^2\), use the feature +\[ + f(a,b)=(e_a,e_b,e_{(a,b)})\in\R^L\oplus\R^L\oplus\R^{L^2}, + \qquad \norm{f(a,b)}=\sqrt3. +\] +For the hashed center state \((a_q,b_q)\), use +\[ + g(q)=(e_{a_q},e_{b_q},-2e_{(a_q,b_q)}), + \qquad \norm{g(q)}=\sqrt6. +\] +Their inner product is +\(\mathbf1_{\{a=a_q\}}+\mathbf1_{\{b=b_q\}} +-2\mathbf1_{\{(a,b)=(a_q,b_q)\}}\), exactly the indicator of the hashed +strict-cross relation. Therefore \Cref{pred:device} compares its actual and +predicted probabilities with error at most +\(4\sqrt{18}(D/K)^{1/4}\), averaged over strict prefixes and the fresh +augmented test. + +For any two fixed true states, hashing preserves an equality, and creates +an equality between two unequal coordinate labels with probability \(1/L\). +Thus a true strict-cross indicator differs from its hashed indicator with +probability at most \(2/L\). This applies both to the actual state pair and +to the predicted pair: conditional on the earlier prefix, the fresh true +test and the chosen vertices, the current hashes are still independent +uniform maps. In particular the predicted point is sampled by +\eqref{pred:posterior-point} using no current hash. Comparing actual true, +actual hashed, predicted hashed, and predicted true failures gives, in that +order, total extra error at most +\(4/L+4\sqrt{18}(D/K)^{1/4}\). Average the current test and current hashes +before fixing a prefix, and use \(4\sqrt{18}<18\). The resulting true-state +expression no longer involves the current hashes. Its history has at most +\((L^2)^{K-1}\le L^{2K}\) values. +\end{proof} + +One form of \Cref{pred:cross} that will be convenient is an integrated +version. Let \(\mathcal T\) be a measurable family of state tests with finite +measure \(W\), and let \(m(\tau)\) bound actual strict-cross failure at +\(\tau\). For \(W>0\), sample \(T\) from the normalized measure +\(\dd\tau/W\). The resulting fixed history satisfies +\begin{equation}\label{pred:integrated-cross} + \int_{\mathcal T}\E_{(p,q)\sim\omega} + [1-\nu_{H(p),\tau}(R_{y_\tau(q)})]\dd\tau + \le\int_{\mathcal T}m(\tau)\dd\tau + +W\bigl(4/L+18(D/K)^{1/4}\bigr). +\end{equation} +For \(W=0\), the assertion holds with the empty history. In the later level +application, \(\mathcal T\) is the common-level set with its weighted level +measure, and \(m\) is the incidence inconsistency probability. Consistency +implies the strict-cross relation, which is the only geometric hypothesis +needed for \eqref{pred:integrated-cross}. We suppress the test subscript and +write \(\nu_h\) when the level being tested is fixed. + +\subsection{Choosing the history length} + +There is no conflict between small prediction error and a subpolynomial +number of histories. Write \(T_0=\log t\), and suppose \(D=o(T_0)\). +For example, take +\[ + K=\left\lceil\sqrt{(D+1)T_0}\right\rceil. +\] +Then \(K\to\infty\), \(K=o(T_0)\), and \(D/K\to0\). +For a fixed alphabet this gives \(M^K=t^{o(1)}\). +For hashing one may additionally take +\[ + L=\left\lfloor\exp\sqrt{T_0/K}\right\rfloor. +\] +Eventually \(L\ge2\), \(L\to\infty\), and +\(K\log L=o(T_0)\), so \(L^{2K}=t^{o(1)}\). +These statements require no quantitative rate for \(D/T_0\). +If a later argument requires an integer \(k\to\infty\) satisfying +\(kK\log L=o(T_0)\) and \(k^{20}\eta_t\to0\) for an additional positive +sequence \(\eta_t\to0\), it suffices to choose +\[ + k=\left\lfloor\min\left\{ + \left(\frac{T_0}{K\log L}\right)^{1/2}, + \eta_t^{-1/40}\right\}\right\rfloor. +\] +Further finitely many slow-growth requirements can be imposed by taking a +smaller such diverging sequence. Thus the prediction lemma supplies both +vanishing error and the small history counts required in the subsequent +geometric arguments. diff --git a/preprints/A-power-saving-for-planar-unit-distances-September-23-2026/build/sections/03-heights.tex b/preprints/A-power-saving-for-planar-unit-distances-September-23-2026/build/sections/03-heights.tex new file mode 100644 index 0000000000000000000000000000000000000000..34cd7715acca9574451a14bce128993dedf5a888 --- /dev/null +++ b/preprints/A-power-saving-for-planar-unit-distances-September-23-2026/build/sections/03-heights.tex @@ -0,0 +1,446 @@ +\section{Heights and coordinate tests}\label{sec:heights} + +We apply the prediction device to the incidence pieces of +\Cref{inc:piece}. Their coordinates are real algebraic numbers. +The first task is to measure the variation of edge directions simultaneously +at all absolute values of a number field. We will prove that this variation +cannot remain bounded. We then construct coordinate labels whose agreement +records a threshold comparison, with a bounded total error. + +Write +\[ + z_x=x_1+i x_2,\qquad w_x=x_1-i x_2 +\] +for a point or center $x=(x_1,x_2)$. For every edge $e=(p,q)$, put +\begin{equation}\label{ht:jumps} + d_e=z_p-z_q,\qquad w_p-w_q=d_e^{-1}. +\end{equation} +The second equality follows from the unit-distance equation, and in +particular $d_e\ne0$. Distinct real points have distinct $z$-coordinates and +distinct $w$-coordinates. These nonzero differences remain nonzero under +every field embedding. At an embedding other than the original one, +we use the two embedded coordinates separately; they need not be complex +conjugates. + +\subsection{A weighted product formula} + +For each configuration choose a finite normal number field $\mathcal N$ +containing all its coordinates and $i$, and let +$d_{\mathcal N}=[\mathcal N:\Q]$. +The following construction allows repeated absolute values; the repetitions +make their weights particularly simple. + +\begin{lemma}[Weighted absolute values]\label{ht:product} +There is a countable collection of absolute-value entries $v$ on +$\mathcal N$, each of weight $1/d_{\mathcal N}$, with the following properties. +Its archimedean entries are $|u|_v=|\sigma(u)|$ for the $d_{\mathcal N}$ embeddings +$\sigma:\mathcal N\to\C$, so their total weight is one. All other entries +are nonarchimedean. For each nonzero $u\in\mathcal N$, only finitely many +nonarchimedean entries have $|u|_v\ne1$, and +\begin{equation}\label{ht:product-formula} + \sum_v^*\log|u|_v=0. +\end{equation} +Here and below a star denotes the sum with the stated weights. +\end{lemma} + +\begin{proof} +Use the classification of number-field places and the product formula +\cite[Theorems~7.14--7.15]{ProductFormula}. At a finite place +\(\mathfrak p\) above a rational prime \(\ell\), normalize the genuine +absolute value by \(|\ell|_{\mathfrak p}=\ell^{-1}\), and put +\(n_{\mathfrak p}=[\mathcal N_{\mathfrak p}:\Q_\ell]\), where +\(\mathcal N_{\mathfrak p}\) is the completion. Use +\(n_{\mathfrak p}\) copies of this absolute value, each of weight +\(1/d_{\mathcal N}\). Milne's local normalization formula +\cite[Lemma~8.6]{ProductFormula} converts his finite-place value to +\(|\cdot|_{\mathfrak p}^{n_{\mathfrak p}}\). His complex-place value +is squared modulus, so we instead use ordinary modulus once for each +embedding, counting conjugate embeddings separately. Taking logarithms +of the product formula and dividing by \(d_{\mathcal N}\) proves +\eqref{ht:product-formula}. There are exactly \(d_{\mathcal N}\) +embeddings, giving archimedean weight one. Finally, the principal +fractional ideal of any nonzero \(u\) has finitely many prime factors +\cite[Theorem~3.20]{ProductFormula}; at every other finite place +\(|u|_v=1\). +\end{proof} + +Define $\log^+r=\max(0,\log r)$ for $r>0$ and $\log^+0=0$. +Set $h(0)=0$, and put +\[ + h(u)=\sum_v^*\log^+|u|_v\qquad(u\ne0). +\] + +\begin{lemma}[Height inequalities]\label{ht:height} +The value $h(u)$ is independent of the chosen finite normal field and of +the extensions of the prime absolute values in \Cref{ht:product}. +For $u\in\mathcal N^\times$, +\begin{equation}\label{ht:height-inversion} + h(u^{-1})=h(u),\qquad + \sum_v^*|\log|u|_v|=2h(u). +\end{equation} +For arbitrary $u,v\in\mathcal N$, +\begin{equation}\label{ht:height-inequalities} + h(uv)\le h(u)+h(v),\qquad + h(u+v)\le h(u)+h(v)+\log2. +\end{equation} +If a random $X\in\mathcal N$ is independent of a uniformly sampled +embedding $\sigma:\mathcal N\to\C$, then, for $B>1$ and $0B\}&\le\frac{\E h(X)}{\log B}, + \label{ht:large-tail}\\ + \Prob\{X\ne0,\ |\sigma(X)|s\}},& + \pi(v,s)&=\Prob_E\{Y_e(v,s)=1\},\\ + f(v,s)&=\min(\pi(v,s),1-\pi(v,s)),& S&=\int f. + \end{aligned} +\end{equation} +Enlarging any $I_v$ further does not change $S$, because all edge bits +agree outside the range of the $n_e(v)$. + +The choice of threshold turns distance between logarithmic profiles into +bit disagreement. For independent uniform edges $e,e'$, Tonelli's theorem +and \eqref{ht:height-inversion} give +\begin{equation}\label{ht:independent-profiles} + 2\E_{e,e'}h(d_e/d_{e'}) + =\E_{e,e'}\sum_v^*|n_e(v)-n_{e'}(v)| + =2\int\pi(1-\pi)\le2S. +\end{equation} + +\subsection{Excluding bounded total variation} + +We first use matching labels to transfer height control from edges to +points. The argument is useful because it bounds heights of differences +before any translation has been chosen. + +\begin{lemma}[Transferring edge heights]\label{ht:transfer} +Suppose a probability law $\omega$ on two finite vertex sets $P\times Q$, +with point marginal $\mu$, has the following property: for all assignments of finite +labels to its vertices, two independent $\mu$-points disagree with +probability at most $C_0$ times the edge disagreement probability. +For any coordinate $U:P\cup Q\to\mathcal N$, +\[ + \E_{p,p'\sim\mu}h(U_p-U_{p'}) + \le C_1\bigl(\E_{(p,q)\sim\omega}h(U_p-U_q)+1\bigr), +\] +where $C_1$ depends only on $C_0$. +\end{lemma} + +\begin{proof} +Fix an entry $v$ and $s\ge0$. At a nonarchimedean entry give two vertices +the same label if their $U$-coordinates differ by at most $e^s$ in +absolute value. The strong triangle inequality makes this an equivalence +relation. At an archimedean entry, use the cell of a square grid in +$\C$ of side $e^s$, shifted by a uniformly random fractional vector. +Only the labels occurring on the finite vertex set matter. + +For complex numbers whose distance is $a$, the probability of different +grid cells is at most +\[ + \min(1,\sqrt2\,a e^{-s}). +\] +Indeed a mismatch in either real coordinate has probability at most its +coordinate distance divided by the side length, and the two coordinate +distances sum to at most $\sqrt2 a$. Its integral over $s\ge0$ is at most +$C(1+\log^+ a)$. At a nonarchimedean entry the integral of mismatch is +exactly $\log^+|U_p-U_q|_v$, with value zero for a zero difference. +Thus the integrated edge mismatch, averaged over shifts at infinity, +is at most $C(\E_{(p,q)\sim\omega}h(U_p-U_q)+1)$. + +Apply the assumed label inequality at each fixed level and shift. +For a point pair at a nonarchimedean entry, mismatch is exactly the +event $|U_p-U_{p'}|_v>e^s$. At infinity, distance greater than +$\sqrt2 e^s$ forces mismatch for every shift. Integration of these +tail events bounds the point-pair height, with an additional +$\log\sqrt2$ from the total archimedean weight. This proves the result. +\end{proof} + +\begin{proposition}[The bounded alternative is impossible]\label{ht:bounded} +For a sequence of the incidence pieces in \Cref{inc:piece}, with the +level spaces and quantities $S$ in \eqref{ht:levels}, no subsequence +has $S$ bounded. Consequently $S\to\infty$. +\end{proposition} + +\begin{proof} +Suppose $S\le S_0$ on a subsequence. All constants in this proof may +depend on the fixed $S_0$. By \eqref{ht:independent-profiles}, there is an +edge value $d_0$ such that +\[ + \E_Eh(d_e/d_0)\le S_0. +\] +Temporarily replace the two coordinates by $z/d_0$ and $d_0w$. +Their edge jumps are $D_e=d_e/d_0$ and $D_e^{-1}$, whose mean heights +are at most $S_0$. \Cref{inc:matching} supplies the label hypothesis of +\Cref{ht:transfer}. Applying it to both coordinates bounds the sum of +the two mean point-pair difference heights. Averaging over the second +point then chooses $p_0\in P$ such that the translated coordinates +\[ + Z_x=(z_x-z_{p_0})/d_0,\qquad W_x=d_0(w_x-w_{p_0}) +\] +satisfy a fixed bound on $\E_\mu(h(Z_p)+h(W_p))$. +Since $Z_q=Z_p-D_{pq}$ and $W_q=W_p-D_{pq}^{-1}$ on an edge, +\Cref{ht:height} also bounds the center means. Thus there is a fixed +$C_h<\infty$ with +\begin{equation}\label{ht:normalized-means} + \E_\mu(h(Z_p)+h(W_p)) + +\E_\kappa(h(Z_q)+h(W_q)) + +\E_E(h(D_e)+h(D_e^{-1}))\le C_h. +\end{equation} +The edge identity is still +$(Z_p-Z_q)(W_p-W_q)=1$. + +We next give the selection argument, keeping the embedding independent +of the points it tests. For \emph{any} fixed finite history map $H$ on +$P$, use the joint law +\begin{equation}\label{ht:coupling} + \mu(p)\,\mu(\widehat p\mid H(\widehat p)=H(p)) + \prod_{j=1}^3\Prob(q_j\mid p), +\end{equation} +and independently choose a uniformly random embedding $\sigma$. +Both $p$ and $\widehat p$ have marginal $\mu$; each $q_j$ has marginal +$\kappa$; and each $(p,q_j)$ has the original edge law. +In particular, the three jumps $D_j=D_{pq_j}$ have the correct individual +edge marginals. No assertion of mutual independence of these jumps is +needed. The height inequality bounds +$\E h(Z_{\widehat p}-Z_p)$ and each $\E h(D_j-D_l)$ using only +\eqref{ht:normalized-means}. These bounds are uniform in $H$. + +Choose a fixed $B\ge2$ so large that, by \eqref{ht:large-tail}, the +probability that any of +\[ + Z_p,W_p,Z_{\widehat p},W_{\widehat p}, + (Z_{q_j},W_{q_j},D_j,D_j^{-1})_{j=1}^3 +\] +has embedded absolute value greater than $B$ is less than $1/100$. +Then choose a fixed $00} + \frac{\sum_{p:H(p)=h}\mu(p)^2}{\mu(H=h)} + \le |H(P)|\max_p\mu(p)=t^{-1+o(1)}. +\end{equation} +For $j\ne l$, the conditional uniform neighbor law gives +$\Prob(q_j=q_l)\le1/\min_p d(p)=o(1)$. +Distinct centers have distinct $Z$-coordinates, so $q_j\ne q_l$ implies +$D_j\ne D_l$. The degree and atom estimates used here follow from +\Cref{inc:piece}. Combining all these estimates gives positive +probability that all three tests pass, all displayed quantities are +bounded by $B$, and +\begin{equation}\label{ht:separation} + |\sigma(Z_{\widehat p}-Z_p)|\ge r,\qquad + |\sigma(D_j-D_l)|\ge r\quad(js$ means +$|d_e|_ve^s$, so exactly $A$ agrees. +For $n_es\}}$, and the coordinate +states $y_x=(A_x,B_{1,x})$ of \Cref{ht:states}. Recall that an edge is +consistent when precisely its $A$ labels match if its bit is $1$, and +precisely its $B_1$ labels match if its bit is $0$. The inconsistency +probability $m(v,s)$ satisfies $\int m=O(1)$. +The purpose of this section is to compare two-edge samples at a point or +a center. Their integrated scores will have a small upper bound from +the product formula and a lower bound recording exceptions to dominant +coordinate labels. These bounds also hold after conditioning point +stars on a short list of histories. + +All level integrals use the weighted measure of \Cref{sec:levels}. +In particular $f=\min(\Prob_E(Y=0),\Prob_E(Y=1))$ and $S=\int f$. +Write $d_{P,\min}=\min_{p\in P}d(p)$ and +$d_{Q,\min}=\min_{q\in Q}d(q)$. + +\subsection{Parameters, rare levels, and histories} + +The parameters must make prediction errors small even after taking +several samples, while keeping the number of possible histories small +relative to a point degree. + +\begin{lemma}[Parameter choice]\label{bud:parameters} +There are integers $K,L,k\to\infty$ such that +\begin{align} + K\log L&=o(\log t),& D/K&=o(1),\label{bud:KL}\\ + kK\log L&=o(\log t),& + k^{20}\left((D/K)^{1/4}+L^{-1}+S^{-1}+d_{Q,\min}^{-1}\right)&=o(1). + \label{bud:hierarchy} +\end{align} +Set $\epsilon=k^{-1/16}$ and +$e_t=(D/K)^{1/4}+L^{-1}$. Then +$k^3/S=o(1)$, $k^{3+1/16}e_t=o(1)$, and $k=t^{o(1)}$. +\end{lemma} + +\begin{proof} +The incidence estimates give $D=o(\log t)$ and $d_{Q,\min}\to\infty$. +For example, first take +$K=\lceil\sqrt{(D+1)\log t}\rceil$ and then choose $L\to\infty$ +with $\log L\sim\sqrt{\log t/K}$. These choices give \eqref{bud:KL}. +Both $K\log L/\log t$ and the parenthesized expression in +\eqref{bud:hierarchy} tend to zero. Choosing $k\to\infty$ sufficiently +slowly gives \eqref{bud:hierarchy}. Its stated consequences follow +directly; in particular $k=o(\log t)$ since $K\log L\to\infty$. +\end{proof} + +Define the common, damaged, and good rare levels, respectively, by +\[ + \mathcal C=\{f\ge\epsilon\},\qquad + \mathcal B=\{f<\epsilon,\ m>1/k\},\qquad + \mathcal G=\{f<\epsilon,\ m\le1/k\}. +\] +Their first two widths satisfy +\begin{equation}\label{bud:widths} + W_c:=\int\mathbf1_{\mathcal C}\le S/\epsilon, + \qquad W_b:=\int\mathbf1_{\mathcal B}\le k\int m=O(k)=o(S). +\end{equation} +At each place, $\mathcal C$ is an interval, possibly empty or of zero +length: the function $s\mapsto\Prob_E(n_e(v)>s)$ is nonincreasing. + +At a good rare level let $r\in\{0,1\}$ be the minority bit, which is +unique for all sufficiently large $t$. On every consistent +majority-bit edge the $B_1$ labels match if $r=1$, and the $A$ labels +match if $r=0$. Thus the mismatch probability of this coordinate is +at most $f+m$. By \Cref{inc:matching} it has a label $\ell$ whose two +marginal exception probabilities have sum $O(f+m)$. Fix one such +label measurably, and let $I_x(v,s)$ indicate that the relevant +coordinate of $x$ differs from $\ell$. Write +\[ + h_P(v,s)=\E_\mu I_p(v,s),\quad + h_Q(v,s)=\E_\kappa I_q(v,s),\quad + H_x^*=\int_{\mathcal G} I_x(v,s). +\] +Here and below $I_x$ is used only on $\mathcal G$. Fixed orderings of +the finite vertex sets settle all choices of labels, so measurability +causes no additional restriction. + +\begin{lemma}[Exception mass and joint exceptions]\label{bud:rare-moments} +Uniformly on $\mathcal G$, +\begin{equation}\label{bud:exception-comparison} + h_P+h_Q\le C(f+m)=o(1),\qquad f\le h_P+h_Q+m. +\end{equation} +Consequently +\begin{equation}\label{bud:first-moments} + \E_\mu H_p^*+\E_\kappa H_q^*=O(S). +\end{equation} +Furthermore, with $(p,q,r)$ the ordinary wedge law, +\begin{equation}\label{bud:joint-exceptions} + \int_{\mathcal G}\Prob_E(I_p=I_q=1)=o(S),\qquad + \int_{\mathcal G}\Prob(I_p=I_r=1)=o(S). +\end{equation} +\end{lemma} + +\begin{proof} +The first inequality was just proved, and its right side is uniformly +$O(\epsilon+1/k)=o(1)$. If both endpoints have the dominant label, +a consistent edge has the majority bit. Every minority-bit edge +therefore has an exceptional endpoint or is inconsistent, proving the +second inequality. Integrating and using $\int m=O(1)$ proves +\eqref{bud:first-moments} because $S\to\infty$. + +For the joint errors, apply \Cref{inc:rare} with the uniform marginal +bound $a_t=C(\epsilon+k^{-1})\to0$ from +\eqref{bud:exception-comparison}. Its edge estimate gives +$\Prob_E(I_p=I_q=1)=o(1)(h_P+h_Q)$, and its ordinary-wedge estimate +gives $\Prob(I_p=I_r=1)=o(1)h_P$. These are exactly the edge and +wedge laws used here. Both multipliers are uniform over all levels +in $\mathcal G$, so they can be taken outside the integrals. +Equation \eqref{bud:first-moments} then gives +\eqref{bud:joint-exceptions}. +\end{proof} + +We next apply the prediction device only to common levels. When +$W_c>0$, sample a fresh level from their normalized measure and use +\Cref{pred:cross}. An inconsistent edge accounts for every true +strict-cross failure. We obtain a fixed history map $H$ with +$|H(P)|\le L^{2K}$ such that, writing $\nu_h$ for the true-state law +of a point of marginal $\mu$ conditional on history $h$ at the fresh +level, +\begin{equation}\label{bud:prediction-error} + k^3\int_{\mathcal C}\E_E[1-\nu_{H(p)}(R_{y_q})]=o(S). +\end{equation} +Indeed the integral before multiplication by $k^3$ is at most +$\int_{\mathcal C}m+O(W_c e_t)$, and +\Cref{bud:parameters} and \eqref{bud:widths} apply. If $W_c=0$, +take the empty history, and the assertion is vacuous. + +For each center $q$, a \emph{template} is an ordered list +$\theta=(h_1,\ldots,h_k)$ obtained by sampling $k$ independent +neighbors of $q$ and recording their histories. Let +$a(q,\theta)=\Prob(\theta\mid q)$. The number $M$ of possible +templates satisfies +\begin{equation}\label{bud:template-count} + M\le L^{2Kk}=t^{o(1)}. +\end{equation} +The history map is already fixed. Thus $a(q,\theta)$ uses no fresh +level. In the joint law +\begin{equation}\label{bud:joint-law} + \omega(p,q,\theta)=\frac{\mathbf1_{\{(p,q)\in E\}}}{|E|} + a(q,\theta), +\end{equation} +the point $p$ and the template are independent conditional on $q$. + +\subsection{Three fixed pair experiments} + +We use three ways to sample a base and two opposite endpoints: +\begin{enumerate}[label=(\roman*)] + \item $Q$: sample $q\sim\kappa$, then $p,r$ independently from the + conditional edge law at $q$; + \item $P$: sample $p\sim\mu$, then $q,q'$ independently from the + conditional edge law at $p$; + \item $P\theta$: sample $(p,\theta)$ from its marginal under + $\omega$, then $q,q'$ independently from $\omega(q\mid p,\theta)$. +\end{enumerate} +Only positive-mass bases are used. In each experiment write $x$ for +the base vertex, $y_1,y_2$ for its opposite endpoints, and $Y_1,Y_2$ +for the corresponding edge bits. The base in the third experiment +includes $\theta$. At a level let +$\rho=\Prob(Y_1=1\mid\text{base})$ under the full indicated law. +These laws are fixed across all levels. Define +\begin{equation}\label{bud:variations} + C_i=\int_{\mathcal C}\E_{\mathrm{base}}[2\rho(1-\rho)], + \qquad i\in\{Q,P,P\theta\}. +\end{equation} + +\begin{lemma}[Marginals and diagonal pairs]\label{bud:pair-laws} +Each edge slot of each experiment, after averaging its base, has the +original uniform law on $E$. For every $p\in P$, the conditional +probability of $q=q'$ in the $P\theta$ experiment, averaged over +$\theta$ conditional on $p$, is at most $M/d(p)$. +\end{lemma} + +\begin{proof} +The first assertion is immediate for ordinary stars. For the third, +the $(p,q)$ marginal of \eqref{bud:joint-law} is the edge law since +$\sum_\theta a(q,\theta)=1$; sampling a conditional slot preserves +this marginal. More explicitly, a slice of positive probability +$w_\theta=\sum_{p,q}\omega(p,q,\theta)$ has one-edge law +$\omega(p,q,\theta)/w_\theta$ and must be averaged with weight +$w_\theta$. One can also attach a template independently at a +$Q$ base: conditional on $(q,\theta)$ its two point neighbors still +have their original law, and its single-slot law in each slice is +the same as for $P\theta$. + +Fix $p$ and put $a_p(\theta)=\Prob(\theta\mid p)$. For +$a_p(\theta)>0$, +\[ + \omega(q\mid p,\theta) + =\frac{\mathbf1_{\{(p,q)\in E\}}a(q,\theta)}{d(p)a_p(\theta)} + \le\frac1{d(p)a_p(\theta)}. +\] +Its collision probability is at most its largest atom. Multiplying +by $a_p(\theta)$ and summing over at most $M$ positive-mass templates +gives the asserted bound pointwise in $p$. +\end{proof} + +We now define scores that reward matching opposite endpoints and +charge differing bits. If $Y_1=Y_2=1$, let $N$ be the event that the +$B_1$ labels of $y_1,y_2$ match; if $Y_1=Y_2=0$, let $N$ instead +mean that their $A$ labels match; set $N$ to be false when the bits +differ. Thus $N$ concerns the coordinate +which a consistent edge of that bit does not share with its base. +For independent bits with parameter $\rho$, equality has probability +$\rho^2+(1-\rho)^2$, whereas disagreement has probability +$2\rho(1-\rho)$. The former is larger except at $\rho=1/2$. +Thus, when equal bits force $N$, a reward slightly above one near +balance and slightly below one elsewhere can give a positive expected +score. The decrease away from balance also makes room for the +increase in the integrated upper bound below. +\Cref{sec:common} will identify the configurations in which equal bits +force the required match. +Set +\[ + \delta=\frac1{100},\qquad \lambda=\frac{\delta^2}{2},\qquad + c(\rho)= + \begin{cases} + 1+\lambda,&|\rho-\tfrac12|\le\delta,\\ + 1-\lambda,&|\rho-\tfrac12|>\delta. + \end{cases} +\] +The scalar $\lambda$ is a score coefficient; it is unrelated to the +edge ratios $\lambda_{pr}$ introduced later. +At a good rare level write $r$ for the minority bit. The score is +\[ + \text{score}= + \begin{cases} + 2I_x\mathbf1_{\{Y_1=Y_2=r,\,N\}},&\text{at a good rare level},\\ + c(\rho)\mathbf1_{\{Y_1=Y_2,\,N\}},&\text{at a common level},\\ + 0,&\text{at a damaged level} + \end{cases} + -\mathbf1_{\{Y_1\ne Y_2\}}. +\] +Let $D_i$ be the expectation of this score integrated over all +levels in experiment $i$. + +The next estimate is the placewise mechanism behind the score +bound. It identifies exactly when the coefficient $1+\lambda$ +can spend more than the available interval length. + +\begin{lemma}[Two intervals and a central window]\label{bud:intervals} +Let $b\le b'$, $T\ge0$, and $J$ be an interval of finite length, +with any choice of included endpoints. Put +$I_-=[b-T,b]$, $I_+=[b',b'+T]$, and +$a=|J\cap I_-|+|J\cap I_+|$, where $|\cdot|$ denotes length. +For $0<\lambda<1$, +\begin{equation}\label{bud:central-envelope} + \big[(1-\lambda)2T+2\lambda a-2T\big]_+ + \le\lambda|J|\, + \mathbf1_{\{b,b'\in\operatorname{int}J\}}. +\end{equation} +If $X$ has any probability law on $\R$, and +$|\Prob(X>s)-1/2|\le\delta$ throughout $J$, then +\begin{equation}\label{bud:interior-mass} + \Prob(X\in\operatorname{int}J)\le2\delta. +\end{equation} +\end{lemma} + +\begin{proof} +The left side of \eqref{bud:central-envelope} is $[2\lambda(a-T)]_+$. +For it to be positive both intersections must have positive length, +since either has length at most $T$. As $J$ is an interval, this +puts both $b,b'$ in its open interior, also when $b=b'$. +Moreover $a\le2T$ and $a\le|J|$, since $I_-$ and $I_+$ have disjoint +interiors. Hence $2(a-T)\le a\le|J|$. +For \eqref{bud:interior-mass}, take $uu)-\Prob(X>v)\le2\delta. +\] +Exhausting the open interior by such intervals proves the claim. +The empty-interior case is immediate. This proof makes no +assumption about atoms at either endpoint of $J$. +\end{proof} + +\Cref{fig:budget-intervals} illustrates why positive excess in +\Cref{bud:intervals} requires both thresholds to lie inside the central +window. + +\begin{figure}[htbp] + \centering + \input{figures/interval-budgets.tex} + \caption{The intervals $I_-=[b-T,b]$ and $I_+=[b',b'+T]$, with + $b\le b'$ and $T\ge0$, flank the bit-difference interval. + The central window $J$ in \Cref{bud:intervals} creates excess only + if its total overlap with the two flanks exceeds $T$; then both + thresholds lie strictly inside $J$.} + \label{fig:budget-intervals} +\end{figure} + +\begin{proposition}[Upper score budget]\label{bud:upper} +For each $i\in\{Q,P,P\theta\}$, +\begin{equation}\label{bud:upper-equation} + D_i\le o(S)+10\delta^2\lambda C_i. +\end{equation} +\end{proposition} + +\begin{proof} +First fix a base and two distinct opposite endpoints, and write +$d,d'$ for their incident jumps. Distinct real endpoints have +distinct $z$ coordinates, so $d\ne d'$. At a place $v$ let +\[ + b=\min(n_e(v),n_{e'}(v)),\quad b'=\max(n_e(v),n_{e'}(v)),\quad + u=-\log|d-d'|_v,\quad \alpha=u-b. +\] +The two opposite-endpoint coordinate differences are $d-d'$ and +$1/d-1/d'$ up to signs. Therefore +\begin{equation}\label{bud:place-identity} + -\log|(d-d')(1/d-1/d')|_v=2\alpha-(b'-b). +\end{equation} +All thresholds lie within the chosen level interval, so the +integrated bit-difference charge at this place is exactly $b'-b$. + +Let $c_v=\log2$ at archimedean entries and $c_v=0$ otherwise. +The triangle inequality gives $\alpha\ge-c_v$, hence +$T_v:=\alpha+c_v\ge0$. Equality of a grid label at infinity bounds +the corresponding difference by $\sqrt2$ times the grid scale; +at a finite place a matching label bounds it by the scale itself. +If both bits are $1$, a positive charge requires $ss_0)<\epsilon$. By monotonicity every common level +and every level with minority bit $0$ lies below $s_0$; none can +produce a positive charge in $I_+$. All minority-$1$ positive +charges lie in $I_-$ by definition. The case of a rare positive +with minority bit $0$ is symmetric. Thus, if any rare positive is +counted, all positive charges are on one side, their coefficients +are at most $2$, and their total is at most $2T_v$. + +Otherwise only common positives remain. For this fixed base put +\[ + J_v=\{s:(v,s)\in\mathcal C,\ |\rho(v,s)-1/2|\le\delta\}. +\] +This is an interval because the conditional edge law is fixed +across levels and both restrictions are intervals. Extend the +coefficient $1-\lambda$ across both full intervals in +\eqref{bud:positive-intervals}, increasing it by $2\lambda$ on +$J_v$. \Cref{bud:intervals} bounds the excess over $2T_v$ by +\[ + \lambda|J_v|\, + \mathbf1_{\{n_e(v),n_{e'}(v)\in\operatorname{int}J_v\}}. +\] +This expression is also a valid excess bound in the rare-positive +case, when no excess is needed. Combining it with the negative +charge and \eqref{bud:place-identity}, then summing over places, +cancels the logarithm by the product formula for the fixed nonzero +element $(d-d')(1/d-1/d')$. The additive allowance is +$2\sum_v^*c_v=2\log2$. Places of zero level width can be included +in this sum: their positive and negative charges vanish and the +same upper bound still holds. + +Now average this bound over the fixed-base independent pair law, +restricting to distinct pairs. Dropping that restriction only in +the nonnegative excess term, \eqref{bud:interior-mass} bounds its +expectation by $4\delta^2\lambda|J_v|$. Boundary atoms are not +counted; distinct jumps with equal thresholds cause no exception +to this argument. On $J_v$, +$2\rho(1-\rho)\ge2(1/4-\delta^2)$. Averaging over bases and places +therefore bounds the distinct-pair contribution by +\[ + O(1)+\frac{4\delta^2\lambda}{2(1/4-\delta^2)}C_i + \le O(1)+10\delta^2\lambda C_i. +\] + +It remains to bound equal endpoints, to which the product formula +was not applied. Their positive score is at most +$2(W_c+H_x^*)$ for a base vertex $x$, and their negative score is +zero. The ordinary collision probabilities are $1/d(q)$ and +$1/d(p)$. For templated stars \Cref{bud:pair-laws} gives the bound +$M/d(p)$ pointwise in $p$. Consequently their total contributions +are at most, respectively, +\[ + \frac{2}{d_{Q,\min}}(W_c+\E_\kappa H_q^*),\quad + \frac{2}{d_{P,\min}}(W_c+\E_\mu H_p^*),\quad + \frac{2M}{d_{P,\min}}(W_c+\E_\mu H_p^*). +\] +Each is $o(S)$ by \Cref{bud:parameters,bud:rare-moments}, +\eqref{bud:template-count}, and +$d_{P,\min}=t^{1+o(a)}$. In particular multiplication by an +unbounded $H_p^*$ requires only its first moment, since the +collision estimate holds for every $p$. Finally $O(1)=o(S)$. +\end{proof} + +The upper budget is insensitive to whether the star lies at a +point or a center. The next lower bound keeps track of that +orientation. Its two main terms will cancel when one point-star +score is added to one center-star score. + +\begin{proposition}[Rare-level lower budget]\label{bud:rare-lower} +For each of the three pair experiments, let $x$ be its star vertex +and $y$ an opposite endpoint. Its score integrated outside +$\mathcal C$ is at least +\begin{equation}\label{bud:rare-lower-equation} + 2\E H_x^*-2\E H_y^*-o(S), +\end{equation} +where both expectations use the original edge marginals on the +appropriate sides. In particular this is +$2\E_\kappa H_q^*-2\E_\mu H_p^*-o(S)$ for $Q$, and its opposite +main term for either $P$ or $P\theta$. +\end{proposition} + +\begin{proof} +At a good rare level denote the inconsistency indicator in slot +$j$ by $M_j$. If $I_x=1$, both other endpoints are normal, and +both edges are consistent, their dominant-coordinate labels +match each other but differ from the base's. Both bits must +therefore be the minority bit and the required nonshared +coordinate matches. The positive score is consequently at least +\[ + 2I_x-2I_x I_{y_1}-2I_x I_{y_2}-2M_1-2M_2. +\] +If both opposite endpoints are normal and both edges consistent, +their bits agree regardless of whether the base is exceptional. +Hence the negative charge is at most +$I_{y_1}+I_{y_2}+M_1+M_2$. +Average these two pointwise bounds. Each slot has the original +edge marginal by \Cref{bud:pair-laws}, even with templates. +The integrals of the joint terms $I_xI_{y_j}$ are $o(S)$ by +\eqref{bud:joint-exceptions}; the inconsistency integrals are +$O(1)$. This gives \eqref{bud:rare-lower-equation} on +$\mathcal G$. On $\mathcal B$ there are no positives and the +negative score is at least $-1$, whose total cost is +$W_b=o(S)$. +\end{proof} diff --git a/preprints/A-power-saving-for-planar-unit-distances-September-23-2026/build/sections/05-common-levels.tex b/preprints/A-power-saving-for-planar-unit-distances-September-23-2026/build/sections/05-common-levels.tex new file mode 100644 index 0000000000000000000000000000000000000000..5afe887e32f9c0af557bb39082c9e2c666a8f895 --- /dev/null +++ b/preprints/A-power-saving-for-planar-unit-distances-September-23-2026/build/sections/05-common-levels.tex @@ -0,0 +1,562 @@ +\section{Common levels and positive rare mass}\label{sec:common} + +We use the histories, templates and three star scores constructed in +\Cref{sec:budgets}. The first objective is to show that common levels contribute +only \(o(S)\) bit variation at a center. The second is to show that the point +exceptions on good rare levels retain a positive fraction of \(S\). Both +conclusions come from comparing the upper and lower star budgets, but the +comparison first requires a precise description of the states permitted by a +template. + +Throughout this section, \(S\to\infty\), the parameters satisfy +\Cref{bud:parameters}, and \(H\) is the fixed history map from +\Cref{bud:prediction-error}. At a level, \(\nu_h\) denotes the distribution of +the true state \((A,B_1)\) of a point sampled from \(\mu\) conditional on +\(H=h\). Dependence on the level is suppressed. For a state \(y=(a,b)\), its +strict cross is +\[ + R_y=\set{(u,v):(u=a,\ v\ne b)\ \text{or}\ (u\ne a,\ v=b)}. +\] +A row means a set with fixed first coordinate; a column has fixed second +coordinate. These conventions do not refer to a geometric direction in the +plane. + +\subsection{From sampled histories to the actual point} + +For a center \(q\), define the mixture of predictions +\[ + \widetilde\nu_q=\E_{p\mid q}\nu_{H(p)}. +\] +For its template \(\theta=(h_1,\ldots,h_k)\), define the finite set of candidate +center states +\begin{equation}\label{com:candidates} + \mathcal Y_\theta + =\set{y\text{ a center state}:\nu_{h_j}(R_y)\ge1-k^{-2} + \text{ for every }1\le j\le k}. +\end{equation} +The actual state \(y_q\) is usually a candidate: the union bound and Markov's +inequality give, after integration on \(\mathcal C\), +\begin{equation}\label{com:actual-candidate} + \int_{\mathcal C}\Prob(y_q\notin\mathcal Y_\theta) + \le k^3\int_{\mathcal C} + \E_E[1-\nu_{H(p)}(R_{y_q})]=o(S). +\end{equation} +We also need every candidate to have large mass under \(\widetilde\nu_q\). +The following elementary sampling bound supplies this uniformity. +For its first assertion, we specialize the \(\varepsilon\)-net +double-sampling argument to strict crosses +\cite[Lemmas~3.4--3.5]{HausslerWelzl1987}. +The strict-cross trace bound, the weighted-law formulation, and the subsequent +posterior-support transfer are established here. + +\Needspace{5\baselineskip} +\begin{lemma}[Sampling strict crosses]\label{com:ghost} +Let \(\nu\) be a probability measure on a finite product set and let +\(X_1,\ldots,X_k\) be independent samples from \(\nu\). For +\(0<\alpha<1\) and \(k\alpha\ge8\), +\[ + \Prob\bigl(\exists y:\nu(R_y)<1-\alpha, + \ X_1,\ldots,X_k\in R_y\bigr) + \le 2(2k+1)^2\,2^{-k\alpha/2}. +\] +Consequently, when \(k\) is sufficiently large, the templates in +\eqref{com:candidates} satisfy +\begin{equation}\label{com:mixture} + \Prob\bigl(\exists y\in\mathcal Y_\theta: + \widetilde\nu_q(R_y)<1-k^{-1/4}\mid q\bigr) + \le 4(2k+1)^2\,2^{-k^{3/4}/2} +\end{equation} +at each level, uniformly in \(q\). +\end{lemma} + +\begin{proof} +Adjoin independent samples \(X_{k+1},\ldots,X_{2k}\). If the first \(k\) +samples admit a cross as in the event, select one by a fixed ordering. For this +selected cross, the number of misses among the new samples has mean +\(k\beta\), where \(\beta=\nu(R_y^c)>\alpha\), and variance at most +\(k\beta\). Thus the probability of fewer than \(k\alpha/2\) misses is at +most \(4/(k\beta)\le1/2\). It suffices to bound twice the probability that +some cross contains the first half and misses at least \(k\alpha/2\) of the +second half. + +Condition on the pooled \(2k\) samples and split their indices uniformly into +two halves. A cross induces at most \((2k+1)^2\) different subsets of these +indices: its first coordinate can agree with one of at most \(2k\) observed +coordinates, or none, and the same holds for its second coordinate. If a +pattern has \(m\) misses, the probability that they all fall in the second +half is zero for \(m>k\), and otherwise is +\[ + \frac{k(k-1)\cdots(k-m+1)}{(2k)(2k-1)\cdots(2k-m+1)} + \le 2^{-m}. +\] +The union bound proves the first assertion. Repeated sampled states cause no +problem, since the argument concerns the indexed samples. + +For the second assertion, after sampling the histories independently given +\(q\), independently sample a state from each corresponding \(\nu_{h_j}\). +Before the histories are revealed these states are independent with law +\(\widetilde\nu_q\). If a candidate violating the required conclusion exists, +select one as a function of the template. Conditional on that template, all +the synthetic states lie in its cross with probability at least +\(1-k\cdot k^{-2}=1-1/k\). Apply the first assertion with +\(\alpha=k^{-1/4}\) and divide by \(1-1/k\). +\end{proof} + +Since \(W_c\le k^{1/16}S\), the integral of the failure probability in +\eqref{com:mixture} is \(o(S)\). A candidate therefore constrains the mixture +of predicted states. To constrain the actual point, we will use only rows, +columns and sets of at most two states. This restriction is essential to the +next argument. + +\begin{lemma}[Transfer for restricted supports]\label{com:transfer} +Fix a level and the joint edge--template law +\[ + \Prob(p,q,\theta)=\Prob_E(p,q)\Prob(\theta\mid q). +\] +Let \(0<\eta<1/2\). As a function of \((q,\theta)\), select a support +\(U(q,\theta)\) that is a row, a column, or a set of at most two states. +On an event \(G\) determined by \((q,\theta)\), suppose +\[ + \widetilde\nu_q(U(q,\theta))\ge1-3\eta^2. +\] +Then the actual point state \(X(p)\) satisfies +\[ + \Prob\bigl(G,\ X(p)\notin U(q,\theta)\bigr)\le8\eta. +\] +The support type and its coordinates may depend on both \(q\) and \(\theta\). +\end{lemma} + +\begin{proof} +Conditional on \((q,\theta)\), the history of the actual point has its +ordinary conditional law given \(q\), because the template is independent of +the point given \(q\). Hence on \(G\), +\[ + \E[\nu_{H(p)}(U(q,\theta)^c)\mid q,\theta]\le3\eta^2. +\] +The event that this support has \(\nu_{H(p)}\)-mass less than \(1-\eta\) +therefore has probability at most \(3\eta\). + +For each history \(h\), define three exceptional sets using only \(\nu_h\). +If a row has mass at least \(1-\eta\), it is unique, and the first exceptional +set is its complement. If no such row exists, the first exceptional set is +empty. Define the second set in the same way for columns. Their masses are +at most \(\eta\) each. + +If a set of at most two states has mass at least \(1-\eta\), put +\[ + M_h=\set{x:\nu_h(x)>\eta} +\] +and take \(M_h^c\) as the third exceptional set. Every set \(U\) of at most +two states and mass at least \(1-\eta\) contains \(M_h\): omitting an atom +of mass greater than \(\eta\) would violate that mass bound. Choosing one +such \(U\) also gives +\[ + \nu_h(M_h^c)\le\nu_h(U^c)+\nu_h(U\setminus M_h) + \le\eta+2\eta=3\eta. +\] +If no such \(U\) exists, the third exceptional set is empty. The union of +the three exceptional sets thus has mass at most \(5\eta\). + +The actual point has marginal law \(\mu\), under which its state conditional +on \(H=h\) has distribution exactly \(\nu_h\). Consequently the probability +of this union of exceptional sets is at most \(5\eta\) under the entire +edge--template law. Outside it and the earlier nonconcentration event, the +actual state belongs to whichever permitted support was selected. Adding +the two error bounds proves the result. +\end{proof} + +In the application, at most three candidates will be chosen by a fixed +ordering at each level and template. Outside the failure event bounded in +\eqref{com:mixture}, the intersection of their crosses has mixture mass at +least \(1-3k^{-1/4}\). When this intersection is contained in a row or a +column, use that row or column as \(U\); when it has size at most two, use the +intersection itself. With \(\eta=k^{-1/8}\), \Cref{com:transfer} costs at most +\[ + 8k^{-1/8}W_c\le8k^{-1/16}S=o(S). +\] +An empty intersection cannot have the asserted mixture mass for large \(k\). +The proof has used the actual point marginal only for exceptional events +defined by the history and level. It has made no assertion about the law of +the actual state after additionally conditioning on a selected support. + +\subsection{Classification and suppression of common variation} + +We now apply this transfer to the finite geometry of strict crosses. Two +distinct center states in one row, say \((a,b)\) and \((a,b')\), have common +cross +\begin{equation}\label{com:row-intersection} + R_{(a,b)}\cap R_{(a,b')} + =\set{(a,v):v\notin\set{b,b'}}. +\end{equation} +A third center state outside that row cuts this intersection down to at most +one point. The corresponding assertions hold for columns. If two states +\((a,b)\) and \((a',b')\) differ in both coordinates, their common cross is +\begin{equation}\label{com:corners} + R_{(a,b)}\cap R_{(a',b')} + =\set{(a,b'),(a',b)}. +\end{equation} +Three states whose first coordinates are all distinct and whose second +coordinates are all distinct have empty common cross. Indeed each of the +two points in \eqref{com:corners} fails to share a coordinate with the third +state. These observations give an exhaustive classification: a candidate +set is empty, is a singleton, contains two distinct states sharing a +coordinate, or consists of states with pairwise distinct coordinates. +In the shared-coordinate case, either all candidates lie in that row or +column, or a third lies outside it. In the pairwise-distinct case there +are either two candidates or at least three. + +\Cref{com:fig-crosses} shows the two-corner case: either endpoint state +together with the bit determines the state at the other endpoint. +\input{figures/strict-crosses} + +Declare a single edge with its template bad at a common level if its state +test is inconsistent, if \(y_q\notin\mathcal Y_\theta\), if +the event inside the probability in \eqref{com:mixture} occurs, or if the +restricted-support transfer fails for +the witnesses specified by this classification. The witnesses are two +states sharing a coordinate, with a third outside that row or column when +one exists; two states differing in both coordinates; or three states with +pairwise distinct coordinates. Empty candidate sets and empty witness +intersections have no good edges. All choices are deterministic functions +of the level and template. The preceding estimates and the integrated +inconsistency bound give +\begin{equation}\label{com:bad-mass} + \int_{\mathcal C}\Prob(p,q,\theta\text{ is bad})=o(S). +\end{equation} + +The comparisons below must use the full conditional bit probability +\(\rho\) in the definition of the scores. We never remove bad edges and +renormalize that law. In particular the discontinuity of \(c(\rho)\) at +\(1/2\pm\delta\) introduces no error. + +Here is the numerical estimate used in the two-corner case. For every +\(0\le\rho\le1\), with \(\delta=0.01\), \(\lambda=\delta^2/2\), and +\(c\) as in \Cref{sec:budgets}, +\begin{align} + c(\rho)\bigl(\rho^2+(1-\rho)^2\bigr)-2\rho(1-\rho) + &= (2\rho-1)^2+ + (c(\rho)-1)\bigl(\rho^2+(1-\rho)^2\bigr)\notag\\ + &\ge\lambda/2.\label{com:score-polynomial} +\end{align} +For \(\abs{\rho-1/2}\le\delta\), discard the first square and use +\(\rho^2+(1-\rho)^2\ge1/2\). Outside that interval the expression is at +least \(4\delta^2-\lambda>\lambda/2\). + +\Needspace{5\baselineskip} +\begin{proposition}[Common variation]\label{com:variation} +For the center-star and point--template-star experiments of +\Cref{bud:pair-laws}, +\begin{equation}\label{com:variation-eq} + C_Q+C_{P\theta}=o(S). +\end{equation} +Moreover their rare exception means satisfy +\begin{equation}\label{com:center-rare} + \E_\kappa H_q^*\le\E_\mu H_p^*+o(S). +\end{equation} +\end{proposition} + +\begin{proof} +Fix a common level. For each template of positive probability, let +\(\pi_\theta=\Prob(\theta)\) and +\(b_\theta=\Prob(\text{bad edge}\mid\theta)\), with both probabilities taken +under the joint law in \Cref{com:transfer}. In the center-star experiment, +condition on \(\theta\) and then sample \(q\) by its actual conditional law. +The two points are still independent with their ordinary law given \(q\). +In the point--template-star experiment, sample \(p\) conditional on +\(\theta\), then two independent centers given \((p,\theta)\). Each slot +in either experiment has the same conditional single-edge law +\(\Prob(p,q\mid\theta)\). Therefore a pair with at least one bad slot has +probability at most \(2b_\theta\) in either experiment. + +Write \(V_Q(\theta)\) and \(V_{P\theta}(\theta)\) for the corresponding +averages of \(2\rho(1-\rho)\) over bases in this slice, and write +\(d_Q(\theta),d_{P\theta}(\theta)\) for its two common-level expected +scores. Each \(V\) is at most \(1/2\), and either score is at least the +negative of its variation. We prove +\begin{equation}\label{com:slice-score} + d_Q(\theta)+d_{P\theta}(\theta) + \ge\frac\lambda4\bigl(V_Q(\theta)+V_{P\theta}(\theta)\bigr) + -C b_\theta +\end{equation} +with an absolute constant \(C\). All coefficients in a common score are +bounded by \(1+\lambda\), so charging a pair with a bad slot always costs +at most an absolute multiple of its probability. + +\emph{An empty candidate set.} +Every edge is bad, so \(b_\theta=1\), and the boundedness of the scores proves +\eqref{com:slice-score} on increasing \(C\). + +\emph{One candidate state.} +At a fixed point--template base, two good edges have the same center state. +Consistency forces their bits to agree, and the nonshared coordinate of +their center states matches. Thus this star score is at least +\(1-\lambda-O(b_\theta)\) after averaging over its bases. The center-star +score is at least \(-1/2\). Their sum is at least +\(1/2-\lambda-O(b_\theta)\), which implies \eqref{com:slice-score} because +the two variations sum to at most one. + +\emph{Two candidates sharing a coordinate, and all candidates in that row +or column.} +The transfer using \eqref{com:row-intersection} places every good point in +the same row or column. If it is a row, every good edge has matching +\(A\)-coordinates and hence bit \(1\), by consistency; in the column case +every good edge has bit \(0\). A bit difference in either pair experiment +therefore requires a bad slot. This gives +\(V_Q(\theta)+V_{P\theta}(\theta)\le4b_\theta\). The lower score bound +\(d_i\ge-V_i\) now proves \eqref{com:slice-score}. + +\emph{Two candidates sharing a coordinate, and a third outside that row +or column.} +The three witness crosses intersect in at most one state. If the +intersection is empty, all edges are bad. Otherwise all good point states +equal that one state. At a center base, two good edges consequently have +equal bits and the required nonshared-coordinate match. Its score is at +least \(1-\lambda-O(b_\theta)\), and the other score is at least \(-1/2\). +The same estimate as in the singleton case applies. + +\emph{No coordinate shared by distinct candidates.} +If there are at least three candidates, their common cross is empty and +all edges are bad. Otherwise there are exactly two candidates. The good +point states lie in their two opposite corners from \eqref{com:corners}. +At either type of base, a good edge's bit determines the state at the other +endpoint. Thus two good edges with equal bits have the required collision. +For the full conditional probability \(\rho\) at that base, their expected +score is at least the left side of \eqref{com:score-polynomial}, less an +absolute multiple of the probability of a bad slot. After averaging bases, +each score is at least \(\lambda/2-O(b_\theta)\), which proves +\eqref{com:slice-score}. + +This completes the classification. Multiply \eqref{com:slice-score} by its +actual slice weight \(\pi_\theta\), sum, and integrate over common levels. +The error is +\(C\int_{\mathcal C}\sum_\theta\pi_\theta b_\theta=o(S)\) by +\eqref{com:bad-mass}. This argument allows individual \(b_\theta\) to be +large; no inverse slice weight or number of templates enters. The common +contributions to \(D_Q+D_{P\theta}\) are therefore at least +\[ + \frac\lambda4(C_Q+C_{P\theta})-o(S). +\] +The lower bounds in \Cref{bud:rare-lower} outside common levels cancel their +main terms, because the two star vertices are on opposite sides. Hence +\[ + \frac\lambda4(C_Q+C_{P\theta})-o(S) + \le D_Q+D_{P\theta} + \le 10\delta^2\lambda(C_Q+C_{P\theta})+o(S), +\] +where the last inequality is \Cref{bud:upper}. Since +\(10\delta^2<1/4\), this proves \eqref{com:variation-eq}. + +Finally the common contribution to \(D_Q\) is at least \(-C_Q\). +\Cref{bud:rare-lower} and \Cref{bud:upper} therefore give +\[ + 2\E_\kappa H_q^*-2\E_\mu H_p^*-C_Q-o(S) + \le D_Q\le10\delta^2\lambda C_Q+o(S). +\] +Use \(C_Q=o(S)\) to obtain \eqref{com:center-rare}. +\end{proof} + +\subsection{A second score forces positive rare mass} + +We have suppressed common variation at centers. To show that point exceptions +remain on rare levels, suppose temporarily that their mean is \(o(S)\). +\Cref{com:variation} then gives the same bound for center exceptions. We will +show that common variation is also small at ordinary point stars. Once both +ordinary star variations are small, expansion forces the edge bit itself to +be almost constant at nearly every level, contradicting \(S=\int f\). + +For a probability measure \(\nu\) on states, let \(\nu_A,\nu_{B_1}\) denote +its coordinate marginals, with squared norms equal to the sums of squares of +their atom masses. Define +\begin{equation}\label{com:T} + T=\int_{\mathcal C}\E_\mu\left[ + \norm{(\nu_{H(p)})_A}_2^2+ + \norm{(\nu_{H(p)})_{B_1}}_2^2-1\right]. +\end{equation} +These norms count coordinate matches between two independent predicted +points with the same history. Unlike the preceding star scores, \(T\) uses +no center. + +\begin{lemma}[The two-prediction score]\label{com:prediction-score} +If \(\E_\mu H_p^*=o(S)\), then \(T\le o(S)\). +\end{lemma} + +\begin{proof} +Choose a history with its marginal probability, then two independent points +\(p',p''\) from \(\mu\) conditional on that history. This is one fixed +coupling throughout all levels, and each point marginal is \(\mu\). At a +level, the expected value of +\[ + F(p',p'')= + \1_{\{A(p')=A(p'')\}}+ + \1_{\{B_1(p')=B_1(p'')\}}-1 +\] +is the integrand in \eqref{com:T}. + +For a distinct pair put \(Z=z_{p'}-z_{p''}\) and +\(W=w_{p'}-w_{p''}\), both nonzero. At a place with level interval +\([l,u]\), the endpoint conditions from \eqref{ht:range} say +\(l\le-\log|Z|_v\) and \(u\ge\log|W|_v\). The \(A\)-match can hold only +for \(s\le-\log|Z|_v+O_v(1)\), and the \(B_1\)-match only for +\(s\ge\log|W|_v-O_v(1)\). Here the constants vanish at nonarchimedean +places and are absolute at infinity. Therefore +\[ + \int_l^u F(p',p'')\dd s + \le-\log|Z|_v-\log|W|_v+O_v(1). +\] +The product formula, and total archimedean weight one, show that the +integral over all levels is at most an absolute constant. + +At a good rare level where neither point is exceptional, their dominant +coordinate labels match, so \(F\ge0\). In general at such a level +\[ + F\ge-\1_{\{p'\text{ exceptional}\}}- + \1_{\{p''\text{ exceptional}\}}. +\] +The contribution on damaged levels is at least \(-W_b\). Removing all +noncommon levels from the full integral consequently gives, for each +distinct pair, +\[ + \int_{\mathcal C}F(p',p'') + \le O(1)+H_{p'}^*+H_{p''}^*+W_b. +\] + +On the diagonal the common-level integral is exactly \(W_c\). Its +probability under the selected coupling is +\[ + \sum_{p\in P}\frac{\mu(p)^2}{\mu(H^{-1}(H(p)))} + \le |H(P)|\max_p\mu(p)=t^{-1+o(1)}. +\] +Here \(|H(P)|\le L^{2K}=t^{o(1)}\) and the marginal degree bounds were +proved in \Cref{sec:incidence}. Since +\(W_c\le k^{1/16}S\) and \(k=t^{o(1)}\), its contribution is \(o(S)\). +Average the bound for distinct pairs, using the two marginals \(\mu\), +\(W_b=o(S)\), and \(S\to\infty\). This proves +\(T\le2\E_\mu H_p^*+o(S)=o(S)\). +\end{proof} + +\begin{proposition}[Positive rare mass]\label{com:rare-positive} +The good rare point exceptions satisfy +\begin{equation}\label{com:rare-positive-eq} + \liminf_{t\to\infty}\frac{\E_\mu H_p^*}{S}>0. +\end{equation} +\end{proposition} + +\begin{proof} +If the conclusion fails, restrict to a subsequence on which +\(\E_\mu H_p^*=o(S)\). By \eqref{com:center-rare}, also +\(\E_\kappa H_q^*=o(S)\), and \Cref{com:prediction-score} gives +\(T\le o(S)\). + +At a common level declare an ordinary edge bad if its test is inconsistent +or if +\(\nu_{H(p)}(R_{y_q})<1-k^{-2}\). Its integrated probability is \(o(S)\), +because Markov's inequality bounds the second contribution by +\(k^2\int_{\mathcal C}\E_E[1-\nu_{H(p)}(R_{y_q})]=o(S)\). +Fix a point \(p\) and this level. Write \(b_p\) for its conditional bad-edge +probability, \(\nu=\nu_{H(p)}\), \(\zeta=k^{-2}\), and +\[ + \tau(\nu)=\norm{\nu_A}_2^2+\norm{\nu_{B_1}}_2^2-1. +\] +Let \(\mathcal Y_p\) be the states of its good neighbors. Every such state +has \(\nu(R_y)\ge1-\zeta\), and the actual point state belongs to each +\(R_y\), by consistency. Let \(d_P(p)\) be the common-level score at this +ordinary point star, using its full bit probability \(\rho_p\). We claim +\begin{equation}\label{com:second-slice-score} + d_P(p)+\tau(\nu) + \ge\frac\lambda4\,2\rho_p(1-\rho_p)-C(b_p+\zeta). +\end{equation} +As before, a pair with a bad slot costs only \(O(b_p)\); its probability is +at most \(2b_p\). The strict-cross classification proves the claim as +follows. + +If \(\mathcal Y_p\) is empty, \(b_p=1\), while \(d_P(p)\ge-1/2\) and +\(\tau(\nu)\ge-1\), so the claim holds for an absolute \(C\). + +If \(\mathcal Y_p=\set{(a,b)}\), put +\(r=\nu_A(a)\), \(s=\nu_{B_1}(b)\). Since +\(r+s\ge\nu(R_{(a,b)})\ge1-\zeta\), +\[ + \tau(\nu)\ge r^2+s^2-1 + \ge\tfrac12(1-\zeta)^2-1\ge-\tfrac12-\zeta. +\] +Two good neighbors have the same state and bit, so +\(d_P(p)\ge1-\lambda-O(b_p)\). Their sum has the positive lower bound +needed for \eqref{com:second-slice-score}. + +If two states share a row, their common cross is in that row, of +\(\nu\)-mass at least \(1-2\zeta\). Thus +\(\tau(\nu)\ge(1-2\zeta)^2-1\ge-4\zeta\). +If every state is in that row, the actual point lies there too, since it +belongs to the crosses of both distinct states. All good edges consequently +have bit \(1\). Their full-law variation is at most \(2b_p\), and +\(d_P(p)\ge-2b_p\), proving the claim in this case. Columns give bit \(0\) +and the same estimates. If there is a third state outside the row or column, +the three crosses have intersection of size at most one and mass at least +\(1-3\zeta\). It must be a singleton for large \(k\). Both marginals then +have an atom of at least that mass, giving +\[ + \tau(\nu)\ge2(1-3\zeta)^2-1\ge1-12\zeta. +\] +Together with \(d_P(p)\ge-1/2\), this is more than sufficient. + +Finally suppose no coordinate is shared by distinct states in +\(\mathcal Y_p\). Three such states are impossible, since the actual point +belongs to all their crosses and their intersection is empty. In the +remaining case there are two states differing in both coordinates. The two +opposite corners carry \(\nu\)-mass at least \(1-2\zeta\). Each coordinate +marginal therefore has at least that mass on a set of size two, so +\[ + \tau(\nu)\ge + 2\cdot\frac{(1-2\zeta)^2}{2}-1\ge-4\zeta. +\] +The actual point is one of those corners. Its bit determines the good +neighbor state; hence \eqref{com:score-polynomial} gives +\(d_P(p)\ge\lambda/2-O(b_p)\). This completes the proof of +\eqref{com:second-slice-score}. + +Average \eqref{com:second-slice-score} over \(p\) and integrate on common +levels. The bad-edge integral is \(o(S)\), and +\(\zeta W_c\le k^{-31/16}S=o(S)\). The result is +\[ + (\text{common contribution to }D_P)+T + \ge\frac\lambda4 C_P-o(S). +\] +The noncommon contribution to \(D_P\) is at least \(-o(S)\) by +\Cref{bud:rare-lower}, under the present assumptions on both exception +means. Therefore +\[ + \frac\lambda4 C_P-o(S) + \le D_P+T\le10\delta^2\lambda C_P+o(S), +\] +and again \(10\delta^2<1/4\) yields \(C_P=o(S)\). + +To finish, let \(V_P,V_Q\) be the ordinary star variations at any fixed +level. Assign to each vertex its conditional majority bit, breaking ties +in a fixed way. The probability that the bit on an incident edge differs +from this label is at most the corresponding average variation, because +\(\min(\rho,1-\rho)\le2\rho(1-\rho)\). Thus the two endpoint labels disagree +on an edge with probability at most \(V_P+V_Q\). When that sum is small, +the matching-label consequence of expansion in \Cref{inc:matching} gives +one bit label with total marginal exception probability +\(O(V_P+V_Q)\). The edge bit differs from it with probability +\(O(V_P+V_Q)\) as well. When the sum is not small the same bound follows +from \(f\le1/2\), with a larger absolute constant. Consequently +\[ + f\le C(V_P+V_Q) +\] +uniformly for all sufficiently large \(t\). Integrating on common levels +gives \(\int_{\mathcal C}f\le C(C_P+C_Q)=o(S)\). + +On good rare levels, the minority-bit probability is bounded by the two +marginal exception probabilities and the inconsistency probability, as in +\Cref{bud:rare-moments}. Its integral is \(o(S)\) under the present +assumptions. Damaged levels contribute at most \(W_b/2=o(S)\). Thus +\(S=\int f=o(S)\), a contradiction. This proves +\eqref{com:rare-positive-eq}. +\end{proof} + +The positive rare mass is a first-moment statement. To extract many +pairs at a single height scale, \Cref{sec:extraction} next controls its +large-height tail and removes errors in height-weighted measure before +selecting a band. Throughout that selection, the center-star variation +bound $C_Q=o(S)$ remains available under the original wedge law. diff --git a/preprints/A-power-saving-for-planar-unit-distances-September-23-2026/build/sections/06-extraction.tex b/preprints/A-power-saving-for-planar-unit-distances-September-23-2026/build/sections/06-extraction.tex new file mode 100644 index 0000000000000000000000000000000000000000..e0669e17c4d862074d8085a7c8e87329f7abbb83 --- /dev/null +++ b/preprints/A-power-saving-for-planar-unit-distances-September-23-2026/build/sections/06-extraction.tex @@ -0,0 +1,455 @@ +\section{Extracting multiplicatively organized pairs}\label{sec:extraction} + +We now use \Cref{com:variation,com:rare-positive}: the ordinary center-star +variation satisfies \(C_Q=o(S)\), whereas the point exception mass satisfies +\(\liminf \E_\mu H_p^*/S>0\). Here \(S\to\infty\), and the parameters, +levels, and exception indicators are those of \Cref{sec:budgets}. +Our objective is a graph on two copies of \(P\) whose edges carry ratios of +unit jumps. These ratios will have large height, while their alternating +products around every complete rectangle have negligible height. + +Throughout this section an \emph{ordinary wedge} is sampled by taking +\(q\) with law \(\kappa\) and then taking \(p,r\) independently and uniformly +among its neighbors in \(E\). Its two point marginals are \(\mu\). +We retain the notation \(W_c\) for the width of common levels, \(W_b\) for +the width of damaged rare levels, and \(m_{pq}\) for the integrated +inconsistency indicator of the edge \(pq\). In particular, +\begin{equation}\label{ext:inputs} + \E_\mu H_p^*+\E_\kappa H_q^*=O(S),\qquad + \E_E m_{pq}=O(1),\qquad + W_c\le S/\epsilon,\qquad W_b=o(S), + \quad \epsilon=k^{-1/16}. +\end{equation} +All integrals over levels include the prescribed place weights. + +\subsection{A pointwise estimate and its tail consequence} + +The first-moment bound in \eqref{ext:inputs} alone does not prevent all +exception mass from being carried by extremely rare points. We obtain the +needed tail control by applying the product formula separately to each +distinct pair of neighbors of a point. + +\begin{lemma}[Pointwise rare-level estimate]\label{ext:pointwise} +For every \(p\in P\) and every pair of distinct neighbors \(q,q'\in Q\), +\begin{equation}\label{ext:pointwise-bound} + 2H_p^*\le + 3\bigl(H_q^*+H_{q'}^*+m_{pq}+m_{pq'}\bigr) + +W_c+W_b+2\log 2. +\end{equation} +\end{lemma} + +\begin{proof} +At a good rare level, let \(X,U,V\) be the exception indicators of +\(p,q,q'\), and let \(I,I'\) be the inconsistency indicators of the two +edges. Write \(P_+\) for the rare positive charge in the point-star score: +it equals two when \(p\) is exceptional, both bits are minority bits, and +the two neighbors have matching nonshared coordinates; otherwise it is +zero. Write \(N=\abs{Y_{pq}-Y_{pq'}}\). +If \(U=V=I=I'=0\), the two bits are the same function of \(X\): they are +both majority bits for \(X=0\), and both minority bits for \(X=1\). +In the latter case the nonshared coordinate of both neighbors has the +dominant label, so \(P_+=2\). Consequently, pointwise, +\[ + P_+\ge 2X-2(U+V+I+I'),\qquad + N\le U+V+I+I', +\] +and hence +\begin{equation}\label{ext:local-rare-score} + P_+-N\ge 2X-3(U+V+I+I'). +\end{equation} +In particular, a simultaneous exception at \(p\) and at a neighbor is +charged directly to the neighbor indicator. + +For completeness, the upper bound for this rare-only score is also +pointwise. Put \(d=z_p-z_q\), \(d'=z_p-z_{q'}\). These jumps are distinct +and nonzero. At a place \(v\), sort their thresholds as \(b\le b'\), and set +\[ + u=-\log\abs{d-d'}_v,\qquad \alpha=u-b,\qquad + T_v=\begin{cases}\alpha+\log 2,&v\text{ archimedean},\\ + \alpha,&v\text{ nonarchimedean}. + \end{cases} +\] +The triangle inequality makes \(T_v\ge0\). Matching coordinates in a +square cell of side length \(r\) gives distance at most \(\sqrt2 r\), and +matching nonarchimedean labels gives distance at most \(r\). +Thus positive charges with both bits one lie within \([b-T_v,b]\), and +positive charges with both bits zero lie within \([b',b'+T_v]\). + +Only one of these two intervals can contain a rare positive charge. +Indeed every level where zero is the minority bit precedes every level +where one is the minority bit. A zero-bit rare positive would require a +threshold at least \(b'\), while a one-bit rare positive would require a +later threshold strictly less than \(b\), which is impossible. +The total rare positive charge at this place is therefore at most +\(2T_v\). The all-level negative charge is \(b'-b\), because the level +interval contains both thresholds. Moreover, +\[ + -\log\abs{(d-d')(1/d-1/d')}_v=2\alpha-(b'-b). +\] +Summing over places, the product formula cancels this last expression. +The archimedean weights sum to one, so the rare positives minus the +all-level negative charge total at most \(2\log2\). + +Integrating \eqref{ext:local-rare-score} gives a lower bound for that same +score. Outside the good rare levels there are no positive terms, and +the negative charge is at most \(W_c+W_b\). Comparing the lower and upper +bounds proves \eqref{ext:pointwise-bound}. +\end{proof} + +\begin{lemma}[Tail control]\label{ext:tail} +Let \(R_0=\epsilon^{-2}=k^{1/8}\). Then +\begin{equation}\label{ext:tail-bound} + \E_\mu\bigl[H_p^*\mathbf1_{\{H_p^*>R_0S\}}\bigr]=o(S). +\end{equation} +\end{lemma} + +\begin{proof} +Choose an absolute \(C_0\) for which each of the two exception expectations +in \eqref{ext:inputs} is at most \(C_0S\), for all sufficiently large +indices. Put +\[ + T=\{p:H_p^*>R_0S\},\qquad + A_t=\E_\mu[H_p^*\mathbf1_T],\qquad + d_{P,\min}=\min_{p\in P}d(p). +\] +On \(T\), the deterministic last three terms of +\eqref{ext:pointwise-bound} have sum at most \(H_p^*\), eventually, since +\[ + \frac{W_c+W_b+2\log2}{R_0S} + \le\epsilon+\frac{W_b+2\log2}{R_0S}=o(1). +\] +After absorbing them, we may cap each neighbor height at \(H_p^*\): +\begin{equation}\label{ext:capped-pointwise} + H_p^*\le3\bigl(\min(H_p^*,H_q^*)+\min(H_p^*,H_{q'}^*) + +m_{pq}+m_{pq'}\bigr),\qquad p\in T, +\end{equation} +for distinct \(q,q'\). If a cap changes a term, that capped term already +equals \(H_p^*\) and suffices for this inequality; otherwise the preceding +inequality is unchanged. + +Sample \(p\) with law \(\mu\) and two neighbors independently and uniformly +given \(p\). Restrict the left side to distinct neighbors. The exact +conditional probability of distinctness is \(1-1/d(p)\). On the +nonnegative right side we can drop this restriction. Thus +\begin{equation}\label{ext:tail-average} + (1-d_{P,\min}^{-1})A_t + \le 6\E_E[\min(H_p^*,H_q^*)\mathbf1_T]+C_m, +\end{equation} +where \(C_m\) is an absolute upper bound for \(6\E_E m_{pq}\). +This calculation uses the distinctness probability at each individual +point, so it remains valid even when its exception mass is very large. + +We estimate the capped expectation by layer cake. For \(u\ge0\) let +\[ + F_u=\{p:H_p^*>\max(u,R_0S)\},\qquad + G_u=\{q:H_q^*>u\}, +\] +and write \(h_P(u)=\mu(F_u)\), \(h_Q(u)=\kappa(G_u)\). +The marginal version of the expansion estimate from \Cref{sec:incidence}, +with \(\eta=\gamma-1/2>0\), is +\[ + \Prob_E(F,G)\le\sqrt{\mu(F)\kappa(G)} + \left[\gamma^{-(1+2\eta)}(\mu(F)\kappa(G))^\eta + +\gamma^{-1}\xi_t\right]. +\] +Because \(h_P(u)\le C_0/R_0\) and \(h_Q(u)\le1\), the same multiplier +\[ + \vartheta_t=\gamma^{-(1+2\eta)}(C_0/R_0)^\eta + +\gamma^{-1}\xi_t=o(1) +\] +works at every threshold \(u\), including thresholds where the center set +has full measure. Tonelli's theorem and Cauchy--Schwarz now give +\begin{align*} + \E_E[\min(H_p^*,H_q^*)\mathbf1_T] + &=\int_0^\infty\Prob_E(F_u,G_u)\dd u\\ + &\le\vartheta_t + \left(\int_0^\infty h_P(u)\dd u\right)^{1/2} + \left(\int_0^\infty h_Q(u)\dd u\right)^{1/2}\\ + &=\vartheta_t\sqrt{A_t\E_\kappa H_q^*} + \le\vartheta_t\sqrt{C_0A_tS}. +\end{align*} +The first integral equals the unknown tail first moment \(A_t\), including +its contribution below the threshold \(R_0S\). +Since \(d_{P,\min}\to\infty\), \eqref{ext:tail-average} implies eventually +\[ + \tfrac12 A_t\le6\vartheta_t\sqrt{C_0A_tS}+C_m + \le\tfrac14 A_t+36C_0\vartheta_t^2S+C_m. +\] +The last step is Young's inequality. Consequently +\[ + \frac{A_t}{S}\le144C_0\vartheta_t^2+\frac{4C_m}{S}\longrightarrow0, +\] +as required. +\end{proof} + +\subsection{Point profiles and a common height band} + +We next express a wedge's jump ratio as a difference of profiles attached +to its two point endpoints. On a good rare level \((v,s)\), let +\(X_x(v,s)=I_x(v,s)\) be the exception indicator of the vertex \(x\) +from \Cref{sec:budgets}, and let +\(\sigma(v,s)=1\) when one is the minority bit and \(-1\) when zero is +the minority bit. Define a real-valued profile, indexed by places, by +\begin{equation}\label{ext:profile-definition} + a_p(v)=\int_{\substack{s:\,(v,s)\text{ is}\\\text{good rare}}} + \sigma(v,s)X_p(v,s)\dd s. +\end{equation} +For a place profile \(b\), write \(\norm{b}_1=\sum_v^*\abs{b(v)}\). + +\begin{lemma}[Wedge profiles]\label{ext:profiles} +For an ordinary wedge \((p,q,r)\), there are nonnegative random variables +\(E_1,E_2\), with \(\E E_1+\E E_2=o(S)\), such that +\begin{align} + \norm{(n_{pq}-n_{rq})-(a_p-a_r)}_1&\le E_1, + \label{ext:profile-error}\\ + \norm{n_{pq}-n_{rq}}_1&\ge H_p^*+H_r^*-E_2. + \label{ext:profile-size} +\end{align} +Here \(n_{pq}(v)=-\log\abs{z_p-z_q}_v\). +\end{lemma} + +\begin{proof} +Let \(\mathcal G\) denote the good rare levels, and let \(I_{pq}\) be +the levelwise inconsistency indicator. At a level in \(\mathcal G\), +write \(b\in\{0,1\}\) for the majority bit. Consistency gives +\[ + Y_{pq}=b+\sigma(X_p+X_q) +\] +unless both endpoints are exceptional or the edge is inconsistent. +In all cases the absolute residual in this identity is at most +\(2(X_pX_q+I_{pq})\). Subtracting the identities for the two edges +cancels the center contribution. At each place, +\[ + n_{pq}(v)-n_{rq}(v)=\int(Y_{pq}-Y_{rq})\dd s, +\] +since both bits are single thresholds inside the level interval. +It follows that \eqref{ext:profile-error} holds with +\[ + E_1=\int_{\mathcal C}\abs{Y_{pq}-Y_{rq}}+W_b + +2\int_{\mathcal G}(X_pX_q+X_rX_q+I_{pq}+I_{rq}). +\] +Its mean is \(o(S)\): the first term averages to \(C_Q=o(S)\), the +joint edge exceptions have integrated expectation \(o(S)\), and the +remaining bounds are in \eqref{ext:inputs}. + +There is also the exact identity +\[ + \norm{n_{pq}-n_{rq}}_1=\int\abs{Y_{pq}-Y_{rq}}. +\] +On a good rare level with exactly one exceptional point endpoint, a +normal center, and two consistent edges, the bits differ. More +explicitly, at every good rare level, +\[ + \abs{Y_{pq}-Y_{rq}}\ge X_p+X_r + -2(X_pX_r+X_pX_q+X_rX_q+I_{pq}+I_{rq}). +\] +If none of the terms subtracted is present, this is the preceding +observation; if any is present, the right side is nonpositive. +Integrating proves \eqref{ext:profile-size} with +\[ + E_2=2\int_{\mathcal G} + (X_pX_r+X_pX_q+X_rX_q+I_{pq}+I_{rq}). +\] +The integrated joint point exceptions on a wedge and joint edge +exceptions are \(o(S)\), by \Cref{bud:rare-moments}. +Thus \(\E E_2=o(S)\) as well. +\end{proof} + +The next selection makes these average error bounds uniform before +choosing a dyadic band. This order matters because the probability of +the eventual band tends to zero. + +\begin{lemma}[A band of accurate wedges]\label{ext:band} +There are a constant \(c>0\), independent of \(t\) along the fixed +sequence, a sequence \(\delta_t\to0\), and scales \(J\to\infty\) such +that a set of ordinary wedges of probability at least +\begin{equation}\label{ext:band-mass} + \frac{c}{R_0\log(2R_0)}=t^{-o(1)} +\end{equation} +satisfies, uniformly on the set, +\begin{equation}\label{ext:band-conditions} + J\le H_p^*+H_r^*\le2J,\qquad + E_1+E_2\le\delta_t(H_p^*+H_r^*). +\end{equation} +Every such wedge has \(p\ne r\). +\end{lemma} + +\begin{proof} +Set \(W=H_p^*+H_r^*\) and \(Z=E_1+E_2\). By +\Cref{com:rare-positive}, choose \(c_0>0\), independent of \(t\) along +the fixed sequence, such that \(\E W\ge4c_0S\) eventually. +Choose \(\delta_t\to0\) sufficiently +slowly that \(\E Z/\delta_t=o(S)\). Before selecting any band, discard +wedges with \(Z>\delta_tW\). Their lost weight is at most +\[ + \E[W\mathbf1_{\{Z>\delta_tW\}}]\le\frac{\E Z}{\delta_t}=o(S). +\] +Also discard \(W2R_0S\), its larger summand exceeds \(R_0S\) and is at least +\(W/2\). Thus \Cref{ext:tail} gives +\[ + \E[W\mathbf1_{\{W>2R_0S\}}] + \le2\E[H_p^*\mathbf1_{\{H_p^*>R_0S\}}] + +2\E[H_r^*\mathbf1_{\{H_r^*>R_0S\}}]=o(S). +\] +The surviving weighted mass is at least \(2c_0S\), whereas every +surviving wedge has \(W\le2R_0S\). Its probability is therefore at +least \(c_0/R_0\). + +Partition \([c_0S,2R_0S]\) into at most +\(1+\lceil\log_2(2R_0/c_0)\rceil=O(\log(2R_0))\) dyadic bands, +using half-open endpoints except at the final endpoint. One band has +the probability in \eqref{ext:band-mass}; if \(J\) is its lower endpoint, +then \(J\ge c_0S\to\infty\), and \eqref{ext:band-conditions} holds. +The parameter hierarchy gives \(k=t^{o(1)}\), so the lower bound in +\eqref{ext:band-mass} is indeed \(t^{-o(1)}\). +Finally, if \(p=r\), the left side of \eqref{ext:profile-size} is zero, +whereas its right side is at least \((1-\delta_t)W>0\) for all +sufficiently large indices. Such a wedge cannot survive. +\end{proof} + +\subsection{Removing common unit circles and choosing witnesses} + +The algebraic argument will require that a retained pair does not lie +on a common unit circle containing many points of \(P\). This exclusion +must concern all real unit circles, including those whose centers do +not belong to \(Q\). + +\begin{lemma}[Popular-circle deletion]\label{ext:circles} +Call a real unit circle \emph{popular} if it contains at least +\(t^{9/10}\) points of \(P\). The ordinary wedge probability of +\(p,r\) lying on a common popular circle is at most +\begin{equation}\label{ext:circle-cost} + t^{-3/5+o(1)}+t^{-9/5+o(1)}+\frac{2}{\min_{q\in Q}d(q)}. +\end{equation} +This is \(o(1/(R_0\log(2R_0)))\). In particular, removing such wedges +from the set in \Cref{ext:band} preserves the lower probability bound +\eqref{ext:band-mass}, after decreasing its constant. +\end{lemma} + +\begin{proof} +Write \(M\) for the number of popular circles. Every pair of distinct +real points lies on at most two unit circles. Counting pairs on popular +circles therefore gives +\[ + M\binom{\lceil t^{9/10}\rceil}{2} + \le2\binom{\abs P}{2},\qquad M\le t^{1/5+o(1)}, +\] +using \(\abs P=t^{1+o(1)}\). +The set \(P_{\rm mult}\) of points on at least two popular circles has +size at most \(2\binom M2\le t^{2/5+o(1)}\), because distinct circles +intersect in at most two points. The degree bounds give +\[ + \max_p\mu(p)\le t^{-1+o(1)},\qquad + \max_q\kappa(q)\le t^{-2+o(1)}. +\] +Consequently wedges hitting \(P_{\rm mult}\) at either point endpoint +have probability at most \(t^{-3/5+o(1)}\). Centers in \(Q\) whose +own unit circle is popular number at most \(M\); wedges with such a +center have probability at most \(t^{-9/5+o(1)}\). + +To bound the remaining bad wedges, fix \(p\notin P_{\rm mult}\) and +a center \(q\) whose own circle is not popular, in the original sampling +law. The point \(p\) belongs to at most one popular circle. If there is +one, it differs from the unit circle centered at \(q\), since the +latter is not popular. Their intersection has at most two points. +In this original law, \(r\) is a uniform neighbor of \(q\), so the +probability that it shares a popular circle with \(p\) is at most +\(2/d(q)\), even before requiring \(r\notin P_{\rm mult}\). +This proves \eqref{ext:circle-cost}. + +For the comparison with the selected band, recall +\(R_0=k^{1/8}\), \(k=t^{o(1)}\), and the explicit requirement +\(k^{20}/\min_qd(q)\to0\) in \Cref{bud:parameters}. +The first two terms of \eqref{ext:circle-cost}, multiplied by +\(R_0\log(2R_0)\), tend to zero by a fixed power of \(t\). +For the third term, +\[ + \frac{R_0\log(2R_0)}{\min_qd(q)} + =\frac{k^{1/8}\log(2k^{1/8})}{k^{20}} + \frac{k^{20}}{\min_qd(q)}\longrightarrow0. +\] +This establishes the required relative loss bound. +\end{proof} + +We can now forget the probability weights and retain an ordinary graph. +The estimates already hold separately for every surviving wedge, so +choosing one center for each endpoint pair preserves them. + +\begin{proposition}[Organized pair graph]\label{ext:pairs} +For the incidence graphs and levels constructed above, there are +\(J\to\infty\), a sequence \(\omega_t\to0\), and a simple bipartite graph +\(\mathcal E\) between two copies of \(P\), each of size at most +\(t^{1+o(1)}\), with at least \(t^{2-o(1)}\) edges, having the following +properties. Each edge \((p,r)\) has distinct endpoints and an assigned +center \(q=q(p,r)\in Q\) with \(pq,rq\in E\). Set +\[ + \lambda_{pr}=\frac{z_p-z_{q(p,r)}}{z_r-z_{q(p,r)}}. +\] +No edge has both endpoints on any real unit circle containing at least +\(t^{9/10}\) points of \(P\). Uniformly over the edges and complete +rectangles of \(\mathcal E\), +\begin{align} + (z_p-z_r)(w_p-w_r)&=2-\lambda_{pr}-\lambda_{pr}^{-1}, + \label{ext:pair-identity}\\ + h(\lambda_{pr})&\ge J/3, \label{ext:pair-height}\\ + h\!\left(\frac{\lambda_{pr}\lambda_{p'r'}} + {\lambda_{pr'}\lambda_{p'r}}\right) + &\le\omega_tJ. \label{ext:rectangle-height} +\end{align} +In the last line all four indicated edges are required to be present; +their assigned centers need not agree. The edge ratios +\(\lambda_{pr}\) are unrelated to the fixed score constant \(\lambda\). +\end{proposition} + +\begin{proof} +Take the band from \Cref{ext:band}, and remove the wedges excluded by +\Cref{ext:circles}. Its probability remains at least +\(c/(R_0\log(2R_0))=t^{-o(1)}\), and all its endpoint pairs are distinct. +For any fixed off-diagonal pair \((p,r)\), its ordinary wedge probability is +\[ + \sum_{q:\,pq,rq\in E}\frac{1}{\abs E\,d(q)} + \le\frac{2}{\abs E\min_qd(q)}. +\] +Here the sum has at most two terms, since two distinct real points have +at most two unit-circle centers. The retained wedges therefore contain +at least +\[ + \frac{c\abs E\min_qd(q)}{2R_0\log(2R_0)}\ge t^{2-o(1)} +\] +distinct ordered endpoint pairs, using +\(\abs E=t^{2+2a+o(a)}\) and \(\min_qd(q)=t^{a+o(a)}\). +These pairs form \(\mathcal E\). Choose one surviving witnessing +center for each pair. + +For its two jumps \(d=z_p-z_q\), \(d'=z_r-z_q\), the exact unit equations +give \(w_p-w_q=1/d\) and \(w_r-w_q=1/d'\). Hence +\[ + (z_p-z_r)(w_p-w_r)=(d-d')(1/d-1/d') + =2-d/d'-d'/d, +\] +proving \eqref{ext:pair-identity}. The product formula gives +\[ + 2h(\lambda_{pr})=\sum_v^*\abs{\log\abs{\lambda_{pr}}_v} + =\norm{n_{pq}-n_{rq}}_1 + \ge(1-\delta_t)J, +\] +so \eqref{ext:pair-height} holds for all sufficiently large indices. + +Finally, put +\[ + b_{pr}(v)=\log\abs{\lambda_{pr}}_v+a_p(v)-a_r(v). +\] +By \eqref{ext:profile-error} and \eqref{ext:band-conditions}, +\(\norm{b_{pr}}_1\le2\delta_tJ\), uniformly on all retained edges. +On a complete rectangle, the unary profiles cancel, leaving +\[ + \log\left|\frac{\lambda_{pr}\lambda_{p'r'}} + {\lambda_{pr'}\lambda_{p'r}}\right|_v + =b_{pr}(v)+b_{p'r'}(v)-b_{pr'}(v)-b_{p'r}(v). +\] +Its weighted absolute sum is at most \(8\delta_tJ\). Another use of the +product formula bounds its height by \(4\delta_tJ\), proving +\eqref{ext:rectangle-height} with \(\omega_t=4\delta_t\). +The popular-circle exclusion holds by construction. +\end{proof} diff --git a/preprints/A-power-saving-for-planar-unit-distances-September-23-2026/build/sections/07-algebra.tex b/preprints/A-power-saving-for-planar-unit-distances-September-23-2026/build/sections/07-algebra.tex new file mode 100644 index 0000000000000000000000000000000000000000..76d39c616ea77c1752d7148cd8349ac89f134441 --- /dev/null +++ b/preprints/A-power-saving-for-planar-unit-distances-September-23-2026/build/sections/07-algebra.tex @@ -0,0 +1,697 @@ +\section{The algebraic obstruction}\label{sec:algebra} + +The preceding section produced many pairs whose edge ratios have large +height, although their multiplicative defects around rectangles have +negligible height. We now show that this is impossible. The argument uses +a fixed complete bipartite pattern. Its positions will be sufficiently +independent to separate several square roots, whereas differentiation of +the rectangle relations will force a dependence between them. + +Here and below a unit circle means a circle of radius exactly one in the +original real plane. For a point \(p=(x_p,y_p)\), retain the coordinates +\(z_p=x_p+iy_p\) and \(w_p=x_p-iy_p\). + +\begin{proposition}[Obstruction to multiplicative rectangles] +\label{alg:obstruction} +There is no sequence with the following properties. At index \(n\), let +\(t_n\to\infty\), let \(J_n\to\infty\), and let +\(P_n\subset\R^2\) be a finite set of algebraic points with +\(\lvert P_n\rvert\le t_n^{1+o(1)}\). Let +\(G_n\subset P_n\times P_n\) be a bipartite graph, using two copies of +\(P_n\), with at least \(t_n^{2-o(1)}\) edges. To each edge \((p,r)\) +assign a nonzero algebraic number \(\lambda_{pr}\). Suppose that, for an +absolute \(c>0\), +\begin{align} + (z_p-z_r)(w_p-w_r) + &=2-\lambda_{pr}-\lambda_{pr}^{-1}, + \label{alg:distance}\\ + h(\lambda_{pr})&\ge cJ_n, + \label{alg:large-height}\\ + h\left(\frac{\lambda_{pr}\lambda_{p'r'}} + {\lambda_{pr'}\lambda_{p'r}}\right)&=o(J_n) + \label{alg:rectangle} +\end{align} +uniformly over edges and complete rectangles, respectively. Finally, +suppose that no edge has both endpoints on a unit circle containing at +least \(t_n^{9/10}\) points of \(P_n\). Here \(h\) is the absolute height +from \Cref{ht:height}. +\end{proposition} + +We prove the proposition through a sequence of algebraic constructions. +Assume its data exist, and suppress the index when this causes no +ambiguity. At each index choose a finite normal number field \(F_n\) +containing all coordinates, ratios and \(i\). By \Cref{ht:height}, this +choice does not affect the height. In the application from +\Cref{ext:pairs}, take the field \(\mathcal N\) already used there: +every ratio is a quotient of coordinate differences. + +\subsection{Negligible height as a field of constants} + +Fix a nonprincipal ultrafilter \(\mathfrak u\) on \(\N\): a collection of +subsets closed under finite intersection and supersets, containing the +cofinite sets and exactly one part of every finite partition. Such a +collection is obtained by extending the cofinite filter to a maximal +proper filter. Call a set of indices \emph{large} if it belongs to +\(\mathfrak u\). Two sequences are identified when they agree on a large +set. Write +\[ + F=\prod_{n\to\mathfrak u} F_n, + \qquad + \mathcal U=\prod_{n\to\mathfrak u}\C, +\] +where \(F_n\) is the number field just chosen. These are characteristic-zero +fields, with \(F\subset\mathcal U\): the inverse of a nonzero class is +defined coordinatewise on its large set of nonzero entries. The field +\(\mathcal U\) is algebraically closed. Indeed, a polynomial of fixed +positive degree has that degree on a large set, and choosing one complex +root at each such index supplies a root in \(\mathcal U\). Complex +conjugation also acts coordinatewise on \(\mathcal U\). + +We use limits only along \(\mathfrak u\). Thus \(a_n=o(J_n)\) means that +\(\{n:\lvert a_n\rvert<\varepsilon J_n\}\) is large for every +\(\varepsilon>0\). A graph count is called \emph{full} +if it is at least \(t_n^{2-o(1)}\) in this sense. Define +\[ + k_0=\bigl\{[u_n]\in F:h(u_n)=o(J_n)\bigr\}. +\] + +\begin{lemma}\label{alg:small-height-field} +The set \(k_0\) is a subfield of \(F\), and every element of \(F\) +algebraic over \(k_0\) belongs to \(k_0\). +\end{lemma} + +\begin{proof} +The height inequalities +\[ + h(uv)\le h(u)+h(v),\qquad + h(u+v)\le h(u)+h(v)+\log 2,\qquad + h(u^{-1})=h(u)\quad(u\ne0) +\] +prove the field assertion, since \(J_n\to\infty\). These statements do +not depend on representatives, and \(h(0)=0\). + +Suppose that \(u\in F\) satisfies +\(u^m+a_{m-1}u^{m-1}+\cdots+a_0=0\) with \(a_r\in k_0\) and fixed +\(m\ge1\). At each finite index where the equation holds, the +nonarchimedean triangle inequality gives +\(\lvert u\rvert_v\le\max(1,\max_r\lvert a_r\rvert_v)\). At infinity +the same estimate holds with the additional factor \(m\): if +\(\lvert u\rvert_v>1\), divide the equation by +\(\lvert u\rvert_v^{m-1}\). Since the total weight of the infinite +places is one, +\[ + h(u)\le\sum_{r=0}^{m-1}h(a_r)+\log m. +\] +This estimate holds on a large set of indices. Its right side is +\(o(J_n)\), proving the claim. +\end{proof} + +In particular, any edge ratio selected from the graphs is transcendental +over \(k_0\), by \eqref{alg:large-height}. The rectangle defects in +\eqref{alg:rectangle}, in contrast, belong to \(k_0^\times\). We next +select finitely many edges whose positions retain enough algebraic +independence to exploit this distinction. + +\subsection{A complete grid with generic positions} + +We first restrict the graph to plane or bounded-degree curve loci of +minimum total dimension among those retaining a full edge count. This +restriction will give uniform bounds for the vertices satisfying additional +polynomial equations, allowing us to choose a complete grid whose limiting +positions have no relations, beyond the locus equations, over the field +generated by their coefficients. + +We use two standard algebraic facts. First, for every field \(K\), +\(K[Z,W]\) is a unique factorization domain +\cite[Lemma~10.120.10]{stacksUFD}. A polynomial in \(W\) with +coefficients in \(K[Z]\) is primitive if its coefficients have no +common nonconstant factor. Gauss's lemma then identifies +factorization of primitive polynomials in \(K[Z][W]\) with +factorization over \(K(Z)\): clear denominators and cancel common +prime factors of the coefficients. In particular, an irreducible +polynomial in \(K[Z,W]\) of positive \(W\)-degree remains irreducible in +\(K(Z)[W]\). Two primitive polynomials that are scalar multiples +over \(K(Z)\) differ by an element of \(K^\times\). + +Second, two distinct irreducible plane curves of degrees at most +\(D\), over an algebraically closed field, have at most \(D^2\) +common points. Indeed their projective closures have the same degrees +and no common component, so B\'ezout's theorem bounds the sum of their +intersection multiplicities by \(D^2\) +\cite[Section~5.3]{fultonCurves}. Every affine intersection contributes +at least one. We will apply this both over \(\C\) at finite indices +and over the algebraically closed field \(\mathcal U\). + +By a locus at an index we mean either the whole \((z,w)\)-plane or the +zero set of a nonconstant irreducible polynomial in \(\C[z,w]\) whose +degree has a fixed bound along the sequence. Its dimension is two or +one, respectively. Among pairs of such locus sequences retaining a full +number of graph edges, choose one of minimal total dimension. A minimum +exists since the pair of planes is available and the possible total +dimensions are \(2,3,4\). Finite partitions let us fix all degrees and +normalize one nonzero coefficient of each curve equation on a large +set. Denote the limiting loci in \(\mathcal U^2\) by \(V_1,V_2\), their +dimensions by \(d_1,d_2\), and the field generated over \(\Q\) by their +normalized equation coefficients and \(i\) by \(B_0\). There are finitely +many such coefficients, so \(B_0\) is countable. It is a subfield of +\(\mathcal U\), and need not be a subfield of \(F\). + +Each limiting curve equation is irreducible over \(\mathcal U\). +Otherwise a factorization into positive-degree factors would give, from +finitely many coefficient sequences, a factorization at a large set of +indices. The degrees of its factors are bounded by the degree of the +equation, so there is no unbounded factorization to consider. + +Each finite curve contains \(t_n^{1-o(1)}\) retained real points: divide +the full edge count by the upper bound for the other vertex set. Its +image under \((z,w)\mapsto(\overline w,\overline z)\) shares all these +points. The bounded intersection fact forces the two curves to be equal +on a large set. The limiting curve has the same symmetry. Neither +coordinate projection is constant, since a curve with constant \(z\) +would be a line \(z=c\), whose symmetric image \(w=\overline c\) is a +different line. Thus every curve equation has positive degree in both +variables. + +The loci cannot both be the same unit circle. To check the exact meaning +of this assertion, suppose their limiting equations, after normalization, +were +\begin{equation}\label{alg:unit-circle} + (z-u)(w-v)=1. +\end{equation} +Equality of its finitely many coefficients holds at a large set of +finite indices. The symmetry gives \(v=\overline u\), again an exact +coordinatewise identity on a large set. Hence the finite loci there are +the same real unit circle. Both vertex sets on it have size +\(t_n^{1-o(1)}>t_n^{9/10}\), contradicting the deletion hypothesis in +\Cref{alg:obstruction}. This reasoning concerns equality in the +ultraproduct; ordinary convergence of a radius to one would not imply +\eqref{alg:unit-circle}. + +Restrict the graph to these loci and prune it to minimum degree +\(t_n^{1-o(1)}\) on both sides. Explicitly, if its initial edge and total +vertex counts are \(e_n,v_n\), repeatedly remove vertices of degree less +than \(e_n/(2v_n)\). Fewer than \(e_n/2\) edges are removed. The +remaining graph still has a full count, both vertex sets have size +\(t_n^{1+o(1)}\), and the asserted minimum degree holds. + +\begin{lemma}[Uniform avoidance and a generic grid] +\label{alg:generic-grid} +For every fixed \(N,j\ge1\), one can choose a complete +\(N\)-by-\(j\) grid in the pruned graphs on a large set of indices +so that its limiting vertices +\(p_1,\ldots,p_N\in V_1\) and \(r_1,\ldots,r_j\in V_2\) have the +following property. Select both coordinates of each vertex on a plane +locus and its \(z\)-coordinate on a curve locus. All the resulting +\(Nd_1+jd_2\) coordinates are algebraically independent over \(B_0\). +The other coordinates on a curve are algebraic over these coordinates +and \(B_0\). In particular, +\begin{equation}\label{alg:joint-degree} + \operatorname{trdeg}_{B_0(r_1,\ldots,r_j)} + B_0(r_1,\ldots,r_j,p_1,\ldots,p_N)=Nd_1. +\end{equation} +The selected coordinates and edge ratios belong to \(F\). +\end{lemma} + +\begin{proof} +First, on a side whose locus is the plane, for each fixed degree bound +\(D\) there is a \(b_D>0\) such that every nonzero polynomial of degree +at most \(D\), even with coefficients depending on \(n\), vanishes at +at most \(t_n^{1-b_D}\) pruned vertices on a large set of indices. +If this failed, choose at each index a polynomial maximizing its number +of zeros among the finite vertex set. A maximum is attained because +there are only finitely many subsets of that set. Failure of every +fixed saving means that its zero count is \(t_n^{1-o(1)}\) along +\(\mathfrak u\). One of at most \(D\) irreducible factors then contains +\(t_n^{1-o(1)}\) vertices. Their minimum degree supplies +\(t_n^{2-o(1)}\) edges after restricting this side to a bounded-degree +curve, contradicting minimality. This proves the uniform assertion. +On a curve side, a nonzero polynomial of degree \(D\) in the selected +\(z\)-coordinate has at most \(D\) zeros among the vertices, since real +points have distinct \(z=x+iy\). + +Let the two pruned set sizes be \(m,n'\) and their edge density be +\(\delta\). Independent uniform vertices +\(p_1,\ldots,p_N\) have a common-neighbor fraction \(X\) on the other +side. Averaging degrees there and applying Jensen gives +\(\E X\ge\delta^N\). Therefore the number of ordered complete grids, +initially allowing repetitions, is +\begin{equation}\label{alg:grid-count} + m^N(n')^j\E X^j + \ge m^N(n')^j\delta^{Nj} + =t_n^{N+j-o(1)}. +\end{equation} + +Consider a fixed nonzero polynomial in the selected coordinates with +coefficients in \(B_0\). Choose representatives of its coefficients. +Its vanishing among all ordered vertex tuples, before requiring edges, +has a power saving: +\begin{equation}\label{alg:test-count} + \#\{\text{tuples at which the test vanishes}\} + \le t_n^{N+j-b} +\end{equation} +on a large set, for some \(b>0\) depending on the test. Here is an +induction that also handles moving coefficients. View the polynomial +as a polynomial in the last point block, and select one nonzero +coefficient polynomial in the earlier blocks. The earlier tuples +annihilating this coefficient have a power saving by induction. At +every other earlier tuple, the remaining polynomial is nonzero, with a +fixed degree bound. The uniform plane estimate, or the bounded root +count for a curve block, bounds its zeros. Multiply by the upper +bounds \(t_n^{1+o(1)}\) for the other block sizes and slightly decrease +the saving. The starting case is exactly the one-block estimates. +The finitely many coefficient identities and nonzero denominators +involved hold on a large set. + +There are countably many nonzero polynomial tests over \(B_0\). +Enumerate them as \(T_1,T_2,\ldots\). For each fixed \(s\), +\eqref{alg:grid-count} and \eqref{alg:test-count}, with a finite union +bound, show that on a large set some grid avoids all the first \(s\) +tests. At index \(n\), choose the largest feasible prefix length +\(s_n\le n\), and choose a grid avoiding that prefix. Include the +validity of all representatives and denominators of a prefix in its +feasibility requirement. If no prefix is feasible, use arbitrary +number-field coordinates at that index. For every fixed \(s\), the +set where \(n\ge s\) and the first \(s\) tests are feasible is large; +hence \(s_n\ge s\) on a large set. Each fixed polynomial test is +therefore nonzero in the limiting grid. Only finite intersections of +large sets were used. + +For a curve equation \(f(z,w)=0\), its positive \(w\)-degree and the +transcendence of \(z\) show that \(w\) is algebraic over \(B_0(z)\). +This proves all the transcendence assertions. The vertices and the +assigned ratios at each chosen finite grid lie in \(F_n\), so their +classes lie in \(F\). +\end{proof} + +For a plane locus \(V\) over a field \(\mathcal B\), define its function +field to be \(\mathcal B(V)=\mathcal B(z,w)\). For a curve with +irreducible equation \(f\), define it to be the fraction field of +\(\mathcal B[z,w]/(f)\). A point on \(V\) is called generic over +\(\mathcal B\) if its coordinates have transcendence degree +\(\dim V\) over that field. + +We record explicitly how generic coordinates identify a function field. +If \(B_0\subset\mathcal B\subset\mathcal U\) and a point on a locus +\(V\) has transcendence degree \(\dim V\) over \(\mathcal B\), then +evaluation at that point embeds \(\mathcal B(V)\) into \(\mathcal U\). +For the plane this is algebraic independence. For a curve, its equation +\(f\) is irreducible even over \(\mathcal U\), hence over +\(\mathcal B\). Its coefficients when viewed as a polynomial in \(w\) +have no common factor of positive degree in \(z\). They cannot all +vanish at an abscissa in \(\mathcal U\): a common root would give a +linear common factor over \(\mathcal U\), contradicting +irreducibility. If \(z\) were algebraic over \(\mathcal B\), the +nonzero specialized equation would make \(w\) algebraic as well. +Thus \(z\) is transcendental. Gauss's lemma makes \(f\) +irreducible over \(\mathcal B(z)\), so every polynomial relation at the +point is a multiple of \(f\). Every nonzero denominator in the function +field consequently evaluates to a nonzero element. The same argument +applies to a point after adjoining the other grid points to +\(\mathcal B\), using \Cref{alg:generic-grid}. + +\subsection{Differentiation and the surviving generic point} + +Take \(N=20\) and \(j=5\) in \Cref{alg:generic-grid}, and write +\(\lambda_{i\ell}\) for the edge ratios. Define +\[ + g_i=\lambda_{i1},\qquad + h_\ell=\frac{\lambda_{11}}{\lambda_{1\ell}},\qquad + \chi_{i\ell}=\frac{\lambda_{i\ell}\lambda_{11}} + {\lambda_{i1}\lambda_{1\ell}}. +\] +The rectangle condition gives \(\chi_{i\ell}\in k_0^\times\), and +\begin{equation}\label{alg:factorization} + \lambda_{i\ell}=g_i h_\ell^{-1}\chi_{i\ell}. +\end{equation} +Repeated rows or columns would violate the algebraic independence just +proved; all vertices on either side are distinct. + +\begin{lemma}\label{alg:derivation} +There is a \(k_0\)-derivation \(\partial:F\to F\) such that +\(\partial\lambda_{i\ell}\ne0\) for every grid edge. +\end{lemma} + +\begin{proof} +A derivation is an additive map satisfying the product rule; being a +\(k_0\)-derivation means that it vanishes on \(k_0\). Each +\(\lambda_{i\ell}\) is transcendental over \(k_0\) by +\Cref{alg:small-height-field}. Put it in a transcendence basis of +\(F/k_0\), prescribe its derivative to be one and those of the other +basis elements to be zero, and extend by the quotient rule to the +rational function field. Since the characteristic is zero, this +derivation extends uniquely over the algebraic extension to \(F\) +\cite[Theorems~1.5 and~4.1]{conradSeparability}. +For an infinite algebraic extension, take the compatible extensions +to its finite subextensions; uniqueness makes them agree on overlaps +\cite[Section~5]{conradSeparability}. + +We have thus obtained a witness derivation nonzero on each chosen +ratio. The derivations form an \(F\)-vector space. A linear combination +of the finitely many witnesses is nonzero on all ratios provided its +coefficients avoid the zeros of a product of finitely many nonzero +linear forms. This product is a nonzero polynomial over the infinite +field \(F\), so such coefficients exist. +\end{proof} + +Put +\[ + A_i=\frac{\partial g_i}{g_i},\qquad + A'_\ell=\frac{\partial h_\ell}{h_\ell},\qquad + H_i=(\partial x_{p_i},\partial y_{p_i}),\qquad + U_\ell=(\partial x_{r_\ell},\partial y_{r_\ell}). +\] +The derivation kills \(i\), which is algebraic over \(\Q\subset k_0\). +Define +\[ + L_\ell(z,w)=(z-z_{r_\ell})(w-w_{r_\ell}), + \qquad R_{i\ell}=\lambda_{i\ell}^{-1}-\lambda_{i\ell}. +\] +Differentiating \eqref{alg:distance} and \eqref{alg:factorization} gives +\begin{align} + 2(p_i-r_\ell)\cdot(H_i-U_\ell) + &=R_{i\ell}(A_i-A'_\ell), + \label{alg:differential}\\ + R_{i\ell}^{\,2} + &=L_\ell(p_i)\bigl(L_\ell(p_i)-4\bigr), + \label{alg:radicand}\\ + A_i-A'_\ell&\ne0. + \label{alg:nonzero-derivative} +\end{align} +Here the dot denotes the bilinear expression \(xu+yv\) in the ordinary +two coordinates, with no conjugation. The second identity follows by +squaring \(\lambda^{-1}-\lambda\), and the third by +\Cref{alg:derivation}. + +We now form a coefficient field containing the right-hand positions and +their derivative data. Let \(K_0\) be the subfield of \(\mathcal U\) +consisting of all elements algebraic over +\[ + B_0(r_1,\ldots,r_j,U_1,\ldots,U_j,A'_1,\ldots,A'_j). +\] +This field is algebraically closed: roots in \(\mathcal U\) of +polynomials over it are still algebraic over the displayed field. +We do not adjoin \(k_0\) to \(K_0\). + +\begin{lemma}\label{alg:surviving-point} +Some \(p=p_i\) has transcendence degree \(d_1\) over \(K_0\), and +evaluation identifies the function field \(K_0(V_1)\) with \(K_0(p)\). +\end{lemma} + +\begin{proof} +Over \(B_0(r_1,\ldots,r_j)\), all the left positions jointly have +transcendence degree \(Nd_1\), by \eqref{alg:joint-degree}. The right +velocities and scalars consist of at most \(3j\) elements. Adjoining +them and then taking algebraic closure can reduce that transcendence +degree by at most \(3j\). Therefore +\begin{equation}\label{alg:dimension-accounting} + \operatorname{trdeg}_{K_0}K_0(p_1,\ldots,p_N) + \ge Nd_1-3j. +\end{equation} +If no individual point had degree \(d_1\), subadditivity would bound +the left side by \(N(d_1-1)\). This contradicts \(N>3j\). +The function-field assertion is the evaluation argument following +\Cref{alg:generic-grid}, with \(\mathcal B=K_0\). +\end{proof} + +The velocity data may depend arbitrarily on the left positions. The +degree count above allows those correlations. No derivative of a locus +equation has been used, and none of its coefficients is being asserted +constant for the original derivation. + +We have reduced the proof to one generic point \(p\). Write +\(M=K_0(V_1)=K_0(p)\), \(R_\ell=R_{i\ell}\), and +\[ + D_\ell=L_\ell(L_\ell-4)\in M. +\] +The next step proves directly over this enlarged field that these +radicands are independent modulo squares. + +\subsection{A separate odd valuation for each radicand} + +An integer-valued valuation of \(M\) is a map +\(v:M^\times\to\Z\) with \(v(ab)=v(a)+v(b)\) and +\(v(a+b)\ge\min\{v(a),v(b)\}\), where \(v(0)=+\infty\). +Our valuations will count the order of a zero at a chosen point or +along an irreducible polynomial. The aim is to give each radicand +one simple zero that none of the others shares. + +\begin{lemma}\label{alg:private-valuations} +For every \(\ell\in\{1,\ldots,j\}\) there is an integer-valued +valuation \(v_\ell\) of \(M\) such that +\[ + v_\ell(D_\ell)=1,\qquad v_\ell(D_m)=0\quad(m\ne\ell). +\] +Consequently the classes of \(D_1,\ldots,D_j\) are independent in +\(M^\times/(M^\times)^2\). +\end{lemma} + +\begin{proof} +An odd valuation detects a nonsquare because the valuation of a square +is even. We construct the stated valuations over \(K_0\) in each +possible locus case. All noncoincidences used in the constructions will +first be proved from the original joint genericity over \(B_0\). + +\emph{The left locus is the plane.} +The polynomial +\[ + Q_\ell=(z-z_{r_\ell})(w-w_{r_\ell})-4 +\] +is irreducible in \(K_0[z,w]\). It is primitive as a polynomial in +\(w\), because its leading coefficient \(z-z_{r_\ell}\) is coprime +to its constant coefficient, and is linear over \(K_0(z)\). The +polynomials \(Q_\ell\) are distinct, as comparison of the coefficients +of \(z\) and \(w\) recovers their centers. None divides any +\(L_m=(z-z_{r_m})(w-w_{r_m})\). The exponent of \(Q_\ell\) in a +rational function is therefore the required valuation. + +\emph{The left locus is a curve, and either the loci differ or its +\(z\)-projection has degree at least two.} +Let \(f(z,w)\) be its irreducible equation and set +\(a=z_{r_\ell}\). The coordinate \(a\) is transcendental over +\(B_0\). Consequently the leading coefficient of \(f\) in \(w\) +does not vanish at \(a\), and \(f(a,w)\) has simple roots: the +coprimality of \(f,f_w\) over \(B_0(z)\), after clearing denominators, +excludes multiple roots at this transcendental abscissa. + +If \(V_2\ne V_1\), the anchor \(r_\ell\) is not on \(V_1\). +Indeed it is generic on \(V_2\), which is either the plane or a +different irreducible curve. Choose any root \(b\) of \(f(a,w)\). +Then \(b\ne w_{r_\ell}\). If \(V_2=V_1\) and the projection degree +is at least two, choose a root other than \(w_{r_\ell}\), again +obtaining \(b\ne w_{r_\ell}\). In both cases +\[ + s_\ell=(a,b)\in V_1,\qquad + f_w(a,b)\ne0,\qquad b\ne w_{r_\ell}. +\] +The ordinate \(b\) is algebraic over \(B_0(a)\). Since \(K_0\) +contains \(a\) and is algebraically closed, \(b\in K_0\). + +For \(m\ne\ell\), original joint genericity implies that +\(r_m\) remains generic on \(V_2\) over \(B_0(s_\ell)\). +Adjoining the sheet ordinate is an algebraic extension of adjoining +\(a\), so it does not reduce the transcendence degree of this +independent anchor. Neither coordinate of \(r_m\) equals the +corresponding coordinate of \(s_\ell\): both projections of a curve +are nonconstant, and the assertion for a plane is immediate. +Thus \(L_m(s_\ell)\ne0\). Also +\(L_m(s_\ell)\ne4\). Otherwise generic evaluation would force +\(V_2\) to lie in +\[ + (z-a)(w-b)=4. +\] +The plane cannot do so. If \(V_2\) is a curve, this irreducible +circle equation must be proportional to its defining equation. +Normalize the coefficient of \(zw\) to one. The coefficient of +\(w\) then recovers \(-a\) from coefficients in \(B_0\), +contradicting the transcendence of \(a\) over \(B_0\). +These are nonvanishing statements in \(\mathcal U\); placing more of their +entries in \(K_0\) cannot change them. + +To obtain the valuation over \(K_0\), introduce an indeterminate +\(T\) and solve +\[ + f(a+T,b+c_1T+c_2T^2+\cdots)=0 +\] +recursively in \(K_0[[T]]\). At each step the coefficient of the new +unknown is \(f_w(a,b)\ne0\), so the solution exists uniquely. +This gives an embedding \(K_0(V_1)\hookrightarrow K_0((T))\). +Indeed \(z\) maps to the transcendental element \(a+T\), and +irreducibility of \(f\) over \(K_0(z)\) shows that its only +polynomial kernel is \((f)\). Now +\[ + L_\ell=T\bigl(b-w_{r_\ell}+O(T)\bigr) +\] +has order one, while \(L_\ell-4\) and all +\(L_m(L_m-4)\), \(m\ne\ell\), have nonzero constant terms. +The order in \(T\) is the required valuation. + +\emph{Both loci are the same curve with projection degree one.} +The symmetry of this curve shows that the other projection has degree +one as well. Its irreducible equation therefore has degree one in +each variable. Solving for \(w\) gives either +\begin{equation}\label{alg:line-form} + w=ez+e',\qquad e\ne0, +\end{equation} +or +\begin{equation}\label{alg:fractional-form} + (z-u)(w-v)=b,\qquad b\ne0. +\end{equation} +The constants here belong to \(B_0\). To see the classification +directly, write the equation as \(azw+cz+dw+e_0=0\). If \(a=0\), +both linear coefficients are nonzero. If \(a\ne0\), complete the +product; its constant is nonzero by irreducibility. In +\eqref{alg:fractional-form}, \(b\ne1\), by the exact unit-circle +exclusion \eqref{alg:unit-circle}. + +In the line case, writing \(a_\ell=z_{r_\ell}\), +\[ + L_\ell=e(z-a_\ell)^2. +\] +Thus \(L_\ell-4\) has two distinct simple zeros. Choose one, say +\(z=a_\ell+c\) with \(c^2=4/e\), and let \(s_\ell\) be the +corresponding point on the line. Its coordinates are algebraic over +\(B_0(a_\ell)\). For an independent anchor \(r_m\), the conditions +\(L_m(s_\ell)=0\) or \(4\) would make \(a_m\) one of finitely +many elements algebraic over this field. This contradicts the joint +genericity. Since \(L_\ell=4\) at the chosen zero, order at that +abscissa in the rational function field \(K_0(z)\) gives the required +valuation. + +In the fractional-linear case, put +\(\zeta=(z-u)/(a_\ell-u)\), where \(a_\ell-u\ne0\). Direct +substitution gives +\begin{equation}\label{alg:fractional-distance} + L_\ell=-b\frac{(\zeta-1)^2}{\zeta}. +\end{equation} +The equation \(L_\ell-4=0\) is equivalent to +\[ + b\zeta^2+(4-2b)\zeta+b=0. +\] +Its discriminant is \(16(1-b)\ne0\), and its two roots are nonzero +because \(b\ne0\). Hence they give simple zeros of \(L_\ell-4\) +at finite points with \(z\ne u\). The corresponding point +\(s_\ell\) is algebraic over \(B_0(a_\ell)\). As a function of +the other anchor abscissa \(a_m\), its squared distance to that +anchor is +\[ + -b\frac{(z_{s_\ell}-a_m)^2} + {(z_{s_\ell}-u)(a_m-u)}. +\] +This is a nonconstant rational function: its numerator has degree two +and its denominator degree one, and \(z_{s_\ell}\ne u\). +Its values cannot be zero or four at the transcendental +\(a_m\) over \(B_0(s_\ell)\). Thus the simple zero is shared by +no other radicand, and order at its abscissa again gives the +valuation over \(K_0(z)\). + +These cases exhaust the possibilities. In each, a nonempty product +\(\prod_{\ell\in I}D_\ell\) has valuation one at any +\(v_\ell\) with \(\ell\in I\), and therefore is not a square. +\end{proof} + +The excluded value \(b=1\) illustrates why the deletion of popular +unit circles was necessary. In that case +\[ + L_\ell(L_\ell-4) + =\left(\frac{\zeta^2-1}{\zeta}\right)^2. +\] +Thus the private odd valuations would fail on a common unit circle. +For all the retained cases they were constructed after passing to +\(K_0\); no preservation of nonsquares under enlargement of the +coefficient field has been assumed. + +\subsection{An unused sign change} + +We use the quadratic case of Kummer theory over the characteristic-zero +field \(M\). If \(D_1,\ldots,D_j\in M^\times\) have independent +classes modulo squares and \(R_\ell^2=D_\ell\), then +\[ + [M(R_1,\ldots,R_j):M]=2^j, +\] +and every choice of signs of the \(R_\ell\) is induced by an +automorphism fixing \(M\) +\cite[Theorem~5.30 and Remark~5.32]{milneFields}. +The hypotheses hold because \(M\) contains both square roots of unity +and has characteristic different from two. Independence of the +square classes is required in this same field \(M\), as proved in +\Cref{alg:private-valuations}. + +Apply this fact with the radicands of +\Cref{alg:private-valuations}, and let +\(\mathcal M=M(R_1,\ldots,R_j)\subset\mathcal U\), using the +actual roots from \eqref{alg:radicand}. Write \(H=H_i\), \(A=A_i\) +for the surviving point. We have the equations +\begin{equation}\label{alg:linear-system} + 2(p-r_\ell)\cdot H-R_\ell A + =2(p-r_\ell)\cdot U_\ell-R_\ell A'_\ell, + \qquad 1\le\ell\le j. +\end{equation} +Their right-hand positions, velocities and scalars belong to \(K_0\). + +First suppose the three rows \(p-r_1,p-r_2,p-r_3\) have rank two. +The determinant of the first three equations in the unknowns +\(H_x,H_y,A\) is a linear combination of \(R_1,R_2,R_3\) over +\(M\). At least one coefficient is nonzero, because these +coefficients, up to nonzero numerical factors and signs, are the +two-by-two minors of the three position rows. Independent sign +changes show that distinct single radicals are linearly independent +over \(M\). Thus the determinant is nonzero. Cramer's rule places +the existing solution \(H_x,H_y,A\) in \(M(R_1,R_2,R_3)\). +The automorphism of \(\mathcal M\) that changes only the sign of +\(R_4\) fixes these unknowns. Apply it to the fourth equation in +\eqref{alg:linear-system} and subtract. Since \(R_4\ne0\), it +follows that \(A=A'_4\), contradicting +\eqref{alg:nonzero-derivative}. + +We verify when the rank condition can fail. If \(V_2\) is a plane +or a curve other than a line, the three independent generic anchors +are noncollinear. Indeed the first two are distinct, and a generic +third point can lie on their line only if its locus is contained +in that line, by generic evaluation. Three noncollinear anchors +make the three position rows have rank two for every \(p\). +If \(V_2\) is a line, its first two anchors are distinct. The rows +have rank two unless \(p\) belongs to that line. Since \(p\) is +generic on \(V_1\) over \(K_0\), this last possibility requires +\(V_1=V_2\) to be the same line. + +In that remaining case choose a nonzero direction vector +\(v\in K_0^2\) for the line, and write +\(p-r_\ell=s_\ell v\), with \(s_\ell\in M^\times\). +Nonzero row factors follow from distinctness of the points. +Only the scalar \(h=v\cdot H\) enters +\eqref{alg:linear-system}. The first two equations in \(h,A\) +have determinant +\[ + 2(s_2R_1-s_1R_2)\ne0, +\] +again by independence of the single radicals. Thus +\(h,A\in M(R_1,R_2)\). Changing only the sign of \(R_3\) in +the third equation gives \(A=A'_3\), the same contradiction. +The line has nonconstant projections, as established above; in +the form \eqref{alg:line-form} its direction has nonzero squared +length proportional to \(e\ne0\). + +This proves \Cref{alg:obstruction}. All sign changes were applied +inside the finite radical field \(\mathcal M\), after Cramer's rule +located the unknowns there. No automorphism of the whole field +\(F\), or compatibility of an automorphism with \(\partial\), is +required. + +\begin{proof}[Proof of \Cref{thm:main}] +If no absolute power saving existed, the sequence reduction in +\Cref{sec:incidence} would give configurations with +\(t^{4-o(1)}\) ordered unit pairs. Specialization and restriction +would give the incidence pieces used throughout the proof. Their +height scales \(S\) have a bounded subsequence or a subsequence +tending to infinity. The bounded alternative is excluded by +\Cref{ht:bounded}. In the unbounded alternative, +\Cref{ext:pairs} supplies the graphs, scales \(J\), edge ratios, +uniform height bounds and popular-circle deletion required by +\Cref{alg:obstruction}. That proposition excludes the other +alternative. Hence some fixed exponent below \(4/3\) admits a +uniform bound. Increasing the exponent to at least one preserves this +bound and keeps it below \(4/3\). Enlarging the positive constant covers +the finitely many remaining sizes \(n\ge2\), and passing from ordered to +unordered pairs only changes the constant. The cases \(n=0,1\) are +immediate. This proves the theorem. +\end{proof} diff --git a/preprints/A-power-saving-for-planar-unit-distances-September-23-2026/paper.pdf b/preprints/A-power-saving-for-planar-unit-distances-September-23-2026/paper.pdf new file mode 100644 index 0000000000000000000000000000000000000000..6fb2a8af0d4037045b1c9ffacce18a32e7657814 --- /dev/null +++ b/preprints/A-power-saving-for-planar-unit-distances-September-23-2026/paper.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:72bfa9153772b2124385cf4e53a5557320ab92e29fe4bdcb35a05a34481aede0 +size 619610 diff --git a/preprints/A-power-saving-for-square-difference-free-sets-September-24-2026/README.md b/preprints/A-power-saving-for-square-difference-free-sets-September-24-2026/README.md new file mode 100644 index 0000000000000000000000000000000000000000..904951c6272c0afc70c3fd55d07961ea6de178e3 --- /dev/null +++ b/preprints/A-power-saving-for-square-difference-free-sets-September-24-2026/README.md @@ -0,0 +1,17 @@ +# [A power saving for square-difference-free sets](paper.pdf) + +OpenAI +September 24, 2026 + +## Citation + +```bibtex +@misc{OAI:A-power-saving-for-square-difference-free-sets-September-24-2026, + author = {{OpenAI}}, + title = {{A power saving for square-difference-free sets}}, + howpublished = {OpenAI Math Release preprint + \href{https://github.com/openai/math/blob/main/preprints/A-power-saving-for-square-difference-free-sets-September-24-2026/paper.pdf}{OAI:A-power-saving-for-square-difference-free-sets-September-24-2026}}, + year = {2026} +} +``` + diff --git a/preprints/A-power-saving-for-square-difference-free-sets-September-24-2026/build/figures/list-regrouping.tex b/preprints/A-power-saving-for-square-difference-free-sets-September-24-2026/build/figures/list-regrouping.tex new file mode 100644 index 0000000000000000000000000000000000000000..d3b7fa1ce463d42df7a602591606093fc0eb012b --- /dev/null +++ b/preprints/A-power-saving-for-square-difference-free-sets-September-24-2026/build/figures/list-regrouping.tex @@ -0,0 +1,28 @@ +\begin{figure}[htbp] +\centering +\[ + \underbrace{\left((Y_{(u,w),i})_{i\in[m_s]}\right)_{w\in\Pi_+}} + _{r\text{ lists, each of length }m_s} + \quad\longleftrightarrow\quad + \underbrace{(Y'_{u,(w,i)})_{(w,i)\in\Pi_+\times[m_s]}} + _{\text{one list of length }m_{s+1}=r m_s} +\] +\[ + \begin{gathered} + j'_v=(w,i)\ \text{ uniform in }\Pi_+\times[m_s] + \quad\xrightarrow{\ \text{restrict to }w=v_b\ }\quad + Y'_{u,(v_b,i)}=Y_{(u,v_b),i},\\[2pt] + \mathbb P(w=v_b)=r^{-1},\qquad + \mathbb P(\text{all }v\in V_+\text{ select their own sublist})=r^{-n}. + \end{gathered} +\] +\caption{The two steps when marking a block $b$. For a new address +$u\in U_{S\cup\{b\}}$, the old plus addresses above it are $(u,w)$, +where $w\in\Pi_+$ is the word in block $b$; the prime on $Y'$ marks +only a relabeling of the same entries. Regrouping the independent lists +preserves their distribution without a loss. For each $v\in V_+$, +with $u=u_{S\cup\{b\}}(v)$, restricting its independent new index to +$w=v_b$ recovers its old emitted label. This second step has the total +probability $r^{-n}$.} +\label{fig:list-regrouping} +\end{figure} diff --git a/preprints/A-power-saving-for-square-difference-free-sets-September-24-2026/build/figures/parameter-roles.tex b/preprints/A-power-saving-for-square-difference-free-sets-September-24-2026/build/figures/parameter-roles.tex new file mode 100644 index 0000000000000000000000000000000000000000..4304561af705ec5eabd77347fce1120f89a2b7b1 --- /dev/null +++ b/preprints/A-power-saving-for-square-difference-free-sets-September-24-2026/build/figures/parameter-roles.tex @@ -0,0 +1,27 @@ +\begin{table}[htbp] +\centering +\small +\begin{tabular}{@{}p{.20\textwidth}p{.71\textwidth}@{}} +\toprule +Parameters & Role and order of choice\\ +\midrule +$h,t_0,\ell,d$ & Fixed cycle length, marking depth, block count, and tuple size +(Section~\ref{sec:tuple-laws}).\\[3pt] +$\gamma,\beta$ & Fixed conductor-decay and rational-frequency cutoff exponents +(Sections~\ref{sec:functional} and~\ref{sec:square-kernel}).\\[3pt] +$P,M_0$ & Prime-law threshold and modulus containing all smaller primes; +fixed before choosing the decay exponent +(Section~\ref{sec:scale-transfer}).\\[3pt] +$\tau,\sigma_0$ & Fixed partition exponent and common error exponent; +chosen in~\eqref{eq:scale-parameters}.\\[3pt] +$\rho,a_0$ & Fixed positive residue proportion and induction exponent; +$a_0$ is chosen only after $M_0$ and $\rho$ in~\eqref{eq:absolute-exponent-choice}.\\[3pt] +$Q,H,\mathcal P_N$ & Scale-dependent cutoffs and prime set, defined from $N$ +in~\eqref{eq:scale-parameters}.\\ +\bottomrule +\end{tabular} +\caption{Parameter roles. Every construction size and exponent is fixed +independently of $N$ and $A$; the Fourier cutoffs and available prime +coordinates change with $N$.} +\label{tab:parameter-roles} +\end{table} diff --git a/preprints/A-power-saving-for-square-difference-free-sets-September-24-2026/build/figures/proof-map.tex b/preprints/A-power-saving-for-square-difference-free-sets-September-24-2026/build/figures/proof-map.tex new file mode 100644 index 0000000000000000000000000000000000000000..f9aa7e1f29e4c9df92af5203468ff454263f4f80 --- /dev/null +++ b/preprints/A-power-saving-for-square-difference-free-sets-September-24-2026/build/figures/proof-map.tex @@ -0,0 +1,33 @@ +\begin{figure}[htbp] +\centering +\begin{tikzpicture}[ + box/.style={draw,rounded corners=2pt,align=center,font=\small, + text width=3.55cm,minimum height=1.1cm,inner sep=5pt}, + arr/.style={-{Stealth[length=5pt]},thick},x=1cm,y=1cm] +\node[box] (graph) at (-4.9,0) {Graph mixing\\Section~\ref{sec:finite-field-graphs}}; +\node[box] (tuple) at (0,0) {Tuple laws\\Section~\ref{sec:tuple-laws}}; +\node[box] (functional) at (4.9,0) {Signed seminorm and\\conductor decay\\Section~\ref{sec:functional}}; +\node[box] (moments) at (-3.5,-2) {Uniform lift moments\\Section~\ref{sec:lift-moments}}; +\node[box] (kernel) at (3.5,-2) {Actual-square kernel\\Section~\ref{sec:square-kernel}}; +\node[box] (transfer) at (-3.5,-4) {Square-step comparison\\Section~\ref{sec:scale-transfer}}; +\node[box] (cancel) at (3.5,-4) {Ordered-interval cancellation\\Section~\ref{sec:ordered-contraction}}; +\node[box,text width=6.4cm] (induct) at (0,-6) {Exceptional-measure expansion,\\weighted recurrence and induction\\Section~\ref{sec:induction}}; +\draw[arr] (graph) -- (tuple); +\draw[arr] (tuple) -- (functional); +\draw[arr] (moments) -- (transfer); +\draw[arr] (moments.south east) -- (cancel.north west); +\draw[arr] (kernel) -- (cancel); +\draw[arr] (functional.east) -- (7.1,0) -- (7.1,-4) -- (cancel.east); +\draw[arr] (functional.south west) -- (1.9,-.95) -- (-6.1,-.95) -- (-6.1,-4) -- (transfer.west); +\draw[arr] (tuple.south) -- (0,-1.25) -- (0,-3.2) -- (cancel.north west); +\draw[arr] (transfer) -- (induct); +\draw[arr] (cancel) -- (induct); +\end{tikzpicture} +\caption{Principal proof inputs. The full tuple law supplies reflection +positivity for the signed seminorm; the normalized nonexceptional law +supplies the strict-square pair law for cancellation. Both have the +conditional mixing used in conductor decay. The two lower branches are +combined only in the weighted recurrence. Arrows indicate dependencies, +not equivalences.} +\label{fig:proof-map} +\end{figure} diff --git a/preprints/A-power-saving-for-square-difference-free-sets-September-24-2026/build/main.tex b/preprints/A-power-saving-for-square-difference-free-sets-September-24-2026/build/main.tex new file mode 100644 index 0000000000000000000000000000000000000000..9e6e55b616713c1489e96563d9fffb8d455d3ce8 --- /dev/null +++ b/preprints/A-power-saving-for-square-difference-free-sets-September-24-2026/build/main.tex @@ -0,0 +1,60 @@ +\documentclass[11pt]{article} +\pdfinfoomitdate=1 +\pdftrailerid{} +\pdfsuppressptexinfo=15 +\usepackage[T1]{fontenc} +\usepackage{lmodern} +\usepackage[margin=1in]{geometry} +\usepackage{amsmath,amssymb,amsthm,mathtools} +\usepackage{microtype} +\usepackage{enumitem} +\usepackage{booktabs} +\usepackage{tikz} +\usetikzlibrary{arrows.meta,positioning} +\usepackage{xcolor} +\usepackage{hyperref} +\hypersetup{colorlinks=true,linkcolor=blue!50!black,citecolor=blue!50!black,urlcolor=blue!50!black, + pdftitle={A power saving for square-difference-free sets},pdfauthor={OpenAI}} +\usepackage[capitalise,noabbrev]{cleveref} +\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{\E}{\mathbb E} +\renewcommand{\P}{\mathbb P} +\newcommand{\R}{\mathbb R} +\newcommand{\C}{\mathbb C} +\newcommand{\F}{\mathbb F} +\newcommand{\Z}{\mathbb Z} +\newcommand{\N}{\mathbb N} +\DeclareMathOperator{\supp}{supp} +\DeclareMathOperator{\lcm}{lcm} +\title{A power saving for square-difference-free sets} +\author{OpenAI} +\date{September 24, 2026} +\begin{document} +\maketitle +\begin{abstract} +We prove that there are absolute constants $c>0$ and $C<\infty$ such that +every set $A\subseteq\{1,\ldots,N\}$ with no nonzero square difference +satisfies $|A|\le C N^{1-c}$. This answers the fixed-power question posed +by Green and Sawhney. +\end{abstract} +\tableofcontents +\input{sections/introduction} +\input{sections/graph} +\input{sections/tuples} +\input{sections/functional} +\input{sections/lifts} +\input{sections/kernel} +\input{sections/transfer} +\input{sections/contraction} +\input{sections/induction} +\bibliographystyle{plain} +\bibliography{references} +\end{document} diff --git a/preprints/A-power-saving-for-square-difference-free-sets-September-24-2026/build/references.bib b/preprints/A-power-saving-for-square-difference-free-sets-September-24-2026/build/references.bib new file mode 100644 index 0000000000000000000000000000000000000000..593e27d3bb6b0fa6c49401bd63a4452a4f079b59 --- /dev/null +++ b/preprints/A-power-saving-for-square-difference-free-sets-September-24-2026/build/references.bib @@ -0,0 +1,203 @@ +@article{Furstenberg1977, + author = {Furstenberg, H.}, + title = {Ergodic behavior of diagonal measures and a theorem of {Szemer{\'e}di} on arithmetic progressions}, + journal = {Journal d'Analyse Math{\'e}matique}, + volume = {31}, + pages = {204--256}, + year = {1977}, + doi = {10.1007/BF02813304}, + url = {https://doi.org/10.1007/BF02813304} +} + +@article{Sarkozy1978I, + author = {S{\'a}rk{\"o}zy, A.}, + title = {On difference sets of sequences of integers. {I}}, + journal = {Acta Mathematica Academiae Scientiarum Hungaricae}, + volume = {31}, + number = {1--2}, + pages = {125--149}, + year = {1978}, + doi = {10.1007/BF01896079}, + url = {https://doi.org/10.1007/BF01896079} +} + +@article{PSS1988, + author = {Pintz, J. and Steiger, W. L. and Szemer{\'e}di, E.}, + title = {On sets of natural numbers whose difference set contains no squares}, + journal = {Journal of the London Mathematical Society}, + series = {2}, + volume = {37}, + number = {2}, + pages = {219--231}, + year = {1988}, + doi = {10.1112/jlms/s2-37.2.219}, + url = {https://doi.org/10.1112/jlms/s2-37.2.219} +} + +@article{BloomMaynard2022, + author = {Bloom, Thomas F. and Maynard, James}, + title = {A new upper bound for sets with no square differences}, + journal = {Compositio Mathematica}, + volume = {158}, + number = {8}, + pages = {1777--1798}, + year = {2022}, + doi = {10.1112/S0010437X22007679}, + url = {https://doi.org/10.1112/S0010437X22007679}, + note = {arXiv version 2, February 24, 2021}, + eprint = {2011.13266}, + archivePrefix = {arXiv}, + primaryClass = {math.NT} +} + +@misc{GreenSawhney2025, + author = {Green, Ben and Sawhney, Mehtaab}, + title = {New bounds for the {Furstenberg--S{\'a}rk{\"o}zy} theorem}, + year = {2025}, + eprint = {2411.17448}, + archivePrefix = {arXiv}, + primaryClass = {math.NT}, + doi = {10.48550/arXiv.2411.17448}, + url = {https://arxiv.org/abs/2411.17448v2}, + howpublished = {\href{https://arxiv.org/abs/2411.17448v2}{arXiv:\allowbreak2411.17448v2}}, + note = {Version 2, January 10, 2025} +} + +@misc{Krachun2026, + author = {Krachun, Dmitry}, + title = {Square-Difference-Free Sets beyond the Three-Quarter Barrier}, + year = {2026}, + eprint = {2608.01325}, + archivePrefix = {arXiv}, + primaryClass = {math.CO}, + doi = {10.48550/arXiv.2608.01325}, + url = {https://arxiv.org/abs/2608.01325v1}, + howpublished = {\href{https://arxiv.org/abs/2608.01325v1}{arXiv:\allowbreak2608.01325v1}}, + note = {Version 1, August 2, 2026} +} + +@article{FILS1978ReflectionI, + author = {Fr{\"o}hlich, J{\"u}rg and Israel, Robert B. and Lieb, Elliott H. and Simon, Barry}, + title = {Phase transitions and reflection positivity. {I}. General theory and long range lattice models}, + journal = {Communications in Mathematical Physics}, + volume = {62}, + number = {1}, + pages = {1--34}, + year = {1978}, + doi = {10.1007/BF01940327}, + url = {https://doi.org/10.1007/BF01940327} +} + +@article{ConlonLee2017Reflection, + 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}, + eprint = {1611.05784}, + archivePrefix = {arXiv}, + primaryClass = {math.CO} +} + +@article{Ruzsa1984, + author = {Ruzsa, I. Z.}, + title = {Difference sets without squares}, + journal = {Periodica Mathematica Hungarica}, + volume = {15}, + number = {3}, + pages = {205--209}, + year = {1984}, + doi = {10.1007/BF02454169}, + url = {https://doi.org/10.1007/BF02454169} +} + +@misc{BeigelGasarch2008, + author = {Beigel, Richard and Gasarch, William}, + title = {Square-Difference-Free Sets of Size {$\Omega(n^{0.7334\ldots})$}}, + year = {2008}, + eprint = {0804.4892}, + archivePrefix = {arXiv}, + primaryClass = {math.CO}, + url = {https://arxiv.org/abs/0804.4892v3}, + howpublished = {\href{https://arxiv.org/abs/0804.4892v3}{arXiv:\allowbreak0804.4892v3}}, + note = {Version 3, May 8, 2008} +} + +@article{Lewko2015, + author = {Lewko, Mark}, + title = {An improved lower bound related to the {Furstenberg--S{\'a}rk{\"o}zy} theorem}, + journal = {Electronic Journal of Combinatorics}, + volume = {22}, + number = {1}, + pages = {Paper P1.32, 6 pp.}, + year = {2015}, + doi = {10.37236/4656}, + url = {https://doi.org/10.37236/4656} +} +@misc{AdajarEtAl2026, + author = {Adajar, Carlo Francisco E. and Agrawal, Rishika and Choudhuri, Mukul Rai and Chuah, Chian Yeong and Fan, Steve and Hegde, Swaroop and Lott, Andrew and Nandakumar, Krishnamohan and Ponagandla, Nagendar Reddy}, + title = {Extensions of the {Furstenberg--S\'ark\"ozy} theorem via the arithmetic level-$d$ inequality}, + year = {2026}, + eprint = {2605.16216}, + archivePrefix = {arXiv}, + primaryClass = {math.NT}, + doi = {10.48550/arXiv.2605.16216}, + url = {https://arxiv.org/abs/2605.16216v2}, + howpublished = {\href{https://arxiv.org/abs/2605.16216v2}{arXiv:\allowbreak2605.16216v2}}, + note = {Version 2, August 28, 2026} +} + +@article{FrohlichLieb1978, + author = {Fr{\"o}hlich, J{\"u}rg and Lieb, Elliott H.}, + title = {Phase transitions in anisotropic lattice spin systems}, + journal = {Communications in Mathematical Physics}, + volume = {60}, + number = {3}, + pages = {233--267}, + year = {1978}, + doi = {10.1007/BF01612891}, + url = {https://doi.org/10.1007/BF01612891} +} +@article{Hatami2010, + author = {Hatami, Hamed}, + title = {Graph norms and {Sidorenko}'s conjecture}, + journal = {Israel Journal of Mathematics}, + volume = {175}, + pages = {125--150}, + year = {2010}, + note = {Theorem locators refer to \href{https://arxiv.org/abs/0806.0047v1}{arXiv:\allowbreak0806.0047v1}}, + doi = {10.1007/s11856-010-0005-1}, + url = {https://doi.org/10.1007/s11856-010-0005-1}, + eprint = {0806.0047}, + archivePrefix = {arXiv} +} + +@article{MontgomeryVaughan1973, + author = {Montgomery, H. L. and Vaughan, R. C.}, + title = {The large sieve}, + journal = {Mathematika}, + volume = {20}, + number = {2}, + pages = {119--134}, + year = {1973}, + doi = {10.1112/S0025579300004708}, + url = {https://doi.org/10.1112/S0025579300004708} +} + +@article{DeLaPenaMontgomerySmith1995, + author = {de la Pe{\~n}a, Victor H. and Montgomery-Smith, Stephen J.}, + title = {Decoupling inequalities for the tail probabilities of multivariate {$U$}-statistics}, + journal = {The Annals of Probability}, + volume = {23}, + number = {2}, + pages = {806--816}, + year = {1995}, + doi = {10.1214/aop/1176988291}, + url = {https://arxiv.org/abs/math/9309211v2}, + eprint = {math/9309211}, + archivePrefix = {arXiv}, + note = {Section locators refer to \href{https://arxiv.org/abs/math/9309211v2}{arXiv:\allowbreak math/9309211v2}} +} diff --git a/preprints/A-power-saving-for-square-difference-free-sets-September-24-2026/build/sections/contraction.tex b/preprints/A-power-saving-for-square-difference-free-sets-September-24-2026/build/sections/contraction.tex new file mode 100644 index 0000000000000000000000000000000000000000..28632fb6f56800962960fe9ce5748c4399404fb7 --- /dev/null +++ b/preprints/A-power-saving-for-square-difference-free-sets-September-24-2026/build/sections/contraction.tex @@ -0,0 +1,313 @@ +\section{Contraction on ordered intervals} +\label{sec:ordered-contraction} + +A tuple of residue classes +\(\boldsymbol s=(s_v)_{v\in V}\in(\Z/M_0\Z)^V\) is called +\emph{good} if +\[ + s_{v_{j+1}}-s_{v_j} + \quad\text{is a strict square class modulo }M_0 + \qquad(0\le j<24), +\] +where \(v_{24}=v_0\). Thus each difference is \(1\) modulo \(8\) +and a nonzero square at every odd prime dividing \(M_0\). Let +\begin{equation} + \rho=M_0^{-d}\bigl|\{\boldsymbol s\in(\Z/M_0\Z)^V: + \boldsymbol s\text{ is good}\}\bigr|. + \label{eq:good-residue-proportion} +\end{equation} +We will bound the contribution of each good tuple when every prime uses +the nonexceptional part of its measure. First we show that these assignments +have a fixed positive proportion; the proof also explains the cycle length +$24$. We retain the parameters and lifts from the preceding section. + +\begin{lemma}[Existence of good residue assignments] +\label{lem:good-residues} +The proportion $\rho$ defined in~\eqref{eq:good-residue-proportion} satisfies +\[ + \rho\ge M_0^{-23}>0. +\] +\end{lemma} + +\begin{proof} +We construct a length-$24$ sequence of admissible increments with sum zero at +each factor of $M_0$. Modulo $8$, take all the increments equal to $1$; their +sum is $24\equiv0\pmod8$. + +Let $p6\beta\), this +length is at least \(10Q^6\). Lemma~\ref{lem:lift-moments} therefore +applies uniformly to every interval lift and every exact-denominator +truncation used below. Its application to \(f_s\), which is bounded by +the fixed constant \(M_0\), changes only the constants. All occurrences +of \(N^{o(1)}\) in this proof are uniform over these interval choices, +residues and sets. + +First consider assignments for which all 24 cycle slots choose the same +interval. Their total probability is +\[ + \sum_{i=1}^K w_i^{24} + \le(\max_i w_i)^{23}\ll N^{-23\tau}. +\] +For every assignment, ordinary H\"older and the uniform single-slot +marginals give +\[ + \left|\int\prod_v G_v(z_v)\,d\Lambda^\circ\right| + \le\prod_v\|G_v\|_d=N^{o(1)}, + \qquad + G_v=\mathcal L_{I(v),\mathcal P_N,Q}f_{s_v}. +\] +Here \(d\) is even, so the required moment is supplied directly by +Lemma~\ref{lem:lift-moments}. The contribution of these assignments is +therefore \(O(N^{-23\tau+o(1)})\); there is no restriction on the interval +choices in the other slots. + +In every remaining assignment there is a directed cycle edge \(u\to v\) +whose first interval precedes its second. Indeed, if the cyclic sequence +of interval indices had no ascent, it would be nonincreasing all the way +around the cycle, and hence constant. Fix one ascending edge, and denote +its two intervals by \(I_1,I_2\) for the rest of this argument. All +integers in \(I_1\) precede all integers in \(I_2\). + +\textit{Eliminating the other slots.}\quad +Retain these two lifts in full. In each of the other \(d-2\) slots, +truncate the lift to exact conductors at most \(T=N^\tau\). +Lemma~\ref{lem:conductor-decay}, applied during a telescoping replacement +of these slots, bounds the total error by +\begin{equation} + O(T^{-\gamma}N^{o(1)})=O(N^{-\gamma\tau+o(1)}). + \label{eq:interval-truncation-error} +\end{equation} +Every norm used here is controlled by Lemma~\ref{lem:lift-moments}: +the retained and discarded pieces are subfamilies of complete exact +denominator pieces. This error is estimated before the Fourier expansion +that follows. + +Each Fourier coefficient in a truncated slot has modulus at most \(M_0\), +and there are at most \(\sum_{q\le T}\varphi(q)\ll T^2\) frequencies. +The product expansion of all other slots consequently has total +coefficient absolute sum at most +\begin{equation} + C_{d,M_0}T^{2(d-2)}\ll N^{2d\tau}. + \label{eq:interval-fourier-cost} +\end{equation} +For one term in this expansion, let \(B\subseteq\mathcal P_N\) be the +union of the prime supports of its characters, and put +\[ + R=\prod_{p\in B}p. +\] +The conductors are squarefree, so their least common multiple satisfies +\begin{equation} + R\le T^{d-2}\le N^{d\tau}. + \label{eq:interval-mode-modulus} +\end{equation} + +Condition on all slot labels at the primes in \(B\). The expanded +characters become constants of modulus one. Since \(\Lambda^\circ\) is +a product over primes, the tuple laws at the remaining primes are +unchanged and independent. The two selected labels now have fixed +coordinates \(a_{1,B},a_{2,B}\) at \(B\), whose difference is a nonzero +square at each prime in \(B\). At every remaining prime their pair +marginal is still uniform on strict square pairs. + +Let \(\mathcal S=\mathcal P_N\setminus B\), let \(s_1=s_u\) and +\(s_2=s_v\), and define functions on \(X_{\mathcal S}\) by +\[ + g_i(z)=\bigl(\mathcal L_{I_i,\mathcal P_N,Q}f_{s_i}\bigr)(a_{i,B},z), + \qquad i=1,2. +\] +With Fourier coefficients normalized for uniform measure on +\(X_{\mathcal S}\), their conditional pair integral is exactly +\begin{equation} + \sum_{\xi\text{ on }X_{\mathcal S}} + \widehat g_1(-\xi)\widehat g_2(\xi)\mathfrak g(\xi). + \label{eq:interval-pair-fourier} +\end{equation} +To see this, simultaneous translation kills character pairs whose +frequencies do not sum to zero. For a matching pair, the multiplier is +the character average of the difference of the labels. At every odd +prime a uniform nonzero square is the square of a uniform nonzero +unit. The Chinese remainder theorem therefore identifies this average +with the unit-square multiplier \(\mathfrak g(\xi)\) used in +Proposition~\ref{prop:square-kernel}. + +\textit{Estimating the selected pair.}\quad +Specialization on a uniform coordinate block of size \(R\) costs at +most \(R^{1/2}\) in \(L^2\): for every function \(G\), +\[ + \|G(a_B,\cdot)\|_2^2\le R\|G\|_2^2. +\] +This inequality holds at every specified fiber, independently of the +law under which that fiber was selected. Hence +\(\|g_i\|_2\le R^{1/2}N^{o(1)}\). By the bound +\(|\mathfrak g(\xi)|\le q(\xi)^{-1/3}\), Parseval and +Cauchy--Schwarz, the part of~\eqref{eq:interval-pair-fourier} with +conductor exceeding \(H\) has modulus at most +\begin{equation} + H^{-1/3}\|g_1\|_2\|g_2\|_2 + \le H^{-1/3}R N^{o(1)}. + \label{eq:interval-pair-tail} +\end{equation} + +We next identify the remaining coefficients with actual integer fibers. +Put \(D=M_0R\). The parameter inequalities +\begin{equation} + d\tau<\frac{\beta\gamma}{1000}<\frac\beta2 + \label{eq:interval-parameter-room} +\end{equation} +imply \(RH\le Q\) and, for sufficiently large \(N\), \(D\le N^\beta\). +For a remaining frequency \(\xi\) with \(q(\xi)\le H\), all frequencies +\(\xi+\eta\), where \(\eta\) ranges over the character group on \(B\), +therefore occur in the original lift. Summing this entire character group +gives +\begin{align} + \widehat g_i(\xi) + &=\sum_{\eta\text{ on }B}\frac1{|I_i|} + \sum_{n\in I_i}f_{s_i}(n)e(-n(\xi+\eta))e(a_{i,B}\eta)\notag\\ + &=\frac{D}{|I_i|} + \sum_{\substack{n\in I_i\\n\equiv a_i\pmod D}} + 1_A(n)e(-n\xi), + \label{eq:interval-selected-fiber} +\end{align} +where \(a_i\pmod D\) combines \(s_i\pmod {M_0}\) and the coordinates +\(a_{i,B}\) by the Chinese remainder theorem. Character orthogonality +supplies the factor \(R\), while the definition of \(f_{s_i}\) supplies +\(M_0\). Goodness of \(\boldsymbol s\) and the constraints at the +conditioned primes show that \(a_2-a_1\) is a strict square class +modulo \(D\). + +The remaining frequencies of conductor at most \(H\) are precisely the +frequencies with squarefree conductor at most \(H\) coprime to \(D\). +Indeed, \(D\) contains every prime less than \(P\); every prime divisor +of such a conductor is consequently an available prime at least \(P\), +at most \(H\le Q\), and outside \(B\). This includes the zero +frequency under the convention \(q(0)=1\). + +Define functions on \([N]\) by +\[ + F_i(n)=1_A(n)1_{I_i}(n)1_{n\equiv a_i\pmod D}. +\] +They are bounded by one, their supports are ordered, and there is no +positive square difference from the first support to the second. +Moreover, \(D\) is eight times an odd squarefree number and +\(D\le N^\beta\). All hypotheses of +Proposition~\ref{prop:square-kernel} hold. If the Fourier coefficients +of \(F_i\) use normalization \(N^{-1}\), then +\eqref{eq:interval-selected-fiber} gives +\(\widehat g_i(\xi)=DN\widehat F_i(\xi)/|I_i|\). Thus the part of +\eqref{eq:interval-pair-fourier} of conductor at most \(H\) is bounded +in modulus by +\begin{align} + &\frac{D^2N^2}{|I_1||I_2|} + \left|\sum_{\substack{q(\xi)\le H,\ q(\xi)\text{ squarefree}\\ + (q(\xi),D)=1}} + \mathfrak g(\xi)\widehat F_1(-\xi)\widehat F_2(\xi)\right| + \notag\\ + &\hspace{2cm}\ll H^{-1/3}D\frac{N^2}{|I_1||I_2|} + \ll H^{-1/3}D N^{2\tau}. + \label{eq:interval-kernel-cost} +\end{align} +The factor \(D\) in Proposition~\ref{prop:square-kernel} has canceled +one of the two fiber-normalization factors. + +Combining~\eqref{eq:interval-pair-tail} +and~\eqref{eq:interval-kernel-cost} bounds each conditional pair integral +by \(O(N^{-\beta/6+(d+2)\tau+o(1)})\). This is uniform over all +conditionings of positive probability. Averaging them, and then summing +the Fourier expansion using~\eqref{eq:interval-fourier-cost}, bounds the +truncated integral for each unequal-cycle interval assignment by +\[ + O\bigl(N^{-\beta/6+(3d+2)\tau+o(1)}\bigr). +\] +Averaging interval assignments incurs no further loss. Including the +equal-cycle contribution and~\eqref{eq:interval-truncation-error}, the +original integral has modulus at most +\begin{equation} + C\left(N^{-23\tau+o(1)}+N^{-\gamma\tau+o(1)} + +N^{-\beta/6+(3d+2)\tau+o(1)}\right). + \label{eq:interval-three-errors} +\end{equation} +Finally, \(23\tau=5888\sigma_0\), \(\gamma\tau=4\sigma_0\), and +\[ + \frac\beta6-(3d+2)\tau + >\beta\left(\frac16-\frac3{64000}\right) + >\frac\beta7>2\sigma_0. +\] +The finitely many uniform moment losses in each term can be bounded by +\(N^{\sigma_0}\), after increasing the absolute threshold in \(N\). +Equation~\eqref{eq:interval-three-errors} is consequently +\(O(N^{-\sigma_0})\), as required. All estimates used absolute values +of signed Fourier lifts; no pointwise positivity of a lift was needed. +\end{proof} diff --git a/preprints/A-power-saving-for-square-difference-free-sets-September-24-2026/build/sections/functional.tex b/preprints/A-power-saving-for-square-difference-free-sets-September-24-2026/build/sections/functional.tex new file mode 100644 index 0000000000000000000000000000000000000000..c1f0572e836f25a7361ef6eafd65d3ec1c0e3ed8 --- /dev/null +++ b/preprints/A-power-saving-for-square-difference-free-sets-September-24-2026/build/sections/functional.tex @@ -0,0 +1,389 @@ +\section{Signed multilinear forms and conductor decay} +\label{sec:functional} + +The tuple laws provide two estimates for functions on products of prime +fields. The first follows the reflection and chessboard method +of~\cite[Theorem~2.2]{FrohlichLieb1978} and +\cite[Theorem~4.1]{FILS1978ReflectionI}; the finite proof below verifies +the required inequality for this particular tuple law. Reflection positivity gives a H\"older inequality for signed +inputs and a seminorm that dominates the uniform mean. The conditional +mixing estimate gives a separate bound for inputs supported on large +Fourier denominators. This second estimate also applies to the +nonexceptional laws. + +Retain the index set \(V=\Pi^\ell\), where \(\Pi\) consists of the +permutation words on \([h]\), \(h=24\), and \(\ell=161\), and put +\(d=|V|\). In particular, \(d\) is even. Fix the prime threshold +\(P\) from Proposition~\ref{prop:tuple-laws} and +Lemma~\ref{lem:conditional-mixing}, and put \(\gamma=1/64\). +For a finite set \(\mathcal P\) of primes at least \(P\), define +\[ + X_{\mathcal P}=\prod_{p\in\mathcal P}\mathbb F_p, + \qquad + \mu_{\mathcal P}=\bigotimes_{p\in\mathcal P}\mu_p. +\] +We regard \(\mu_{\mathcal P}\) as a probability law on +\(X_{\mathcal P}^V\) by collecting, for each slot \(v\), its labels +at all primes into \(z_v\in X_{\mathcal P}\). Every single-slot +marginal is uniform on \(X_{\mathcal P}\). Unless another measure +is specified, norms and expectations on \(X_{\mathcal P}\) use +uniform probability measure. Set +\begin{equation} + \mathcal Z_{\mathcal P}((g_v)_{v\in V}) + =\int\prod_{v\in V}g_v(z_v)\,d\mu_{\mathcal P}(z), + \qquad + Z_{\mathcal P}(g)=\mathcal Z_{\mathcal P}((g)_{v\in V}). + \label{eq:functional-definition} +\end{equation} +For an empty prime set, \(X_{\varnothing}\) is a singleton and +the product law has mass one; thus \(Z_{\varnothing}(g)=g^d\). + +\subsection{A finite signed reflection inequality} + +\begin{proposition}[Signed H\"older inequality] +\label{prop:signed-holder} +For every finite prime set \(\mathcal P\) as above and real functions +\(g,g_v:X_{\mathcal P}\to\mathbb R\), one has +\begin{equation} + Z_{\mathcal P}(g)\ge0, + \qquad + \left|\mathcal Z_{\mathcal P}((g_v)_{v\in V})\right| + \le\prod_{v\in V}Z_{\mathcal P}(g_v)^{1/d}. + \label{eq:signed-holder} +\end{equation} +The inequality includes inputs with zero diagonal integral. +\end{proposition} + +\begin{proof} +Across any reflection cut, the matrix of \(\mu_{\mathcal P}\) is +the tensor product of the corresponding matrices of the \(\mu_p\). +It is therefore symmetric positive semidefinite. This also holds for +the empty prime set. We first record precisely what a reflection does +to an assignment of functions. + +Fix a block and two distinct symbols. Let \(\theta\) exchange those +symbols in that block, and let \(V_+\) be the vertices in which the +first symbol occurs before the second. Put \(V_-=\theta V_+\). +Define the two fold maps \(\phi_+,\phi_-:V\to V\) by +\[ + \phi_+(v)= + \begin{cases}v,&v\in V_+,\\\theta v,&v\in V_-,\end{cases} + \qquad + \phi_-(v)= + \begin{cases}\theta v,&v\in V_+,\\v,&v\in V_-.\end{cases} +\] +For an assignment \(a:V\to\mathbb R^{X_{\mathcal P}}\), abbreviate +\(\mathcal Z_{\mathcal P}((a(v))_{v\in V})\) by \(\Lambda(a)\). +The bilinear form of the cut matrix is evaluated on the two real +functions +\[ + F(x)=\prod_{v\in V_+}a(v)(x_v), + \qquad + G(x)=\prod_{v\in V_+}a(\theta v)(x_v). +\] +Its mixed value is \(\Lambda(a)\); its two quadratic values are +\(\Lambda(a\circ\phi_+)\) and \(\Lambda(a\circ\phi_-)\). +Cauchy--Schwarz for this positive semidefinite form gives +\begin{equation} + |\Lambda(a)|^2 + \le\Lambda(a\circ\phi_+)\Lambda(a\circ\phi_-), + \qquad + \Lambda(a\circ\phi_\pm)\ge0. + \label{eq:folding-cauchy-schwarz} +\end{equation} +This remains valid when the form has a nontrivial null space. +For a constant assignment, \(F=G\); hence \(Z_{\mathcal P}(g)\ge0\). + +We next specify a finite sequence of folds whose composition is constant. +For a word \(w\) in block \(b\), write \(p_i=w^{-1}(i)\) for the +position of symbol \(i\). Let \(\phi_{b,i}\) be the fold onto +\(p_i0\), choose a +maximizer. Each of its folded assignments still takes values in the +same finite list, so each folded integral is at most \(M\) in absolute +value. Equation~\eqref{eq:folding-cauchy-schwarz} says that those two +integrals are nonnegative and their product is at least \(M^2\). +They must therefore both equal \(M\). In particular, either fold of +any maximizing assignment is again maximizing. + +Apply the folds in reverse sorting order, +\(\phi_T,\phi_{T-1},\ldots,\phi_1\). Pullbacks compose on the +right, so the resulting assignment is +\(a\circ\phi_T\circ\cdots\circ\phi_1\), which is constant by +\eqref{eq:constant-fold-composition}. Every intermediate integral is +\(M\), and the last is one of the diagonal values \(Z_{\mathcal P}(f_i)\). +Conversely, every constant assignment was included in the maximum. +It follows that +\begin{equation} + M=\max_{1\le i\le k}Z_{\mathcal P}(f_i). + \label{eq:finite-folding-maximum} +\end{equation} +When \(M=0\), all these diagonal values are zero, so the same identity +holds. This is a finite argument and requires no limiting sequence of +folds. The maximizing-assignment argument is analogous to the proof of +the abstract chessboard estimate in +\cite[Theorem~4.1]{FILS1978ReflectionI}. + +Finally, for \(\epsilon>0\), apply +\eqref{eq:finite-folding-maximum} to the list +\[ + f_v=\frac{g_v}{(Z_{\mathcal P}(g_v)+\epsilon)^{1/d}}, + \qquad v\in V. +\] +Every denominator is positive and \(Z_{\mathcal P}(f_v)\le1\). +Multilinearity consequently gives +\[ + \left|\mathcal Z_{\mathcal P}((g_v)_{v\in V})\right| + \le\prod_{v\in V}(Z_{\mathcal P}(g_v)+\epsilon)^{1/d}. +\] +Letting \(\epsilon\downarrow0\) proves +\eqref{eq:signed-holder}, including zero diagonal values. +\end{proof} + +\subsection{The seminorm and its symmetries} + +Proposition~\ref{prop:signed-holder} allows us to define, for real \(g\), +\[ + \|g\|_{\mathcal P}=Z_{\mathcal P}(g)^{1/d}. +\] +For \(B\subseteq\mathcal P\), let \(\mathbb E_Bg\) denote +uniform averaging over the coordinates in \(B\), holding the other +coordinates fixed. We may regard this either as a function on +\(X_{\mathcal P}\) independent of \(B\), or as a function on +\(X_{\mathcal P\setminus B}\). + +\begin{lemma}[Seminorm properties] +\label{lem:functional-properties} +The map \(g\mapsto\|g\|_{\mathcal P}\) is a seminorm on the real +functions on \(X_{\mathcal P}\), and +\begin{equation} + |\mathbb E g|\le\|g\|_{\mathcal P}. + \label{eq:functional-mean} +\end{equation} +For \(a_p\in\mathbb F_p\) and \(r_p\in Q_p^*\), let +\(A(x)_p=r_px_p+a_p\). Then +\begin{equation} + \|g\circ A\|_{\mathcal P}=\|g\|_{\mathcal P}. + \label{eq:functional-affine} +\end{equation} +For every \(B\subseteq\mathcal P\), +\begin{equation} + \|\mathbb E_Bg\|_{\mathcal P\setminus B} + =\|\mathbb E_Bg\|_{\mathcal P} + \le\|g\|_{\mathcal P}. + \label{eq:functional-projection} +\end{equation} +\end{lemma} + +\begin{proof} +Nonnegativity is already proved. Since \(d\) is even, homogeneity of +the diagonal form gives \(\|cg\|_{\mathcal P}=|c|\|g\|_{\mathcal P}\) +for every real \(c\). For real functions \(g_1,g_2\), expand +\(Z_{\mathcal P}(g_1+g_2)\) multilinearly and apply +\eqref{eq:signed-holder} to each term. With +\(a=\|g_1\|_{\mathcal P}\) and \(b=\|g_2\|_{\mathcal P}\), this gives +\[ + 0\le Z_{\mathcal P}(g_1+g_2) + \le\sum_{S\subseteq V}a^{|S|}b^{d-|S|}=(a+b)^d. +\] +Taking the nonnegative \(d\)-th root proves the triangle inequality, +also when one of the seminorms vanishes. + +The probability normalization gives \(Z_{\mathcal P}(1)=1\). +Insert \(g\) in one slot and \(1\) in all others in +\eqref{eq:signed-holder}. The uniform single-slot marginal identifies +the left side with \(|\mathbb E g|\), proving +\eqref{eq:functional-mean}. The simultaneous affine invariance of each +\(\mu_p\) proves \eqref{eq:functional-affine}; its parameters may be +chosen independently at different primes. + +Let \(G_B\) be the additive subgroup of \(X_{\mathcal P}\) consisting +of vectors supported on \(B\). Then +\[ + (\mathbb E_Bg)(x)=\frac1{|G_B|}\sum_{t\in G_B}g(x+t). +\] +The triangle inequality and translation invariance show that this average +has seminorm at most \(\|g\|_{\mathcal P}\). Since the averaged +function is independent of \(B\), integration over the prime laws in +\(B\), each of mass one, gives +\[ + Z_{\mathcal P}(\mathbb E_Bg) + =Z_{\mathcal P\setminus B}(\mathbb E_Bg). +\] +This proves \eqref{eq:functional-projection}. All statements include +the empty prime set, for which the seminorm is absolute value. +\end{proof} + +\subsection{Conductor decay} + +The characters of \(X_{\mathcal P}\) are indexed by +\(\xi\in\mathbb Q/\mathbb Z\) whose reduced denominator \(q(\xi)\) +divides \(\prod_{p\in\mathcal P}p\), with \(q(0)=1\). We write +such a character as \(e(z\xi)\): it means \(e(n\xi)\) for any integer +\(n\) satisfying \(n\equiv z_p\pmod p\) for all \(p\in\mathcal P\). +This is well defined by the Chinese remainder theorem. The Fourier +coefficient of \(g\) is +\[ + \widehat g(\xi)=\mathbb E_{z\in X_{\mathcal P}}g(z)e(-z\xi). +\] +For \(U\subseteq\mathcal P\), put \(q_U=\prod_{p\in U}p\), +including \(q_{\varnothing}=1\). The exact prime support of a +character is the set of primes at which it is nontrivial; its denominator +is exactly the product of these primes. + +Recall the exceptional mass and the normalized nonexceptional law, +\[ + \eta_p=\varepsilon_p(\mathbb F_p^V), + \qquad + \mu_p^\circ=\frac{\mu_p-\varepsilon_p}{1-\eta_p}. +\] +Proposition~\ref{prop:tuple-laws} gives \(\eta_p\le p^{-4}<1\). + +\begin{lemma}[Conductor decay] +\label{lem:conductor-decay} +Let \(\mathcal P\) be a finite set of primes at least \(P\), and +at each prime choose \(\rho_p\) to be either \(\mu_p\) or +\(\mu_p^\circ\). Let \(R\ge1\), and let +\(g_w:X_{\mathcal P}\to\mathbb C\), \(w\in V\), be functions. +Suppose that for one slot \(v\), +\(\widehat g_v(\xi)=0\) whenever \(q(\xi)\le R\). Then +\begin{equation} + \left|\int\prod_{w\in V}g_w(z_w)\, + d\!\bigotimes_{p\in\mathcal P}\rho_p(z)\right| + \le R^{-\gamma}\|g_v\|_2 + \prod_{w\ne v}\|g_w\|_{2(d-1)}. + \label{eq:conductor-decay} +\end{equation} +All norms on the right use uniform probability measure on +\(X_{\mathcal P}\). The constant in the inequality is one. +\end{lemma} + +\begin{proof} +Fix the distinguished slot \(v\). For each \(p\), let +\(Y_p=\mathbb F_p^{V\setminus\{v\}}\), let \(\lambda_p\) be +the marginal law of the other labels under \(\rho_p\), and define +\[ + T_pf(y)=\mathbb E_{\rho_p}[f(z_v)\mid (z_w)_{w\ne v}=y]. +\] +This is an operator from \(L^2(\mathbb F_p,\mathrm{unif})\) to +\(L^2(Y_p,\lambda_p)\); its values on marginal-null points are +irrelevant. It preserves constants, and the tower property gives +\begin{equation} + T_p1=1, + \qquad + \int T_pf\,d\lambda_p=\mathbb E_{x\in\mathbb F_p}f(x). + \label{eq:conditional-centering} +\end{equation} +For uniform-mean-zero \(f\), Lemma~\ref{lem:conditional-mixing} +gives +\begin{equation} + \|T_pf\|_{L^2(\lambda_p)} + \le p^{-\gamma}\|f\|_{L^2(\mathrm{unif})}. + \label{eq:local-conductor-contraction} +\end{equation} +Let \(H_p^0\) be the mean-zero subspace of +\(L^2(\mathbb F_p,\mathrm{unif})\), and let \(K_p^0\) be the +mean-zero subspace of \(L^2(Y_p,\lambda_p)\). Thus +\[ + L^2(\mathbb F_p,\mathrm{unif})=\mathbb C1\mathbin{\oplus}H_p^0, + \qquad + L^2(Y_p,\lambda_p)=\mathbb C1\mathbin{\oplus}K_p^0, +\] +with orthogonal sums. Equation~\eqref{eq:conditional-centering} +says that \(T_pH_p^0\subseteq K_p^0\). Notice that centering in +the codomain refers to \(\lambda_p\), which need not be uniform. + +For \(U\subseteq\mathcal P\), let \(H_U\) be the tensor subspace +with factor \(H_p^0\) when \(p\in U\) and factor \(\mathbb C1\) +otherwise; define \(K_U\) in the same way using \(K_p^0\). +The \(H_U\) form an orthogonal direct sum of +\(L^2(X_{\mathcal P},\mathrm{unif})\). Distinct \(K_U,K_W\) +are also orthogonal: at a prime in the symmetric difference of \(U,W\), +one tensor factor is constant and the other has mean zero. Inner products +of pure tensors therefore vanish, and linearity gives the assertion for +the whole subspaces. + +Under the product law \(\bigotimes_p\rho_p\), the marginal of all +the other labels is \(\lambda=\bigotimes_p\lambda_p\). Conditional +independence across primes shows that the operator from the distinguished +slot to all other labels is +\[ + T=\bigotimes_{p\in\mathcal P}T_p. +\] +For completeness, on an input \(\prod_pf_p(z_{v,p})\), the conditional +expectation is \(\prod_pT_pf_p\), since the conditional law is a +product over primes. Such pure tensors span the finite-dimensional +domain, proving the operator identity. Consequently +\begin{equation} + T(H_U)\subseteq K_U, + \qquad + \|T|_{H_U}\|\le\prod_{p\in U}p^{-\gamma}=q_U^{-\gamma}. + \label{eq:exact-support-contraction} +\end{equation} +The norm bound follows by applying +\eqref{eq:local-conductor-contraction} one tensor factor at a time; +on the constant factors the norm is one. + +Nontrivial characters at \(p\) form an orthonormal basis for \(H_p^0\). +Thus \(H_U\) consists precisely of Fourier sums with exact denominator +\(q_U\). Decompose the distinguished input as +\(g_v=\sum_{q_U>R}g_{v,U}\), with \(g_{v,U}\in H_U\). The +images of distinct summands are orthogonal by +\eqref{eq:exact-support-contraction} and the orthogonality of the +\(K_U\). Hence +\begin{align*} + \|Tg_v\|_{L^2(\lambda)}^2 + &=\sum_{q_U>R}\|Tg_{v,U}\|_{L^2(\lambda)}^2\\ + &\le\sum_{q_U>R}q_U^{-2\gamma}\|g_{v,U}\|_2^2 + \le R^{-2\gamma}\|g_v\|_2^2. +\end{align*} +Only distinct exact supports need have orthogonal images; no such +assertion is made for different characters within a single support. + +Finally condition the multilinear integral on all labels except those +in slot \(v\), and apply Cauchy--Schwarz. Its absolute value is at most +\[ + R^{-\gamma}\|g_v\|_2 + \left\|\prod_{w\ne v}g_w(z_w)\right\|_{L^2(\lambda)}. +\] +H\"older's inequality with \(d-1\) factors and the uniform marginal +in each remaining slot give +\[ + \left\|\prod_{w\ne v}g_w(z_w)\right\|_{L^2(\lambda)} + \le\prod_{w\ne v} + \left(\mathbb E_{\lambda}|g_w(z_w)|^{2(d-1)}\right)^{1/(2(d-1))} + =\prod_{w\ne v}\|g_w\|_{2(d-1)}. +\] +This proves \eqref{eq:conductor-decay}. If \(\mathcal P\) is empty, +the support hypothesis forces \(g_v=0\), so the assertion holds there +too. No reflection positivity of the laws \(\rho_p\) was used. +\end{proof} diff --git a/preprints/A-power-saving-for-square-difference-free-sets-September-24-2026/build/sections/graph.tex b/preprints/A-power-saving-for-square-difference-free-sets-September-24-2026/build/sections/graph.tex new file mode 100644 index 0000000000000000000000000000000000000000..aadc33676b3e7345814f7092c09c435f99e71584 --- /dev/null +++ b/preprints/A-power-saving-for-square-difference-free-sets-September-24-2026/build/sections/graph.tex @@ -0,0 +1,159 @@ +\section{Finite-field square constraints} +\label{sec:finite-field-graphs} + +This section controls systems of square constraints on independent field +variables. A single constraint is close to a constant kernel in operator +norm. Replacing constraints one at a time then counts any fixed directed +graph without loops or opposite edges. Applying this count to a four-cycle +of two lists gives the operator estimate needed for the tuple measures. + +Throughout, finite sets carry uniform probability measure, and +\(e(t)=\exp(2\pi i t)\). For an odd prime \(p\), write +\[ + Q_p^*=\{u^2:u\in\mathbb F_p^\times\}, + \qquad + A_p(t)=1_{Q_p^*}(t), + \qquad + \delta_p=\frac{\sqrt p+1}{2p}\le p^{-1/2}. +\] +For finite sets \(X,Y\), the operator associated with a kernel +\(K:X\times Y\to\mathbb R\) is +\[ + T_K f(x)=\mathbb E_{y\in Y}K(x,y)f(y). +\] +We use \(\|K\|_{\mathrm{op}}\) to mean the norm of \(T_K\) from +\(L^2(Y)\) to \(L^2(X)\). A constant kernel \(c\) therefore denotes +the operator \(f\mapsto c\,\mathbb E f\). + +\begin{lemma}[Counting directed square constraints] +\label{lem:graph-count} +Let \(p\) be an odd prime and let \(G=(W,E)\) be a finite directed graph +with no loops and at most one edge on each unordered pair of distinct +vertices. Put \(m=|E|\), and let \((z_w)_{w\in W}\) be independent +uniform elements of \(\mathbb F_p\). Then +\[ + \left| + \mathbb E\prod_{(u,v)\in E}A_p(z_v-z_u)-2^{-m} + \right| + \le m\delta_p. +\] +The assertion includes the empty graph, whose product is \(1\). +\end{lemma} + +\begin{proof} +We first show that the kernel +\(b_p(x,y)=A_p(y-x)-1/2\) has operator norm at most \(\delta_p\). +The characters \(x\mapsto e(kx/p)\), \(k\in\mathbb F_p\), are an +orthonormal eigenbasis. The constant eigenvalue is +\[ + \frac{|Q_p^*|}{p}-\frac12=-\frac1{2p}. +\] +For \(k\ne0\), put \(G_p(k)=\sum_{u\in\mathbb F_p}e(ku^2/p)\). +The corresponding eigenvalue is +\[ + \frac1p\sum_{t\in Q_p^*}e(kt/p) + =\frac{G_p(k)-1}{2p}. +\] +Here the subtraction of \(1\) removes the zero square, and division +by \(2\) removes the multiplicity of every nonzero square. Since +\((u,v)\mapsto(u-v,u+v)\) is a bijection on \(\mathbb F_p^2\), +\[ + |G_p(k)|^2 + =\sum_{u,v\in\mathbb F_p}e\bigl(k(u^2-v^2)/p\bigr) + =\sum_{a,b\in\mathbb F_p}e(kab/p) + =p. +\] +Thus all eigenvalues have absolute value at most \(\delta_p\). +In particular, +\begin{equation} + \left| + \mathbb E_{x,y} f(x)g(y)\bigl(A_p(y-x)-\tfrac12\bigr) + \right| + \le\delta_p\|f\|_2\|g\|_2. + \label{eq:single-square-mixing} +\end{equation} +The same bound holds with \(x-y\) in place of \(y-x\), since the +transposed operator has the same norm. + +Now replace the \(m\) edge indicators one at a time by \(1/2\). +In a telescoping error for an edge \((u,v)\), fix every variable +except \(z_u,z_v\). No other edge has both these endpoints. +Consequently all the other factors separate as +\(C f(z_u)g(z_v)\), where \(|C|,|f|,|g|\le1\). +Equation~\eqref{eq:single-square-mixing} bounds the conditional error +by \(\delta_p\). Averaging the fixed variables and summing over +the edges proves the assertion. +\end{proof} + +We next divide the variables into two independent lists. Only +constraints between the lists enter the following kernel; constraints +within either list must be treated separately. + +\begin{lemma}[Mixing between two lists] +\label{lem:list-mixing} +Let \(I,J\) be finite disjoint sets, and let \(G\) be a directed graph +on \(I\sqcup J\) whose \(m\) edges all join \(I\) to \(J\), with at +most one edge on each unordered pair. The directions may vary from +edge to edge. For +\(x\in\mathbb F_p^I\), \(y\in\mathbb F_p^J\), write \(z_i=x_i\) for +\(i\in I\), \(z_j=y_j\) for \(j\in J\), and set +\[ + K(x,y)=\prod_{(u,v)\in E(G)}A_p(z_v-z_u). +\] +Then +\begin{equation} + \|K-2^{-m}\|_{\mathrm{op}} + \le (32m)^{1/4}p^{-1/8}. + \label{eq:list-mixing} +\end{equation} +For \(m=0\), both sides are zero. In particular, the implied +constant for any fixed pair of lists is independent of \(p\). +\end{lemma} + +\begin{proof} +Put \(c=2^{-m}\) and \(B=K-c\). With respect to the normalized +point-mass bases, \(T_B\) has matrix entries +\(B(x,y)/\sqrt{|\mathbb F_p^I|\,|\mathbb F_p^J|}\). +If \(s_r\) are its singular values, then +\begin{align} + \|B\|_{\mathrm{op}}^4 + &\le \sum_r s_r^4 + =\operatorname{tr}\bigl((T_BT_B^*)^2\bigr) \notag\\ + &=\mathbb E_{x^0,x^1,y^0,y^1} + B(x^0,y^0)B(x^0,y^1)B(x^1,y^0)B(x^1,y^1), + \label{eq:list-four-cycle} +\end{align} +where the four lists are independent and uniform. The kernel is +real, so no conjugates appear in the last expression. + +Let \(\Omega=\{0,1\}^2\). Expanding the last line gives +\[ + \sum_{F\subseteq\Omega}(-c)^{4-|F|} + \mathbb E\prod_{(a,b)\in F}K(x^a,y^b). +\] +For each \(F\), its product is the constraint indicator of a graph +on +\[ + (I\times\{0,1\})\sqcup(J\times\{0,1\}). +\] +Every base edge between \(i\in I\) and \(j\in J\) is copied, in its +original direction, between \((i,a)\) and \((j,b)\) for every +\((a,b)\in F\). The two endpoints determine the base unordered +pair and the corner \((a,b)\). Hence this graph has no loops or +repeated unordered edges, and has exactly \(m|F|\) edges. +Lemma~\ref{lem:graph-count} therefore gives +\[ + \mathbb E\prod_{(a,b)\in F}K(x^a,y^b) + =c^{|F|}+E_F, + \qquad |E_F|\le m|F|\delta_p. +\] +The main terms sum to \((c-c)^4=0\). The remaining terms have +absolute value at most +\[ + m\delta_p\sum_{t=0}^4\binom4t t c^{4-t} + =4m(1+c)^3\delta_p + \le32m p^{-1/2}. +\] +Combining this with \eqref{eq:list-four-cycle} and taking fourth +roots proves \eqref{eq:list-mixing}. +\end{proof} diff --git a/preprints/A-power-saving-for-square-difference-free-sets-September-24-2026/build/sections/induction.tex b/preprints/A-power-saving-for-square-difference-free-sets-September-24-2026/build/sections/induction.tex new file mode 100644 index 0000000000000000000000000000000000000000..5e0aaf6a30574ed2be62980dda4b06d1c0fc9430 --- /dev/null +++ b/preprints/A-power-saving-for-square-difference-free-sets-September-24-2026/build/sections/induction.tex @@ -0,0 +1,318 @@ +\section{Exceptional terms and the power-saving induction} +\label{sec:induction} + +We now combine the scale comparison, which applies to every small-residue +assignment, with the cancellation obtained for good assignments. For good +assignments the nonexceptional contribution is already negligible; we control +the exceptional parts by expanding the prime measures. The other assignments +are bounded directly by the scale comparison. Expanding measures rather than +comparing integrands pointwise keeps the argument valid for signed Fourier +lifts throughout. + +The finite set $V$ and the constants $P,M_0,\beta,\gamma,\tau,\sigma_0$ +have now been fixed. Lemma~\ref{lem:good-residues} has also established the +fixed positive proportion $\rho$ of good small-residue assignments. +The prime threshold $P$ was chosen so that +\begin{equation} + \prod_{p\ge P}(1+p^{-3})-1\le\frac14. + \label{eq:exceptional-euler-budget} +\end{equation} +For every positive integer $n$, define +\begin{equation} + \mathcal Y(n)= + \max_{\substack{A\subseteq[n]\\ A\text{ square-difference-free}}}Y(n,A). + \label{eq:maximal-functional} +\end{equation} +This is a finite nonnegative number, by Proposition~\ref{prop:signed-holder}. +Write $\eta_p=\varepsilon_p(\F_p^V)$ for the mass of the exceptional measure, +so $0\le \eta_p\le p^{-4}$. For +$B\subseteq\mathcal P_N$, put +\begin{equation} + q_B=\prod_{p\in B}p,\qquad + \eta_B=\prod_{p\in B}\eta_p,\qquad + n_B=\left\lceil\frac{N}{(M_0q_B)^2}\right\rceil. + \label{eq:exceptional-subset-notation} +\end{equation} +Empty products have value $1$, so $q_\varnothing=\eta_\varnothing=1$ and +$n_\varnothing=\lceil N/M_0^2\rceil$. + +\begin{proposition}\label{prop:induction-recurrence} +There are absolute constants $K<\infty$ and $N_1\ge2$ such that, for every +integer $N\ge N_1$, +\begin{align} + \mathcal Y(N) + &\le K N^{-\sigma_0} + +(1+u_N)\left((1-\rho)\mathcal Y(n_\varnothing) + +\rho\! + \sum_{\substack{\varnothing\ne B\subseteq\mathcal P_N\\q_B\le N^\tau}} + \eta_B\mathcal Y(n_B)\right), + \label{eq:functional-recurrence}\\ + u_N&=(1+M_0^2N^{2\tau-1})^d-1. + \label{eq:uniform-ceiling-error} +\end{align} +In particular, $u_N\to0$ as $N\to\infty$. The constants $K,N_1$ do not +depend on any proposed induction bound for $\mathcal Y$. +\end{proposition} + +\begin{proof} +Fix a square-difference-free set $A\subseteq[N]$. Write +\[ + G_s=\mathcal L_{[N],\mathcal P_N,Q}f_s, + \qquad + \Phi_{\mathbf s}((z_v)_{v\in V})=\prod_{v\in V}G_{s_v}(z_v), +\] +where $\mathbf s=(s_v)_{v\in V}$ is a small-residue assignment. By +\eqref{eq:small-residue-input}, $1_A=\E_s f_s$. The lift is linear, and the +product-space integral is multilinear, so +\begin{equation} + Y(N,A)=\E_{\mathbf s} + \int\Phi_{\mathbf s}\,d\bigotimes_{p\in\mathcal P_N}\mu_p. + \label{eq:residue-average-functional} +\end{equation} + +For $B\subseteq\mathcal P_N$, define the nonnegative finite measure +\[ + \kappa_B= + \bigotimes_{p\in B}\varepsilon_p + \otimes\bigotimes_{p\in\mathcal P_N\setminus B}\mu_p. +\] +Its mass is $\eta_B$. The finite product expansion gives the following +identity of signed measures: +\begin{equation} + \bigotimes_{p\in\mathcal P_N}\mu_p + -\bigotimes_{p\in\mathcal P_N}(\mu_p-\varepsilon_p) + =\sum_{\varnothing\ne B\subseteq\mathcal P_N} + (-1)^{|B|+1}\kappa_B. + \label{eq:signed-measure-inclusion-exclusion} +\end{equation} +For any real integrand $\Phi$, integration followed by the triangle inequality +therefore yields +\begin{equation} + \int\Phi\,d\bigotimes_p\mu_p + \le + \left|\int\Phi\,d\bigotimes_p(\mu_p-\varepsilon_p)\right| + +\sum_{B\ne\varnothing}\left|\int\Phi\,d\kappa_B\right|. + \label{eq:exceptional-signed-bound} +\end{equation} +This step makes no sign assumption on $\Phi$. Notice also that the measure +outside $B$ in each summand is the full measure $\mu_p$. + +We first bound the summands with $q_B>N^\tau$. A summand of zero mass is +zero, so assume $\eta_B>0$. Each normalized exceptional measure +$\varepsilon_p/\eta_p$ has uniform single-label marginals: its invariance under +simultaneous translation implies that any one label is uniformly distributed +on $\F_p$. The same uniformity holds for $\mu_p$. Consequently every +single-slot marginal of $\kappa_B/\eta_B$ is uniform on +$X_{\mathcal P_N}$. No independence between different slots is needed. +Ordinary H\"older's inequality gives +\begin{equation} + \left|\int\Phi_{\mathbf s}\,d\kappa_B\right| + \le\eta_B\prod_{v\in V}\|G_{s_v}\|_{L^d(X_{\mathcal P_N})} + \le C_\epsilon\eta_BN^\epsilon + \qquad(\epsilon>0). + \label{eq:exceptional-holder-moment} +\end{equation} +The last inequality is Lemma~\ref{lem:lift-moments}, applied at the +fixed even order $d$ and with the loss in each factor chosen sufficiently +small. The bound $|f_s|\le M_0$ only introduces a fixed factor. The constant +$C_\epsilon$ is independent of $B$, $A$, $N$, and $\mathbf s$. + +The exceptional masses are summable with the extra factor $q_B$: +\begin{equation} + \sum_{\varnothing\ne B\subseteq\mathcal P_N}q_B\eta_B + =\prod_{p\in\mathcal P_N}(1+p\eta_p)-1 + \le\prod_{p\ge P}(1+p^{-3})-1\le\frac14. + \label{eq:weighted-exceptional-mass} +\end{equation} +It follows from \eqref{eq:exceptional-holder-moment} that +\[ + \sum_{\substack{\varnothing\ne B\subseteq\mathcal P_N\\q_B>N^\tau}} + \left|\int\Phi_{\mathbf s}\,d\kappa_B\right| + \le\frac{C_\epsilon}{4}N^{-\tau+\epsilon}. +\] +Choose $\epsilon=\tau/2$. Since $\sigma_0=\gamma\tau/4<\tau/2$, we obtain +\begin{equation} + \sum_{q_B>N^\tau} + \left|\int\Phi_{\mathbf s}\,d\kappa_B\right| + \le K_1N^{-\sigma_0}, + \label{eq:large-exceptional-tail} +\end{equation} +uniformly in the residue assignment. Thus the number of exceptional subsets +has already been accounted for by the convergent product +\eqref{eq:weighted-exceptional-mass}. + +Now suppose $q_B\le N^\tau$ and $\eta_B>0$. Condition on all the labels at +primes in $B$ under $\kappa_B/\eta_B$. The remaining prime law is exactly +$\bigotimes_{p\in\mathcal P_N\setminus B}\mu_p$. For each realization, the +function in slot $v$ is $G_{s_v}$ specialized at its prescribed labels in $B$. +The mixed assertion of Proposition~\ref{prop:scale-transfer} applies to these +arbitrary slotwise residues and labels. With +\[ + D_B=M_0q_B,\qquad \lambda_B=\frac{D_B^2n_B}{N}, +\] +it gives +\begin{equation} + \left|\int\Phi_{\mathbf s}\,d\kappa_B\right| + \le\eta_B\left(\lambda_B^d\mathcal Y(n_B)+K_2N^{-\sigma_0}\right). + \label{eq:small-exceptional-transfer} +\end{equation} +The same inequality is trivially true when $\eta_B=0$. Crucially, the error +and its lower threshold for $N$ are uniform in $B$ and in every conditioned +tuple. No induction hypothesis has been used. + +The ceiling in the smaller scale produces a uniform multiplicative error. +For $q_B\le N^\tau$, +\[ + 1\le\lambda_B\le1+\frac{D_B^2}{N} + \le1+M_0^2N^{2\tau-1}, +\] +and hence $\lambda_B^d\le1+u_N$. Moreover, +$\sum_{B\ne\varnothing}\eta_B\le1/4$ by +\eqref{eq:weighted-exceptional-mass}. Summing the errors in +\eqref{eq:small-exceptional-transfer} is therefore harmless. + +If $\mathbf s$ is good, Proposition~\ref{prop:nonexceptional-contraction} +bounds the first term on the right of \eqref{eq:exceptional-signed-bound} by +$K_3N^{-\sigma_0}$. Combining this with +\eqref{eq:large-exceptional-tail} and +\eqref{eq:small-exceptional-transfer}, we obtain +\[ + \int\Phi_{\mathbf s}\,d\bigotimes_p\mu_p + \le K_4N^{-\sigma_0} + +(1+u_N) + \sum_{\substack{\varnothing\ne B\subseteq\mathcal P_N\\q_B\le N^\tau}} + \eta_B\mathcal Y(n_B). +\] +For a nongood $\mathbf s$, apply the mixed scale comparison directly with +$B=\varnothing$: +\[ + \left|\int\Phi_{\mathbf s}\,d\bigotimes_p\mu_p\right| + \le(1+u_N)\mathcal Y(n_\varnothing)+K_2N^{-\sigma_0}. +\] +Averaging these two estimates in +\eqref{eq:residue-average-functional} gives +\eqref{eq:functional-recurrence}, since the good assignments have probability +$\rho$. The estimate is uniform in $A$, so we may take its maximum. +Finally, $2\tau<1$ by the fixed parameter choice, and $M_0,d$ are fixed, so +$u_N\to0$. All constants in the proof came from the preceding estimates and +fixed parameters; none depended on an induction constant. +\end{proof} + +\begin{theorem}\label{thm:functional-decay} +There are absolute constants $a_0>0$ and $C_0<\infty$ such that, for every +positive integer $N$ and every square-difference-free set $A\subseteq[N]$, +\[ + Y(N,A)\le C_0N^{-a_0}. +\] +\end{theorem} + +\begin{proof} +We make all remaining parameter choices before starting the induction. +The finite construction fixes $h,t_0,\ell,d$; the conditional and kernel estimates +fix $\gamma$ and $\beta$. The threshold $P$ was then chosen, and with it +$M_0$, $\tau$, and $\sigma_0$. In particular the probability $\rho$ of +Lemma~\ref{lem:good-residues} is already a fixed positive number. We shall +not enlarge $P$ after choosing the exponent below. + +Put $r=1-3\rho/4$, so $1/4\le r<1$, and choose +\begin{equation} + a_0=\min\left\{\frac14,\frac{\sigma_0}{2}, + \frac{\log(1/r)}{4\log M_0}\right\}. + \label{eq:absolute-exponent-choice} +\end{equation} +Table~\ref{tab:parameter-roles} summarizes the roles of the parameters; +among the listed quantities, only those in the last row vary with +the interval length. +\input{figures/parameter-roles} + +Every quantity in~\eqref{eq:absolute-exponent-choice} is fixed and positive. Thus +$00$. Choose an integer $N_*\ge N_1$ so large that, +for every $N\ge N_*$, +\begin{equation} + \theta u_N\le\kappa,\qquad + K N^{a_0-\sigma_0}\le\kappa, + \label{eq:induction-threshold-choice} +\end{equation} +where $K,N_1$ are supplied by +Proposition~\ref{prop:induction-recurrence}. This is possible because +$u_N\to0$ and $a_0<\sigma_0$. The choice of $N_*$ is independent of $C_0$. +Choose $C_0\ge1$ large enough that +\begin{equation} + \mathcal Y(n)\le C_0 n^{-a_0}\qquad(1\le n0$ are absolute constants for which +$|A|\le CN^{1-c}$ for every such $N$ and $A$. +\end{proof} diff --git a/preprints/A-power-saving-for-square-difference-free-sets-September-24-2026/build/sections/introduction.tex b/preprints/A-power-saving-for-square-difference-free-sets-September-24-2026/build/sections/introduction.tex new file mode 100644 index 0000000000000000000000000000000000000000..2c67ac5653ac2144749b4396052036d666042059 --- /dev/null +++ b/preprints/A-power-saving-for-square-difference-free-sets-September-24-2026/build/sections/introduction.tex @@ -0,0 +1,253 @@ +\section{Introduction} +\label{sec:introduction} + +For a positive integer $N$, write $[N]=\{1,\ldots,N\}$. A set +$A\subseteq[N]$ is \emph{square-difference-free} if +\[ + (A-A)\cap\{m^2:m\in\N,\ m\ge1\}=\varnothing. +\] +Thus the restriction concerns differences in the integers. Define +\[ + s(N)=\max\{|A|:A\subseteq[N]\text{ is square-difference-free}\}. +\] +Our main result is a fixed power saving in this extremal problem. + +\begin{theorem} +\label{thm:main} +There are absolute constants $c>0$ and $C<\infty$ such that, for every +integer $N\ge1$ and every square-difference-free set $A\subseteq[N]$, +\[ + |A|\le C N^{1-c}. +\] +Equivalently, every subset of $[N]$ larger than $C N^{1-c}$ contains +distinct elements whose absolute difference is the square of a positive +integer. +\end{theorem} + +The exponent obtained here is positive and absolute, but extremely small. +The argument does not yield a useful numerical value or an optimal exponent. +All finite constructions in the proof are fixed independently of $N$ and +$A$; their size affects the constants and the exponent. + +\begin{remark}[Square-difference colorings] +Theorem~\ref{thm:main} gives a polynomial bound for the classical +square-difference coloring problem~\cite[Section~4]{BeigelGasarch2008}. +Fix its constants $c,C$, reducing $c$ so that $0(Cr)^{1/c}$, every $r$-coloring +of $[N]$ has same-colored elements $a,a+d^2$ with $d\ge1$. +Indeed, a largest color class has size at least $N/r>C N^{1-c}$. +Only the two endpoints must share a color; no color restriction is +imposed on $d$. Equivalently, if $G_N$ is the graph on $[N]$ joining +distinct integers whose absolute difference is a square, then +\[ + \chi(G_N)\ge \frac{N}{s(N)}\ge C^{-1}N^c, +\] +since its independent sets are exactly the square-difference-free sets. +\end{remark} + +\subsection{Previous work} + +The problem originates in a question of Lov\'asz. +Furstenberg~\cite[Theorem~1.2]{Furstenberg1977} and S\'ark\"ozy~\cite{Sarkozy1978I} +proved independently that $s(N)=o(N)$, using ergodic theory and the +circle method, respectively. S\'ark\"ozy's quantitative estimate has the form +$s(N)\le N/(\log N)^{1/3+o(1)}$; see also the historical account +in~\cite{BloomMaynard2022}. Pintz, Steiger and +Szemer\'edi~\cite[Theorem~1]{PSS1988} strengthened this to +\[ + s(N)\ll \frac{N}{(\log N)^{c_{\rm PSS}\log\log\log\log N}} +\] +for an absolute $c_{\rm PSS}>0$. Their argument combines concentration on +arithmetic progressions with an iteration that increases the collection +of large Fourier coefficients. Bloom and Maynard~\cite{BloomMaynard2022} +subsequently obtained +\[ + s(N)\ll \frac{N}{(\log N)^{c_{\rm BM}\log\log\log N}} +\] +for an absolute $c_{\rm BM}>0$, using bounds for the additive energy of +rational numbers with small denominators to improve the density-increment +argument. + + +Green and Sawhney~\cite[Theorem~1.1]{GreenSawhney2025} prove +\[ + s(N)\ll N\exp\bigl(-c_1\sqrt{\log N}\bigr) +\] +for an absolute $c_1>0$. Their arithmetic level-$d$ inequality transfers +product-space hypercontractive estimates to rational Fourier coefficients. +Theorem~\ref{thm:main} answers affirmatively the fixed-power question +posed in~\cite[Section~1.1]{GreenSawhney2025}. +Adajar et al.~\cite[Theorem~1.1]{AdajarEtAl2026} extend the arithmetic +level-$d$ approach to fixed integer polynomials having a root modulo +every positive integer. For degree at least two they obtain a density +saving $\exp(-c(\log N)^\mu)$ for every fixed $0<\mu<1/2$, with +constants depending on the polynomial and $\mu$. + +In the opposite direction, Ruzsa~\cite[Theorems~1 and~2]{Ruzsa1984} +constructed square-difference-free sets of size +$\gg N^{\gamma_0}$, where +\[ + \gamma_0=\frac12\left(1+\frac{\log 7}{\log 65}\right) + =0.733077\ldots. +\] +His construction uses a set of residues with no square differences +modulo a squarefree integer to restrict selected digits of an integer +expansion. Beigel and Gasarch~\cite[Lemma~3.6 and the proof of Theorem~3.7]{BeigelGasarch2008} and +Lewko~\cite[Theorem~3]{Lewko2015} obtained the improved exponent +\[ + \gamma_1=\frac12\left(1+\frac{\log 12}{\log 205}\right) + =0.733411797\ldots +\] +by exhibiting suitable sets of twelve residues modulo $205$. +Krachun's construction~\cite[Theorem~1]{Krachun2026} passes the +three-quarter threshold and gives +\[ + \liminf_{N\to\infty}\frac{\log s(N)}{\log N} + \ge 0.752796455874514\ldots. +\] +A substantial gap remains between this lower exponent and the upper +exponent $1-c$ in Theorem~\ref{thm:main}. + +Reflection positivity yields multilinear inequalities through the chessboard +method of Fr\"ohlich and Lieb~\cite[Theorem~2.2]{FrohlichLieb1978} and its +abstract formulation by Fr\"ohlich, Israel, Lieb and +Simon~\cite[Theorem~4.1]{FILS1978ReflectionI}. +Hatami~\cite[Theorem~2.8 of the arXiv version]{Hatami2010} characterizes +graph seminorms through mixed H\"older inequalities. Conlon and +Lee~\cite[Theorems~1.2 and~1.3]{ConlonLee2017Reflection} use finite +reflection groups to obtain such inequalities for absolute-value inputs, +and also for real-valued inputs when the two defining sets of simple +reflections are disjoint. +The signed mixed inequality is a classical feature of the chessboard +method; its role here is to permit Fourier lifts that can change sign. +We construct a finite-field probability law with the particular cycle +support, reflection positivity and conditional mixing needed for the +integer problem, and prove the signed inequality for that law directly. + +The rational-frequency energy estimates of Bloom and +Maynard~\cite{BloomMaynard2022} and the product-space methods of Green and +Sawhney~\cite{GreenSawhney2025} are methodological precedents for the +arithmetic part. The argument retains rational Fourier analysis and +rescaling along square-step progressions. Here the quantity compared +between scales is a multilinear functional of a signed Fourier lift. + +\subsection{Outline of the proof} + +The proof constructs a nonnegative quantity that dominates a fixed power +of the density, and then proves that this quantity decreases by a power +of the interval length. The comparison with shorter intervals is available +for every residue specification. A separate estimate makes a fixed positive +proportion of those specifications negligible; these two estimates produce +the contraction. + +Here is the quantity to be controlled. Choose a fixed even integer $d$ +and a fixed set $V$ of $d$ labels. For a finite set $\mathcal P$ of sufficiently +large primes, put $X_{\mathcal P}=\prod_{p\in\mathcal P}\F_p$. +We will construct a probability law for a tuple $(z_v)_{v\in V}$ in +$X_{\mathcal P}$, and for a real function $g$ on this space define +\[ + Z_{\mathcal P}(g)=\E\prod_{v\in V}g(z_v). +\] +Its $d$-th root will be a seminorm satisfying +$|\E g|\le Z_{\mathcal P}(g)^{1/d}$. To apply it to $A\subseteq[N]$, +take $\mathcal P$ to be the primes between a fixed threshold $P$ and +$N^\beta$, where $\beta>0$ is fixed and small. Let $G$ retain the +rational Fourier coefficients of $1_A$ at frequencies of squarefree +reduced denominator at most $N^\beta$ whose prime factors all belong +to $\mathcal P$, including the zero frequency. +Section~\ref{sec:lift-moments} defines this \emph{Fourier lift} precisely. +It is real and its mean is exactly $|A|/N$, although its values may have +either sign. With these choices, write $Y(N,A)=Z_{\mathcal P}(G)$. +The seminorm bound gives +\[ + (|A|/N)^d\le Y(N,A). +\] +Thus a fixed power decay for $Y$ proves the theorem. + +The tuple law is constructed separately over each prime field, then +multiplied over the primes, collecting the field labels into vectors +$z_v\in X_{\mathcal P}$. Every individual field label is uniform. +The law has two further properties with distinct purposes. +First, it is reflection positive: specified involutions of $V$ exchange +two halves, and the matrix coupling the half-tuples is symmetric positive +semidefinite. Repeated Cauchy--Schwarz across these cuts proves the signed +mixed inequality and seminorm bound +(Proposition~\ref{prop:signed-holder}). Second, outside a part of mass +at most $p^{-4}$, a designated directed cycle of $24$ labels has nonzero +square differences in $\F_p$, and each of its edge pairs is uniform +among pairs with such a difference. This exact pair law supplies the +arithmetic input below. The full reflection-positive law and its normalized +nonexceptional part also satisfy conditional mixing, which makes inputs +with large Fourier denominators negligible +(Lemma~\ref{lem:conductor-decay}). Section~\ref{sec:tuple-laws} constructs +these laws from the graph estimates of Section~\ref{sec:finite-field-graphs}. + +The approximation errors require control of signed lifts beyond their +means. Every fixed moment of $G$ grows more slowly than any fixed positive +power of $N$, uniformly in the prime set and in every retained family of +complete denominator classes. Lemma~\ref{lem:lift-moments} proves this +by a conditional square-function estimate and symmetrization. The same +bounds, with an explicit fiber-size cost, remain available after specifying +some prime coordinates. Combined with conductor decay, they justify the +Fourier truncations used in both estimates that follow. + +For the change of scale, fixing prime coordinates whose product is at +most a sufficiently small fixed power of $N$ makes the low Fourier +coefficients exact averages on an integer residue class modulo an integer +$D$. Splitting that class into progressions of common difference $D^2$ +and rescaling preserves square-difference avoidance: a square difference +in a rescaled set would give a square difference in $A$. +The seminorm's square-affine symmetries therefore compare the specialized +functional with the same functional at length $\lceil N/D^2\rceil$, +up to a factor tending to one and a power-decaying error +(Proposition~\ref{prop:scale-transfer}). This comparison applies to all +specified residues, without using the exceptional splitting. + +The arithmetic cancellation uses the normalized nonexceptional prime laws. +Set aside the finitely many primes below the tuple-law threshold, together +with modulus $8$, in one fixed modulus $M_0$. Independently choose one +residue modulo $M_0$ for each slot. A fixed positive proportion of these +assignments has strict square differences along the designated cycle: +they are $1$ modulo $8$ and nonzero squares modulo every odd prime +dividing $M_0$. +For such an assignment, divide $[N]$ into ordered intervals. Unless all +cycle slots choose the same interval, one directed edge goes from an +earlier interval to a later one. Truncating and expanding the other slots +reduces its contribution to the selected pair law. A signed kernel +supported on actual shifts $m^2$ then bounds that contribution using the +absence of integer square differences +(Propositions~\ref{prop:square-kernel} and~\ref{prop:nonexceptional-contraction}). +The event that all cycle slots choose one interval is itself small. + +Finally, Section~\ref{sec:induction} expands the exceptional submeasures +by inclusion--exclusion. Exceptional prime sets of large product are +bounded directly by moment estimates. A smaller exceptional prime set +with product $q_B$ leads to the shorter scale approximately $N/(M_0q_B)^2$. +Its mass remains summable even after multiplication by $q_B$; this absorbs +the factor $q_B^{2a_0}$ incurred by a sufficiently small proposed decay +exponent $a_0$. Combining this bound with the universal scale comparison +and the cancellation on a positive proportion of small-residue assignments +gives a strict weighted recurrence. Induction proves +$Y(N,A)\ll N^{-a_0}$, and the density inequality above gives +Theorem~\ref{thm:main} with $c=a_0/d$. +Figure~\ref{fig:proof-map} records the principal inputs to this assembly. +\input{figures/proof-map} + +\subsection{Conventions} + +All averages on finite sets are with respect to uniform probability +measure unless another measure is displayed. All $L^r$ norms of functions +on $X_{\mathcal P}$ use its uniform product measure. Empty products are +$1$; the product over an empty prime set is a one-point space. We write +$e(t)=\exp(2\pi i t)$ and identify rational frequencies modulo $1$. +The \emph{conductor} of a rational frequency is its reduced positive +denominator; the conductor of $0$ is $1$. + +For nonnegative quantities, $U\ll V$ means $U\le C V$ for a constant +with the dependencies stated in context. A uniform bound $U=N^{o(1)}$ +means that for every $\varepsilon>0$ there is a constant +$C_\varepsilon$ such that $U\le C_\varepsilon N^\varepsilon$ for all +allowed data. Fixed tuple sizes and fixed moment orders may affect +$C_\varepsilon$; sets, intervals, chosen residues and retained frequency +families may not. All final choices of parameters are made independently +of $N$ and $A$. diff --git a/preprints/A-power-saving-for-square-difference-free-sets-September-24-2026/build/sections/kernel.tex b/preprints/A-power-saving-for-square-difference-free-sets-September-24-2026/build/sections/kernel.tex new file mode 100644 index 0000000000000000000000000000000000000000..de55c38993bc586787cff2d224da13fc5847b4d8 --- /dev/null +++ b/preprints/A-power-saving-for-square-difference-free-sets-September-24-2026/build/sections/kernel.tex @@ -0,0 +1,568 @@ +\section{A signed kernel for actual square differences} +\label{sec:square-kernel} + +The preceding tuple measures impose square-difference constraints in residue fields. +This section supplies the passage to actual positive integer squares. +We construct a signed measure supported on such squares and approximate +its Fourier transform uniformly by a sum over small rational frequencies. +Because every actual square shift vanishes between the supports under +consideration, the signs of the measure cause no difficulty. The weighted +square sums and major/minor-arc analysis have a direct antecedent +in~\cite[Section~5]{GreenSawhney2025}. Here the signed divisor weight +is chosen to match the strict-square pair law of our finite-field tuples. + +For the remainder of the paper, fix +\begin{equation} + \beta=10^{-5}. + \label{eq:kernel-beta} +\end{equation} +Write $e(t)=\exp(2\pi i t)$, and let $q(\xi)$ be the reduced denominator +of $\xi\in\mathbb Q/\mathbb Z$, with $q(0)=1$. +For every rational frequency define the normalized unit quadratic sum +\begin{equation} + \mathfrak g(\xi) + =\frac{1}{\varphi(q(\xi))} + \sum_{m\in(\mathbb Z/q(\xi)\mathbb Z)^\times}e(\xi m^2), + \qquad \mathfrak g(0)=1. + \label{eq:kernel-unit-gauss} +\end{equation} +Here $\varphi$ is Euler's totient function. +If $D=8d_0$ with $d_0$ odd and squarefree, call a residue modulo $D$ +a \emph{strict square class} if it is $1$ modulo $8$ and a nonzero +square modulo every prime dividing $d_0$. + +\begin{proposition}[Ordered square-kernel estimate] +\label{prop:square-kernel} +Let $N\ge1$ be an integer, put $H=N^{\beta/2}$, and let +$D=8d_0\le N^\beta$, where $d_0$ is odd and squarefree. +Suppose $F,J:[N]\to\mathbb C$ satisfy $|F|,|J|\le1$ and are supported +in residue classes $a,b$ modulo $D$, respectively. Assume that $b-a$ +is a strict square class, that $x3$, a primitive character averages to zero +on that class. + +Apply these local facts to the decomposition +$\alpha_0=\alpha_D+\alpha_1$ already used in +\eqref{eq:kernel-model-coefficient}. Chinese remaindering shows that +the factor at primes dividing $D$ is +\begin{equation} + \mathbb E_m\rho_D(m)e(\alpha_Dm^2) + =\begin{cases} + e(\alpha_Dr),&q(\alpha_D)\mid D,\\ + 0,&q(\alpha_D)\nmid D. + \end{cases} + \label{eq:kernel-root-law} +\end{equation} +The expectation here is the uniform average over any common period. +The normalized odd-root restrictions, and the normalized oddness +restriction at two, are precisely responsible for the factor $D$ +in the model kernel. In particular, choosing only one odd root +introduces no additional factor in \eqref{eq:kernel-root-law}. + +\paragraph{The coprime divisor sieve.} +Every prime of an admissible $u$ is coprime to $D$. +If such a prime does not divide $q_1$, its mean-zero factor $w_p$ +is independent of the exponential and all other factors by the +Chinese remainder theorem. Thus $A_u=0$ unless +$u\mid\operatorname{rad}(q_1)$. + +Suppose first that $q_1\le H$. All divisors of +$\operatorname{rad}(q_1)$ then occur in the sum. By +\eqref{eq:kernel-complete-sieve}, the divisor weight is the density of +the uniform unit law modulo $q_1$ relative to the uniform law. +If $p^j$ exactly divides $q_1$ with +$j\ge2$, its local primitive unit quadratic average vanishes. +Indeed, conditional on a unit $x$ modulo $p^{j-1}$, the $p$ lifts +$x+tp^{j-1}$ have phase factors differing by $e(2atx/p)$, +where $p\nmid 2ax$. Their sum is zero. +If $q_1$ is squarefree, the unit average is exactly +$\mathfrak g(\alpha_1)$. +Together with \eqref{eq:kernel-root-law}, this proves +\begin{equation} + c(\alpha_0)=c_0(\alpha_0)\qquad(q_1\le H). + \label{eq:kernel-small-coefficient} +\end{equation} + +For $q_1>H$ the model coefficient is zero, and we need a bound for +the truncated divisor sum. We give the prime-power calculation +explicitly. For an odd prime $p$, an integer $j\ge1$, and $p\nmid a$, +write +\[ + G_j(a)=\frac1{p^j}\sum_{m\bmod p^j}e(am^2/p^j), + \qquad G_0(a)=1. +\] +Because two is invertible modulo $p^j$, the substitution +$(m,n)\mapsto(m-n,m+n)$ is a bijection. Consequently +\begin{equation} + |G_j(a)|^2 + =\frac1{p^{2j}}\sum_{u,v\bmod p^j}e(auv/p^j) + =p^{-j}. + \label{eq:kernel-prime-power-gauss} +\end{equation} +Let +$T_j=\mathbb E_{m\bmod p^j}\mathbf1_{p\mid m}e(am^2/p^j)$. +For $j=1$ one has $T_1=1/p$; for $j\ge2$, substituting $m=pt$ +gives $T_j=p^{-1}G_{j-2}(a)$. +It follows that +\begin{equation} + B_j:=\mathbb E_{m\bmod p^j}w_p(m)e(am^2/p^j) + =\frac{pT_j-G_j(a)}{p-1}, + \qquad |B_j|\le2p^{-j/2}. + \label{eq:kernel-weighted-prime-power-gauss} +\end{equation} +For $j\ge2$, the last bound follows from +$(p+1)/(p-1)\le2$ and \eqref{eq:kernel-prime-power-gauss}. +For $j=1$, the bound obtained from the formula is +$(1+p^{-1/2})/(p-1)$; its ratio to $p^{-1/2}$ is +$1/(\sqrt p-1)\le2$. Thus the exponent-one case is included. + +For a fixed divisor $u\mid\operatorname{rad}(q_1)$, the coprime +local mean is a product of factors from +\eqref{eq:kernel-prime-power-gauss} and +\eqref{eq:kernel-weighted-prime-power-gauss}. Its modulus is at most +$2^{\omega(u)}q_1^{-1/2}$. +The factor at primes of $D$ has modulus at most one by +\eqref{eq:kernel-root-law}. Dropping the restriction $u\le H$ +after taking absolute values therefore gives +\begin{equation} + |c(\alpha_0)| + \le q_1^{-1/2}\sum_{u\mid\operatorname{rad}(q_1)}2^{\omega(u)} + =3^{\omega(q_1)}q_1^{-1/2} + \ll q_1^{-1/3} + \le H^{-1/3}. + \label{eq:kernel-large-coefficient} +\end{equation} +The penultimate implied constant is absolute. For example, one may use +\[ + C_*= + \prod_{\substack{p<729\\p\text{ odd prime}}}\max(1,3p^{-1/6}), +\] +since $3\le p^{1/6}$ for every prime $p\ge729$. +Repeated prime powers only increase $q_1^{1/6}$. +This argument covers all prime powers in $q_1$, and if +$q(\alpha_D)\nmid D$, every coefficient under discussion is already +zero by \eqref{eq:kernel-root-law}. + +\paragraph{Completion of the uniform comparison.} +On the minor arcs, \eqref{eq:kernel-minor-actual} and +\eqref{eq:kernel-minor-model} apply. On a major arc, +\eqref{eq:kernel-major-model}, \eqref{eq:kernel-major-actual}, +\eqref{eq:kernel-small-coefficient} and +\eqref{eq:kernel-large-coefficient} apply, with $|V_N(y)|\le1$. +For clarity, every approximation error is bounded here using the fixed +value \eqref{eq:kernel-beta}: +\begin{align*} + DHN^{-1/20}\sqrt{\log N} + &\le N^{-0.049985}\sqrt{\log N},\\ + DH^2N^{-1/10} + &\le N^{-0.09998},\\ + qD^2H^3/N + &\le N^{-0.899965},\\ + \frac{qD^2H^2(1+|y|N)}{\sqrt N} + &\le3N^{-0.29997}. +\end{align*} +In the last two lines $q\le N^{1/10}$ and +$|y|\le2N^{-9/10}$. +Each bound is $O(N^{-\beta/6})=O(H^{-1/3})$ with an absolute constant. +Together with the coefficient bound this proves +\eqref{eq:kernel-uniform-approximation} for all sufficiently large $N$. +All size and separation thresholds used above are absolute. +For the bounded remaining range, the elementary bounds +$\|S\|_\infty\ll DH$ and $\|S_0\|_\infty\le DH^2$ +absorb that range by increasing the absolute constant. +Thus \eqref{eq:kernel-uniform-approximation} is uniform in all the +data of the proposition. + +\paragraph{The directional bilinear identity.} +It remains to pass from the Fourier comparison to the supports in the +statement. Extend $F$ and $J$ by zero to $\mathbb Z$. +For any finitely supported kernel $h$ define +\[ + \mathcal B_h(F,J)=\frac1N\sum_{x\in\mathbb Z} + \sum_{n\in\mathbb Z}h(n)F(x)J(x+n). +\] +Both sums use counting measure; the displayed $1/N$ normalizes only +the base variable. Let +\[ + \widetilde F(\alpha)=\sum_{x\in\mathbb Z}F(x)e(-x\alpha), + \qquad + \widetilde J(\alpha)=\sum_{y\in\mathbb Z}J(y)e(-y\alpha), + \qquad + S_h(\alpha)=\sum_{n\in\mathbb Z}h(n)e(\alpha n). +\] +Expanding and integrating the characters imposes $y=x+n$, giving +the bilinear identity +\[ + \mathcal B_h(F,J)=\frac1N\int_0^1 + S_h(\alpha)\widetilde F(-\alpha)\widetilde J(\alpha)\,d\alpha. +\] +There are no conjugates in this identity. +Cauchy--Schwarz and counting-measure Plancherel now give +\begin{align} + |\mathcal B_{k-k_0}(F,J)| + &\le\frac{\|S-S_0\|_\infty}{N} + \|F\|_{\ell^2(\mathbb Z)}\|J\|_{\ell^2(\mathbb Z)} + \ll H^{-1/3}, + \label{eq:kernel-plancherel-error} +\end{align} +because each of the last two norms is at most $\sqrt N$. + +Every term in $\mathcal B_k(F,J)$ contains $F(x)J(x+m^2)$ with +$m\ge1$, and this product is zero by hypothesis. +Thus $\mathcal B_k(F,J)=0$ term by term, regardless of the signed +weights in \eqref{eq:kernel-actual}. +For a pair $x,y$ with $F(x)J(y)\ne0$, the ordering assumption gives +$1\le y-x\le N-1$, and the residue conditions give +$y-x\equiv r\pmod D$. +Consequently both indicators in \eqref{eq:kernel-model} are automatic +for every contributing pair. Its expansion is exactly +\begin{align*} + \mathcal B_{k_0}(F,J) + &=\frac{D}{N^2}\sum_{x,y\in[N]}F(x)J(y) + \sum_{\xi\in\Xi}\mathfrak g(\xi)e(-\xi(y-x))\\ + &=D\sum_{\xi\in\Xi}\mathfrak g(\xi) + \widehat F(-\xi)\widehat J(\xi). +\end{align*} +Combining this with \eqref{eq:kernel-plancherel-error} proves +\eqref{eq:kernel-conclusion}. +\end{proof} + +\begin{remark} +\label{rem:kernel-interval-normalization} +The normalization for later interval restrictions follows directly +from the proposition. Suppose $I_1,I_2\subseteq[N]$ are nonempty +intervals with $\max I_1<\min I_2$, and let $\Xi$ be the frequency +set in \eqref{eq:kernel-conclusion}. Suppose $F_i$ is the indicator of the restriction of a +square-difference-free set to $I_i$ and a class $a_i$ modulo $D$, +where $a_2-a_1$ is a strict square class. The coefficients +\[ + c_i(\xi)=\frac D{|I_i|}\sum_{n\in I_i}F_i(n)e(-n\xi) + =\frac{DN}{|I_i|}\widehat F_i(\xi) +\] +satisfy +\[ + \left|\sum_{\xi\in\Xi}\mathfrak g(\xi)c_1(-\xi)c_2(\xi)\right| + \ll H^{-1/3}\frac{DN^2}{|I_1||I_2|}. +\] +This conclusion uses the integer supports of the restrictions; +it requires no pointwise positivity of a Fourier lift. +\end{remark} diff --git a/preprints/A-power-saving-for-square-difference-free-sets-September-24-2026/build/sections/lifts.tex b/preprints/A-power-saving-for-square-difference-free-sets-September-24-2026/build/sections/lifts.tex new file mode 100644 index 0000000000000000000000000000000000000000..afed1b9c9183d4a94dde0396093c7f5653907336 --- /dev/null +++ b/preprints/A-power-saving-for-square-difference-free-sets-September-24-2026/build/sections/lifts.tex @@ -0,0 +1,411 @@ +\section{Uniform moment bounds for Fourier lifts} +\label{sec:lift-moments} + +We now transfer a bounded function on an integer interval to the product +of its prime residue spaces. Keeping rational frequencies of bounded +denominator need not preserve pointwise bounds or positivity. The +replacement we require is a uniform bound for every fixed moment, valid +also after retaining arbitrary complete denominator classes. We first +control a square function of the exact prime supports, then recover the +sum by symmetrization. For related arithmetic moment estimates and Fourier +lifting, see \cite[Theorem~2 of the arXiv version]{BloomMaynard2022} and +\cite[Section~3]{GreenSawhney2025}; the uniform estimate used here is +proved below. + +Let \(\mathcal P\) be a finite set of primes, and give +\[ + X_{\mathcal P}=\prod_{p\in\mathcal P}\mathbb F_p +\] +uniform probability measure. A character is indexed by +\(\xi=\sum_{p\in\mathcal P}a_p/p\pmod1\), where +\(a_p\in\{0,\ldots,p-1\}\), and we write +\[ + e(z\xi)=\prod_{p\in\mathcal P}e(a_pz_p/p), + \qquad e(t)=\exp(2\pi i t). +\] +Its exact prime support is \(\{p:a_p\ne0\}\), and its reduced +denominator is the product of these primes. In particular \(q(0)=1\). +Summing a geometric progression in each coordinate shows that these +characters form an orthonormal basis. +For \(U\subseteq\mathcal P\), put \(q_U=\prod_{p\in U}p\), with +\(q_\varnothing=1\). + +Let \(I\) be a nonempty consecutive interval of integers, of length +\(L\), and let \(f:I\to\mathbb C\). Define +\[ + \widehat f_I(\xi)=\frac1L\sum_{n\in I}f(n)e(-n\xi), + \qquad + f_U(z_U)=\sum_{q(\xi)=q_U}\widehat f_I(\xi)e(z_U\xi). +\] +For \(Q\ge1\), the Fourier lift is +\[ + \mathcal L_{I,\mathcal P,Q}f + =\sum_{\substack{U\subseteq\mathcal P\\q_U\le Q}}f_U. +\] +If \(\mathcal F\) is a subfamily of these supports, write +\(\mathcal L^{\mathcal F}_{I,\mathcal P,Q}f=\sum_{U\in\mathcal F}f_U\). +Thus retaining a support means retaining all frequencies of its exact +denominator. Each \(f_U\) depends only on \(z_U\) and has mean zero +in each coordinate belonging to \(U\). Real \(f\) gives real +\(f_U\), because the summation set is closed under negation. + +\begin{remark}[A lift need not be nonnegative] +Take distinct primes $p,q$ with $p,q\le QH,B} + =\{U\in\mathcal F:q_{U\setminus B}>H\}. +\] +After specialization their Fourier supports lie, respectively, at +remaining denominators at most \(H\) and greater than \(H\). +\end{lemma} + +\begin{proof} +The specified fiber has uniform probability \(1/q_B\). Its +contribution to \(\mathbb E|G|^p\) is +\[ + q_B^{-1}\mathbb E_{z_{\mathcal P\setminus B}} + |G(z_B,z_{\mathcal P\setminus B})|^p. +\] +Discarding all other nonnegative contributions proves +\eqref{eq:lift-specialization}. +Apply Lemma~\ref{lem:lift-moments} to each displayed family. +Every character of \(f_U\), when specialized on \(B\), retains +exactly the coordinate support \(U\setminus B\). Combining such +characters can cancel coefficients but cannot create frequencies +outside the asserted remaining-denominator ranges. This proves the +last assertion. +\end{proof} + +In particular, for \(d\ge2\), a multilinear estimate using one \(L^2\) norm and +\(d-1\) norms in \(L^{2(d-1)}\), with the same coordinates fixed +in every slot, incurs the total specialization factor +\[ + q_B^{1/2}\left(q_B^{1/(2(d-1))}\right)^{d-1}=q_B. +\] +Two specialized \(L^2\) factors likewise incur \(q_B\). +Multiplying the original bounded function by a fixed constant simply +multiplies every norm bound by the absolute value of that constant. + +For later uniform estimates one can make the subpower dependence +entirely explicit. Put +\[ + k_*(Q)=\max\{j\in\mathbb Z_{\ge0}:(j+1)!\le Q\}, + \qquad + \mathfrak M_r(Q)=2\bigl(k_*(Q)+1\bigr)A_r^{k_*(Q)}. +\] +The moment bound is at most \(M\mathfrak M_r(Q)\), and for every +fixed \(r\) and \(\epsilon>0\), one has +\(\mathfrak M_r(Q)\le Q^\epsilon\) for sufficiently large \(Q\). +The threshold may depend on \(r,\epsilon\), but on none of the +intervals, functions, fibers, or retained families. Thus any fixed +finite product of the envelopes \(\mathfrak M_r(Q)\) remains subpower, +even when their fixed moment orders are very large. The displayed +input bounds and specialization powers remain explicit. diff --git a/preprints/A-power-saving-for-square-difference-free-sets-September-24-2026/build/sections/transfer.tex b/preprints/A-power-saving-for-square-difference-free-sets-September-24-2026/build/sections/transfer.tex new file mode 100644 index 0000000000000000000000000000000000000000..04791f0777463799b8c887e746a27d119ea98e9e --- /dev/null +++ b/preprints/A-power-saving-for-square-difference-free-sets-September-24-2026/build/sections/transfer.tex @@ -0,0 +1,399 @@ +\section{Comparison along square-step progressions} +\label{sec:scale-transfer} + +We next compare the product-space functional with the same functional on +shorter intervals. Fixing a few prime coordinates makes its low Fourier +coefficients exact averages on a residue class. Splitting that class into +progressions with square step preserves the forbidden integer differences +and permits a change of scale. The comparison proved here uses no induction +hypothesis. Restriction to square-step progressions is already a key +feature of quantitative square-difference arguments; see the Main Lemma +of~\cite{PSS1988} and~\cite[Lemma~6.2]{GreenSawhney2025}. Here it is +used to compare the signed multilinear functional itself. + +Keep the constants \(\gamma=1/64\) and \(\beta=10^{-5}\) fixed as above. +Enlarge the fixed prime threshold \(P\), if necessary, so that the tuple +measures are available for every prime \(p\ge P\) and +\begin{equation} + \prod_{\substack{p\ge P\\p\ {\rm prime}}}(1+p^{-3})-1\le\frac14, + \qquad + |\mathfrak g(\xi)|\le q(\xi)^{-1/3} + \label{eq:prime-threshold-conditions} +\end{equation} +whenever \(q(\xi)\) is squarefree and all its prime factors are at least +\(P\). Such a choice is possible: the product converges, while the prime +Gauss estimate used in Proposition~\ref{prop:square-kernel} gives +\[ + \left|\frac1{p-1}\sum_{x\in\F_p^\times}e(ax^2/p)\right| + \le\frac{\sqrt p+1}{p-1} + =\frac1{\sqrt p-1}\le p^{-1/3} +\] +for all sufficiently large \(p\) and all \(a\ne0\) in \(\F_p\). +Chinese remaindering then gives the second inequality in +\eqref{eq:prime-threshold-conditions}, including \(q(\xi)=1\). +Define the fixed integer +\begin{equation} + M_0=8\prod_{\substack{3\le p0\), this lower bound is at least \(N^{1/2}\) +for sufficiently large \(N\). As \(D\ge8\), we have \(N'H}\) according to the denominator on the +remaining prime coordinates \(\mathcal S\). +Before specialization, these pieces retain exactly the denominator +families +\[ + \{U\subseteq\mathcal P_N:q_U\le Q,\ q_{U\setminus B}\le H\}, + \qquad + \{U\subseteq\mathcal P_N:q_U\le Q,\ q_{U\setminus B}>H\}. +\] +Thus Lemma~\ref{lem:lift-moments} applies to both pieces. +For a function \(u\) on \(X_B\times X_{\mathcal S}\), uniform counting +gives the specialization bound +\[ + \|u(z_B,\cdot)\|_r + \le q_B^{1/r}\|u\|_r + \qquad(r\ge1); +\] +this is also the estimate in Lemma~\ref{lem:lift-specialization}. +Since \(|f_s|\le M_0\), it follows that, for +\(g_*\in\{g,g_{\le H},g_{>H}\}\) and every fixed even \(r\), +\begin{equation} + \|g_*\|_r\le M_0q_B^{1/r}N^{o(1)}. + \label{eq:specialized-lift-moments} +\end{equation} +All subpower bounds in this proof are uniform over the indicated +sets and fibers. + +Telescope the \(d\) inputs in the diagonal integral from \(g\) to +\(g_{\le H}\). Each difference has one input \(g_{>H}\). +Lemma~\ref{lem:conductor-decay} and +\eqref{eq:specialized-lift-moments} imply, for every fixed +\(\varepsilon>0\), +\begin{equation} + |Z_{\mathcal S}(g)-Z_{\mathcal S}(g_{\le H})| + \le C_\varepsilon H^{-\gamma}q_BN^\varepsilon. + \label{eq:scale-transfer-first-error} +\end{equation} +Indeed the specialization powers in that lemma add to +\[ + \frac12+\frac{d-1}{2(d-1)}=1. +\] +The factors \(d\) and \(M_0^d\) are fixed and are included in +\(C_\varepsilon\). Applying the moment lemma with sufficiently small +individual exponents makes the aggregate moment loss at most +\(N^\varepsilon\). + +\smallskip +\noindent\emph{Splitting the exact fiber into square-step progressions.} +Let \(a\bmod D\) combine \(s\) and \(z_B\). By +Lemma~\ref{lem:exact-fiber}, for every remaining \(\xi\) with +\(q(\xi)\le H\), +\begin{equation} + \widehat g(\xi)=\frac D N + \sum_{\substack{n\le N\\n\equiv a\pmod D}}1_A(n)e(-n\xi). + \label{eq:scale-transfer-fiber} +\end{equation} +Let +\[ + \begin{gathered} + J_a=\{j\in[D^2]:j\equiv a\pmod D\},\qquad + c_j=j-D^2,\\ + A'_j=\{n'\in[N']:D^2n'+c_j\in A\}. + \end{gathered} +\] +There are exactly \(D\) elements of \(J_a\). +The map \((n',j)\mapsto D^2(n'-1)+j\) bijects +\([N']\times[D^2]\) with \([D^2N']\). +Restricting \(j\) to \(J_a\) selects exactly the residue class +\(a\bmod D\). The definition of \(A'_j\) assigns zero to all padded +points greater than \(N\), so this decomposition incurs no additive +error. + +Each \(A'_j\) is square-difference-free: a difference \(m^2>0\) +between its elements would give the difference +\(D^2m^2=(Dm)^2\) between the corresponding elements of \(A\). +This argument concerns actual integer squares. + +For \(p\in\mathcal S\), the integer \(D\) is invertible modulo \(p\). +Define +\[ + T_j(z)_p=D^{-2}(z_p-c_j)\pmod p,\qquad + h_j^H=\mathcal L_{[N'],\mathcal S,H}1_{A'_j}. +\] +Each \(T_j\) is a translation followed by multiplication by the +nonzero square \((D^{-1})^2\). +Multiplication of a remaining frequency by \(D^2\) preserves its +exact denominator, so changing variables \(\eta=D^2\xi\) in the +Fourier expansion of \eqref{eq:scale-transfer-fiber} gives the exact +identity +\begin{equation} + g_{\le H}(z) + =\frac{\lambda}{D}\sum_{j\in J_a}h_j^H(T_j(z)). + \label{eq:scale-transfer-affine-average} +\end{equation} +For clarity, the coefficient of \(e(z\xi)\) on the right is +\[ + \frac{DN'}N\sum_{j\in J_a}e(-c_j\xi) + \left(\frac1{N'}\sum_{n'\le N'} + 1_{A'_j}(n')e(-D^2n'\xi)\right), +\] +which is exactly \eqref{eq:scale-transfer-fiber}. + +The seminorm triangle inequality and affine invariance in +Lemma~\ref{lem:functional-properties} now imply +\[ + Z_{\mathcal S}(g_{\le H})^{1/d} + \le \frac{\lambda}{D} + \sum_{j\in J_a}Z_{\mathcal S}(h_j^H)^{1/d} + \le \lambda\max_{j\in J_a}Z_{\mathcal S}(h_j^H)^{1/d}. +\] +Consequently, +\begin{equation} + Z_{\mathcal S}(g_{\le H}) + \le\lambda^d\max_{j\in J_a}Z_{\mathcal S}(h_j^H). + \label{eq:scale-transfer-progressions} +\end{equation} +The averaging by \(D\) is essential: the number of progressions +causes no further loss. + +\smallskip +\noindent\emph{Recovering the smaller-scale functional.} +Set +\[ + h_j^{Q'}=\mathcal L_{[N'],\mathcal S,Q'}1_{A'_j}. +\] +The difference \(h_j^{Q'}-h_j^H\) has conductor greater than \(H\). +By \eqref{eq:scale-transfer-cutoffs}, the moment lemma applies at +scale \(N'\) to these lifts and their pieces. A second telescoping +application of Lemma~\ref{lem:conductor-decay}, this time without +specialization, gives +\begin{equation} + |Z_{\mathcal S}(h_j^H)-Z_{\mathcal S}(h_j^{Q'})| + \le C_\varepsilon H^{-\gamma}(N')^\varepsilon + \label{eq:scale-transfer-second-error} +\end{equation} +uniformly in \(j\). + +The lift \(h_j^{Q'}\) depends only on coordinates in +\(\mathcal S'=\mathcal P_{N'}\setminus B\), because every prime in a +denominator at most \(Q'\) is itself at most \(Q'\). +On the other hand, the full lift +\[ + u'_j=\mathcal L_{[N'],\mathcal P_{N'},Q'}1_{A'_j} +\] +becomes exactly \(h_j^{Q'}\) after uniform averaging over +\(B\cap\mathcal P_{N'}\). Character orthogonality removes precisely +the denominators containing one of those primes. +By Lemma~\ref{lem:functional-properties}, coordinate averaging contracts +the seminorm; unused probability factors integrate to one. Therefore +\begin{equation} + Z_{\mathcal S}(h_j^{Q'}) + =Z_{\mathcal S'}(h_j^{Q'}) + \le Z_{\mathcal P_{N'}}(u'_j) + =Y(N',A'_j). + \label{eq:scale-transfer-projection} +\end{equation} +This also covers primes of \(B\) larger than \(Q'\), which are absent +from \(\mathcal P_{N'}\) already. + +Combining \eqref{eq:scale-transfer-first-error}, +\eqref{eq:scale-transfer-progressions}, +\eqref{eq:scale-transfer-second-error}, and +\eqref{eq:scale-transfer-projection} proves +\eqref{eq:scale-transfer-diagonal} with error +\begin{equation} + C_\varepsilon\left( + H^{-\gamma}q_BN^\varepsilon+ + \lambda^dH^{-\gamma}(N')^\varepsilon\right). + \label{eq:scale-transfer-explicit-error} +\end{equation} + +\smallskip +\noindent\emph{Uniform error bounds and mixed inputs.} +The definition of \(N'\) gives +\(1\le\lambda<1+D^2/N\), and +\[ + D^2/N\le M_0^2N^{-1+2\tau}=o(1). +\] +Since \(d\) is fixed, these inequalities prove +\eqref{eq:scale-transfer-factor} and show that \(\lambda^d\) +is bounded for large \(N\). +Choose \(\varepsilon=\beta\gamma/8\) in +\eqref{eq:scale-transfer-explicit-error}. Its two terms, before +the bounded factor \(\lambda^d\), are at most +\[ + N^{-3\beta\gamma/8+\tau} + \quad\hbox{and}\quad N^{-3\beta\gamma/8}, +\] +respectively. As +\(\tau\le\beta\gamma/1000\) and +\(\sigma_0=\gamma\tau/4\le\tau/4\), they are +\(O(N^{-\sigma_0})\). This establishes the diagonal bound with +constants independent of any inductive estimate for \(Y\). + +Every \(g_v\) in the mixed statement is real and satisfies the same +diagonal bound. Its right-hand side is nonnegative and independent +of \(v\). Proposition~\ref{prop:signed-holder} therefore gives +\eqref{eq:scale-transfer-mixed}. +\end{proof} + +In particular, the mixed estimate holds pointwise after specifying all +labels at primes in \(B\), regardless of the law used to choose those +labels. It can therefore be integrated against any such probability +law. When the prime laws form a product, the remaining coordinates +still carry the full laws used in Proposition~\ref{prop:scale-transfer}. +No reflection positivity is required of the law on the specified labels. diff --git a/preprints/A-power-saving-for-square-difference-free-sets-September-24-2026/build/sections/tuples.tex b/preprints/A-power-saving-for-square-difference-free-sets-September-24-2026/build/sections/tuples.tex new file mode 100644 index 0000000000000000000000000000000000000000..d8795d7457bab0c7b91b6ac434348ae26b9ac7b0 --- /dev/null +++ b/preprints/A-power-saving-for-square-difference-free-sets-September-24-2026/build/sections/tuples.tex @@ -0,0 +1,855 @@ +\section{Finite reflection-positive probability laws} +\label{sec:tuple-laws} + +We construct probability laws on a fixed set of field labels. Their +nonexceptional parts enforce a directed cycle of square differences. +The full laws also satisfy positivity across a family of reflection cuts. +To obtain both properties, we add a finite hierarchy of measures in which +successively more blocks of the index set are marked. A marked block +provides a positive quadratic form that dominates the small negative part +at the corresponding unmarked cut. + +\subsection{The index set and the required properties} + +Fix the integers +\begin{equation} + h=24,\qquad t_0=80,\qquad \ell=2t_0+1=161. + \label{eq:tuple-fixed-parameters} +\end{equation} +Let \(\Pi\) be the set of permutation words on \([h]=\{1,\ldots,h\}\), +viewed as bijections from positions to symbols. Write +\[ + K=h!,\qquad r=K/2,\qquad V=\Pi^\ell,\qquad d=|V|=K^\ell, + \qquad n=d/2. +\] +The coordinates of a vertex \(v\in V\) are called its blocks. +Fix a cycle \(c\) of length \(h\) on the positions. Composition on +the right, \(w\mapsto wc\), cyclically shifts a word. Define +\(v_0,\ldots,v_{h-1}\in V\) by giving every block of \(v_j\) the +word \(c^j\), and put \(v_h=v_0\). + +A reflection is specified by a block \(b\in[\ell]\) and two +distinct symbols \(\alpha,\beta\). It acts on \(V\) by swapping +these symbols on the left in block \(b\); denote this involution by +\(\theta\). Its associated cut is +\[ + V_+=\{v:\alpha\text{ precedes }\beta\text{ in block }b\}, + \qquad V_-=\theta V_+. +\] +The reflection has no fixed vertices, and both halves have size \(n\). + +For a finite alphabet \(X\), a measure \(\nu\) on \(X^V\) defines +a matrix whose row index is \((z_v)_{v\in V_+}\) and whose column +index is \((z_{\theta v})_{v\in V_+}\). We say that \(\nu\) is +\emph{reflection positive} if this matrix is symmetric positive +semidefinite for every cut above. Equivalently, symmetry holds and +\[ + \int F((z_v)_{v\in V_+})F((z_{\theta v})_{v\in V_+})\,d\nu(z) + \ge0 +\] +for every real function \(F:X^{V_+}\to\mathbb R\). + +Recall that \(Q_p^*=\{u^2:u\in\mathbb F_p^\times\}\) for an odd +prime \(p\). The following proposition summarizes the construction. + +\begin{proposition}[Tuple laws] +\label{prop:tuple-laws} +There is an absolute integer \(P\) such that, for every prime +\(p\ge P\), there are a probability measure \(\mu_p\) on +\(\mathbb F_p^V\) and a nonnegative submeasure \(\varepsilon_p\le\mu_p\) +whose mass satisfies +\[ + \eta_p=\varepsilon_p(\mathbb F_p^V)\le p^{-4}. +\] +Writing +\[ + \mu_p^\circ=\frac{\mu_p-\varepsilon_p}{1-\eta_p}, +\] +the following properties hold. +\begin{enumerate} +\item The full law \(\mu_p\) is reflection positive. Both +\(\mu_p\) and \(\varepsilon_p\) are invariant under simultaneous +changes \(z_v\mapsto qz_v+a\), where \(a\in\mathbb F_p\) and +\(q\in Q_p^*\). Every single-label marginal of \(\mu_p\) and +\(\mu_p^\circ\) is uniform on \(\mathbb F_p\). +\item On the support of +\(\mu_p^\circ\), +\[ + z_{v_{j+1}}-z_{v_j}\in Q_p^*\qquad(0\le j\ell\), +which is impossible. Each unordered pair of list entries therefore +supports at most one directed edge in \(\mathcal G_S\). +Lemma~\ref{lem:graph-count} applies to this graph and all its induced +subgraphs. + +Set +\[ + \delta=\frac1{10h^2}. +\] +For distinct \(v,w\in V\), define a symmetric interaction +\(H^S_{vw}\) as follows. It is zero unless \(v,w\) differ by a +symbol transposition in exactly one marked block. In that case, +if the positions of the swapped symbols have gap \(g\), put +\(H^S_{vw}=\delta^g\). Independently of the lists, choose all +indices \(j_v\in[m_s]\) uniformly and independently, and give their +joint distribution the unnormalized weight +\begin{equation} + W_S(j)=\exp\left( + \sum_{\{v,w\}\subseteq V}H^S_{vw}1_{\{j_v=j_w\}}\right). + \label{eq:tuple-index-weight} +\end{equation} +Every sum over \(\{v,w\}\) is over unordered pairs of distinct +vertices, so each interaction is counted once. + +Define the finite measure \(\nu_{p,S}\) by the identity +\begin{equation} + \int G(z)\,d\nu_{p,S}(z) + =\mathbb E_Y\mathbb E_j I_S(Y)W_S(j) + G\bigl((Y_{u_S(v),j_v})_{v\in V}\bigr) + \label{eq:tuple-component-law} +\end{equation} +for all functions \(G\) on \(\mathbb F_p^V\). Both expectations +on the right are uniform probability averages. In particular, +\(\nu_{p,S}\) has not been normalized to have mass one. +We also write \(\nu(1)\) for the total mass of a finite measure \(\nu\). + +Each vertex has at most \(s\binom h2\) neighbors with nonzero +interaction, and each interaction is at most \(\delta\). Thus, +with the fixed constant +\begin{equation} + B=\exp\left(\frac{d(t_0+1)\binom h2\delta}{2}\right), + \label{eq:tuple-weight-bound} +\end{equation} +we have +\begin{equation} + 1\le W_S(j)\le B,\qquad \nu_{p,S}(\mathbb F_p^V)\le B. + \label{eq:tuple-component-mass-upper} +\end{equation} + +For a cut with reflection \(\theta\), define the quadratic form +\begin{equation} + Q_{p,S}^{b}(F)= + \int F((z_v)_{v\in V_+}) + F((z_{\theta v})_{v\in V_+})\,d\nu_{p,S}(z). + \label{eq:tuple-cut-form} +\end{equation} +The notation suppresses the two symbols that specify the cut. +Every component has a symmetric matrix at every cut, as we now verify. + +\begin{lemma}[Symmetry of the component matrices] +\label{lem:tuple-component-symmetry} +For every allowed mark set \(S\) and every reflection \(\theta\), +the measure \(\nu_{p,S}\) is invariant under the permutation +\((z_v)\mapsto(z_{\theta v})\). Its matrix at that cut is symmetric. +\end{lemma} + +\begin{proof} +The array \(H^S\) is invariant under simultaneous application of +\(\theta\) to both endpoints. In the reflected block, conjugating +a symbol transposition changes only the names of the swapped symbols, +not their positional gap. Interactions in other blocks retain their +gap as well. Hence relabeling indices by \(j_v\mapsto j_{\theta v}\) +preserves their uniform law and weight. + +If the reflected block is marked, \(u_S(\theta v)=u_S(v)\), so +no relabeling of lists is needed. If it is unmarked, \(\theta\) +acts on addresses. Left symbol permutations commute with the right +position shift \(c\); therefore it preserves the directed address +relation \eqref{eq:tuple-address-edge}, including its orientation. +Relabeling lists by this address involution preserves their uniform +law and \(I_S\). In either case these bijections in +\eqref{eq:tuple-component-law} send the output label at \(v\) to +the label at \(\theta v\), proving invariance. At the cut this +invariance exchanges the two matrix indices, which is symmetry. +\end{proof} + +\subsection{Strict positivity at a marked cut} + +The next estimate is the source of domination between successive +mark sets. It must remain valid when different index assignments +produce identical field labels. + +\begin{lemma}[Marked-cut lower bound] +\label{lem:marked-positive} +Put +\[ + \kappa=\frac9{10}\delta^{h-1},\qquad + \lambda=(e^\kappa-1)^n,\qquad + c_* =\lambda m_{t_0+1}^{-n}>0. +\] +For every \(S\) with \(|S|\le t_0+1\), every cut in a marked block +\(b\in S\), and every real \(F:\mathbb F_p^{V_+}\to\mathbb R\), +\begin{equation} + Q_{p,S}^{b}(F) + \ge \lambda m_s^{-n}\, + \mathbb E_{Y,j_+}I_S(Y) + F((Y_{u_S(v),j_v})_{v\in V_+})^2. + \label{eq:tuple-marked-lower} +\end{equation} +Here the expectation uses all uniform lists and only the uniform +indices on \(V_+\). In particular, the coefficient in +\eqref{eq:tuple-marked-lower} is at least \(c_*\), independently +of the prime and of the field-list values. +\end{lemma} + +\begin{proof} +Index an \(n\times n\) matrix by \(x,y\in V_+\) and set +\[ + \mathsf A_{xy}=H^S_{x,\theta y}. +\] +The invariance of \(H^S\) used in +Lemma~\ref{lem:tuple-component-symmetry}, together with its symmetry, +gives \(\mathsf A_{xy}=\mathsf A_{yx}\). + +Suppose the two cut symbols occupy positions \(ij\), or move +the second from \(j\) to a position \(k0\), Lemma~\ref{lem:list-mixing}, with its operator +transposed if necessary, gives +\begin{equation} + \|a-c_v\|_2\le\Delta_v,\qquad + \|b_f\|_2\le\Delta_v\|f\|_2, + \qquad + \Delta_v=(32m_v)^{1/4}p^{-1/8}. + \label{eq:conditional-numerator-bounds} +\end{equation} +The incident constraints form a cross-list graph between the single +variable \(x\) and the other labels. Incoming and outgoing edges are +both allowed by that lemma. The first inequality applies the centered +kernel to the constant function, while the second uses +\(\mathbb E f=0\). All norms in +\eqref{eq:conditional-numerator-bounds} use the uniform base measures. + +Put \(Z_0=\nu_{p,\varnothing}(1)\), and let \(T_0\) be the +conditional operator for the normalized leading component. Its +other-label marginal is \(Z_0^{-1}L(y)a(y)\,dy\), and therefore +\begin{equation} + \|T_0f\|_2^2 + =Z_0^{-1}\mathbb E_y L(y)\frac{|b_f(y)|^2}{a(y)}. + \label{eq:conditional-leading-norm} +\end{equation} +The ratio is defined as zero where \(a(y)=0\), a set of zero marginal +mass. On \(\{a\ge c_v/2\}\), its contribution to +\eqref{eq:conditional-leading-norm} is at most +\(2c_0^{-1}c_v^{-1}\Delta_v^2\|f\|_2^2\). +On the complementary set, Chebyshev's inequality gives +\[ + \mathbb P_y(a1. +\] +Thus \(K\) violates \eqref{eq:proposed-bound} in dimension twenty. +\end{theorem} + +The exact witness in Theorem~\ref{thm:counterexample} provides an +elementary illustration of the failure of the simplex maximum. The +optimal upper constant and its maximizing bodies remain separate +questions. + +The useful mechanism is a product test: for full-dimensional polytopes +\(A\subset\R^r\) and \(B\subset\R^s\), with \(r,s\ge2\), +\[ + R_{r+s}(A\times B)=R_r(A)R_s(B). +\] +Consequently, the product \(c_rc_s\) of the simplex values is a necessary +lower bound for any universal upper constant in dimension \(r+s\). The +dimension-twenty counterexample comes from comparing two copies of the +ten-dimensional simplex value with the twenty-dimensional simplex value. +All identities used in that comparison are proved below. + +The growth of upper constants with dimension is a further quantitative +question. For centrally symmetric bodies, Henk gives the lower bound +\(2^d(9/8)^{\lfloor d/3\rfloor}\) for the optimal upper +constant~\cite{Henk}. Iterating our product identity gives an exponential +excess over the simplex benchmark: for some fixed \(\lambda>1\) and +every sufficiently large \(n\), a product of simplices +\(K_n\subset\R^n\) satisfies \(R_n(K_n)\ge\lambda^n c_n\). +Corollary~\ref{cor:exponential-excess} proves this consequence by +repeating ten-dimensional factors and using one additional simplex to +reach each dimension. + +Section~\ref{sec:facets} derives the facet formula, the product identity, +and affine covariance. Section~\ref{sec:simplex} computes the simplex +value by expressing its projection body as a cube plus one segment. +Section~\ref{sec:counterexample} combines the identities and gives the +exact integer comparison, then proves the exponential excess in all +sufficiently large dimensions. + +\section{Facet vectors and Cartesian products}\label{sec:facets} + +We first express projection bodies of polytopes as sums of segments. +The facets of a Cartesian product then separate into two families, +which will make its normalized projection-body volume multiplicative. +The facet formula is the polytopal case of Cauchy's projection formula; +see \cite[Section~1]{Saroglou}. We include its proof +to fix the factor \(1/2\) on which the subsequent constants depend. + +We use Minkowski addition \(C+D=\{c+d:c\in C,\ d\in D\}\). +Support functions add under this operation and determine compact convex +sets uniquely. For the latter assertion, if \(x\notin C\) and \(y\in C\) +is nearest to \(x\), minimization along each segment \([y,z]\subset C\) +gives \(\langle x-y,z-y\rangle\le0\) for every \(z\in C\). The linear +functional with vector \(x-y\) therefore separates \(x\) from \(C\). + +\begin{samepage} +\begin{lemma}\label{lem:facets} +Let \(P\subset\R^d\), \(d\ge2\), be a full-dimensional convex polytope. For each facet +\(F\), let \(s_F\) be its \((d-1)\)-dimensional volume and \(\nu_F\) its +outward unit normal. Then +\begin{equation}\label{eq:facet} + h_{\Pi P}(u)=\frac12\sum_Fs_F|\langle\nu_F,u\rangle|, + \qquad u\in\R^d, +\end{equation} +and therefore +\begin{equation}\label{eq:zonotope} + \Pi P=\sum_F + \left[-\frac{s_F\nu_F}{2},\frac{s_F\nu_F}{2}\right]. +\end{equation} +\end{lemma} +\end{samepage} + +\begin{proof} +First take \(u\) to be a unit vector. Orthogonal projection from the +hyperplane of \(F\) to \(u^\perp\) has absolute Jacobian +\(|\langle\nu_F,u\rangle|\). Indeed, for an orthonormal tangent basis +\(v_1,\ldots,v_{d-1}\), that Jacobian is +\(|\det(v_1,\ldots,v_{d-1},u)|\), which equals the stated inner product +in absolute value. + +Apart from a set of \((d-1)\)-dimensional measure zero, a point in the +projection of \(P\) lies on the projections of exactly two facet +interiors: the entry and exit points of its line parallel to \(u\). +To see that the exceptions are null, discard the boundary of the +projection and the projections of all faces of dimension at most +\(d-2\). Also discard projections of facets whose hyperplanes are parallel to +\(u\); these projections have dimension at most \(d-2\). +Every remaining line meets \(P\) in a nondegenerate interval with its +two endpoints in facet interiors. Integrating this multiplicity gives +\eqref{eq:facet} for unit \(u\), and homogeneity gives the general case. + +The support function of \([-v/2,v/2]\) is +\(\tfrac12|\langle v,u\rangle|\). Adding these support functions proves +\eqref{eq:zonotope}. +\end{proof} + +\begin{proposition}\label{prop:product} +Let \(A\subset\R^r\) and \(B\subset\R^s\) be full-dimensional convex +polytopes, where \(r,s\ge2\). In the orthogonal product space +\(\R^{r+s}=\R^r\times\R^s\), +\begin{equation}\label{eq:product-body} + \Pi(A\times B)=(|B|\Pi A)\times(|A|\Pi B). +\end{equation} +In particular, +\begin{equation}\label{eq:multiplicative} + R_{r+s}(A\times B)=R_r(A)R_s(B). +\end{equation} +\end{proposition} + +\begin{proof} +The face exposed by a linear functional \((u,v)\) on \(A\times B\) +is the product of the faces exposed by \(u\) and \(v\). +Dimensions add under Cartesian products. A nonzero linear functional +exposes a proper face of a full-dimensional body, so if both \(u\) and +\(v\) are nonzero, the product face has codimension at least two. +A facet therefore has one facet factor and one whole factor. +Thus the facets are \(F\times B\) +and \(A\times G\), with area-normal vectors +\[ + (|B|s_F\nu_F,0) + \quad\hbox{and}\quad + (0,|A|s_G\nu_G), +\] +respectively. The area factors are products because the two coordinate +spaces are orthogonal. Lemma~\ref{lem:facets} now gives +\[ + h_{\Pi(A\times B)}(u,v) + =|B|h_{\Pi A}(u)+|A|h_{\Pi B}(v). +\] +This is the support function of the right side of +\eqref{eq:product-body}, proving that identity. Hence +\[ + \begin{split} + R_{r+s}(A\times B) + &=\frac{|B|^r|A|^s|\Pi A||\Pi B|} + {(|A||B|)^{r+s-1}}\\ + &=\frac{|\Pi A|}{|A|^{r-1}} + \frac{|\Pi B|}{|B|^{s-1}}, + \end{split} +\] +as required. +\end{proof} + +The product identity reduces our comparison to the values of its factors. +To use simplices as a dimension-dependent benchmark, we also need to +know that all full-dimensional simplices of a fixed dimension have the +same value. The following affine covariance formula supplies that fact. + +\begin{lemma}\label{lem:affine} +Let \(P\subset\R^d\), \(d\ge2\), be a convex body. For every +invertible linear map \(L\colon\R^d\to\R^d\), +\[ + \Pi(LP)=|\det L|L^{-t}\Pi P,\qquad R_d(LP)=R_d(P), +\] +where \(L^{-t}\) denotes the inverse transpose. Translations also +preserve \(R_d\). +\end{lemma} + +\begin{proof} +First suppose that \(P\) is a polytope. +The transformed outward unit normal of \(F\) is +\(L^{-t}\nu_F/\|L^{-t}\nu_F\|\). Its area is +\[ + s_{LF}=|\det L|\,\|L^{-t}\nu_F\|\,s_F. +\] +For the area formula, take a prism of height one over \(F\), extending +in direction \(\nu_F\). Its image has volume \(|\det L|s_F\) and height +\(1/\|L^{-t}\nu_F\|\) over \(LF\); dividing volume by height gives the +formula. Thus each area-normal vector transforms by +\(|\det L|L^{-t}\), including when \(\det L<0\). +Apply \eqref{eq:zonotope}. The absolute determinant of this map is +\(|\det L|^{d-1}\), exactly the scaling of \(|LP|^{d-1}\). +Translation invariance follows directly from projection volumes. + +For completeness, we justify the passage to an arbitrary convex body +and the existence of its projection body. Translate an interior point +of \(P\) to the origin, choose \(R>0\) with \(P\subset RB_2^d\), and +take increasing inscribed polytopes \(P_m\) containing a fixed ball +\(aB_2^d\), \(a>0\), such that +\[ + P_m\subset P\subset P_m+\varepsilon_mB_2^d + \subset(1+\varepsilon_m/a)P_m, + \qquad \varepsilon_m\longrightarrow0. +\] +Take, for example, convex hulls of cumulative finite nets in \(P\) and +a fixed inscribed polytope containing \(aB_2^d\). Put +\(\lambda_m=1+\varepsilon_m/a\), and for unit \(u\) write +\(F(u)=\vol_{d-1}(\proj_{u^\perp}P)\) and +\(f_m(u)=h_{\Pi P_m}(u)\). Projection and volume monotonicity give +\[ + 0\le F(u)-f_m(u) + \le (\lambda_m^{d-1}-1)\kappa_{d-1}R^{d-1}. +\] +Thus \(f_m\to F\) uniformly. The increasing bodies \(\Pi P_m\) lie +in a fixed ball and contain \(\kappa_{d-1}a^{d-1}B_2^d\), so their +union closure is a convex body with support function \(F\), namely +\(\Pi P\). The support inequalities also give +\[ + \Pi P_m\subset\Pi P\subset\lambda_m^{d-1}\Pi P_m, + \qquad + |\Pi P_m|\le|\Pi P|\le\lambda_m^{d(d-1)}|\Pi P_m|. +\] +Likewise \(|P_m|\le|P|\le\lambda_m^d|P_m|\). Applying \(L\) to +the original sandwich gives the same convergence for \(LP_m\). +Passing to the limit in the polytope identity proves covariance and, +by these volume inequalities, the asserted invariance of \(R_d\). +\end{proof} + +\section{The simplex value}\label{sec:simplex} + +We now compute the value to which the product will be compared. +Only one extra segment beyond a coordinate cube is needed to describe +the projection body of a simplex. + +\begin{proposition}\label{prop:simplex} +For \(d\ge2\), the standard simplex +\(T_d=\conv(0,e_1,\ldots,e_d)\subset\R^d\) satisfies +\[ + |T_d|=\frac1{d!},\qquad + |\Pi T_d|=\frac{d+1}{((d-1)!)^d},\qquad + R_d(T_d)=c_d. +\] +Every full-dimensional \(d\)-simplex has the same value \(c_d\). +\end{proposition} + +\begin{proof} +The simplex volume follows by iterated base and height. Put +\(w=e_1+\cdots+e_d\). The coordinate facets have area-normal vectors +\(-e_i/(d-1)!\). The remaining facet has unit normal \(w/\sqrt d\). +Dropping its last coordinate projects it onto the standard +\((d-1)\)-simplex with Jacobian \(1/\sqrt d\); its area-normal vector is +therefore \(w/(d-1)!\). In the centered segments of +Lemma~\ref{lem:facets}, changing the sign of a generator changes no +segment. Translating each segment to start at the origin therefore shows +that \(\Pi T_d\) is a translate of +\[ + \frac{[0,1]^d+[0,w]}{(d-1)!}. +\] + +To compute the numerator's volume, let \(Q=[0,1]^d\). +Every nonempty fiber of \(Q\) parallel to \(w\) is an interval, and +adding \([0,w]\) increases its length by \(\|w\|\). +The base projection is unchanged. Fubini therefore gives +\[ + |Q+[0,w]| + =|Q|+\|w\|\vol_{d-1}(\proj_{w^\perp}Q) + =1+h_{\Pi Q}(w). +\] +The cube has two facets of area one in each coordinate direction, +so \eqref{eq:facet} gives \(h_{\Pi Q}(w)=d\). +Thus \(|Q+[0,w]|=d+1\); Figure~\ref{fig:extrusion} illustrates the +two-dimensional case. Dilation by \(1/(d-1)!\) and division by +\(|T_d|^{d-1}\) now yield +\[ + R_d(T_d)=\frac{(d+1)(d!)^{d-1}}{((d-1)!)^d} + =\frac{(d+1)d^d}{d!}. +\] +Every full-dimensional simplex is an invertible affine image of +\(T_d\), so Lemma~\ref{lem:affine} proves the final assertion. +\end{proof} + +\begin{figure}[tb] +\centering +\begin{tikzpicture}[scale=1.5, line join=round] + \fill[black!7] (0,0) rectangle (1,1); + \fill[blue!13] (1,0)--(2,1)--(2,2)--(1,1)--cycle; + \fill[yellow!23] (0,1)--(1,1)--(2,2)--(1,2)--cycle; + \draw[thick] (0,0)--(1,0)--(2,1)--(2,2)--(1,2)--(0,1)--cycle; + \draw[thin] (0,1)--(1,1)--(1,0); + \draw[thin] (1,1)--(2,2); + \node at (.5,.5) {\(Q\)}; + \node at (1.5,1) {\(1\)}; + \node at (1,1.5) {\(1\)}; + \node[below left] at (0,0) {\(0\)}; + \node[above right] at (2,2) {\((2,2)\)}; + \draw[->,thick] (2.6,.3)--(3.6,1.3); + \node[right] at (3.15,.65) {\(w=(1,1)\)}; +\end{tikzpicture} +\caption{The planar extrusion \(Q+[0,(1,1)]\), with +\(Q=[0,1]^2\), consists of the unit square and two parallelograms of +area one. Its area is \(3\). The fiber argument in +Proposition~\ref{prop:simplex} proves the corresponding volume \(d+1\) +in every dimension.} +\label{fig:extrusion} +\end{figure} + +\section{The counterexample and exponential excess}\label{sec:counterexample} + +The geometric work is complete. The product formula turns the comparison +in dimension twenty into a calculation with two simplex constants. + +\begin{proof}[Proof of Theorem~\ref{thm:counterexample}] +The product \(K=T_{10}\times T_{10}\) is compact and convex with +nonempty interior in \(\R^{20}\). Propositions~\ref{prop:product} +and~\ref{prop:simplex} give +\[ + R_{20}(K)=R_{10}(T_{10})^2 + =\left(\frac{11\cdot10^{10}}{10!}\right)^2. +\] +Consequently, +\[ + \frac{R_{20}(K)}{c_{20}} + =\frac{121\cdot10^{20}\cdot20!} + {(10!)^2\cdot21\cdot20^{20}} + =\frac{121\binom{20}{10}}{21\cdot2^{20}} + =\frac{22\,355\,476}{22\,020\,096}>1. +\] +Here \(\binom{20}{10}=184\,756\), and the exact integer certificate is +\[ + 121\cdot184\,756-21\cdot1\,048\,576=335\,380>0. +\] +This is an exact strict violation of \eqref{eq:proposed-bound}. +\end{proof} + +More generally, for \(r,s\ge2\), the same two propositions give the +explicit product test +\[ + \frac{R_{r+s}(T_r\times T_s)}{c_{r+s}} + =\frac{(r+1)(s+1)}{r+s+1} + \binom{r+s}{r}\frac{r^rs^s}{(r+s)^{r+s}}. +\] +Thus any universal upper constant in dimension \(r+s\) must be at least +\(c_rc_s\). + +Iterating the product test gives the following quantitative consequence. + +\begin{corollary}\label{cor:exponential-excess} +There exist a constant \(\lambda>1\) and an integer \(N\ge20\) such that, for every +integer \(n\ge N\), some full-dimensional convex polytope +\(K_n\subset\R^n\) satisfies +\[ + \frac{R_n(K_n)}{R_n(T_n)}\ge\lambda^n. +\] +The polytopes \(K_n\) may be chosen as Cartesian products of simplices. +\end{corollary} + +\begin{proof} +We use repeated ten-dimensional factors to obtain exponential growth, +and one factor of dimension between ten and nineteen to reach every +sufficiently large dimension. For \(r\in\{0,\ldots,9\}\), put +\(d_r=10+r\). Every integer \(n\ge20\) has a unique representation +\(n=10k+d_r\) with \(k\ge1\). In the orthogonal coordinate product +space, take +\[ + K_n=T_{10}^{\,k}\times T_{d_r}. +\] +Here \(T_{10}^{\,k}\) denotes the \(k\)-fold Cartesian product. All +factors and their successive products are full-dimensional convex +polytopes, with each factor dimension at least ten. Thus +Propositions~\ref{prop:product} and~\ref{prop:simplex} apply iteratively +and give +\[ + R_n(K_n)=c_{10}^k c_{d_r}. +\] + +To compare this value with \(c_n\), we bound the simplex constants by +\(n+1\) times a fixed exponential, whose rate is smaller than that +supplied by the ten-dimensional factors. Put \(\beta=11/4\). +The exponential series gives +\[ + e=\frac52+\sum_{j=3}^{\infty}\frac1{j!} + <\frac52+\frac16\sum_{j=0}^{\infty}3^{-j} + =\beta, +\] +since the successive ratios in the tail are strictly less than \(1/3\). +The positive term \(n^n/n!\) in the series for \(e^n\) is strictly +smaller than the full sum, so +\[ + c_n<(n+1)e^n<(n+1)\beta^n\qquad(n\ge2). +\] +For the fixed ten-dimensional block, exact arithmetic yields +\[ + c_{10}=\frac{17\,187\,500}{567},\qquad + A:=\frac{c_{10}}{\beta^{10}} + =\frac{1\,638\,400\,000\,000}{1\,336\,956\,340\,797}>1. +\] +Choose a fixed \(\lambda\) with \(1<\lambdac_{d_r}\beta^{-d_r}\lambda^{-d_r} + \frac{(A/\lambda^{10})^k}{10k+d_r+1}. +\] +For each fixed \(r\), the right side tends to infinity as \(k\to\infty\), +because \(A/\lambda^{10}>1\). A threshold can therefore be chosen in +each of the ten residue classes; their maximum, enlarged to be at least +twenty, gives one \(N\) for all \(n\ge N\). +\end{proof} + +\input{references} +\end{document} diff --git a/preprints/A-product-counterexample-to-the-simplex-maximum-for-projection-body-volume-September-24-2026/build/references.tex b/preprints/A-product-counterexample-to-the-simplex-maximum-for-projection-body-volume-September-24-2026/build/references.tex new file mode 100644 index 0000000000000000000000000000000000000000..b5ca002e7cd37b5cfbf11c9869d5e6bb92f6b701 --- /dev/null +++ b/preprints/A-product-counterexample-to-the-simplex-maximum-for-projection-body-volume-September-24-2026/build/references.tex @@ -0,0 +1,48 @@ +\begin{thebibliography}{7} + +\bibitem{Brannen} +N.~S.~Brannen, +\emph{Volumes of projection bodies}, +Mathematika \textbf{43} (1996), no.~2, 255--264. +\href{https://doi.org/10.1112/S002557930001175X}{doi:10.1112/S002557930001175X}. + +\bibitem{ChenEtAl} +S.~Chen, Y.~Feng, Y.~Li, D.~Xi, and L.~Xu, +\emph{Petty's conjectured projection inequality in dimension three}, +preprint, 2026, MathSciDoc:2608.23001. +\href{https://archive.ymsc.tsinghua.edu.cn/pacm_download/741/12780-pettyinequality.pdf}{Full text}. + +\bibitem{FengEtAl} +Y.~Feng, S.~Hu, W.~Liu, and L.~Xu, +\emph{On the reverse projection inequality}, +preprint, 2026, MathSciDoc:2608.23002. +\href{https://archive.ymsc.tsinghua.edu.cn/pacm_download/743/12781--2026.8.26.pdf}{Full text}. + +\bibitem{Henk} +M.~Henk, +\emph{Note on projection bodies of zonotopes with \(n+1\) generators}, +Acta Mathematica Scientia \textbf{45} (2025), 96--103. +\href{https://doi.org/10.1007/s10473-025-0107-9}{doi:10.1007/s10473-025-0107-9}. + +\bibitem{Ludwig} +M.~Ludwig, +\emph{Projection bodies and valuations}, +Advances in Mathematics \textbf{172} (2002), no.~2, 158--168. +\href{https://doi.org/10.1016/S0001-8708(02)00021-X}{doi:10.1016/S0001-8708(02)00021-X}. + +\bibitem{Petty} +C.~M.~Petty, +\emph{Projection bodies}, +in \emph{Proceedings of the Colloquium on Convexity (Copenhagen, 1965)}, +K\o benhavns Universitet Matematiske Institut, Copenhagen, 1967, +pp.~234--241. + +\bibitem{Saroglou} +C.~Saroglou, +\emph{On the shape of a convex body with respect to its second projection body}, +Advances in Applied Mathematics \textbf{67} (2015), 55--74. +\href{https://doi.org/10.1016/j.aam.2015.03.004}{doi:10.1016/j.aam.2015.03.004}. +Preprint: +\href{https://arxiv.org/abs/1409.4347v2}{arXiv:1409.4347v2}. + +\end{thebibliography} diff --git a/preprints/A-product-counterexample-to-the-simplex-maximum-for-projection-body-volume-September-24-2026/paper.pdf b/preprints/A-product-counterexample-to-the-simplex-maximum-for-projection-body-volume-September-24-2026/paper.pdf new file mode 100644 index 0000000000000000000000000000000000000000..1a61e07b03308e5f07e64b4ccc51e224c81383d9 --- /dev/null +++ b/preprints/A-product-counterexample-to-the-simplex-maximum-for-projection-body-volume-September-24-2026/paper.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:1d2211bb8d175f40fb55554586f02538fc9f4c04ecdc9793999a6f7ec514720a +size 264039 diff --git a/preprints/A-projective-fourfold-with-large-fundamental-group-and-non-Stein-universal-cover-October-5-2026/README.md b/preprints/A-projective-fourfold-with-large-fundamental-group-and-non-Stein-universal-cover-October-5-2026/README.md new file mode 100644 index 0000000000000000000000000000000000000000..e9327da5fbe0fb8f3d002165cc4c95d213ec9390 --- /dev/null +++ b/preprints/A-projective-fourfold-with-large-fundamental-group-and-non-Stein-universal-cover-October-5-2026/README.md @@ -0,0 +1,18 @@ +# [A projective fourfold with large fundamental group and non-Stein universal cover](large-fundamental-group-non-stein.pdf) + +**Author:** OpenAI + +**Date:** October 5, 2026 + +## Citation + +```bibtex +@misc{OAI:A-projective-fourfold-with-large-fundamental-group-and-non-Stein-universal-cover-October-5-2026, + author = {{OpenAI}}, + title = {{A projective fourfold with large fundamental group and non-Stein universal cover}}, + howpublished = {OpenAI Math Release preprint + \href{https://github.com/openai/math/blob/main/preprints/A-projective-fourfold-with-large-fundamental-group-and-non-Stein-universal-cover-October-5-2026/large-fundamental-group-non-stein.pdf}{OAI:A-projective-fourfold-with-large-fundamental-group-and-non-Stein-universal-cover-October-5-2026}}, + year = {2026} +} +``` + diff --git a/preprints/A-projective-fourfold-with-large-fundamental-group-and-non-Stein-universal-cover-October-5-2026/build/figures/arrangement.tex b/preprints/A-projective-fourfold-with-large-fundamental-group-and-non-Stein-universal-cover-October-5-2026/build/figures/arrangement.tex new file mode 100644 index 0000000000000000000000000000000000000000..3d10e8c0aa3d9c460eb092b1cb8a61d6d2b6e9fc --- /dev/null +++ b/preprints/A-projective-fourfold-with-large-fundamental-group-and-non-Stein-universal-cover-October-5-2026/build/figures/arrangement.tex @@ -0,0 +1,64 @@ +\begin{figure}[htbp] +\centering +\begingroup +\definecolor{arrred}{RGB}{165,58,37} +\definecolor{arrblue}{RGB}{31,87,145} +\begin{tikzpicture}[x=1cm,y=1cm,>=Latex, + every node/.style={font=\small}, + redline/.style={arrred,line width=.8pt}, + blueline/.style={arrblue,dashed,line width=.8pt}, + redpoint/.style={circle,fill=arrred,draw=arrred,inner sep=1.8pt}, + bluepoint/.style={rectangle,fill=arrblue,draw=arrblue,inner sep=1.8pt}] + \node at (2.55,4.55) {The six marked directions}; + \draw[->,black!65] (0,2.6)--(5.1,2.6); + \node[right] at (5.1,2.6) {$\lambda$}; + \draw[black!45,rounded corners=9pt] (1.35,2.12) rectangle (3.75,3.08); + \foreach \x in {1.7,2.25,2.8,3.35} + \node[redpoint] at (\x,2.6) {}; + \node[bluepoint] at (.35,2.6) {}; + \node[bluepoint] at (4.75,2.6) {}; + \node[align=center,arrred] at (2.55,3.6) + {four red points\\$|\lambda_i|<0.01$}; + \node[align=center,arrblue] at (2.55,1.45) + {two blue points: $|\lambda_i|>4$}; + \node[align=center] at (2.55,.45) + {$C\longrightarrow\mathbb P^1_{\lambda}$\\double cover branched at these points}; + + \begin{scope}[shift={(9.15,2.25)}] + \node at (0,2.3) {One pencil and its couplings}; + \draw[black!40,line width=.6pt] (0,0) circle (2.05); + \foreach \angle in {-13,-4,4,13} + \draw[redline] (\angle:2.05)--({\angle+180}:2.05); + \foreach \angle in {78,102} + \draw[blueline] (\angle:2.05)--({\angle+180}:2.05); + \draw[line width=1pt] (-1.78,-1.017)--(-1.78,1.017); + \draw[line width=1pt] (1.78,-1.017)--(1.78,1.017); + \foreach \angle in {-13,-4,4,13}{ + \fill[arrred] (1.78,{1.78*tan(\angle)}) circle (1.9pt); + \fill[arrred] (-1.78,{-1.78*tan(\angle)}) circle (1.9pt); + } + \fill (0,0) circle (2pt); + \node[fill=white,inner sep=1.5pt,below=8pt] at (0,0) + {$L_z$}; + \node[align=center,below=4pt,fill=white,inner sep=1.5pt] at (-1.78,-1.017) + {$z_1=-c$}; + \node[align=center,below=4pt,fill=white,inner sep=1.5pt] at (1.78,-1.017) + {$z_1=c$}; + \node[arrblue,fill=white,inner sep=1pt] at (.86,1.64) {blue}; + \node[arrred,fill=white,inner sep=1pt] at (.92,.57) {red}; + \node at (0,-2.55) {$z_2=\lambda_i z_1$}; + \end{scope} +\end{tikzpicture} +\endgroup +\caption{The finite arrangement in one coordinate pair. Left: a + schematic placement of the six branch points on the real direction + axis, grouped into four red circles and two blue squares. Right: + the slice $w=0$, $z_1,z_2\in\mathbb R$ of the $z$ pencil in the + ball; solid red slopes meet both coupling + hyperplanes, whereas dashed blue slopes meet neither. The red + angular separation is exaggerated to show these incidences. + The center represents the slice of $L_z$. The $w$ family is an + identical construction in the other coordinate pair; the full + arrangement lies in complex dimension four.} +\label{fig:arrangement} +\end{figure} diff --git a/preprints/A-projective-fourfold-with-large-fundamental-group-and-non-Stein-universal-cover-October-5-2026/build/figures/blowdown.tex b/preprints/A-projective-fourfold-with-large-fundamental-group-and-non-Stein-universal-cover-October-5-2026/build/figures/blowdown.tex new file mode 100644 index 0000000000000000000000000000000000000000..023ca4df65ce7749550f254182937c7f3c79f04c --- /dev/null +++ b/preprints/A-projective-fourfold-with-large-fundamental-group-and-non-Stein-universal-cover-October-5-2026/build/figures/blowdown.tex @@ -0,0 +1,53 @@ +\begin{figure}[htbp] +\centering +\begingroup +\definecolor{blowaccent}{RGB}{22,91,119} +\begin{tikzpicture}[x=1cm,y=1cm,>=Latex, + every node/.style={font=\small}, + marked/.style={blowaccent,line width=1.5pt}] + \coordinate (sw) at (3.6,.65); + \coordinate (se) at (7.8,.65); + \coordinate (nw) at (4.5,3.25); + \coordinate (ne) at (8.7,3.25); + \fill[black!3] (sw)--(se)--(ne)--(nw)--cycle; + \foreach \t in {.25,.5,.75}{ + \draw[black!23] ($(sw)!\t!(nw)$)--($(se)!\t!(ne)$); + \draw[black!23] ($(sw)!\t!(se)$)--($(nw)!\t!(ne)$); + } + \draw[black!55] (sw)--(se)--(ne)--(nw)--cycle; + \coordinate (el) at ($(sw)!.5!(nw)$); + \coordinate (er) at ($(se)!.5!(ne)$); + \draw[marked] (el)--(er); + \node[above=6pt] at (6.65,3.25) + {$G=\mathbb P^1_{\mathrm{curve}}\times\mathbb P^1_{\mathrm{normal}}$}; + \node[fill=white,inner sep=2pt,text=blowaccent] at (6.15,1.95) + {$\widehat Y\cap G=E\times\{q\}$}; + \node[below=4pt] at (5.7,.65) {curve direction}; + + \draw[marked] (0,.7)--(2.05,.7); + \node[below=5pt] at (1.025,.7) {$E\subset Y_1\subset T$}; + \draw[->,line width=.75pt] (3.45,1.65)--(1.65,1.05); + \node[align=center] at (1.6,2.35) + {original blowdown\\$G\longrightarrow\mathbb P^1_{\mathrm{curve}}$}; + + \draw[black!65,line width=.8pt] (11.1,.75)--(11.7,2.65); + \fill[blowaccent] (11.4,1.7) circle (2.8pt); + \node[right=5pt,align=left] at (11.4,1.7) {$q$\\in $Y_0$}; + \draw[->,line width=.75pt] (8.7,1.9)--(10.7,1.9); + \node[align=center] at (10.35,3.65) + {other blowdown\\$G\longrightarrow\mathbb P^1_{\mathrm{normal}}$}; + \node[below=6pt,align=center] at (11.1,.6) + {image curve\\in $T'$}; +\end{tikzpicture} +\endgroup +\caption{The two rulings of the last exceptional divisor in + $\widehat T$. The original blowdown contracts the normal-direction + fibers and leaves the curve $E$ in $Y_1$ unchanged. The other + blowdown, whose target threefold $T'$ is constructed below, + contracts the horizontal ruling, including + $\widehat Y\cap G=E\times\{q\}$, so the surface undergoes its + point blowdown $Y_1\to Y_0$. The parallelogram records only the + product structure of $G$, whose normal bundle is + $\mathcal O_G(-1,-1)$.} +\label{fig:blowdown} +\end{figure} diff --git a/preprints/A-projective-fourfold-with-large-fundamental-group-and-non-Stein-universal-cover-October-5-2026/build/figures/path-ladder.tex b/preprints/A-projective-fourfold-with-large-fundamental-group-and-non-Stein-universal-cover-October-5-2026/build/figures/path-ladder.tex new file mode 100644 index 0000000000000000000000000000000000000000..6243a9d6c28f86c77481834f01cea717d64edf9c --- /dev/null +++ b/preprints/A-projective-fourfold-with-large-fundamental-group-and-non-Stein-universal-cover-October-5-2026/build/figures/path-ladder.tex @@ -0,0 +1,52 @@ +\begin{figure}[htbp] +\centering +\begingroup +\definecolor{ladderaccent}{RGB}{22,91,119} +\begin{tikzpicture}[x=1cm,y=1cm,>=Latex, + every node/.style={font=\small}, + guide/.style={line width=.9pt}, + oldrung/.style={densely dashed,line width=.65pt,black!65}, + newrung/.style={line width=1.1pt,ladderaccent}] + \coordinate (am) at (0,2.05); + \coordinate (ai) at (5.4,2.05); + \coordinate (ap) at (10.8,2.05); + \coordinate (bm) at (0,.2); + \coordinate (bo) at (6.7,.2); + \coordinate (bn) at (5.4,.2); + \coordinate (bp) at (10.8,.2); + \draw[guide] (am)--(ap); + \draw[guide] (bm)--(bp); + \draw[oldrung] (am)--node[left] {$\le h$} (bm); + \draw[oldrung] (ap)--node[right] {$\le h$} (bp); + \draw[oldrung] (ai)--node[pos=.50,right,fill=white,inner sep=1.5pt] + {old: $\le h$} (bo); + \draw[newrung] (ai)--node[pos=.55,left,fill=white,inner sep=1.5pt] + {new: $\le 2\Delta$} (bn); + \foreach \p in {am,ai,ap,bm,bo,bp} + \fill (\p) circle (1.7pt); + \draw[newrung,fill=white] (bn) circle (2.2pt); + \node[above=4pt] at (am) {$x_{i-1}$}; + \node[above=4pt] at (ai) {$x_i$}; + \node[above=4pt] at (ap) {$x_{i+1}$}; + \node[below=5pt] at (bm) {$b_{i-1}$}; + \node[below=5pt] at (bn) {$b_i'$}; + \node[below=5pt] at (bo) {$b_i$}; + \node[below=5pt] at (bp) {$b_{i+1}$}; + \node[left=18pt] at (am) {$\alpha$}; + \node[left=18pt] at (bm) {$\beta$}; + \draw[decorate,decoration={brace,amplitude=4pt}] + (0,2.76)--node[above=5pt] {length in $[L,2L]$} (5.4,2.76); + \draw[decorate,decoration={brace,amplitude=4pt}] + (5.4,2.76)--node[above=5pt] {length in $[L,2L]$} (10.8,2.76); +\end{tikzpicture} +\endgroup +\caption{Narrowing a ladder in a single chart. Both displayed guide + pieces read as geodesics. The interior mark $x_i$ is far from the + end rungs, so quadrangle slimness supplies a new mate $b_i'$ on + the opposite side, within $2\Delta$ of $x_i$. The old scheduled + mate $b_i$ is at distance at most $h+2\Delta$ from $b_i'$ along + that side. The drawing is + schematic: the long side intervals and the two rung bounds use + different visual scales.} +\label{fig:path-ladder} +\end{figure} diff --git a/preprints/A-projective-fourfold-with-large-fundamental-group-and-non-Stein-universal-cover-October-5-2026/build/main.tex b/preprints/A-projective-fourfold-with-large-fundamental-group-and-non-Stein-universal-cover-October-5-2026/build/main.tex new file mode 100644 index 0000000000000000000000000000000000000000..108f28530b3b54c1d4e2a2b168d5c8b3cd9bb48a --- /dev/null +++ b/preprints/A-projective-fourfold-with-large-fundamental-group-and-non-Stein-universal-cover-October-5-2026/build/main.tex @@ -0,0 +1,30 @@ +\newcommand{\DoNotLoadEpstopdf}{} +\documentclass[11pt,reqno]{amsart} +\input{preamble} +\title[Large fundamental group and non-Stein universal cover]{A projective fourfold with large fundamental group\protect\\ +and non-Stein universal cover} +\author{OpenAI} +\date{October 5, 2026} +\subjclass[2020]{32Q30, 14F35, 20F67, 14K12} +\keywords{Shafarevich conjecture, universal cover, large fundamental group, +complex hyperbolic arrangement, Stein manifold} +\begin{document} +\begin{abstract} +We construct a smooth projective complex fourfold with large fundamental group +whose universal cover is not Stein. The universal cover contains no +positive-dimensional compact complex-analytic subvariety, so this disproves +Shafarevich's holomorphic-convexity conjecture even under the +large-fundamental-group hypothesis. +\end{abstract} +\maketitle +\tableofcontents +\input{sections/01-introduction} +\input{sections/02-reduction} +\input{sections/03-paths} +\input{sections/04-geometry} +\input{sections/05-models} +\input{sections/06-image} +\input{sections/07-projective} +\bibliographystyle{amsplain} +\bibliography{references} +\end{document} diff --git a/preprints/A-projective-fourfold-with-large-fundamental-group-and-non-Stein-universal-cover-October-5-2026/build/preamble.tex b/preprints/A-projective-fourfold-with-large-fundamental-group-and-non-Stein-universal-cover-October-5-2026/build/preamble.tex new file mode 100644 index 0000000000000000000000000000000000000000..2346b95f9d53464f0adf9e12cd7e5990610dd16a --- /dev/null +++ b/preprints/A-projective-fourfold-with-large-fundamental-group-and-non-Stein-universal-cover-October-5-2026/build/preamble.tex @@ -0,0 +1,58 @@ +\usepackage[T1]{fontenc} +\usepackage{lmodern} +\usepackage{amsmath,amssymb,amsthm,mathtools} +\usepackage[margin=1.1in]{geometry} +\usepackage{microtype} +\usepackage{enumitem} +\usepackage{booktabs,array} +\usepackage{xcolor} +\usepackage{tikz} +\usetikzlibrary{arrows.meta,calc,positioning,decorations.pathreplacing,patterns} +\usepackage{flafter} +\usepackage{placeins} +\usepackage{hyperref} +\setcounter{tocdepth}{1} +\hypersetup{colorlinks=true,linkcolor=blue!45!black,citecolor=blue!45!black, + bookmarksdepth=2, + urlcolor=blue!45!black,pdfauthor={OpenAI}, + pdftitle={A projective fourfold with large fundamental group and non-Stein universal cover}, + pdfsubject={A counterexample to the holomorphic-convexity conjecture}, + pdfkeywords={Shafarevich conjecture, large fundamental group, Stein manifold}} +\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{construction}[theorem]{Construction} +\newtheorem{example}[theorem]{Example} +\theoremstyle{remark} +\newtheorem{remark}[theorem]{Remark} +\newtheorem{convention}[theorem]{Convention} +\newcommand{\C}{\mathbb C} +\newcommand{\R}{\mathbb R} +\newcommand{\Q}{\mathbb Q} +\newcommand{\Z}{\mathbb Z} +\newcommand{\BB}{\mathbb B} +\newcommand{\PP}{\mathbb P} +\newcommand{\cD}{\mathcal D} +\newcommand{\cS}{\mathcal S} +\DeclareMathOperator{\im}{im} +\DeclareMathOperator{\id}{id} +\DeclareMathOperator{\diam}{diam} +\DeclareMathOperator{\dist}{dist} +\DeclareMathOperator{\Sym}{Sym} +\DeclareMathOperator{\Jac}{Jac} +\DeclareMathOperator{\Pic}{Pic} +\DeclareMathOperator{\End}{End} +\DeclareMathOperator{\Span}{span} +\DeclareMathOperator{\Gal}{Gal} +\DeclareMathOperator{\Aut}{Aut} +\DeclareMathOperator{\Hom}{Hom} +\DeclareMathOperator{\Spec}{Spec} +\DeclareMathOperator{\codim}{codim} +\setlist[enumerate]{label=\textup{(\arabic*)},leftmargin=*,itemsep=3pt} +\setlist[itemize]{leftmargin=*,itemsep=3pt} +\emergencystretch=1.2em +\allowdisplaybreaks[1] diff --git a/preprints/A-projective-fourfold-with-large-fundamental-group-and-non-Stein-universal-cover-October-5-2026/build/references.bib b/preprints/A-projective-fourfold-with-large-fundamental-group-and-non-Stein-universal-cover-October-5-2026/build/references.bib new file mode 100644 index 0000000000000000000000000000000000000000..8fd405193a1edac2e5d4f6be9cb44418d7933205 --- /dev/null +++ b/preprints/A-projective-fourfold-with-large-fundamental-group-and-non-Stein-universal-cover-October-5-2026/build/references.bib @@ -0,0 +1,338 @@ +@book{BridsonHaefliger, + author = {Bridson, Martin R. and Haefliger, Andr{\'e}}, + title = {Metric Spaces of Non-Positive Curvature}, + series = {Grundlehren der mathematischen Wissenschaften}, + volume = {319}, + publisher = {Springer-Verlag}, + address = {Berlin}, + year = {1999}, + doi = {10.1007/978-3-662-12494-9} +} + +@incollection{Gromov1987, + author = {Gromov, Mikhail}, + title = {Hyperbolic groups}, + booktitle = {Essays in Group Theory}, + editor = {Gersten, S. M.}, + series = {Mathematical Sciences Research Institute Publications}, + volume = {8}, + pages = {75--263}, + publisher = {Springer-Verlag}, + address = {New York}, + year = {1987}, + doi = {10.1007/978-1-4613-9586-7_3} +} + +@article{BonkSchramm2000, + author = {Bonk, Mario and Schramm, Oded}, + title = {Embeddings of {Gromov} hyperbolic spaces}, + journal = {Geometric and Functional Analysis}, + volume = {10}, + number = {2}, + pages = {266--306}, + year = {2000}, + doi = {10.1007/s000390050009} +} + +@article{BorelHarishChandra1962, + author = {Borel, Armand and {Harish-Chandra}}, + title = {Arithmetic subgroups of algebraic groups}, + journal = {Annals of Mathematics}, + series = {2}, + volume = {75}, + number = {3}, + pages = {485--535}, + year = {1962}, + doi = {10.2307/1970210} +} + +@article{BailyBorel1966, + author = {Baily, Jr., Walter L. and Borel, Armand}, + title = {Compactification of arithmetic quotients of bounded symmetric domains}, + journal = {Annals of Mathematics}, + series = {2}, + volume = {84}, + number = {3}, + pages = {442--528}, + year = {1966}, + doi = {10.2307/1970457} +} + +@book{Goldman1999, + author = {Goldman, William M.}, + title = {Complex Hyperbolic Geometry}, + series = {Oxford Mathematical Monographs}, + publisher = {The Clarendon Press, Oxford University Press}, + address = {New York}, + year = {1999}, + doi = {10.1093/oso/9780198537939.001.0001} +} + +@article{Zarhin, + author = {Zarhin, Yuri G.}, + title = {Hyperelliptic {Jacobians} without complex multiplication}, + journal = {Mathematical Research Letters}, + volume = {7}, + number = {1}, + pages = {123--132}, + year = {2000}, + doi = {10.4310/MRL.2000.v7.n1.a11} +} + +@article{StoverToledo, + author = {Stover, Matthew and Toledo, Domingo}, + title = {Residual finiteness for central extensions of lattices in {PU(n,1)} and negatively curved projective varieties}, + journal = {Pure and Applied Mathematics Quarterly}, + volume = {18}, + number = {4}, + pages = {1771--1797}, + year = {2022}, + doi = {10.4310/PAMQ.2022.v18.n4.a15} +} + +@article{LlosaPy, + author = {Llosa Isenrich, Claudio and Py, Pierre}, + title = {Groups with exotic finiteness properties from complex {Morse} theory}, + journal = {Journal of Topology}, + volume = {18}, + number = {1}, + pages = {e70013}, + year = {2025}, + doi = {10.1112/topo.70013} +} + +@article{HammLe, + author = {Hamm, Helmut A. and {L{\^e} D{\~u}ng Tr{\'a}ng}}, + title = {Lefschetz theorems on quasi-projective varieties}, + journal = {Bulletin de la Soci{\'e}t{\'e} Math{\'e}matique de France}, + volume = {113}, + pages = {123--142}, + year = {1985}, + doi = {10.24033/bsmf.2023} +} + +@article{AndreottiFrankel1959, + author = {Andreotti, Aldo and Frankel, Theodore}, + title = {The {Lefschetz} theorem on hyperplane sections}, + journal = {Annals of Mathematics}, + series = {2}, + volume = {69}, + number = {3}, + pages = {713--717}, + year = {1959}, + doi = {10.2307/1970034} +} + +@book{MilnorMorse, + author = {Milnor, John}, + title = {Morse Theory}, + series = {Annals of Mathematics Studies}, + volume = {51}, + publisher = {Princeton University Press}, + address = {Princeton, NJ}, + year = {1963} +} + +@incollection{Totaro1999, + author = {Totaro, Burt}, + title = {The {Chow} ring of a classifying space}, + booktitle = {Algebraic $K$-Theory ({Seattle}, {WA}, 1997)}, + series = {Proceedings of Symposia in Pure Mathematics}, + volume = {67}, + pages = {249--281}, + publisher = {American Mathematical Society}, + address = {Providence, RI}, + year = {1999}, + doi = {10.1090/pspum/067/1743244} +} + +@incollection{Serre1958, + author = {Serre, Jean-Pierre}, + title = {Sur la topologie des vari{\'e}t{\'e}s alg{\'e}briques en caract{\'e}ristique $p$}, + booktitle = {Symposium Internacional de Topolog{\'i}a Algebraica}, + pages = {24--53}, + publisher = {Universidad Nacional Aut{\'o}noma de M{\'e}xico and UNESCO}, + address = {Mexico City}, + year = {1958} +} + +@book{GrauertRemmert1979, + author = {Grauert, Hans and Remmert, Reinhold}, + title = {Theory of {Stein} Spaces}, + series = {Grundlehren der mathematischen Wissenschaften}, + volume = {236}, + publisher = {Springer-Verlag}, + address = {New York}, + year = {1979}, + note = {Translated by Alan Huckleberry}, + doi = {10.1007/978-1-4757-4357-9} +} + +@book{Kollar1995, + author = {Koll{\'a}r, J{\'a}nos}, + title = {Shafarevich Maps and Automorphic Forms}, + series = {M. B. Porter Lectures}, + publisher = {Princeton University Press}, + address = {Princeton, NJ}, + year = {1995}, + doi = {10.1515/9781400864195} +} + +@article{Kollar1993, + author = {Koll{\'a}r, J{\'a}nos}, + title = {Shafarevich maps and plurigenera of algebraic varieties}, + journal = {Inventiones Mathematicae}, + volume = {113}, + pages = {177--215}, + year = {1993}, + doi = {10.1007/BF01244307} +} + +@article{Campana1994, + author = {Campana, Fr{\'e}d{\'e}ric}, + title = {Remarques sur le rev{\^e}tement universel des vari{\'e}t{\'e}s k{\"a}hl{\'e}riennes compactes}, + journal = {Bulletin de la Soci{\'e}t{\'e} Math{\'e}matique de France}, + volume = {122}, + number = {2}, + pages = {255--284}, + year = {1994}, + doi = {10.24033/bsmf.2232} +} + +@article{Napier1990, + author = {Napier, Terrence}, + title = {Convexity properties of coverings of smooth projective varieties}, + journal = {Mathematische Annalen}, + volume = {286}, + pages = {433--479}, + year = {1990}, + doi = {10.1007/BF01453583} +} + +@article{Katzarkov1997, + author = {Katzarkov, Ludmil}, + title = {Nilpotent groups and universal coverings of smooth projective varieties}, + journal = {Journal of Differential Geometry}, + volume = {45}, + number = {2}, + pages = {336--348}, + year = {1997}, + doi = {10.4310/jdg/1214459801} +} + +@article{KatzarkovRamachandran1998, + author = {Katzarkov, Ludmil and Ramachandran, Mohan}, + title = {On the universal coverings of algebraic surfaces}, + journal = {Annales scientifiques de l'{\'E}cole Normale Sup{\'e}rieure}, + series = {4}, + volume = {31}, + number = {4}, + pages = {525--535}, + year = {1998}, + doi = {10.1016/S0012-9593(98)80105-5} +} + +@article{Eyssidieux2004, + author = {Eyssidieux, Philippe}, + title = {Sur la convexit{\'e} holomorphe des rev{\^e}tements lin{\'e}aires r{\'e}ductifs d'une vari{\'e}t{\'e} projective alg{\'e}brique complexe}, + journal = {Inventiones Mathematicae}, + volume = {156}, + pages = {503--564}, + year = {2004}, + doi = {10.1007/s00222-003-0345-0} +} + +@article{EKPR2012, + author = {Eyssidieux, Philippe and Katzarkov, Ludmil and Pantev, Tony and Ramachandran, Mohan}, + title = {Linear {Shafarevich} conjecture}, + journal = {Annals of Mathematics}, + series = {2}, + volume = {176}, + number = {3}, + pages = {1545--1581}, + year = {2012}, + doi = {10.4007/annals.2012.176.3.4} +} + +@misc{BBT2024, + author = {Bakker, Benjamin and Brunebarbe, Yohan and Tsimerman, Jacob}, + title = {The linear {Shafarevich} conjecture for quasiprojective varieties and algebraicity of {Shafarevich} morphisms}, + year = {2024}, + note = {Preprint, arXiv:2408.16441; author manuscript revised September 9, 2026}, + url = {https://arxiv.org/abs/2408.16441} +} + +@article{Atiyah1958, + author = {Atiyah, Michael F.}, + title = {On analytic surfaces with double points}, + journal = {Proceedings of the Royal Society of London. Series A}, + volume = {247}, + number = {1249}, + pages = {237--244}, + year = {1958}, + doi = {10.1098/rspa.1958.0181} +} + +@book{MilnorSingular, + author = {Milnor, John}, + title = {Singular Points of Complex Hypersurfaces}, + series = {Annals of Mathematics Studies}, + volume = {61}, + publisher = {Princeton University Press}, + address = {Princeton, NJ}, + year = {1968} +} + +@book{Hartshorne, + author = {Hartshorne, Robin}, + title = {Algebraic Geometry}, + series = {Graduate Texts in Mathematics}, + volume = {52}, + publisher = {Springer-Verlag}, + address = {New York}, + year = {1977}, + doi = {10.1007/978-1-4757-3849-0} +} + +@article{BogomolovKatzarkov1998, + author = {Bogomolov, Fedor and Katzarkov, Ludmil}, + title = {Complex projective surfaces and infinite groups}, + journal = {Geometric and Functional Analysis}, + volume = {8}, + number = {2}, + pages = {243--272}, + year = {1998}, + doi = {10.1007/s000390050055} +} + +@article{Nakano1971, + author = {Nakano, Shigeo}, + title = {On the inverse of monoidal transformation}, + journal = {Publications of the Research Institute for Mathematical Sciences}, + volume = {6}, + number = {3}, + pages = {483--502}, + year = {1971}, + doi = {10.2977/PRIMS/1195193917} +} + +@article{FujikiNakano1972, + author = {Fujiki, Akira and Nakano, Shigeo}, + title = {Supplement to ``{On} the inverse of monoidal transformation''}, + journal = {Publications of the Research Institute for Mathematical Sciences}, + volume = {7}, + number = {3}, + pages = {637--644}, + year = {1972}, + doi = {10.2977/PRIMS/1195193401} +} + +@article{Noohi2014, + author = {Noohi, Behrang}, + title = {Fibrations of topological stacks}, + journal = {Advances in Mathematics}, + volume = {252}, + pages = {612--640}, + year = {2014}, + doi = {10.1016/j.aim.2013.11.008} +} diff --git a/preprints/A-projective-fourfold-with-large-fundamental-group-and-non-Stein-universal-cover-October-5-2026/build/sections/01-introduction.tex b/preprints/A-projective-fourfold-with-large-fundamental-group-and-non-Stein-universal-cover-October-5-2026/build/sections/01-introduction.tex new file mode 100644 index 0000000000000000000000000000000000000000..b22ab200c8645dd010f2ea16acff4a617c41dbcd --- /dev/null +++ b/preprints/A-projective-fourfold-with-large-fundamental-group-and-non-Stein-universal-cover-October-5-2026/build/sections/01-introduction.tex @@ -0,0 +1,191 @@ +\section{Introduction}\label{sec:introduction} + +Let $X$ be a smooth connected projective complex variety, and let +$\widetilde X$ denote its simply connected topological universal cover, +equipped with the lifted complex structure. Shafarevich's +holomorphic-convexity conjecture asks whether $\widetilde X$ is +holomorphically convex. Its particularly direct form for varieties with +large fundamental group asks whether $\widetilde X$ is Stein when it +contains no positive-dimensional compact complex analytic subvariety. + +Recall that $X$ has \emph{large fundamental group} if, for every +positive-dimensional closed irreducible subvariety $Z\subset X$, the image +of $\pi_1(Z^\nu)\to\pi_1(X)$ is infinite; here $Z^\nu$ is the +normalization. This is equivalent to the absence of positive-dimensional +compact analytic subvarieties in $\widetilde X$. We prove the following. + +\begin{theorem}\label{thm:main} +There exist a smooth connected projective complex fourfold $X$ and a +simple abelian surface $A\subset X$ such that +\begin{enumerate} +\item $X$ has large fundamental group; +\item $\im\bigl(\pi_1(A)\to\pi_1(X)\bigr)\simeq\Z$. +\end{enumerate} +The universal cover $\widetilde X$ contains no positive-dimensional +compact complex analytic subvariety and is not Stein. +\end{theorem} + +The obstruction to Steinness is topological and occurs in a closed +complex surface inside $\widetilde X$. Since $\pi_1(A)\simeq\Z^4$, +the kernel of the homomorphism in Theorem~\ref{thm:main} has rank three. +The corresponding cover of $A$ is therefore diffeomorphic to +$\R\times(S^1)^3$ and has nonzero third homology. A Stein surface has +the homotopy type of a CW complex of real dimension at most two. +This contradiction rules out Steinness of the lifted surface and hence +of $\widetilde X$. Simplicity of $A$ ensures that this topological +obstruction introduces no compact complex curve in that covering. + +\subsection{Historical context} +Shafarevich's question extends the uniformization viewpoint for curves +to universal covers of projective varieties. Koll\'ar's account +\cite[Introduction, Section~0.3]{Kollar1995} explains both the +holomorphic-convexity conjecture and its formulation for large +fundamental groups. These formulations are related by Remmert +reduction: a holomorphically convex complex manifold admits a proper +holomorphic map to a Stein space with connected compact analytic +fibers \cite[Introduction, p.~1546]{EKPR2012}. If there are no positive-dimensional +compact analytic subvarieties, all those fibers are points and the +reduction is an isomorphism. Theorem~\ref{thm:main} therefore also +disproves the holomorphic-convexity conjecture. + +An important development was the construction of almost-holomorphic reduction maps by +Campana and Koll\'ar \cite{Campana1994,Kollar1993,Kollar1995}. +They describe, through a very general point, the normalized subvarieties +whose fundamental groups have finite image in the ambient group. +Their existence provides an algebraic +and geometric description of the part of the variety that such +subvarieties fill. Holomorphic convexity asks in addition for a +proper holomorphic reduction of the universal cover to a Stein space. +The large case makes this additional analytic requirement especially +visible: the expected reduction has no positive-dimensional fibers. + +Linearity of the fundamental group supports strong affirmative +results. Katzarkov and Ramachandran proved holomorphic convexity for +projective surfaces whose fundamental group admits a faithful complex +representation with reductive Zariski closure +\cite{KatzarkovRamachandran1998}. Eyssidieux treated the reductive +linear case in arbitrary dimension \cite{Eyssidieux2004}. +Eyssidieux, Katzarkov, Pantev, and +Ramachandran proved the linear Shafarevich conjecture for smooth +projective varieties \cite{EKPR2012}. More recently, Bakker, +Brunebarbe, and Tsimerman developed linear Shafarevich theory for +normal algebraic spaces and quasiprojective varieties +\cite{BBT2024}. The construction here concerns the unrestricted +fundamental group. Its infinite-order argument uses compatible +local covers of an arrangement, so it does not require a faithful +linear representation of that group. + +A closely related line of inquiry appears in Bogomolov and Katzarkov's +constructions of projective surfaces and their potential counterexamples +to the Shafarevich conjecture +\cite[Section~4]{BogomolovKatzarkov1998}. Their proposed obstruction +uses infinite chains of compact curves and depends on group-infiniteness +statements left conjectural there. +Here the obstruction is a rank-three lattice cover of a simple abelian +surface. The group-theoretic task is to prove that its cyclic image +survives all global relations; the local path criterion supplies that +infiniteness argument. + +Several established tools make the construction possible. Arithmetic +ball quotients supply arbitrarily large regions with exactly prescribed +finite hyperplane configurations. Stover and Toledo's virtual +ramified-cover theorem supplies the required sign covers +\cite{StoverToledo,LlosaPy}, and Zarhin's theorem supplies a +genus-two curve with simple Jacobian \cite{Zarhin}. Standard +hyperbolic geometry \cite{BridsonHaefliger} underlies the local path +criterion. Hamm--L\^e's quasi-projective Lefschetz theorem +\cite{HammLe}, together with a deformation argument, allows us to +retain a prescribed surface while passing +from an orbifold to a smooth projective ambient variety. Each use is +stated with its hypotheses at the relevant point below. + +\subsection{The central construction} +The principal task is to embed a simple abelian surface in a smooth +projective variety so that its fundamental group has infinite cyclic +image. Once such an embedding is available, an ample complete +intersection in a product with a sufficiently large abelian variety +gives Theorem~\ref{thm:main}. The complete intersection contains the +prescribed surface, and its other fibers over the added abelian variety +are finite. This separates the two reasons that a subvariety has +infinite fundamental-group image: it either moves in the added abelian +variety or lies in the simple surface. Section~\ref{sec:reduction} +proves this reduction first. + +To obtain the cyclic image, we begin with a genus-two curve $C$ whose +Jacobian is simple. Its presentation as a double cover of $\PP^1$ +uses four branch points in one small disk and two outside that disk. +We call the corresponding meridians red and blue. A complex hyperbolic +arrangement realizes this marked line as an exceptional fiber. Extra +hyperplanes meeting the red part of the pencil impose relations that +identify all four red meridians. The sphere relation then identifies +the two blue meridians. The image of the genus-two surface group is +therefore cyclic: it lies in the cyclic subgroup generated by the product +of the resulting red and blue involutions. + +These relations alone give only an upper bound. The central difficulty +is to prove that the cyclic image is infinite after all global +identifications and fillings. We use covers defined separately on +large local models of the arrangement. We model their monodromies +using the infinite dihedral group generated by two involutions $r,b$; +its even words $(rb)^n$ carry the integer translation label $n$. +These local monodromies need not +define a representation of the full projective fundamental group. +Instead, they agree on the endpoint tests needed to read short paths +in overlapping charts. The graphs of these local covers are uniformly +Gromov hyperbolic. A local path criterion then prevents a loop detected +by one chart from becoming trivial through global relations. + +The edges of these graphs are paths with bounded \emph{projected +diameter}, rather than bounded length. Consequently every power of a +fixed loop near the distinguished fiber is a single edge. The local +criterion detects every nonzero power separately and gives the +infinite-order lower bound. This feature is essential to the argument; +neither a global coloring nor a family of finite quotients is needed +for that lower bound. + +Two copies of the pencil give a distinguished fiber $C\times C$. +An order-four symmetry exchanges the factors and rotates their normal +directions. Resolving its coarse quotient over this fiber produces a +map from a point blowup of +$\Sym^2 C$, itself the blowup of $\Jac(C)$ at a point. The same local +argument detects the sum of the two translation labels. We then +realize the resulting surface map in an ordinary smooth projective +ambient variety and remove its point blowups by projective +modifications. Throughout these operations we preserve the actual +homomorphism from the surface group. + +The local path criterion and the ambient blowdown construction are +formulated separately from the example. The former converts bounded +chart compatibility and hyperbolicity into nontriviality of loops. +The latter realizes a point blowdown of an embedded surface inside a +new smooth projective ambient variety, preserving the ambient +fundamental group and the induced surface homomorphism. These two +statements isolate the group-theoretic and projective-geometric +mechanisms of the construction. + +\subsection{Structure of the proof} +Section~\ref{sec:paths} proves the local path criterion independently +of the geometric application. Its hypotheses are bounded-range path +readings, compatibility of endpoint equality, and uniform hyperbolicity +of the reading graphs; no local compactness is required. +Section~\ref{sec:geometry} constructs the arithmetic arrangement, +its sign covers, and the distinguished surface map. +Section~\ref{sec:models} defines the local monodromy labels, proves +their buffered compatibility, and establishes hyperbolicity of the +associated graphs. Section~\ref{sec:image} verifies the path +criterion for the constructed orbifold and proves that the surface +image is infinite cyclic. Section~\ref{sec:projective} carries this +image into an ordinary smooth projective variety, removes the point +blowups, and completes the proof using Section~\ref{sec:reduction}. + +\subsection{Conventions} +All varieties are over $\C$ unless a field is specified, and all +fundamental groups are taken in the usual topology. For a smooth +complex orbifold, equivalently a smooth Deligne--Mumford stack in the +constructions below, the fundamental group means that of its +classifying space. An \emph{ordinary} point has trivial stabilizer. +Fundamental-group image statements are understood up to a common +choice of basepoints and connecting paths. Projective bundles +parametrize lines. Complex hyperbolic distance is normalized so that +a point of Euclidean radius $\tanh r$ in the unit ball has distance +$r$ from its center. diff --git a/preprints/A-projective-fourfold-with-large-fundamental-group-and-non-Stein-universal-cover-October-5-2026/build/sections/02-reduction.tex b/preprints/A-projective-fourfold-with-large-fundamental-group-and-non-Stein-universal-cover-October-5-2026/build/sections/02-reduction.tex new file mode 100644 index 0000000000000000000000000000000000000000..ddce76b6320d67d77f398bf3c14eea6066fcd8c8 --- /dev/null +++ b/preprints/A-projective-fourfold-with-large-fundamental-group-and-non-Stein-universal-cover-October-5-2026/build/sections/02-reduction.tex @@ -0,0 +1,191 @@ +\section{Reduction to an abelian surface with cyclic image} +\label{sec:reduction} + +The main construction will produce a simple abelian surface inside a +smooth projective variety, with infinite cyclic image on fundamental +groups. We first explain why that is enough. This separates the +geometric obstruction to Steinness from the group-theoretic construction +that occupies the intervening sections. + +\begin{proposition}\label{prop:large-reduction} +Let $A$ be a simple abelian surface and let $A\hookrightarrow S'$ be a +closed embedding in a smooth connected projective complex variety. +Suppose that +\[ + \im\bigl(\pi_1(A)\longrightarrow\pi_1(S')\bigr)\simeq\Z. +\] +There exist a smooth connected projective fourfold $X$ with large +fundamental group and a closed embedding $A\hookrightarrow X$ with +infinite cyclic image on fundamental groups, such that the universal +cover of $X$ is not Stein. +\end{proposition} + +We begin with the equivalence between largeness and the condition on +the universal cover used in Theorem~\ref{thm:main}. + +\begin{lemma}\label{red:compact-criterion} +Let $X$ be a smooth connected projective complex variety. Its universal +cover contains no positive-dimensional compact complex-analytic +subvariety if and only if $X$ has large fundamental group. +\end{lemma} + +\begin{proof} +Suppose that $Z\subset X$ has positive dimension and that +$\pi_1(Z^\nu)$ has finite image in $\pi_1(X)$. The covering of $Z^\nu$ +corresponding to the kernel has finite degree and is compact. Its map +to $X$ lifts holomorphically to $\widetilde X$. The image of the lift +is a positive-dimensional compact analytic subvariety, by the proper +mapping theorem. + +Conversely, let $K\subset\widetilde X$ be a positive-dimensional +irreducible compact analytic subvariety. Its image $Z$ in $X$ is +analytic by the same theorem and algebraic by Chow's theorem. The +restriction $K\to Z$ is finite: it is proper and its fibers are discrete, +since the covering map is locally biholomorphic. Thus +$K^\nu\to Z^\nu$ is finite and surjective. Such a map has finite-index +image on fundamental groups. To see the latter assertion, remove a +proper analytic subset of the target so that the map becomes a finite +unramified cover of smooth connected spaces. On these open sets the +index is its covering degree. Their fundamental groups surject onto +those of the normal spaces, by general position for loops; passing to +these quotients can only decrease the index. But the composite +$\pi_1(K^\nu)\to\pi_1(X)$ is zero, because $K^\nu$ maps to +$\widetilde X$. Hence $\pi_1(Z^\nu)$ has finite image in $\pi_1(X)$. +\end{proof} + +The next elementary interpolation observation allows us to control all +fibers of a general complete intersection simultaneously. + +\begin{lemma}\label{red:interpolation} +Let $W$ be a projective variety, $F\subset W$ a closed subscheme, and +$L$ a very ample line bundle. Fix a positive integer $\ell$. For all +sufficiently large $m$ and every set of distinct points +$p_1,\ldots,p_\ell\in W\setminus F$, the evaluation map +\[ + H^0(W,\mathcal I_F\otimes L^m) + \longrightarrow \bigoplus_{i=1}^{\ell}L^m|_{p_i} +\] +is surjective. The same bound on $m$ works for every such set. +\end{lemma} + +\begin{proof} +Choose $m_0$ so that $\mathcal I_F\otimes L^{m_0}$ is globally +generated. For each $i$, choose a section $a_i$ nonzero at $p_i$. +For every $j\ne i$, very ampleness gives a section of $L$ vanishing at +$p_j$ and nonzero at $p_i$. Their product $b_i$ vanishes at all the +$p_j$ with $j\ne i$ and is nonzero at $p_i$. The sections $a_ib_i$ +therefore give a diagonal basis for evaluation in degree +$m_0+\ell-1$. Multiplying each by a section of the remaining power +of $L$ nonzero at $p_i$ gives the assertion in every higher degree. +\end{proof} + +\begin{proof}[Proof of Proposition~\ref{prop:large-reduction}] +Put $d=\dim S'$, choose an abelian variety $B$ of dimension $g=6$, and +write +\[ + W=S'\times B,\qquad F=A\times\{0\},\qquad + c=d+g-4=d+2. +\] +Fix a very ample line bundle $L$ on $W$. We shall take $X$ to be the +common zero locus of $c$ general sections of +$\mathcal I_F\otimes L^m$, for a sufficiently large $m$. +There are two requirements: $X$ must be smooth with the expected +fundamental group, and its fibers over $B$ must be finite away from $F$. + +\smallskip +\noindent\emph{Smoothness and the fundamental group.} +For large $m$, the sections generate the ideal away from $F$ and their +normal derivatives generate +$N^*_{F/W}\otimes L^m|_F$. Bertini's theorem gives smoothness away +from $F$. Along $F$, smoothness is the independence of $c$ general +vectors in a bundle of rank $d+g-2=c+2$. The rank-deficient matrices +have codimension $3$, larger than $\dim F=2$, so no rank loss occurs +for a general tuple. The same count for each prefix of the tuple +shows that the successive intersections are smooth. They are ample +hypersurfaces in their predecessors, with final dimension four. +The Lefschetz hyperplane theorem consequently gives connectedness and +an isomorphism, induced by inclusion, +\begin{equation}\label{red:fundamental-group} + \pi_1(X)\xrightarrow{\ \sim\ } + \pi_1(S'\times B)=\pi_1(S')\times\pi_1(B). +\end{equation} +Here and below we use sufficiently high powers of a very ample line +bundle to obtain both ideal generation and generation of the indicated +normal derivatives; these are the usual applications of Serre +vanishing and Bertini's theorem \cite{Hartshorne}. + +\smallskip +\noindent\emph{Fibers away from the fixed surface.} +We claim that a general such tuple excludes four distinct points of +$W\setminus F$ in any one fiber of $W\to B$. Excluding four points +is enough to exclude every positive-dimensional fiber component +outside $F$; the choice $g=6$ makes the following incidence count +strict. The parameter space of +ordered quadruples in one fiber has dimension at most $4d+g=4d+6$. +By Lemma~\ref{red:interpolation}, requiring all $c$ equations to +vanish at such a quadruple imposes exactly $4c=4d+8$ independent +linear conditions on the space of tuples of sections. The incidence +variety of tuples and quadruples thus has dimension strictly less than +the section parameter space. Its constructible image has closure of +strictly smaller dimension as well. A general tuple lies outside this +closure. This condition is compatible with the open smoothness +conditions already imposed. + +It follows that each fiber of $X\to B$ contains at most three points +outside $F$. In particular, every positive-dimensional subvariety of +a fiber is contained in $F$. + +\smallskip +\noindent\emph{Largeness.} +Let $Z\subset X$ be a positive-dimensional closed irreducible +subvariety. If its projection to $B$ is nonconstant, then +$\pi_1(Z^\nu)\to\pi_1(B)$ has infinite image. Otherwise its image, +a finite subgroup of the torsion-free group $\pi_1(B)$, would be +trivial. The map $Z^\nu\to B$ would then lift holomorphically to +$\C^g$. Every holomorphic function on the connected compact normal +space $Z^\nu$ is constant, a contradiction. + +If the projection is constant, the fiber property gives $Z\subset F$. +For $Z=F$, its image in $\pi_1(X)$ is infinite cyclic by +\eqref{red:fundamental-group} and the hypothesis. The only other +possibility is an irreducible curve in $A$. Let $D$ be its smooth +normalization. After a translation in $A$, the nonconstant map +$D\to A$ extends to a homomorphism $\Jac(D)\to A$. Its image is a +nonzero abelian subvariety, and therefore all of $A$ by simplicity. +A surjective homomorphism of complex tori carries the source integral +lattice to a finite-index subgroup of the target lattice. Consequently +$\pi_1(D)$ has finite-index image in $\pi_1(A)\simeq\Z^4$, and its +image under the homomorphism onto the infinite cyclic group remains +infinite. This proves largeness in every case. Lemma~\ref{red:compact-criterion} +then gives the required absence of compact analytic subvarieties in +$\widetilde X$. + +\smallskip +\noindent\emph{The obstruction to Steinness.} +Let $K$ be the kernel of $\pi_1(A)\to\pi_1(X)$. Its quotient is +infinite cyclic, so $K\simeq\Z^3$. A connected component of the +inverse image of $F$ in $\widetilde X$ is the connected covering +\[ + A_K=\C^2/K. +\] +It is a closed complex submanifold of $\widetilde X$: the full inverse +image of $F$ is closed, and its components are closed and locally +separated in covering charts. The real span of $K$ has dimension +three, so, as a real manifold, +\[ + A_K\simeq(\R^3/\Z^3)\times\R, + \qquad H_3(A_K,\Z)\simeq\Z. +\] +By the Andreotti--Frankel theorem \cite{AndreottiFrankel1959}, a Stein +manifold of complex dimension two has the homotopy type of a CW complex of real dimension at most +two; see \cite[Theorem~7.2]{MilnorMorse}. Thus $A_K$ is not Stein. +A closed complex submanifold of a Stein manifold is Stein, so +$\widetilde X$ cannot be Stein either. +\end{proof} + +Proposition~\ref{prop:large-reduction} leaves one concrete task: embed a +simple abelian surface in a smooth projective variety so that its +fundamental group has infinite cyclic image. We first construct the +corresponding map from a point blowup of the surface to a smooth +orbifold, and then convert it to the required embedding in +Section~\ref{sec:projective}. diff --git a/preprints/A-projective-fourfold-with-large-fundamental-group-and-non-Stein-universal-cover-October-5-2026/build/sections/03-paths.tex b/preprints/A-projective-fourfold-with-large-fundamental-group-and-non-Stein-universal-cover-October-5-2026/build/sections/03-paths.tex new file mode 100644 index 0000000000000000000000000000000000000000..adada096b18d1c769c36a31d79abc75471460454 --- /dev/null +++ b/preprints/A-projective-fourfold-with-large-fundamental-group-and-non-Stein-universal-cover-October-5-2026/build/sections/03-paths.tex @@ -0,0 +1,478 @@ +\section{A local criterion for nontrivial loops}\label{sec:paths} + +The geometric construction will attach group labels to paths only within +bounded regions. We therefore need a criterion that detects a nontrivial +loop using such local information. The criterion below uses hyperbolic +graphs as the local models. It does not require a global map to a model, +local finiteness of a graph, or a bound on the information carried by one +edge. + +A model to keep in mind is a graph whose vertices are points of a space +and whose edges are paths confined to small regions. A reading lifts +such paths to a covering space chosen near the initial vertex; its +endpoint records both the actual endpoint and the sheet of the cover. +The argument follows the coarse local-to-global viewpoint of hyperbolic +geometry \cite{Gromov1987}; we prove here the version for bounded path +readings in compatible charts. + +\subsection{Paths and their readings} + +All graphs have oriented edges with specified reverses; multiple edges and +loops are allowed. Each edge has length one. Paths are finite edge paths, +and their lengths count edges. A constant path has length zero. The path +groupoid of a graph is obtained by cancelling consecutive reverse edges. +We use the path metric on each connected component of a graph. + +Fix a positive integer $N$. At each vertex $v$ of a graph $G$, suppose that +we are given a pointed graph $(D_v,d_v)$ and a rule for reading every path +of length at most $N$ starting at $v$ as an edge path in $D_v$ starting at +$d_v$. Write $[p]_v$ for the terminal vertex of the reading of $p$. +The following are the required properties. +\begin{enumerate} +\item[\textnormal{(P1)}] Readings respect prefixes and reverses of edges, +and read a backtrack as a backtrack. If $p$ starts at $v$ and $|p|2\Delta$ from +either end connector. Quadrangle slimness therefore puts it within +$2\Delta$ of a vertex $b'_i$ on the opposite geodesic, the corresponding +piece of $\beta$. Its old mate $b_i$ was within $h$ of $x_i$, so +geodesicity gives +\begin{equation}\label{path:mate-shift} + |b'_i-b_i|_{\beta}\le h+2\Delta. +\end{equation} +Here the left side denotes distance in the parameter of the path +$\beta$. The new mates remain in order, since the difference between +consecutive new positions is at least +\[ + L-2h-2(h+2\Delta)=L-4h-4\Delta>0. +\] +The same estimate at the ends puts them strictly between the unchanged +endpoint mates. Lift short paths from $x_i$ to $b'_i$ to obtain new +rungs of length at most $2\Delta$; use empty rungs at $i=0,m$. + +The new rungs agree with the old calculation: following an old rung and +the relevant portion of $\beta$ has the same read endpoint as the new +rung. On a marked interval this equality can be checked together with +both adjacent marked intervals. Their union has first-side length at +most $6L$, so the preceding bound and the short-strip calculation apply. +Thus between successive new mates both sides are geodesic, and their +end rungs have length at most $2\Delta$. Apply +\eqref{path:geodesic-comparison}, lift the comparison rungs, and retain +the specified rungs at the marked endpoints. Each elementary circuit +closes in this chart and hence at its own corner. The resulting width +is at most $30\Delta2\Delta$. +Also, the distances from $x_0$ increase at every mark by at least +$L-2\Delta$, since +\[ + d(x_0,x_{i+1})-d(x_0,x_i) + =d(x_i,x_{i+1})-2(x_0\mid x_{i+1})_{x_i}. +\] +The analogous estimate holds from the other endpoint. + +Let $A,B$ be the read endpoints of the chosen paths at the two ends of +the block. They are at distance at most $H$ from $x_0,x_q$, +respectively. By \eqref{path:block-products}, every interior $x_i$ +has a mate on a geodesic $[x_0,x_q]$ at distance at most $12\Delta$. +Such a mate is farther than $2\Delta$ from either of the geodesic +connectors $[x_0,A]$ and $[x_q,B]$: its distance from those connectors +is at least $L-H-14\Delta$. Quadrangle slimness therefore gives a +vertex $y_i$ on $[A,B]$ with +\begin{equation}\label{path:block-mates} + d(x_i,y_i)\le20\Delta0. +\] +The same inequalities keep them strictly between $A$ and $B$. + +Lift short paths from the interior $x_i$ to the $y_i$ to replace the +interior choices $c_i$, keeping the two outer choices fixed. These are +legal choices by \eqref{path:block-mates}. For them the sum of the +successive distances equals $d(A,B)$, since their endpoints occur in +order on a geodesic. Transport of distances shows that this is also +their sum of single-interval objectives $F_i$. Minimality of the +original total objective implies that its sum on this block is at most +$d(A,B)$: changing only interior choices leaves every term outside the +block unchanged. The triangle inequality gives the reverse inequality. +Thus the original joins concatenate geodesically on the block, as +claimed. + +Each join $\beta_i$ has length at least $L-2H$. A subpath of $\beta$ +of length at most $k=20L$ therefore meets fewer than $100$ consecutive +joins: apart from at most two end joins, every join it meets is entirely +contained in that subpath. The block result and transport of +geodesicity prove that $\beta$ is a guide. + +Finally compare $\alpha_i$ and $\beta_i$ in their single-interval +chart. Their corresponding endpoints are joined by the given paths +$c_i,c_{i+1}$ of length at most $H$. Formula~\eqref{path:geodesic-comparison} +gives comparison rungs of length at most $10(H+\Delta)$; retain the +given choice paths as rungs at the marks. Lift these comparisons and +concatenate them. At the last mark the rung is $e$; let the first +side traverse $e$ and finish with an empty rung. This gives the required +ladder of width at most $50H$. +\end{proof} + +\subsection{The invariant of a path} + +We can now finish the local criterion. Fix an initial vertex $v$ and +consider guides starting at $v$, modulo width-$H$ ladders. By +Lemma~\ref{path:extension}, appending an edge $e$ defines a continuation +of such a class. It is independent of both choices involved. If +$\alpha,\alpha'$ represent the same class, append $e$ to their +width-$H$ ladder. If $\beta,\beta'$ are any two guides supplied by +Lemma~\ref{path:extension}, composition gives a ladder between them of +width at most +\[ + 50H+H+50H=101H. +\] +Lemma~\ref{path:narrowing} reduces this to width $H$. The same argument +covers different choices of the extension for a fixed guide. + +Let $q$ be a loop of length at most three that closes on reading. +It has a ladder of width at most three with the constant path. One +explicit schedule traverses $q$ on the first side while the second +waits: after a noninitial position use the remaining suffix of $q$ as +the rung, and use empty first and last rungs. The first circuit closes +because $q$ does, and the subsequent ones cancel consecutive reverse +edges. The short-chart rules justify the same assertion at each corner. +Appending this ladder after any guide $\alpha$ gives a width-three +ladder between $\alpha q$ and $\alpha$. + +Successively applying Lemma~\ref{path:extension} along $q$ produces a +guide $\beta$ with a ladder to $\alpha q$ of width at most $150H$. +To see the bound, at each step append the remaining common trailing +edges to the comparison and compose; each of at most three steps adds +$50H$. Composing with the preceding width-three ladder and narrowing +shows that $\beta$ and $\alpha$ represent the same class. The same +reasoning applies to a backtrack, which closes on reading by (P1). + +Starting from the empty guide at $v$ and continuing along a path thus +gives a class unchanged by every defining relation of $\mathcal Q$. +Insertion of a relation in the middle of a path causes no problem: +the classes agree after the inserted loop, and the well-defined +continuations along the remaining edges preserve that agreement. + +\begin{proof}[Proof of Lemma~\ref{thm:path-lemma}] +Let $e$ be the edge in the statement. Its reading has distinct +endpoints and length one, so $e$ itself is a guide. Its class is +therefore the continuation of the empty guide along $e$. If $e$ were +trivial in $\mathcal Q$, the invariant just constructed would give a +width-$H$ ladder between $e$ and the constant path. This entire +comparison calculates in the chart at $v$: its sides have lengths one +and zero and its individual rungs have length at most $H$. +Equation~\eqref{path:strip-calculation}, with the empty final rung, +would then give $[e]_v=d_v$, a contradiction. +\end{proof} diff --git a/preprints/A-projective-fourfold-with-large-fundamental-group-and-non-Stein-universal-cover-October-5-2026/build/sections/04-geometry.tex b/preprints/A-projective-fourfold-with-large-fundamental-group-and-non-Stein-universal-cover-October-5-2026/build/sections/04-geometry.tex new file mode 100644 index 0000000000000000000000000000000000000000..577f2d4f7872029c78e03736340de7d22d7fab62 --- /dev/null +++ b/preprints/A-projective-fourfold-with-large-fundamental-group-and-non-Stein-universal-cover-October-5-2026/build/sections/04-geometry.tex @@ -0,0 +1,526 @@ +\section{An arithmetic arrangement and a distinguished surface}\label{sec:geometry} + +We construct a smooth proper effective orbifold $\cS$ with projective +coarse space and a morphism $f:Y\to\cS$, where $Y$ is obtained from a +simple abelian surface by point blowups. The image of $f$ will lie in +the ordinary locus of $\cS$. The arrangement constructed here provides +both the relations and the local covering spaces used to prove that +$f_*\pi_1(Y)$ is infinite cyclic. The fundamental-group calculation is +made in Theorem~\ref{thm:orbifold-cyclic}. + +\subsection{Two pencils and their coupling hyperplanes}\label{geo:prototype} + +The two pencils will use the same six marked directions on $\PP^1$. +We first choose these markings so that their double cover has simple +Jacobian, and then choose the arithmetic ball containing the pencils. +Choose six distinct real algebraic numbers $\lambda_1,\ldots,\lambda_6$ with +\begin{equation}\label{geo:slopes} + |\lambda_i|<0.01\quad(1\le i\le4), + \qquad \lambda_5<-4,\quad \lambda_6>4, +\end{equation} +and such that the smooth double cover +\begin{equation}\label{geo:curve} + \pi:C\longrightarrow\PP^1, + \qquad y^2=\prod_{i=1}^6(t-\lambda_i), +\end{equation} +has simple Jacobian. We call the first four slopes \emph{red} and the +last two \emph{blue}. These requirements can be imposed simultaneously, +as follows. + +Start with six disjoint real open intervals satisfying +\eqref{geo:slopes}. Monic real polynomials with one root in each interval +form a nonempty open set in coefficient space. At three sufficiently +large distinct primes prescribe squarefree degree-six reductions with +factor degrees +\[ + (6),\qquad(5,1),\qquad(2,1,1,1,1). +\] +Weak approximation produces a monic polynomial in $\Q[t]$, integral at +these primes, with these reductions and with its real coefficients in +the chosen open set. Its Galois group contains a six-cycle, a five-cycle, +and a transposition, by the Frobenius cycle criterion. The six-cycle +makes the group transitive. The stabilizer of the fixed point of the +five-cycle is transitive on the remaining five points, so the group is +two-transitive. Conjugating its transposition then gives every +transposition, and the Galois group is $S_6$. Let $F$ be its splitting +field. Then $F$ is totally real, $F\ne\Q$, and all six slopes belong +to $F$. Zarhin's theorem gives +$\End(\Jac(C))=\Z$ over an algebraic closure, hence $\Jac(C)$ is simple +\cite[Theorem~2.1]{Zarhin}. + +Use the inclusion $F\subset\R$ containing the chosen roots as the +distinguished real embedding. Choose a CM extension $E/F$ and an +element $a\in F$ which is positive at this embedding and negative at +every other real embedding; weak approximation supplies $a$. On $E^5$ put +\begin{equation}\label{geo:hermitian} + h=\operatorname{diag}(1,1,1,1,-a). +\end{equation} +At the distinguished embedding the negative lines form complex +hyperbolic space of dimension four. In the affine chart with last +homogeneous coordinate one, write this ball as +\[ + \BB=\{(z,w)\in\C^2\times\C^2:|z|^2+|w|^2a$. +Every intersection involving the two families has a local product +description in the $z$ and $w$ variables. In particular, the phrase +``product'' here concerns local analytic coordinates, rather than a +product decomposition of the hyperbolic metric. + +Figure~\ref{fig:arrangement} shows the incidences in one family; the +second family is obtained by using the other coordinate pair. +\input{figures/arrangement} + +\subsection{Exact models in congruence quotients}\label{geo:separation} + +The arithmetic lattice theorem of Borel--Harish-Chandra +\cite[Corollary~12.4]{BorelHarishChandra1962}, applied to +\eqref{geo:hermitian}, gives a +cocompact arithmetic lattice of simplest type in $\mathrm{PU}(4,1)$. +Indeed all the other archimedean factors are compact, and the form is +anisotropic over $E$: a nonzero $E$-rational isotropic vector would +remain isotropic at a definite conjugate embedding. Torsion-free +principal congruence subgroups therefore give compact complex +hyperbolic manifolds, which are projective by the Baily--Borel theorem +\cite[Theorem~10.11]{BailyBorel1966}. One can +also obtain projectivity from the positive canonical bundle of a +compact ball quotient. We use principal congruence subgroups normalized +by $\sigma$, and may take their levels arbitrarily deep. These standard +arithmetic facts are recalled in \cite[Section~3.1]{StoverToledo}. + +Fix positive normal vectors $n_i\in E^5$ for the members of $\cD$, +scaled to have integral coordinates. We always label a translate of +the $i$th hyperplane by $i$, and use the prescribed normal $\gamma n_i$ +for its translate by $\gamma$. Congruence subgroups may be chosen to +avoid the finite scalar kernel of the projective action. + +\begin{proposition}[Exact local arrangements]\label{prop:arrangement-local} +Given $W>0$ and $R_0>0$, there is a torsion-free principal congruence +subgroup $\Gamma_0$, normalized by $\sigma$, with the following +properties. Let $\mathcal A_0$ be the union of all $\Gamma_0$-translates +of the labeled hyperplanes of $\cD$. +\begin{enumerate} +\item The arrangement $\mathcal A_0$ is locally finite. The hyperplanes +meeting any ball of radius $W$ have distinct labels and are carried to +the corresponding subarrangement of $\cD$ by a holomorphic isometry +induced by a unitary map sending each prescribed translated normal to +its prototype normal. +\item The injectivity radius of $\Gamma_0\backslash\BB$ is greater than +$R_0$. +\item In the ball of radius $R_0$ about $o$, the arrangement is exactly +$\cD$ restricted to that ball. The analogous assertion holds in every +$\Gamma_0$-translate of this ball. +\end{enumerate} +All these conclusions concerning the lifted arrangement remain valid +on any subsequent finite cover of $\Gamma_0\backslash\BB$, provided one +uses the full inverse image of the arrangement. +\end{proposition} + +\begin{proof} +First fix a radius $W$ and a finite collection of hyperplanes meeting a +$W$-ball. At its viewpoint, represented by a unit negative +vector $x$, the distance to the hyperplane with positive normal $n$ +satisfies +\[ + \sinh d(x,H_n)=\frac{|h(n,x)|}{\sqrt{h(n,n)}}. +\] +For a normal of one of our finitely many fixed lengths, intersection +with the $W$-ball bounds $|h(n,x)|$. Write $n=n_++\alpha x$ with +$n_+\in x^\perp$. Since +\[ + h(n_+,n_+)=h(n,n)+|\alpha|^2, +\] +the positive component is bounded as well. Cauchy--Schwarz in +$x^\perp$ now bounds every Gram entry of two normals visible from this +ball, by a constant depending only on $W$ and their labels. + +At every other archimedean place, definiteness and the fixed normal +lengths bound the same Gram entries. All entries belong to one fixed +fractional ideal of $E$. Its image under the archimedean embeddings +is a lattice, so only finitely many Gram entries satisfy these bounds. +If $\gamma,\gamma'$ are in a sufficiently deep integer principal +congruence subgroup, then +\[ + h(\gamma n_i,\gamma'n_j)\equiv h(n_i,n_j) +\] +modulo the level, in this fractional ideal. Choose the level to +separate all the finitely many possible nonzero differences. We obtain +exact equality: +\begin{equation}\label{geo:gram} + h(\gamma n_i,\gamma'n_j)=h(n_i,n_j) +\end{equation} +whenever the corresponding hyperplanes are visible together. + +Here equality of Gram matrices gives the asserted isometry even for +dependent collections. Every $E$-rational subspace of $E^5$ is +nondegenerate for $h$: a nonzero radical, being defined by linear +equations over $E$, would persist at a definite embedding. Consequently +the kernel of a Gram matrix is exactly the relation space of its +vectors. Equality \eqref{geo:gram} thus defines an isometry between +their spans, which extends to a unitary isometry of the ambient +Hermitian space. Its projectivization is a holomorphic ball isometry. +Two visible copies of the same label must in fact have the same normal: +their difference has norm zero by \eqref{geo:gram}, and anisotropy over +$E$ makes that difference zero. Two different labels cannot coincide, +since this would contradict equality of their relation spaces with +those of the distinct prototypes. + +Local finiteness follows from the same boundedness argument applied to +normals themselves at a fixed viewpoint. The normals have integral +coordinates; their distinguished components are bounded by the distance +bound and their conjugate components by definiteness. Only finitely +many can meet a fixed bounded ball. + +To obtain exact agreement at $o$, choose an auxiliary ball about $o$ +large enough to contain the prescribed $R_0$-ball and to meet every +prototype hyperplane. Apply the Gram argument to this ball. The +prototype normals span $E^5$: the slope normals span the four positive +coordinate directions, and a coupling normal supplies the last +homogeneous direction. An additional translated normal visible in +this ball has, by \eqref{geo:gram}, the same inner products with this +spanning set as its prototype. It therefore equals that prototype. +This proves the third assertion. + +Finally fix a torsion-free congruence subgroup in advance and a compact +fundamental set for its action on $\BB$. A group element moving a point +of this set a bounded distance belongs to a finite set. A nested +sequence of normal principal congruence subgroups has trivial +intersection, after the scalar kernel has been removed. Deep enough +levels avoid every nonidentity element of the finite set. Normality +allows an arbitrary viewpoint to be moved into the compact fundamental +set without changing subgroup membership. This proves the required +uniform lower bound on injectivity radius. Increasing the level +accommodates all the preceding requirements simultaneously. Passing +to a subgroup preserves them when the arrangement is pulled back in +full. +\end{proof} + +We next record the consequences for the divisors and centers that we +will blow up. In $M_0=\Gamma_0\backslash\BB$, each label gives a closed +embedded smooth totally geodesic divisor, possibly disconnected. Its +lifts are locally finite and have disjoint sheets by +Proposition~\ref{prop:arrangement-local}. The divisors are algebraic +because $M_0$ is projective. + +Within one family the intersection centers have complex codimension +two. A center is either a common intersection of at least two slopes, +or an intersection of a red slope with a coupling hyperplane. At a +slope center only a subset of the six slopes may occur. The subset is +constant along a connected center: if another slope meets it, the +local model shows that it contains the intersection, and the +containment extends along its lifts. Distinct centers within one +family are disjoint; otherwise all participating hyperplanes would be +visible near a point of intersection, contradicting the prototype +pattern. Coincident centers are counted once. Thus these centers are +closed embedded smooth subvarieties. Centers from different families +have the product structure already described. These assertions apply +to the full pulled-back arrangement after any finite cover. + +\subsection{Two global sign covers}\label{geo:signs} + +For the later local models we need global parity characters on the +arrangement complement. We obtain these from the following theorem +of Stover--Toledo, in the form stated by Llosa Isenrich--Py +\cite[Theorem~4.2]{LlosaPy}; see also \cite{StoverToledo}. + +\begin{theorem}[Virtual cyclic ramified covers]\label{geo:stover-toledo} +Let $M_0$ be a compact ball quotient of complex dimension at least two +by a torsion-free congruence arithmetic lattice of simplest type. Let +$D$ be a nonempty totally geodesic divisor whose components are smooth, +embedded, and pairwise disjoint. For every integer $d\ge2$ there is a +finite unramified cover $M_1\to M_0$ such that $M_1$ admits a cyclic +cover of degree $d$, branched over the full inverse image of $D$. +\end{theorem} + +Apply Theorem~\ref{geo:stover-toledo} with $d=2$ to each of the sixteen +label divisors separately, on the initial congruence quotient $M_0$. +Each application satisfies the disjoint-component hypothesis. We do +not apply the theorem to the union of the intersecting divisors. Take +a common finite unramified cover of the resulting base covers. The +sum, in $\Z/2$, of the eight characters for the $z$ labels gives a +character $\chi_z$ on the complement of the $z$ arrangement. It has +value one on a meridian of every $z$ hyperplane. Pull it back to the +complement of both families. The same procedure gives a $w$ character. +Subsequent finite covers need not be congruence: every use of +Theorem~\ref{geo:stover-toledo} has already taken place on $M_0$. + +We arrange equivariance before fixing the signs. First intersect the +finitely many $\sigma$-conjugates of the subgroup defining our common +cover, so that $\sigma$ acts on it. Since $\sigma^2$ preserves the +$z$-family, the character +\[ + \delta=\chi_z+\sigma^{2*}\chi_z +\] +is zero on all meridians. The kernel of the map from the fundamental +group of a divisor complement to that of the ambient smooth variety +is normally generated by meridians. Hence $\delta$ factors through +the fundamental group of the compact ball quotient. Pass to a further +finite cover by intersecting the kernel of this character with its +$\sigma$-conjugates. On that cover the pulled-back $z$ character +satisfies $\sigma^{2*}\chi_z=\chi_z$. Define +\begin{equation}\label{geo:sign-equivariance} + \chi_w=\sigma^*\chi_z. +\end{equation} +Then $\sigma$ exchanges the two characters. They have the prescribed +meridian values in their respective families, and each is defined +using only that family. + +Write $M=\Gamma\backslash\BB$ for this final finite cover, still with +the full pulled-back arrangement. The image of $o$ in $M$, also denoted +$o$, is fixed by $\sigma$. It is an isolated component even of the +fixed locus of $\sigma^2$, because the derivative of $\sigma^2$ at $o$ +is $-\id$. No assertion of this kind is needed for the other fixed +loci of the symmetry on $M$. + +\subsection{Filling the arrangement}\label{geo:fill} + +Blow up all intersection centers within the $z$-family, and then the +transforms of those within the $w$-family. Product coordinates show +that the operations for the two families commute. The branch rule +for the $z$ character is as follows. Every strict transform of a +$z$ hyperplane is a branch divisor of order two. If $k$ such +hyperplanes pass through a center, its exceptional meridian is the +product of their meridians, and therefore has sign $k\bmod2$. +Accordingly the exceptional divisor is branched precisely when $k$ +is odd. In that case blow up each meeting of this exceptional divisor +with a strict transform of a hyperplane. The two meeting divisors +both have sign one; their new exceptional divisor has sign zero. +The branch components within this family are now smooth and disjoint. +Apply the identical rule to the $w$ character. This prescription +includes partial pencils. In particular an ordinary red--coupling +pair has $k=2$, so its first exceptional divisor is unbranched. + +Let $\widehat M$ denote the resulting smooth projective base. Extend +the two sign covers from the complement by normalization over +$\widehat M$, and take them together. Finite topological covers of +smooth complex algebraic varieties algebraize, and normalization in +the resulting finite extensions gives finite algebraic covers; thus +this procedure is projective. Locally along a branch divisor the +normalization is the smooth double-cover model $t=u^2$. Branch +divisors within a family are disjoint, and across the families the +local equations and the two signs are independent. Their combined +normalization is therefore smooth, including at crossings, where it +has product equations $(t_1,t_2)=(u_1^2,u_2^2)$. We obtain a smooth +projective variety +\begin{equation}\label{geo:V} + V\longrightarrow\widehat M +\end{equation} +with deck group $(\Z/2)^2$. It is connected: a meridian at a general +point of a $z$ divisor has sign $(1,0)$, and a meridian at a general +point of a $w$ divisor has sign $(0,1)$, so the combined monodromy is +surjective. + +Near $o$ all six slopes in each pencil occur and the coupling +hyperplanes are absent after the neighborhood has been made small. +The blowups are the product of the blowups of the two coordinate +planes at their origins. The exceptional fiber of $\widehat M\to M$ +is $\PP^1\times\PP^1$. Each exceptional line has six branch markings +and has even meridian sign. Its inverse image in $V$ is consequently +\begin{equation}\label{geo:Fstar} + F_*=C\times C. +\end{equation} +Put +\[ + L=\pi^*\mathcal O_{\PP^1}(-1), + \qquad N=N_{F_*/V} + =\operatorname{pr}_1^*L\oplus\operatorname{pr}_2^*L. +\] +In fact a neighborhood of $F_*$ has the product line-bundle description +provided by these two tautological lines. Indeed, in each factor, after +the unbranched radial disks have been filled, their disk bundle retracts onto its +zero section. Away from the six marked directions its cover is thus +pulled back from the direction line. Normalization gives the +six-branch cover in that direction variable, with the radial line +bundle pulled back unchanged. + +The equivariance \eqref{geo:sign-equivariance} makes the kernel of the +combined sign character invariant, so $\sigma$ lifts on the complement. +The equivariant blowups and normalization extend the lift to an +automorphism $s$ of $V$. Its restriction to $F_*$ exchanges the two +copies of $C$, possibly followed by either hyperelliptic deck +involution. Composing with a deck transformation makes this +restriction the pure swap. To determine the lift in the whole local +model, note that $u,v$ are actual tautological vectors: the covering +map in these coordinates is +\[ + (p,q;u,v)\longmapsto(\pi(p),\pi(q);u,v). +\] +The pure swap determines the lifted direction maps on the connected +local cover. Since the base map sends the radial vectors to $(v,-u)$, +there is no further radial multiplier. Thus +\begin{equation}\label{geo:line-action} + s(p,q;u,v)=(q,p;v,-u). +\end{equation} +Its fourth power is a deck transformation fixing $F_*$ pointwise, +so $s^4=1$; a nonidentity element of $(\Z/2)^2$ does not fix $C\times C$ +pointwise. Its square is the identity on $F_*$ and is multiplication +by $-1$ on both normal lines. + +\subsection{An ordinary neighborhood of the distinguished fiber} +\label{geo:resolution} + +The quotient stack $[V/\langle s\rangle]$ is a smooth proper orbifold +with projective coarse space, but a surface mapping into its distinguished +fiber would +meet stabilizers. We now replace just that fiber by an ordinary +smooth resolution. Retaining the quotient stack elsewhere allows +all other fixed loci to be left in place. + +First blow up $F_*$ in $V$, and take the coarse quotient by +$\langle s^2\rangle$. The exceptional divisor is +\begin{equation}\label{geo:P} + P=\PP_{F_*}(N), +\end{equation} +where projectivization parametrizes lines. Before taking the quotient, +$s^2$ acts trivially on $P$ and as $t\mapsto-t$ on its transverse +coordinate. The quotient is thus smooth near $P$, with transverse +coordinate $t^2$. + +The residual involution induced by $s$ has fixed points near $P$ only +in $P$: away from the fiber, a fixed point would project to a +nontrivial fixed point of the symmetry near $o$ in $M$. Its fixed +points in $P$ project to the diagonal of $C\times C$. On the +projective normal line over that diagonal, it acts by +\[ + [u:v]\longmapsto[v:-u]. +\] +There are exactly two eigensections, with eigenvalues $i$ and $-i$ +before projectivization. These give two disjoint smooth fixed curves. +On the three normal directions to either curve the involution acts +by $-1$: one direction is antisymmetric in the base, one is tangent +to the projective fiber, and the transverse coordinate to $P$ has +eigenvalue $(\pm i)^2=-1$. Blow up both fixed curves and take the +coarse quotient by the residual involution. This quotient is smooth +near the distinguished fiber. Indeed, for the linear local model +\[ + (x,t_1,t_2,t_3)\longmapsto(x,-t_1,-t_2,-t_3), +\] +blowing up the fixed curve $\{t_1=t_2=t_3=0\}$ makes the action a +reflection in the coordinate normal to the exceptional divisor, and +the coarse quotient is smooth. Finite-order holomorphic actions are +locally linearizable, so this model applies. + +All centers just used are closed and lie over $o$. The blowups and +finite coarse quotients can therefore be performed globally, +producing a projective coarse space $S_0$ which modifies +$V/\langle s\rangle$ only over $o$. The intermediate coarse spaces +may retain quotient singularities elsewhere; the preceding calculation +establishes smoothness on a neighborhood of the distinguished fiber. +To form $\cS$, use this smooth scheme near that fiber and use +$[V/\langle s\rangle]$ away from it. This is an algebraic gluing. +More explicitly, in $M$ remove all components of the nontrivial fixed +loci other than the isolated point $o$, and take the resulting invariant +Zariski neighborhood. Outside $o$ in this neighborhood the action is +free, so both constructions restrict to the same ordinary quotient. +Their gluing is a smooth connected effective orbifold $\cS$ with +coarse space $S_0$. The map $\cS\to S_0$ is separated and proper: +these properties are local on $S_0$, where it is either the coarse map of a finite-group +quotient stack or the identity. Since $S_0$ is projective, +$\cS$ is separated and proper over $\C$. In particular the whole distinguished +fiber lies in the ordinary smooth locus of $\cS$. + +\subsection{The surface mapping into the ordinary fiber}\label{geo:surface} + +Let $A=\Jac(C)$. For a genus-two curve, the Abel map +\[ + \Sym^2C\longrightarrow\Pic^2(C)\simeq A +\] +is the blowup at the canonical class, after choosing the displayed +translation. To recall the geometry, Riemann--Roch gives a unique +effective divisor in every degree-two class except $K_C$; the latter +has the pencil $E_C=|K_C|\simeq\PP^1$. For the degree-two quotient +$q:C\times C\to\Sym^2 C$, the inverse image of $E_C$ is the graph +$\Gamma_\iota$ of the hyperelliptic involution. The map $q$ is generically +unramified along this graph, so $q^*E_C=\Gamma_\iota$ as divisors; +ramification at its six diagonal points does not change the generic +multiplicity. The projection formula gives +\[ + 2E_C^2=(q^*E_C)^2=\Gamma_\iota^2=2-2g(C)=-2. +\] +The Abel map is therefore a birational +morphism between smooth surfaces with one exceptional $(-1)$-curve, +and is the stated point blowup. + +Choose a nonzero rational section $\tau$ of $L$. On the open subset +of $C\times C$ where both values are defined and nonzero, put +\begin{equation}\label{geo:rational-section} + (p,q)\longmapsto[\tau(p),i\tau(q)]\in P. +\end{equation} +This section is equivariant for swapping the two factors and the +involution on $P$. Indeed \eqref{geo:line-action} gives +\[ + [\tau(p),i\tau(q)]\longmapsto[i\tau(q),-\tau(p)] + =[\tau(q),i\tau(p)]. +\] +Away from the diagonal it avoids the two fixed curves, and thus gives +a rational map from $\Sym^2C$ to the smooth resolution constructed +above. All the bundles and maps in this description are algebraic: +the normal-bundle identification is algebraic as well as analytic, +by GAGA on the projective fiber. Resolve the rational map by a finite +sequence of point blowups of its smooth projective source, using the +projective coarse space as target. The resolved image remains in its +closed distinguished fiber, where the coarse space agrees with the +ordinary locus of $\cS$. We therefore obtain a morphism +\begin{equation}\label{geo:Ymap} + f:Y\longrightarrow\cS +\end{equation} +whose image lies entirely over $o$. Thus $Y$ is an iterated point +blowup of the simple abelian surface $A$, and $f(Y)$ lies in the +ordinary locus. + +\begin{proposition}[The geometric construction]\label{prop:orbifold-surface} +After any prescribed finite lower bounds on the arrangement radii and +injectivity radius, the construction above gives a smooth connected +proper effective orbifold $\cS$ with projective coarse space, a simple +abelian surface $A=\Jac(C)$, an iterated point blowup $Y\to A$, and a +morphism $f:Y\to\cS$ with image in its ordinary locus. The distinguished +fiber is obtained from the product $C\times C$, its normal bundle +$\operatorname{pr}_1^*L\oplus\operatorname{pr}_2^*L$, and the action +\eqref{geo:line-action} by the explicit blowups and coarse quotients +above. Away from that fiber the orbifold is $[V/\langle s\rangle]$, +with $V$ given by the two parity covers and the filling rules of +Section~\ref{geo:fill}. +\end{proposition} + +All parameters used in the finite arrangement have now been fixed. +The depth of $\Gamma_0$ has not: Proposition~\ref{prop:arrangement-local} +permits its choice after the constants required for the finite-model +graphs in the next section. Passing to the finite covers which produce +the signs preserves these estimates. This order of choices will +allow the local calculations to establish +\[ + \im(f_*:\pi_1(Y)\to\pi_1(\cS))\simeq\Z. +\] diff --git a/preprints/A-projective-fourfold-with-large-fundamental-group-and-non-Stein-universal-cover-October-5-2026/build/sections/05-models.tex b/preprints/A-projective-fourfold-with-large-fundamental-group-and-non-Stein-universal-cover-October-5-2026/build/sections/05-models.tex new file mode 100644 index 0000000000000000000000000000000000000000..d7203bd262be463058b6d38422c5f403f0feaa49 --- /dev/null +++ b/preprints/A-projective-fourfold-with-large-fundamental-group-and-non-Stein-universal-cover-October-5-2026/build/sections/05-models.tex @@ -0,0 +1,699 @@ +\section{Local covers and hyperbolic path graphs}\label{sec:models} + +The geometric construction gives relations among the loops in the +surface. To show that these relations leave an element of infinite +order, we will read paths in auxiliary covers of finite arrangements. +This section constructs those covers and proves the two properties +needed for Lemma~\ref{thm:path-lemma}: their kernels agree on sufficiently +buffered overlaps, and their path graphs are uniformly hyperbolic. +All constants in this section depend only on the finite arrangement +$\cD$. In particular, they are fixed before choosing the arithmetic +level. + +\subsection{Dihedral labels on a finite arrangement} + +For a subset $T\subseteq\cD$, set +\[ + U_T=\BB\setminus\bigcup_{H\in T}H. +\] +There are characters $\epsilon_z,\epsilon_w:\pi_1(U_T)\to\Z/2$, +with $\epsilon_z$ equal to one on a meridian of a $z$ hyperplane and +zero on a meridian of a $w$ hyperplane, and conversely for +$\epsilon_w$. One can define each character by the parity of the +winding number of the product of defining affine equations of the +hyperplanes in its family. Let $E_T\to U_T$ denote the associated +four-sheet cover; it may be disconnected. These signs restrict +canonically to any ball patch, up to a choice of sheets. Indeed, +meridians normally generate the complement group in a simply connected +ball, as is seen by filling a loop with a disk transverse to the +hypersurfaces. Thus their values determine a sign character there. + +We refine these signs using the infinite dihedral group +\[ + D_\infty=\langle r,b\mid r^2=b^2=1\rangle, + \qquad \epsilon(r)=\epsilon(b)=1. +\] +Its even subgroup is the infinite cyclic translation subgroup +$\langle rb\rangle$. Conjugation in $D_\infty$ preserves this subgroup +and changes its integer coordinate at most by a sign. + +We first construct $q_z:\pi_1(U_T)\to D_\infty$. The construction +ignores all $w$ hyperplanes. If $T$ contains fewer than all six $z$ +slope hyperplanes, define +\[ + q_z(\gamma)=r^{\epsilon_z(\gamma)}. +\] +Suppose instead that the whole $z$ pencil is present. With only its +six hyperplanes removed, projection to the direction +$\lambda=z_2/z_1$ gives a map to the six-punctured projective line. +Choose its standard meridians so that the four red punctures occur +first, based together in a small red disk, and the two blue punctures +occur last. Assign $r$ to each red meridian and $b$ to each blue +meridian. The sphere relation is respected because $r^4b^2=1$. +We fix this choice of the direction-line representation once, and use +the same choice for every complete pencil and for both families. + +It remains to add the coupling hyperplanes that belong to $T$. +They lie over the red-direction disk: on $z_1=\pm c$ in the ball, +\[ + |\lambda|<0.7. +\] +Over $|\lambda|<1.5$, the pencil local system has a reduction of +structure group to $\langle r\rangle$. In this reduction, multiply +its transition functions by the two-sheet systems defined, for the +coupling hyperplanes that are present, by +\begin{equation}\label{mod:coupling-functions} + (1-z_1/c)^{1/2},\qquad (1+z_1/c)^{1/2}. +\end{equation} +This means adding their winding parities to the exponent of $r$. +Over $|\lambda|>1$ one has $|z_1|0$ there is $B_1(R)$ with the following property. +In any finite model with a complete $z$ pencil, if a loop in the +sign cover projects into $B(x,R)$ and has nonzero $z$ translation, +then +\[ + \dist(x,L_z)\le B_1(R). +\] +The analogous assertion holds for $w$. The same bounds hold if +the loops must avoid any additional analytic subsets. +\end{lemma} +\begin{proof} +If the ball meets at most one hyperplane of the relevant family, +its complement for that family has trivial or cyclic fundamental +group generated by the one meridian. One way to see the latter +assertion is to use the convexity of the ball and a transverse +complex coordinate for the hyperplane: its projection has an +ellipse as image, and the centers of the convex fibers give a +continuous section. The complement thus has the homotopy type +of a punctured ellipse. Thus its dihedral image lies in a +conjugate of an order-two subgroup. It has zero translation on +the sign lift. Omitting hyperplanes from the other family does +not alter this conclusion, because the representation factors +through the complement of the relevant family. + +If two distinct slope hyperplanes meet $B(x,R)$, take a unit +negative homogeneous vector representing $x$. Its inner products +with their fixed positive normals are bounded in terms of $R$: +for a normal $n$ the normalized absolute inner product is +$\sinh\dist(x,H_n)$. The two normals form a basis of the positive +normal plane of $L_z$. Their inner-product bounds therefore bound +the positive projection of $x$ onto that plane, and hence bound +$\dist(x,L_z)$. There are only finitely many pairs of normals, +so this bound is uniform. + +For the remaining case, suppose toward a contradiction that there +are balls supporting nonzero translation whose centers leave every +bounded neighborhood of $L_z$. After passing to a subsequence, +at least one fixed coupling hyperplane meets each ball: otherwise +the preceding two cases apply. The centers tend to a boundary +point $\xi$, and a fixed-radius hyperbolic ball converges to the +same boundary point as its center. Thus $\xi$ belongs to the +closure of that coupling hyperplane. The closures of the two +coupling hyperplanes are disjoint. At $\xi$ the direction coordinate +is defined and lies strictly inside the red disk, since $z_1=\pm c$ +and the boundary estimate is uniform: +\[ + |z_2/z_1|\le\frac{\sqrt{a-c^2}}{c} + <\frac{\sqrt{1-0.85^2}}{0.85}<0.62<0.7. +\] +Eventually the whole ball lies over this red disk and misses the +other coupling hyperplane. Its label system consequently reduces +to $\langle r\rangle$, contradicting the assumed nonzero +translation. This proves the bound. Restricting the class of +allowed loops does not weaken it. +\end{proof} + +\begin{proposition}[Compatibility of kernels]\label{prop:kernel-compatibility} +For every $R>0$ there are constants $B(R)>R$ and $J_0(R)$, depending +only on $\cD$, with the following properties. +\begin{enumerate} +\item Suppose two finite-model descriptions, with their prescribed +normal matches, agree on all hyperplanes meeting $B(x,B(R))$. +Identify their sign covers there by a fixed choice of sheets. +On loops in this sign cover that project into $B(x,R)$, their +individual $z$ kernels agree and their individual $w$ kernels +agree. More precisely, for each family the corresponding integer +homomorphisms agree up to sign. +The descriptions may include additional hyperplanes outside the +buffered ball. +\item In the full model, if $\dist(x,o)>J_0(R)$, then on such loops +\[ + \ker(\ell_z+\ell_w)=\ker\ell_z\cap\ker\ell_w. +\] +\end{enumerate} +Both statements remain valid after restricting the loops to avoid +any further analytic subsets. The first statement is equivariant +under the symmetry interchanging the families. +\end{proposition} +\begin{proof} +Choose $B(R)>R+B_1(R)+1$, enlarging it if needed to include the +corresponding bound for both families. Use the subset of hyperplanes +visible in the buffered ball as an intermediate description. +If a larger description has an incomplete pencil, so does this +subset, and both corresponding translation labels vanish. +If the larger pencil is complete but the buffered pencil is not, +a nonzero translation on an $R$-ball would, by +Lemma~\ref{mod:buffer-bound}, put its center within $B_1(R)$ of the +common pencil intersection. Since every slope hyperplane contains +that intersection, all six slopes would then meet the buffered +ball, a contradiction. Thus again both labels vanish on these +loops. If both pencils are complete, the normal match fixes the +direction-line system, and extra coupling hyperplanes can be +omitted by the square-root trivialization just proved. The only +remaining change is conjugation of a dihedral label, which changes +a translation coordinate at most by sign. This proves the first +assertion, including its equivariance and its compatibility with +the chosen sign sheets. + +The sets +\[ + \{x:\dist(x,L_z)\le B_1(R)\},\qquad + \{x:\dist(x,L_w)\le B_1(R)\} +\] +have bounded intersection. Indeed, in homogeneous coordinates +the two normal planes together form the positive four-dimensional +space. Bounds on both positive projections bound the distance +from $o$. Choose $J_0(R)$ beyond that intersection. Outside it, +at least one of $\ell_z,\ell_w$ vanishes on every loop over +$B(x,R)$, by Lemma~\ref{mod:buffer-bound}. The displayed equality +of kernels follows. Every argument concerns the same individual +loops before and after imposing additional omissions, so those +omissions preserve the conclusions. +\end{proof} + +\subsection{Graphs defined by projected diameter} + +We now turn the local covers into graphs to which the path lemma +applies. Over each connected component of $E_T$, let +$\widehat E_T$ be the connected regular cover defined by +$\ker\ell_z\cap\ker\ell_w$. For the full arrangement we also use +the cover defined by $\ker(\ell_z+\ell_w)$ and its lifted-$s$ +quotient. Restrict all these covers to the open arrangement +complement when defining graph vertices. + +For $u>0$, vertices are all points of the relevant covering space. +An oriented edge is a parametrized continuous path whose projection +to $\BB$ has diameter strictly less than $u$; its reverse is the +reverse path. Multiple edges are retained. In the central case +we take the quotient graph by $\widehat s$. The symmetry acts +freely on the open arrangement complement, so edges out of a +quotient vertex correspond bijectively to edges out of any chosen +representative. Denote these graphs by $\mathcal G_T(u)$, with +$\mathcal G_{\mathrm{cen}}(u)$ for the central quotient. + +The use of diameter has an important consequence. A word of +arbitrary length in loops supported in one fixed bounded region +can be one edge. It is this feature that prevents the unbounded +translation labels from producing large flat regions in the +graphs. + +\begin{proposition}[Hyperbolicity of the models]\label{prop:model-hyperbolicity} +There are $u>0$ and $\delta\ge0$, depending only on $\cD$, such that +every connected component of every $\mathcal G_T(u)$ and of +$\mathcal G_{\mathrm{cen}}(u)$ is $\delta$-hyperbolic. +The same $\delta$ can be used if an arbitrary locally finite union +of proper complex analytic subsets of $\BB$ is additionally +omitted, with invariant omissions in the central case. On the +vertices retained after such an omission, graph distances are +unchanged. +\end{proposition} + +The proof has three parts. First, paths carrying arbitrary deck +translations inside a fixed bounded region give bounded graph diameter +over every bounded radial region. Next, an arrangement-preserving +collar identifies the remaining vertices with points of a covering of +the sphere complement, together with a radial coordinate; the angular +diameter permitted by a graph edge decays exponentially with that +coordinate. Finally, comparison with the cone graph defined below +proves uniform hyperbolicity. No properness or local finiteness of the +graphs is required. + +\begin{lemma}\label{mod:bounded-core} +One can choose $u$, simultaneously for all the finite models, so +that the vertices projecting into any fixed bounded radial region +have bounded graph diameter within each connected component. +\end{lemma} +\begin{proof} +The deck group over a component of $E_T$ is a subgroup of $\Z^2$, +or of $\Z$ for the sum cover, and is finitely generated. At a +basepoint choose loops representing a finite set of its generators. +There are only finitely many sign sheets; also choose finitely +many paths joining the sheets that belong to the same component. +All these projected paths lie in a fixed bounded region. Choose +$u$ greater than its diameter, for each of the finitely many models. +An arbitrary word in the generator loops then defines a single +edge. Thus all deck translates over a basepoint have uniformly +bounded graph distance, including the finitely many sign choices. + +For a fixed radial bound, any point in the ball region can be +joined to a chosen basepoint by a uniformly bounded chain of small +balls. Within each ball, paths can avoid the finite complex +hypersurface arrangement: its complement is path connected by +general position. Choose successive junctions off the arrangement +in the overlaps. The resulting paths have uniformly bounded +edge count and lift to the covering. Any discrepancy in their +terminal sheet is corrected using the preceding generator loops. +This gives the asserted diameter bound. One can take the chain +bound uniformly, for example by covering the compact closure of +the radial region and a path to the basepoint by finitely many +small balls with connected overlap graph. No positive lower +bound on distance to the arrangement is needed. +\end{proof} + +\subsection{The arrangement at the sphere at infinity} + +Rescale the ball to unit radius and write +\[ + x=\tanh(r)\xi,\qquad |\xi|=1. +\] +Let $\Sigma$ be its unit sphere. Each affine intersection of the +hyperplanes that reaches $\Sigma$ is transverse to $\Sigma$. +Indeed its homogeneous Hermitian subspace is nondegenerate by the +arithmetic normal-space property in the proof of +Proposition~\ref{prop:arrangement-local}. More explicitly, write +an affine intersection as $a_0+V_0$ with $a_0\perp V_0$. On its +homogeneous span, the Hermitian form is +\[ + |v|^2+(|a_0|^2-1)|t|^2. +\] +Nondegeneracy excludes $|a_0|=1$, precisely the tangency case. +If the intersection reaches the sphere, then $|a_0|<1$, and the +intersection is transverse there. This also excludes intersections +supported only on the boundary. + +There is an arrangement-preserving product collar near $\Sigma$. +Here is a direct construction, including the estimate needed for +the graph metric. At each $\xi\in\Sigma$, choose a constant real +vector tangent to all affine hyperplanes through $\xi$ and with +positive outward radial component. Transversality of their +intersection supplies this vector. Use a small neighborhood +missing every other hyperplane. A partition of unity combines +these local vectors into a smooth vector field tangent to each +hyperplane and transverse to the spheres. Normalize it so that +the derivative of the Euclidean radius is one. Its flow yields, +for $r\ge r_0$ with $r_0$ sufficiently large, diffeomorphisms +between the ideal arrangement complement and the complement on +the radius-$r$ sphere. They preserve all arrangement strata. +Because the vector field is bounded on a compact collar, their +angular displacement from the original ideal point is +\begin{equation}\label{mod:collar-shift} + O(1-\tanh r)=O(e^{-2r}) + \quad\hbox{in Euclidean distance.} +\end{equation} +In the central model, average the field under $\sigma$ before +normalizing. This retains tangency and positive radial component +and makes the collar equivariant. The collar lifts to each +covering; the lifted end is a radial product with a possibly +disconnected covering $I$ of the ideal complement. + +On $\Sigma$ use the metric +\[ + d_K(\xi,\eta)=|1-\langle\xi,\eta\rangle|^{1/2}. +\] +For completeness, it satisfies the triangle inequality. If +$a=d_K(\xi,\zeta)$ and $b=d_K(\zeta,\eta)$, then +$|\xi-\zeta|\le\sqrt2a$ and +$|\zeta-\eta|\le\sqrt2b$. Expanding the Hermitian inner product +around $\zeta$ gives +\[ + |1-\langle\xi,\eta\rangle| + \le a^2+b^2+|\xi-\zeta|\,|\eta-\zeta| + \le(a+b)^2. +\] +Positivity and symmetry are immediate. In particular, +\eqref{mod:collar-shift} is an $O(e^{-r})$ displacement in $d_K$. + +The ball distance in our normalization satisfies +\begin{equation}\label{mod:ball-distance} + \cosh d(x,y)= + \frac{|1-\langle x,y\rangle|} + {\sqrt{1-|x|^2}\sqrt{1-|y|^2}}. +\end{equation} +Consequently, for points of radii $r+O(1)$, bounded hyperbolic +distance is equivalent, with uniform constants, to angular +$d_K$ distance $O(e^{-r})$. To check both implications, write +$x=\rho\xi$, $y=\tau\eta$, where $\rho=\tanh r$ and +$\tau=\tanh r'$. Then +\[ + 1-\langle x,y\rangle + =1-\rho\tau+\rho\tau(1-\langle\xi,\eta\rangle). +\] +The second parenthesis has nonnegative real part. Thus the +absolute value is at least $\rho\tau d_K(\xi,\eta)^2$ and at most +$1-\rho\tau+d_K(\xi,\eta)^2$. For $|r-r'|$ bounded and both +radii large, the denominator in \eqref{mod:ball-distance} and +$1-\rho\tau$ are comparable to $e^{-2r}$. These inequalities +prove the assertion. Conversely a bounded ball distance bounds +$|r-r'|$ by the triangle inequality for radial distance, so no +separate radial hypothesis is needed in that direction. + +On the lifted ideal complement $I$, define +\begin{equation}\label{mod:ideal-metric} + d_I(\xi',\eta')= + \min\left\{1,\inf_\alpha\diam_{d_K}(\pi\alpha)\right\}, +\end{equation} +where $\alpha$ runs over paths in $I$ joining $\xi'$ to $\eta'$ +and $\pi$ is projection to $\Sigma$. Put $d_I=1$ for points in +different components. The triangle inequality follows by +concatenation, because the projected paths meet and the diameter +of their union is at most the sum of their diameters. Distinct +projected endpoints give positivity directly. For distinct lifts +of the same endpoint, take a small evenly covered neighborhood +in the ideal complement. A path changing the lift must leave +this neighborhood, so its projected diameter has a positive +lower bound. Therefore $d_I$ is a metric, bounded by one. + +\subsection{A cone over an arbitrary bounded metric space} + +We isolate the metric calculation, a discrete form of the familiar +hyperbolic-cone construction; compare \cite[Section~7]{BonkSchramm2000}. +We give the proof for arbitrary bounded metric spaces, as no +compactness hypothesis is available here. For a metric space $(I,d_I)$ +with $d_I\le1$, let $\mathcal C(I)$ have vertices $(\xi',t)$, +$\xi'\in I$, $t\in\Z_{\ge0}$. Add vertical unit edges between +successive heights at the same point, and horizontal unit edges +at height $t$ whenever $d_I(\xi',\eta')\le e^{-t}$. +In particular all vertices at height zero are pairwise adjacent. + +\begin{lemma}\label{mod:cone} +The graph $\mathcal C(I)$ is hyperbolic with an absolute constant, +independent of the bounded metric space $I$. More precisely, put +\[ + j=\min\{t,t',-\log d_I(\xi',\eta')\}, + \qquad -\log0=+\infty. +\] +Its distance $D$ satisfies +\begin{equation}\label{mod:cone-distance} + t+t'-2j\le + D\bigl((\xi',t),(\eta',t')\bigr) + \le t+t'-2j+3. +\end{equation} +\end{lemma} +\begin{proof} +If $\xi'=\eta'$, both the formula without its additive error and +the claim are immediate from vertical distance. Otherwise, +descend to height $\lfloor j\rfloor$, take one horizontal edge, +and ascend. This gives the upper bound. + +For the lower bound, let any connecting path have minimum height +$m$ and $n\ge1$ horizontal edges. The number of its vertical +edges is at least $t+t'-2m$. By the triangle inequality in $I$, +\[ + d_I(\xi',\eta')\le n e^{-m}. +\] +Since also $m\le t,t'$, this implies $m\le j+\log n$. +The path length is therefore at least +\[ + t+t'-2j+n-2\log n\ge t+t'-2j. +\] +The last inequality holds for every integer $n\ge1$ (indeed its +minimum over positive real $n$ is $2-2\log2>0$). + +To deduce hyperbolicity, fix a height-zero basepoint $p$. +Its distance to $(\xi',t)$ differs from $t$ by at most one. +It follows from \eqref{mod:cone-distance} that the Gromov product +of two vertices differs by at most $3/2$ from their corresponding +$j$ value. For three base points, the metric triangle inequality +implies +\[ + -\log d_I(\xi',\eta') + \ge\min\{-\log d_I(\xi',\zeta'), + -\log d_I(\zeta',\eta')\}-\log2. +\] +Taking the minimum with the relevant heights preserves this +inequality. Thus the Gromov products satisfy the hyperbolicity +inequality with an absolute additive constant. The standard +Gromov-product characterization of hyperbolicity for graphs +completes the proof; see \cite[Chapter~III.H]{BridsonHaefliger}. +\end{proof} + +\subsection{Comparison with the model graphs} + +We can now complete the proof of +Proposition~\ref{prop:model-hyperbolicity}. Work first before the +finite symmetry quotient, in a connected component. Using the +collar coordinates, send a cone vertex $(\xi',t)$ to the covering +point of radius $r_0+t$ above its ideal coordinate. A horizontal +edge at positive height is represented by a path whose projected +ideal diameter is at most $2e^{-t}$: the factor two allows the +infimum in \eqref{mod:ideal-metric} to be unattained. Place that +path at radius $r_0+t$ with the collar. The angular estimates and +\eqref{mod:ball-distance} give a uniform bound on its projected +ball diameter. Vertical edges likewise have uniformly bounded +projected diameter. Enlarge $u$ to exceed these bounds strictly. +At height zero, even when ideal components are different, +Lemma~\ref{mod:bounded-core} gives a uniform bound on the graph +distance of the two images. The cone map therefore sends adjacent +vertices to pairs at uniformly bounded graph distance. + +In the opposite direction, round a point of radius at least $r_0$ +to the nearest integer layer, retaining its lifted ideal coordinate. +Map the bounded radial part to one fixed height-zero vertex. +The two maps are coarse inverses. Radial rounding has uniformly +bounded graph displacement, and the bounded-core lemma handles +all other points. To verify that this inverse also sends edges +to pairs of bounded distance, consider an edge whose projection +lies sufficiently far out. If one endpoint has radius $r$, its +whole projected path has radii in $[r-u,r+u]$. Formula +\eqref{mod:ball-distance} bounds the angular diameter of this path +by $O_u(e^{-r})$. The collar changes this by another term of the +same order. Following the lifted path in the collar consequently +gives +\[ + d_I(\xi',\eta')\le C_u e^{-r}. +\] +Its rounded heights differ by a bounded amount, so +\eqref{mod:cone-distance} bounds their cone distance uniformly. +If an edge meets the bounded radial part, both endpoints have +bounded radius and the same conclusion follows through height +zero. Hence the two graphs are quasi-isometric. Hyperbolicity +of the cone and quasi-isometry invariance for geodesic spaces +\cite[Chapter~III.H]{BridsonHaefliger} prove hyperbolicity of the +model graph. + +For the central model, the collar and the comparison maps on the +end are equivariant under $\widehat s$. The lifted action preserves +$d_I$, since its projection is unitary and hence preserves $d_K$. +For the finite group $H=\langle\widehat s\rangle$, the formula +\[ + \overline d_I(H\xi',H\eta') + =\min_{h\in H}d_I(\xi',h\eta') +\] +defines a metric on $I/H$. Positivity uses finiteness of $H$, and +the triangle inequality follows by translating and concatenating +minimizers. The quotient of the cone graph is the cone graph of this quotient +metric, possibly with multiple edges and loops, which do not affect +vertex distances. +The preceding comparison thus descends, while the quotient radial +core remains bounded. Lemma~\ref{mod:cone} proves hyperbolicity +also in the central case. There are only finitely many subsets +$T$ and one central model, so one choice of $u$ and one +hyperbolicity constant work for all of them. + +Finally, let an additional locally finite union of proper complex +analytic subsets be omitted. Removing vertices and restricting +edges cannot decrease distances. For the reverse inequality, +take a finite edge path whose endpoints are retained. Perturb +its finitely many junctions and constituent paths, relative to +the endpoints, to miss the omitted loci. General position permits +this because proper complex analytic subsets have real codimension +at least two. Locally finite omissions cause no extra difficulty: +a compact path meets only finitely many members, and the +perturbation can be made in finitely many coordinate neighborhoods. +Make this perturbation as a homotopy of the whole concatenated +chain in the existing arrangement complement, relative to its two +overall endpoints. Homotopy lifting preserves those two endpoints +in the cover; the intermediate junctions move coherently. Each +original projected path has diameter strictly smaller than $u$, so a +sufficiently small perturbation preserves that strict inequality. +The perturbed path has exactly the same number of edges. Thus +all distances between retained vertices are unchanged. For a +central quotient edge path, lift the whole chain to the cover, perturb it relative to its overall endpoints, and project +back; invariance of the omitted set ensures that the projected +chain avoids it. The four-point inequality for hyperbolicity +restricts to the retained vertex set with the same constant, proving the +last assertion of the proposition. + +We now fix such a value of $u$, a common hyperbolicity constant, +and a reading radius $N$ supplied by Lemma~\ref{thm:path-lemma}. +Only after these choices will we choose the arithmetic level and +the geometric sizes of the charts. diff --git a/preprints/A-projective-fourfold-with-large-fundamental-group-and-non-Stein-universal-cover-October-5-2026/build/sections/06-image.tex b/preprints/A-projective-fourfold-with-large-fundamental-group-and-non-Stein-universal-cover-October-5-2026/build/sections/06-image.tex new file mode 100644 index 0000000000000000000000000000000000000000..cd47a90422558200927134d6fe98f4a75797ebac --- /dev/null +++ b/preprints/A-projective-fourfold-with-large-fundamental-group-and-non-Stein-universal-cover-October-5-2026/build/sections/06-image.tex @@ -0,0 +1,577 @@ +\section{An infinite cyclic image}\label{sec:image} + +We now compute the image of the surface constructed in +Proposition~\ref{prop:orbifold-surface}. The relations supplied by a +coupling hyperplane show that this image is cyclic. Proving that the +cyclic group is infinite requires a separate argument: the translation +labels of the preceding section exist in charts, and need not define a +homomorphism on the fundamental group of the whole orbifold. + +\begin{theorem}\label{thm:orbifold-cyclic} +For a sufficiently deep choice of the arithmetic level in the +construction of Proposition~\ref{prop:orbifold-surface}, the map +$f:Y\longrightarrow\cS$ satisfies +\[ + \im\bigl(f_*:\pi_1(Y)\longrightarrow\pi_1(\cS)\bigr)\cong\Z. +\] +Here $Y$ is an iterated point blowup of the simple abelian surface +$\Jac(C)$, and its image lies in the ordinary locus of $\cS$. +\end{theorem} + +Throughout the proof, fundamental groups are based at points in the +ordinary locus. Changes of basepoint are made along specified common +paths; in particular, the comparisons below concern homomorphisms with +one common conjugation, rather than separate conjugations of individual +generators. + +\subsection{Surface loops and the cyclic upper bound} + +Recall the distinguished fiber $F_*=C\times C$ in $V$, the order-four +lift $s$, and the rational section used to construct $Y$ in +Section~\ref{geo:surface}. Put +\[ + \mathcal K=[V/\langle s\rangle]. +\] +There is a natural homomorphism +\begin{equation}\label{app:comparison-map} + \pi_1(\mathcal K)\longrightarrow\pi_1(\cS). +\end{equation} +Indeed, the substack $[F_*/\langle s\rangle]$ has complex codimension +two in the smooth stack $\mathcal K$. Removing it does not change the +orbifold fundamental group, and its complement is the open substack on +which the construction of $\cS$ leaves $\mathcal K$ unchanged. + +Choose a general point $q_0\in C$. The map +\[ + C\longrightarrow\Sym^2 C\longrightarrow\Jac(C), + \qquad p\longmapsto p+q_0, +\] +is, up to translation, an Abel map. Its map on fundamental groups +surjects onto $\pi_1(\Jac(C))=H_1(C,\Z)$. Since point blowups preserve +fundamental groups, loops from $C\times\{q_0\}$ supply generators of +$\pi_1(Y)$. They may be represented by a finite bouquet avoiding the +branch points, $p=q_0$, the zeros and poles of the rational section, +and the finitely many points where the rational map or its resolution +must be avoided. Removing finitely many points from a smooth curve is +surjective on fundamental groups, so these restrictions do not lose any +generators. + +\begin{lemma}\label{app:surface-comparison} +With common choices of basing, the images in $\pi_1(\cS)$ of the +preceding generators of $\pi_1(Y)$ equal the images, under +\eqref{app:comparison-map}, of the corresponding loops in +$C\times\{q_0\}\subset V$. The loops can also be represented in the +sign-covered arrangement complement, arbitrarily close to $F_*$, with +both normal coordinates nonzero. +\end{lemma} + +\begin{proof} +Lift the bouquet by the rational section to the exceptional divisor of +$\operatorname{Bl}_{F_*}V$. Over this bouquet the section has the form +\[ + [\tau(p),\sqrt{-1}\,\tau(q_0)], +\] +with both entries nonzero. It avoids the fixed curves involved in the +last resolution step, because the bouquet avoids the diagonal of +$C\times C$. + +A complex line bundle on a finite graph is topologically trivial. +Thus the normal line to this exceptional divisor admits one continuous +nonvanishing section over the entire lifted bouquet. In the product normal coordinates, this push can be written +explicitly as +\[ + \varepsilon\bigl(\tau(p),\sqrt{-1}\,\tau(q_0)\bigr). +\] +One sufficiently small $\varepsilon>0$ works over the compact bouquet +and gives a closed based bouquet off the exceptional divisor. On blowing down to $V$, the pushed bouquet is +homotopic to the original one in $F_*$. On passing to the resolution +neighborhood in $\cS$, it is homotopic to the bouquet obtained from +$Y$. These homotopies use the same push of the common basepoint. + +The two components of the displayed section are nonzero, and the +chosen directions avoid the marked slopes. Taking the push small +therefore puts its projection off the arrangement. The distinguished +point is an isolated component of the fixed locus of every nonidentity +power of $\sigma$ in a sufficiently small neighborhood. Hence the +pushed bouquet can also avoid the inverse images of all those fixed +loci. This proves all the assertions, including the common basing. +\end{proof} + +We next compute an upper bound before taking the quotient by $s$. +Write $H=(\Z/2)^2$ for the sign deck group. The map +\[ + V\longrightarrow[V/H] +\] +is an orbifold covering, so it induces an injection of $\pi_1(V)$ into +$\pi_1([V/H])$. This remains true over branch points: the target is the +quotient stack, which retains their isotropy groups. + +The pencil through $L_z$ gives the orbifold line +$[C/(\Z/2)]$, with six order-two points. Number the four red points +first. Its fundamental group has the presentation +\begin{equation}\label{app:pencil-group} + Q=\langle h_1,\ldots,h_6\mid + h_i^2=1,\ h_1h_2h_3h_4h_5h_6=1\rangle. +\end{equation} +The kernel of the parity character sending every $h_i$ to $1\in\Z/2$ +is $\pi_1(C)$. With the $w$ sign point fixed over a general direction, +the map of this orbifold pencil to $[V/H]$ identifies the homomorphism +from this even subgroup with the homomorphism from +$\pi_1(C\times\{q_0\})$ to $\pi_1(V)$, followed by the injection just +described. + +\begin{lemma}\label{app:cyclic-upper} +The image of $\pi_1(C\times\{q_0\})$ in $\pi_1(V)$ is cyclic. +Consequently the image of $\pi_1(Y)$ in $\pi_1(\cS)$ is cyclic. +\end{lemma} + +\begin{proof} +We show that the four red meridians in \eqref{app:pencil-group} have +the same image in $\pi_1([V/H])$. First move the $w$ coordinate a little +off its exceptional direction line, keeping it fixed, generic, and +small. Push the red pencil meridians off the $z$ exceptional divisor. +They can be represented with +\[ + t=z_1\ne0\quad\hbox{fixed and small},\qquad + \lambda=z_2/z_1 +\] +and with the red meridians based by paths in a small disk containing +exactly the four red values of $\lambda$. + +Slide this calculation in the $t$ variable to the coupling hyperplane +$t=c$. The disk of red directions can be chosen small enough that +this entire motion, with the chosen small $w$, remains inside the +ball: the red slopes have modulus less than $0.01$, whereas +$c/\sqrt a<0.9$. The motion and the finitely many paths used here lie +in a fixed compact part of the full finite model. We include this +compact set among the regions that must agree with the actual +arrangement when the arithmetic level is chosen. + +Before blowing up the pair intersections, the relevant local divisor +is +\[ + \{t=c\}\ \cup\ \bigcup_{i=1}^4\{\lambda=\lambda_i\}. +\] +Let $k$ be the meridian in the $t$ variable, with one common choice of +basing. The product structure makes $k$ commute with each red +meridian. More precisely, the same $t$ circle can be transported along +each of the based paths in the direction disk; this gives one element +$k$, not four unrelated conjugates. + +Blowing up $\{t=c,\lambda=\lambda_i\}$ introduces an exceptional +divisor whose meridian is $kh_i$: in the two transverse coordinates, +the exceptional meridian circles both coordinates once. The filling +rules of Section~\ref{geo:fill} impose no branching along this +exceptional divisor, since two branch components met there. Thus its +meridian is killed. Therefore +\[ + kh_i=1\qquad(1\le i\le4). +\] +The strict transforms retain order two, so $k^2=h_i^2=1$. All four +red images are consequently one involution, denoted $r$. + +The product relation in \eqref{app:pencil-group} now gives +$h_5h_6=1$, and the two blue images are one involution $b$. The image +of $Q$ is thus a quotient of +\[ + \langle r,b\mid r^2=b^2=1\rangle. +\] +Every even word in $r,b$ is a power of $rb$. Hence the image of the +even subgroup $\pi_1(C)$ is cyclic. Injectivity of +$\pi_1(V)\to\pi_1([V/H])$ gives the assertion in $\pi_1(V)$. +Lemma~\ref{app:surface-comparison} and the surjectivity from the chosen +$C$ loops onto $\pi_1(Y)$ give the final assertion. +\end{proof} + +This argument allows the cyclic image to be finite. We will exclude +that possibility by applying Lemma~\ref{thm:path-lemma} to paths in a +dense open subset of $\cS$. + +\subsection{Paths in the ordinary open set} + +Let $O\subset\cS$ be the sign-covered arrangement complement, +quotiented by $s$, after also removing the full inverse image of the +fixed loci in $M$ of all nonidentity powers of $\sigma$. The +modifications defining $\cS$ are isomorphisms over this set. In +particular $O$ is a dense connected smooth variety, and all quotient +actions used over it are free. + +There is a useful space for specifying lifts. Over the complement in +$\BB$ of the lifted arrangement and the lifted omitted fixed loci, +pull back the sign data from $V$, and call the resulting space +$O^\#$. Then +\[ + O^\#\longrightarrow O +\] +is an ordinary covering. Changes of lift are generated by the ball +deck group $\Gamma$ and by $\sigma$, the latter acting on the sign data +through $s$. These are precisely the transformations used to form +$O$; we do not divide out the two sign deck transformations. The group generated by $\Gamma$ and $\sigma$ acts freely on the +retained ball region. + +Fix the diameter bound $u$ supplied by +Proposition~\ref{prop:model-hyperbolicity}. Define a graph $G$ whose +vertices are the points of $O$. An edge is a parametrized continuous +path in $O$ whose lifted projection to $\BB$ has diameter strictly +less than $u$. All graphs use the same parametrization and reversal +conventions. The diameter condition is independent of the lift, +because changes of lift act by ball isometries. No bound is imposed +on path length or winding. In particular, arbitrarily high powers of +a loop can be individual edges when their projections stay in one +small region. + +\subsection{Choosing and comparing the charts} + +We make the order of the parameters explicit. First fix $u$, then a +common hyperbolicity constant for the model graphs, and then the +integer $N$ required by Lemma~\ref{thm:path-lemma}. Set +\begin{equation}\label{app:radius} + R=(2N+100)(u+1). +\end{equation} +A lift of a graph path with at most $N$ edges stays within distance +$Nu$ of its initial ball point, regardless of the lengths of the +individual parametrized paths. + +The special points of $\BB$ are the points of $\Gamma o$, the lifts +of the distinguished point of $M$. Choose $J>10R$ so large that, in +the central model, both of the following hold at every point of +distance at least $J-3R$ from $o$: +\begin{enumerate} +\item On sign loops projected into the $R$ ball about that point, the +kernel of the sum translation equals the intersection of the two +individual translation kernels. +\item For $j=1,2,3$, the displacement by $\sigma^j$ exceeds $10R$. +\end{enumerate} +The first requirement follows from +Proposition~\ref{prop:kernel-compatibility}. For the second, none of +$\sigma,\sigma^2,\sigma^3$ has a boundary fixed point. The ball +distance formula then implies that its displacement tends uniformly +to infinity as the viewpoint tends to the boundary. + +Let $B(R)$ be the buffer radius in +Proposition~\ref{prop:kernel-compatibility}, enlarged so that +$B(R)>R$, and choose +\begin{equation}\label{app:chart-radius} + K>10\bigl(J+R+B(R)\bigr). +\end{equation} +All these constants depend only on the finite models. Now take the +initial arithmetic level deep enough that +Proposition~\ref{prop:arrangement-local} holds on balls of radius +$10K$, that such balls inject into the torsion-free ball quotient, +that the full finite arrangement agrees with the lifted arrangement +in these neighborhoods of every special point, and that distinct +special points have distance greater than $100K$. Enlarge this last +choice, if necessary, to include the fixed compact paths used in +Lemma~\ref{app:cyclic-upper}. Subsequent finite covers preserve all +these requirements. + +For a vertex $v\in G$, choose a lift with ball coordinate $x$ and its +actual sign data. Its chart has one of two forms. +\begin{description} +\item[Central chart.] If $\dist(x,\Gamma o)5$. Removing a subset of complex codimension at least two +from a smooth orbifold does not change its orbifold fundamental group: +paths and their homotopies can be moved off that subset in smooth +charts, or in a smooth frame presentation. Also, the projective-space +bundle $\mathcal P\to\mathcal T$ induces an isomorphism on fundamental +groups, since its fibers are simply connected. Thus the natural maps +give +\begin{equation}\label{proj:ordinary-group} + \pi_1(U)\simeq\pi_1(\mathcal P)\simeq\pi_1(\mathcal T). +\end{equation} + +It remains to cut down inside $U$ while retaining the embedded surface. +Put $N=\dim P$. In a sufficiently high projective embedding of $P$, +take $N-5$ general hyperplane sections through $Y$. Their linear systems +are free off $Y$ and generate the normal differentials along $Y$. +The locus where $r$ general normal differentials fail to be independent +has expected codimension +\begin{equation}\label{proj:rank-loss} + (N-2)-r+1 +\end{equation} +on the surface. For $r=N-5$ this is $4>2$, so the final intersection +is smooth along $Y$. Bertini gives smoothness elsewhere in $U$. +Since $P\setminus U$ has codimension greater than five and is disjoint +from $Y$, a general constrained intersection avoids it. We obtain a +smooth projective fivefold $S\subset U$ containing $Y$. + +To verify the assertion about $\pi_1$, first take unconstrained general +hyperplane sections in this same embedding. The quasi-projective +Lefschetz theorem of Hamm and L\^e +\cite[Theorem~1.1.3(ii)]{HammLe} says that a smooth open variety of +complex dimension $n$ is obtained, up to homotopy, from a general +hyperplane section by attaching cells of dimensions at least $n$. +At every stage here the dimension before cutting is at least six. +The successive inclusions therefore preserve connectedness and induce +isomorphisms on fundamental groups. The final intersection avoids +$P\setminus U$, by the same dimension count. + +Now consider the full product of the hyperplane parameter spaces, +without a containment requirement. The tuples whose intersections +have dimension five, are smooth, and avoid $P\setminus U$ form a +nonempty Zariski-open set $\Omega$. It contains both a general +unconstrained tuple and the constrained tuple just constructed, and it +is path connected. The universal complete intersection over $\Omega$ +is a smooth proper family with a map to $U$. Along a path in $\Omega$, +Ehresmann's theorem identifies its fibers by diffeomorphisms and gives +a homotopy of their inclusions into $U$. Thus the isomorphism on +fundamental groups for an unconstrained fiber also holds for $S$. +Composing $S\hookrightarrow U\subset\mathcal P$ with the bundle +projection gives $h$, with $h\circ i=f$, and +\eqref{proj:ordinary-group} proves the lemma. +\end{proof} + +Applied to $Y\to\cS$, this gives a smooth connected projective ambient +variety $S$ with an embedding of $Y$ and +\begin{equation}\label{proj:blownup-image} + \im\bigl(\pi_1(Y)\longrightarrow\pi_1(S)\bigr)\simeq\Z. +\end{equation} +The remaining issue is that Proposition~\ref{prop:large-reduction} +requires $A$ itself. We address one point blowup at a time. + +\subsection{Realizing a surface blowdown in a new ambient variety} + +\begin{lemma}\label{lem:ambient-blowdown} +Let $Y_1\hookrightarrow S$ be an embedding of a smooth projective +surface in a smooth connected projective variety, and let +$b:Y_1\to Y_0$ be the blowdown of a $(-1)$-curve to a smooth projective +surface. There exist a smooth connected projective variety $S_0$, an +embedding $Y_0\hookrightarrow S_0$, and an isomorphism +$\theta:\pi_1(S)\xrightarrow{\sim}\pi_1(S_0)$ such that +\[ + \theta\circ(\pi_1(Y_1)\to\pi_1(S)) + = (\pi_1(Y_0)\to\pi_1(S_0))\circ b_*. +\] +The equality is understood with compatible basepoints, or up to +conjugation. +\end{lemma} + +The proof first places $Y_1$ in a projective threefold and adjusts the +normal bundle of its exceptional curve to +$\mathcal O(-1)\oplus\mathcal O(-1)$. Blowing up that curve gives an +exceptional $\PP^1\times\PP^1$. Contracting the other ruling then +performs $b$ on the surface. This is the classical local two-ruling +construction; compare \cite{Atiyah1958}. The essential global point +here is to construct a polarization that makes this second contraction +projective. + +\begin{proof} +Write $E\subset Y_1$ for the exceptional curve. Replace $S$ by its +product with a projective space and embed $Y_1$ by the graph of +\[ + Y_1\xrightarrow{b}Y_0\hookrightarrow\PP^r. +\] +The new ambient fundamental group is naturally $\pi_1(S)$. Let $Q$ +be the pullback of $\mathcal O_{\PP^r}(1)$; it is nef, and +$Q|_{Y_1}=b^*L_0$ for a very ample line bundle $L_0$ on $Y_0$. +Further projective-space factors allow us to make the ambient dimension +as large as convenient. + +\smallskip +\noindent\emph{A threefold containing the unchanged surface.} +Cut to a smooth projective fourfold $W$ containing $Y_1$, using high +ample degrees. For each successive cut the rank-loss count +\eqref{proj:rank-loss}, with final dimension four, is at least three; +thus it exceeds $\dim Y_1=2$. Together with Bertini, this gives smooth +successive cuts, and the Lefschetz theorem preserves the fundamental +group throughout. + +For a final threefold, the rank-loss codimension is now two, equal to +the dimension of the surface. We therefore allow isolated singularities +in the last cut and resolve them while leaving $Y_1$ unchanged. +Take one more sufficiently ample hypersurface $Z\subset W$ through +$Y_1$. On the surface, its two normal derivatives form a general +section of the rank-two bundle +$N^*_{Y_1/W}\otimes\mathcal O_W(Z)|_{Y_1}$. In sufficiently high +degree, generation of its first jets makes the zeros transverse and +isolated; since $\dim E=1$, they can also avoid $E$. Bertini gives +smoothness away from $Y_1$. At a zero choose coordinates with +$Y_1=(x=y=0)$. Writing the equation as $xa+yb=0$, transversality says +that $a,b$ restrict to coordinates on $Y_1$. Hence $x,y,u=a,v=b$ are +coordinates on $W$ and the hypersurface equation is +\begin{equation}\label{proj:node} + xu+yv=0,\qquad Y_1=(x=y=0). +\end{equation} +Thus $Z$ is a connected normal threefold with only ordinary double +points, all disjoint from $E$. + +We require the small resolution of these nodes that leaves $Y_1$ +unchanged. It exists projectively as follows. Choose a high-degree +Cartier divisor on $Z$ containing $Y_1$, general in its ideal at the +finitely many nodes. At each node its equation can be taken to be +$x$, after a change of the coordinates in \eqref{proj:node}. This +divisor is the sum of $Y_1$ and a residual effective Weil divisor $R$. +The latter is Cartier away from the nodes, and at a node its ideal is +$(x,v)$. Blow up this coherent ideal. The blowup is projective, is an +isomorphism away from the nodes, and is smooth at every node. Indeed +its two charts are +\[ + x=tv,\quad y=-tu + \qquad\text{and}\qquad + v=sx,\quad u=-sy, +\] +with coordinates $(t,u,v)$ and $(s,x,y)$, respectively. The strict +transform of $Y_1$ is $t=0$ in the first chart and misses the second +chart. It therefore maps isomorphically to the original $(u,v)$-plane. +Denote the resulting smooth projective threefold by $T$ and keep the +notation $Y_1\subset T$. + +These operations preserve the required fundamental group. For the +last cut, deform $Z$ to a smooth ample hypersurface in $W$. Away from +disjoint small neighborhoods of its nodes the family is locally +trivial. A three-dimensional ordinary double point has simply +connected link and simply connected Milnor fiber, and the small +resolution neighborhood retracts to $\PP^1$. For completeness, after +writing the node as $\sum_{j=0}^3 z_j^2=0$, its link is the space of +orthonormal two-frames in $\R^4$, hence the unit tangent bundle of +$S^3\simeq\mathrm{SU}(2)$, which is $S^3\times S^2$. Its smoothing +retracts to $S^3$; writing $z=x+iy$ identifies it with the tangent +bundle of $S^3$ before truncation. These are also the elementary +quadratic case of the Milnor-fiber theorem \cite{MilnorSingular}. +Van Kampen therefore gives the same fundamental group for the nodal +hypersurface, its smoothing, and its small resolution, compatibly with +the maps to $W$. Lefschetz for the smoothing yields +$\pi_1(T)\simeq\pi_1(W)\simeq\pi_1(S)$, with the prescribed map +from $\pi_1(Y_1)$. + +\smallskip +\noindent\emph{Adjusting the normal bundle.} +The hypersurface degree can be chosen so large that +\[ + N_{Y_1/T}|_E\simeq\mathcal O_E(-m),\qquad m\ge1. +\] +In fact there are no nodes on $E$, and the normal exact sequence there +subtracts $\deg\mathcal O_W(Z)|_E$ from the fixed degree of +$\det N_{Y_1/W}|_E$. The sequence +\[ + 0\longrightarrow\mathcal O_E(-1) + \longrightarrow N_{E/T} + \longrightarrow\mathcal O_E(-m)\longrightarrow0 +\] +splits because +$H^1(E,\mathcal O_E(m-1))=0$. Blow up the threefold along $E$. +Since $E$ is a Cartier divisor on $Y_1$, the strict transform of $Y_1$ +is isomorphic to $Y_1$. Its normal line bundle becomes +$N_{Y_1/T}\otimes\mathcal O_{Y_1}(-E)$, so its degree on $E$ increases +by one, because $E^2=-1$. Repeating $m-1$ times gives a smooth +projective threefold, still denoted $T$, in which +\[ + N_{E/T}\simeq\mathcal O_E(-1)\oplus\mathcal O_E(-1), +\] +with the first summand the tangent-normal direction inside $Y_1$. +At each step the same extension vanishing justifies this splitting. + +Blow up $E$ once more, obtaining $\widehat T$, and let $\widehat Y$ +be the strict transform of $Y_1$. The exceptional divisor is +\begin{equation}\label{proj:exceptional-quadric} + G=\PP^1_{\mathrm{curve}}\times\PP^1_{\mathrm{normal}},\qquad + N_{G/\widehat T}=\mathcal O_G(-1,-1),\qquad + \widehat Y\cap G=\PP^1_{\mathrm{curve}}\times\{q\}. +\end{equation} +The intersection is transverse. Moreover +$\mathcal O_{\widehat T}(\widehat Y)|_G=\mathcal O_G(0,1)$, and +$N_{\widehat Y/\widehat T}|_E=\mathcal O_E$. +We continue to denote by $Q$ the pullback of the earlier nef bundle; +it is trivial on $G$ and restricts on $\widehat Y\simeq Y_1$ to +$b^*L_0$. Figure~\ref{fig:blowdown} shows the two rulings and the +curve on the surface that the second contraction must remove. + +\input{figures/blowdown} + +\smallskip +\noindent\emph{A polarization for the other contraction.} +We seek an ample line bundle $H$ restricting to +$\mathcal O_G(\alpha,\beta)$, with $\beta>\alpha>0$. +Since $\mathcal O_{\widehat T}(G)|_G=\mathcal O_G(-1,-1)$, +adding $\alpha G$ will then give the restriction +$\mathcal O_G(0,\beta-\alpha)$: it is trivial on the curves to be +contracted and positive on the retained factor. +Choose an ample integral line bundle $H_0$ on $\widehat T$, with +$H_0|_G=\mathcal O_G(\alpha,\beta_0)$, where $\alpha,\beta_0>0$. +Adding $k\widehat Y$ increases the second degree by $k$ and leaves the +first unchanged. Choose $k\ge0$ with $\beta:=\beta_0+k>\alpha$. +On $\widehat Y$, the line bundle $H_0+k\widehat Y$ still has positive +degree on $E$, because $N_{\widehat Y/\widehat T}|_E$ is trivial. +It is therefore relatively ample for $b:\widehat Y\to Y_0$, whose +only positive-dimensional fiber is $E$. Adding a sufficiently large +multiple $lQ$ makes its restriction to $\widehat Y$ ample. + +We claim that +\[ + H=H_0+k\widehat Y+lQ +\] +is ample on all of $\widehat T$. Here $H_0+lQ$ is ample since $Q$ +is nef. We use the following consequence of the Nakai--Moishezon +criterion: if $A_0$ is ample, $D_0$ is an effective Cartier divisor, +$k\ge0$, and $(A_0+kD_0)|_{D_0}$ is ample, then $A_0+kD_0$ is ample. +Indeed, for an irreducible $r$-dimensional subvariety $V$ not contained +in $D_0$, the difference of top intersections is +\[ + \bigl((A_0+kD_0)^r-A_0^r\bigr)\cdot V + = kD_0\cdot V\cdot + \sum_{j=0}^{r-1}(A_0+kD_0)^j A_0^{r-1-j}\ge0. +\] +Every term on the right is a mixed intersection of ample restrictions +on the effective cycle $D_0\cap V$. For $V\subset D_0$, positivity +follows directly from the restriction hypothesis. Nakai--Moishezon +now applies. Taking $D_0=\widehat Y$ proves the claim. We have obtained +\begin{equation}\label{proj:ample-degrees} + H|_G=\mathcal O_G(\alpha,\beta),\qquad \beta>\alpha>0. +\end{equation} +The relative ampleness, Serre vanishing, and numerical ampleness facts +used here are standard projective criteria; see \cite{Hartshorne}. + +Set $D=H+\alpha G$. We shall prove directly that a high multiple of +$D$ contracts $G$ to its second factor and has smooth projective image. +For a sufficiently large integer $n$, put +$L_j=nD-jG$. Then +\[ + L_{n\alpha}=nH, + \qquad L_j|_G=\mathcal O_G\bigl(j,n(\beta-\alpha)+j\bigr). +\] +Serre vanishing gives $H^1(\widehat T,L_{n\alpha})=0$. +For $0\le j\abs{N_2^+(v)}$ for every $v\in V(D)$. +Choose $D$ with minimum order and, subject to that, minimum number of +arcs among all graphs with this property. Then $D$ is strongly connected, +every vertex has positive indegree, and +\begin{equation} +\label{eq:subset-deficit} + \abs{F^2(S)\setminus F(S)}<\abs{F(S)\setminus S} + \qquad\text{for every }\varnothing\ne S\subsetneq V(D). +\end{equation} +\end{lemma} + +\begin{proof} +A sink satisfies the second-neighborhood inequality, so $D$ has no sink +and has more than one vertex. If $D$ were not strongly connected, take +a strongly connected component with no outgoing arc to another component. +All first and second out-neighborhoods of its vertices lie in that +component. Its induced graph would therefore be a smaller counterexample, +contrary to the choice of $D$. Thus $D$ is strongly connected, and every +vertex has an in-neighbor. + +Fix a nonempty proper subset $S$ and put +\[ + I=F(S)\cap S,\qquad E=F(S)\setminus S, + \qquad g(T)=\abs{F(T)\setminus(I\cup T)}\quad(T\subseteq E). +\] +Strong connectivity gives $E\ne\varnothing$. We will enlarge $T$ from +$\varnothing$ to $E$ so that the increase in $g(T)$ is strictly smaller +than the number of newly adjoined vertices at every step. + +For $T\subsetneq E$, let $D_T$ be obtained from the original graph $D$ +by deleting all arcs from $S$ to $E\setminus T$. Since every vertex of +$E\setminus T$ receives an arc from $S$, at least one arc is deleted. +Minimality of the number of arcs gives a vertex $v$ satisfying +$\abs{N_{1,D_T}^+(v)}\le\abs{N_{2,D_T}^+(v)}$. +If $v$ had lost no first out-neighbor, deleting arcs could only shrink +its exact second out-neighborhood, so it would still violate this +inequality. Consequently $v\in S$, and the set +\[ + Q=F(\{v\})\cap(E\setminus T) +\] +is nonempty. It is exactly the set of first out-neighbors that $v$ loses. +The original first out-neighborhood of $v$ is contained in $I\cup T\cup Q$, +and the remaining first out-neighbors lie in $I\cup T$. + +We compare the exact second out-neighborhoods before and after deletion. +A two-step path from $v$ in $D_T$ through a vertex of $I\subseteq S$ +ends in $I\cup T$, since all arcs from $S$ to $E\setminus T$ were +deleted. A path through a vertex of $T$ ends in $F(T)$. Thus +\begin{equation} +\label{eq:deleted-path-endpoints} + F_{D_T}^2(\{v\})\subseteq I\cup T\cup F(T). +\end{equation} +Every newly counted exact second out-neighbor must have been a first +out-neighbor in $D$: deletion creates no paths, and $v$ itself is never +an exact second out-neighbor. Such a vertex lies in $Q$, and +\eqref{eq:deleted-path-endpoints}, together with +$Q\cap(I\cup T)=\varnothing$, places it in $Q\cap F(T)$. + +On the other hand, define +\[ + C=F(Q)\setminus\bigl(I\cup T\cup Q\cup F(T)\bigr). +\] +Every vertex of $C$ was reached from $v$ in two steps through $Q$, and +none was a first out-neighbor. Also $v\notin F(Q)$: an arc from a +vertex of $Q$ back to $v$ would oppose the arc from $v$ to that vertex. +Hence $C\subseteq N_{2,D}^+(v)$. By +\eqref{eq:deleted-path-endpoints}, no vertex of $C$ remains reachable +in two steps in $D_T$. + +Write $d_1=\abs{N_{1,D}^+(v)}$ and +$d_2=\abs{N_{2,D}^+(v)}$. The gains and losses just identified imply +\[ + d_1-\abs{Q} + \le \abs{N_{2,D_T}^+(v)} + \le d_2-\abs{C}+\abs{Q\cap F(T)}. +\] +The change in $g$ is exact: adjoining $Q$ removes the points +$Q\cap F(T)$ from the previously counted image and adds precisely $C$. +Since $d_2n$, replace each vertex by a +transitive tournament on $m$ vertices. For every arc $v\to w$ of $D$, +put all arcs from the block at $v$ to the block at $w$, and put no other +arcs between distinct blocks. The resulting graph $G$ is oriented, +all its indegrees are positive, and it satisfies +\eqref{eq:strict-subset-growth}. +\end{lemma} + +\begin{proof} +Label the vertices within each block by $1,\ldots,m$, with arcs from +smaller to larger labels. Let $f$ be the image operator within this +transitive tournament. For a nonempty set $T$ of labels with least +element $a$, we have $f(T)=\{a+1,\ldots,m\}$. Thus +\begin{equation} +\label{eq:internal-block-bound} + \abs{T}-\abs{f(T)}\le1,\qquad + f^2(T)\subseteq f(T),\qquad + \abs{T}-2\abs{f(T)}+\abs{f^2(T)}\le1. +\end{equation} +The construction produces an oriented graph. Every vertex receives +arcs from an incoming base block, so all indegrees in $G$ are positive. +Call an arc external if it joins distinct blocks, and internal otherwise. + +Fix $\varnothing\ne U\subsetneq V(G)$. Write $T_v$ for the labels +selected by $U$ in the block at $v$, and define its support and a set +of base vertices by +\[ + S=\{v\in V(D):T_v\ne\varnothing\},\qquad + H_0=F(S)\cup F^2(S). +\] +Every block indexed by $F(S)$ is filled by $F_G(U)$. In each remaining +block the first image consists exactly of $f(T_v)$, so +\[ + \abs{F_G(U)}=m\abs{F(S)} + +\sum_{v\in S\setminus F(S)}\abs{f(T_v)}. +\] +A two-step path using two external arcs ends in an $F^2(S)$-block. +One using exactly one external arc ends in an $F(S)$-block, regardless +of the order of its internal and external steps. Paths using only +internal arcs contribute $f^2(T_v)$. Therefore +\[ + \abs{F_G^2(U)}\le m\abs{H_0} + +\sum_{v\in S\setminus H_0}\abs{f^2(T_v)}. +\] +Combining these estimates gives the sharper bound +\begin{align} +\label{eq:blowup-sharp} + \abs{U}-2\abs{F_G(U)}+\abs{F_G^2(U)} + &\le m\bigl(\abs{H_0}-2\abs{F(S)}\bigr) + +\sum_{v\in S}\abs{T_v} \notag\\ + &\quad -2\sum_{v\in S\setminus F(S)}\abs{f(T_v)} + +\sum_{v\in S\setminus H_0}\abs{f^2(T_v)}. +\end{align} +Since $F(S)\subseteq H_0$, extending the last sum to +$S\setminus F(S)$ only adds nonnegative terms. This may count internal +second images in blocks already bounded by a full block in +\eqref{eq:blowup-sharp}; it remains a valid upper bound. Now use +$\abs{T_v}\le m$ on $S\cap F(S)$ and +\eqref{eq:internal-block-bound} on $S\setminus F(S)$ to obtain +\begin{align*} + \abs{U}-2\abs{F_G(U)}+\abs{F_G^2(U)} + &\le m\bigl(\abs{H_0}-2\abs{F(S)} + +\abs{S\cap F(S)}\bigr) + +\abs{S\setminus F(S)}\\ + &\le m\bigl(\abs{F^2(S)\setminus F(S)} + -\abs{F(S)\setminus S}\bigr)+n. +\end{align*} +If $S\subsetneq V(D)$, the parenthesized difference is an integer +strictly less than zero by \eqref{eq:subset-deficit}. The last bound +is consequently at most $-m+n<0$. + +It remains to consider $S=V(D)$, for which the subset-deficit hypothesis +does not apply. Positive indegrees in $D$ give $F(S)=V(D)$, so +$F_G(U)=V(G)$. Positive indegrees in $G$ then give +$F_G^2(U)=V(G)$ as well. In this case +\[ + \abs{U}-2\abs{F_G(U)}+\abs{F_G^2(U)} + =\abs{U}-\abs{V(G)}<0, +\] +because $U$ is proper. This proves the required inequality for every +allowed $U$. +\end{proof} + +\begin{proof}[Proof of Proposition~\ref{prop:counterexample-reduction}] +Choose a counterexample of minimum order and then minimum number of +arcs. Lemma~\ref{lem:subset-deficit} supplies positive indegrees and +the subset deficit, so Lemma~\ref{lem:blowup}, with any integer +$m>\abs{V(D)}$, supplies the required graph $G$. +\end{proof} diff --git a/preprints/A-proof-of-Seymours-second-neighborhood-conjecture-September-23-2026/build/source/02-pruning.tex b/preprints/A-proof-of-Seymours-second-neighborhood-conjecture-September-23-2026/build/source/02-pruning.tex new file mode 100644 index 0000000000000000000000000000000000000000..4fe3b01285d67d6fb5a5e7a8798a3c12f432ff76 --- /dev/null +++ b/preprints/A-proof-of-Seymours-second-neighborhood-conjecture-September-23-2026/build/source/02-pruning.tex @@ -0,0 +1,356 @@ +\section{Pruning two families of pairs}\label{sec:pruning} + +We now prove a deletion bound for two families of ordered pairs. The bound +charges the deleted members to two sets of pairs: one determined by the +original conflicts, and the other consisting of points left uncovered after +deletion. This dependence on the surviving families will be essential in +Section~\ref{sec:incompatibility}. The result holds for every finite binary +relation; no orientation assumption is needed. + +\begin{lemma}[Pruning]\label{lem:pruning} +Let $X$ be a finite set with a binary relation denoted by $\to$, and let +$\mathcal R,\mathcal C\subseteq X\times X$ be two families, regarded as +separate copies even when their coordinates agree. Say that +$(p,j)\in\mathcal R$ and $(i,s)\in\mathcal C$ \emph{conflict} if +\[ + p\to i\qquad\text{and}\qquad s\to j. +\] +Define +\begin{align*} + Z&=\{(i,j):\text{some }(p,j)\in\mathcal R + \text{ conflicts with some }(i,s)\in\mathcal C\},\\ + H&=\{(p,s):\text{some }(p,j)\in\mathcal R + \text{ conflicts with some }(i,s)\in\mathcal C\}. +\end{align*} +A point $(i,j)$ is \emph{covered} by a member $(p,j)\in\mathcal R$ when +$p\to i$, and by a member $(i,s)\in\mathcal C$ when $s\to j$. +These are the only ways a member covers a point. + +There are subfamilies $\mathcal R'\subseteq\mathcal R$ and +$\mathcal C'\subseteq\mathcal C$ with no conflicts between them such that +every initially conflict-free member is retained and +\begin{equation}\label{eq:pruning-bound} + \abs{\mathcal R\setminus\mathcal R'}+ + \abs{\mathcal C\setminus\mathcal C'} + \leq \abs H+ + \abs{\{z\in Z:z\text{ is covered by no member of } + \mathcal R'\cup\mathcal C'\}}. +\end{equation} +The two sides are counted separately throughout. +\end{lemma} + +The proof will bound a matching in an augmented bipartite graph by using +matrix ranks. We first record the elementary rank fact that makes the +argument work. For finite index sets $I,J$, a matrix supported on +$T\subseteq I\times J$ means a real matrix whose entries outside $T$ are +zero; entries in $T$ are allowed to vanish. + +\begin{lemma}[Kernels at maximum support rank]\label{lem:kernel-support} +Let $T\subseteq I\times J$, where $I$ and $J$ are finite. Suppose that +$D:\R^J\to\R^I$ has maximum rank among all real matrices supported on $T$. +Then, for every $x\in\ker D$, every $y\in\ker D^{\mathsf T}$, and every +$(s,j)\in T$, one has $x_jy_s=0$. +\end{lemma} + +\begin{proof} +Suppose instead that $x_jy_s\ne0$. For a nonzero real $\tau$, the matrix +$D_\tau=D+\tau e_s e_j^{\mathsf T}$ is still supported on $T$, and +\[ + D_\tau x=\tau x_j e_s. +\] +Thus $e_s\in\im D_\tau$. Moreover, for every $w\in\R^J$, +\[ + Dw=D_\tau w-\tau w_j e_s\in\im D_\tau, +\] +so the entire old image $\im D$ is contained in $\im D_\tau$. +On the other hand, $y$ annihilates $\im D$ and $y_s\ne0$, so +$e_s\notin\im D$. Therefore $\rank D_\tau>\rank D$, a contradiction. +\end{proof} + +\begin{proof}[Proof of Lemma~\ref{lem:pruning}] +If there are no conflicts, retain every member; then $Z=H=\varnothing$ +and the assertion is immediate. In what follows, all vector spaces have +their usual coordinate bases. Empty minors have determinant $1$, so the +rank arguments also apply when a matching or an index set is empty. + +\medskip\noindent +\emph{The augmented graph and the required matching bound.} +Form a bipartite graph $\Gamma$ with parts +\[ + \mathcal V_L=\mathcal R\sqcup Z_L, + \qquad + \mathcal V_R=\mathcal C\sqcup Z_R, +\] +where $Z_L$ and $Z_R$ are separate copies of $Z$, disjoint also from the +two original families. Join conflicting original members. Also join +each original member to the opposite copy of every point of $Z$ that it +covers. There are no edges between $Z_L$ and $Z_R$. +Figure~\ref{fig:pruning} displays the three edges forced by one conflict. + +\begin{figure}[!ht] + \centering + \input{figures/pruning.tex} + \caption{A conflict between $r=(p,j)$ and $c=(i,s)$ forces this path, + where $z=(i,j)\in Z$. The copies $z_L$ and $z_R$ are distinct vertices. + The vertices alternate between the two sides of the bipartition. + Different conflicts can share vertices and edges; the full augmented + graph need not be a disjoint union of such paths.} + \label{fig:pruning} +\end{figure} + +An initially conflict-free member is isolated in $\Gamma$. +Indeed, if $(p,j)\in\mathcal R$ covered $(i,j)\in Z$, an original +conflict witnessing $(i,j)\in Z$ would supply a member +$(i,s)\in\mathcal C$ with $s\to j$. This member would conflict with +$(p,j)$. The same argument with the sides exchanged applies to +$\mathcal C$. + +It suffices to find a vertex cover $K$ of $\Gamma$ of size at most +$\abs Z+\abs H$, omitting isolated vertices. Delete precisely its +original members. No conflict then survives. If $q$ points of $Z$ are +uncovered by the surviving members, each of the other $\abs Z-q$ points +has a surviving member adjacent to one of its two copies. That copy +must lie in $K$. Distinct points require distinct new vertices, whence +\begin{equation}\label{eq:cover-charge} + \abs{K\cap(\mathcal R\sqcup\mathcal C)} + \leq \abs K-(\abs Z-q)\leq\abs H+q. +\end{equation} +Thus the cover gives exactly~\eqref{eq:pruning-bound} and retains every +initially conflict-free member. + +We will obtain this cover by proving that a maximum matching in $\Gamma$ +has size at most $\abs Z+\abs H$ and then giving the usual alternating-path +construction of the cover. Choose a maximum matching $\mathcal M$ with +the fewest edges between original members. Write $k$ for the number of +these edges, and let $\alpha$ and $\beta$ be its edges in +$\mathcal R\times Z_R$ and $Z_L\times\mathcal C$, respectively. Put +\[ + \ell=\abs\alpha,\qquad t=\abs\beta, + \qquad\abs{\mathcal M}=k+\ell+t. +\] + +Each of $\alpha$ and $\beta$ is a maximum matching in its own subgraph. +For example, if the $\mathcal R$--$Z_R$ subgraph had a larger matching, +its symmetric difference with $\alpha$ would contain an +$\alpha$-augmenting path, starting at an $\alpha$-unmatched member of +$\mathcal R$ and ending at an $\alpha$-unmatched vertex of $Z_R$. +The latter vertex is unmatched in $\mathcal M$, and every internal vertex +of the path is matched by $\alpha$. If the starting member is unmatched +in $\mathcal M$, augmenting gives a larger matching in $\Gamma$. +Otherwise it lies on an original--original edge of $\mathcal M$; +remove that edge and augment along the path. This preserves the total +size and reduces $k$. Both conclusions contradict the choice of +$\mathcal M$. The proof for $\beta$ is identical with the sides +exchanged. Denote by $R_\alpha\subseteq\mathcal R$ and +$C_\beta\subseteq\mathcal C$ the original vertices matched by these +two matchings. + +\medskip\noindent +\emph{Four maps and a simultaneous choice of coefficients.} +To bound $k+\ell+t$, we encode the two maximum matchings by matrix ranks. +For each arrow $x\to y$, introduce independent variables $a_{xy}$ and +$b_{xy}$, with the two arrays independent of one another. Set both +entries to zero when $x\not\to y$. We shall choose real values for all +the variables shortly. For any real assignment, define four maps by +\[ + \begin{array}{ccc} + \R^{\mathcal R}&\xrightarrow{\ L\ }&\R^Z\\[3pt] + {\scriptstyle B}\big\downarrow&&\big\downarrow{\scriptstyle N}\\[3pt] + \R^H&\xrightarrow{\ A\ }&\R^{\mathcal C} + \end{array} +\] +with entries +\begin{equation}\label{eq:pruning-maps} + \begin{aligned} + L_{(i,j),(p,j)}&=a_{pi},& + N_{(i,s),(i,j)}&=b_{sj},\\ + B_{(p,s),(p,j)}&=b_{sj},& + A_{(i,s),(p,s)}&=a_{pi}. + \end{aligned} +\end{equation} +The row and column indices in each formula range over its indicated +coordinate sets, and every entry not of a listed form is zero. +The maps $L,A$ change the first coordinate forward along $p\to i$; +$N,B$ change the second coordinate backward from $j$ to $s$ along $s\to j$. +For a conflict, the two routes from $(p,j)$ to $(i,s)$ change the +coordinates in opposite orders, through $(i,j)\in Z$ or $(p,s)\in H$. +For a fixed column $(p,j)\in\mathcal R$ and row +$(i,s)\in\mathcal C$, the only possible intermediate index in $NL$ is +$(i,j)$, and the only possible one in $AB$ is $(p,s)$. If the two +endpoints conflict, these indices belong to $Z$ and $H$, respectively, +and both products have entry $a_{pi}b_{sj}$. If they do not conflict, +at least one factor is zero in both products. Hence +\begin{equation}\label{eq:pruning-commute} + NL=AB. +\end{equation} + +We seek an assignment for which +\[ + \rank L=\ell,\qquad \rank N=t,\qquad \dim\ker A\geq k. +\] +These three conditions give the desired matching bound. Indeed, +rank--nullity and $NL=AB$ would give +\begin{align} + \abs H + &=\dim\ker A+\rank A\notag\\ + &\geq k+\rank A + \geq k+\rank(NL)\notag\\ + &\geq k+\ell+t-\abs Z. + \label{eq:pruning-rank-bound} +\end{align} +The last inequality follows by restricting $N$ to $\im L$: +\[ + \rank(NL) + =\dim\im L-\dim(\im L\cap\ker N) + \geq\ell-(\abs Z-t). +\] +The first two rank conditions will come from $\alpha$ and $\beta$. +For the third, we will lift the $k$ original left endpoints to vectors +$u^r\in\ker L$ and show that their images $Bu^r$ are independent. +The identity $AB=NL$ places these images in $\ker A$. + +Our choice of coefficients uses the correspondence between generic matrix +rank and maximum matching size~\cite[Section~5, Theorem~1]{Edmonds1967}. +We apply the determinant argument within coordinate blocks and verify that +all required minors can be nonzero at the same assignment. +We choose the coefficients so that $\rank L=\ell$, with its +$R_\alpha$ columns a basis of $\im L$, and +$\rank N^{\mathsf T}=t$, with its $C_\beta$ columns a basis of +$\im N^{\mathsf T}$. To justify this choice, a nonzero minor of $L$ +requires a matching between its column and row indices: a nonzero term +in its determinant supplies the matching. Since $\alpha$ is maximum, +$\rank L\leq\ell$ for every coefficient assignment. The square minor +using precisely the endpoints of $\alpha$ is, up to row and column +ordering, block diagonal by the second coordinate $j$. In each block +all allowed entries are distinct variables $a_{pi}$. The matching +gives a determinant monomial that cannot cancel, because different +permutations use different sets of these variables. Each block +determinant is therefore a nonzero polynomial, and so is their product. +Variables may occur in more than one block, but a product of nonzero +polynomials remains nonzero. This proves the required independence +when that minor does not vanish. For $N^{\mathsf T}$ the same argument +uses blocks with the first coordinate $i$ fixed and distinct variables +$b_{sj}$ within each block. + +One additional finite collection of rank conditions will be needed. +For every arrow $p\to i$, define +\[ + J_p=\{j:(p,j)\in\mathcal R\},\qquad + S_i=\{s:(i,s)\in\mathcal C\},\qquad + D^{p,i}=(b_{sj})_{s\in S_i,\ j\in J_p}. +\] +We require $D^{p,i}$ to have maximum rank among all matrices supported +on +\[ + T^{p,i}=\{(s,j)\in S_i\times J_p:s\to j\}. +\] +These matrices will let Lemma~\ref{lem:kernel-support} compare restrictions +of vectors in $\ker B$ and $\ker N^{\mathsf T}$. +The allowed entries of each individual $D^{p,i}$ are independent +variables. A minor attaining its maximum possible rank at some +assignment is therefore a nonzero polynomial in the $b$ variables. +Choose one such minor for each arrow, taking the empty minor when the +maximum rank is zero. Multiply these finitely many polynomials with +the two matching minors just described. The product is nonzero in the +polynomial ring generated by all $a$ and $b$ variables, even though +variables are shared between different matrices. A nonzero real +polynomial is nonzero at some real assignment, as follows by induction +on its number of variables. Fix such an assignment once and for all. +It gives these support ranks and the two basis properties simultaneously. + +For every edge $(r,c)$ of $\mathcal M$ between original members, the +basis properties now give vectors +\begin{equation}\label{eq:pruning-lifts} + \begin{aligned} + u^r&=e_r+\sum_{a\in R_\alpha}\lambda_a^r e_a + \in\ker L,\\ + v^c&=e_c+\sum_{b\in C_\beta}\mu_b^c e_b + \in\ker N^{\mathsf T}. + \end{aligned} +\end{equation} +The original--original endpoints are disjoint from $R_\alpha$ and +$C_\beta$, respectively. In particular, the $u^r$ restrict to distinct +unit vectors on the $k$ original left endpoints, and $v^c_c=1$. + +\medskip\noindent +\emph{The kernel restriction and the matching bound.} +It remains to show that the $k$ lifts just constructed give independent +vectors $Bu^r$. The key assertion, at our fixed coefficient assignment, is +\begin{equation}\label{eq:conflict-kernel} + u_{pj}v_{is}=0 + \quad\text{for every conflict }((p,j),(i,s)), + \quad u\in\ker B,\quad v\in\ker N^{\mathsf T}. +\end{equation} +Here the assertion quantifies over \emph{all} vectors in the two kernels. + +To prove it, fix an arrow $p\to i$. For $u\in\ker B$, consider the +vector $(u_{pj})_{j\in J_p}$. If a row $s\in S_i$ of $D^{p,i}$ has +an allowed entry, choose $j\in J_p$ with $s\to j$. Then +$(p,j)$ and $(i,s)$ conflict, so $(p,s)\in H$. The complete equation +of $Bu=0$ at this coordinate is +\begin{equation}\label{eq:pruning-row-equation} + \sum_{j\in J_p}b_{sj}u_{pj}=0. +\end{equation} +If the row has no allowed entry, this equation holds because all its +coefficients are zero. Thus it holds for every $s\in S_i$, including +rows for which $(p,s)$ is absent from $H$. Consequently +$(u_{pj})_{j\in J_p}\in\ker D^{p,i}$. + +Similarly, let $v\in\ker N^{\mathsf T}$. If a column $j\in J_p$ +of $D^{p,i}$ has an allowed entry, choose $s\in S_i$ with $s\to j$. +The resulting conflict implies $(i,j)\in Z$, and the complete equation +of $N^{\mathsf T}v=0$ at this coordinate is +\begin{equation}\label{eq:pruning-column-equation} + \sum_{s\in S_i}b_{sj}v_{is}=0. +\end{equation} +Columns with no allowed entries satisfy the same equation +automatically. Hence $(v_{is})_{s\in S_i}\in\ker(D^{p,i})^{\mathsf T}$. +Since $D^{p,i}$ has maximum rank on its allowed support, +Lemma~\ref{lem:kernel-support} gives $u_{pj}v_{is}=0$ whenever +$s\to j$, proving~\eqref{eq:conflict-kernel}. +The hypothetical single-entry perturbation in that lemma concerns only +$D^{p,i}$; it need not preserve any other rank condition. The conclusion +holds for every pair of kernel vectors at the fixed, original assignment. + +For each original--original matched edge $(r,c)$, substitute $v=v^c$ +in~\eqref{eq:conflict-kernel}. Since $v^c_c=1$, every vector in +$\ker B$ vanishes at $r$. It follows that the vectors $Bu^r$, as $r$ +ranges over the $k$ original left endpoints, are linearly independent. +Indeed, a relation +\[ + \sum_r\theta_r Bu^r=0 +\] +puts the single vector $u=\sum_r\theta_r u^r$ in $\ker B$. +Its coordinate at each such endpoint $r$ is $u_r=\theta_r$, by +\eqref{eq:pruning-lifts}. All these coordinates vanish, so every +$\theta_r$ is zero. Furthermore, +\[ + A(Bu^r)=NLu^r=0 +\] +by~\eqref{eq:pruning-commute}. Thus $\dim\ker A\geq k$, establishing +the third rank condition. Equation~\eqref{eq:pruning-rank-bound} now gives +$\abs{\mathcal M}=k+\ell+t\leq\abs Z+\abs H$, as required. + +\medskip\noindent +\emph{From the matching bound to a pruning.} +For completeness, we give the matching-to-cover argument in +K\H{o}nig's theorem~\cite{konig1931}. Starting at all unmatched vertices +of $\mathcal V_L$, follow edges outside $\mathcal M$ from left to +right and edges in $\mathcal M$ from right to left. Let $W_L,W_R$ +be the reached vertices on the two sides. No unmatched right vertex +is reached, since an alternating path to one would augment the maximum +matching. The endpoints of a matched edge are either both reached or +both unreached: a reached right endpoint leads to its mate, and a +matched left endpoint can only have been reached through its mate. +It follows that +\[ + K=(\mathcal V_L\setminus W_L)\cup W_R +\] +covers every edge. Indeed, an edge from a reached left vertex to an +unreached right vertex could be neither an unmatched edge, which would +be traversed, nor a matched edge, whose endpoints have the same +reachability. The cover contains exactly one endpoint of each matched +edge and no unmatched vertices. Hence +$\abs K=\abs{\mathcal M}\leq\abs Z+\abs H$; it also omits isolated +vertices. Applying~\eqref{eq:cover-charge} to this $K$ proves the lemma. +\end{proof} diff --git a/preprints/A-proof-of-Seymours-second-neighborhood-conjecture-September-23-2026/build/source/03-incompatibility.tex b/preprints/A-proof-of-Seymours-second-neighborhood-conjecture-September-23-2026/build/source/03-incompatibility.tex new file mode 100644 index 0000000000000000000000000000000000000000..76a3df4e1ff7fae7a4a41d1512d28bafa5349dac --- /dev/null +++ b/preprints/A-proof-of-Seymours-second-neighborhood-conjecture-September-23-2026/build/source/03-incompatibility.tex @@ -0,0 +1,148 @@ +\section{The extremal contradiction}\label{sec:incompatibility} + +We use Lemma~\ref{lem:pruning} to rule out the graph supplied by +Proposition~\ref{prop:counterexample-reduction}. The proof converts the strict +subset inequalities into a strict increase of an extremal quantity associated +with two families of ordered pairs. + +\begin{proposition}\label{prop:no-concavity} +There is no nonempty finite oriented graph $G$ in which every vertex has an +in-neighbor and whose one-step image operator +\[ + F(U)=\{y\in V(G):x\to y\text{ for some }x\in U\} +\] +satisfies +\begin{equation}\label{eq:impossible-concavity} + \abs{U}+\abs{F^2(U)}<2\abs{F(U)} + \qquad(\varnothing\ne U\subsetneq V(G)), +\end{equation} +where $F^2=F\circ F$. +\end{proposition} + +\begin{proof} +Suppose that such a graph exists, and put $X=V(G)$. For $S\subseteq X\times X$, +define the two coordinate image operators by +\[ + \begin{aligned} + F_1S&=\{(i,j): (p,j)\in S\text{ and }p\to i + \text{ for some }p\in X\},\\ + F_2S&=\{(i,j): (i,s)\in S\text{ and }s\to j + \text{ for some }s\in X\}. + \end{aligned} +\] +Write $F_1^2=F_1\circ F_1$, $F_2^2=F_2\circ F_2$, and +$\Delta=\{(v,v):v\in X\}$. Call $P,Q\subseteq X\times X$ \emph{compatible} +if $F_1P\cap F_2Q=\varnothing$. Equivalently, no left member $(p,j)\in P$ +and right member $(i,s)\in Q$ satisfy both $p\to i$ and $s\to j$; this is +precisely the conflict relation in Lemma~\ref{lem:pruning}. + +The pair $P=Q=\Delta$ is compatible: a point $(i,j)$ in both images would +require $j\to i$ and $i\to j$, contrary to orientation. Thus we may choose, +among all compatible pairs with $\Delta\subseteq P\cap Q$, one maximizing +$M+d$, where +\[ + M=\abs{P}+\abs{Q},\qquad + d=\abs{X}^2-\abs{F_1P}-\abs{F_2Q}. +\] +Such a maximum exists because $X$ is finite. Compatibility makes $d$ the +number of points outside $F_1P\cup F_2Q$. Including these uncovered points +in the objective lets the points left uncovered by pruning offset the +corresponding part of the deletion cost. + +Every column $P_j=\{p:(p,j)\in P\}$ is nonempty, since $j\in P_j$. +It is also proper. Indeed, positive indegrees give $F(X)=X$, so $P_j=X$ +would imply $X\times\{j\}\subseteq F_1P$. Choose an in-neighbor $w$ of +$j$. Then $(w,j)\in F_2\Delta\subseteq F_2Q$, contradicting compatibility. +Likewise every row $Q_i=\{s:(i,s)\in Q\}$ contains $i$. If $Q_i=X$, +then $\{i\}\times X\subseteq F_2Q$; choosing $w\to i$ gives +$(i,w)\in F_1\Delta\subseteq F_1P$, again a contradiction. +We can therefore sum~\eqref{eq:impossible-concavity} over the columns of $P$ +and over the rows of $Q$ to obtain +\begin{equation}\label{eq:strict-fiber-sums} + \abs{P}+\abs{F_1^2P}<2\abs{F_1P},\qquad + \abs{Q}+\abs{F_2^2Q}<2\abs{F_2Q}. +\end{equation} + +Define new left and right families by taking complements in $X\times X$: +\[ + \mathcal R=(X\times X)\setminus F_2^2Q, + \qquad + \mathcal C=(X\times X)\setminus F_1^2P. +\] +The two strict inequalities in~\eqref{eq:strict-fiber-sums} give +\begin{equation}\label{eq:complement-gain} + \begin{aligned} + \abs{\mathcal R}+\abs{\mathcal C} + &=2\abs{X}^2-\abs{F_2^2Q}-\abs{F_1^2P}\\ + &>2\abs{X}^2+M-2\bigl(\abs{F_1P}+\abs{F_2Q}\bigr) + =M+2d. + \end{aligned} +\end{equation} +These families need not be compatible. We will apply the pruning lemma, +first verifying that the diagonal survives and then bounding the cost. + +Both families contain $\Delta$. If $(v,v)\in F_2^2Q$, there is an $s\to v$ +with $(v,s)\in F_2Q$. But $(s,s)\in P$ gives $(v,s)\in F_1P$, a +contradiction. If $(v,v)\in F_1^2P$, there is a $p\to v$ with +$(p,v)\in F_1P$. Now $(p,p)\in Q$ gives $(p,v)\in F_2Q$, the same +contradiction. + +Moreover, each diagonal member is conflict-free in the new families. +A conflict between the left member $(v,v)$ and a right member +$(i,s)\in\mathcal C$ would give the path $s\to v\to i$. +Starting at $(s,s)\in P$ and expanding its first coordinate twice would +then put $(i,s)$ in $F_1^2P$, contrary to its membership in $\mathcal C$. +A conflict between a left member $(p,j)\in\mathcal R$ and the right member +$(v,v)$ would give $p\to v\to j$. Expanding the second coordinate of +$(p,p)\in Q$ twice would put $(p,j)$ in $F_2^2Q$, again a contradiction. +The preservation clause of Lemma~\ref{lem:pruning} thus applies to every +diagonal member on both sides. + +Let $H,Z\subseteq X\times X$ be the sets associated with +$\mathcal R,\mathcal C$ in that lemma. Thus each conflict +\[ + (p,j)\in\mathcal R,\quad (i,s)\in\mathcal C,\quad + p\to i,\quad s\to j +\] +contributes $(p,s)$ to $H$ and $(i,j)$ to $Z$. +We claim that +\begin{equation}\label{eq:old-uncovered-bound} + H\subseteq (X\times X)\setminus(F_1P\cup F_2Q), + \qquad\text{and hence}\qquad \abs{H}\le d. +\end{equation} +To see this, fix such a conflict. If $(p,s)\in F_1P$, choose +$(u,s)\in P$ with $u\to p$. The path $u\to p\to i$ then puts +$(i,s)$ in $F_1^2P$, contrary to $(i,s)\in\mathcal C$. +If $(p,s)\in F_2Q$, choose $(p,t)\in Q$ with $t\to s$. +The path $t\to s\to j$ puts $(p,j)$ in $F_2^2Q$, contrary to +$(p,j)\in\mathcal R$. This proves~\eqref{eq:old-uncovered-bound}. + +Apply Lemma~\ref{lem:pruning} to obtain conflict-free families +$P'\subseteq\mathcal R$ and $Q'\subseteq\mathcal C$. +They still contain $\Delta$, and their lack of conflicts means +$F_1P'\cap F_2Q'=\varnothing$. Set +\[ + d'=\abs{X}^2-\abs{F_1P'}-\abs{F_2Q'}. +\] +A point is covered by a kept left member exactly when it belongs to +$F_1P'$, and by a kept right member exactly when it belongs to $F_2Q'$. +Consequently at most $d'$ points of $Z$ are uncovered after pruning. +By Lemma~\ref{lem:pruning} and~\eqref{eq:old-uncovered-bound}, at most $d+d'$ members +were deleted, counting the two sides separately. Using +\eqref{eq:complement-gain}, we conclude that +\[ + \begin{aligned} + \abs{P'}+\abs{Q'}+d' + &\ge \abs{\mathcal R}+\abs{\mathcal C}-(d+d')+d'\\ + &>M+2d-(d+d')+d'=M+d. + \end{aligned} +\] +This contradicts the choice of $P,Q$ and proves the proposition. +\end{proof} + +\begin{proof}[Proof of Theorem~\ref{thm:main}] +If the theorem were false, Proposition~\ref{prop:counterexample-reduction} +would produce a nonempty finite oriented graph with positive indegrees +satisfying~\eqref{eq:impossible-concavity} for every nonempty proper vertex +set. Proposition~\ref{prop:no-concavity} rules out such a graph. +\end{proof} diff --git a/preprints/A-proof-of-Seymours-second-neighborhood-conjecture-September-23-2026/build/source/04-weighted-consequences.tex b/preprints/A-proof-of-Seymours-second-neighborhood-conjecture-September-23-2026/build/source/04-weighted-consequences.tex new file mode 100644 index 0000000000000000000000000000000000000000..82870248391de0112d0df454bbe8b8b6d9c69c53 --- /dev/null +++ b/preprints/A-proof-of-Seymours-second-neighborhood-conjecture-September-23-2026/build/source/04-weighted-consequences.tex @@ -0,0 +1,70 @@ +\section{Weighted consequences}\label{sec:weighted-consequences} + +The following corollary applies Theorem~\ref{thm:main} through Seacrest's +established equivalences of the corresponding assertions over all nonempty +finite oriented graphs~\cite{Seacrest2015}. The equivalences themselves are +not new here. + +\begin{corollary}[Weighted second-neighborhood forms]\label{cor:weighted-forms} +Let \(D\) be a nonempty finite oriented graph, and write \(A(D)\) for its +arc set. For a vertex weighting \(\eta:V(D)\to[0,\infty)\), write +\(\eta(S)=\sum_{x\in S}\eta(x)\). Then the following hold. +\begin{enumerate}[label=\textup{(\roman*)}] +\item For every vertex weighting \(\eta:V(D)\to[0,\infty)\), there is a +vertex \(v\in V(D)\) such that +\[ + \eta(N_1^+(v))\leq\eta(N_2^+(v)). +\] + +\item There is a vertex weighting \(\eta:V(D)\to[0,\infty)\) with +\(\sum_{x\in V(D)}\eta(x)=1\) such that +\[ + \eta(N_1^+(v))\leq\eta(N_2^+(v))\qquad\text{for every }v\in V(D). +\] +The entries of this weighting need not all be positive. + +\item For every arc weighting \(w:A(D)\to[0,\infty)\), extend the notation +by setting \(w(vs)=0\) when \(vs\notin A(D)\). For \(v,s\in V(D)\), put +\[ + \beta_v^w(s)=\max\Bigl(\{0\}\cup + \{w(us)-w(vs):u\in V(D),\ v\to u\to s\}\Bigr). +\] +Then there is a vertex \(v\in V(D)\) such that +\[ + \sum_{s\in V(D)}\beta_v^w(s) + \geq \sum_{u\in N_1^+(v)}w(vu). +\] +The maximum is zero if there is no indicated two-step path. Such paths +are determined by the arcs of \(D\), so an arc of weight zero is still +present and is not treated as a missing arc. +\end{enumerate} +\end{corollary} + +\begin{proof} +Seacrest's Proposition~2 proves that his arc-weighted Conjecture~4 is +equivalent to the unweighted conjecture, so Theorem~\ref{thm:main} gives +\textup{(iii)}. + +To obtain \textup{(i)}, set $w(us)=\eta(s)$ on every arc. For each $v,s$, +\[ + \beta_v^w(s)= + \begin{cases} + \eta(s),&s\in N_2^+(v),\\ + 0,&s\notin N_2^+(v). + \end{cases} +\] +Indeed, when $v\to s$, every eligible difference $w(us)-w(vs)$ is zero. +At an exact second out-neighbor, $w(vs)=0$ and the maximum is $\eta(s)$; +no other endpoint contributes. Thus the two sides of \textup{(iii)} +become $\eta(N_2^+(v))$ and $\eta(N_1^+(v))$. This is the +specialization noted immediately after Seacrest's Proposition~2. + +For \textup{(ii)}, let $M_D$ have entry $1$ at $(v,s)$ when +$s\in N_2^+(v)$, entry $-1$ when $s\in N_1^+(v)$, and zero otherwise. +Reversing all arcs transposes this matrix: +$M_{\overleftarrow D}=M_D^{\mathsf T}$. Seacrest's Theorem~1 applies +Farkas' lemma to give either a nonzero $\eta\ge0$ with $M_D\eta\ge0$, +or a $\xi\ge0$ with $M_D^{\mathsf T}\xi<0$ at every coordinate. +The latter contradicts \textup{(i)} on $\overleftarrow D$. +Normalize the former weighting to total weight one, proving \textup{(ii)}. +\end{proof} diff --git a/preprints/A-proof-of-Seymours-second-neighborhood-conjecture-September-23-2026/build/source/05-directed-cycle-consequences.tex b/preprints/A-proof-of-Seymours-second-neighborhood-conjecture-September-23-2026/build/source/05-directed-cycle-consequences.tex new file mode 100644 index 0000000000000000000000000000000000000000..9ee0c1b766f81a904bb24db980a62a9c49691447 --- /dev/null +++ b/preprints/A-proof-of-Seymours-second-neighborhood-conjecture-September-23-2026/build/source/05-directed-cycle-consequences.tex @@ -0,0 +1,72 @@ +\section{Directed-cycle consequences}\label{sec:directed-cycle-consequences} + +In an oriented graph with no directed triangle, every exact second +out-neighbor is a non-neighbor. Combining this observation with +Theorem~\ref{thm:main} gives degree and order bounds. +The first assertion below is the oriented-graph form of Thomass\'e's +non-neighbor statement recorded by +Sullivan~\cite[Conjecture~6.12]{Sullivan2006}. +The consequence using both minimum indegree and minimum outdegree is also +noted in \cite[Section~3]{Sullivan2006}. The regular case gives the +girth-four lower bound in the Behzad--Chartrand--Wall minimum-order +conjecture~\cite{BehzadChartrandWall1970}, also recorded by +Sullivan~\cite[Section~4]{Sullivan2006}. + +\begin{corollary}[Non-neighbors and directed cycles]\label{cor:directed-cycle-consequences} +Let \(D\) be a nonempty finite oriented graph of order \(n\). For +\(v\in V(D)\), write +\[ + N_1^-(v)=\{u:u\to v\},\qquad + d^\pm(v)=\abs{N_1^\pm(v)},\qquad + \delta^\pm(D)=\min_{x\in V(D)}d^\pm(x). +\] +Define the non-neighbors of \(v\), excluding \(v\) itself, by +\[ + Z(v)=V(D)\setminus + \bigl(\{v\}\cup N_1^+(v)\cup N_1^-(v)\bigr). +\] +Then the following hold. +\begin{enumerate}[label=\textup{(\roman*)}] +\item If \(D\) has no directed \(3\)-cycle, then some vertex \(v\) satisfies +\[ + d^+(v)\leq\abs{Z(v)},\qquad + n-1\geq 2d^+(v)+d^-(v). +\] +\item If \(\delta^+(D)\geq n/3\) and \(\delta^-(D)\geq n/3\), then \(D\) +contains a directed \(3\)-cycle. +\item If \(r\geq1\) is an integer, \(d^+(v)=d^-(v)=r\) for every vertex, +and \(D\) has directed girth exactly four, then \(n\geq3r+1\). +\end{enumerate} +Here directed girth is the length of a shortest directed cycle. +\end{corollary} + +\begin{proof} +Suppose that \(D\) has no directed \(3\)-cycle. If \(w\in N_2^+(v)\), +choose a path \(v\to u\to w\). The exact-distance definition excludes +\(w=v\) and \(w\in N_1^+(v)\). If \(w\in N_1^-(v)\), the arc \(w\to v\) +would close a directed \(3\)-cycle. Thus +\(N_2^+(v)\subseteq Z(v)\) for every \(v\). Theorem~\ref{thm:main} +now gives a vertex with +\[ + d^+(v)\leq\abs{N_2^+(v)}\leq\abs{Z(v)}. +\] +Since \(D\) is oriented, its first in- and out-neighborhoods are disjoint, +and their cardinalities are the corresponding degrees. Hence +\[ + n-1=d^+(v)+d^-(v)+\abs{Z(v)} + \geq 2d^+(v)+d^-(v), +\] +proving \textup{(i)}. + +Under the hypotheses of \textup{(ii)}, absence of a directed \(3\)-cycle +would therefore give +\[ + n-1\geq2\delta^+(D)+\delta^-(D)\geq n, +\] +a contradiction. The integer degree thresholds are +\(\delta^\pm(D)\geq\lceil n/3\rceil\), so this includes the endpoint +when \(3\) divides \(n\). Finally, directed girth four excludes directed +\(3\)-cycles; in the regular case the count in \textup{(i)} becomes +\(n-1\geq3r\), proving \textup{(iii)}. +\end{proof} + diff --git a/preprints/A-proof-of-Seymours-second-neighborhood-conjecture-September-23-2026/build/source/figures/pruning.tex b/preprints/A-proof-of-Seymours-second-neighborhood-conjecture-September-23-2026/build/source/figures/pruning.tex new file mode 100644 index 0000000000000000000000000000000000000000..ba64fc5cf8ea4785d83063b7daa63f62150296d8 --- /dev/null +++ b/preprints/A-proof-of-Seymours-second-neighborhood-conjecture-September-23-2026/build/source/figures/pruning.tex @@ -0,0 +1,19 @@ +\begin{tikzpicture}[ + every node/.style={font=\small}, + original/.style={circle,draw,fill=black!7,minimum size=8mm,inner sep=1pt}, + copy/.style={rectangle,draw,minimum size=8mm,inner sep=3pt}, + edge label/.style={font=\footnotesize,above=4pt}, + family label/.style={font=\footnotesize,below=6pt} +] + \node[copy] (zl) at (0,0) {$z_L$}; + \node[original] (c) at (3,0) {$c$}; + \node[original] (r) at (6,0) {$r$}; + \node[copy] (zr) at (9,0) {$z_R$}; + \draw[dashed] (zl) -- node[edge label] {covers $z$} (c); + \draw (c) -- node[edge label] {conflict} (r); + \draw[dashed] (r) -- node[edge label] {covers $z$} (zr); + \node[family label] at (zl.south) {$Z_L$}; + \node[family label] at (c.south) {$\mathcal C:\ (i,s)$}; + \node[family label] at (r.south) {$\mathcal R:\ (p,j)$}; + \node[family label] at (zr.south) {$Z_R$}; +\end{tikzpicture} diff --git a/preprints/A-proof-of-Seymours-second-neighborhood-conjecture-September-23-2026/build/source/main.tex b/preprints/A-proof-of-Seymours-second-neighborhood-conjecture-September-23-2026/build/source/main.tex new file mode 100644 index 0000000000000000000000000000000000000000..88c55a92e2a94599fa322c8bad03501e4f7b00b6 --- /dev/null +++ b/preprints/A-proof-of-Seymours-second-neighborhood-conjecture-September-23-2026/build/source/main.tex @@ -0,0 +1,47 @@ +\documentclass[11pt]{article} +\usepackage[T1]{fontenc} +\usepackage{lmodern} +\pdfglyphtounicode{parenleftbig}{0028} +\pdfglyphtounicode{parenrightbig}{0029} +\pdfglyphtounicode{parenleftBig}{0028} +\pdfglyphtounicode{parenrightBig}{0029} +\pdfgentounicode=1 +\usepackage[margin=1.05in]{geometry} +\usepackage{amsmath,amssymb,amsthm,mathtools} +\usepackage{microtype} +\usepackage{enumitem} +\usepackage{tikz} +\usetikzlibrary{arrows.meta,positioning,calc} +\usepackage[hidelinks]{hyperref} +\hypersetup{pdftitle={A proof of Seymour's second-neighborhood conjecture}, + pdfauthor={OpenAI}, pdfsubject={Oriented graphs and finite relations}} +\pdftrailerid{} +\newtheorem{theorem}{Theorem}[section] +\newtheorem{lemma}[theorem]{Lemma} +\newtheorem{proposition}[theorem]{Proposition} +\newtheorem{corollary}[theorem]{Corollary} +\theoremstyle{remark} +\newtheorem{remark}[theorem]{Remark} +\newcommand{\abs}[1]{\lvert #1\rvert} +\DeclareMathOperator{\rank}{rank} +\DeclareMathOperator{\im}{im} +\newcommand{\R}{\mathbb R} +\title{A proof of Seymour's second-neighborhood conjecture} +\author{OpenAI} +\date{September 23, 2026} +\begin{document} +\maketitle +\begin{abstract} +We prove that every nonempty finite oriented graph has a vertex with at least +as many vertices at directed distance two as at directed distance one. +This resolves Seymour's second neighborhood conjecture positively. +\end{abstract} +\input{00-introduction} +\input{01-reduction} +\input{02-pruning} +\input{03-incompatibility} +\input{04-weighted-consequences} +\input{05-directed-cycle-consequences} +\bibliographystyle{plain} +\bibliography{references} +\end{document} diff --git a/preprints/A-proof-of-Seymours-second-neighborhood-conjecture-September-23-2026/build/source/references.bib b/preprints/A-proof-of-Seymours-second-neighborhood-conjecture-September-23-2026/build/source/references.bib new file mode 100644 index 0000000000000000000000000000000000000000..f4ea21a4eb97842bb0a2441266e237b8c4a56878 --- /dev/null +++ b/preprints/A-proof-of-Seymours-second-neighborhood-conjecture-September-23-2026/build/source/references.bib @@ -0,0 +1,193 @@ +@article{Fisher1996, + author = {Fisher, David C.}, + title = {Squaring a tournament: A proof of {Dean}'s conjecture}, + journal = {Journal of Graph Theory}, + volume = {23}, + number = {1}, + year = {1996}, + pages = {43--48}, + doi = {10.1002/(SICI)1097-0118(199609)23:1<43::AID-JGT4>3.0.CO;2-K}, + url = {https://doi.org/10.1002/(SICI)1097-0118(199609)23:1<43::AID-JGT4>3.0.CO;2-K}, + note = {\href{https://doi.org/10.1002/(SICI)1097-0118(199609)23:1<43::AID-JGT4>3.0.CO;2-K}{doi}} +} + +@article{HavetThomasse2000, + author = {Havet, Fr{\'e}d{\'e}ric and Thomass{\'e}, St{\'e}phan}, + title = {Median orders of tournaments: A tool for the second neighborhood problem and {Sumner}'s conjecture}, + journal = {Journal of Graph Theory}, + volume = {35}, + number = {4}, + year = {2000}, + pages = {244--256}, + doi = {10.1002/1097-0118(200012)35:4<244::AID-JGT2>3.0.CO;2-H}, + url = {https://doi.org/10.1002/1097-0118(200012)35:4<244::AID-JGT2>3.0.CO;2-H}, + note = {\href{https://doi.org/10.1002/1097-0118(200012)35:4<244::AID-JGT2>3.0.CO;2-H}{doi}} +} + +@misc{Seacrest2019, + author = {Seacrest, Tyler}, + title = {{Seymour}'s Second Neighborhood Conjecture for Subsets of Vertices}, + year = {2019}, + eprint = {1808.06293v3}, + archivePrefix = {arXiv}, + primaryClass = {math.CO}, + doi = {10.48550/arXiv.1808.06293}, + url = {https://arxiv.org/abs/1808.06293v3}, + note = {\href{https://arxiv.org/abs/1808.06293v3}{arXiv:1808.06293v3} [math.CO], revised 12 April 2019} +} + +@article{Seacrest2015, + author = {Seacrest, Tyler}, + title = {The Arc-Weighted Version of the Second Neighborhood Conjecture}, + journal = {Journal of Graph Theory}, + volume = {78}, + number = {3}, + year = {2015}, + pages = {219--228}, + doi = {10.1002/jgt.21800}, + url = {https://doi.org/10.1002/jgt.21800}, + note = {Preprint (numbering used here): \href{https://arxiv.org/abs/1212.1883v1}{arXiv:1212.1883v1}} +} + +@article{konig1931, + author = {K{\H{o}}nig, D{\'e}nes}, + title = {Graphok {\'e}s matrixok}, + journal = {Matematikai {\'e}s Fizikai Lapok}, + volume = {38}, + year = {1931}, + pages = {116--119}, + url = {https://real-j.mtak.hu/7307/}, + note = {In Hungarian} +} + +@article{DeanLatka1995, + author = {Dean, Nathaniel and Latka, Brenda J.}, + title = {Squaring the tournament---an open problem}, + journal = {Congressus Numerantium}, + volume = {109}, year = {1995}, pages = {73--80} +} +@article{FidlerYuster2007, + author = {Fidler, Dror and Yuster, Raphael}, + title = {Remarks on the second neighborhood problem}, + journal = {Journal of Graph Theory}, + volume = {55}, number = {3}, year = {2007}, pages = {208--220}, + doi = {10.1002/jgt.20229}, + url = {https://doi.org/10.1002/jgt.20229} +} +@article{Ghazal2012, + author = {Ghazal, Salman}, + title = {{Seymour}'s second neighborhood conjecture for tournaments missing a generalized star}, + journal = {Journal of Graph Theory}, + volume = {71}, number = {1}, year = {2012}, pages = {89--94}, + doi = {10.1002/jgt.20634}, + url = {https://doi.org/10.1002/jgt.20634} +} +@article{DaraFrancisJacobNarayanan2022, + author = {Dara, Suresh and Francis, Mathew C. and Jacob, Dalu and Narayanan, N.}, + title = {Extending some results on the second neighborhood conjecture}, + journal = {Discrete Applied Mathematics}, + volume = {311}, year = {2022}, pages = {1--17}, + doi = {10.1016/j.dam.2021.12.034}, + url = {https://doi.org/10.1016/j.dam.2021.12.034} +} +@article{DaamouchAlMninyGhazal2025, + author = {Daamouch, Moussa and Al-Mniny, Darine and Ghazal, Salman}, + title = {About the second neighborhood conjecture for tournaments missing two stars or disjoint paths}, + journal = {Contributions to Discrete Mathematics}, + volume = {20}, number = {2}, year = {2025}, pages = {363--383}, + doi = {10.55016/ojs/cdm.v20i2.77499}, + url = {https://cdm.ucalgary.ca/article/download/77499/58529/271837} +} +@article{KanekoLocke2001, + author = {Kaneko, Yoshihiro and Locke, Stephen C.}, + title = {The minimum degree approach for {Paul Seymour}'s distance 2 conjecture}, + journal = {Congressus Numerantium}, + volume = {148}, year = {2001}, pages = {201--206} +} +@misc{SadhukhanSandeepSen2026, + author = {Sadhukhan, Arpan and Sandeep, R. B. and Sen, Sagnik}, + title = {A proof of {Seymour}'s second neighborhood conjecture for oriented graphs with minimum out-degree equal to 7}, + year = {2026}, eprint = {2606.30588v1}, + archivePrefix = {arXiv}, primaryClass = {math.CO}, + url = {https://arxiv.org/abs/2606.30588v1}, + note = {arXiv:2606.30588v1, June 29, 2026} +} +@article{ChenShenYuster2003, + author = {Chen, Guantao and Shen, Jian and Yuster, Raphael}, + title = {Second neighborhood via first neighborhood in digraphs}, + journal = {Annals of Combinatorics}, + volume = {7}, number = {1}, year = {2003}, pages = {15--20}, + doi = {10.1007/s000260300001}, + url = {https://doi.org/10.1007/s000260300001} +} +@misc{HuangPeng2024, + author = {Huang, Hao and Peng, Fei}, + title = {An improved bound on {Seymour}'s second neighborhood conjecture}, + year = {2024}, eprint = {2412.20234v1}, + archivePrefix = {arXiv}, primaryClass = {math.CO}, + url = {https://arxiv.org/abs/2412.20234v1}, + note = {arXiv:2412.20234v1, December 28, 2024} +} +@article{BotlerMouraNaia2023, + author = {Botler, F{\'a}bio and Moura, Phablo F. S. and Naia, T{\'a}ssio}, + title = {{Seymour}'s second neighborhood conjecture for orientations of (pseudo)random graphs}, + journal = {Discrete Mathematics}, + volume = {346}, number = {12}, year = {2023}, pages = {113583}, + doi = {10.1016/j.disc.2023.113583}, + url = {https://doi.org/10.1016/j.disc.2023.113583} +} +@article{EspunyDiazEtAl2025, + author = {Espuny D{\'i}az, Alberto and Gir{\~a}o, Ant{\'o}nio and Granet, Bertille and Kronenberg, Gal}, + title = {{Seymour}'s second neighbourhood conjecture: random graphs and reductions}, + journal = {Random Structures \& Algorithms}, + volume = {66}, number = {1}, year = {2025}, pages = {e21251}, + doi = {10.1002/rsa.21251}, + url = {https://doi.org/10.1002/rsa.21251} +} +@misc{Brukhman2026, + author = {Brukhman, Jake}, + title = {A dense-case theorem for {Seymour}'s second neighborhood conjecture}, + year = {2026}, eprint = {2608.11530v1}, + archivePrefix = {arXiv}, primaryClass = {math.CO}, + url = {https://arxiv.org/abs/2608.11530v1}, + note = {arXiv:2608.11530v1, August 12, 2026} +} +@article{BrantnerBrockmanKaySnively2009, + author = {Brantner, James and Brockman, Greg and Kay, Bill and Snively, Emma}, + title = {Contributions to {Seymour}'s second neighborhood conjecture}, + journal = {Involve}, + volume = {2}, number = {4}, year = {2009}, pages = {387--395}, + url = {https://msp.org/involve/2009/2-4/involve-v2-n4-p02-s.pdf} +} +@article{Edmonds1967, + author = {Edmonds, Jack}, + title = {Systems of distinct representatives and linear algebra}, + journal = {Journal of Research of the National Bureau of Standards, Section B}, + volume = {71B}, number = {4}, year = {1967}, pages = {241--245}, + doi = {10.6028/jres.071B.033}, + url = {https://doi.org/10.6028/jres.071B.033} +} + +@misc{Sullivan2006, + author = {Sullivan, Blair D.}, + title = {A Summary of Results and Problems Related to the {Caccetta--H{\"a}ggkvist} Conjecture}, + year = {2006}, + howpublished = {American Institute of Mathematics}, + url = {https://aimath.org/WWN/caccetta/caccetta.pdf}, + note = {AIM workshop report dated April 14, 2006; + \href{https://aimath.org/WWN/caccetta/caccetta.pdf}{report}; + Section~3 and Conjecture~6.12} +} + +@article{BehzadChartrandWall1970, + author = {Behzad, Mehdi and Chartrand, Gary and Wall, Curtiss E.}, + title = {On minimal regular digraphs with given girth}, + journal = {Fundamenta Mathematicae}, + volume = {69}, + number = {3}, + year = {1970}, + pages = {227--231}, + doi = {10.4064/fm-69-3-227-231}, + url = {https://doi.org/10.4064/fm-69-3-227-231}, + note = {\href{https://doi.org/10.4064/fm-69-3-227-231}{doi}} +} diff --git a/preprints/A-proof-of-Seymours-second-neighborhood-conjecture-September-23-2026/paper.pdf b/preprints/A-proof-of-Seymours-second-neighborhood-conjecture-September-23-2026/paper.pdf new file mode 100644 index 0000000000000000000000000000000000000000..c02c62f213301d94265ba52b50f26e7a95729d97 --- /dev/null +++ b/preprints/A-proof-of-Seymours-second-neighborhood-conjecture-September-23-2026/paper.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:20be859799f83e3412dba6fd645a72a0d929cbe05ca546807c591f001f9e6e82 +size 387437 diff --git a/preprints/A-quadratic-bound-for-Jacobsthals-function-September-25-2026/README.md b/preprints/A-quadratic-bound-for-Jacobsthals-function-September-25-2026/README.md new file mode 100644 index 0000000000000000000000000000000000000000..24b07a14149b713cb19cd784c68834ac2c5ba7f2 --- /dev/null +++ b/preprints/A-quadratic-bound-for-Jacobsthals-function-September-25-2026/README.md @@ -0,0 +1,19 @@ +# [A quadratic bound for Jacobsthal's function](paper.pdf) + +OpenAI +September 25, 2026 + +## Citation + +To cite this manuscript: + +```bibtex +@misc{OAI:A-quadratic-bound-for-Jacobsthals-function-September-25-2026, + author = {{OpenAI}}, + title = {{A quadratic bound for Jacobsthal's function}}, + howpublished = {OpenAI Math Release preprint + \href{https://github.com/openai/math/blob/main/preprints/A-quadratic-bound-for-Jacobsthals-function-September-25-2026/paper.pdf}{OAI:A-quadratic-bound-for-Jacobsthals-function-September-25-2026}}, + year = {2026} +} +``` + diff --git a/preprints/A-quadratic-bound-for-Jacobsthals-function-September-25-2026/build/figures/proof-route.tex b/preprints/A-quadratic-bound-for-Jacobsthals-function-September-25-2026/build/figures/proof-route.tex new file mode 100644 index 0000000000000000000000000000000000000000..ad47cdf98fcb819453d62c0bfde9c4b19dad671f --- /dev/null +++ b/preprints/A-quadratic-bound-for-Jacobsthals-function-September-25-2026/build/figures/proof-route.tex @@ -0,0 +1,47 @@ +\begin{figure}[tbp] +\centering +\begingroup +\setlength{\unitlength}{1mm} +\setlength{\fboxsep}{2mm} +\small +\begin{picture}(154,98) + \put(77,89){\makebox(0,0){\fbox{\parbox{58mm}{\centering + Expanded lower node $d$\\[-1pt] + every available child is included}}}} + \put(77,79){\vector(0,-1){15}} + \put(81,71){\makebox(0,0)[l]{$p$}} + \put(77,54){\makebox(0,0){\fbox{\parbox{58mm}{\centering + Included upper node $dp$\\[-1pt] + test each prospective lower child}}}} + \put(69,45){\vector(-4,-1){43}} + \put(77,44){\vector(0,-1){12}} + \put(85,45){\vector(4,-1){43}} + \put(46,37){\makebox(0,0){$q$}} + \put(81,38){\makebox(0,0)[l]{$q'$}} + \put(110,37){\makebox(0,0){$u$}} + \put(26,19){\makebox(0,0){\fbox{\parbox{42mm}{\centering + Included lower node $dpq$\\[-1pt] + chosen stop: retain\\[-1pt] + $S_{dpq}(x(q))$ exactly}}}} + \put(77,19){\makebox(0,0){\fbox{\parbox{40mm}{\centering + Included lower node $dpq'$\\[-1pt] + continue its\\[-1pt] + lower-bound expansion}}}} + \put(105,9.5){\dashbox{1}(46,19){\parbox{40mm}{\centering + Prospective child $dpu$\\[-1pt] + omitted by the\\[-1pt] + admission rule}}} + \put(51,3){\makebox(0,0){$q,q'$ satisfy \eqref{eq:admission}}} + \put(128,3){\makebox(0,0){$u$ fails \eqref{eq:admission}}} +\end{picture} +\endgroup +\caption{Admission and stopping have different roles. Only selected branches +are drawn, with $q,q',u0$ such that +\begin{equation}\label{eq:assembly-reference} + P_\ee(r_0,B)\ge\frac{c_L}{B^2} +\end{equation} +for all sufficiently large $z$. Let $C_{\rm abs}$ be an absolute +constant such that +\[ + \left|N_d/\mu_d-1\right|\le C_{\rm abs} +\] +at every included node; this follows from +\eqref{eq:tree-small-counts}. + +For nodes with large gap, the relative error in +\eqref{eq:tree-small-counts} is $O(e^{-cr_d})$ after a fixed lower +threshold on $r_d$. The tail estimate in +Proposition~\ref{prop:prime-occupation} permits us to choose a fixed +$K$ so large that +\begin{equation}\label{eq:assembly-tail} + \sum_{\substack{d\text{ included}\\r_d>K}} + |N_d-\mu_d| + \le\frac{c_L}{8}\frac{YV_0}{B^2} +\end{equation} +eventually. Standard paths contain the included paths, so the +nonnegative tail estimate applies. This choice depends only on the +fixed reference margin and the absolute small-prime error constants. + +The same proposition gives a constant $C_K$ for the total harmonic +mass of nodes with $r_d\le K$. Choose $\eps>0$ such that +$\eps C_K0$ so small that +\begin{equation}\label{eq:assembly-beta} + C_{\rm abs}\beta<\frac{c_L}{8}. +\end{equation} +We claim that the stopped tree can be chosen so that the retained +bad compact nodes have total harmonic mass at most $\beta/B^2$. +The following choices give this claim without requiring any constant +to be chosen again at an earlier stage. + +First use Lemma~\ref{lem:edge-witness} to fix a larger threshold +$K_*$, a positive edge-discrepancy threshold, and a fixed maximum +number $H_e$ of candidate positions. Choose the adjustable regularity +exceptions in Proposition~\ref{prop:regular-boxes} small enough to +cost less than a prescribed small fraction of $\beta/B^2$. +This fixes the path-length bound and the isolated-prime interval. +For the moment leave the exception for near-threshold even prefixes +to the eventual choice of bin width. + +The enlarged regular boxes have total harmonic mass at most +$C'_K/B^2$, with $C'_K$ independent of sufficiently small fixed +$\xi$. Choose the exception fraction $\sigma$ in +Proposition~\ref{prop:inverse} sufficiently small that +\[ + H_eC'_K\sigma +\] +is less than another prescribed small fraction of $\beta$. +The inverse proposition now fixes a positive alignment fraction and +a list bound $F$, independently of the eventual sufficiently small +fixed width. Each list is attached to its entire candidate box. + +Next choose the variance exception fraction in +Proposition~\ref{prop:structured-variance} small enough relative to +$H_e$, $C'_K$, $F$ and $\beta$. A prime-count fraction in the isolated +bin converts to a harmonic fraction at a bounded absolute cost. +After this conversion and the finite unions over candidates and +list elements, the total variance exception costs another prescribed +small fraction of $\beta/B^2$. The variance proposition fixes a +constant $C_s$. Corollary~\ref{cor:hard-endpoints} then shows that +all remaining bad compact endpoints have a hard rational satisfying +the global eligibility bounds and alignment of every factor with +exponent greater than $C_s$. + +Choose the absolute owner threshold $\eta$ required by +Proposition~\ref{prop:stopped-comparison}. Choose $b_0$ sufficiently +large in terms of $C_s$ and $K$ and for the local reference and prime +estimates in that proposition. Finally choose $b_1/b_0$ sufficiently +large that Proposition~\ref{prop:hard-stopping} bounds the hard +endpoints that remain outside stopped subtrees by the remaining +allocated small fraction of $\beta/B^2$. This uses the marked-visit +estimate with a prescribed failure probability, as well as the +vanishing owner and wrong-tag exceptions for the fixed candidate +lists. + +All constants except the bin width have now been fixed. Choose a +fixed $\xi>0$ sufficiently small for the interval-freezing errors, +candidate-size slack, representative stopping margins and all +earlier small-width requirements. In particular, the near-threshold +exception, of cost $O(\Delta)+o(1)$ in the compact-history +probability measure with $\Delta=2C_{\rm len}\xi$, can be made as +small as the error budget requires. Then take $z$ sufficiently large +for every preceding fixed choice. Asymptotic bounds whose thresholds +depend on $\xi$ are used only at this final stage. + +The sum of all these finitely many allocated costs is at most +$\beta/B^2$, proving the claim. More explicitly, after removing the +regularity exceptions, every retained bad endpoint has a witnessing +edge. Outside the inverse exceptions it has an alignment in the +fixed list. Outside the variance exceptions it is hard. Outside +the hard-stopping exceptions it lies in a stopped subtree and is +therefore not retained. This exhausts the cases. Fractional +exceptions are summed using $H_eC'_K/B^2$, not the number of boxes. + +The claim and \eqref{eq:assembly-beta} show that retained bad compact +nodes contribute at most $(c_L/8)YV_0/B^2$ in absolute error. +Together with \eqref{eq:assembly-tail} and +\eqref{eq:assembly-good-compact}, this gives +\begin{equation}\label{eq:assembly-retained} + \sum_{d\in\mathcal T}|N_d-\mu_d| + \le\frac{3c_L}{8}\frac{YV_0}{B^2} +\end{equation} +for the retained expanded nodes of the chosen stopped tree. +Proposition~\ref{prop:stopped-comparison} gives +\[ + \sum_{d\in\mathcal S} + \bigl(S_d(b_d)-\mu_dP_\ee(r_d,b_d)\bigr)\ge0. +\] +Apply \eqref{eq:stopped-comparison-identity}, +Lemma~\ref{lem:tree-lower-bound} and +\eqref{eq:assembly-reference}. We obtain +\[ + S_1(B)\ge\left(1-\frac38\right)c_L\frac{YV_0}{B^2} + \ge\frac{c_L}{2}\frac{YV_0}{B^2}. +\] +Every constant is independent of the prescribed residue classes. +This proves Theorem~\ref{thm:survivors}. +\end{proof} + +\subsection{Removing the remaining divisor primes} + +The factor $\log w$ in the survivor estimate allows a cutoff smaller +than $k\log k$. We choose it so that the loss from the at most $k$ +remaining divisor primes is a fixed fraction of the surviving count. + +\begin{proof}[Proof of Theorem~\ref{thm:main}] +By Theorem~\ref{thm:survivors} and Mertens' formula, there is an +absolute $c_0>0$ such that, for every sufficiently large real $z$, +every assignment of one forbidden class at each prime $p\le z$ +leaves at least +\begin{equation}\label{eq:assembly-survivor-lower} + c_0\frac{Y\log w}{L^2} +\end{equation} +integers in $[1,Y]$. Indeed, +$V_0/B^2\sim e^{-\gamma}\log w/L^2$. + +Let $k$ be sufficiently large, put +$\ell=\log k$ and $M=\log\log k$, and take +\begin{equation}\label{eq:assembly-jacobsthal-cutoff} + z=A_0\frac{k\log k}{\log\log k}, +\end{equation} +where the absolute constant $A_0>1$ will be fixed below, before $k$ +tends to infinity. With the parameters of \eqref{eq:parameters}, +\begin{equation}\label{eq:assembly-jacobsthal-scales} + L\sim\ell,\qquad \log w\sim M,\qquad + Y\sim A_0^2\frac{k^2}{M^2},\qquad + X:=\frac Yz\sim A_0\frac{k}{\ell M},\qquad + \log X\sim\ell. +\end{equation} +In particular $z\to\infty$ and $X\to\infty$ for every such fixed +$A_0$. + +Let $n$ be any positive integer with $\omega(n)\le k$, and let the +given interval of $Y$ consecutive integers begin at an arbitrary +integer $a$. Represent its integers as $a+j-1$, $1\le j\le Y$. +For each prime $p\mid n$ with $p\le z$, prescribe the forbidden class +\[ + j\equiv1-a\pmod p. +\] +At the other primes $p\le z$, prescribe any class. The estimate +\eqref{eq:assembly-survivor-lower} counts integers that avoid every +divisor prime at most $z$, together with these additional restrictions. + +Fix a prime divisor $q>z$ of $n$. Its divisibility class has at most +$1+Y/q\le1+X$ representatives in the coordinate interval. Enclose +them in a progression segment whose coordinate length is +$\lceil X\rceil+2$. Fix a sufficiently small absolute $\theta>0$ +and sieve this segment by primes at most $X^\theta$. This cutoff is +below $z$ for large $k$, the step $q$ is invertible modulo every sieve +prime, and the integers already counted in +\eqref{eq:assembly-survivor-lower} avoid its prescribed class there. +Lemma~\ref{lem:small-sieve} and Mertens' formula therefore give +\begin{equation}\label{eq:large-prime-deletion} + \#\{\text{these survivors divisible by }q\} + \le C_1\left(1+\frac X{\log X}\right) +\end{equation} +for an absolute $C_1$, uniform in $q$, $A_0$, the classes, and the +translation once the displayed length conditions hold. More +explicitly, the enclosing coordinate length is comparable to $X$ +and at least $(X^\theta)^{1/(2\theta)}$ for large $X$, so the fixed +exponent upper bound in Lemma~\ref{lem:small-sieve} applies. +Enclosing a shorter or empty progression only increases the count; +in particular the argument also covers $q>Y$. + +There are at most $k$ such further divisor primes. For each fixed +$A_0$, \eqref{eq:assembly-jacobsthal-scales} gives +\begin{equation}\label{eq:all-large-deletions} + \frac{k(1+X/\log X)}{Y\log w/L^2} + =\frac1{A_0}+o(1). +\end{equation} +For the second term in the numerator, the quotient is +$kL^2/(z\log X\log w)\to1/A_0$; the term $k$ is negligible since +$Y\log w/L^2\asymp_{A_0}k^2/(\ell^2M)$. +Choose once and for all $A_0>2C_1/c_0$. Equations +\eqref{eq:large-prime-deletion} and \eqref{eq:all-large-deletions} +then show that fewer than the number in +\eqref{eq:assembly-survivor-lower} are removed, for all sufficiently +large $k$. At least one integer in the original interval is therefore +coprime to $n$. + +This argument is uniform in the prime divisors of $n$ and in every +integer $a$, including negative starting points and intervals containing +zero. Prime multiplicities do not enter. The length estimate in +\eqref{eq:assembly-jacobsthal-scales}, with $A_0$ now absolute, gives +\[ + h(k)\ll\frac{k^2}{(\log\log k)^2} + \ll\frac{k^2}{(\log\log(3k))^2} +\] +for all sufficiently large $k$, since +$\log\log(3k)/\log\log k\to1$. + +To include the remaining finitely many values of $k$, suppose $n$ has +$r\le k$ distinct prime divisors. Inclusion--exclusion gives at least +\[ + m\prod_{p\mid n}\left(1-\frac1p\right)-2^r + \ge\frac m{k+1}-2^k +\] +coprime integers in every interval of $m$ consecutive integers. +For the product bound, order the distinct primes as +$p_1<\cdots\cdots>p_m>w$, write +\[ + x_j=x(p_j),\qquad + r_j=r_0-\sum_{i=1}^j x_i,\qquad + v_j=\log_w J_{p_1\cdots p_j}=r_j+a_\star-2. +\] +An endpoint means a tuple, together with its length, in the included +tree of Section~\ref{sec:tree}. Its prefixes are the preceding tuples +on the same branch. For a nonempty tuple its strict cutoff is +$b_d=x_m$; the root has the weak cutoff $B$. +For a family $\mathcal A$ of endpoints, put +\begin{equation}\label{eq:boxes-harmonic-measure} + \mathfrak m(\mathcal A)=\sum_{d\in\mathcal A}\frac1d. +\end{equation} +Different lengths are counted separately. This is the natural measure +because $\mu_d=YV_0/d$. + +Here is the error accounting that the construction will serve. Let +$\mathcal S$ be any prefix-free collection of included lower nodes at +which the original tree is stopped, and let $\mathcal T$ be its retained +nonstopped nodes. Expanding the reference tree only as far as these +stops and then using the original tree identity gives +\begin{align} + S_1(B)\ &\ge YV_0 P_\ee(r_0,B) + +\sum_{d\in\mathcal S} + \bigl(S_d(b_d)-\mu_dP_\ee(r_d,b_d)\bigr) + -\sum_{d\in\mathcal T}|N_d-\mu_d| . + \label{eq:boxes-error-ledger} +\end{align} +All stopped nodes have even length, so their exact-subtree corrections +have a positive sign. By Lemma~\ref{lem:small-sieve} and the +exponentially weighted tail estimate in +Proposition~\ref{prop:prime-occupation}, +\begin{equation}\label{eq:boxes-large-gap-error} + \lim_{K\to\infty}\limsup_{z\to\infty} + \frac{B^2}{YV_0} + \sum_{\substack{d\text{ included}\\r_d>K}}|N_d-\mu_d|=0. +\end{equation} +Also, for each fixed $K$, +\begin{equation}\label{eq:boxes-small-error} + \sum_{\substack{d\text{ included},\ r_d\le K\\ + |N_d/\mu_d-1|\le\eps}} + |N_d-\mu_d| + \le C_K\eps\,\frac{YV_0}{B^2}. +\end{equation} +These estimates are over the full included tree, so they remain valid +after any stops have been imposed. We must control the remaining +compact endpoints, at which the relative error exceeds $\eps$. +At all included endpoints Lemma~\ref{lem:small-sieve} also gives +$|N_d/\mu_d-1|\ll_{a_\star}1$. Thus discarding a family +$\mathcal A$ of compact endpoints costs +$O_{a_\star}(YV_0\mathfrak m(\mathcal A))$ in +\eqref{eq:boxes-error-ledger}. + +\subsection{A recent witnessing edge} + +\begin{lemma}[Witnessing edge]\label{lem:edge-witness} +Fix $K$ and $0<\eps<1$. One can choose a fixed +$K_*>K+10$ such that, with +\[ + H_e=\lceil 2K_*+10\rceil,\qquad + \delta=\frac{\eps}{4H_e}, +\] +the following holds for all sufficiently large $z$, uniformly in the +prescribed residue classes. If an included endpoint $d$ has +$r_d\le K$ and $|N_d/\mu_d-1|>\eps$, then one of its last $H_e$ edges, +with parent $d'$ and child prime $u$, satisfies +\begin{equation}\label{eq:boxes-witness} + \left|N_{d'u}-\frac{N_{d'}}u\right| + >\delta\frac{\mu_{d'}}u, + \qquad + x(u),\ \log_w J_{d'}\le 2K_*+3, + \qquad + \log_w J_{d'u}\ge a_\star. +\end{equation} +\end{lemma} + +\begin{proof} +Choose $K_*$ so that Lemma~\ref{lem:small-sieve} gives +$|N_e/\mu_e-1|<\eps/2$ whenever $\log_w J_e\ge K_*$. Along the +given branch the numbers $v_j$ decrease strictly, by more than one +at each step. Let $i$ be the first index for which $v_i\delta$. +\end{proof} + +\subsection{Bins, boxes, and regular endpoints} + +Fix for the moment $0<\xi<1$. Partition $(w,z]$ into right-closed +bins of equal logarithmic width, using +\[ + n_{\rm bin}=\left\lceil\frac{\log(z/w)}{\log(1+\xi)}\right\rceil + \quad\text{bins},\qquad + h=\frac{\log(z/w)}{n_{\rm bin}\log w} +\] +for their width in the exponent coordinate $x=\log_w p$. +Thus $h\asymp\xi/\log w$ for fixed $\xi$, and the ratio of the upper +and lower endpoints of a bin is at most $1+\xi$. A bin with lower +endpoint $U$ has +\begin{equation}\label{eq:boxes-bin-count} + n(U)\asymp\frac{\xi U}{\log U} +\end{equation} +primes, uniformly for $w\le U0$ to be chosen below, a bin wholly +contained in $[\alpha_{\min}B,\alpha_{\max}B]$ in exponent coordinates will be +called an isolated-bin candidate. When at least one such bin is +used, designate the first such position as the \emph{isolated +position}; its prime will be denoted by $p$. + +\begin{proposition}[Regular boxes]\label{prop:regular-boxes} +Fix $K$ and $K_*>K+10$. For every $\vartheta>0$ there are fixed +constants $C_{\rm len}$ and $\alpha_{\min}>0$, followed by constants +$\xi_0>0$ and $C_{\rm th}<\infty$, with the following properties. +They are chosen before $z$ tends to infinity, and $C_{\rm th}$ is +independent of $0<\xi\le\xi_0$. Put +\begin{equation}\label{eq:boxes-delta} + \Delta=2C_{\rm len}\xi. +\end{equation} +For each fixed $0<\xi\le\xi_0$ and sufficiently large $z$, there is +a family $\mathcal G$ of regular included endpoints with $r_d\le K$ +such that +\begin{equation}\label{eq:boxes-exception} + \mathfrak m\bigl(\{d\text{ included}:r_d\le K\}\setminus\mathcal G\bigr) + \le \frac{\vartheta+C_{\rm th}\Delta+o_\xi(1)}{B^2}. +\end{equation} +The term $o_\xi(1)$ tends to zero after all fixed parameters, +including $\xi$, have been chosen. The statement is uniform in +the prescribed residue classes. The regular endpoints and their +covering full boxes have these additional properties. + +\begin{enumerate}[label=\textup{(\roman*)}] +\item Every regular endpoint has length at most +$C_{\rm len}\log B$ and has an isolated position. Every used +search bin, including its isolated bin, has multiplicity one. +Every position satisfying +\[ + x_j\le2K_*+3,\qquad r_{j-1}\le2K_*+5 +\] +also has multiplicity one. The isolated position precedes every +edge supplied by Lemma~\ref{lem:edge-witness}. + +\item For every nonempty even prefix of a regular endpoint, +\begin{equation}\label{eq:boxes-threshold-clearance} + r_j-\mathcal H(x_j)>10\Delta. +\end{equation} +Moving prime choices anywhere within their fixed bins changes any +prefix gap by at most $\Delta$, and changes a threshold expression +$r_j-\mathcal H(x_j)$ by at most $\Delta+2h$. In particular the +representative gaps and exponents of every regular box satisfy +\begin{equation}\label{eq:boxes-representative-safety} + r_j^{\rm rep}\ge \mathcal H(x_j^{\rm rep})+3\Delta + \quad\text{at every nonempty even prefix}. +\end{equation} + +\item Let $\mathfrak B$ be the distinct full boxes containing at +least one regular endpoint. There is a constant $C_K^{\rm box}$, +depending only on a fixed enlargement of the endpoint compact +range and the allowed starting ratios, such that +\begin{equation}\label{eq:boxes-expanded-mass} + \sum_{\mathcal B\in\mathfrak B}\mathfrak m(\mathcal B) + \le\frac{C_K^{\rm box}}{B^2}. +\end{equation} +In particular this constant is independent of $\xi$, $\vartheta$, and +the total number of boxes. Every tuple in these boxes is a +standard path from initial gap $r_0+1$, ends at gap at most $K+2$ +for that starting gap, and has its actual final +$\log_wJ\ge3a_\star/4$. + +\item In a regular box, declare an edge a candidate if it is +among the last $H_e=\lceil2K_*+10\rceil$ positions, is after the +isolated position, has multiplicity one, and can satisfy the size +bounds in \eqref{eq:boxes-witness} for some tuple in the box. +There are at most $H_e$ candidates, and every witness for a regular +endpoint is among them. Throughout the whole box of each candidate, +writing its parent as $d'=pq$ and its prime as $u$, one has +\begin{equation}\label{eq:boxes-candidate-slack} + \log_w\frac{Y}{pq}\le2K_*+4, + \qquad + \log_w\frac{Y}{pqu}\ge\frac{3a_\star}{4}. +\end{equation} +The bins for $p$ and $u$ have lower endpoints $R$ and $U$ satisfying +\begin{equation}\label{eq:boxes-prime-ranges} + z^{\alpha_{\min}}\le R\le z^{\alpha_{\max}}, + \qquad w\le U\le w^{2K_*+4}. +\end{equation} +The choices of $p$, $q$, $u$, and the descendants after this edge +are independent under the full-box measure. All their prime +factors are disjoint. If $S_q$ is the product of the bin upper +endpoints for the factors of $q$, with their specified +multiplicities, then +\begin{equation}\label{eq:boxes-sq} + 1\le\frac{S_q}{q}\le(1+\xi)^{C_{\rm len}\log B}. +\end{equation} + +\item Every regular box has at least $c_7\log w$ search +positions, for an absolute $c_7>0$ and sufficiently large $z$. +These positions have distinct bins. All its nonsearch positions +follow all its search positions. +\end{enumerate} +\end{proposition} + +\begin{proof} +We first establish the exclusions leading to \eqref{eq:boxes-exception}. +The compact-history estimate in +Proposition~\ref{prop:prime-occupation} permits each geometric +exception to be estimated in the tilted prime chain, at a cost at +most $C_KB^{-2}$ times its occurrence probability. Histories with +a ratio exceeding $S_*=(\log B)^2$ cost $O_A(B^{-A})$ in the +original measure, by Lemma~\ref{lem:high-states}. On the remaining +histories, Proposition~\ref{prop:prime-coupling} compares the prime +chain with the continuous chain up to +\[ + N_*=O((\log B)^3) +\] +steps. Its failure probability tends to zero, and its matched +states satisfy $|s-\widetilde s|\le h_0$ and +$|\log(r/\widetilde r)|\ll N_*h_0$, with $h_0$ smaller than every +fixed negative power of $\log B$. These errors remain negligible +in all polynomially many opportunities below. We work with the +continuous comparison where useful, retaining the indicated slack +when returning to prime paths. + +\paragraph{Length.} +The iid regeneration cycles from Lemma~\ref{lem:regeneration} have +finite mean length and positive finite mean log decrement. Their +exponential moments imply finite variances, so Chebyshev's inequality +applied to the iid cycle sums gives their respective weak laws. +Inverting the positive mean decrement and bounding the last incomplete +cycle then gives a positive limiting log decrement per step in +probability. Indeed, the largest length among $O(\log B)$ cycles is +$o(\log B)$ in probability by the exponential moment. The initial +segment is uniformly tight for the allowed starting ratios. +Choose a fixed $C_{\rm len}$ larger than twice the reciprocal of +this speed. The probability that more than $C_{\rm len}\log B$ +steps occur before the accumulated decrement exceeds +$\log r_0+O(1)$ tends to zero. An included compact endpoint has +$r_d\ge2$, so its accumulated decrement is at most that quantity. +The coupling transfers this conclusion. The excluded harmonic +mass is $o(B^{-2})$. + +\paragraph{An isolated position.} +Put $\theta=\alpha_{\max}/4$. A path from the initial gap to a +fixed compact must cross below $\theta B$. After the first step, +each transition of a standard path leaves at least half of the +old gap: an even parent has next exponent at most half its gap, +and an odd parent with an admitted even child has next exponent +at most one third of its gap. The first step cannot itself make +this crossing when $z$ is large. Thus the crossing arrival has +\[ + .4\theta B\le r\le\theta B. +\] +Consider the slightly enlarged gap band +$[.3\theta B,1.1\theta B]$. By Lemma~\ref{lem:band-bounds}, the +probability of any arrived-at state in this band with ratio +$s>T$ tends to zero uniformly in $B$ as $T\to\infty$. +Indeed the expected count is bounded by a constant times the +tail integrals of the invariant densities, together with the +corresponding first-step density; all these tails tend to zero. + +Choose $T$ so that this probability costs less than $\vartheta/2$ in +units of $B^{-2}$. At the crossing arrival with $s\le T$, its last +exponent is $x=r/s$, so +\[ + \frac{.4\theta}{T}B\le x\le\theta B. +\] +Choose $0<\alpha_{\min}<.2\theta/T$. The entire bin of this +factor then lies in $[\alpha_{\min}B,\alpha_{\max}B]$ for large +$z$, also after the coupling errors. Designate the first such +used bin. Since $\alpha_{\min}B\to\infty$, its position precedes +every possible witnessing edge of +Lemma~\ref{lem:edge-witness}. Also +$\alpha_{\min}B>w^{1/4}+h$ eventually, so its bin is a search bin. + +\paragraph{Repeated search bins.} +If two positions have the same bin label, some two adjacent +positions have that label. Write $x'$ for the earlier exponent, +$x$ for the next one, $s=r/x'$ for the ratio at their parent, and +$t=r/x-1$ for the next ratio. Then +\begin{equation}\label{eq:boxes-repeat-width} + 0\le t-(s-1)=r\left(\frac1x-\frac1{x'}\right) + \le\frac{s h}{x}. +\end{equation} +For a search bin, $x\ge w^{1/4}$. On low-state histories, the +continuous comparison must therefore draw into a neighborhood of +its lower support endpoint of width at most +\[ + O\left((S_*+1) + \left(\frac{h}{w^{1/4}}+N_*h_0\right)\right). +\] +The densities of the kernels in \eqref{eq:tilted-kernel} on such +pieces are bounded by a polynomial in $S_*$, as proved in the +prime-coupling argument. Summing this conditional probability +over at most $N_*$ draws gives $o(1)$. Here every polynomial in +$\log B$ is negligible compared with $w^{1/4}$, and the same +polynomials times $h_0$ tend to zero. We discard all these repeats. + +\paragraph{Repeated bins at a possible terminal edge.} +Next discard any repetition involving a position with +$x_j\le2K_*+3$ and $r_{j-1}\le2K_*+5$. It suffices to compare +with an adjacent position in the same bin. If that neighbor is +the preceding position, the comparing draw is the edge at $j$; +if it is the following position, the comparing draw is at $j+1$. +In either case the parent gap is already in a fixed compact. +Since all used exponents exceed 1, its ratios and those of the +comparing draw are bounded in terms of $K_*$. Formula +\eqref{eq:boxes-repeat-width} now gives a width $O_{K_*}(h)$, +enlarged under coupling to $O_{K_*}(h+N_*h_0)$. +Only boundedly many draws can have parent gap in this compact +before cutoff exit, because each such step decreases the gap by +more than 1. The kernel densities are bounded there. The +probability of this exception is consequently $o_\xi(1)$. +This verifies all multiplicity assertions in part~(i), and every +witness has the required singleton bin because its parent +$r\le2K_*+5$. + +\paragraph{Near-threshold even prefixes.} +Discard paths having a nonempty even prefix for which +$0\le r_j-\mathcal H(x_j)\le10\Delta$. This event is estimated before +imposing the length restriction, so it suffices to bound its +probability by $O(\Delta)+o(1)$. +At its preceding odd state, let the gap be $R$ and the next ratio +be $t$. Then $x=R/(t+1)$ and $r_j=Rt/(t+1)$. If $x\ge2$, +\[ + r_j-\mathcal H(x)=\frac{R(t-2)}{t+1}, +\] +so the allowed ratios lie within $O(\Delta/R)$ of 2. If $12). +\] +Then $\omega(u)\to e^{-\gamma}$ as $u\to\infty$, where +$\gamma=\lim_{n\to\infty}(H_n-\log n)$ and +$H_n=\sum_{j=1}^n1/j$. +\end{lemma} + +\begin{proof} +Successive integration on unit intervals constructs a unique positive +continuous function, locally absolutely continuous and with bounded +one-sided derivatives on each such interval. For $u>2$ its equation +gives, almost everywhere, +\begin{equation}\label{eq:elementary-buchstab-derivative} + u\omega'(u)=\omega(u-1)-\omega(u) + =-\int_{u-1}^{u}\omega'(v)\,dv. +\end{equation} +Put $M_n=\mathop{\rm ess\,sup}_{nM_{n-1}$ this inequality forces $M_n=0$, a contradiction; +otherwise it gives $M_n\le M_{n-1}/n$. Thus $M_n\le1/n!$. +In particular $\omega'$ is integrable on $[1,\infty)$, so $\omega$ +is bounded and has a finite limit, say $\ell$. + +It remains to determine $\ell$. For $t>0$ define +\[ + W(t)=\int_1^\infty\omega(u)e^{-tu}\,du. +\] +Boundedness justifies differentiation and integration by parts. +The derivative of $u\omega(u)$ is zero on $(1,2)$; integrating it +on $[1,\infty)$, including the boundary value at $1$, gives +\[ + -e^{-t}-tW'(t) + =\int_2^\infty\omega(u-1)e^{-tu}\,du + =e^{-t}W(t). +\] +Since $W(t)\to0$ as $t\to\infty$, solving this differential equation +yields +\begin{equation}\label{eq:elementary-buchstab-transform} + 1+W(t)=\exp\!\left(\int_t^\infty\frac{e^{-s}}s\,ds\right). +\end{equation} +The constant in this formula can be evaluated without a Tauberian +argument. Integration by parts shows that +\[ + \int_t^\infty\frac{e^{-s}}s\,ds+\log t + =(1-e^{-t})\log t+\int_t^\infty e^{-s}\log s\,ds. +\] +Moreover, +\begin{equation}\label{eq:elementary-euler-integral} + \int_0^\infty e^{-s}\log s\,ds=-\gamma. +\end{equation} +For an elementary verification, substitution $s=nv$ and integration +of the geometric sum give +\[ + \int_0^n(1-s/n)^n\log s\,ds + =\frac{n}{n+1}\bigl(\log n-H_{n+1}\bigr). +\] +Indeed, +\[ + \int_0^1(1-v)^n\log v\,dv + =-\frac1{n+1}\int_0^1\frac{1-(1-v)^{n+1}}v\,dv + =-\frac{H_{n+1}}{n+1}. +\] +The integrands in the first identity, extended by zero for $s>n$, +are bounded in absolute value by $e^{-s}|\log s|$. Dominated +convergence proves \eqref{eq:elementary-euler-integral}. +Consequently \eqref{eq:elementary-buchstab-transform} gives +$tW(t)\to e^{-\gamma}$ as $t\downarrow0$. +On the other hand, substituting $v=tu$ and using bounded convergence +gives +\[ + tW(t)=\int_t^\infty\omega(v/t)e^{-v}\,dv\longrightarrow\ell. +\] +This identifies $\ell=e^{-\gamma}$. +\end{proof} + +This derivation supplies precisely the independently normalized limit +needed to rule out a jump at $2$ in the proof of +Lemma~\ref{lem:future-integrals}. It makes no use of the auxiliary +Markov chain or of the future-integral identities proved there. + +\subsection{The additive large sieve at the required order} + +The sharp large sieve is due to Montgomery and Vaughan +\cite{MontgomeryVaughan1973}; see also \cite{VaughanLargeSieve}. +The following separated-arcs argument gives the order of +\eqref{eq:large-sieve-input} that is used in the inverse estimate. + +\begin{lemma}\label{lem:elementary-large-sieve} +Let $x_1,\ldots,x_R\in\R/\Z$ have pairwise circular distance at least +$\delta$, where $0<\delta\le1$. Let $J\ge1$ be an integer. If complex +coefficients $b_j$ are supported on $J$ consecutive integers, then +\[ + \sum_{r=1}^{R}\left|\sum_j b_j e(jx_r)\right|^2 + \ll (J+\delta^{-1})\sum_j|b_j|^2 +\] +with an absolute implied constant. Hence +\eqref{eq:large-sieve-input} holds for every real $Q\ge1$. +\end{lemma} + +\begin{proof} +Shifting the frequencies multiplies their sum by a complex number of +modulus one. We may therefore write +$T(x)=\sum_{j=0}^{J-1}b_j e(jx)$ and $f(x)=|T(x)|^2$. +Let $I_r$ be the arc of length $\delta$ centred at $x_r$. +Their interiors are disjoint. For $t\in I_r$, the fundamental theorem +of calculus along that arc gives +\[ + f(x_r)\le f(t)+\int_{I_r}|f'(x)|\,dx. +\] +Average over $t\in I_r$ and sum over $r$ to obtain +\[ + \sum_r f(x_r) + \le\delta^{-1}\int_0^1|T(x)|^2\,dx + +\int_0^1|f'(x)|\,dx. +\] +Since $|f'|\le2|T'||T|$, Cauchy--Schwarz and Parseval give +\[ + \int_0^1|f'(x)|\,dx + \le2\|T'\|_2\|T\|_2 + \le4\pi(J-1)\sum_{j=0}^{J-1}|b_j|^2. +\] +This proves the asserted estimate. Distinct reduced fractions modulo +one with denominators at most $Q$ have circular distance at least +$Q^{-2}$: subtracting two such fractions, or subtracting an integer +from their difference, leaves a nonzero integer numerator over the +product of their denominators. Apply the estimate with +$\delta=Q^{-2}$. When there is just one fraction, the same arc +argument applies directly. +\end{proof} + +\subsection{A degree-uniform plane-curve intersection bound} + +The appeals to B\'ezout's theorem in Lemma~\ref{lem:nonlinear-points} +require a bound uniform in both degrees and coefficients. Here is the +needed affine form, proved by the elementary resultant argument. +Only distinct points are counted; multiplicities are unnecessary. + +\begin{lemma}\label{lem:elementary-bezout} +Let $F,G\in\mathbb C[X,Y]$ be nonzero polynomials of respective total +degrees $m,n\ge1$, with no common nonconstant factor. Then they have +at most $mn$ common zeros in $\mathbb C^2$. +\end{lemma} + +\begin{proof} +First choose linear coordinates in which the coefficients of $Y^m$ +in $F$ and $Y^n$ in $G$ are nonzero constants. Such coordinates exist: +choose the direction of the $Y$-axis outside the zeros of the two +highest homogeneous parts. Each part excludes only finitely many +directions in the projective line. Scaling the polynomials makes +these leading coefficients one. + +Put $K=\mathbb C(X)$. The two polynomials are coprime in $K[Y]$. +Indeed, clearing denominators in a common factor and removing common +factors of its coefficients would give a common polynomial factor +in $\mathbb C[X,Y]$. This is the elementary Gauss lemma: the product +of two polynomials whose coefficients have no common divisor again +has coefficients with no common divisor, as follows by reduction +modulo any irreducible divisor in $\mathbb C[X]$. + +Consider the linear map over $K$ +\begin{equation}\label{eq:elementary-sylvester-map} + (A,B)\longmapsto AF+BG, + \qquad \deg_Y A0$ such that, for every integer +$q\ge1$, every integer $a$ with $(a,q)=1$, and every real $x$ and +$H>0$ with $J=H/q\ge J_0$, +\[ + \#\{x2),\qquad + (sF(s))'=f(s-1)\quad(s>3). +\] +These prescriptions determine the functions successively on intervals of +length one. We shall also use \(F(s)=A/s\) on \(0.95\le s<1\). +Only at a starting lower state, and when defining its derivative weight, +we continue \(f\) below \(2\) by +\begin{equation}\label{eq:functions-starting-extension} + f_{\rm ext}(s)=\frac{A\log(s-1)}s\qquad(1.98\le s<2). +\end{equation} +This is the same expression as the ordinary \(f\) on \(2\le s\le4\). +An omitted lower child of ratio less than \(2\) always has benchmark +\(f=0\); the continuation \eqref{eq:functions-starting-extension} is not +used at such a child. + +Put \(f_\ee=f\) and \(f_\oo=F\). The continuous benchmark at a state +of type \(i\), gap \(r\), and cutoff \(b\) is \(f_i(r/b)/b\). +For fixed \(r\), the delay equations give +\begin{equation}\label{eq:functions-benchmark-cutoff} + \frac{d}{db}\left(\frac{f_i(r/b)}b\right) + =-\frac{f_{i'}(r/b-1)}{b^2}, +\end{equation} +where \(i'\) is the opposite type. On the right, the value at an +omitted lower child is defined to be zero, including when its formal +ratio is negative. On the ordinary ranges the identity holds for +\(r/b>2\) at a lower state and \(r/b>1\) at an upper state. +At an upper state the right side is zero when \(r/b-1\le2\), exactly +as required for an omitted lower child. Integrating with respect to the +cutoff therefore gives the standard continuous tree recurrence above +exponent \(2\), with the boundary value at cutoff \(2\) left to be +specified. The explicit starting extensions satisfy the same identity +in the interiors of their ranges, with one-sided derivatives at the +endpoints; the extension of \(f\) is used at the parent lower state, +never at an omitted child. + +The root lies at \(r_0/B=2-a_\star/B+o(B^{-1})\), where the ordinary +lower-sieve function vanishes. Its continued benchmark is +\[ + \frac{f_{\rm ext}(r_0/B)}B + =-a_\star e^\gamma B^{-2}+o(B^{-2}), +\] +since \(f'_{\rm ext}(2)=e^\gamma\). Thus a positive reference margin +must come from the finite boundary, after its discrepancy has been +propagated through all larger exponents. To evaluate that contribution, +we will use the positive derivative weights of \(f,F\) to normalize the +harmonic transitions. The first step is to establish positivity and +decay in a form that justifies this change of measure. + +Let \(\rho\) be the Dickman function, determined by +\[ + \rho(u)=1\quad(0\le u\le1),\qquad + u\rho'(u)=-\rho(u-1)\quad(u>1). +\] +For the unextended functions put +\[ + a(s)=1-f(s)\quad(s\ge0),\qquad + c(s)=F(s)-1\quad(s\ge1),\qquad + c_0(s)=\frac{A\rho(s-1)}s\quad(s\ge1). +\] + +\begin{lemma}[Future integrals]\label{lem:future-integrals} +The functions \(a,c\) are positive and satisfy, with absolute comparison +constants, +\begin{align} + a(s)+c(s)&=c_0(s),& + a(s)&\asymp c_0(s),& + c(s)&\asymp c_0(s)\qquad(s\ge1),\label{eq:functions-deficits}\\ + s a(s)&=\int_{s-1}^{\infty}c(t)\,dt + &&(s\ge2),\label{eq:functions-future-a}\\ + s c(s)&=\int_{s-1}^{\infty}a(t)\,dt + &&(s\ge1).\label{eq:functions-future-c} +\end{align} +For \(u\ge2\), +\begin{equation}\label{eq:functions-dickman-ratio} + \frac1{4u^2}\le\frac{\rho(u)}{\rho(u-1)}\le\frac1u . +\end{equation} +Moreover, for each fixed \(\lambda>0\), there is \(U_\lambda\) such that +\begin{equation}\label{eq:functions-dickman-logderivative} + -\frac{\rho'(u)}{\rho(u)}\ge\lambda\qquad(u\ge U_\lambda). +\end{equation} +\end{lemma} + +\begin{proof} +The delay equation gives the integral identity +\begin{equation}\label{eq:functions-dickman-integral} + u\rho(u)=\int_{u-1}^{u}\rho(v)\,dv\qquad(u\ge1). +\end{equation} +For example, both sides agree at \(u=1\), and their derivatives agree +where the delay equation applies. The identity and the delay equation +give positivity and monotonicity: a first zero of \(\rho\) would contradict +the positive integral of its immediately preceding values. +Monotonicity gives the upper bound in +\eqref{eq:functions-dickman-ratio}. Integrating first over the lower +half of each of two unit intervals gives +\[ + \rho(u)\ge\frac{\rho(u-\tfrac12)}{2u} + \ge\frac{\rho(u-1)}{4u(u-\tfrac12)} + \ge\frac{\rho(u-1)}{4u^2}. +\] +The upper ratio bound also shows that \(\rho\) tends to zero faster +than any exponential. + +The negative logarithmic derivative is at least \(1\). If it is at +least \(\lambda\) on a tail, then, on a later tail, +\[ + \int_{u-1}^{u}\rho(v)\,dv + \le \rho(u-1)\frac{1-e^{-\lambda}}{\lambda}. +\] +Equation \eqref{eq:functions-dickman-integral} consequently improves +the lower bound to \(\lambda/(1-e^{-\lambda})\). +Iteration tends to infinity, since this map is strictly larger than its +argument and has no finite positive fixed point. This proves +\eqref{eq:functions-dickman-logderivative}. + +Directly from the delay equations and the initial conditions, +\begin{equation}\label{eq:functions-gap-sum} + F(s)-f(s)=c_0(s),\qquad + F(s)+f(s)=A\omega_{\rm Bu}(s)\qquad(s\ge1). +\end{equation} +Here the continuous Buchstab function is determined by +\(s\omega_{\rm Bu}(s)=1\) on \([1,2]\) and +\((s\omega_{\rm Bu}(s))'=\omega_{\rm Bu}(s-1)\) above \(2\); +its limit is stated in Section~\ref{sec:inputs}. +For the first equality, the equation for +\(s(F-f)\) is the negative delay equation for \(A\rho(s-1)\); +for the second, it is the Buchstab equation with initial value \(A/s\). +We next prove the individual positivity and future integrals; these do +not follow merely by taking the difference in +\eqref{eq:functions-gap-sum}. + +To prove these properties, we use an auxiliary absorbing chain. This +chain establishes positivity of the deficits; the harmonic-path chain +will instead be defined from the derivative weights below. +For \(s\ge2\), give the auxiliary chain transition density +\begin{equation}\label{eq:functions-auxiliary-kernel} + \frac{c_0(t)}{s c_0(s)}\,\1_{\{t\ge s-1\}}\,dt, +\end{equation} +and absorb it on reaching \([1,2)\). Its normalization is +\[ + \int_{s-1}^{\infty}c_0(t)\,dt + =A\int_{s-2}^{\infty}\frac{\rho(v)}{v+1}\,dv + =A\rho(s-1)=s c_0(s). +\] +The integration at infinity is justified by the decay already proved. +The excess of the next state above \(s-1\) has tail +\[ + \frac{\rho(s-1+h)}{\rho(s-1)}\qquad(h\ge0). +\] +For large \(s\), \eqref{eq:functions-dickman-logderivative} bounds this +tail by \(e^{-\lambda h}\), with \(\lambda\) as large as desired. +For \(\lambda>1\), therefore, +\[ + \E(e^{s_{\rm next}}\mid s) + \le e^{s-1}\frac{\lambda}{\lambda-1}. +\] +Taking \(\lambda=2\) makes this a strict contraction of \(V(s)=e^s\), +with factor \(2/e<1\), outside a fixed compact. +Stopped drift implies almost sure return to that compact. From any +state in the compact, successive draws in +\([s-1,s-1+1/4]\) reach \([1,2)\) within a bounded number of steps, +with probability bounded below independently of the initial state +in the compact. The densities in these intervals have a positive +compact lower bound. Repeating the attempt after each return proves +almost sure absorption. + +Alternate labels \(a,c\) at every draw. At the terminal state \(t\) +assign fractions \(1/c_0(t)\) and \(1-1/c_0(t)\) to the two labels, +respectively. Since \(c_0(t)=A/t\) on \([1,2)\), both terminal +fractions lie in a fixed compact subinterval of \((0,1)\). +The expected terminal fraction, multiplied by \(c_0(s)\), defines +positive functions \(a^\circ(s),c^\circ(s)\) with +\[ + a^\circ+c^\circ=c_0,\qquad + a^\circ,c^\circ\asymp c_0, +\] +and terminal values \(a^\circ=1,\ c^\circ=c_0-1\) on \([1,2)\). +The first-step equations give +\[ + s a^\circ(s)=\int_{s-1}^{\infty}c^\circ(t)\,dt,\qquad + s c^\circ(s)=\int_{s-1}^{\infty}a^\circ(t)\,dt + \qquad(s\ge2). +\] +In particular these functions are continuous from the right at \(2\). + +Set \(U=(2+c^\circ-a^\circ)/A\). The last two equations imply +\((sU(s))'=U(s-1)\) above \(2\), in integrated form, while +\(sU(s)=1\) on \([1,2)\). +Also \(U(s)\to 2/A=e^{-\gamma}\), because \(a^\circ,c^\circ\) +are bounded by \(c_0\). +There can be no jump at \(2\) relative to the continuous Buchstab +solution. Indeed, a jump \(\delta\) in \(sU(s)\) there would give +a difference \(\delta H(s)\), where \(H(s)=1/s\) on \([2,3]\) and +\((sH(s))'=H(s-1)\) thereafter. The bound \(H\ge1/3\) on +\([2,3]\) persists by induction: if it holds up to an integer \(n\ge3\), +then for \(n\le s\le n+1\), +\[ + sH(s)=nH(n)+\int_n^s H(v-1)\,dv\ge n/3+(s-n)/3. +\] +For either sign of \(\delta\ne0\), this contradicts +\(U(s)-\omega_{\rm Bu}(s)\to0\), using the independently proved Buchstab limit in +Lemma~\ref{lem:elementary-buchstab}. +Thus \(U=\omega_{\rm Bu}\). Together with +\(a^\circ+c^\circ=c_0\) and \eqref{eq:functions-gap-sum}, this identifies +\(a^\circ=a,\ c^\circ=c\), proving +\eqref{eq:functions-deficits} and both future integrals for \(s\ge2\). + +Finally \(c(s)=A/s-1\) and \(a=1\) below \(2\). +The equation at \(s=2\) gives +\(\int_1^\infty a=A-2\); hence for \(1\le s\le2\), +\[ + \int_{s-1}^{\infty}a(t)\,dt=2-s+A-2=A-s=s c(s). +\] +This proves the remaining range of \eqref{eq:functions-future-c}. +\end{proof} +\subsection{Derivative weights and change of measure} + +The deficits are now positive and have the required decay. We use +derivatives of the sieve functions to normalize the harmonic transitions +of the standard tree. + +Define +\[ + \phi_\ee(s)=s^2f'(s)\quad(s\ge2),\qquad + \phi_\oo(s)=-s^2F'(s)\quad(s\ge0.95). +\] +At \(s=2\) use the right derivative, and on \(1.98\le s<2\) use +the derivative of \eqref{eq:functions-starting-extension}. +The state weight will be \(\phi_i(r/b)/r^2\); its reciprocal at the +endpoint is the factor that appears when we convert a probability +estimate back to harmonic weight. For paths from the root, this gives +the scale \(B^{-2}\) when the endpoint gap and ratio remain in fixed +compact ranges. +Put +\[ + W_0(t)=\frac{t+1}{t^2},\qquad + m_\ee(s)=s-1,\qquad m_\oo(s)=\max(2,s-1). +\] + +\begin{lemma}[Derivative weights]\label{lem:derivative-weights} +The derivative weights are positive and nonincreasing on their indicated +ranges. If \(i'\) denotes the type opposite to \(i\), then +\begin{equation}\label{eq:functions-derivative-integrals} + \phi_i(s)=\int_{m_i(s)}^{\infty}W_0(t)\phi_{i'}(t)\,dt. +\end{equation} +This holds also on the specified starting extensions. +On a tail, +\begin{equation}\label{eq:functions-phi-dickman} + \phi_\ee(s)\asymp\rho(s-2),\qquad + \phi_\oo(s)\asymp\rho(s-2). +\end{equation} +There are absolute constants \(C,q\) such that the absolute logarithmic +derivatives, where they exist, and the ratios +\(\phi_{i'}(m_i(s))/\phi_i(s)\) are at most \(C(1+s)^q\). +In particular, for \(u,v\le S\) in the indicated domain of the +same weight, with \(S\ge3\), +\[ + |\log\phi_i(u)-\log\phi_i(v)| + \le C(1+S)^q|u-v|. +\] +For each fixed \(\lambda>0\), the negative logarithmic derivatives +of both weights are at least \(\lambda\) on a sufficiently remote tail. +\end{lemma} + +\begin{proof} +On the ordinary ranges the delay equations give +\begin{equation}\label{eq:functions-phi-deficits} + \phi_\ee(s)=s\bigl(a(s)+c(s-1)\bigr),\qquad + \phi_\oo(s)=s\bigl(c(s)+a(s-1)\bigr). +\end{equation} +The second expression uses \(a=1\) on \([0,1]\). +On the extensions, the explicit formulas are +\[ + \phi_\ee(s)=A\left(\frac{s}{s-1}-\log(s-1)\right),\qquad + \phi_\oo(s)=A\quad(0.95\le s\le3). +\] +These formulas and Lemma~\ref{lem:future-integrals} prove positivity. +Differentiation gives +\begin{align} + \phi_\ee'(s)&=-\frac{s}{(s-1)^2}\phi_\oo(s-1) + &&(s\ge1.98),\label{eq:functions-phi-e-derivative}\\ + \phi_\oo'(s)&=-\frac{s}{(s-1)^2}\phi_\ee(s-1) + &&(s>3),\label{eq:functions-phi-o-derivative} +\end{align} +at points of differentiability; the odd weight is constant below \(3\). +Equation \eqref{eq:functions-phi-deficits} and +\eqref{eq:functions-deficits} give \eqref{eq:functions-phi-dickman}, +and in particular both weights vanish at infinity. +Integrating \eqref{eq:functions-phi-e-derivative} and +\eqref{eq:functions-phi-o-derivative} backwards from infinity gives +\eqref{eq:functions-derivative-integrals}. +For the even extension, the lower endpoint lies below \(1\), where +the explicitly constant odd weight gives exactly the displayed +continuation. For odd \(s\le3\), the lower endpoint is \(2\), so the +integral is constant, as required. + +On a tail, \eqref{eq:functions-dickman-ratio} and +\eqref{eq:functions-phi-dickman} give +\[ + \frac{\phi_{i'}(s-1)}{\phi_i(s)}=O(s^2). +\] +The derivative formulas then bound the absolute logarithmic +derivatives by a polynomial (in fact \(O(s)\) on a tail). +On the remaining compact ranges the explicit positive functions +and their piecewise derivatives give fixed bounds. +The weights are locally absolutely continuous by +\eqref{eq:functions-derivative-integrals}; integrating the +logarithmic derivative gives the stated Lipschitz bound. +Finally the same comparison gives a positive constant times +\[ + \frac{\rho(s-3)}{s\rho(s-2)} + =\frac{s-2}{s}\left(-\frac{\rho'(s-2)}{\rho(s-2)}\right) +\] +as a lower bound for either negative logarithmic derivative on a +tail. Apply \eqref{eq:functions-dickman-logderivative}. +\end{proof} + +A \emph{standard} path replaces the admission condition +\eqref{eq:admission} by \(r-x\ge2x\) at an upper node. +Thus it contains every path in the original tree, and the two rules +agree for \(x\ge2\). +At a state of type \(i\), gap \(r\), cutoff \(b\), and ratio \(s=r/b\), +make a draw \(t\) with probability density +\begin{equation}\label{eq:tilted-kernel} + K_i(s,dt)= + \frac{W_0(t)\phi_{i'}(t)}{\phi_i(s)} + \1_{\{t\ge m_i(s)\}}\,dt. +\end{equation} +Lemma~\ref{lem:derivative-weights} proves that this is a probability +kernel. The next exponent, gap, and ratio are +\[ + x=\frac r{t+1},\qquad r'=\frac{rt}{t+1},\qquad s'=t. +\] +The lower support endpoint ensures \(x\le b\), and the upper-to-lower +transition also satisfies \(s'\ge2\). +Put \(g(t)=\log(1+1/t)\), so \(r'=r e^{-g(t)}\). + +The harmonic transition \(dx/x\), after reversing orientation in +\(t=r/x-1\), is \(dt/(t+1)\). Dividing this by +\eqref{eq:tilted-kernel} gives +\[ + \frac{t^2}{(t+1)^2}\frac{\phi_i(s)}{\phi_{i'}(t)} + =\left(\frac{r'}r\right)^2 + \frac{\phi_i(s)}{\phi_{i'}(t)}. +\] +Consequently, for every finite prefix, +\begin{equation}\label{eq:path-weight} + \text{original harmonic weight} + = + \text{probability weight}\, + \left(\frac{r_{\rm end}}{r_{\rm in}}\right)^2 + \frac{\phi_{i_{\rm in}}(s_{\rm in})} + {\phi_{i_{\rm end}}(s_{\rm end})}. +\end{equation} +For a nonempty prefix its last exponent is +\(x=r_{\rm end}/s_{\rm end}\). Since successive cutoffs are +nonincreasing, restriction of all exponents to \(x>\ell\) is +equivalent to the single arrival condition +\begin{equation}\label{eq:functions-arrival-cutoff} + s_{\rm end}1\) such that +\(\E(V_{\rm next}\mid i,s)\le V(i,s)/\alpha\) outside \(C_0\). +Stopping the nonnegative supermartingale +\(\alpha^{n\wedge\sigma}V(i_{n\wedge\sigma},s_{n\wedge\sigma})\), +where \(\sigma\) is the first entrance to \(C_0\), gives +\[ + \E_{i,s}\alpha^\sigma\le V(i,s) +\] +for starts outside \(C_0\), by Fatou's lemma. +In particular the entrance time is almost surely finite. + +From \(C_0\) there are an integer \(m\) and a probability \(p>0\) +such that regeneration is reached within \(m\) steps with probability +at least \(p\), uniformly in the starting state. +To see this, whenever \(s>3\) choose the next state in the interval +of width \(1/4\) immediately above its support minimum; this lowers +the state by at least \(3/4\). +An even state with \(s\le3\) then draws into the regenerating odd +range on the same choice. +All prescribed intervals lie in a fixed compact and have uniformly +positive transition probabilities. + +Run such an \(m\)-step attempt, and on failure wait for the next +entrance to \(C_0\) before retrying. Let \(D\) be the duration of +one failed attempt together with this return. The stopped estimate +and the uniform bound on the \(V\)-moment after the \(m\) attempted +steps show that, for some \(\alpha_1>1\), +\[ + \sup_{(i,s)\in C_0}\E_{i,s} + [\alpha_1^D\1_{\{\text{failure}\}}]<\infty. +\] +For \(0<\theta<1\), Hölder's inequality bounds the same expression +with \(\alpha_1^\theta\) by +\[ + \left(\sup_{C_0}\E[ + \alpha_1^D\1_{\{\text{failure}\}}]\right)^\theta + (1-p)^{1-\theta}. +\] +For sufficiently small \(\theta>0\) this is less than \(1\). +Summing over successive failed attempts proves an exponential +moment for the hitting time of regeneration from \(C_0\), and then +from \eqref{eq:functions-start-range}. +The same reasoning applies to a return after a regenerating start: +its outgoing common draw has a bounded \(V\)-moment. + +The common outgoing law proves independence and identical +distribution of the complete cycles. During a cycle all odd +states are at least \(1\), and all even states are at least \(2\), +so each \(g(s_j)\le\log2\). +Thus the exponential moment of \(\tau\) supplies an exponential +moment of \(G\) for a sufficiently small parameter. +The terminal odd state lies in \([1,3]\), so its last decrement +alone is at least \(g(3)>0\). + +For nonarithmeticity, consider a two-step return whose even state +\(t\) lies in a fixed subinterval of \((2,3)\) and whose next odd +state \(u\) lies strictly between \(t-1\) and \(3\). +There is a two-dimensional region of such pairs on which the +joint density is positive. On that region +\(G=g(t)+g(u)\) varies continuously and has a nonzero derivative +in \(u\). Its distribution therefore has a nonzero absolutely +continuous component, and cannot be supported on an arithmetic +lattice. +\end{proof} + +\begin{lemma}[Invariant densities and cycle occupation] +\label{lem:invariant} +On the ordinary even and odd state spaces, respectively, the finite +raw invariant densities are +\begin{equation}\label{eq:functions-invariant-densities} + \pi_\ee(s)=\phi_\ee(s)\quad(s\ge2),\qquad + \pi_\oo(s)=(1-s^{-2})\phi_\oo(s)\quad(s\ge1). +\end{equation} +Put +\[ + \beta=\int_1^3\pi_\oo(s)\,ds=\frac{4A}{3}. +\] +For every nonnegative measurable state function \(Q\), +\begin{equation}\label{eq:functions-cycle-occupation} + \E\sum_{j=1}^{\tau}Q(i_j,s_j) + =\frac1\beta\sum_i\int Q(i,s)\pi_i(s)\,ds. +\end{equation} +In particular, if +\[ + M_g=\sum_i\int\pi_i(s)g(s)\,ds, +\] +then \(\E G=M_g/\beta\), and +\begin{equation}\label{eq:mg-bound} + 00\). Taking \(Q(i,s)=g(s)\) in +\eqref{eq:functions-cycle-occupation} identifies \(\E G\). +\end{proof} + +\subsection{Renewal limits and uniform band estimates} + +We use tests in the log-gap coordinate \(u=\log r\). +An admissible compact test \(F_i(u,s)\) is bounded and continuous on +the extended state ranges, with support in a fixed compact rectangle +in \((u,s)\). We also permit multiplication by indicators of fixed +intervals in \(u\) or \(s\), and by +\(\1_{\{sb\), +\[ + |H_0(v)|\le + \|F\|_\infty + \E\!\left[\tau\, + \1_{\{\tau\ge(v-b)/\log2\}}\right]. +\] +The exponential moment in Lemma~\ref{lem:regeneration} gives an +exponentially decreasing upper bound as \(v\to\infty\). + +The function \(H_0\) is continuous, including for the permitted +indicator tests. To verify this, condition on the history before +the \(j\)-th draw and on the event \(j\le\tau\), which is determined +before that draw. The next state \(t=s_j\) has a density. +Writing \(D=e^{v-S_{j-1}}>0\), its gap is +\[ + e^{v-S_j}=\frac{Dt}{t+1}. +\] +Equality at a fixed positive gap edge therefore specifies at most +one value of \(t\); equality at a fixed state edge does likewise. +For the arrival boundary, +\[ + t=e^{v-S_j}/\ell + \quad\Longleftrightarrow\quad + t+1=D/\ell. +\] +Thus each boundary has conditional probability zero. +For any fixed number of cycle steps dominated convergence gives +continuity. The remaining steps are uniformly bounded in expectation +by \(\|F\|_\infty\E[\tau\1_{\{\tau>N\}}]\), which tends to zero. +This proves continuity of \(H_0\). +Continuity, the vanishing left tail, and the exponential right +envelope show that \(H_0\) is directly Riemann integrable. + +Let \(U_G\) be the renewal measure of successive complete-cycle +decrements, including the boundary at zero: +\[ + U_G=\sum_{k\ge0}\mathcal L(G_1+\cdots+G_k). +\] +The expectation of the complete future test from a regeneration is +\begin{equation}\label{eq:functions-renewal-convolution} + R_F(v)=\int_{[0,\infty)}H_0(v-y)\,U_G(dy). +\end{equation} +By Lemma~\ref{lem:regeneration}, \(G\) is nonarithmetic and has +finite positive mean. The key renewal theorem +\eqref{eq:renewal-input} therefore gives +\[ + R_F(v)\longrightarrow \frac1{\E G}\int_\R H_0(u)\,du. +\] +Fubini is justified by the bounded test, its compact log-gap +support, and \(\E\tau<\infty\). Translating the log-gap integral +within each cycle and applying +\eqref{eq:functions-cycle-occupation} gives +\[ + \int_\R H_0(u)\,du + =\E\sum_{j=1}^{\tau}\int_\R F_{i_j}(u,s_j)\,du + =\frac1\beta\sum_i\int\!\!\int F_i(u,s)\pi_i(s)\,ds\,du. +\] +Since \(\E G=M_g/\beta\), this is the limit in +\eqref{eq:functions-occupation-limit}. + +We record a boundedness fact needed to restore the initial segment. +Every complete cycle has decrement at least \(g(3)>0\). +Consequently any interval of fixed length contains a bounded number +of cycle boundaries on each sample path, and +\begin{equation}\label{eq:functions-renewal-band} + \sup_{y\in\R}U_G([y,y+h])\le + 1+\left\lfloor\frac{h}{g(3)}\right\rfloor + \qquad(h>0). +\end{equation} +The compact bound and the exponential envelope for \(H_0\), summed +over unit intervals, show from +\eqref{eq:functions-renewal-convolution} that +\(\sup_v|R_F(v)|<\infty\). + +Now let \(\tau_0\) be the first regenerating time from an allowed +even start, and \(D_0=T_{\tau_0}\). +All initial per-step decrements are bounded by +\(c_*=\log(1+1/0.98)\). Lemma~\ref{lem:regeneration} gives a +uniform exponential moment for \(\tau_0\), hence a uniformly +tight, indeed exponentially bounded, family of \(D_0\)'s. +The portion of the test through time \(\tau_0\) tends to zero +uniformly as \(v\to\infty\): a contribution requires +\(c_*\tau_0\ge v-b\), and its absolute size is at most +\(\|F\|_\infty(\tau_0+1)\). +Conditional on the initial history, the future after that state +has expectation \(R_F(v-D_0)\). +It is the same function \(R_F\) for every history, because the +outgoing regenerating law is common. +For a fixed \(M\), convergence of \(R_F(y)\) as \(y\to\infty\) +is uniform over \(y=v-d,\ 0\le d\le M\). +Uniform tightness of \(D_0\) and boundedness of \(R_F\) then give +the claimed uniform limit. +\end{proof} + +\begin{lemma}[Uniform band bounds]\label{lem:band-bounds} +For every fixed \(h>0\) there is \(C_h<\infty\) with the following +properties. For any nonnegative measurable state function \(Q\) +and any \(v\in\R\), a regenerating start satisfies +\begin{equation}\label{eq:functions-band-from-regeneration} + \E\sum_{n\ge1}Q(i_n,s_n) + \1_{\{T_n\in[v,v+h]\}} + \le C_h\sum_i\int Q(i,s)\pi_i(s)\,ds. +\end{equation} +From an even start in \eqref{eq:functions-start-range}, the same +bound holds for the sum over \(n\ge2\), uniformly in the starting +ratio. The density of the first odd state is at most +\[ + C\,\1_{\{s\ge0.98\}}\phi_\oo(s). +\] +In particular, uniformly in \(r_{\rm in}>0\), \(R>0\), and +\(\ell\in\{1,2\}\), +\begin{equation}\label{eq:functions-inverse-phi-band} + \E\sum_{n\ge1} + \frac{\1_{\{R\le r_n\le eR,\ s_n0\). +Since \(t+1\ge3\), it follows that +\[ + \int_1^{t+1}W_0(u)\int_2^{u+1}W_0(y)\,dy\,du + \gg \int_2^{t+1}W_0(u)\,du + \gg \int_{s_0-1}^{t+1}W_0(u)\,du, +\] +uniformly for \(s_0\in[1.99,2.3]\). +The last comparison uses the bounded integral over +\([0.99,2]\) and the positive lower bound for +\(\int_2^{t+1}W_0\). +The starting denominator \(\phi_\ee(s_0)\) also has a positive +compact lower bound. Thus +\begin{equation}\label{eq:functions-initial-density-domination} + p_{s_0,2}(t)\le C p_{{\rm reg},3}(t)\qquad(t\ge2). +\end{equation} + +To use this marginal comparison, the correlated elapsed decrement +must be removed explicitly. Let \(I=[v,v+h]\), +\(C_2=2c_*\), and \(C_3=3\log2\). +The initial two decrements sum to at most \(C_2\), so an event +in \(I\) at time \(n\ge2\) has, measured from state \(2\), a future +decrement in +\[ + I_2=[v-C_2,v+h]. +\] +Condition on that state, discard the initial decrement by this +enlargement, and apply +\eqref{eq:functions-initial-density-domination}. +For a path started at regeneration, its first three decrements +sum to at most \(C_3\); a future decrement in \(I_2\) measured +from state \(3\) therefore gives a total decrement in +\[ + [v-C_2,v+h+C_3]. +\] +The resulting expectation is bounded by a constant times the +regenerating occupation sum in this enlarged band, which is +covered by \eqref{eq:functions-band-from-regeneration}. +This proves the bound from even starts, including their state +at time \(2\), without imposing independence on the initial +state and elapsed decrement. + +At step \(1\), the density is +\[ + \frac{W_0(s)\phi_\oo(s)}{\phi_\ee(s_0)} + \1_{\{s\ge s_0-1\}}. +\] +Both \(W_0(s)\) for \(s\ge0.99\) and +\(1/\phi_\ee(s_0)\) on the allowed starts are bounded. +This proves the asserted first-step estimate. + +The condition \(R\le r_n\le eR\) is a band of length \(1\) in +\(T_n\). For \(n\ge2\), use the preceding result with +\[ + Q(i,s)=\frac{\1_{\{s\ell\) and \(r_n\le K\), +\eqref{eq:functions-arrival-cutoff} confines the state to a +fixed compact with a positive lower endpoint. Hence +\(r_n^2/\phi_{i_n}(s_n)\) is bounded. +Moreover, each such transition decreases the gap by more than +\(\ell\); there are only \(O_K(1)\) such compact arrivals on +one path. Formula \eqref{eq:path-weight} therefore bounds the +total original harmonic weight of compact prefixes by +\(O_K(r_{\rm in}^{-2})\). +If the counted prefixes can occur only on an event \(\mathcal A\) +of the tilted path, the bound is +\(O_K(r_{\rm in}^{-2})\Pbb(\mathcal A)\). + +Retain the restriction \(x_n>\ell\), equivalently +\(s_n0\) and \(\eps>0\). There is a constant +\(C_\eps>10\), independent of \(b_0\), such that, whenever +\(b_1\ge C_\eps b_0\) is fixed, the following event has +probability at least \(1-\eps-o(1)\) as \(r_{\rm in}\to\infty\), +uniformly in \eqref{eq:functions-start-range}. +At a regenerating odd state with pre-draw gap in +\([10b_0,b_1]\), the next draw is an even arrival satisfying +\[ + s\in[2.08,2.14],\qquad x\in[2b_0,b_1/2]. +\] +The proof yields the smaller ratio interval \([2.09,2.13]\); +thus the stated ratio window has positive slack. +The arriving gap exceeds \(6b_0\), so, if \(6b_0>K\), such a visit +precedes every subsequent arrival of gap at most \(K\). +\end{lemma} + +\begin{proof} +Mark a complete regeneration cycle if its first draw lies in +\([2.09,2.13]\). The common outgoing kernel gives a fixed +probability +\[ + p=\frac1A\int_{2.09}^{2.13}W_0(t)\phi_\ee(t)\,dt + \in(0,1). +\] +The complete cycle objects, consisting of their state sequences +and their decrements, are iid. +Consequently the spacings \(W\) between successive marked cycle +beginnings are iid: each consists of one cycle conditioned on +being marked, followed by a geometric number of cycles conditioned +on being unmarked. More explicitly, the number \(N\) of intervening +unmarked cycles has +\(\Pbb(N=m)=(1-p)^m p\), \(m\ge0\); conditional on these marks, +the cycle objects are independent with their respective +conditional laws. +The next marked cycle is excluded from the preceding spacing +and begins the next such block. This proves independence of +the spacings, even though a mark depends on its own cycle. +It also gives +\[ + \E W= + \E(G\mid\text{marked}) + +\frac{1-p}{p}\E(G\mid\text{unmarked}) + =\frac{\E G}{p}<\infty. +\] +Furthermore \(W\ge g(3)>0\). + +From an allowed initial start, the delay \(D\) to the first +marked beginning consists of the initial segment to regeneration +and a geometric number of unmarked complete cycles. +The first segment has uniformly bounded mean by +Lemma~\ref{lem:regeneration}; the latter sum has finite mean by +the preceding conditional-cycle description. +Thus \(D\) is uniformly tight over the allowed initial states. +The spacings after that beginning are independent of the delay. + +We give the required wide-interval bound using only +\(\E W<\infty\). +Let \(U_W\) be the renewal measure for the marked spacings, +with an initial renewal at zero. +The lower bound on \(W\) gives a uniform bound \(C\) for its +mass in any interval of length one. +For \(q\ge0\), if there is no marked renewal in \([q,q+H]\), +some renewal at or before \(q\) has its next spacing overshoot +\(q+H\). Independence of a renewal time and its next spacing +therefore gives +\[ + \Pbb(\text{no renewal in }[q,q+H]) + \le\int_{[0,q]}\Pbb(W>q+H-t)\,U_W(dt) + \le C\sum_{j\ge0}\Pbb(W>H+j). +\] +Changing one endpoint in the unit-interval partition only changes +the harmless absolute constant or shifts \(H\) by \(1\). +The last quantity tends to zero as \(H\to\infty\), since \(W\) +has finite mean. It is uniform in \(q\). +For the delayed process, condition on \(D\le v\) and apply the +same bound with \(q=v-D\). The additional error +\(\Pbb(D>v)\) tends uniformly to zero as \(v\to\infty\). + +Apply this with +\[ + v=\log(r_{\rm in}/b_1),\qquad + H=\log\bigl(b_1/(10b_0)\bigr). +\] +Choose \(H\) sufficiently large in terms of \(\eps\), equivalently +choose \(b_1/b_0\) sufficiently large, and then let +\(r_{\rm in}\to\infty\). +A marked beginning in this log-decrement interval has pre-draw +gap \(r\in[10b_0,b_1]\). +For its marked draw \(t\in[2.09,2.13]\), +\[ + \frac{10b_0}{3.13}\le x=\frac r{t+1} + \le\frac{b_1}{3.09}. +\] +This lies inside \([2b_0,b_1/2]\), and the arriving ratio is \(t\). +The new gap satisfies +\[ + r'=\frac{rt}{t+1}\ge + 10b_0\frac{2.09}{3.09}>6b_0. +\] +Gaps strictly decrease along the path, so the last assertion +follows. +\end{proof} diff --git a/preprints/A-quadratic-bound-for-Jacobsthals-function-September-25-2026/build/sections/hyperbola.tex b/preprints/A-quadratic-bound-for-Jacobsthals-function-September-25-2026/build/sections/hyperbola.tex new file mode 100644 index 0000000000000000000000000000000000000000..35e26f7b52320aceb39e3852c56eef06d7529d6e --- /dev/null +++ b/preprints/A-quadratic-bound-for-Jacobsthals-function-September-25-2026/build/sections/hyperbola.tex @@ -0,0 +1,299 @@ +\section[A modular hyperbola count]{A modular hyperbola count with residue restrictions} +\label{sec:hyperbola} + +The large-modulus case of Proposition~\ref{prop:structured-variance} +requires a rectangle count with a congruence restriction on the unit +coordinate of a modular hyperbola. We give the completion argument with +an explicit error, retaining multiplicities when an interval is longer +than the modulus. + +\begin{lemma}[Completion with a unit residue restriction] +\label{lem:hyperbola} +Let $H,T_0,T_1$ be positive integers such that $H\ge2$, $T_0$ is +squarefree, every prime divisor of $T_0$ divides $H$, and +$(T_1,HT_0)=1$. Put $N=HT_0$. Let $C_0\in\Z$ satisfy $(C_0,H)=1$, +and prescribe the residue pattern +\begin{equation} +\label{eq:hyperbola-pattern} + p_0\in(\Z/T_0\Z)^\times,\qquad k_0\in\Z/T_0\Z,\qquad + p_1\in(\Z/T_1\Z)^\times,\qquad m_1\in\Z/T_1\Z. +\end{equation} +The residue classes modulo $1$ have their usual unique interpretation. +Let $I,J$ be bounded real intervals of lengths $X,Y_2\ge0$, with either +endpoint independently included or excluded. For a positive integer +$l$ with $(l,HT_1)=1$, let $C_l$ be the number of integer pairs $(p,m)$ +satisfying +\begin{equation} +\label{eq:hyperbola-counted-set} +\begin{gathered} + p\in I,\qquad m\in J,\qquad l\mid p,\\ + p\equiv p_0\pmod{T_0},\qquad + p\equiv p_1\pmod{T_1},\qquad + m\equiv m_1\pmod{T_1},\\ + pm\equiv C_0+Hk_0\pmod N. +\end{gathered} +\end{equation} +Here $p$ is an integer variable, without a primality condition. +Uniformly in all these data, +\begin{equation} +\label{eq:hyperbola-explicit-count} + C_l= + \frac{XY_2\varphi(N)}{lT_1^2N^2\varphi(T_0)} + +O\left( + \tau(N)^2\log^2(2N)\sqrt{NT_0} + \left(1+\frac{X}{lN}+\frac{Y_2}{N}\right) + \right), +\end{equation} +with an absolute implied constant. + +In particular, suppose that $R\to\infty$ and the data range uniformly +over families satisfying +\begin{equation} +\label{eq:hyperbola-application-range} + R^{4/5}0$ the error in +\eqref{eq:hyperbola-application-count} is $O_\delta(R^{3/5+\delta})$ +for sufficiently large $R$ in that family. No lower bound for $X/l$ +is required. +\end{lemma} + +\begin{proof} +Write $e_n(t)=\exp(2\pi i t/n)$. We use the classical complete +Kloosterman estimate, in the form recorded in +\cite[Lemma~7.1]{Lichtman2022}: +\begin{equation} +\label{eq:hyperbola-weil} + \Kl(a,b;n):=\sum_{\substack{u\bmod n\\(u,n)=1}} + e_n(au+b\bar u), + \qquad + |\Kl(a,b;n)|\ll\tau(n)n^{1/2}(a,b,n)^{1/2}. +\end{equation} +This bound permits arbitrary composite $n$ and arbitrary integer +frequencies, including zero frequencies. The inverse $\bar u$ is +taken modulo $n$. + +\paragraph{Changing the first coordinate.} +Choose a representative of $k_0$ and put $c=C_0+Hk_0$ modulo $N$. +The integers $H$ and $N$ have the same prime support. Since +$(C_0,H)=1$, it follows that $(c,N)=1$. The hypothesis on $l$ also +implies $(l,NT_1)=1$. On setting $p=lv$, define +\[ + \begin{gathered} + U=X/l,\qquad V=Y_2,\qquad c'=c\bar l\pmod N,\\ + a_0=p_0\bar l\pmod{T_0},\qquad + a_1=p_1\bar l\pmod{T_1}. + \end{gathered} +\] +Every inverse here is taken in its indicated modulus. The interval +$I_l=\{x/l:x\in I\}$ has length $U$ and the induced endpoint +conventions. The pairs now satisfy +\[ + \begin{gathered} + v\in I_l,\quad v\equiv a_1\pmod{T_1},\qquad + m\in J,\quad m\equiv m_1\pmod{T_1},\\ + v\equiv a_0\pmod{T_0},\qquad vm\equiv c'\pmod N. + \end{gathered} +\] +In particular $v$ and $m$ are automatically units modulo $N$. + +\paragraph{Fourier weights and their divisor-weighted sums.} +For $h,k\bmod N$, define the unnormalized weights +\begin{equation} +\label{eq:hyperbola-fourier-weights} + A(h)=\sum_{\substack{v\in I_l\cap\Z\\v\equiv a_1\ (T_1)}}e_N(-hv), + \qquad + B(k)=\sum_{\substack{m\in J\cap\Z\\m\equiv m_1\ (T_1)}}e_N(-km). +\end{equation} +These are sums over the integers in the progressions, so a residue +modulo $N$ is counted with its full multiplicity. They therefore apply +unchanged when $U$ or $V$ exceeds $N$. Counting the progressions gives +\begin{equation} +\label{eq:hyperbola-zero-weights} + A(0)=U/T_1+O(1),\qquad B(0)=V/T_1+O(1). +\end{equation} +For a nonzero residue $h\bmod N$, let +$|h|_N=\min_{j\in\Z}|h+jN|$. A geometric-progression sum, with ratio +$e_N(-hT_1)\ne1$, gives +\begin{equation} +\label{eq:hyperbola-geometric} + |A(h)|,\ |B(h)|\ll\frac{N}{|hT_1|_N}\qquad(h\not\equiv0\pmod N). +\end{equation} +The right side is independent of the number of complete periods and +of the locations of the intervals. + +Multiplication by $T_1$ permutes the nonzero residues modulo $N$ and +preserves their greatest common divisors with $N$. Thus +\begin{equation} +\label{eq:hyperbola-gcd-sum} +\begin{split} + \sum_{h\ne0\ (N)}\frac{(h,N)^{1/2}}{|hT_1|_N} + &=\sum_{r\ne0\ (N)}\frac{(r,N)^{1/2}}{|r|_N}\\ + &\le\sum_{d\mid N}d^{1/2} + \sum_{\substack{r\ne0\ (N)\\d\mid r}}\frac1{|r|_N}\\ + &\ll\log(2N)\sum_{d\mid N}d^{-1/2} + \le\tau(N)\log(2N). +\end{split} +\end{equation} +Indeed the inner sum is at most +$2d^{-1}\sum_{1\le j\le N/(2d)}j^{-1}$, with a harmless possible +overcount of the midpoint. Without the gcd weight the corresponding +sum is $O(\log(2N))$. In conjunction with +\eqref{eq:hyperbola-geometric}, this proves +\begin{equation} +\label{eq:hyperbola-fourier-sums} +\begin{gathered} + \sum_{h\ne0\ (N)}|A(h)|\ll N\log(2N),\qquad + \sum_{h\ne0\ (N)}|A(h)|(h,N)^{1/2} + \ll N\tau(N)\log(2N), +\end{gathered} +\end{equation} +and the identical estimates with $B$ in place of $A$. + +\paragraph{Completion with the unit restriction.} +Define +\[ + S(h,k)= + \sum_{\substack{u\bmod N\\(u,N)=1\\u\equiv a_0\ (T_0)}} + e_N(hu+kc'\bar u). +\] +Fourier inversion, with the multiplicities in +\eqref{eq:hyperbola-fourier-weights}, gives the exact identity +\begin{equation} +\label{eq:hyperbola-completion-identity} + C_l=\frac1{N^2}\sum_{h,k\bmod N}A(h)B(k)S(h,k). +\end{equation} +For example, summing $A(h)e_N(hu)/N$ over $h$ counts all allowed +integers $v\equiv u\pmod N$, which verifies the normalization directly. + +Orthogonality modulo $T_0$ also gives the exact formula +\begin{equation} +\label{eq:hyperbola-restricted-sum} + S(h,k)=\frac1{T_0}\sum_{j\bmod T_0} + e_{T_0}(-ja_0)\Kl(h+jH,kc';N). +\end{equation} +The average over $j$ does not cost a factor $T_0$ in the following +absolute bounds. When $k\ne0\pmod N$, the fact that $(c',N)=1$ +and \eqref{eq:hyperbola-weil} give +\begin{equation} +\label{eq:hyperbola-second-nonzero} + |S(h,k)|\ll\tau(N)\sqrt N\,(k,N)^{1/2}, + \qquad k\ne0\pmod N, +\end{equation} +uniformly also when $h=0$. For $h\ne0$ and $k=0$, observe that +\begin{equation} +\label{eq:hyperbola-shifted-gcd} + (h+jH,N)\le T_0(h,N). +\end{equation} +In fact any common divisor of $h+jH$ and $N$ divides $T_0h$, because +$T_0H=N$, and $(T_0h,N)\le T_0(h,N)$. Consequently +\begin{equation} +\label{eq:hyperbola-first-nonzero} + |S(h,0)|\ll\tau(N)\sqrt{NT_0}\,(h,N)^{1/2}, + \qquad h\ne0\pmod N. +\end{equation} +If a shift $h+jH$ vanishes modulo $N$, it is still covered by +\eqref{eq:hyperbola-shifted-gcd}; no shifted zero frequency is omitted. +No squarefreeness of $H$ or $N$ has been used. + +At the two original zero frequencies, +\begin{equation} +\label{eq:hyperbola-complete-zero} + S(0,0)=\frac{\varphi(N)}{\varphi(T_0)},\qquad + \varphi(N)=T_0\varphi(H). +\end{equation} +For the first equality, reduction from the units modulo $N$ onto +the units modulo $T_0$ is surjective with equal fibers. Explicitly, +if $t^a$ exactly divides $H$ and $t\mid T_0$, then the local modulus +in $N$ is $t^{a+1}$, and every unit modulo $t$ has $t^a$ unit lifts. +At a prime of $H$ outside $T_0$ there is no prescribed local class. +The Chinese remainder theorem proves the first equality. The same +prime-power calculation gives the second. + +\paragraph{The four frequency cases.} +The contribution of $h=k=0$ to +\eqref{eq:hyperbola-completion-identity} is +\begin{equation} +\label{eq:hyperbola-main-term} + \frac{A(0)B(0)\varphi(N)}{N^2\varphi(T_0)} + =\frac{UV\varphi(N)}{T_1^2N^2\varphi(T_0)} + +O\left(\frac{U+V+1}{N}\right), +\end{equation} +where \eqref{eq:hyperbola-zero-weights} handles every endpoint +convention. This is the stated main term since $UV=XY_2/l$. + +For $h\ne0$, $k\ne0$, use +\eqref{eq:hyperbola-second-nonzero} and +\eqref{eq:hyperbola-fourier-sums}. The total absolute contribution is +\[ + \ll\frac{\tau(N)\sqrt N}{N^2} + \bigl(N\log(2N)\bigr) + \bigl(N\tau(N)\log(2N)\bigr) + =\tau(N)^2\sqrt N\log^2(2N). +\] +For $h=0$, $k\ne0$, the same argument with the zero weight retained +gives +\[ + \ll A(0)\tau(N)^2N^{-1/2}\log(2N). +\] +For $h\ne0$, $k=0$, instead use +\eqref{eq:hyperbola-first-nonzero}; this contribution is +\[ + \ll B(0)\tau(N)^2T_0^{1/2}N^{-1/2}\log(2N). +\] +Finally, $A(0)\ll U/T_1+1$ and $B(0)\ll V/T_1+1$. +These three estimates and the rounding term in +\eqref{eq:hyperbola-main-term} are all bounded by the error displayed +in \eqref{eq:hyperbola-explicit-count}. This proves the general +assertion, with an absolute implied constant. + +\paragraph{The size range needed in the variance argument.} +Under \eqref{eq:hyperbola-application-range}, +\[ + R^{4/5}0$, +$\mathcal Z=\exp(L/\log L)$, and +\[ + T_0=\prod_{\substack{t\le w\\t\mid H}}t,\qquad + T_1=\prod_{\substack{t\le w\\t\nmid H}}t. +\] +Then $\log(T_0T_1)=O(w)=o(\log R)$ and +$\log\mathcal Z=o(\log R)$, uniformly in the isolated bin. +Thus the uniform family conditions in +\eqref{eq:hyperbola-application-range} follow from the parameters of +the main argument. A fixed bin-width parameter may affect the point +at which these asymptotics apply, but it does not enter the absolute +constant in \eqref{eq:hyperbola-explicit-count}. +\end{remark} diff --git a/preprints/A-quadratic-bound-for-Jacobsthals-function-September-25-2026/build/sections/inputs.tex b/preprints/A-quadratic-bound-for-Jacobsthals-function-September-25-2026/build/sections/inputs.tex new file mode 100644 index 0000000000000000000000000000000000000000..0500d8dc33f8b23e2e3137f684bba8049aa416ab --- /dev/null +++ b/preprints/A-quadratic-bound-for-Jacobsthals-function-September-25-2026/build/sections/inputs.tex @@ -0,0 +1,206 @@ +\section{Sieve and prime-distribution inputs}\label{sec:inputs} + +We state the forms of the classical estimates used in the proof. +Appendix~\ref{sec:elementary-inputs} supplies elementary proofs of the +Buchstab normalization, the required additive large-sieve bound and the +plane-curve intersection estimate, and derives the progression upper +bound from the fundamental lemma. The inverse and stopping arguments +in the later sections are proved directly. + +\begin{lemma}[Fundamental lemma of the sieve]\label{lem:fundamental} +Let $\mathcal P$ be any subset of the primes at most $v$. Suppose that +intersection counts for bad conditions indexed by squarefree products +$l$ of primes in $\mathcal P$ have the form +\[ + Xg(l)+r_l, +\] +where $g$ is multiplicative and +\[ + 0\le g(p)\le\frac2p,\qquad g(p)<1,\qquad g(2)\le\frac12 + \quad\text{if }2\in\mathcal P. +\] +There are absolute constants $c,C>0$ and $s_0$ such that for $s\ge s_0$ +the count avoiding all the bad conditions is +\begin{equation}\label{eq:fundamental-input} + X\prod_{p\in\mathcal P}(1-g(p))\bigl(1+O(e^{-cs})\bigr) + +O\!\left( + \sum_{\substack{l\le v^s\\l\text{ squarefree}\\p\mid l\Rightarrow p\in\mathcal P}} + \tau(l)^C|r_l|\right). +\end{equation} +The same statement holds for nonnegative weighted counts. A local +condition that makes the sifted set empty may instead be removed from +consideration as an exactly empty case. +\end{lemma} + +This is a standard dimension-two form of the fundamental lemma; see +\cite[Lemma~2.8, p.~S137]{Sofos2023}, which records the stronger remainder sum +without divisor weights and attributes the result to +\cite[Corollary~6.10]{FriedlanderIwaniec2010}. To check the uniform +dimension condition here, put $g(p)=0$ outside $\mathcal P$. For $p\ge3$, +\[ + (1-2/p)^{-1}=(1-1/p)^{-2}\bigl(1+O(p^{-2})\bigr). +\] +Mertens' theorem therefore gives +\[ + \prod_{u\le p0$ such that +\[ + \pi(v)=\li(v)+O\!\left(\frac v{\log v}e^{-c\sqrt{\log v}}\right), + \qquad + \prod_{p\le v}(1-1/p) + =\frac{e^{-\gamma}}{\log v} + \bigl(1+O(e^{-c\sqrt{\log v}})\bigr). +\] +Decreasing $c$ absorbs harmless powers of $\log v$. +\item For every fixed $A,M>0$, uniformly for +$q\le(\log v)^A$ and $(a,q)=1$, +\[ + \pi(v;q,a)=\frac{\li(v)}{\varphi(q)} + +O_{A,M}\!\left(\frac v{(\log v)^M}\right). +\] +The implied constants may be ineffective. +\item For $(a,q)=1$ and $H/q$ exceeding a suitable absolute constant, +\[ + \#\{x0$, uniformly when $J\ge u^\eta$, its sifted count +$N$ satisfies +\begin{equation}\label{eq:small-upper} + N\ll_\eta JV_{\mathrm{sm}}(u). +\end{equation} +There are absolute $c_1,C_1>0$ and a sufficiently large fixed $v_0$ +such that, writing $v=\log_uJ$, uniformly for $v\ge v_0$, +\begin{equation}\label{eq:small-relative} + \left|\frac{N}{JV_{\mathrm{sm}}(u)}-1\right|\le C_1e^{-c_1v}. +\end{equation} +All estimates are independent of the progression's initial point, +step and forbidden classes. +\end{lemma} + +\begin{proof} +For any squarefree $l$ supported on the small primes, the simultaneous +bad conditions specify one class of the progression coordinate modulo +$l$. Their count is $J/l+O(1)$, with an absolute error independent of +the step and the classes. Apply Lemma~\ref{lem:fundamental} at level +$J^{1/2}$. Its sieve parameter is $v/2$, and the remainder sum is +\[ + \ll J^{1/2}(1+\log J)^{C_2} +\] +for an absolute fixed $C_2$. After division by $JV_{\mathrm{sm}}(u)$, this is +\[ + \ll J^{-1/2}(1+\log J)^{C_2}\log u. +\] +For all sufficiently large $u$ this is at most $e^{-c_1v}$ uniformly +for $v\ge v_0$, on decreasing $c_1$ and increasing $v_0$ if necessary. +The relative fundamental-lemma error has the same form. This proves +\eqref{eq:small-relative}. + +For \eqref{eq:small-upper}, sieve only by primes at most $u^\theta$, +where $0<\theta<1$ is fixed sufficiently small in terms of $\eta$. +At level $J^{1/2}$ the sieve parameter is at least +$\eta/(2\theta)$ and can therefore be made larger than $s_0$. +The upper fundamental lemma gives +\[ + N\ll JV_{\mathrm{sm}}(u^\theta)+J^{1/2}(1+\log J)^{C_2} + \ll_\eta JV_{\mathrm{sm}}(u), +\] +using Mertens and $J\ge u^\eta$. The same calculation covers either +endpoint convention and nonintegral $J$, since each intersection still +has an $O(1)$ endpoint error. +\end{proof} + +We will repeatedly apply \eqref{eq:small-upper} to short difference +pieces created by freezing an interval length. If $\xi>0$ is fixed and +the coordinate length of such a piece is comparable to $\xi J$, with +$\xi J\ge u^\eta$, its cost is $O_\eta(\xi JV_{\mathrm{sm}}(u))$. The constant does +not depend on $\xi$; the threshold from which the length inequality +holds may depend on that fixed width. The same observation applies to +a specified residue subsequence of the piece. diff --git a/preprints/A-quadratic-bound-for-Jacobsthals-function-September-25-2026/build/sections/introduction.tex b/preprints/A-quadratic-bound-for-Jacobsthals-function-September-25-2026/build/sections/introduction.tex new file mode 100644 index 0000000000000000000000000000000000000000..9fbc0cb6a34f8b13f6973fe5e281a9f3f55b2dae --- /dev/null +++ b/preprints/A-quadratic-bound-for-Jacobsthals-function-September-25-2026/build/sections/introduction.tex @@ -0,0 +1,269 @@ +\section{Introduction} + +For a positive integer $n$, let $j(n)$ be the least positive integer $m$ +such that every interval of $m$ consecutive integers contains an integer +coprime to $n$. Write $\omega(n)$ for the number of distinct prime divisors +of $n$, with $\omega(1)=0$, and put +\[ + h(k)=\sup_{\substack{n\ge1\\\omega(n)\le k}}j(n),\qquad k\ge1. +\] +The starting point of the interval is arbitrary, and replacing $n$ by +the product of its distinct prime divisors does not change $j(n)$. +Thus $h(k)-1$ is the greatest length of an interval that can be covered +by the divisibility classes of at most $k$ primes. + +Jacobsthal studied this function in a series of papers beginning in +1960 \cite{Jacobsthal1960}. Erd\H{o}s subsequently recorded Jacobsthal's question +whether $h(k)\ll k^2$ \cite[p.~163, equations (1) and (3)]{Erdos1962}. +In Erd\H{o}s's notation, $C(k)+1$ is the maximum of $j(n)$ over integers +with exactly $k$ distinct prime divisors. This equals our $h(k)$: +adjoining prime divisors cannot decrease $j(n)$. +We prove the following bound, which gives an affirmative answer and a +further iterated-logarithmic saving. + +\begin{theorem}\label{thm:main} +There is an absolute constant $C>0$ such that +\[ + h(k)\le C\frac{k^2}{(\log\log(3k))^2} + \qquad\text{for every integer }k\ge1. +\] +\end{theorem} + +All logarithms are natural. In particular the theorem gives a +uniform quadratic bound for arbitrary sets of prime divisors, with +no restriction on their sizes or on the interval's translation. + +\subsection*{Historical context and related work} + +The development of the upper bound is closely tied to sieve theory. +Erd\H{o}s observed that Brun's method gives $h(k)\ll k^{C_0}$ for some +absolute exponent $C_0$ \cite[p.~163]{Erdos1962}. +Iwaniec's work on the error term in the linear sieve gave an estimate +of order $k^2\log^2 k$ when the prime divisors are the first $k$ primes +\cite{Iwaniec1971}; this primorial case is distinguished explicitly in +\cite[p.~329]{Vaughan1977} and \cite[pp.~225--226]{Iwaniec1978}. +Vaughan then proved the uniform estimate +\[ + j(n)\ll\omega(n)^2\bigl(\log(2\omega(n))\bigr)^4 + \qquad(n>1) +\] +\cite[Theorem, p.~329]{Vaughan1977}. +Iwaniec's shifted-sieve argument removed two logarithmic factors and +established $h(k)\ll k^2\log^2 k$ for arbitrary prime sets +\cite[Theorem and Corollary, p.~226]{Iwaniec1978}. +Theorem~\ref{thm:main} improves this bound and, in particular, reaches +the quadratic scale asked for by Jacobsthal. + +Vaughan explained the exponent $2$ through the one-dimensional sieve's +restriction to moduli near the square root of the interval length +\cite[p.~329]{Vaughan1977}. In the usual linear-sieve notation this +appears as the vanishing of the lower function $f$ at sieve parameter +$2$. Our proof also reaches this limiting parameter. It keeps a +boundary contribution that disappears from the leading linear-sieve +term, and then controls the discrepancy between this reference +calculation and the actual prescribed classes. The latter comparison +requires the inverse estimate and the stopped counts developed below; +positivity of the reference calculation alone does not give a lower +bound for the original sieve. + +A separate question concerns which prime sets give the largest gap. +Let $p_k$ denote the $k$th prime and $P_k=\prod_{i=1}^k p_i$. +Jacobsthal suggested that $j(P_k)=h(k)$ for every $k$. +Hajdu and Saradha proved equality for $k\le23$, but found +\[ + j(P_{24})=234<236=h(24) +\] +\cite[Theorem~1.2]{HajduSaradha2012}. +Thus the primorial extremality conjecture is distinct from the +quadratic-bound question, and a theorem only about $j(P_k)$ does not +settle the uniform problem. The computational literature reflects +this distinction. Costello and Watts obtained recursive lower bounds +for the minimum number of integers coprime to $P_k$ in an interval, +and used them to compute upper bounds for $j(P_k)$ +\cite[Theorem~4.4 and Section~5]{CostelloWatts2015}. +Ziller computed the maximum over arbitrary prime sets through +$43$ prime factors \cite[Section~3]{Ziller2019}. +These finite computations concern explicit values, whereas the +constant in Theorem~\ref{thm:main} is not made explicit. + +Primorials nevertheless give the strongest lower bounds relevant to +the order of $h(k)$. They arise from constructions of long covered +intervals, which also produce large gaps between consecutive primes. +Erd\H{o}s's account already places Rankin's lower-bound construction +beside Brun's polynomial upper bound \cite[p.~163, equation (2)]{Erdos1962}. +The later construction of Ford, Green, Konyagin, Maynard and Tao gives +\begin{equation}\label{eq:historical-lower-bound} + j(P_k)\gg + k(\log k)^2\frac{\log\log\log k}{\log\log k} + \qquad(k\longrightarrow\infty) +\end{equation} +\cite[Section~1.1]{FGKMT2018}. +Indeed, their covered-interval bound at prime cutoff $x$ is +\[ + \gg x\log x\,\log\log\log x/\log\log x; +\] +take $x=p_k$ and use +$p_k\sim k\log k$. +Since $j(P_k)\le h(k)$, this supplies a lower bound for the uniform +maximum as well. It is an existence construction and does not assert +the same lower bound for every integer with $k$ prime divisors. +The gap between \eqref{eq:historical-lower-bound} and our upper bound +remains substantial. Vaughan suggested the stronger possible estimate +$j(n)\ll_\eps\omega(n)^{1+\eps}$ for $n>1$ and each fixed $\eps>0$ +\cite[p.~329]{Vaughan1977}. + +\subsection*{A quantitative covering estimate} + +The proof works with one prescribed residue class at each prime below +a size cutoff. This is another form of the same covering problem: +for a finite prime set, the Chinese remainder theorem supplies an +integer $b$ with $b\equiv a_p\pmod p$ at every prime in the set, and +\[ + n\not\equiv a_p\pmod p\ \text{for every such }p + \quad\Longleftrightarrow\quad + \gcd\left(n-b,\prod p\right)=1. +\] +This equivalence explains why arbitrary classes and arbitrary +translations occur together; see also +\cite[Lemma~2.1]{HajduSaradha2012}. + +For sufficiently large real $z$, set +\begin{equation}\label{eq:parameters} +\begin{gathered} + L=\log z,\qquad Y=\left\lfloor\frac{z^2}{L^2}\right\rfloor, + \qquad w=\frac{L}{(\log L)^2},\qquad B=\frac{L}{\log w},\\ + V_0=\prod_{p\le w}\left(1-\frac1p\right). +\end{gathered} +\end{equation} +Products indexed by a letter explicitly called prime always range over +primes. The two cutoffs separate the small primes, which are sieved +inside arithmetic progressions, from the primes in $(w,z]$, which +index the expansion used in the proof. The quantity $B$ is the +logarithmic size of the upper cutoff in base $w$. + +\begin{theorem}\label{thm:survivors} +There are absolute constants $c>0$ and $z_0$ with the following property. +For every real $z\ge z_0$ and every choice of a residue class +$a_p\pmod p$ for each prime $p\le z$, +\begin{equation}\label{eq:main-survivors} + \#\{1\le n\le Y:n\not\equiv a_p\pmod p\text{ for every }p\le z\} + \ge c\frac{YV_0}{B^2}. +\end{equation} +\end{theorem} + +Mertens' theorem identifies the size of this lower bound: +\begin{equation}\label{eq:survivor-scale} + \frac{YV_0}{B^2}\sim e^{-\gamma}\frac{Y\log w}{L^2}. +\end{equation} +The number of survivors, rather than positivity alone, permits the +passage from a prime cutoff to a bound in terms of the number of prime +divisors. For sufficiently large $k$, if $n$ has at most $k$ distinct +prime divisors, apply +Theorem~\ref{thm:survivors} at a cutoff +$z=A_0 k\log k/\log\log k$, where $A_0$ is a sufficiently large fixed +constant. A progression upper sieve bounds the total number of these +survivors removed by the remaining divisor primes $q>z$ by +$O(k(1+X/\log X))$, where $X=Y/z$. +The lower bound in \eqref{eq:survivor-scale} dominates this loss for +that choice of $A_0$, while $Y\asymp k^2/(\log\log k)^2$. +Section~\ref{sec:assembly} gives the comparison with constants and +handles the bounded range of $k$. + +The quantitative estimate also relates to the interval-sieve quantity +studied by Banks, Ford and Tao \cite[Sections~1.3--1.4]{BanksFordTao2023}. +They minimize the survivor count over all choices of forbidden classes +at primes up to $(y/\log y)^{1/2}$ in an interval of length $y$. +At that cutoff a classical lower sieve gives a bound of order +$y\log\log y/(\log y)^2$. +Theorem~\ref{thm:survivors} retains this order of magnitude at the +larger cutoff $z\asymp\sqrt Y\log Y$. +The classes here are arbitrary; no random choice of the forbidden +classes is assumed in this theorem. + +\subsection*{Outline of the argument} + +The proof has two parts: a reference calculation for a decreasing-prime +expansion, and a comparison with the actual residue classes. +Section~\ref{sec:tree} partitions discarded integers by the least +prime at which they hit a forbidden class. Iteration gives tuples of +strictly decreasing primes. +At even depths the expansion is a lower bound, and at odd depths it +is an upper bound. An admission rule limits the progressions retained +in this expansion so that each is long enough for an absolute upper +sieve. Replacing its small-prime count by progression coordinate +length times $V_0$ defines the reference tree. + +The leading reference term is governed by the classical linear-sieve +functions $f,F$ \cite[Section~21.2.2]{MontgomeryVaughanII}, with the +root tending to the critical parameter $2$. The lower exponent cutoff +leaves a boundary contribution on the smaller scale $B^{-2}$. +Each prime path has weight equal to the reciprocal product of its +primes. Replacing reciprocal-prime sums by harmonic integrals $dx/x$ +gives a continuous reference process. To evaluate the boundary +contribution, Section~\ref{sec:functions} uses derivative weights of +$f,F$ to normalize its transitions to a probability kernel. +Its regeneration cycles give limiting occupation measures, +which Section~\ref{sec:prime-paths} transfers to prime tuples. +Section~\ref{sec:reference} then evaluates the boundary contribution +and bounds the losses imposed by the admission rule. A positive +margin remains after these losses and the negative contribution from +the root's small displacement below parameter $2$. + +The actual small-prime counts agree with their reference values in +long progressions, by the fundamental lemma of sieve theory. +At a short progression, a large relative discrepancy must already be +visible along one of the last few edges of its path. +Section~\ref{sec:boxes} places these paths in boxes in which selected +prime factors vary independently. An inverse estimate in +Section~\ref{sec:inverse} shows that many discrepant edges force +alignment with one of finitely many rationals $A/D$: a prime $p$ +aligns when $Da_p\equiv A\pmod p$. + +The inverse step is related to the principle that modular +concentration forces algebraic structure, developed by Helfgott and +Venkatesh and by Walsh +\cite{HelfgottVenkatesh2009,Walsh2012}. +Our argument uses partial incidence constraints in a single prime bin +and obtains a rational list fixed for the whole box. +It combines polynomial interpolation with Jarn\'{i}k's +convex-arc lattice-point argument \cite{Jarnik1926} and the +pair-counting principle of Gallagher's larger sieve +\cite{Gallagher1971}. +Variance estimates in Section~\ref{sec:variance} then show that a +rational can account for many discrepant endpoints only when its +effective modulus and intercept are small. The pair-moment calculation +is a truncated form of mean singular-series averaging +\cite{Gallagher1976}; the paper proves the required uniformity and +stability under the changes of measure used here. + +For the endpoints remaining after this argument, many earlier prime +factors align with the same rational. Section~\ref{sec:stops} stops +their branches at suitable even nodes. The stop rule depends on prime +bins in a way that permits one aligned prime to vary over a large +subset of its bin. Averaging over that prime gives more exact +survivors than the reference subtree requires. A marked-renewal +argument shows that all but an arbitrarily small amount of the +exceptional path mass reaches one of these stops. + +The comparison identity \eqref{eq:stopped-comparison-identity} joins +these parts. Proposition~\ref{prop:reference-margin} supplies a positive +reference term; the inverse and variance estimates leave only the +endpoints of Corollary~\ref{cor:hard-endpoints}; and +Propositions~\ref{prop:stopped-comparison} and~\ref{prop:hard-stopping} +supply the stopped comparison and its coverage. +Section~\ref{sec:assembly} bounds the total remaining discrepancy, +proves Theorem~\ref{thm:survivors}, and deduces Theorem~\ref{thm:main}. + +\subsection*{Conventions} + +The constants in the proof are absolute after the explicitly stated +parameters have been chosen. They need not be effective. Whenever a +bin width $\xi$ occurs, it is fixed before $z$ tends to infinity. +Constants independent of sufficiently small fixed $\xi$ do not assert +uniformity for a width $\xi(z)$ tending to zero. This order matters +in the rational-list and stopping arguments. + +We use $\varphi$ for Euler's totient, $\tau$ for the divisor function, +$\rad(m)$ for the product of the distinct prime divisors of $m$, and +$\bar u$ for a multiplicative inverse in a displayed modulus. All +rationals $A/D$ used for alignment are reduced, with $D>0$. diff --git a/preprints/A-quadratic-bound-for-Jacobsthals-function-September-25-2026/build/sections/inverse.tex b/preprints/A-quadratic-bound-for-Jacobsthals-function-September-25-2026/build/sections/inverse.tex new file mode 100644 index 0000000000000000000000000000000000000000..896d203e2e47402f7de8934a5bc3ea5bc3b02f95 --- /dev/null +++ b/preprints/A-quadratic-bound-for-Jacobsthals-function-September-25-2026/build/sections/inverse.tex @@ -0,0 +1,866 @@ +\section{An inverse estimate for a witnessing edge} +\label{sec:inverse} + +The exceptional compact nodes will be controlled by varying the isolated +prime in a regular box. The first step identifies the only obstruction +to this variation: a positive proportion of the relevant primes must +share a rational intercept. The use of modular concentration to obtain +algebraic structure is related to the inverse-sieve methods of +\cite{HelfgottVenkatesh2009,Walsh2012}. More specifically, the +small-height polynomial and prime-product strategy is related to Walsh's +algebraicity method \cite[Section~3]{Walsh2014}. The partial-incidence +modular-line form needed in a prime bin, including its uniformity across +the independent boxes, is proved here rather than invoked from that work. + +\begin{definition} +A prime \(p\) \emph{aligns} with a reduced rational \(A/D\), where +\(A\in\Z\), \(D\in\N\), and \(\gcd(A,D)=1\), if +\[ + D a_p\equiv A\pmod p. +\] +This condition implies \(p\nmid D\). +For a squarefree product \(q\) of primes carrying prescribed classes, +write +\[ + q_E(A/D)= + \prod_{\substack{\ell\mid q\\ + \ell\text{ does not align with }A/D}}\ell . +\] +\end{definition} + +Throughout this section put +\begin{equation} + \mathcal Z=\exp\!\left(\frac{L}{\log L}\right). + \label{eq:inverse-z} +\end{equation} +As usual, all parameters other than \(z\) are fixed before taking a limit +in \(z\). In particular, a statement valid for every sufficiently small +fixed \(\xi>0\) allows its eventual lower threshold on \(z\) to depend on +\(\xi\). + +Fix one candidate edge in a full regular box from +\cref{prop:regular-boxes}. Write its parent as \(p q\), with \(p\) the +isolated prime, and denote its edge prime by \(u\). Let +\(\mathcal P,\mathcal U\) be their respective prime bins, let \(R,U\) be +the lower endpoints of these bins, and put +\[ + n_p=\#\mathcal P,\qquad n_u=\#\mathcal U. +\] +Both bins are right-closed and have upper-to-lower endpoint ratio at most +\(1+\xi\). Their logarithmic widths are comparable to \(\xi\); hence, +for each fixed \(\xi>0\), +\begin{equation} + n_p\asymp \frac{\xi R}{\log R}, + \qquad + n_u\asymp \frac{\xi U}{\log U}. + \label{eq:inverse-bin-counts} +\end{equation} +The constants here can be chosen uniformly for \(0<\xi\le1\), once \(z\) +is sufficiently large depending on \(\xi\). + +Let \(\mathcal Q\) be the set of possible products \(q\). Such a product +uniquely determines its sorted prime choices. Denote by \(S_q\) the +product of the upper endpoints of its specified bins, with the +prescribed multiplicities; \(S_q\) is therefore fixed for the box. +The hypotheses supplied by \cref{prop:regular-boxes} include +\begin{align} + &z^{\alpha_{\min}}\le R\le z^{\alpha_{\max}}, + \qquad \alpha_{\max}=\frac1{100}, + \qquad w\le U\le w^{2K_*+4}, \label{eq:inverse-box-ranges}\\ + &\log_w\frac{Y}{p q}\le2K_*+4, + \qquad + \log_w\frac{Y}{p q u}\ge\frac{3a_\star}{4}, \label{eq:inverse-length-ranges}\\ + &1\le \frac{S_q}{q}\le + (1+\xi)^{C_{\rm len}\log B}. \label{eq:inverse-q-range} +\end{align} +The bins of \(p\) and \(u\) each have multiplicity one in the full box. +Thus the choices of \(p,q,u\), and all descendants after the edge are +independent in the full box, and \(p,u\) divide none of the other +products involved. In particular, +\[ + \gcd(p,q)=\gcd(pq,u)=1 . +\] +Writing \(\alpha=\log R/L\), the parent-length bounds imply +\begin{equation} + q=z^{\,2-\alpha+o(1)},\qquad + S_q=z^{\,2-\alpha+o(1)}, + \qquad + \alpha_{\min}\le\alpha\le\alpha_{\max}. + \label{eq:inverse-q-size} +\end{equation} +All the \(o(1)\)'s in this section are uniform over boxes satisfying +these fixed-parameter bounds, for each permitted fixed \(\xi\). + +The normalized \emph{harmonic measure} on the box assigns mass +proportional to the reciprocal of the product of its chosen primes. +On \(p,q,u\) this is the product measure with respective weights +\(1/p,1/q,1/u\). Descendant choices may be integrated out whenever an +event depends only on \(p,q,u\). + +\begin{proposition}[Inverse estimate for an edge] +\label{prop:inverse} +Fix the parameters in \eqref{eq:inverse-box-ranges}--% +\eqref{eq:inverse-q-range}, the witnessing threshold \(\delta>0\), and +\(0<\sigma<1\). +There are constants \(c_p>0\), a list-size bound \(F<\infty\), and +\(\xi_{\rm inv}>0\), independent of the eventual choice of fixed +\(0<\xi\le\xi_{\rm inv}\), with the following property for all +sufficiently large \(z\). + +The full candidate box has a list \(\mathcal A\) of at most \(F\) +reduced rationals \(A/D\) satisfying +\begin{equation} + D\le R\mathcal Z^{10},\qquad + |A|\le S_qR\mathcal Z^{10},\qquad + \#\{p\in\mathcal P:p\text{ aligns with }A/D\}\ge c_p n_p . + \label{eq:inverse-list} +\end{equation} +Outside a set of normalized harmonic box measure at most \(\sigma\), +every occurrence of +\begin{equation} + \left|N_{pqu}-\frac{N_{pq}}u\right| + >\delta\,\frac{YV_0}{pqu} + \label{eq:inverse-bad-edge} +\end{equation} +has a rational \(A/D\in\mathcal A\) for which +\begin{equation} + p\text{ aligns with }A/D, + \qquad D q_E(A/D)\le R\mathcal Z^{10}. + \label{eq:inverse-structure} +\end{equation} +The exceptional measure is relative to the whole box, rather than to +the subset on which \eqref{eq:inverse-bad-edge} holds. +\end{proposition} + +The proof fixes the whole-box rational list before arguing by +contradiction. If too many bad edges fail \eqref{eq:inverse-structure}, +sampling their frozen residue tests produces many parent pairs but few +permissible slopes. An integer polynomial with small coefficient height +then vanishes on the corresponding integer points. The lemma below bounds +the contribution of its nonlinear components. A remaining rational line +has an intercept in the fixed list, forcing most selected pairs on that +line to satisfy \eqref{eq:inverse-structure} after all. + +The lattice-point estimate below combines algebraic decomposition with +Jarn\'{i}k's convex-arc argument \cite[Sections~1--2]{Jarnik1926}. +We give the proof with explicit degree dependence because the degree +grows with the prime bin. The coefficient-independent intersection +bound used in its proof is established in +Lemma~\ref{lem:elementary-bezout}. + +\begin{lemma}[Degree-uniform nonlinear lattice-point bound] +\label{lem:nonlinear-points} +Let \(P\in\R[X_1,X_2]\) be nonzero and of degree at most \(D_0\ge1\), +and let \(S\ge1\). +The number of lattice points in \([0,S]^2\) lying on at least one +nonlinear irreducible component of \(P=0\), with irreducibility taken +over \(\mathbb C\), is +\[ + O\!\left(D_0^4(1+S^{2/3})\right). +\] +The implied constant is absolute, independently of the coefficients +of \(P\). No bound is asserted for points lying only on line +components. +\end{lemma} + +\begin{proof} +Consider first one nonlinear irreducible factor \(Q\) of degree \(d\). +If \(Q\) is not proportional to a real polynomial, \(Q\) and its +coefficientwise conjugate are distinct irreducible polynomials. Every +real zero of \(Q\) is a common zero, so B\'ezout's theorem gives at most +\(d^2\) such points. + +Suppose instead that \(Q\) has been scaled to have real coefficients. +Neither \(Q_{X_1}\) nor \(Q_{X_2}\) vanishes identically: otherwise +irreducibility over \(\mathbb C\) would force \(Q\) to be linear. +Their degrees are smaller than \(d\), so their intersections with +\(Q=0\) give \(O(d^2)\) points in total. These include all singularities +and all horizontal or vertical tangencies. +There are also \(O(d)\) intersections with the boundary of the square. + +On a smooth graph \(X_2=f(X_1)\) with \(Q_{X_2}\ne0\), the second +derivative vanishes precisely when +\begin{equation} + \mathcal I_Q + =Q_{X_1X_1}Q_{X_2}^{\,2} + -2Q_{X_1X_2}Q_{X_1}Q_{X_2} + +Q_{X_2X_2}Q_{X_1}^{\,2} + =0. + \label{eq:inverse-inflection} +\end{equation} +Indeed \(f''=-\mathcal I_Q/Q_{X_2}^3\) on the graph. +If \(Q\nmid\mathcal I_Q\), B\'ezout's theorem adds \(O(d^2)\) +inflection points, because \(\deg\mathcal I_Q\le3d-4\). +If \(Q\mid\mathcal I_Q\), any nonspecial interior smooth graph arc has +\(f''=0\) throughout and therefore lies on a line. Vanishing on that +arc would make the line a factor of \(Q\), a contradiction. +Thus in this latter case all real points in the square are already +among the critical or boundary points counted above. + +In the remaining case, mark the abscissae of all critical, boundary, +and inflection points, together with \(0,S\). +There are \(O(d^2)\) marked abscissae. +Between consecutive marked abscissae the interior real zero set is a +union of at most \(d\) smooth graphs extending across that open +interval. To justify extension, the implicit function theorem applies +because \(Q_{X_2}\ne0\); continuation cannot end inside the interval +without a critical point or an intersection with the boundary. +The bound \(d\) follows from the degree of a vertical fiber. +Each graph is monotone, and its derivative is strictly monotone, +because neither \(Q_{X_1}\) nor \(\mathcal I_Q\) vanishes on it. + +Consider the lattice points of one such graph, ordered by increasing +abscissa. Consecutive difference vectors have positive integer first +coordinate and distinct slopes, by strict convexity or strict +concavity. They are therefore distinct integer vectors. +Their total \(\ell^1\)-length is at most \(2S\), since both graph +coordinates are monotone and range over intervals of length at most +\(S\). There are \(O(T^2)\) integer vectors of \(\ell^1\)-length at most +\(T\). Consequently \(m\) distinct such vectors have total length +\(\gg m^{3/2}\): take \(T\) to be a sufficiently small fixed multiple +of \(m^{1/2}\), so that at least half of the vectors have length greater +than \(T\). It follows that this graph contains +\(O(1+S^{2/3})\) lattice points. + +The marked vertical fibers contain at most \(d\) points each, since +an identically vanishing fiber would give a vertical line factor of +\(Q\). Summing over the \(O(d^3)\) graphs and the marked fibers proves +the bound \(O(d^3(1+S^{2/3}))\) for this component. +Finally, sum over the distinct nonlinear factors of \(P\). Their +degrees sum to at most \(D_0\), so the asserted, slightly looser, +bound with \(D_0^4\) follows. +\end{proof} + +\begin{proof}[Proof of \cref{prop:inverse}] +\textbf{The fixed whole-box list.} +To make the order of choices explicit, choose an integer \(h_1\) such +that +\[ + \frac{a_\star h_1}{2}>12, +\] +and set +\begin{equation} + \beta=\frac{\sigma}{16},\qquad + \gamma_{\rm inc}=\frac{\beta}{2} + \left(\frac{\beta}{4}\right)^{h_1},\qquad + c_5=\frac{\gamma_{\rm inc}}{8},\qquad + c_p=\frac{\gamma_{\rm inc}}{32}. + \label{eq:inverse-fixed-constants} +\end{equation} +These constants are fixed before \(\xi\) is chosen and before any +failure set, cell, or sample is selected. +For each candidate box let \(\mathcal A\) be the set of \emph{all} +reduced rationals satisfying \eqref{eq:inverse-list}. +This is a finite set by the height restrictions. + +If two distinct list rationals \(A_1/D_1,A_2/D_2\) align at a prime, +that prime divides the nonzero determinant \(A_1D_2-A_2D_1\). +Its absolute value is at most +\[ + 2 S_q R^2\mathcal Z^{20}=z^{\,2+\alpha+o(1)}. +\] +Every prime of \(\mathcal P\) is at least \(z^{\alpha_{\min}}\). +Thus two distinct rationals have at most \(C_{\alpha_{\min}}\) +common aligning primes, with a fixed constant independent of \(\xi\). +If \(M=\#\mathcal A\), and \(m_p\) counts the list rationals aligning +at \(p\), Cauchy's inequality gives +\[ + M^2c_p^2 n_p + \le\sum_{p\in\mathcal P}m_p^2 + \le M n_p+C_{\alpha_{\min}}M(M-1). +\] +Once \(n_p\ge2C_{\alpha_{\min}}/c_p^2\), this implies +\[ + M\le \frac{2}{c_p^2}. +\] +We may therefore take \(F=\lceil2/c_p^2\rceil\). +This bound is independent of \(\xi\); the threshold on \(z\) may depend +on \(\xi\) through \(n_p\). + +Call a pair \((p,q)\) \emph{structured} if +\eqref{eq:inverse-structure} holds for some member of this fixed +whole-box list. A triple \((p,q,u)\) is called \emph{bad} if its parent +pair is nonstructured and \eqref{eq:inverse-bad-edge} holds. +Assume, towards a contradiction, that bad triples occupy more than a +\(\sigma\) fraction of the full harmonic box measure. Descendants can +now be integrated out. + +\medskip +\textbf{From bad triples to sampled incidences.} +We will select a fixed number of edge primes so that many nonstructured +parent pairs satisfy all their frozen residue tests, while the total +number of permitted slopes over the isolated primes is small. +First we restrict \(q\) to a +sufficiently large cell. For \(q\in\mathcal Q\), let +\(b_q\in[1,q]\cap\Z\) be its +Chinese remainder theorem (CRT) representative for the prescribed hits. Put +\[ + M_0=\prod_{t\le w}t , +\] +where the product is over primes. By the prime number theorem, +\[ + \log M_0=O(w) + =O\!\left(\frac{L}{(\log L)^2}\right)=o(\log\mathcal Z). +\] +In particular \(M_0^2=\mathcal Z^{o(1)}\). +Partition \(\mathcal Q\) into cells by a value interval of ratio at +most \(1+\xi\), and by the pair +\((q,b_q)\bmod M_0\). By \eqref{eq:inverse-q-range} the number of value +intervals needed is \(O(C_{\rm len}\log B+1)\), so the total number of +cells is \(\mathcal Z^{o(1)}\). + +We record the cardinality bound +\begin{equation} + \#\mathcal Q\ge \frac{S_q}{\mathcal Z^{o(1)}}. + \label{eq:inverse-many-q} +\end{equation} +Indeed, a bin with upper endpoint \(P_j\) and multiplicity \(m_j\) +contains \(\gg_\xi P_j/\log P_j\) primes. Even in the smallest bin +this number is much larger than \(C_{\rm len}\log B\), for fixed +\(\xi>0\) and large \(z\). +If \(n_j\) is its prime count and +\(k=\sum_jm_j\le C_{\rm len}\log B=O(\log L)\), then +\[ + \prod_j\binom{n_j}{m_j} + \ge \prod_j\frac{(n_j/2)^{m_j}}{m_j!} + \ge S_q\exp\!\bigl(-O_\xi((\log L)^2)\bigr). +\] +The first inequality holds once \(n_j\ge2m_j\). +In the second, the losses from \(\log P_j\), fixed bin-width constants, +and the factorials have total logarithm \(O_\xi((\log L)^2)\). +Unique factorization identifies these choices with distinct \(q\)'s. +Since \((\log L)^2=o(\log\mathcal Z)\), this proves +\eqref{eq:inverse-many-q}. + +Discard cells containing fewer than \(S_q/\mathcal Z\) points. +They contain at most \(S_q\mathcal Z^{-1+o(1)}\) points in total. +The weights \(1/q\) vary globally by at most +\((1+\xi)^k=\mathcal Z^{o(1)}\), so their total normalized harmonic +mass is \(\mathcal Z^{-1+o(1)}=o(1)\). +For sufficiently large \(z\), a retained initial cell \(\mathcal C\) +therefore has conditional harmonic density of bad triples at least +\(\sigma/2\), and +\begin{equation} + \#\mathcal C\ge S_q/\mathcal Z. + \label{eq:inverse-cell-size} +\end{equation} +Within \(\mathcal P,\mathcal C,\mathcal U\) each reciprocal weight +varies by a factor at most \(1+\xi\). +As long as \(\xi\le1\), converting their product measure to uniform +counting costs at most \((1+\xi)^3\le8\). +The count density of bad triples in +\(\mathcal P\times\mathcal C\times\mathcal U\) is consequently at least +\(\beta=\sigma/16\), with \(\beta\) fixed in +\eqref{eq:inverse-fixed-constants}. + +\medskip +\emph{Freezing the parent interval.} +We now place the small-prime count on an interval that is fixed as +\(q\) varies within \(\mathcal C\). +For fixed \(p\in\mathcal P\) and \(q\in\mathcal C\), put +\begin{equation} + s_p(q)=(a_p-b_q)\overline q\pmod p, + \qquad 0\le s_p(q)0\) this length +is eventually much larger than \(w^{a_\star/2}\); the corresponding +progression step \(pqu\) is a unit at every prime at most \(w\). +The same reasoning applies to the unconditioned difference interval. +The upper-sieve constants are independent of fixed \(\xi\), although +the threshold at which these lengths suffice may depend on \(\xi\). + +The edge hit selects the class +\begin{equation} + e_u=e_u(p,q)=(a_u-b_q-q s_p(q))\overline{pq}\pmod u . + \label{eq:inverse-edge-class} +\end{equation} +Choose \(\xi_{\rm inv}\le1\) sufficiently small, depending on +\(\delta\) and the fixed upper-sieve constants. +The preceding replacement estimates show that +\eqref{eq:inverse-bad-edge} implies +\begin{equation} + |m_{e_u,u}-m/u|>\delta'\frac{JV_0}{u}, + \label{eq:inverse-frozen-deviation} +\end{equation} +for a fixed \(\delta'>0\). +For example, after decreasing \(\xi_{\rm inv}\) the choice +\(\delta'=\delta/4\) suffices. +Define \(E_{p,u}(v)\) to be the set of residues \(e\bmod u\) satisfying +\eqref{eq:inverse-frozen-deviation}. +These sets depend on \(p\), the initial cell, and \(v\), and no +independence in those variables is asserted. + +\medskip +\emph{The large-sieve saving.} +We bound the average proportion of residues passing an exceptional test. +For fixed \(p,v\), let \(\mathcal S\subset I_p\) be the surviving +sequence, so \(\#\mathcal S=m\), and put +\[ + S(\theta)=\sum_{j\in\mathcal S}\exp(2\pi i j\theta). + \] +Additive orthogonality gives the exact identity +\begin{equation} + u\sum_{e\bmod u}(m_{e,u}-m/u)^2 + =\sum_{a=1}^{u-1}|S(a/u)|^2. + \label{eq:inverse-orthogonality} +\end{equation} +For prime \(u\), all fractions \(a/u\) on the right are reduced. +Fractions belonging to distinct such primes are distinct, and the +whole family is separated modulo one by at least +\(((1+\xi)U)^{-2}\ge(4U^2)^{-1}\). +Lemma~\ref{lem:elementary-large-sieve} therefore gives +\[ + \sum_{u\in\mathcal U} + u\sum_{e\bmod u}(m_{e,u}-m/u)^2 + \ll (J+U^2)m; +\] +see \cite{MontgomeryVaughan1973,VaughanLargeSieve}. +No condition other than interval support is imposed on the sequence +\(\mathcal S\). +Each exceptional residue contributes more than +\((\delta'JV_0)^2/u\) to the corresponding left side. +Using \eqref{eq:inverse-m-upper}, we obtain +\begin{align} + \frac1{n_u}\sum_{u\in\mathcal U}\frac{|E_{p,u}(v)|}{u} + &\ll + \frac{(J+U^2)m}{n_u(JV_0)^2} \notag\\ + &\ll_\xi(\log w)^2\left(\frac1U+\frac UJ\right) + \le w^{-a_\star/2}. + \label{eq:inverse-large-sieve} +\end{align} +For the last inequality, \(\log U=O_{K_*}(\log w)\), +\(V_0\asymp1/\log w\), \(U\ge w\), and +\(J/U\gg w^{3a_\star/4}\). +The fixed factor depending on \(\xi\), as well as the logarithmic +factors, is absorbed between the exponents \(3a_\star/4\) and +\(a_\star/2\) by taking \(z\) large. +This proves \eqref{eq:inverse-large-sieve} uniformly in \(p,v\) and +the chosen cell, in exactly the fixed-\(\xi\) sense required here. + +\medskip +\emph{Selecting the sampled relation.} +For \((p,q)\in\mathcal P\times\mathcal C\), let \(f(p,q)\) be the +fraction of \(u\in\mathcal U\) for which \((p,q,u)\) is bad. +Its uniform count average is at least \(\beta\). +At least a \(\beta/2\) fraction of pairs therefore have +\(f(p,q)\ge\beta/2\): the complementary bound follows at once from +\(0\le f\le1\). + +Sample an ordered \(h_1\)-tuple of distinct primes +\(\mathbf u=(u_1,\ldots,u_{h_1})\) uniformly from \(\mathcal U\). +For this sample define the pair relation +\[ +\begin{split} + \mathcal I_{\mathbf u}=\bigl\{(p,q)\in\mathcal P\times\mathcal C: + &\ (p,q)\text{ is nonstructured, and}\\ + &\ e_{u_i}(p,q)\in E_{p,u_i}(s_p(q)\bmod M_0) + \text{ for every }1\le i\le h_1\bigr\}, +\end{split} +\] +and put \(I(\mathbf u)=\#\mathcal I_{\mathbf u}/(n_p\#\mathcal C)\). +Members of \(\mathcal I_{\mathbf u}\) will be called \emph{sampled +incidences}. By \eqref{eq:inverse-frozen-deviation}, a pair for which +every sampled triple is bad belongs to \(\mathcal I_{\mathbf u}\). +This implication suffices for the lower bound on its expected density. +Once \(n_u\ge4h_1/\beta\), each pair with \(f(p,q)\ge\beta/2\) has probability +at least \((\beta/4)^{h_1}\) that all sampled primes witness +\eqref{eq:inverse-bad-edge}. +Consequently +\begin{equation} + \E I(\mathbf u)\ge + \frac{\beta}{2}\left(\frac{\beta}{4}\right)^{h_1} + =\gamma_{\rm inc}. + \label{eq:inverse-incidence-expectation} +\end{equation} + +Write \(W=\prod_{i=1}^{h_1}u_i\) and refine \(\mathcal C\) by fixing +\((q,b_q)\bmod W\). For any such refinement \(\mathcal C'\), and +fixed \(p\), let \(\mathcal T_p(\mathcal C')\subset[0,p)\cap\Z\) +consist of those integers \(s\) satisfying all the sampled tests, +using \(v=s\bmod M_0\). +For a fixed \(v\), \eqref{eq:inverse-edge-class} is an invertible +affine function of \(s\bmod u_i\), with coefficient +\(-\overline p\bmod u_i\). +Its other coefficients are fixed by the refinement. +Thus it restricts \(s\bmod u_i\) to exactly +\(|E_{p,u_i}(v)|\) residues. +CRT, applied also to \(s\equiv v\pmod {M_0}\), gives +\begin{align} + |\mathcal T_p(\mathcal C')| + &\le K_p(\mathbf u),\notag\\ + K_p(\mathbf u) + &:= + \frac{p}{M_0}\sum_{v\bmod M_0} + \prod_{i=1}^{h_1}\frac{|E_{p,u_i}(v)|}{u_i} + +M_0 W . + \label{eq:inverse-crt-slopes} +\end{align} +Indeed, for each \(v\) the number of allowed residues modulo \(M_0W\) +is \(\prod_i|E_{p,u_i}(v)|\); an interval of length \(p\) contains at +most \(p/(M_0W)+1\) integers in each residue. +Summing the rounding terms over \(v\) gives at most \(M_0W\). +All CRT factors are coprime, and every sampled \(u_i\) is coprime to +\(pq\) by the singleton-bin hypotheses. +Crucially, the upper bound \(K_p(\mathbf u)\) is independent of the +refinement centers. + +For any numbers \(0\le t_u\le1\), their product averaged over distinct +ordered \(h_1\)-tuples is at most +\[ + \frac{n_u^{h_1}}{(n_u)_{h_1}} + \left(\frac1{n_u}\sum_{u\in\mathcal U}t_u\right)^{h_1}, + \qquad + (n_u)_{h_1}=n_u(n_u-1)\cdots(n_u-h_1+1). +\] +The prefactor tends to one since \(h_1\) is fixed. +Applying \eqref{eq:inverse-large-sieve} for each \(p,v\) in +\eqref{eq:inverse-crt-slopes} yields +\begin{equation} + \E\frac{K_p(\mathbf u)}p + \le 2w^{-a_\star h_1/2}+z^{-c'} + \label{eq:inverse-slopes-expectation} +\end{equation} +for some fixed \(c'>0\). +Here \(M_0W=z^{o(1)}\), since \(h_1\) and the edge-bin exponent range are +fixed, whereas \(p\ge z^{\alpha_{\min}}\). +Since \(p/R\le1+\xi\le2\), the same estimates imply +\[ + \E\left(\frac1{n_pR}\sum_{p\in\mathcal P}K_p(\mathbf u)\right) + \ll w^{-a_\star h_1/2}+z^{-c'} + =o(L^{-6}). +\] +The last step uses \(w=L/(\log L)^2\) and \(a_\star h_1/2>12\). +By Markov's inequality, the probability that +\(\sum_pK_p>n_pR/L^6\) is \(o(1)\). +As \(0\le I(\mathbf u)\le1\), removing those samples changes the +expectation in \eqref{eq:inverse-incidence-expectation} by \(o(1)\). +Hence there exists a sample for which +\begin{equation} + I(\mathbf u)\ge\gamma_{\rm inc}/2, + \qquad + \sum_{p\in\mathcal P}K_p(\mathbf u)\le\frac{n_pR}{L^6}. + \label{eq:inverse-good-sample} +\end{equation} +Fix this sample and abbreviate its bounds by \(K_p\). + +There are at most \(W^2=w^{O(1)}=\mathcal Z^{o(1)}\) refined cells. +Discard those with fewer than \(S_q/\mathcal Z^3\) points. +Relative to \(\mathcal C\), whose size satisfies +\eqref{eq:inverse-cell-size}, their total count fraction is at most +\(W^2/\mathcal Z^2=o(1)\). +Thus some remaining refinement \(\mathcal C'\) satisfies +\begin{equation} + \#\mathcal C'\ge S_q/\mathcal Z^3 + \label{eq:inverse-refined-size} +\end{equation} +and satisfies +\[ + \frac{\#\bigl(\mathcal I_{\mathbf u}\cap + (\mathcal P\times\mathcal C')\bigr)} + {n_p\#\mathcal C'}\ge\frac{\gamma_{\rm inc}}4. +\] +At least a \(\gamma_{\rm inc}/8=c_5\) fraction of its \(q\)-points consequently +have at least \(c_5 n_p\) primes \(p\) with +\((p,q)\in\mathcal I_{\mathbf u}\). +Call these points \emph{rich}, and let \(\mathcal X\) be their set of +integer points \((q,b_q)\). We have therefore fixed a sample, a refinement +\(\mathcal C'\), and slope sets \(\mathcal T_p(\mathcal C')\) with +\(\#\mathcal T_p(\mathcal C')\le K_p\). For large \(z\), their bounds are +\begin{equation} +\begin{gathered} + |\mathcal X|\ge \frac{S_q}{\mathcal Z^4},\\ + \#\{p\in\mathcal P:(p,q)\in\mathcal I_{\mathbf u}\}\ge c_5n_p + \quad\bigl((q,b_q)\in\mathcal X\bigr),\\ + \sum_{p\in\mathcal P}K_p\le\frac{n_pR}{L^6}. +\end{gathered} + \label{eq:inverse-rich-points} +\end{equation} +The point-count bound follows from \eqref{eq:inverse-refined-size}; +the slope sum is \eqref{eq:inverse-good-sample}. Every sampled incidence +with \(q\in\mathcal C'\) uses the slope +\(s_p(q)\in\mathcal T_p(\mathcal C')\). +The constants in these bounds are precisely those fixed in +\eqref{eq:inverse-fixed-constants}, without choosing them after the +list or after the failure set. + +\medskip +\textbf{Algebraic concentration and the fixed list.} +We will find a rational line containing many rich points and show that +its intercept belongs to \(\mathcal A\). Almost all sampled incidences +on that line will then be structured, giving the contradiction. +First the small slope sum lets us construct a common polynomial equation +for the rich points. +Put +\[ + D_0=\left\lfloor\frac{R}{L^2}\right\rfloor, + \qquad N_0=\binom{D_0+2}{2}. +\] +We construct a nonzero integer polynomial \(P(X_1,X_2)\) of total +degree at most \(D_0\) that modulo each \(p\in\mathcal P\) vanishes +identically on every line +\[ + X_2=a_p-sX_1,\qquad s\in\mathcal T_p(\mathcal C'). +\] +The coefficients of the polynomial after substitution in a line give +at most \(D_0+1\) homogeneous congruences. +The lattice of coefficient vectors in \(\Z^{N_0}\) satisfying all +these congruences consequently has index at most +\[ + \prod_{p\in\mathcal P} + p^{(D_0+1)|\mathcal T_p(\mathcal C')|} + \le + \exp\!\left((D_0+1)\sum_{p\in\mathcal P}K_p\log p\right). +\] +Dependencies among congruences can only decrease this index. +Applying the pigeonhole principle to a box of coefficient vectors, +and taking the difference of two vectors in the same lattice coset, +gives a nonzero \(P\) with maximum absolute coefficient \(H_P\) satisfying +\begin{equation} + \log(2H_P) + \ll 1+\frac1{D_0}\sum_{p\in\mathcal P}K_p\log p + =O(R/L^4)=o(R). + \label{eq:inverse-polynomial-height} +\end{equation} +For the final equality use \eqref{eq:inverse-good-sample}, +\(n_p\ll R/\log R\), and \(\log p\asymp\log R\). +More explicitly, a coefficient box with more vectors than the +lattice index has two in the same coset; taking side length equal +to the ceiling of the index to the power \(1/N_0\) proves the stated +height bound. + +All the points \((q,b_q)\) under consideration lie in \([0,S_q]^2\). +If \(P(q,b_q)\ne0\), then +\begin{equation} + \log|P(q,b_q)| + \le \log N_0+\log H_P+D_0\log\max(1,S_q) + =O(R/L)=o(R). + \label{eq:inverse-polynomial-value} +\end{equation} +At a rich point the integer \(P(q,b_q)\) is divisible by every prime +of its at least \(c_5n_p\) distinct sampled incidences. +Their product has logarithm at least +\[ + c_5n_p\log R\gg_{\xi,c_5}R. +\] +For each fixed \(\xi>0\), this contradicts +\eqref{eq:inverse-polynomial-value} unless \(P(q,b_q)=0\). +Every point of \(\mathcal X\) therefore lies on the zero set of \(P\). + +\medskip +\emph{Extraction of a rational line.} +By \cref{lem:nonlinear-points}, the number of points of \(\mathcal X\) +on nonlinear irreducible components is at most +\[ + O\!\left(D_0^4(1+S_q^{2/3})\right) + =o(S_q/\mathcal Z^4). +\] +To check this uniformly, the principal ratio is bounded by +\[ + R^4S_q^{-1/3}\mathcal Z^4 + =z^{(13\alpha-2)/3+o(1)}=o(1), +\] +using \(\alpha\le1/100\); the term without \(S_q^{2/3}\) is smaller. +There are at most \(D_0\) line components. +Discard any line containing fewer than +\[ + N_{\min}=\frac{S_q}{R\mathcal Z^6} +\] +selected rich points. +All discarded lines together contain at most +\[ + D_0N_{\min}\le\frac{S_q}{L^2\mathcal Z^6} + =o(S_q/\mathcal Z^4) +\] +such points. +Vertical lines have at most one selected point because \(q\) uniquely +determines \(b_q\), and \(N_{\min}\to\infty\). +A line not proportional to a real line likewise has at most one real +point. Thus a remaining nonvertical real line contains \(N\ge +N_{\min}\) selected rich points. + +Two distinct lattice points on this line give it a rational slope. +It can be written +\begin{equation} + D'b_q=A'q+B',\qquad + D'>0,\quad \gcd(A',D')=1,\quad A',B',D'\in\Z. + \label{eq:inverse-rich-line} +\end{equation} +For its lattice points, consecutive distinct \(q\)'s differ by an +integer multiple of \(D'\). Their \(b_q\)'s differ by the corresponding +multiple of \(A'\). +Their coordinate ranges are at most \(S_q\), so +\begin{equation} + D',|A'|\ll \frac{S_q}{N}\ll R\mathcal Z^6, + \qquad + |B'|\ll S_qR\mathcal Z^6. + \label{eq:inverse-line-heights} +\end{equation} +The assertion for \(A'\) also holds for a horizontal line, when +\(A'=0,D'=1\). + +Reduce the intercept \(B'/D'\) to \(A/D\), putting +\[ + g=\gcd(B',D'),\qquad D'=gD,\qquad B'=gA. +\] +Reduction of \eqref{eq:inverse-rich-line} modulo \(g\) gives +\(g\mid A'q\), and \(\gcd(A',g)=1\); hence +\begin{equation} + g\mid q + \quad\text{at every selected point of the line}. + \label{eq:inverse-g-divides} +\end{equation} +If a prime \(\ell\mid q\) does not divide \(g\), division of +\eqref{eq:inverse-rich-line} by \(g\) modulo \(\ell\) gives +\[ + D b_q\equiv A\pmod\ell. +\] +Because \(b_q\equiv a_\ell\pmod\ell\), the prime \(\ell\) aligns with +the intercept. This congruence also implies \(\ell\nmid D\), since +\(\gcd(A,D)=1\). +All nonaligned prime factors of \(q\) therefore divide \(g\). +As \(q\) is squarefree and \(g\mid q\), their product divides \(g\), +and +\begin{equation} + Dq_E(A/D)\le Dg=D'\ll R\mathcal Z^6. + \label{eq:inverse-line-structure} +\end{equation} +Neither \(D\) nor \(D'\) has been assumed squarefree. +In particular, primes common to \(D\) and \(g\) cause no difficulty in +\eqref{eq:inverse-line-structure}. +The height bounds in \eqref{eq:inverse-list} follow from +\eqref{eq:inverse-line-heights} with ample room in the additional +\(\mathcal Z^4\) factors. +It remains to establish the alignment-count condition for this +intercept. + +\medskip +\emph{The intercept belongs to the list.} +We use the pair-counting principle of Gallagher's larger sieve +\cite{Gallagher1971}; see \cite[Section~2]{HelfgottVenkatesh2009} for +the difference-product argument. +Ignore the isolated-bin primes dividing \(D'\). +There are at most +\[ + \frac{\log D'}{\log R}=O(1) +\] +of them by \eqref{eq:inverse-line-heights}. +Let \(\mathcal P_{\rm un}\) consist of the other primes of +\(\mathcal P\) that do not align with \(A/D=B'/D'\). +For \(p\in\mathcal P_{\rm un}\), substitution of the line equation +into the incidence equation gives +\[ + (A'/D'+s)q\equiv a_p-B'/D'\pmod p, + \qquad s\in\mathcal T_p(\mathcal C'). + \] +The right side is nonzero because \(p\) is unaligned. +Each allowed \(s\) therefore admits either no solution or exactly one +class of \(q\bmod p\). +Thus those of the \(N\) selected points on the line that satisfy the +sampled constraints for \(p\) occupy at most \(K_p\) classes of \(q\bmod p\). +Let their number be \(N_p\), and let \(n_{p,c}\) denote the numbers +in these occupied classes. Here \(N_p\) counts all tested points on the +line, so it dominates the number of pairs +\((p,q)\in\mathcal I_{\mathbf u}\) there; the latter pairs also require +the parent to be nonstructured. + +For two distinct selected points, \(q_1-q_2\ne0\) and +\(|q_1-q_2|\le S_q\). +The number of isolated-bin primes dividing this difference is at +most +\[ + \frac{\log S_q}{\log R}=O_{\alpha_{\min}}(1). +\] +Counting ordered equal-class pairs, with the diagonal separated, +therefore gives +\[ + \sum_{p\in\mathcal P_{\rm un}}\sum_c n_{p,c}^2 + \le C_{\alpha_{\min}}N^2+ + \sum_{p\in\mathcal P_{\rm un}}N_p. +\] +Cauchy's inequality, first over all occupied classes and primes, +yields +\begin{equation} + \left(\sum_{p\in\mathcal P_{\rm un}}N_p\right)^2 + \le + \left(\sum_{p\in\mathcal P}K_p\right) + \left(C_{\alpha_{\min}}N^2+ + \sum_{p\in\mathcal P_{\rm un}}N_p\right). + \label{eq:inverse-collisions} +\end{equation} +Use \(\sum_p N_p\le n_pN\), the good-sample bound, and +\(N\ge S_q/(R\mathcal Z^6)\). +After division by \(n_p^2N^2\), the right side of +\eqref{eq:inverse-collisions} is +\begin{equation} + O\!\left(\frac{R}{n_pL^6}+\frac{R}{NL^6}\right) + =O_\xi(L^{-5})+o(1)=o(1). + \label{eq:inverse-collision-normalization} +\end{equation} +For the first term use \eqref{eq:inverse-bin-counts} and +\(\log R\le L\); for the second use +\(R^2\mathcal Z^6/S_q=z^{3\alpha-2+o(1)}=o(1)\). +Thus unaligned primes supply only \(o(n_pN)\) tested pairs on the line, +and hence at most this many sampled incidences. + +Each of its \(N\) selected points has at least \(c_5n_p\) +sampled incidences by \eqref{eq:inverse-rich-points}. +The ignored divisors of \(D'\) supply at most \(O(N)\). +The remaining incidences force at least +\((c_5-o(1))n_p\) primes of \(\mathcal P\) to align with the intercept. +Since \(c_p=c_5/4\), it satisfies the last condition in +\eqref{eq:inverse-list}, and hence \(A/D\in\mathcal A\). +At every selected point of this line, +\eqref{eq:inverse-line-structure} implies the required bound +\(Dq_E(A/D)\le R\mathcal Z^{10}\) for large \(z\). +Every tested pair on the line whose isolated prime aligns is therefore +structured. Since all members of \(\mathcal I_{\mathbf u}\) are +nonstructured, the sampled incidences must consequently come from the +unaligned primes or the \(O(1)\) ignored primes; those +contribute only \(o(n_pN)+O(N)\), contradicting their lower bound +\(c_5n_pN\). + +This contradiction proves the exceptional-fraction assertion. +The constants in \eqref{eq:inverse-fixed-constants} and the list bound +were fixed before \(\xi\). +Only interval freezing required an upper restriction +\(\xi\le\xi_{\rm inv}\); all losses that depend on a fixed positive +\(\xi\), including prime counts, polynomial divisibility, and +\eqref{eq:inverse-collision-normalization}, are absorbed by taking +\(z\) sufficiently large afterwards. +This completes the prescribed order of choices. +\end{proof} diff --git a/preprints/A-quadratic-bound-for-Jacobsthals-function-September-25-2026/build/sections/prime-paths.tex b/preprints/A-quadratic-bound-for-Jacobsthals-function-September-25-2026/build/sections/prime-paths.tex new file mode 100644 index 0000000000000000000000000000000000000000..9bdb0bd7af8e0cde61502281b011a963c5217e8e --- /dev/null +++ b/preprints/A-quadratic-bound-for-Jacobsthals-function-September-25-2026/build/sections/prime-paths.tex @@ -0,0 +1,372 @@ +\section{Passage from harmonic paths to primes}\label{sec:prime-paths} + +We now transfer the path estimates to actual primes. Throughout this +section, a \emph{standard prime path} starts at a lower state with cutoff +$B$ and gap $r_{\mathrm{st}}=s_{\mathrm{st}}B$, where +$s_{\mathrm{st}}\in[1.99,2.3]$. Its primes decrease strictly after the +first step. At the first step the upper endpoint is included; all +subsequent upper endpoints are excluded. The next exponent is required +to exceed $\ell$, where $\ell\in\{1,2\}$. A lower state includes every +available child, and an upper state includes precisely the children for +which $r-x\geq 2x$. Thus these paths contain the paths admitted by +\eqref{eq:admission}, and agree with them above exponent $2$. + +For a prefix $\mathfrak p=(p_1,\ldots,p_m)$ write +\[ + h(\mathfrak p)=\prod_{j=1}^m\frac1{p_j},\qquad + r(\mathfrak p)=r_{\mathrm{st}}-\sum_{j=1}^m x(p_j),\qquad + s(\mathfrak p)=\frac{r(\mathfrak p)}{x(p_m)}. +\] +The empty prefix has weight $1$. Sums over prefixes below include all +allowed lengths, and therefore count a path once at each of its visited +states. At a nonempty prefix, $x(p_m)>\ell$ is equivalent to +$s(\mathfrak p)0$, eventually. + +\begin{lemma}[Harmonic prime measure]\label{lem:prime-harmonic} +There are absolute positive constants $C,c$ such that, on any exponent +interval with lower endpoint $u\geq1/2$, +\begin{equation}\label{eq:prime-harmonic} + \sum_{x(p)\ \mathrm{in\ the\ interval}}\frac1p + =\int_{\mathrm{interval}}\frac{dx}{x}+O(E(u)), + \qquad E(u)=C\exp(-c\sqrt{u\log w}). +\end{equation} +The error is uniform in the upper endpoint and in either endpoint +convention. In particular, the total weight of all decreasing prime +prefixes with exponents in $(\ell,B]$, with no admission restrictions, +is $O(B)$. +\end{lemma} + +\begin{proof} +Apply partial summation to the prime number theorem in +Lemma~\ref{lem:prime-inputs}, between $w^u$ and the upper endpoint. +The main term is $\int dt/(t\log t)=\int dx/x$. The integrated error, +including its lower-endpoint term, is bounded by a constant times +$\exp(-c\sqrt{\log(w^u)})$, after reducing $c$. The bound remains valid +if the upper endpoint is sent to infinity in the error integral. A +change of endpoint convention contributes at most one reciprocal prime, +which is absorbed by $E(u)$. Finally, +\[ + \sum_{\mathfrak p}h(\mathfrak p) + =\prod_{w^\ell0$ and $A>0$. The total original harmonic weight of prefixes +ending at gap at most $K$ and having some earlier or terminal ratio +greater than $S$ is $O_{K,A}(B^{-A})$. More generally, for any fixed +$c'>0$, the sum of +$h(\mathfrak p)\exp(-c'r(\mathfrak p))$ over prefixes having such a +high state is $O_{A,c'}(B^{-A})$. Both conclusions also hold in the +continuous harmonic model, uniformly in the starting ratios under +consideration. +\end{lemma} + +\begin{proof} +Suppose a prefix reaches a state of gap $R$, cutoff $b$, and ratio +$R/b>S$. Every subsequent factor has exponent at most $b$. Reducing +the gap below $R/2$ therefore requires at least $S/2$ additional factors. +If $T=\log B+O(1)$ is an upper bound for the remaining harmonic +intensity, the sum of weights of these continuations is at most +\[ + \sum_{j\geq S/2}\frac{T^j}{j!}. +\] +Since $S=(\log B)^2$, this tail is smaller than every fixed negative +power of $B$, even after multiplication by $B$. To sum over all +possible high ancestors, use Lemma~\ref{lem:prime-harmonic}: their +total prefix weight is $O(B)$. This argument is an upper bound even +when it counts a terminal prefix more than once. + +At a high ancestor $R>Sb>S\ell$. Thus a prefix ending at fixed gap +$K$ must reduce this gap by more than half for all sufficiently large +$B$. For the weighted assertion, prefixes which do not do so have +terminal gap greater than $S\ell/2$, and their reward is +$O(\exp(-c'S\ell/2))$. Their total unweighted prefix mass is $O(B)$, +so they too contribute less than every $B^{-A}$. In the continuous +model, the ordered integrals of length $j$ are bounded by $T^j/j!$, +and their total is $e^T\ll B$; the proof is unchanged. +\end{proof} + +For the rest of the discrete approximation, retain only states of ratio +at most $S$. There are at most $N=C S\log B$ arrived-at states, with +a fixed sufficiently large $C$. Indeed a transition to ratio $t\leq S$ +decreases $\log r$ by $g(t)=\log(1+1/t)\gg1/S$. Moreover the gap at a +retained nonempty state is bounded below by $0.99\ell$, and the starting +gap is $O(B)$. This also proves the same deterministic horizon for a +continuous path up to its first exit from these conditions. + +\begin{proposition}[Coupling the tilted paths]\label{prop:prime-coupling} +Choose sufficiently small absolute positive constants in the following +order. For a permitted prime child of a state of type $i$, ratio $s$, +and gap $r$, put $t=r/x(p)-1$ and give that child mass +\begin{equation}\label{eq:prime-tilt} + q(p\mid r,s,i)=\frac1p\left(\frac{t+1}{t}\right)^2 + \frac{\phi_{i'}(t)}{\phi_i(s)}. +\end{equation} +Only children with $x(p)>\ell$ and $t\leq S$ are retained. For +$\epsilon_w=\exp(-c_2\sqrt{\log w})$ their total mass is at most +$1+\epsilon_w$. Dividing by $1+\epsilon_w$ and assigning the remaining +mass to a cemetery state defines a killed Markov chain. + +This chain can be coupled with the continuous chain +\eqref{eq:tilted-kernel}, killed at the first draw with $x\leq\ell$ or +$t>S$, so that the probability of failure is at most +$\exp(-c_3\sqrt{\log w})$. On success the chains have the same number +of arrived-at states and matching exits. At corresponding states, +\begin{equation}\label{eq:prime-coupling-errors} + |s-\widetilde s|\leq h,\qquad + \left|\log\frac r{\widetilde r}\right|\leq C N h,\qquad + h=\frac1{\lceil\exp(c_4\sqrt{\log w})\rceil}. +\end{equation} +All statements are uniform in $s_{\mathrm{st}}\in[1.99,2.3]$ and in the +endpoint conventions stated above. For a length-$m$ retained prefix, +if $\mathbb P_w(\mathfrak p)$ denotes its probability in the killed +chain, then the exact weight identity is +\begin{equation}\label{eq:prime-weight} + h(\mathfrak p)=\mathbb P_w(\mathfrak p)(1+\epsilon_w)^m + \left(\frac{r(\mathfrak p)}{r_{\mathrm{st}}}\right)^2 + \frac{\phi_\ee(s_{\mathrm{st}})}{\phi_i(s(\mathfrak p))}, + \qquad (1+\epsilon_w)^m=1+o(1). +\end{equation} +\end{proposition} + +\begin{proof} +Under $t=r/x-1$ we have $|dx/x|=dt/(t+1)$. Multiplying by the +non-prime part of \eqref{eq:prime-tilt} gives +$W_0(t)\phi_{i'}(t)/\phi_i(s)$, exactly the continuous transition +density. Also $r_{\mathrm{child}}/r=t/(t+1)$, so multiplication over a +prefix telescopes and proves \eqref{eq:prime-weight}. The last +assertion there follows from $m\leq N$ and \eqref{eq:prime-scales}. + +Partition the ratio axis into bins $[jh,(j+1)h)$. For a bin meeting +the allowed next range, temporarily extend it to the full bin when +seeking an upper bound. Its corresponding $x$-interval has logarithmic +width comparable to $h/(t+1)$ and lower endpoint at least $\ell/2$, +for sufficiently large $w$. Lemma~\ref{lem:derivative-weights} +bounds logarithmic derivatives of the weights and backward opposite +weight ratios by fixed polynomials in $S$. Therefore a full interior +bin has discrete mass equal to its continuous mass with relative error +\begin{equation}\label{eq:prime-bin-error} + O\left(\operatorname{poly}(S) + \left(h+\frac{E(1/2)}h\right)\right). +\end{equation} +Here the first term controls variation across the bin, and the second +is \eqref{eq:prime-harmonic} divided by its harmonic main mass. +The density itself is bounded by $\operatorname{poly}(S)$ on any +bin meeting the support. There are only a bounded number of partial +support bins, each with mass at most +$\operatorname{poly}(S)h$. The fixed even-child boundary $2$ is a +mesh endpoint. Summing the interior relative errors and the partial +bin upper bounds proves that the total mass is at most +$1+\epsilon_w$, after choosing $c_4$ and then $c_2$ sufficiently small. + +Suppose the chains have been matched through the previous draw. Their +lower support endpoints are $s-1$ or $\max(2,s-1)$, and consequently +differ by at most $h$. The upper endpoint imposed by $x>\ell$ is +$r/\ell-1$. It needs comparison only when it is at most $S$, up to a +mesh error; in that case $r=O(S+1)$, and its discrepancy is at most +$O((S+1)Nh)$. Thus a large starting gap does not enter this support +error. Ratios of the two density denominators are +$1+O(\operatorname{poly}(S)h)$. The total mass in mismatched or partial +bins, together with the total variation distance between full bin +labels and the discrepancy in exit probabilities after normalization, +is at most $\exp(-c'\sqrt{\log w})$. + +Maximally couple these bin labels and the cemetery label at each step. +Inside a matched nonexit bin the two next ratios differ by at most $h$. +Since $g'(t)=-1/(t(t+1))$ is bounded on the used ranges, each matched +step changes the logarithmic-gap discrepancy by $O(h)$. Iteration up +to $N$ steps gives \eqref{eq:prime-coupling-errors}; a union bound and +\eqref{eq:prime-scales} give the stated failure probability. Single +endpoint primes have already been included in the uniform estimate +\eqref{eq:prime-harmonic}. The initial lower ratios slightly below $2$ +are covered by the extended derivative-weight identities in +Lemma~\ref{lem:derivative-weights}. +\end{proof} + +The total variation coupling by itself is insufficient for rewards +containing $1/\phi_i(s)$ far out in the ratio tail. The following +pointwise majorants supply the required additional control. + +\begin{lemma}[Discrete marginal majorants]\label{lem:discrete-marginals} +For the killed chain of Proposition~\ref{prop:prime-coupling}, the +probability of a nonexit bin with left endpoint $t$ at step $2\leq n\leq N$ +is bounded, with an absolute constant, by +\begin{equation}\label{eq:prime-marginals} + \begin{cases} + C h\phi_\ee(t),&n\ \hbox{even},\\[2mm] + C h\phi_\oo(t)\bigl(1-t^{-2}+3hW_0(t)\bigr),&n\ \hbox{odd}. + \end{cases} +\end{equation} +At step $1$ the bound is $C hW_0(t)\phi_\oo(t)$, on the slightly +extended range $t\geq0.98$. The constants are uniform in the starting +ratios and in $w$ sufficiently large. +\end{lemma} + +\begin{proof} +The bin argument in the preceding proof gives the conditional upper +bound +\begin{equation}\label{eq:prime-conditional-bin} + (1+\delta_w)hW_0(t)\frac{\phi_{i'}(t)}{\phi_i(s)}, + \qquad \delta_w=\exp(-c'\sqrt{\log w}), +\end{equation} +for every feasible child bin. Division by $1+\epsilon_w$ cannot +increase it. The first-step assertion follows because the starting +denominator is bounded away from zero. At the second step, canceling +the intermediate derivative weight leaves an incoming sum bounded by +\[ + C\sum_{s_{\mathrm{st}}-1\leq u\leq t+1+h}hW_0(u) + \ll \log(t+2). +\] +Since $t\geq2$ and $W_0(t)\log(t+2)$ is bounded, this proves the even +majorant at step $2$. + +For the induction, replacing a weight at an actual state by its value +at the bin's left endpoint costs a relative +$1+O(\operatorname{poly}(S)h)$. Given the even majorant, the incoming +bins for an odd child of left endpoint $t$ start at $2$ and end at most +at $t+1+h$. Their total mesh length is at most $t-1+3h$. Multiplication +by $W_0(t)$ gives +\[ + W_0(t)(t-1+3h)=1-t^{-2}+3hW_0(t), +\] +as required. In the reverse direction, the incoming odd bins start at +$1$; their bracket sums satisfy +\begin{align*} + \sum_{1\leq u\leq t+1+h}h + \bigl(1-u^{-2}+3hW_0(u)\bigr) + &\leq\int_1^{t+1}(1-u^{-2})\,du+O(h\log(t+2))\\ + &=\frac{t^2}{t+1}+O(h\log(t+2)). +\end{align*} +The new factor $W_0(t)$ cancels the leading expression, and the error +is $O(h)$ for $t\geq2$. The points $1$ and $2$ are mesh aligned, and +the $3hW_0$ term covers the endpoint mesh interval where the stationary +odd factor vanishes. Each iteration enlarges the constant by at most +$1+O(\delta_w+\operatorname{poly}(S)h)$. Its product over $N$ steps is +$1+o(1)$ by \eqref{eq:prime-scales}, completing the proof. +\end{proof} + +\begin{proposition}[Prime occupation estimates]\label{prop:prime-occupation} +The following assertions hold uniformly for +$s_{\mathrm{st}}\in[1.99,2.3]$ and $\ell\in\{1,2\}$. +\begin{enumerate}[label=\textup{(\roman*)}] +\item For fixed $K$, the original harmonic mass of prefixes with +terminal gap at most $K$ is $O_K(B^{-2})$. +\item Consider any selection of such compact-ending prefixes whose +low-state histories imply an event $\mathcal A$ in the killed prime +chain. Their total weight is at most +\begin{equation}\label{eq:prime-compact-event} + C_K B^{-2}\mathbb P_w(\mathcal A)+O_{K,A}(B^{-A}) +\end{equation} +for every fixed $A>0$. One may take $\mathcal A$ to be the event that +at least one selected compact prefix occurs. +\item For every fixed $c'>0$, +\begin{equation}\label{eq:prime-tail} + \lim_{K\to\infty}\limsup_{w\to\infty} + B^2\sum_{\substack{\mathfrak p\ \mathrm{standard}\\r(\mathfrak p)>K}} + h(\mathfrak p)e^{-c'r(\mathfrak p)}=0. +\end{equation} +The same compact and tail assertions hold for continuous original +harmonic weights. +\item Let $H_i(r,s)$ be bounded continuous functions supported in a +fixed compact positive gap interval and a fixed compact ratio interval. +Fixed interval boundaries and the indicator $s\ell s$ and $\ell\geq1$, cancellation of $\phi_i$ bounds the +expected reward from $r>K_B$ by a constant times +\begin{equation}\label{eq:prime-far-tail} + \sum_{t=jh\geq0.97}h + \sup_{u\geq\max(t,K_B)}u^2e^{-c'u} + \ll (1+K_B)^3e^{-c'K_B}. +\end{equation} +Increasing $C'$ makes this $o(N^{-1})$. The first-step majorant gives +the same bound, because $W_0$ is bounded on its extended range. The +empty prefix contributes $O(B^2e^{-c''B})$. + +On $KK/2,\quad \widetilde s<\widetilde r/\ell. +\] +Here the indicator follows from matching the nonexit arrivals, and +the comparison of weights follows from their logarithmic derivative +bounds and \eqref{eq:prime-coupling-errors}. On a band +$R\leq\widetilde r\leq eR$, Lemma~\ref{lem:band-bounds} bounds the +expected sum after cancellation of $\phi_i$ by $O(R+1)$. Thus this +band contributes at most $C(R+1)R^2e^{-c'R/2}$. Summing the geometric +sequence of bands above $K/2$ proves \eqref{eq:prime-tail}. High-state +histories were already negligible by Lemma~\ref{lem:high-states}. +The continuous proof uses the same band bound directly, along with +the continuous version of that lemma. + +Finally, for (iv) use \eqref{eq:prime-weight} with the compact test +$r^2H_i(r,s)/\phi_i(s)$. The correcting factor is uniformly $1+o(1)$, +and the number of compact-gap visits is bounded. Coupling therefore +transfers this test by uniform continuity. Boundary indicators cause +no difficulty: the limiting continuous occupation measure gives zero +mass to their boundaries, and continuous upper and lower +approximations suffice. Continuous removal of high states is +harmless as well. For example, the drift estimate in +Lemma~\ref{lem:regeneration} gives a uniformly bounded exponential +moment of the ratio at each step, so a union bound up to the low-state +horizon gives $O(Ne^{-S})=o(1)$ for reaching a high state before exit. +Alternatively one can use the continuous original-weight conclusion +of Lemma~\ref{lem:high-states} on compacts. Apply +Proposition~\ref{prop:occupation} to the transferred test and use its +uniformity in the starting state. This gives +\eqref{eq:prime-occupation} and completes the proof. +\end{proof} diff --git a/preprints/A-quadratic-bound-for-Jacobsthals-function-September-25-2026/build/sections/reference.tex b/preprints/A-quadratic-bound-for-Jacobsthals-function-September-25-2026/build/sections/reference.tex new file mode 100644 index 0000000000000000000000000000000000000000..f5b44cf2b256fe70534007c1f2d901a0427064ef --- /dev/null +++ b/preprints/A-quadratic-bound-for-Jacobsthals-function-September-25-2026/build/sections/reference.tex @@ -0,0 +1,582 @@ +\section{Evaluation of the reference tree}\label{sec:reference} + +We now evaluate the boundary contribution left open by the continuous +benchmark in Section~\ref{sec:functions}. Above exponent $2$, the +reference tree follows the standard admission rule. At cutoff $2$, +its remaining finite polynomial differs from the linear-sieve benchmark. +The occupation estimates of Section~\ref{sec:prime-paths} transport this +boundary discrepancy to the root, where it contributes at scale $B^{-2}$. + +The sign requires a separate calculation. We first compare with the +full product over the boundary exponents, whose signed contribution can +be evaluated exactly. The stronger admission rule below exponent $2$ +omits lower children and reduces this contribution. Expanding at the +first omitted child will bound the loss and leave enough positive mass +to overcome the root's negative benchmark. The resulting lower bound +is Proposition~\ref{prop:reference-margin}. We also obtain a local +reference estimate for the later stopped-subtree comparison. + +We begin by replacing the continuous factor $1/b$ in the benchmark by +the corresponding prime product and controlling the resulting quadrature +error. After this error is removed, only the cutoff-$2$ discrepancy +remains to be evaluated. + +For an available cutoff $b$, let $V(b)$ be the product of $1-1/p$ over +the available primes greater than $w$. The root cutoff includes its +upper endpoint, whereas a cutoff at a previously chosen prime excludes +that prime. Write $V_x$ for the latter product when the new prime has +exponent $x$. The auxiliary cutoff $2$ includes the boundary primes +$w2$, subtracting constants by the identity +\[ + V(b)=V(2)-\sum_{22$ from the root, +\begin{equation}\label{eq:ref-residual-sum} + \sum_{\mathfrak p}h(\mathfrak p) + \left|R_{i(\mathfrak p)}^{(w)} + (r(\mathfrak p),b(\mathfrak p))\right|=o(B^{-2}). +\end{equation} +The same assertion is uniform for lower starting ratios in +$[1.99,2.3]$ and starting cutoffs $B$ tending to infinity with the +parameters of \eqref{eq:parameters}. +\end{lemma} + +\begin{proof} +The absolute next deviation in \eqref{eq:ref-residual} is nondecreasing +as a function of $x$. Thus partial summation on an exponent interval +with lower endpoint $u\geq2$ bounds the error in replacing +\[ + \sum\frac1p\frac{g_{i'}(r/x(p)-1)}{x(p)} + \quad\hbox{by}\quad + \int\frac{g_{i'}(r/x-1)}{x^2}\,dx +\] +by $O(E(u)D/u)$, where $D$ is the maximum absolute next deviation on +that interval. To justify the variation bound despite the additional +factor $1/x$, use the product-variation inequality: the total variation +is at most a constant times $D/u$. By \eqref{eq:ref-product}, replacing +the products introduces endpoint errors of the same kind and a sum +error at most $O((\log B)E(u)D)$. These estimates apply to subintervals +as well, and their continuous contributions cancel by +\eqref{eq:ref-differentiation}. + +At a state of ratio $s$, all the absolute parent, boundary, and next +deviations under discussion are $O(\phi_i(s))$. For example, +\[ + a(s)\leq\phi_\ee(s)/s,\qquad + c(s-1)\leq\phi_\ee(s)/s, +\] +and the analogous inequalities follow from +$\phi_\oo(s)=s(c(s)+a(s-1))$ on its normal range. Monotonicity bounds +the later arguments by these earliest ones. The bounded extensions +at the initial states obey the same inequalities with an absolute +constant. An omitted lower deviation is just $1$ in absolute value, +which occurs where the relevant upper derivative weight is bounded +away from zero. + +First restrict to histories with all ratios at most +$S=(\log B)^2$. If $b\leq(\log B)^4$, then $r\leq(\log B)^6$, and +the preceding partial summation bounds imply, for some fixed $C$, +\begin{equation}\label{eq:ref-small-cutoff-error} + \frac{r^2|R_i^{(w)}(r,b)|}{\phi_i(s)} + \ll(\log B)^C E(2). +\end{equation} +This is smaller than every negative power of $\log B$. + +If $b>(\log B)^4$, split the exponent interval at $\sqrt b$. Above +this point $E(\sqrt b)$ is smaller than every $B^{-A}$: indeed +$\sqrt{\sqrt b\log w}\gg(\log B)^{3/2}$. Below it every relevant +next ratio is at least $s\sqrt b-1$, and the boundary ratio is at +least $s\sqrt b$. The Dickman bounds from +Lemma~\ref{lem:derivative-weights}, together with the backward-ratio +bounds there, show that these deviations divided by $\phi_i(s)$ +are smaller than every $B^{-A}$, uniformly for $s\leq S$. One can see +the uniformity by comparing the arguments to $s$: their separation +is at least a constant times $(\log B)^2$, while the derivative weights +have exponentially decreasing ratios beyond any fixed compact set. +Any fixed polynomial losses are absorbed. Since $r=O(B)$ along a +path, the same assertion holds after multiplication by $r^2$. + +Apply \eqref{eq:prime-weight} and sum over at most +$N=O((\log B)^3)$ steps. The starting factor +$\phi_\ee(s_{\mathrm{st}})/r_{\mathrm{st}}^2$ is $O(B^{-2})$, and +\eqref{eq:ref-small-cutoff-error} and its large-cutoff counterpart show +that the low-state contribution to \eqref{eq:ref-residual-sum} is +$o(B^{-2})$. + +It remains to bound histories having a ratio above $S$. The crude +bound for every residual is $O(\log B)$, since all deviations in its +definition are bounded and the total harmonic prime sum is +$O(\log B)$. We claim the sharper bound +\begin{equation}\label{eq:ref-high-gap-error} + |R_i^{(w)}(r,b)|=O_A(B^{-A})\qquad(r\geq S/2) +\end{equation} +without a ratio restriction. In the preceding error estimates split +instead at +$Q=(\log B)\sqrt{\log\log B}$. For $x\geq Q$, the PNT saving has +exponent at least a constant times +$(\log B)(\log\log B)^{1/4}$, and hence beats every $B^{-A}$. For +$x\leq Q$, each relevant argument $r/x$ or $r/x-1$ is at least a +constant times $\log B/\sqrt{\log\log B}$. The Dickman bound +$\rho(v)\leq\exp(-v\log v+O(v))$ makes its deviation smaller than every +$B^{-A}$. The endpoint errors fall into the same alternatives. +This proves \eqref{eq:ref-high-gap-error}. + +The total prefix weight is $O(B)$, so +\eqref{eq:ref-high-gap-error} deals with the high-history prefixes +whose terminal gaps are at least $S/2$. A high ancestor on the +$x>2$ tree has gap greater than $2S$. Any remaining prefix with +terminal gap below $S/2$ has reduced that ancestor's gap by more than +half. The factorial-tail argument in Lemma~\ref{lem:high-states}, +multiplied by the crude $O(\log B)$ residual bound, deals with those +prefixes. This completes the proof. +\end{proof} + +\subsection{The boundary anomaly} + +For the actual discrete reference tree at cutoff $2$, put +\begin{equation}\label{eq:ref-anomaly} + \delta_i^{(w)}(r)=P_i(r,2)-V(2)f_i(r/2), + \qquad r\geq4\ (i=\ee),\quad r\geq2\ (i=\oo). +\end{equation} +These domains contain every state reached when iterating the root +recurrence only through exponents $x>2$. A possible first upper state +arising from a starting ratio below $2$ still has gap tending to +infinity, so it causes no exception. + +Let $Q_i(r,b)$ denote the continuous reference polynomial, obtained +from the same admission rule by replacing prime sums by ordered +integrals in $dx/x$, with exponents above $1$. On the displayed +domains define +\begin{equation}\label{eq:ref-cont-anomaly} + \delta_i(r)=Q_i(r,2)-\tfrac12 f_i(r/2). +\end{equation} + +\begin{lemma}[Boundary convergence and decay]\label{lem:ref-anomaly} +On every fixed compact subset of their respective domains, +$\delta_i^{(w)}$ converges uniformly to the continuous function +$\delta_i$. Uniformly for sufficiently large $w$, +\begin{equation}\label{eq:ref-anomaly-decay} + |\delta_i^{(w)}(r)|+|\delta_i(r)| + \ll\exp(-c r\log(r+2)). +\end{equation} +\end{lemma} + +\begin{proof} +On a fixed compact gap interval the admitted polynomial has bounded +length, because every exponent exceeds $1$. For each such length, +Lemma~\ref{lem:prime-harmonic} gives convergence of the ordered prime +sum to its harmonic integral. Ties, moving upper endpoints, and +admission equalities are confined to sets of continuous measure zero. +The same argument works along any convergent sequence of gap +parameters. The continuous integrals depend continuously on those +parameters by dominated convergence, so compactness yields uniform +convergence. Also $V(2)\to1/2$ by \eqref{eq:ref-product}. + +For decay, compare the cutoff polynomial with the full product $V(2)$. +At a first omitted lower leaf after $m$ boundary exponents, all of +which are at most $2$, the failure of admission requires +$m>(r-4)/2$. The total discrepancy is bounded by the sum of the +weights of such first omissions times their exact subsequent survivor +product, which is at most $1$. Thus it is bounded by +\[ + \sum_{m>(r-4)/2}\frac{T_2^m}{m!},\qquad + T_2=\sum_{w2$. At every visited state, including the +root, insert its anomaly \eqref{eq:ref-anomaly} and its residual +\eqref{eq:ref-residual}; propagate an insertion with the original +prefix weight and sign $(-1)^m$ at depth $m$. More explicitly, +subtracting the benchmark from the recurrence gives its cutoff-$2$ +anomaly minus the child differences, plus the residual; repeated +substitution gives +\begin{equation}\label{eq:ref-root-decomposition} + P_\ee(r_0,B)=V(B)f_{\mathrm{ext}}(r_0/B) + +\mathcal D_w+o(B^{-2}), + \qquad + \mathcal D_w=\sum_{\mathfrak p}(-1)^{|\mathfrak p|} + h(\mathfrak p)\delta_{i(\mathfrak p)}^{(w)} + (r(\mathfrak p)), +\end{equation} +where Lemma~\ref{lem:ref-residual} bounds the residual sum absolutely. + +Compact convergence in Lemma~\ref{lem:ref-anomaly}, the compact +occupation limit in Proposition~\ref{prop:prime-occupation}, and its +exponentially weighted tail estimate justify passing to the limit in +$\mathcal D_w$. The limiting stationary densities are +$\pi_\ee(t)=\phi_\ee(t)$ and +$\pi_\oo(t)=(1-t^{-2})\phi_\oo(t)$, and the arrived-at cutoff +restriction is $t0$ on $[1,2]$, and $A_2>2$. +Consequently the first omitted child occurs at even $m$ for a lower +start, and at odd $m$ for an upper start, exactly when +\begin{equation}\label{eq:ref-omission-window} + A'/2\leq y\frac83-1-0.66-0.52-0.34>0.14. +\end{equation} + +\begin{proposition}[Positive reference margin]\label{prop:reference-margin} +At the root of the reference tree, +\begin{equation}\label{eq:ref-margin} + \frac{B^2P_\ee(r_0,B)}{e^\gamma} + \longrightarrow -a_\star+\frac{2I}{M_g}>0.02. +\end{equation} +In particular, there is an absolute $c_L>0$ such that +$B^2P_\ee(r_0,B)\geq c_L$ for all sufficiently large $z$. +\end{proposition} + +\begin{proof} +The parameter definitions imply +\[ + r_0=2B-a_\star+2-\frac{2\log L}{\log w}+o(1) + =2B-a_\star+o(1). +\] +Since $f'_{\mathrm{ext}}(2)=e^\gamma$, \eqref{eq:ref-product} gives +\[ + \frac{B^2V(B)f_{\mathrm{ext}}(r_0/B)}{e^\gamma} + \longrightarrow-a_\star. +\] +Equations \eqref{eq:ref-root-decomposition}, +\eqref{eq:ref-correction-limit}, and \eqref{eq:ref-I} give the limit +in \eqref{eq:ref-margin}. Finally $M_g\leq4e^\gamma$ by +\eqref{eq:mg-bound}. The elementary bound +$\gamma\leq\sum_{j=1}^6j^{-1}-\log6<0.66$ implies $e^\gamma<1.95$. +Together with $a_\star=0.01$ and \eqref{eq:ref-I-positive}, this yields +\[ + -a_\star+\frac{2I}{M_g} + >-0.01+\frac{0.28}{4(1.95)}>0.02. +\] +The eventual positive constant $c_L$ follows. +\end{proof} + +\begin{remark}[A sharper bound for the correction integral] +\label{rem:ref-sharper-margin} +The same analytic majorant gives $I>0.2229793730$ using rational +arithmetic. To record the bound explicitly, put +\[ + b=\frac{693147180559946}{10^{15}},\qquad + J(t)=\frac{517}{432}-\frac{17t}{12}+\frac{2t^2}{3}, +\] +\[ + t_m(t)=\frac{(m+1)^2}{2m!} + \left[\left(1-\frac1m\right)t^{m-2} + +\frac{3}{2m}t^{m-1}\right]. +\] +The expansion +$\log2=2\sum_{j\ge0}((2j+1)3^{2j+1})^{-1}$ shows that +$0.69<\lambda=\log2J(b)-\sum_{m=4}^{12}t_m(b) + -\frac{t_{13}(b)}{1-225b/2744} + >\frac{222979373}{10^9}. +\] +\end{remark} + +\subsection{A uniform local comparison} + +The stopped-node argument needs a different consequence of the same +boundary analysis. Its ratios lie slightly above $2$, and its cutoffs +lie in a large but fixed window. We need a uniform upper comparison +with $f(r/b)$ there, rather than the signed root limit. The following +estimate permits the cutoff window to be chosen before the prime limit; +its $O(1/b)$ constant is independent of that window. + +\begin{lemma}[Local reference estimate]\label{lem:local-reference} +For the continuous reference polynomial, +\begin{equation}\label{eq:ref-local-cont} + Q_\ee(r,b)=\frac{f(r/b)}b+O(b^{-2}) + \qquad\left(b\longrightarrow\infty,\quad2\leq r/b\leq2.22\right), +\end{equation} +uniformly in the displayed ratios. On every fixed cutoff window +$[b_0,b_1]$ with $b_0$ sufficiently large, the corresponding discrete +estimate is +\begin{equation}\label{eq:ref-local-prime} + \frac{P_\ee(r,b)}{V(b)} + =f(r/b)+O(1/b)+o_{w\to\infty}(1), + \qquad b\in[b_0,b_1],\quad2\leq r/b\leq2.22. +\end{equation} +The constant in $O(1/b)$ is absolute, the last convergence is uniform +on the fixed window, and either prime endpoint convention is allowed. +\end{lemma} + +\begin{proof} +In the continuous model the benchmark is $f(r/b)/b$ and the quadrature +residual is identically zero by \eqref{eq:ref-differentiation}. Its +correction is the sum of the boundary anomalies $\delta_i$ along +the standard paths above $2$, with parity signs. The continuous +version of \eqref{eq:path-weight} supplies the starting factor +$\phi_\ee(r/b)/r^2\ll b^{-2}$, uniformly for $r/b\in[2,2.22]$. +The expected sum of the absolute remaining reward +$R^2|\delta_i(R)|/\phi_i(s)$ is bounded uniformly in the start. +To verify this, use bounded compact-gap visits on a fixed compact +range and, outside it, sum the unit logarithmic gap bands from +Lemma~\ref{lem:band-bounds}. After canceling $\phi_i$, a band +$R\leq r'\leq eR$ costs at most a fixed polynomial in $R$ times +$\exp(-cR\log(R+2))$, by \eqref{eq:ref-anomaly-decay}; the resulting +series converges. The empty prefix and the explicitly bounded first +step satisfy the same estimate. This proves +\eqref{eq:ref-local-cont}. + +For fixed $b\in[b_0,b_1]$ and the indicated ratios, the starting gap +is bounded and the ordered reference polynomial has uniformly bounded +length. Lemma~\ref{lem:prime-harmonic} therefore gives uniform +convergence of the discrete polynomial to $Q_\ee$: along any +convergent parameter sequence, tie, cutoff, and admission boundaries +have continuous measure zero, and dominated convergence applies to +each bounded-length integral. Compactness makes this uniform on +the whole window. Endpoint inclusion or exclusion makes no +difference to the limit. Finally \eqref{eq:ref-product} gives +$V(b)\to1/b$ uniformly there. Divide the continuous estimate by +$1/b$, and transfer it to the primes, to obtain +\eqref{eq:ref-local-prime}. +\end{proof} diff --git a/preprints/A-quadratic-bound-for-Jacobsthals-function-September-25-2026/build/sections/stops.tex b/preprints/A-quadratic-bound-for-Jacobsthals-function-September-25-2026/build/sections/stops.tex new file mode 100644 index 0000000000000000000000000000000000000000..abcd8a18bb7a270ceb5385cb8f2f1eb420763426 --- /dev/null +++ b/preprints/A-quadratic-bound-for-Jacobsthals-function-September-25-2026/build/sections/stops.tex @@ -0,0 +1,780 @@ +\section{Stops for endpoints with small effective size} +\label{sec:stops} + +The inverse and variance estimates leave compact endpoints whose large +prime factors all align to a rational of small effective size. We now +stop the tree earlier on such branches. At these stops we compare the +exact survivor counts with their reference subtrees after averaging over +one large prime. All prime cutoffs at a nonempty node in this section +are strict: if its last prime is $P$, the local survivor count avoids +the forbidden classes at the primes $t0$, is \emph{eligible} if +\begin{equation}\label{eq:stops-eligible} + D\le w^{C_s},\qquad |A|\le Yw^{C_s+2}. +\end{equation} +Fix a constant $\eta>0$, to be bounded by an absolute constant below. +A search bin has an \emph{owner} if some eligible rational aligns to +at least a $1-\eta$ fraction of its primes, by prime count. The owner +is that rational. Along a descending-prime path, its \emph{tag prime} +is the first search-bin factor that aligns to the owner of its bin, +if such a factor occurs. Its \emph{tag rational} is this owner and +is retained for the rest of the path. +\end{definition} + +The owner is unique for all sufficiently large $z$, provided +$\eta<1/2$. Indeed, if $A_1/D_1\ne A_2/D_2$ are eligible, every +prime aligning to both divides the nonzero integer +$A_1D_2-A_2D_1$, whose absolute value is at most +\[ + 2Yw^{2C_s+2}=z^{2+o(1)}. +\] +Consequently, in a bin with lower prime endpoint $R$, their common +aligning primes number at most +\begin{equation}\label{eq:stops-overlap} + O\left(\frac{L}{\log R}+1\right). +\end{equation} +For a search bin, $\log R\ge w^{1/4}\log w$, while its prime count +is $\asymp\xi R/\log R$. The bound in +\eqref{eq:stops-overlap} is a uniformly negligible fraction of this +count. Two subsets of relative size at least $1-\eta$ would have an +intersection of relative size at least $1-2\eta$, proving uniqueness. +Also, every prime of a search bin eventually exceeds every eligible +denominator. + +Choose fixed constants $b_0,b_1$ with $b_1>b_0$, taking $b_0$ +sufficiently large compared with $C_s+1$. Further conditions on these +constants will be stated below. For a prefix with bin representatives +$x_1^{\rm rep},\ldots,x_j^{\rm rep}$, write +\[ + r_j^{\rm rep}=r_0-\sum_{i=1}^j x_i^{\rm rep}. +\] + +\begin{definition}\label{def:stops-rule} +An included, nonempty, even node is a \emph{stop} if it is the first +node along its branch satisfying all the following conditions: +\begin{enumerate}[label=(\roman*)] +\item Its length is at most $C_{\rm len}\log B$. Its tag has been +attained, the tag bin has multiplicity one in this prefix, and every +prime of the prefix aligns to its tag rational. +\item Every nonempty even prefix ending at position $j$ satisfies +\begin{equation}\label{eq:stops-safety} + r_j^{\rm rep}\ge \mathcal H(x_j^{\rm rep})+3\Delta. +\end{equation} +\item At the present node, its last representative exponent and +representative ratio satisfy +\begin{equation}\label{eq:stops-window} + b_0\le x_j^{\rm rep}\le b_1, + \qquad + 2.06\le r_j^{\rm rep}/x_j^{\rm rep}\le2.16. +\end{equation} +\end{enumerate} +The empty root is not subjected to \eqref{eq:stops-safety}. +\end{definition} + +This is a prefix-free choice of lower nodes. The rule applies to the +entire included tree; the inverse and variance exceptions classify +endpoints but do not restrict this rule. +The representative safety +condition ensures that varying prime choices inside their bins, while +preserving multiplicities and distinctness, preserves admission of all +even prefixes by \eqref{eq:admission}. One may see this directly from +the right-endpoint convention: the actual gap is at least the +representative gap and the actual last exponent is at most its +representative. Alternatively, the movement bound and the +$2$-Lipschitz property of $\mathcal H$ give the same conclusion with room to +spare. Odd children of lower nodes require no admission test. + +For sufficiently small fixed $\xi$, every stop has actual last +exponent $b$ and actual ratio satisfying +\begin{equation}\label{eq:stops-actual-window} + b_0-o(1)\le b\le b_1, + \qquad 2.04\le r_d/b\le2.18. +\end{equation} +Here and throughout this section, the eventual bounds may depend on all +the fixed constants, including $\xi$. The tag prime lies in a search +bin, whose exponent tends to infinity. Thus it is different from the +last prime $P=w^b$ at every stop, once $z$ is sufficiently large. + +\subsection{The average over a large prime bin} + +At a stop, we will fix all prime factors except the tag prime. The +arithmetic estimate below first lets that prime range over its whole +bin, imposing the aligned hit class even where the actual class does +not align. Its lower average and absolute individual upper bound will +then permit removal of the small nonaligning subset. + +For a prime $P$, write +\[ + V_{\rm all}(P)=\prod_{tb_0$ be fixed. For every sufficiently small fixed $\xi>0$, the +following holds for all sufficiently large $z$. +Let $A/D$ be eligible in the sense of \eqref{eq:stops-eligible}, let +$P=w^b$ be a prime with $b\in[b_0-1,b_1]$, and let $q'$ be a squarefree +product of primes at most $z$ whose least prime factor is $P$ and all of whose prime factors +align to $A/D$. Let $\mathcal P$ be a search bin, none of whose primes +divides $q'$. Suppose that, for every $p\in\mathcal P$, +\begin{equation}\label{eq:stops-length-exponents} + J_p=\frac{Y}{pq'},\qquad 2.03\le\log_PJ_p\le2.19. +\end{equation} +Define $T_p$ to count the integers $1\le n\le Y$ satisfying +$Dn\equiv A\pmod{pq'}$ and avoiding the prescribed classes at every +prime $tP$ prime. +The prime factors of $d_0$ are chosen among the nonaligned primes +less than $P$ that do not divide $D$. For such a prime $t$, divisibility +of $d_0$ by $t$ makes the avoidance condition automatic; otherwise +one of the $t-1$ unit classes for the varying prime is forbidden. +We isolate the resulting estimate before proving the arithmetic lemma. +For any subset $E$ of the primes less than $P$, put +\begin{equation}\label{eq:stops-euler-weight} + W_E(d_0)=\prod_{\substack{t\in E\\t\nmid d_0}} + \left(1-\frac1{t-1}\right). +\end{equation} +The next lemma is uniform in $E$, including when $2\in E$. + +\begin{lemma}\label{lem:stops-euler-probability} +Uniformly over all subsets $E$ of the primes less than $P$, +\begin{equation}\label{eq:stops-euler-lower} + \sum_{\substack{d_0\le P^{.95}\\t\mid d_0\Rightarrow t\in E}} + \frac{W_E(d_0)}{d_0} + \ge e^{-\gamma}(.94-o(1)). +\end{equation} +\end{lemma} + +\begin{proof} +For $t\in E$, let $V_t$ be independent nonnegative integer-valued +random variables with probabilities +\[ + \Pbb(V_t=0)=\frac{t-2}{t-1},\qquad + \Pbb(V_t=j)=t^{-j}\quad(j\ge1). +\] +These masses sum to one, and the probability that +$\prod_{t\in E}t^{V_t}=d_0$ is exactly $W_E(d_0)/d_0$. This +interpretation includes $t=2$: the mass at zero is then zero and +$\sum_{j\ge1}2^{-j}=1$. In particular the zero factors in +\eqref{eq:stops-euler-weight} cause no exception to the probability +interpretation. + +Let $U_t$ instead have the ordinary geometric probabilities +$\Pbb(U_t=j)=(1-1/t)t^{-j}$ for $j\ge0$. For $j\ge1$, +\[ + \Pbb(V_t\ge j)=\frac{t^{1-j}}{t-1} + \ge t^{-j}=\Pbb(U_t\ge j). +\] +Couple $U_t,V_t$ monotonically, independently over $t$. Their mean +difference is +\[ + \E(V_t-U_t) + =\frac{t}{(t-1)^2}-\frac1{t-1} + =\frac1{(t-1)^2}. +\] +It follows that the excess log-product +$Z_E=\sum_{t\in E}(V_t-U_t)\log t$ has uniformly bounded mean, +since $\sum_t\log t/(t-1)^2<\infty$. Extend the independent ordinary +variables to all primes $t.01\log P)=O(1/\log P)$, uniformly in $E$. Outside this +exception the preceding event implies +$\prod_{t\in E}t^{V_t}\le P^{.95}$. Subtracting the exceptional +probability proves \eqref{eq:stops-euler-lower}; no independence +between these last two events is required. +\end{proof} + +\begin{proof}[Proof of \cref{lem:stops-aligned-average}] +Every prime in the search bin eventually exceeds both $D$ and $P$. +The alignment of the prime factors of $q'$ implies $(D,q')=1$; +hence $(D,pq')=1$ for every $p\in\mathcal P$. The hypothetical hit +condition and integrality of $n$ are therefore equivalent to +\begin{equation}\label{eq:stops-m-coordinate} + Dn-A=pq'm, + \qquad + 1\le\frac{A+pq'm}{D}\le Y, + \qquad pq'm\equiv-A\pmod D. +\end{equation} +We count a subfamily of these integers in which $m$ has one prime +factor exceeding $P$. Averaging over $p$ then imposes the remaining +avoidance conditions. + +\paragraph{A common interval for the auxiliary integer.} +Partition $\mathcal P$ into +$\lceil w^{2C_s+10}\rceil$ equal-length subbins in prime coordinates. +In one such subbin let $R'$ be its lower endpoint, let $V$ be its +length, and put +\[ + J^\circ=\frac{Y}{R'q'},\qquad s^\circ=\log_P J^\circ. +\] +Uniformly over these subbins, +\begin{equation}\label{eq:stops-subbin-size} + \frac{V}{R'}\asymp\xi w^{-2C_s-10}, + \qquad J_p=J^\circ(1+o(1)). +\end{equation} +Let $I$ be the intersection, over the real values of $p$ in the +subbin, of the real $m$-intervals specified by the middle condition +in \eqref{eq:stops-m-coordinate}. The endpoints of one such interval +are +\[ + \frac{D-A}{pq'},\qquad \frac{DY-A}{pq'}. +\] +Relative to $DJ^\circ$, the variation of either endpoint is at most +\[ + O\left(\xi w^{-2C_s-10} + \left(1+\frac{|A|}{DY}\right)\right)=o(1), +\] +by \eqref{eq:stops-eligible}. Consequently +\begin{equation}\label{eq:stops-common-interval} + |I|=(1-o(1))DJ^\circ. +\end{equation} +The height of this interval satisfies +\[ + \sup_{m\in I}|m| + \le (D+|A|/Y)J^\circ(1+o(1)) + \le P^{s^\circ+.02}, +\] +provided $b_0$ is sufficiently large in terms of $C_s$. Indeed the +extra factor is bounded by a constant times $w^{C_s+2}$, which is +absorbed by $P^{.02}$ after such a choice. All these statements are +uniform in the aligned progression family, its eligible rational, +and the subbin. + +Among primes $tP\ \hbox{prime}, +\end{equation} +with $m\in I$. Such a representation is unique: its factor $d_0$ +is less than $P$, while the distinguished prime factor is greater than +$P$. Furthermore, $(m,D)=1$, since $d_0$ has no prime factor dividing +$D$, and $DP^{1.04}, +\] +using \eqref{eq:stops-length-exponents} and +\eqref{eq:stops-common-interval}. Its length divided by its upper +endpoint is at least $(1-o(1))DP^{-.04}$, and hence at least +$P^{-.05}$. If its upper endpoint is $v$, then $v>P^{1.04}$ implies +\[ + vP^{-.05}\ge v^{9/10}. +\] +The uniform short-interval prime number theorem in +\cref{lem:prime-inputs} therefore applies. Every upper endpoint has +logarithm at most $(s^\circ+.02)\log P$. Summing the resulting +lower bounds for the retained sign pieces proves +\eqref{eq:stops-prime-factor-count}. In particular their prime factors +automatically exceed $P$. + +\paragraph{Sieving the varying prime.} + +Fix one of the integers $m$ just counted. The last congruence in +\eqref{eq:stops-m-coordinate} prescribes a single unit class for $p$ +modulo $D$. This uses $(A,D)=(q'm,D)=1$. For each prime $t\mid D$, +increasing $p$ by $D$ changes the integer $n$ by $q'm$, which is a +unit modulo $t$. Thus avoiding $a_t$ removes exactly one of the $t$ +lifts of this base class modulo $Dt$. Taken together, the remaining +classes modulo +\[ + Q=D\rad(D) +\] +form a set $\mathcal A$ of +$\varphi(\rad(D))$ unit classes. Since +$\varphi(Q)=\varphi(D)\rad(D)$, their proportion among all unit +classes modulo $Q$ is exactly +\begin{equation}\label{eq:stops-lift-fraction} + \frac{|\mathcal A|}{\varphi(Q)}=\frac1D. +\end{equation} +For $D=1$ this means the single unrestricted class and the same formula +holds. + +For $t\in E$, avoiding the original class is equivalent to +\[ + pq'm\not\equiv Da_t-A\pmod t, +\] +with nonzero right side. If $t\mid d_0$, this is automatic. If +$t\nmid d_0$, then $m$ is a unit modulo $t$ because $\ell'>P$, and +exactly one nonzero $p$-class modulo $t$ is forbidden. Thus the +product of the remaining survival factors is +$W_E(d_0)$ from \eqref{eq:stops-euler-weight}. +Let $N_{\mathcal I}$ be the number of primes in the present subbin +$\mathcal I$. We claim that for any fixed $\gamma_1>0$, for all +sufficiently large $z$, +\begin{equation}\label{eq:stops-tag-prime-count} + \#\{p\in\mathcal I:\ p\ \hbox{gives a local survivor for this }m\} + \ge(1-\gamma_1)\frac{N_{\mathcal I}}D W_E(d_0), +\end{equation} +uniformly in all the data. If the product vanishes, the claim is +trivial. In every other case, the restricted primes are at least $3$; +their sieve densities $g(t)=1/(t-1)$ satisfy the hypotheses of +\cref{lem:fundamental}. + +Here are the required progression and remainder bounds. Choose a fixed +$A_2$ large enough, in terms of $\gamma_1$, to make the relative +fundamental-lemma error small, and use sieve level +$S=P^{A_2}$. For a squarefree intersection modulus $l\le S$ on +the restricted primes and one class in $\mathcal A$, the forbidden +conditions specify a single reduced class modulo $Ql$, since +$(l,Q)=1$. Siegel--Walfisz from \cref{lem:prime-inputs} gives its +count as +\[ + \frac{\li(R'+V)-\li(R')}{\varphi(Q)\varphi(l)} + +O_{M_2}\left(\frac{R'}{(\log R')^{M_2}}\right). +\] +The exponent $M_2$ can be any sufficiently large fixed constant. To +check the modulus range, observe that +\begin{equation}\label{eq:stops-sw-size-bounds} + Q\le w^{2C_s},\qquad |\mathcal A|\le w^{C_s},\qquad + S\le w^{b_1A_2},\qquad \log R'\ge w^{1/4}\log w. +\end{equation} +Thus $Ql$, the level, and the lift count are each bounded by fixed +powers of $\log R'$. The main term, for each base class, has the +required multiplicative intersection density +$1/\varphi(l)=\prod_{t\mid l}1/(t-1)$. + +For completeness, let $C$ be the fixed divisor-weight exponent in the +remainder of \cref{lem:fundamental}. Summing over lifts and using the +crude bound $\sum_{l\le S}\tau(l)^C\ll S^{C+1}$, the total remainder +is at most +\[ + O_{M_2}\left( + |\mathcal A|S^{C+1}\frac{R'}{(\log R')^{M_2}} + \right). +\] +In a nonzero case, uniformly over subsets of restricted primes, +\[ + W_E(d_0)\gg\frac1{\log P}. +\] +Indeed the product over all $3\le t.423. +\] +Choose $\gamma_1$ as a sufficiently small absolute constant; for +example $\gamma_1=.05$ leaves a limiting coefficient greater than +$.402$. Since $J_p=J^\circ(1+o(1))$ uniformly within a subbin, +summing over all subbins proves \eqref{eq:stops-all-primes-lower}. +All relative errors used here are uniform, so the growing number of +subbins and the number of pairs $(d_0,m)$ do not accumulate additional +errors. + +For the upper bound, the hit conditions give one ordinary progression +of step $pq'$ and coordinate length $J_p+O(1)$. +Every sieve prime below $P$ is +coprime to this step. Apply the upper form of +\cref{lem:fundamental} using primes up to $P^\theta$, for a +sufficiently small fixed absolute $\theta>0$, and a level such as +$P^{1/2}$. The relative sieve parameter is a sufficiently large +absolute constant; the summed $O(1)$ intersection remainders are +$P^{1/2+o(1)}$, whereas $J_p\ge P^2$. Mertens' theorem gives +\[ + \prod_{t0$ with the following property. +Fix $C_s$ and the regular-box parameters. Take $0<\eta\le\eta_0$, +then take $b_0$ sufficiently large in terms of $C_s$, and take any +fixed $b_1>b_0$. For all sufficiently small fixed $\xi>0$ and all +sufficiently large $z$, the stops of \cref{def:stops-rule} satisfy +\begin{equation}\label{eq:stops-comparison} + \sum_{d\ \mathrm{stopped}} S_d(x(P_d)) + \ge + \sum_{d\ \mathrm{stopped}}\mu_d P_\ee(r_d,x(P_d)), +\end{equation} +where $P_d$ denotes the last prime of the tuple and both cutoffs are +strict below $P_d$. +\end{proposition} + +\begin{proof} +We first explain exactly which prime is averaged. Partition the stopped +nodes by their ordered bin labels, their tag position, and all prime +choices other than the tag prime $p$. In a nonempty group, let $A/D$ +be the owner of its tag bin. Then $p$ ranges over the entire subset of +that bin aligning to $A/D$. + +To justify this assertion, replace $p$ by any other aligning prime in +the bin. All earlier choices remain fixed, so no earlier tag appears. +At the designated position the new prime aligns to the same owner; +hence the tag position and tag rational remain unchanged. Every +subsequent all-prefix alignment test has the same truth value, since +the other primes are fixed and the replacement aligns. The +representative tests, lengths, and bin multiplicities depend only on +the labels. Admission is preserved by \eqref{eq:stops-safety}, and +the tag-bin multiplicity one prevents a change in prime ordering or a +collision with another factor. These observations apply to each +earlier prefix as well, so no earlier stop is introduced. This proves +the assertion about the entire aligning subset. + +In this group put $q'=d/p$. The last prime $P$ is a factor of $q'$ +and is fixed. All primes of $q'$ are at least $P$, and they align to +$A/D$. First allow \emph{all} primes $p$ in the tag bin, assigning +to $p$ the hypothetical hit class aligned to $A/D$. These hypothetical +choices need not be actual stops; their local counts are auxiliary. +Let $\mathcal P$ denote this tag bin, and use the hypothetical counts +$T_p$ of \cref{lem:stops-aligned-average}. On its aligning subset, +$T_p=S_{pq'}(x(P))$. + +For all choices of $p$ in the tag bin, put $b=x(P)$ and use the gap +$r_{pq'}=\log_w(Y/(pq'))-a_\star+2$. Thus +\[ + \log_PJ_p=\frac{r_{pq'}}b+\frac{a_\star-2}{b}. +\] +The bin labels and representative safety tests are fixed, so the +movement bound applies also to these hypothetical choices. By +\eqref{eq:stops-actual-window}, after enlarging $b_0$ if necessary, +all of them satisfy \eqref{eq:stops-length-exponents}. The search +bin is disjoint from the prime factors of $q'$ because the tag bin +has multiplicity one. All hypotheses of +\cref{lem:stops-aligned-average} therefore hold. + +For sufficiently large $z$, the actual cutoff lies in the fixed +interval $[b_0-1,b_1]$. +Applying \cref{lem:local-reference} on this interval, together with the +product identity $V_{\rm all}(P)=V_0V(b)$, gives +\[ + \frac{\mu_{pq'}P_\ee(r_{pq'},b)}{J_pV_{\rm all}(P)} + \le f(r_{pq'}/b)+O(1/b)+o(1). +\] +The constants are uniform for the fixed enlarged window $[b_0-1,b_1]$ and for +the ratio window in \eqref{eq:stops-actual-window}. On that ratio +window, the near-$2$ formula and monotonicity of $f$ give +\[ + f(r_{pq'}/b)\le f(2.18) + <\frac{2(1.95)(.166)}{2.18}<.30. +\] +Here $e^\gamma<1.95$ follows, for example, from +$\gamma\le\sum_{j=1}^6j^{-1}-\log6<.66$, and +$\log(1.18)<.166$ follows from +$\log(1+u)\le u-u^2/2+u^3/3$ at $u=.18$. +Thus first choosing $b_0$ large and subsequently taking $z$ large gives +\begin{equation}\label{eq:stops-reference-upper} + \mu_{pq'}P_\ee(r_{pq'},b)\le .33 J_pV_{\rm all}(P) +\end{equation} +uniformly over the group and over its hypothetical choices. + +Apply \eqref{eq:stops-all-primes-lower} to the full tag bin. +Let $\mathcal G$ be the aligning subset in the tag bin. Its complement +contains at most an $\eta$ fraction of the bin primes. Since the +$J_p$'s vary by at most $1+\xi$ within the whole bin, for $\xi\le1$ +we have +\[ + \sum_{p\notin\mathcal G}T_p + \le 2C_U\eta V_{\rm all}(P) + \sum_{p\ \mathrm{in\ bin}}J_p. +\] +Fix an absolute $\eta_0<1/2$ so small that +$2C_U\eta_0<.05$. For $0<\eta\le\eta_0$, subtracting this bound +from \eqref{eq:stops-all-primes-lower} and using +\eqref{eq:stops-reference-upper} gives +\[ + \begin{split} + \sum_{p\in\mathcal G}S_{pq'}(x(P)) + &=\sum_{p\in\mathcal G}T_p\\ + &\ge .34V_{\rm all}(P)\sum_{p\ \mathrm{in\ bin}}J_p\\ + &\ge\sum_{p\in\mathcal G}\mu_{pq'}P_\ee(r_{pq'},b). + \end{split} +\] +The last inequality uses $.34>.33$ and positivity of $J_p$; it does +not require a distribution hypothesis for the survivors inside the +owner subset. Every stopped node belongs to exactly one of the +groups considered. Summing the group inequalities proves +\cref{prop:stopped-comparison}. + +\end{proof} + +\subsection{Almost every hard endpoint lies below a stop} + +We next explain why the same stops capture all but an arbitrarily small +scaled harmonic mass of the hard compact endpoints. In what follows, +the harmonic mass of a family of tuples is $\sum_d1/d$. + +\begin{proposition}[Stopping hard endpoints] +\label{prop:hard-stopping} +Fix a compact gap bound $K$, the regular-box parameters, the bounded +candidate lists from \cref{prop:inverse}, and the exponent $C_s$ +from \cref{cor:hard-endpoints}. Fix +$0<\eta\le\eta_0$ as in \cref{prop:stopped-comparison}. +For every $\delta>0$, take $b_0$ sufficiently large in terms of $C_s$ +and $K$, then take $b_1/b_0$ sufficiently large in terms of the desired +error, and finally take $\xi$ sufficiently small. For these fixed +choices, the regular included hard endpoints with $r_d\le K$ having +no stopped prefix have harmonic mass at most +\begin{equation}\label{eq:stops-hard-mass} + (\delta+o(1))B^{-2}. +\end{equation} +The eventual $o(1)$ may depend on all the fixed choices. In particular, +$b_1$ and $\xi$ are fixed before $z$ tends to infinity. +\end{proposition} + +\begin{proof} +For a hard endpoint, \cref{cor:hard-endpoints} gives a rational on one +of its candidate lists that satisfies \eqref{eq:stops-eligible} and +aligns to every path factor of exponent greater than $C_s$. We need +two facts about this path: its tag must be this rational, and +it must have an earlier even visit in the stop window. We establish +the tag statement in the full-box product measure, and the visit +statement in the unconditioned path process. Their exceptional masses +will then be added. + +\paragraph{Identifying the tag.} +Except for $o(B^{-2})$ mass, the path acquires this same rational as +its tag during its search positions, as we now show. + +Fix one regular full candidate box and one eligible rational $A/D$ +on its list. Each of its search bins has multiplicity one, and +\cref{prop:regular-boxes} supplies at least $c_7\log w$ such bins. +For clarity, the count follows because all nonsearch factors together +decrease the gap by at most +\[ + C_{\rm len}\log B\,(w^{1/4}+h)=o(w^{1/2}). +\] +The search factors come first and must therefore decrease the gap from +$r_0\asymp B\asymp w\log w$ to at most $w^{1/2}$. A standard +transition decreases the gap by at most a bounded multiplicative +factor, including at the initial extended start. At least a constant +multiple of $\log w$ search factors is necessary. + +At a hard endpoint for the chosen rational, all these search primes +align to it. If none of their bins is near-full for this rational, +each of the bins has more than an $\eta$ fraction of nonaligning +primes by count. For primes in $[R,(1+\xi)R]$, this is a harmonic +fraction at least $\eta/(1+\xi)$. The search choices are independent +in the full product box, so the fraction of its harmonic measure with +all those alignments is at most +\[ + \left(1-\frac{\eta}{1+\xi}\right)^{c_7\log w}=o(1). +\] +This bounds the relevant hard event in the full box without asserting +independence after conditioning on hardness. + +Otherwise some visited search bin is near-full for this rational, +which is therefore its owner. The hard endpoint's choice in this bin +aligns to its owner, so a tag has been acquired by that position. If +the first tag is a different rational, the prime at the earlier tag +position aligns both to $A/D$ and to the distinct eligible owner of +that bin. By \eqref{eq:stops-overlap}, its possible choices form a +uniformly negligible fraction, even after summation over +$C_{\rm len}\log B$ positions. Indeed in a search bin this fraction +is at most a constant, depending on fixed $\xi$, times +\[ + \frac{L+\log R}{R}, +\] +and $R\ge\exp(w^{1/4}\log w)$. + +There are at most the fixed list bound times the fixed candidate count +such rational choices per box. The union of the preceding exceptions +therefore costs $o(1)$ of the full candidate-box harmonic measure. +The total mass of the enlarged boxes is $O_K(B^{-2})$, independently +of sufficiently small $\xi$, by \cref{prop:regular-boxes}. Summing +over the boundedly many candidates gives a total $o(B^{-2})$ +exception. Outside it, a hard endpoint has the correct tag by the end +of its search positions. + +\paragraph{Finding an even visit before the compact endpoint.} +Apply +\cref{lem:marked-visits} to the continuous standard path. For any +prescribed probability error $\varepsilon_1>0$, with $b_0$ fixed, +choosing $b_1/b_0$ sufficiently large ensures, uniformly from the +allowed starting ratios as the initial gap tends to infinity, an even +arrival with ratio in $[2.08,2.14]$ and last exponent in +$[2b_0,b_1/2]$, except with probability at most +$\varepsilon_1+o(1)$. The proof of that lemma marks the first draw +from an odd regeneration when its ratio lies in $[2.09,2.13]$, using +a pre-draw gap in $[10b_0,b_1]$. In particular the arrival has the +stronger interior margins +\begin{equation}\label{eq:stops-marked-slack} + \frac{10b_0}{3.13}\le x\le\frac{b_1}{3.09}, + \qquad 2.09\le s\le2.13. +\end{equation} +The choice of $b_1$ depends on the probability budget and fixed $b_0$, +not on $z$ or on $\xi$. + +Use \cref{prop:prime-coupling} for standard prime paths with lower +exponent cutoff $1$. Actual included paths are a subset of these +standard paths; their rules agree above exponent $2$. High-ratio +histories ending in the fixed compact window have negligible scaled +mass by \cref{lem:high-states}. On the remaining histories the +coupling matches states and exits, with a failure probability tending +to zero. The interior margins in \eqref{eq:stops-marked-slack} +therefore give the asserted slightly wider visit windows on matched +prime paths. + +This visit lies in the segment being compared whenever the matched +prime history ends in the compact window. In fact its arrival gap is +at least a fixed multiple of $b_0$. Taking $b_0$ sufficiently large +relative to $K$, that gap exceeds the matched compact endpoint gap. +Since gaps decrease along the path, a raw continuous marked visit +cannot occur after this endpoint. Matching exits puts the visit +before any cutoff exit as well. Thus the coupling cannot lose the +needed visit merely by stopping the compared segment at the compact +endpoint. + +The compact-history event estimate in \cref{prop:prime-occupation} +now bounds the harmonic mass of compact endpoints missing such a visit +by +\[ + C_K(\varepsilon_1+o(1))B^{-2}+o(B^{-2}), +\] +where $C_K$ depends on the fixed compact window and the starting bounds, +not on the later bin width. Choose $\varepsilon_1$ sufficiently small +in terms of $\delta/C_K$, and choose $b_1$ accordingly. + +\paragraph{Verifying the stop conditions.} +Consider a regular hard endpoint with the correct tag and with this +even visit. Every factor up to the visit has exponent at least the +visit's last exponent, which exceeds $C_s$ by the choice of $b_0$. +Hence all these factors align to its tag rational. Its search +positions have already passed: their lower exponents are at least +$w^{1/4}$, whereas the visit exponent is bounded by fixed $b_1$. +The tag is therefore attained by the visit, and its bin has multiplicity +one. The visit-prefix length is at most the regular endpoint length +$C_{\rm len}\log B$. + +By regularity, every earlier nonempty even prefix, including the visit, +has actual margin greater than $10\Delta$ in the admission test. +The representative movement is at most $\Delta$, and changing the +last exponent changes $\mathcal H$ by at most $2h$. For sufficiently large +$z$, the representative margin is consequently at least +$10\Delta-\Delta-2h>3\Delta$. Thus +\eqref{eq:stops-safety} holds. Finally the ratio and exponent slack +in the visit windows, the prefix movement bound, and a sufficiently +small fixed $\xi$ imply \eqref{eq:stops-window}. All stop conditions +hold at this visit. The branch has therefore stopped there or earlier. + +Combining the $o(B^{-2})$ incorrect-tag exception with the arbitrarily +small missing-visit mass proves \eqref{eq:stops-hard-mass}. +\end{proof} + +The order of choices in these two propositions is worth making +explicit. The constant $C_U$ in +\eqref{eq:stops-individual-upper} is absolute, so the owner threshold +$\eta$ can be fixed before either stop-window endpoint. The choice +of $b_0$ absorbs $C_s$, the local reference error, the interval-height +margin, and the compact gap range. Only the ratio $b_1/b_0$ must then +be enlarged to make the marked-visit error as small as desired. +Afterwards $\xi$ is chosen for representative movements and all earlier +freezing conditions. Every progression modulus and prime-sieve level +used here remains a fixed power of $\log R'$ for these choices, however +large the fixed $b_1$ is. The corresponding asymptotic thresholds may +therefore wait until this final fixed $\xi$ has been chosen. diff --git a/preprints/A-quadratic-bound-for-Jacobsthals-function-September-25-2026/build/sections/tree.tex b/preprints/A-quadratic-bound-for-Jacobsthals-function-September-25-2026/build/sections/tree.tex new file mode 100644 index 0000000000000000000000000000000000000000..ff67712177b5ccfacbef14dfeb9e81a788e952e3 --- /dev/null +++ b/preprints/A-quadratic-bound-for-Jacobsthals-function-September-25-2026/build/sections/tree.tex @@ -0,0 +1,148 @@ +\section{The decreasing-prime tree}\label{sec:tree} + +Fix the residue classes in Theorem~\ref{thm:survivors}, and use +\eqref{eq:parameters}. Set +\[ + a_\star=\frac1{100},\qquad x(p)=\frac{\log p}{\log w}. +\] +Thus $x(p)$ measures a prime's exponent in base $w$, and +$12$ for large $z$, giving the same conclusion +for every first child. Included lower nodes also have gap exceeding +2 by their admission rule. Hence all included nodes have +$\log_wJ_d=r_d-2+a_\star\ge a_\star$. +\end{proof} + +Figure~\ref{fig:admission-and-stopping} separates the admission decision +at an upper node from the later choice of lower stops. +\input{figures/proof-route} + +In particular, uniformly at included nodes, +\begin{equation}\label{eq:tree-small-counts} + \frac{N_d}{\mu_d}\ll_{a_\star}1, + \qquad + \left|\frac{N_d}{\mu_d}-1\right| + \ll e^{-c_1\log_wJ_d} + \quad\text{when }\log_wJ_d\ge v_0. +\end{equation} +The absolute upper bound remains valid on the slightly enlarged +progressions with $J_d\ge w^{a_\star/2}$ that will arise in boxes. + +\subsection{The reference tree} + +Replace $N_d$ by $\mu_d$ in the unstopped tree. Divide a subtree value +by its own $\mu_d$, and denote the result by $P_\ee(r,b)$ or +$P_\oo(r,b)$ according to parity. Dependence on $w$ and on the +endpoint convention is left implicit. Since $\mu_{dp}=\mu_d/p$, +the recursions are +\begin{equation}\label{eq:reference-tree} +\begin{split} + P_\ee(r,b)&=1-\sum_{p\text{ available}}\frac1p + P_\oo(r-x(p),x(p)),\\ + P_\oo(r,b)&=1- + \sum_{\substack{p\text{ available}\\r-x(p)\ge \mathcal H(x(p))}} + \frac1p P_\ee(r-x(p),x(p)). +\end{split} +\end{equation} +The reference value at the root is $YV_0P_\ee(r_0,B)$. +Auxiliary starting states used below follow the same recursions on +their declared domains; no actual forbidden classes are needed to +define them. + +A path's \emph{harmonic weight} is the reciprocal product of its +chosen primes. Dropping all admission restrictions shows that the +sum of the weights of all decreasing-prime prefixes is +\begin{equation}\label{eq:total-prefix-mass} + \prod_{w0\), from the candidate list in +\cref{prop:inverse}. All of \(q_f,A,D\) remain fixed as \(p\) varies. +As before, let \(q_{f,E}\) be the product of the nonaligned prime factors +of \(q_f\), and put \(q_{f,A}=q_f/q_{f,E}\). + +If the structured-parent conclusion of \cref{prop:inverse} holds, the +product of the candidate edge and the later factors is at most a fixed +power of \(w\), by the candidate parent-gap bound. Hence, for large \(z\), +\begin{equation}\label{eq:variance-structure-size} + H:=Dq_{f,E}\le R\mathcal Z^{11}, + \qquad + |A|\le S_qR\mathcal Z^{10}. +\end{equation} +Here \(q\) is the product of the other factors at the candidate parent, +\(S_q\) is the corresponding product of bin upper endpoints, and +\[ + q_f\ge q,\qquad S_q/q\le\mathcal Z^{o(1)}. +\] +The additional factor \(\mathcal Z\) in the bound for \(H\) absorbs any +fixed power of \(w\). + +Let \(b\in[1,q_f]\) be the integer CRT representative of the hit +conditions at the primes of \(q_f\), and define +\begin{equation}\label{eq:variance-effective} + C_0=\frac{Db-A}{q_{f,A}}, + \qquad T=\max\left(H,\frac{|C_0|}{R}\right). +\end{equation} +The numerator is divisible by every aligned factor of \(q_f\), so \(C_0\) +is an integer. To see the role of these parameters, write the hit +progression at \(q_f\) as \(n=b+q_f v\). Then +\[ + Dn-A=q_{f,A}(C_0+Hv). +\] +Thus removing the aligned factors leaves an arithmetic progression of +step \(H\) and initial value \(C_0\). Alignment at the isolated prime +will require this remaining value to be divisible by \(p\). +The scale \(T\) measures both its step and the size of its initial value +after division by a prime in the bin \(p\asymp R\). + +We also have +\begin{equation}\label{eq:variance-unit} + (C_0,H)=1. +\end{equation} +Indeed, \(q_{f,A}\) is coprime to \(Dq_{f,E}\). At a prime \(t\mid D\), +reducedness gives \(Db-A\equiv-A\not\equiv0\pmod t\); at a prime +\(t\mid q_{f,E}\) not dividing \(D\), nonalignment gives +\(Db-A\equiv Da_t-A\not\equiv0\pmod t\). +This also covers common prime divisors of \(D\) and \(q_{f,E}\). +Prime powers in \(D\) cause no further restriction. +In particular, \(C_0=0\) can occur only when \(H=1\). +The height bound gives +\[ + \frac{|C_0|}{RH} + =\frac{|Db-A|}{RDq_f} + \le \frac1R+\frac{|A|}{RDq_f} + \le \frac1R+\frac{S_q}{q}\mathcal Z^{10}, +\] +and therefore +\begin{equation}\label{eq:variance-effective-ratio} + T/H\le\mathcal Z^{12} +\end{equation} +for sufficiently large \(z\). + +For any \(p\in\mathcal P\) not dividing \(H\), temporarily impose the +aligned class at \(p\), whether or not its actual prescribed class aligns. +Since \(p\nmid q_f\), the additional hit condition is +\(C_0+Hv\equiv0\pmod p\). There is a unique +\(k\in\{0,\ldots,p-1\}\) satisfying this congruence, and all its +solutions have the form \(v=k+pj\). Define +\[ + m_0=\frac{C_0+Hk}{p}. +\] +Equivalently, \(m_0\) is the unique integer satisfying +\begin{equation}\label{eq:variance-m-coordinate} + m_0\in[C_0/p,C_0/p+H), + \qquad pm_0\equiv C_0\pmod H. +\end{equation} +The interval has length \(H\), and the congruence fixes one residue +modulo \(H\), which also verifies uniqueness in these coordinates. +The resulting simultaneous hit progression is exactly +\begin{equation}\label{eq:variance-progression} + n=b+q_f(k+pj),\qquad j\ge0, + \qquad + Dn-A=pq_{f,A}(m_0+Hj). +\end{equation} +Its initial representative belongs to \([1,pq_f]\), so the allowable +\(j\)'s in \(1\le n\le Y\) form an initial integer interval of length +\(Y/(pq_f)+O(1)\). + +An included endpoint has \(J_d\ge w^{a_\star}\). The within-box movement +of \(\log_w J_d\) is at most \(\Delta\), by +\cref{prop:regular-boxes}. Taking \(\Delta0\), its length eventually exceeds \(w^{a_\star/2}\); +this is why the slack in \eqref{eq:variance-common-interval} is retained. +The constant in this survivor bound is independent of sufficiently small +fixed \(\xi\), although the eventual threshold may depend on it. +The reference count changes by \(O(\xi JV_0)\) as well. +Consequently, when \(\xi\) is small relative to a fixed \(\eps>0\), it +suffices to control +\begin{equation}\label{eq:variance-bad-common} + \left|S-JV_0\right|>\frac{\eps}{2}JV_0, +\end{equation} +where \(S\) counts the small-prime survivors on \(\mathcal J\). + +For a prime \(t\le w\), the avoidance condition on \(\mathcal J\) is +\begin{equation}\label{eq:variance-avoidance} + \begin{cases} + pq_{f,A}(m_0+Hj)\not\equiv Da_t-A\pmod t,&t\nmid H,\\ + k+pj\not\equiv(a_t-b)\overline{q_f}\pmod t,&t\mid H. + \end{cases} +\end{equation} +In the first line \(t\nmid D\), so multiplication by \(D\) is reversible. +In the second line \(q_f\) is invertible modulo \(t\). +All factors of \(q_f\) exceed \(w\); hence the primes \(t\le w\) dividing +\(H\) are precisely the small primes dividing \(D\). +Put +\begin{equation}\label{eq:variance-small-products} + T_0=\prod_{\substack{t\le w\\t\mid H}}t, + \qquad + T_1=\prod_{\substack{t\le w\\t\nmid H}}t. +\end{equation} +Thus \((T_1,HT_0)=1\), every prime of \(T_0\) divides \(H\), and +\begin{equation}\label{eq:variance-pattern-size} + T_0T_1=\exp(O(w))=R^{o(1)}. +\end{equation} + +\subsection{Variance of independent classes} + +The next second-moment argument is a truncated, two-point form of +mean-one singular-series averaging; compare Gallagher +\cite[equation~(3) and Section~2]{Gallagher1976}. +We include the proof to establish the uniform range of interval lengths +and the stability under changes of measure needed below. + +\begin{lemma}\label{lem:random-variance} +Choose, independently at each prime \(t\le w\), one uniformly distributed +forbidden residue class. Let \(S\) count the integers in an interval +\(\mathcal J\) of cardinality \(J\) that avoid all these classes. +For every fixed \(\delta>0\), uniformly for \(J\ge w^\delta\), +\[ + \E S=JV_0,\qquad + \E(S-JV_0)^2=o((JV_0)^2) + \quad(w\longrightarrow\infty). + \] +The assertion has the following stable version. Let \(\mu\) be a finite +positive measure carrying indicators \(I_j\), \(j\in\mathcal J\), and let +\(S=\sum_j I_j\). Suppose, with an absolute fixed implied constant, +\[ + \mu(\Omega)=1+O(\zeta),\qquad + \int I_j\,d\mu=V_0(1+O(\zeta)), + \] +and, for distinct \(i,j\), the joint probability has the upper bound +\[ + \int I_iI_j\,d\mu + \le(1+O(\zeta))\,\Pbb_{\rm ind}(I_i=I_j=1). + \] +Here \(\Pbb_{\rm ind}\) denotes the independent forbidden-class model +in the first part of the lemma. A zero probability on the right requires +exact zero on the left. +For \(0\le\zeta\le1/2\), +\[ + \int(S-JV_0)^2\,d\mu + \ll\bigl(\zeta+o(1)\bigr)(JV_0)^2, + \] +uniformly in the same range of \(J\). +\end{lemma} + +\begin{proof} +The single-point survival probability is \(V_0\). +For a nonzero spacing \(h\), the pair probability divided by \(V_0^2\) +equals +\begin{equation}\label{eq:variance-pair-factor} + 2\1_{2\mid h} + \prod_{22}\left(1+\frac1{t(t-2)}\right)<\infty. +\] +Dominated convergence applies even though the available primes increase +with \(w\). The limiting triangular average of +\eqref{eq:variance-pair-factor} is 1, because +\[ + \prod_{t>2}\left(1-\frac1{(t-1)^2}\right) + \prod_{t>2}\left(1+\frac1{t(t-2)}\right)=1. +\] +The convergence is uniform for \(J\ge w^\delta\): first truncate the +convergent divisor series at fixed \(l\), and then use \(J\to\infty\) +uniformly in this range. The diagonal contribution to +\(\E S^2/(JV_0)^2\) is \(1/(JV_0)=o(1)\), by Mertens' theorem. +This proves the independent-model assertion. + +For the stable version, sum the assumed pair upper bounds and the +single-point bounds to obtain +\[ + \int S^2\,d\mu\le + (1+O(\zeta)+o(1))(JV_0)^2. +\] +Use also the lower bound for \(\int S\,d\mu\) and the upper bound for +\(\mu(\Omega)\) in the identity +\[ + \int(S-JV_0)^2\,d\mu + =\int S^2\,d\mu-2JV_0\int S\,d\mu+(JV_0)^2\mu(\Omega). +\] +This gives the stated result without normalizing \(\mu\). +\end{proof} + +\subsection{The effective-size alternative} + +\begin{proposition}\label{prop:structured-variance} +Fix the compact-box parameters, the inverse-estimate parameters and a +relative-error threshold \(\eps>0\). For every fixed \(\sigma_2>0\) there +is a fixed sufficiently large \(C_s\), independent of sufficiently small +fixed \(\xi\), with the following property. +For every candidate box, every choice of its factors other than the +isolated prime, and every rational on its fixed candidate list for which +\eqref{eq:variance-structure-size} holds, suppose +\[ + T>w^{C_s}. +\] +For sufficiently large \(z\), fewer than \(\sigma_2 n_p\) primes +\(p\in\mathcal P\) both align to \(A/D\) and satisfy +\[ + |N_{pq_f}/\mu_{pq_f}-1|>\eps. +\] +The estimate is uniform in the box, the fixed prime choices, the rational +and the forbidden classes. The threshold on \(z\) may depend on fixed +\(\xi\). The fraction is measured against all primes of \(\mathcal P\). +\end{proposition} + +\begin{proof} +It is enough to impose the aligned hit at every \(p\) hypothetically and +bound \eqref{eq:variance-bad-common}. The primes dividing \(H\) may be +discarded: there are \(O(1)\) of them in \(\mathcal P\), since +\(H\le R\mathcal Z^{11}\), and their fraction tends to zero. +For \(H\le R^{4/5}\), Brun--Titchmarsh controls each residue pattern +after we fix \(m_0\): the possible primes lie in a progression of +modulus \(HT_0T_1\). The condition that \(T\) is large then gives a +long enough interval in \(m_0\) for its small-prime conditions to have +the required first and second moments. For larger \(H\), we instead +count integer pairs \((p,m_0)\) on a modular hyperbola and then sieve +the \(p\)-coordinate. This gives domination by the uniform distribution +on all small-prime patterns. In both ranges the target is a small +exceptional fraction among all primes of the isolated bin. + +\paragraph{The range \(H\le R^{4/5}\).} +As \(p\) varies in \(\mathcal P\), the coordinate \(m_0\) lies in an +enclosing interval \(\mathcal I\) of length +\begin{equation}\label{eq:variance-small-rectangle} + Y_1\asymp H+\xi|C_0|/R,\qquad (m_0,H)=1. +\end{equation} +For a fixed \(m_0\), possible \(p\)'s lie in an interval of length at most +a constant times +\[ + X_1=\min\left(\xi R,\frac{R^2H}{|C_0|}\right), +\] +using the first entry if \(C_0=0\). +Indeed, \(C_0/p\) must lie in an interval of length \(H\), while its +derivative has magnitude comparable to \(|C_0|/R^2\) on the bin. +This argument is unchanged when \(C_0\) is negative. +Writing \(c=|C_0|/R\), the two cases \(c\le H/\xi\) and \(c>H/\xi\) give +\begin{equation}\label{eq:variance-small-area} + X_1Y_1\ll\xi RH,\qquad + Y_1\gg\xi T,\qquad + X_1\ge R/\mathcal Z^{O(1)}. +\end{equation} +The last inequality follows from \eqref{eq:variance-effective-ratio}; +fixed factors depending on \(\xi\) are harmless here. + +Specify a unit pattern for \(p\bmod T_1\) and an arbitrary pattern for +\(k\bmod T_0\). Together with fixed \(m_0\), these conditions specify one +unit class for \(p\) modulo \(HT_0T_1\): the condition at \(HT_0\) is +\[ + pm_0\equiv C_0+Hk\pmod{HT_0}. +\] +Both \(m_0\) and the right side are units modulo \(HT_0\), and \(T_1\) is +coprime to this modulus. In particular, +\[ + \varphi(HT_0T_1)=\varphi(H)T_0\varphi(T_1). +\] +Since \(HT_0T_1\le R^{4/5+o(1)}\) and \(X_1=R^{1-o(1)}\), +Brun--Titchmarsh, in the form of \cref{lem:prime-inputs}, bounds the +number of primes for each such choice by +\begin{equation}\label{eq:variance-bt} + \ll\frac{X_1}{\log R\,\varphi(H)T_0\varphi(T_1)}. +\end{equation} +We may enclose a shorter actual interval by one of the displayed +upper-bound length before applying the theorem. + +Define a positive measure on choices consisting of an integer +\(m\in\mathcal I\), \((m,H)=1\), a unit \(p\)-pattern modulo \(T_1\), +and an arbitrary \(k\)-pattern modulo \(T_0\), as follows: +\begin{equation}\label{eq:variance-small-measure} + \text{weight of each }m=\frac{H}{\varphi(H)Y_1}, + \quad + p\bmod T_1\text{ and }k\bmod T_0 + \text{ independent and uniform}. +\end{equation} +The number of the latter patterns is \(T_0\varphi(T_1)\). +Dividing \eqref{eq:variance-bt} by +\(n_p\gg\xi R/\log R\), and using +\eqref{eq:variance-small-area}, shows that the fraction of actual primes +whose associated choices lie in any event is at most a fixed absolute +constant times the measure of that event. This constant is independent +of sufficiently small fixed \(\xi\). Each actual prime has one associated +\(m_0\), so this upper count introduces no missing multiplicity. + +We verify the hypotheses of the stable part of +\cref{lem:random-variance} for the measure +\eqref{eq:variance-small-measure}. +Take one or two fixed \(j\)'s and let \(\nu_t\in\{1,2\}\) be the number of +their distinct residues modulo \(t\). +At \(t\mid T_0\), reducing the congruence for \(p\) modulo \(t\) shows +that \(p\bmod t\) is already determined by \(m\), independently of the +lift \(k\bmod t\). Uniform \(k\bmod t\) in the second line of +\eqref{eq:variance-avoidance} therefore gives exactly the joint factor +\(1-\nu_t/t\). + +At primes \(t\mid T_1\), condition on the unit pattern \(p\bmod T_1\). +The first line of \eqref{eq:variance-avoidance}, as a condition on \(m\), +excludes exactly \(\nu_t\) residue classes, since \(H\) is invertible +there. For a squarefree product \(l\) of these primes, counting the +intersection of its bad classes together with \((m,H)=1\) gives +\begin{equation}\label{eq:variance-small-intersections} + Y_1\frac{\varphi(H)}H\prod_{t\mid l}\frac{\nu_t}{t} + +O\left(\tau(H)\prod_{t\mid l}\nu_t\right). +\end{equation} +To see this, expand the coprimality condition over squarefree divisors of +\(H\). Each chosen combination of classes modulo \(l\) combines with +each divisor condition by CRT, because \((l,H)=1\); the interval count +has error \(O(1)\). This gives both the main term and the error in +\eqref{eq:variance-small-intersections}. + +Fix a small \(\zeta>0\). Apply \cref{lem:fundamental} at level +\(w^{A_1}\), first choosing the fixed \(A_1\) large enough that its +fundamental-lemma error is smaller than a prescribed constant times +\(\zeta\). If a local product is zero, the conditions are exactly empty +and need no sieve estimate. Otherwise the possible prime 2 has density +at most \(1/2\), and all other densities are at most \(2/t\), as required. +For some fixed \(M>0\), chosen after \(A_1\), the remainder sum is +\(O(\tau(H)w^M)\). Since \(H\ll Y_1\), the elementary divisor and +totient bounds give +\[ + \tau(H)\frac{H}{\varphi(H)}\ll Y_1^{1/4}. +\] +Every nonzero surviving product is \(\gg(\log w)^{-2}\). Thus the +relative remainder is bounded by a constant times +\[ + \begin{aligned} + \frac{\tau(H)H}{Y_1\varphi(H)}w^M(\log w)^2 + &\ll Y_1^{-3/4}w^M(\log w)^2\\ + &\ll_{\xi}w^{M-3C_s/4}(\log w)^2=o(1), + \end{aligned} +\] +because \(Y_1\gg\xi T>\xi w^{C_s}\) and we may choose +\(C_s>2M+10\). Thus \(A_1\), then \(M\), then \(C_s\) are fixed +before the eventual choice of \(\xi\). These estimates are uniform in +the pattern, the positions, the interval location and \(H\); a fixed +small \(\xi\) changes only the threshold on \(z\). + +After multiplication by the normalization in +\eqref{eq:variance-small-measure}, the single-point and pair probabilities +thus equal their independent-model values with relative error +\(O(\zeta)+o(1)\), whenever those values are nonzero. Factors at \(T_0\) +are exact. The total mass is \(1+o(1)\), either by the same argument with +no bad classes or by the elementary count +\[ + \#\{m\in\mathcal I:(m,H)=1\} + =Y_1\varphi(H)/H+O(\tau(H)). +\] +The stable variance lemma and Chebyshev's inequality show that the +measure of \eqref{eq:variance-bad-common} is +\(O_\eps(\zeta)+o(1)\). Choose \(\zeta\) sufficiently small in terms of +\(\sigma_2,\eps\) and the domination constant. This proves the desired +prime fraction bound in the first range. + +\paragraph{The range \(R^{4/5}0\) is sufficiently small, and use level +\(R^{1/10}\) in \cref{lem:fundamental}. +The intersection density supplied by +\eqref{eq:variance-hyperbola-count} is \(1/l\). +The primes dividing \(HT_1\) are already excluded from the \(p\)-coordinate +by the unit conditions. The main sieve product is at most +\begin{equation}\label{eq:variance-large-sieve-product} + \ll\frac1{\log R}\frac H{\varphi(H)}\frac{T_1}{\varphi(T_1)}. +\end{equation} +Including factors of \(H\) larger than the sieve cutoff only increases +this upper bound. +The fundamental-lemma parameter is \((1/10)/c_6\), so a fixed small +choice of \(c_6\) suffices. + +The weighted sum of the errors in +\eqref{eq:variance-hyperbola-count} over \(l\le R^{1/10}\) is +\(R^{4/5+o(1)}\), and in particular is \(O(R^{9/10})\) for large \(z\). +Because every prime of \(T_0\) divides \(H\), +\(\varphi(N)=\varphi(H)T_0\). Multiplying the main term at \(l=1\) by +\eqref{eq:variance-large-sieve-product} and using \(Y_2\le3H\) therefore +gives, for each pattern and block, +\begin{equation}\label{eq:variance-pattern-primes} + \ll \frac{X}{\log R\,T_0\varphi(T_0)T_1\varphi(T_1)} + +O(R^{9/10}). +\end{equation} +There are \(R^{o(1)}\) blocks and \(R^{o(1)}\) patterns, so all error +terms together are +\[ + R^{9/10+o(1)}=o(n_p). +\] +Here division by \(n_p\asymp\xi R/\log R\) is harmless for each fixed +\(\xi>0\). Summing the main term over the blocks uses +\(\sum X\asymp\xi R\). Thus the fraction of actual primes associated to +any set of bad full patterns is bounded by a fixed constant times its +fraction among all \(M_{\rm pat}\) patterns, plus \(o(1)\). +The bound counts pairs, which is sufficient: every actual prime has +exactly one \(m_0\) from \eqref{eq:variance-m-coordinate}. + +Under independent uniform choices in \eqref{eq:variance-full-pattern}, +the forbidden \(j\)-classes in \eqref{eq:variance-avoidance} are exactly +independent and uniform. Conditional on the unit patterns, use uniform +\(k_0\bmod t\) at each \(t\mid T_0\); at the other primes use uniform +\(m_1\bmod t\). The coefficients of these variables are units. +CRT supplies independence across the primes. +The event \eqref{eq:variance-bad-common} depends only on this full +pattern and on the already fixed \(\mathcal J,q_f,A,D,b\). +By \cref{lem:random-variance}, its uniform-pattern fraction is \(o(1)\). +The domination just proved gives the required prime fraction bound in +the second range. + +Finally, the first range chose \(\zeta\), then \(A_1\), then \(C_s\); +the second range needs no further increase of \(C_s\). +These choices are independent of sufficiently small fixed \(\xi\). +Choose \(\xi\) small enough for the earlier interval freezing and then +let \(z\) tend to infinity. This proves all asserted uniformities. +\end{proof} + +\subsection{Hard endpoints and global eligibility} + +\begin{corollary}\label{cor:hard-endpoints} +Fix the compact range and \(\eps>0\). +For every prescribed positive \(\rho\), the inverse exception parameter +and then the variance exception parameter can be chosen so that, for a +fixed \(C_s\) independent of sufficiently small fixed \(\xi\), the +following holds for all sufficiently large \(z\). +Consider the actual included regular compact endpoints covered by the +full boxes of \cref{prop:regular-boxes}. Except for a set of these +endpoints of total harmonic mass at most \(\rho B^{-2}\), every such +bad endpoint +\[ + |N_d/\mu_d-1|>\eps +\] +has a witnessing candidate and a rational on that candidate's fixed +list for which the parent is structured, the isolated prime aligns, and +\begin{equation}\label{eq:variance-hard} + Dq_{f,E}\le w^{C_s}, + \qquad |C_0|/R\le w^{C_s}. +\end{equation} +For each such endpoint all its prime factors with exponent +\(x(t)>C_s\) align to this same rational, including the isolated prime. +The rational satisfies the global bounds +\begin{equation}\label{eq:variance-global-height} + D\le w^{C_s}, + \qquad |A|\le Yw^{C_s+2}. +\end{equation} +The enlarged full boxes are used only to dominate measure: the assertion +concerns the actual included endpoints, rather than every tuple in +those enlarged boxes. +The exceptional mass can be bounded before choosing \(\xi\); only the +eventual threshold on \(z\) is allowed to depend on fixed \(\xi\). +\end{corollary} + +\begin{proof} +Every actual included bad endpoint under consideration has a witness +from the bounded candidate set, by +\cref{lem:edge-witness,prop:regular-boxes}. +For each candidate, \cref{prop:inverse} costs at most a prescribed +\(\sigma\) fraction of its full harmonic box and supplies a fixed list +of at most \(F\) rationals. Both \(F\) and the list definition are +independent of the choice of the isolated prime. + +To apply \cref{prop:structured-variance}, fix all the other prime choices +and one rational on this list. Then \(H,C_0,T\) are fixed as \(p\) varies. +If the structured-parent condition is possible in this slice, +\eqref{eq:variance-structure-size} holds throughout the fixed slice. +When \(T>w^{C_s}\), the proposition bounds all bad aligning \(p\)'s, +and hence also those that supply the particular witness. +Since the isolated bin has multiplicity one, its prime choice is +independent of all the other bin choices in full-box measure. +The weights \(1/p\) vary by a bounded factor within this bin, uniformly +for sufficiently small \(\xi\). Therefore the count bound +\(\sigma_2n_p\) converts to at most \(C\sigma_2\) of its harmonic +measure, with fixed \(C\). + +Union over the list and over the at most \(H_e\) candidates bounds the +discarded fraction of each full box by +\[ + H_e(\sigma+CF\sigma_2). +\] +The total enlarged-box mass is at most \(C'_K B^{-2}\), where \(C'_K\) +is independent of sufficiently small \(\xi\), by +\cref{prop:regular-boxes}. First choose \(\sigma\) sufficiently small; +this fixes \(F\). Then choose \(\sigma_2\) sufficiently small in terms +of \(F\) and \(\rho\), and finally choose \(C_s\) from +\cref{prop:structured-variance}. This makes the discarded mass at most +\(\rho B^{-2}\), without multiplying by the number of boxes. +For every remaining bad endpoint, some structured rational has +\(T\le w^{C_s}\), which is \eqref{eq:variance-hard}. + +The bound on \(H\) controls the nonaligned factors. Each nonaligned +prime factor of \(q_f\) divides \(q_{f,E}\le w^{C_s}\), so it is at +most \(w^{C_s}\). +The isolated prime already aligns by the structured-parent conclusion; +its exponent tends to infinity, and in particular exceeds \(C_s\). +This proves the alignment assertion. +The bound on \(|C_0|/R\) controls the numerator of the rational on a +scale common to all boxes. We have \(b\le q_f\), \(q_{f,A}\le q_f\), +\(R\le p\), and +\(pq_f=d\le Y\), since these are included endpoints with \(J_d>1\). +Consequently, +\[ + |A|\le Dq_f+|C_0|q_{f,A} + \le w^{C_s}q_f(1+R) + \le 2Yw^{C_s}. +\] +This implies the second inequality of +\eqref{eq:variance-global-height} for large \(z\); the first follows +directly from \eqref{eq:variance-hard}. +\end{proof} diff --git a/preprints/A-quadratic-bound-for-Jacobsthals-function-September-25-2026/paper.pdf b/preprints/A-quadratic-bound-for-Jacobsthals-function-September-25-2026/paper.pdf new file mode 100644 index 0000000000000000000000000000000000000000..1307187a7dd67ee966ffc37ff2c7b02a2c7c2761 --- /dev/null +++ b/preprints/A-quadratic-bound-for-Jacobsthals-function-September-25-2026/paper.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:f80267eebc82a9b8905057b45b35ef8734485b2b0b7b809a3051c06240cf0b1e +size 824106 diff --git a/preprints/A-radial-continuum-phase-transition-with-algebraic-decay-September-24-2026/README.md b/preprints/A-radial-continuum-phase-transition-with-algebraic-decay-September-24-2026/README.md new file mode 100644 index 0000000000000000000000000000000000000000..5685bd5f4b71c108779959314b96720d243ff5aa --- /dev/null +++ b/preprints/A-radial-continuum-phase-transition-with-algebraic-decay-September-24-2026/README.md @@ -0,0 +1,17 @@ +# [A radial continuum phase transition with algebraic decay](paper.pdf) + +OpenAI +September 24, 2026 + +## Citation + +```bibtex +@misc{OAI:A-radial-continuum-phase-transition-with-algebraic-decay-September-24-2026, + author = {{OpenAI}}, + title = {{A radial continuum phase transition with algebraic decay}}, + howpublished = {OpenAI Math Release preprint + \href{https://github.com/openai/math/blob/main/preprints/A-radial-continuum-phase-transition-with-algebraic-decay-September-24-2026/paper.pdf}{OAI:A-radial-continuum-phase-transition-with-algebraic-decay-September-24-2026}}, + year = {2026} +} +``` + diff --git a/preprints/A-radial-continuum-phase-transition-with-algebraic-decay-September-24-2026/build/figures/support_geometry.tex b/preprints/A-radial-continuum-phase-transition-with-algebraic-decay-September-24-2026/build/figures/support_geometry.tex new file mode 100644 index 0000000000000000000000000000000000000000..48d5cc917ff5e327625f8ac8cd47a3935589d8b9 --- /dev/null +++ b/preprints/A-radial-continuum-phase-transition-with-algebraic-decay-September-24-2026/build/figures/support_geometry.tex @@ -0,0 +1,48 @@ +\begin{figure}[tb] +\centering +\begin{tikzpicture}[>=Latex,font=\small] + \begin{scope}[scale=.48] + \draw[->,black!55] (-.45,0) -- (13.1,0) + node[below=2pt,text=black] {\(e/p\)}; + \draw[->,black!55] (0,-.35) -- (0,5.0) + node[above,text=black] {\(d/p\)}; + \fill[blue!8] (0,0) -- (.75,1) -- (12,4) -- cycle; + \draw[blue!60!black,thick] (0,0) -- (.75,1) -- (12,4) -- cycle; + \fill (0,0) circle (2.4pt) node[below left] {\(v_1\)}; + \fill (.75,1) circle (2.4pt) node[above left] {\(v_2\)}; + \fill (12,4) circle (2.4pt) node[above right] {\(v_3\)}; + \fill[orange!75!black] (4.25,1.666667) circle (2.4pt); + \draw[orange!85!black,thick] (4.25,1.666667) circle (.115); + \node[above left] at (4.0,2.25) {\(g\)}; + \draw[black!55] (3.93,2.12) -- (4.18,1.80); + \draw[black!55,dashed] (3.75,1.15) rectangle (4.75,2.2); + \end{scope} + \draw[black!55,->] (2.28,1.05) to[bend left=14] (7.48,2.3); + \begin{scope}[shift={(8.9,1.65)}] + \fill[orange!8] (0,0) circle (1.35); + \draw[orange!85!black,thick] (0,0) circle (1.35); + \draw[black!55,dashed] (-1.7,0) -- (1.7,0); + \draw[very thick,blue!65!black] (-.93,0) -- (.93,0); + \fill[blue!65!black] (-.93,0) circle (2pt) + node[below=5pt] {\((s_1,\rho)\)}; + \fill[blue!65!black] (.93,0) circle (2pt) + node[below=5pt] {\((s_2,\rho)\)}; + \fill (0,0) circle (1.6pt) node[above=4pt] {\(g\)}; + \node at (0,1.67) {\(\overline B(g,\gamma)\)}; + \node at (0,-1.72) {\(\rho=5p/3,\quad s_10$ and repeat a maximizing configuration in $[0,s]^3$ by +translations in $(s+1)\mathbb Z^3$. Points in distinct translates are +separated by at least one: in a coordinate in which the translates +differ, their coordinate separation is at least $(s+1)-s=1$. +Taking all complete translates contained in $[0,L]^3$ gives +\[ + \liminf_{L\to\infty}\frac{M_L}{L^3} + \ge \frac{M_s}{(s+1)^3}. +\] +Letting $s\to\infty$ along a sequence realizing the upper asymptotic +density makes the right side tend to that upper density. The lower +and upper limits therefore agree. +\end{proof} + +Set $A=3/(2p)$ and $\Delta=10^{-3}$. The parameter $t>1$ below is a +construction parameter, distinct from inverse temperature $\beta$. +We shall eventually choose one sufficiently large $t$ and keep it +fixed. Put +\[ + l_t=\ee^{-t},\qquad I_t=[l_t,(1+\Delta)l_t], + \qquad 2(1+\Delta)l_t<1. +\] +Define $\psi_t:[0,\infty)\to[0,t^2]$ by +\begin{equation}\label{f:core} + \begin{aligned} + \psi_t(r)&=t^2 + &&\text{on }[0,l_t]\cup[(1+\Delta)l_t,1],\\ + \psi_t(r)&=0 + &&\text{on }[(1+\Delta/3)l_t,(1+2\Delta/3)l_t] + \cup[1+1/t,\infty), + \end{aligned} +\end{equation} +and make it affine on each of the three intervening intervals. Thus +$\psi_t$ is continuous, including at zero, nonnegative, and supported +in $[0,2]$. + +Fix a radial nonnegative $C^1$ function $w$ on $\R^3$, supported in the +closed unit ball, with $\int_{\R^3}w(x)\,\dd x=1$. For example, one +may normalize $((1-\abs{x}^2)_+)^2$. Write +$w_R(x)=R^{-3}w(x/R)$. When its argument is a displacement in $\R^3$, +we also write $\psi_t$ for the radial lift $x\mapsto\psi_t(\abs{x})$. +The family of potentials used below is +\begin{equation}\label{f:family} + \varphi(x)=\psi_t(\abs{x})-S(x),\qquad + S(x)=t\sum_j\alpha_j w_{R_j}(x),\qquad + R_j\ge1,\quad \alpha_j\ge0,\quad \sum_j\alpha_j\le A+1. +\end{equation} +The index set may be empty, finite, or countable. Since +$\norm{w_{R_j}}_\infty\le\norm{w}_\infty$, the sum converges +uniformly in the countable case. Every such potential is therefore +bounded, continuous, radial, and nonpositive when $\abs{x}\ge2$. + +The following lemma combines an elementary geometric obstruction with +an extremal graph estimate. The weighted edge maximization in its +proof is the Motzkin--Straus argument for Tur\'an's bound +\citep[Theorem~1]{MotzkinStraus1965}. We include it to make the +quadratic core estimate explicit. + +\begin{lemma}[The shell graph]\label{f:shell-graph} +For any finite configuration in $\R^3$, join two particles when their +distance belongs to $I_t$. This graph has no clique of size five. +On any $m$ vertices it has at most $3m^2/8$ edges. +\end{lemma} + +\begin{proof} +If five points formed a clique, rescale their distances by $l_t^{-1}$ +and take the four difference vectors from one of them. Their Gram +matrix differs entrywise by at most $2\Delta+\Delta^2$ from the matrix +with diagonal one and off-diagonal entries $1/2$. For a diagonal entry +this follows from $1\le r^2\le(1+\Delta)^2$; for an off-diagonal entry +use polarization, +$\ip{v_i}{v_j}=(\abs{v_i}^2+\abs{v_j}^2-\abs{v_i-v_j}^2)/2$. +The reference matrix has smallest eigenvalue $1/2$. The perturbation +has operator norm at most $4(2\Delta+\Delta^2)<1/2$, so the Gram matrix +would be positive definite and have rank four. Four vectors in +$\R^3$ cannot have this Gram matrix. + +For the edge bound, maximize $\sum_{\{i,j\}\in E}u_i u_j$ over +$u_i\ge0$, $\sum_i u_i=1$, choosing a maximizer with the smallest +possible support. If two nonadjacent vertices have positive weights, +the objective is affine in their two weights with their sum fixed. +Putting their combined weight on one of the vertices cannot decrease +the objective, contradicting minimality of the support. The support +is therefore a clique of some size $r\le4$. Its objective is +\[ + \frac12\left(1-\sum_i u_i^2\right) + \le\frac12(1-1/r)\le\frac38. +\] +For $m>0$, apply this bound to equal weights $1/m$. The empty graph +case is immediate. +\end{proof} + +\subsection{Quadratic control of cell counts} + +We now convert the exclusion of five-particle shell cliques into a +lower bound that controls arbitrary particle numbers, including +coincident particles. This is a concrete superstable estimate in the +sense of \citet{Ruelle1970}; the constants will be uniform over all +attraction ranges. + +Partition space into half-open cubes $C_i$ of side $b=1/4$, with any +fixed translation of the grid. When working in a bounded region, +replace each cell by its intersection with that region and omit empty +cells. Let $n_i$ be the number of particles in cell $i$. If +\[ + J_{ij}=\sup_{x\in C_i,\,y\in C_j}S(x-y), +\] +then $J$ is symmetric and +\begin{equation}\label{f:row-bound} + \sup_i\sum_jJ_{ij}\le M:=c_b t(A+1), + \qquad c_b=\norm{w}_\infty(2/b+5)^3. +\end{equation} +Indeed, a cell interacts through $w_R$ with at most $(2R/b+5)^3$ +cells, each contributing at most $R^{-3}\norm{w}_\infty$. Since +$R\ge1$, their row sum is at most $c_b$. Summing the coefficients +proves \eqref{f:row-bound}, also for a countable sum and cropped cells. + +We shall need the same estimates after removing some interactions. +More precisely, in any bounded region allow symmetric measurable +pair interactions +\begin{equation}\label{f:modified-energy} + v(x,y)=c(x,y)\psi_t(\abs{x-y})-K(x,y), + \qquad 0\le c\le1,\quad K\ge0, +\end{equation} +where $c(x,y)=1$ for points in the same small cell, $K$ is bounded, +and the symmetric matrix +$J_{ij}=\sup_{x\in C_i,y\in C_j}K(x,y)$ has row sums at most $M$. +The coefficients are fixed functions of the positions, independent of +the particle configuration. This includes partial deletion of any +attractive pairs, partial deletion of core pairs between distinct +cells, and their linear interpolations whenever they share the stated +row bound. No translation invariance is required for these finite +volume estimates. Write $U^v(X)=\sum_{r4(A+1),\qquad c=1/3456. +\end{equation} +Then, in any axis-parallel cube of side $L\ge1$, with $V=L^3$, +\begin{equation}\label{f:density-stability} + U^v\ge ct^2\frac{N^2}{V}-\frac{t^2}{2}N. +\end{equation} +In particular all members of \eqref{f:family} are stable with the +same constant $B_0$. +\end{proposition} + +\begin{proof} +The diameter $\sqrt3 b$ of a small cell is less than one. By +\Cref{f:shell-graph}, among $m$ particles in one cell at least +$m(m-1)/2-3m^2/8=m^2/8-m/2$ pairs have distance outside $I_t$. +Each of these pairs contributes $t^2$, while all remaining core +contributions are nonnegative. Summing over cells proves the first +bound, and also proves it when core terms between cells are deleted. +For the attractive contribution, including extra diagonal self terms +only enlarges the bound, and symmetry gives +\[ + \sum_{r4(A+1)$ +keeps the density bound coercive even after adding the quadratic +attractive weight in the next grand-sum estimate. + +\subsection{Canonical limits and pressure duality} + +The local quadratic bound controls configurations of high density. +We next use it to establish the thermodynamic functions required by +the attraction transform. The proof applies directly to the infinite +attraction sum: separated boxes interact only attractively, so no +finite-range approximation is needed to construct these limits. + +For $\Lambda_L=[0,L]^3$, define the canonical partition functions +and their normalization by +\begin{equation}\label{f:canonical-definition} + \begin{aligned} + Z^\varphi_{L,n}(\beta) + &=\frac1{n!}\int_{\Lambda_L^n}\exp(-\beta U^\varphi(X))\,\dd X, + & Z^\varphi_{L,0}&=1,\\ + H^\varphi_{L,n}(\beta)&=-\frac1{tL^3}\log Z^\varphi_{L,n}(\beta). + \end{aligned} +\end{equation} + +\begin{lemma}[Finite volume canonical bounds]\label{f:canonical-bounds} +There is an absolute constant $C<\infty$, independent of $t$ and all +attraction ranges, such that for every admissible $t$, $L\ge1$, +$n\ge0$, $\beta>0$, and $d_L=n/L^3$, +\begin{equation}\label{f:finite-bounds} + \beta t(cd_L^2-d_L/2)+\frac{d_L\log d_L-d_L}{t} + \le H^\varphi_{L,n}(\beta) + \le \beta Ct d_L^2+\frac{d_L\log d_L}{t}, +\end{equation} +where $0\log0=0$. +\end{lemma} + +\begin{proof} +The lower bound follows by bounding the integrand from above using +\eqref{f:density-stability} and using $n!\ge(n/\ee)^n$ for $n\ge1$. +For the other bound, let $X_1,\ldots,X_n$ be independent uniform +points in $\Lambda_L$. Since $U^\varphi\le U^{\psi_t}$ and +$\int_{\R^3}\psi_t(\abs{x})\,\dd x\le t^2\abs{B(0,2)}$, +\[ + \mathbb E U^\varphi + \le\frac{n(n-1)}{2L^3}\int_{\R^3}\psi_t(\abs{x})\,\dd x + \le Ct^2n^2/L^3. +\] +Jensen's inequality gives +$\log Z^\varphi_{L,n}\ge n\log L^3-\log n!-\beta Ct^2n^2/L^3$. +Now use $n!\le n^n$. At $n=0$ both bounds are equalities. +\end{proof} + +The following bound controls the entire large-particle-number tail. +The additional quadratic factor is included because it will also +occur in the range comparison. + +\begin{lemma}[Grand sum tails]\label{f:grand-tail} +Fix an admissible $t$ and a compact $\mathcal D\subset\Omega$, where +$\Omega=(0,\infty)\times\R$ has coordinates $(\beta,h)$. +There are constants $D\ge1$ and $a_0>0$, depending only on $t$ and +$\mathcal D$, such that, for every $\varphi$ in \eqref{f:family}, +$L\ge1$, and $0\le\zeta\le2(A+1)$, +\begin{equation}\label{f:grand-tail-bound} + \sum_{n>DL^3}\ee^{thn}Z^\varphi_{L,n}(\beta) + \exp\left(\frac{\beta t\zeta n^2}{2L^3}\right) + \le\frac{\exp[-a_0(\lfloor DL^3\rfloor+1)]}{1-\ee^{-a_0}} + \qquad ((\beta,h)\in\mathcal D). +\end{equation} +In particular the tail tends uniformly to zero, and is at most one +for all sufficiently large $L$. These constants require no local +occupation estimate. +\end{lemma} + +\begin{proof} +Enclose $\mathcal D$ in $\beta\in[\beta_-,\beta_+]$ and +$\abs{h}\le H$. Put $V=L^3$. By \eqref{f:finite-bounds}, the logarithm +of the summand at $n\ge1$ is at most +\[ + -\beta t(ct-\zeta/2)\frac{n^2}{V} + +(th+\beta t^2/2+1)n-n\log(n/V). +\] +The choice \eqref{f:t-requirements} implies +$ct-\zeta/2\ge ct/2$. Set +$\kappa=\beta_-ct^2/2$ and $B=tH+\beta_+t^2/2+1$. +For $n/V\ge1$ the displayed expression is at most +$-\kappa n^2/V+Bn$. Choose $D\ge\max(1,2B/\kappa)$. +For $n>DV$ this is at most +$-\kappa n^2/(2V)\le-(\kappa D/2)n$. +Summing this geometric majorant proves the assertion with +$a_0=\kappa D/2$. +\end{proof} + +\begin{proposition}[Thermodynamic limits and duality] +\label{f:thermodynamics} +For every member $\varphi$ of \eqref{f:family}, including countably +infinite sums, and every $\beta>0$, $d\ge0$, the limit +\begin{equation}\label{f:canonical-limit} + H_\varphi(\beta,d) + =\lim_{\substack{L\to\infty\\ n/L^3\to d}} + H^\varphi_{L,n}(\beta) +\end{equation} +exists along every such sequence of real sides and integer counts. +It is finite, convex and continuous in $d\in[0,\infty)$, and concave +in $\beta>0$ for each fixed $d$. The bounds \eqref{f:finite-bounds} +hold with $H_\varphi(\beta,d)$ and $d$ in place of their finite volume +counterparts. + +The grand partition function and pressure satisfy +\begin{equation}\label{f:pressure-definition} + \Xi^\varphi_L(\beta,h) + =\sum_{n\ge0}\ee^{thn}Z^\varphi_{L,n}(\beta), + \qquad + Q_\varphi(\beta,h) + =\lim_{L\to\infty}\frac{\log\Xi^\varphi_L(\beta,h)}{tL^3}. +\end{equation} +The pressure is finite, nonnegative, and jointly convex on $\Omega$. +For $\beta>0$, +\begin{equation}\label{f:duality} + \begin{split} + Q_\varphi(\beta,h) + &=\sup_{d\ge0}\{hd-H_\varphi(\beta,d)\},\\ + H_\varphi(\beta,d) + &=\sup_{h\in\R}\{hd-Q_\varphi(\beta,h)\},\qquad d>0. + \end{split} +\end{equation} +For fixed admissible $t$, pressures of the whole family are uniformly +bounded on each compact subset of $\Omega$. +\end{proposition} + +\begin{proof} +We first prove \eqref{f:canonical-limit}, without using a pressure +limit or an approximation by finite-range potentials. At $d=0$, +both bounds in \eqref{f:finite-bounds} tend to zero. Fix $d>0$ and +$\beta>0$. Let $H_*$ be the lower limit of +$H^\varphi_{L,n}(\beta)$ as $L\to\infty$ and $n/L^3\to d$, and choose +$s_j\to\infty$ and integers $n_j$ with +\[ + n_j/s_j^3\longrightarrow d,\qquad + H^\varphi_{s_j,n_j}(\beta)\longrightarrow H_*. +\] +The finite volume bounds make $H_*$ finite. Consider any other +sequence $L\to\infty$, $n/L^3\to d$. For each fixed, sufficiently +large $j$, place +$k=\lfloor L/(s_j+2)\rfloor^3$ cubes of side $s_j$ in $\Lambda_L$, +separated by gaps of width at least two. For any nonnegative integers +$m_1,\ldots,m_k$ summing to $n$, restriction to configurations with +these box counts gives +\begin{equation}\label{f:canonical-product} + Z^\varphi_{L,n}(\beta) + \ge\prod_{i=1}^k Z^\varphi_{s_j,m_i}(\beta). +\end{equation} +Indeed, interactions between different boxes are nonpositive. There +are $n!/\prod_i m_i!$ disjoint assignments of particle labels to the +boxes, so the global factor $1/n!$ leaves exactly the product of +their factors $1/m_i!$. + +We retain the favorable count $n_j$ in most boxes and correct the +total particle number in the remaining boxes. Only a vanishing fraction +of boxes will change, and \eqref{f:finite-bounds} will bound their +canonical costs uniformly. To make this exact for every target count, put +\[ + q_j=\lfloor(4d+4)s_j^3\rfloor, + \qquad c_d=\min(d/2,1). +\] +For all sufficiently large $j$, +$n_j\ge c_ds_j^3$ and $q_j-n_j\ge c_ds_j^3$. +Start with $n_j$ particles in every box. If +$E=n-kn_j\ge0$, fill boxes successively to $q_j$, with at most one +last box given an intermediate integer count. If $E<0$, empty boxes +successively, again with at most one last intermediate count. This +changes at most $r$ boxes, where +\begin{equation}\label{f:integer-adjustment} + \frac rk\le\frac1{c_d} + \abs{\frac{n}{ks_j^3}-\frac{n_j}{s_j^3}}+\frac1k. +\end{equation} +For fixed large $j$ and then sufficiently large $L$, the target lies +between $0$ and $kq_j$: its mean count divided by $s_j^3$ tends to +$d(1+2/s_j)^30$. Convexity of the finite function +$H_\varphi(\beta,\cdot)$ gives a supporting slope $h$ there. Explicitly, +one may choose $h$ between +\[ + \sup_{0\le vd}\frac{H_\varphi(\beta,v)-H_\varphi(\beta,d)}{v-d}. +\] +The left endpoint is at most the right endpoint by convexity, and +both are finite by comparison with secants through fixed points on +either side of $d$. The resulting supporting inequality holds for +all $v\ge0$, so the first duality gives +$Q_\varphi(\beta,h)=hd-H_\varphi(\beta,d)$. For all other $h$ the +same first duality gives the opposite variational inequality. This +proves the second equality in \eqref{f:duality} without a differentiability +or uniqueness assumption. +\end{proof} diff --git a/preprints/A-radial-continuum-phase-transition-with-algebraic-decay-September-24-2026/build/sections/initial.tex b/preprints/A-radial-continuum-phase-transition-with-algebraic-decay-September-24-2026/build/sections/initial.tex new file mode 100644 index 0000000000000000000000000000000000000000..50e8b0aa4e3efa780d89383354c5a0d8e4fadefd --- /dev/null +++ b/preprints/A-radial-continuum-phase-transition-with-algebraic-decay-September-24-2026/build/sections/initial.tex @@ -0,0 +1,458 @@ +\section{An initial triple of pressure slopes} +\label{i:section} + +The narrow zero-energy shell in the core permits clusters of four particles, +with relative volume of order \(l_t^9=\ee^{-9t}\). This entropy cost, +together with the packing density \(p\), produces three competing branches +after the first ideal attraction. We show that, for one sufficiently large +finite \(t\), the pressure subdifferential throughout a parameter neighborhood +is generated by actual supports near the three limiting slopes. The +subsequent iteration will preserve this separation and find a parameter +where all three groups are present. + +In this section alone we let \(t\) tend to infinity to obtain the required +estimates; we fix its finite value at the end. + +\subsection{Clusters and the bare-core pressure} + +\begin{proposition}[Pressure of the bare core] +\label{i:bare-pressure} +Let \(q^{(t)}=Q_{\psi_t}\), with the pressure normalization and the core +of \Cref{f:thermodynamics,f:core}. Then, locally uniformly on +\(\Omega=(0,\infty)\times\R\), +\begin{equation} + q^{(t)}(\beta,h)\longrightarrow + q_*(\beta,h):=\max\{0,ph,4ph-9p\}. + \label{i:bare-limit} +\end{equation} +\end{proposition} + +\begin{proof} +Fix a compact parameter set \(K\subset\Omega\), and choose +\(\beta_->0\) and \(H<\infty\) such that +\(\beta\ge\beta_-\) and \(\abs{h}\le H\) on \(K\). +Throughout the proof \(V=L^3\), and +\(I_t=[l_t,(1+\Delta)l_t]\). + +\emph{Free configurations and the upper bound.} +Call a pair bad when its distance is less than one and is not in \(I_t\). +A configuration is free if it has no bad pair. In a free configuration, +join distinct particles whose distance is less than one. Every edge then +has length at most \((1+\Delta)l_t\). The endpoints of a path of two +edges are therefore at distance at most \(2(1+\Delta)l_t<1\), and hence +are joined. Each connected component is consequently a clique with all +pair distances in \(I_t\). By \Cref{f:shell-graph}, its size is at most +four. Choosing one representative +from every component gives a set with pairwise distances at least one, +so the component count is at most \(M_L\). + +Consider the grand integral over free configurations with their energy +discarded: +\[ + \Xi_{L,\mathrm{free}}(h) + =\sum_{n\ge0}\frac{\ee^{thn}}{n!} + \int_{\Lambda_L^n}\mathbf 1_{\mathrm{free}}(x_1,\ldots,x_n) + \dd x_1\cdots\dd x_n. +\] +For an ordered list of component sizes \(m_1,\ldots,m_k\), with +\(m_i\in\{1,2,3,4\}\) and \(\sum_i m_i=n\), the number of ordered +partitions of the particle labels into these blocks is +\(n!/\prod_i m_i!\). On summing over all ordered size lists, division +by \(k!\) counts each unordered family of label blocks once. This +remains true when sizes repeat: the label blocks themselves are distinct +and have exactly \(k!\) orderings. + +In a block of size \(m\), choose its least label as the root. Its position has volume +at most \(V\), and each of the other \(m-1\) particles lies in a ball of +volume \(C l_t^3\) about it, where \(C=4\pi(1+\Delta)^3/3\). +After cancellation of the global factor \(1/n!\), the block therefore +contributes at most \(V(C l_t^3)^{m-1}/m!\). Discarding all further +restrictions within and between components and summing their sizes gives +\begin{equation} + \Xi_{L,\mathrm{free}}(h) + \le\sum_{k=0}^{M_L}\frac{(V\Theta_t(h))^k}{k!}, + \qquad + \Theta_t(h)=\sum_{m=1}^4 + \frac{\ee^{thm}(C l_t^3)^{m-1}}{m!}. + \label{i:rooted-count} +\end{equation} + +To pass from free configurations to all configurations, let \(B(X)\) +be the set of labels incident to a bad pair. There are at least +\(\abs{B(X)}/2\) bad pairs, and each has core energy exactly \(t^2\). +All other core terms are nonnegative, and \(X\setminus B(X)\) is free. +Thus the following pointwise bound holds: +\begin{equation} + \ee^{-\beta U^{\psi_t}(X)} + \le\sum_{J\subset\{1,\ldots,n\}} + \ee^{-\beta t^2\abs{J}/2} + \mathbf 1_{\mathrm{free}}(X\setminus J). + \label{i:removal-pointwise} +\end{equation} +Indeed, the summand \(J=B(X)\) already bounds the left side. +For \(r=\abs{J}\) and \(m=n-r\), the label factor is +\(\binom{m+r}{r}/(m+r)!=1/(r!m!)\). The removed positions integrate +to \(V^r\). Tonelli's theorem and the grand sum now give the exact +factorization of the upper bound +\begin{equation} + \Xi_L^{\psi_t}(\beta,h) + \le + \exp\!\left(V\ee^{th-\beta t^2/2}\right) + \Xi_{L,\mathrm{free}}(h). + \label{i:removal-factor} +\end{equation} +Shell endpoints are included in \(I_t\), although their energy may be +positive. Pairs at distance exactly one are not edges of the component +graph. Both conventions merely discard nonnegative energy in the upper +bound, so these arguments hold pointwise, including at the endpoints. + +Set \(a_t(h)=t^{-1}\log\Theta_t(h)\). The four exponential rates in +\(\Theta_t\) are +\[ + h,\qquad 2h-3,\qquad 3h-6,\qquad 4h-9. +\] +The middle two are convex combinations of the first and last. Their +positive prefactors in \eqref{i:rooted-count} are fixed, so +\begin{equation} + a_t(h)=\max\{h,4h-9\}+O(t^{-1}), + \label{i:cluster-rates} +\end{equation} +uniformly for \(\abs{h}\le H\). For \(k\ge1\), the inequality +\(k!\ge(k/\ee)^k\), with \(x=k/V\), gives +\[ + \frac1{tV}\log\frac{(V\Theta_t)^k}{k!} + \le x a_t(h)+\frac{x(1-\log x)}t + \le \frac{M_L}{V}\max\{0,a_t(h)\}+\frac1t, +\] +since \(x(1-\log x)\le1\) for \(x>0\). The same final bound covers +\(k=0\). There are only \(M_L+1=O(1+V)\) terms in +\eqref{i:rooted-count}. First letting \(L\to\infty\), with \(t\) +fixed, and using \Cref{f:packing,f:thermodynamics}, we obtain +\begin{equation} + q^{(t)}(\beta,h) + \le p\max\{0,a_t(h)\}+\frac1t + +\frac1t\ee^{th-\beta t^2/2}. + \label{i:upper-pressure} +\end{equation} +The final term is at most +\(t^{-1}\ee^{tH-\beta_-t^2/2}\) on \(K\). Thus +\eqref{i:cluster-rates} proves the required uniform upper bound. + +\emph{Zero-energy configurations and the lower bound.} +Fix \(s>0\) and \(0<\nu\le1\). Dilate a maximizing packing in a +side-\(s\) cube by \(1+\nu\), and center it in a cube of side +\[ + P=(s+2)(1+\nu). +\] +Repeat this pattern in adjacent period cubes. Centers in one period +have separation at least \(1+\nu\); centers in different periods have +at least this separation as well. Every center has distance at least +\(1+\nu\) from the boundary of its period cube. A cube of side \(mP\) +contains exactly \(k=m^3M_s\) centers. + +Choose \(r\in(0,\nu/16)\) and let \(b_0=4\pi r^3/3\). +At every center \(z\), put the root particle in \(B(z,r)\). +There are two constructions, with cluster size \(j=1\) or \(j=4\). +For \(j=4\), choose vectors \(a_2,a_3,a_4\in\R^3\) such that +\(0,a_2,a_3,a_4\) are the vertices of a regular tetrahedron of side +\(1+\Delta/2\). Fix \(\delta\in(0,\Delta/24)\) and require +\[ + x_i-x_1\in B(l_t a_i,\delta l_t),\qquad i=2,3,4. +\] +A root-to-vertex distance differs from +\((1+\Delta/2)l_t\) by at most \(\delta l_t\), and another edge +differs by at most \(2\delta l_t\). All six distances consequently +lie strictly inside +\[ + ((1+\Delta/3)l_t,(1+2\Delta/3)l_t), +\] +where the core vanishes. Write \(b_1=4\pi\delta^3/3>0\). +The change from positions to the root and the three relative vectors +has Jacobian one. For one prescribed label ordering the integration +volume at each center is therefore +\(b_0(b_1l_t^3)^3\), with positive relative volume \(b_1^3l_t^9\). +For \(j=1\), the corresponding volume is \(b_0\). + +Every particle lies within \(r+2l_t\) of its center. For all sufficiently +large \(t\), depending on \(\nu,r\) but not on \(m,\beta,h\), these +center neighborhoods are disjoint and lie inside their period cubes, +and intercluster distances are greater than +\(1+\nu/2>1+1/t\). Hence all configurations just described have +exactly zero core energy. + +The centers are distinguishable. Assign an unordered set of \(j\) +particle labels to each of the \(k\) centers, and retain just one +deterministic ordering of each assigned set, for example the increasing +order. There are \((jk)!/(j!)^k\) such assignments. Different +assignments send some label to different disjoint center neighborhoods, +so their integration events are disjoint. Possible alternative choices +of the root within the same set are never summed. Dividing by the +canonical factorial \((jk)!\) gives +\begin{equation} + Z^{\psi_t}_{mP,jk}(\beta) + \ge\left(\frac{b_0(b_1l_t^3)^{j-1}}{j!}\right)^k, + \qquad j\in\{1,4\}. + \label{i:cluster-lower} +\end{equation} +Using this term in the grand sum and first letting \(m\to\infty\), we +find +\begin{equation} + q^{(t)}(\beta,h) + \ge c_{s,\nu}\left(jh-3(j-1) + +\frac1t\log\frac{b_0b_1^{j-1}}{j!}\right), + \qquad + c_{s,\nu}=\frac{M_s}{((s+2)(1+\nu))^3}. + \label{i:lower-pressure} +\end{equation} +The vacuum also gives \(q^{(t)}\ge0\). The constants in +\eqref{i:lower-pressure} have no dependence on \(\beta,h\), and +\(c_{s,\nu}\) can be made arbitrarily close to \(p\) by taking \(s\) +large and \(\nu\) small. More explicitly, on \(\abs{h}\le H\) the +two functions \(jh-3(j-1)\), \(j=1,4\), have a common finite absolute +bound. Given any error tolerance, first choose \(s,\nu\) so that +replacing \(c_{s,\nu}\) by \(p\) changes both branches by less than +that tolerance, and then take \(t\) large enough to make the two +logarithmic terms small. Taking the maximum of these branches and zero +preserves the uniform error, even where a nonvacuum branch is negative. +Together with \eqref{i:upper-pressure}, this proves +\eqref{i:bare-limit} on \(K\). +\end{proof} + +\subsection{The limiting attraction and its active densities} + +We next apply the ideal attraction with mass \(A=3/(2p)\). Let +\(P_1^t=T_Aq^{(t)}\), and define +\begin{equation} + \begin{split} + L_1(\beta,h)&=0,\\ + L_2(\beta,h)&=ph+\frac{\beta A p^2}{2},\\ + L_3(\beta,h)&=4ph-9p+\frac{\beta A(4p)^2}{2},\\ + P_*(\beta,h)&=\max_{1\le i\le3}L_i(\beta,h), + \qquad D_*=\{0,p,4p\}. + \end{split} + \label{i:three-planes} +\end{equation} + +\begin{lemma}[The limiting transform and its optimizer gap] +\label{i:limiting-transform} +The functions \(P_1^t\) converge locally uniformly on \(\Omega\) to +\(P_*\). For every compact \(K\subset\Omega\), all their active +optimizers lie in one interval \([0,U]\) for all sufficiently large +\(t\), and +\[ + \sup_{\lambda\in K}\ \sup_{u\ \mathrm{active\ for}\ P_1^t(\lambda)} + \dist(u,D_*)\longrightarrow0. +\] +In particular, this holds at parameters where several limiting planes +meet. If \(\beta_-=\min_{(\beta,h)\in K}\beta\), the limiting +objective obeys the explicit bound +\begin{equation} + F_*(\lambda,u):=q_*(\beta,h+\beta Au)-\frac{\beta A u^2}{2} + \le P_*(\lambda)-\frac{\beta_-A}{2}\dist(u,D_*)^2 + \quad(\lambda\in K,\ u\in\R). + \label{i:objective-gap} +\end{equation} +\end{lemma} + +\begin{proof} +Apply the uniform-in-\(t\) bound in \Cref{k:transform-properties} to +the bare core \(\psi_t\), mass \(A\), and parameter compact \(K\). +For all sufficiently large \(t\), every active shift lies in one fixed +interval \([0,U]\). Enlarge \(U\), if necessary, so that \(4p\le U\). + +Write \(d_1=0,d_2=p,d_3=4p\). Completing the square gives the identity +\begin{equation} + F_*(\lambda,u) + =\max_{1\le i\le3} + \left\{L_i(\lambda)-\frac{\beta A}{2}(u-d_i)^2\right\}. + \label{i:limiting-parabolas} +\end{equation} +Its supremum over \(u\) is \(P_*(\lambda)\), and all its maximizing +values belong to \(D_*\). The same identity proves +\eqref{i:objective-gap}. + +The shifted points +\[ + K'=\{(\beta,h+\beta Au):(\beta,h)\in K,\ 0\le u\le U\} +\] +form a compact subset of \(\Omega\). Put +\(E_t=\sup_{K'}\abs{q^{(t)}-q_*}\), which tends to zero by +\Cref{i:bare-pressure}. Both transforms can be maximized over +\([0,U]\), so +\[ + \sup_K\abs{P_1^t-P_*}\le E_t. +\] +If \(u\) is active for \(P_1^t(\lambda)\), approximation of its +objective and of the maximum shows +\(F_*(\lambda,u)\ge P_*(\lambda)-2E_t\). By +\eqref{i:objective-gap}, +\begin{equation} + \dist(u,D_*) + \le 2\sqrt{\frac{E_t}{\beta_-A}}. + \label{i:active-density-error} +\end{equation} +This estimate is uniform on \(K\) and requires no strict inequality +between the three affine branches. +\end{proof} + +\subsection{Uniform separation of generating slopes} + +The optimizer gap locates the density coordinates of active supports. +We now identify neighborhoods of the full slopes that will remain separated +under the iteration. The slopes of the three limiting planes and their +barycenter are +\begin{equation} + \begin{gathered} + d_1=0,\qquad d_2=p,\qquad d_3=4p,\qquad + v_i=(Ad_i^2/2,d_i),\\ + v_1=(0,0),\qquad v_2=(3p/4,p),\qquad v_3=(12p,4p),\\ + g=\frac{v_1+v_2+v_3}{3}=(17p/4,5p/3),\qquad + \lambda_*=(1,-3/4). + \end{gathered} + \label{i:slopes} +\end{equation} +All three planes vanish at \(\lambda_*\), and +\(\det(v_2-v_1,v_3-v_1)=-9p^2\ne0\). Thus \(g\) is strictly +inside their slope triangle. + +\begin{lemma}[Robust slope geometry] +\label{i:geometry} +There are constants \(0<\tau

0\) with the +following properties. Every choice +\(w_i\in\clB{v_i}{\tau}\), \(i=1,2,3\), satisfies +\begin{equation} + \clB{g}{\gamma}\subset\conv\{w_1,w_2,w_3\}. + \label{i:robust-triangle} +\end{equation} +For each distinct pair \(i,j\), +\begin{equation} + g\notin\conv\bigl(\clB{v_i}{\tau}\cup\clB{v_j}{\tau}\bigr). + \label{i:two-groups} +\end{equation} +Moreover, for every \(\lambda\in\Omega\), +\begin{equation} + P_*(\lambda)-P_*(\lambda_*) + -\ip{g}{\lambda-\lambda_*} + \ge\gamma\norm{\lambda-\lambda_*}. + \label{i:linear-growth} +\end{equation} +\end{lemma} + +\begin{proof} +Because \(g\) is an interior point of the nondegenerate triangle, the +two numbers +\[ + \eta=\min_{\norm{z}=1}\max_i\ip{v_i-g}{z}>0, + \qquad + \zeta=\min_{i0 +\] +are positive. For the first assertion, one may also see positivity +directly: if all three inner products were nonpositive, their sum would +be zero, so all would vanish, contradicting noncollinearity. +Choose +\[ + 0<\tau<\min\{p/16,\eta/4,\zeta/2\}, + \qquad \gamma=\eta/2. +\] +For any unit vector \(z\) and any allowed vertices \(w_i\), +\[ + \max_i\ip{w_i-g}{z}\ge\eta-\tau>\gamma. +\] +The support-function characterization of inclusion of compact convex +sets proves \eqref{i:robust-triangle}. Every point of the hull of two +radius-\(\tau\) balls lies within distance \(\tau\) of the segment +joining their centers. Since \(\tau<\zeta\), this also proves +\eqref{i:two-groups}. Finally, all three planes agree at +\(\lambda_*\), and hence for \(z=\lambda-\lambda_*\), +\[ + P_*(\lambda)-P_*(\lambda_*)-\ip{g}{z} + =\max_i\ip{v_i-g}{z}\ge\eta\norm{z}\ge\gamma\norm{z}. +\] +\end{proof} + +The geometry of \Cref{i:geometry} is illustrated in +\Cref{fig:support-geometry}. +\input{figures/support_geometry} + +Define the closed parameter neighborhoods +\begin{equation} + W(r)=\clB{\lambda_*}{r},\qquad W_1=W(1/4)\subset\Omega. + \label{i:neighborhoods} +\end{equation} +The next proposition gives a statement about actual supporting slopes +throughout \(W_1\), as required for the subsequent iteration. + +\begin{proposition}[Initial separation of generating slopes] +\label{i:initial-separation} +For all sufficiently large finite \(t\), the size conditions in +\Cref{f:stability,k:transform-properties} hold and +\begin{equation} + \sup_{\lambda\in W_1}\abs{P_1^t(\lambda)-P_*(\lambda)} + <\frac{\gamma}{64}. + \label{i:initial-pressure-error} +\end{equation} +At every \(\lambda\in W_1\) one also has the exact identity +\begin{equation} + \partial P_1^t(\lambda) + =\conv\left( + \partial P_1^t(\lambda)\cap + \bigcup_{i=1}^3\clB{v_i}{\tau/2}\right). + \label{i:initial-hull} +\end{equation} +Fix one such \(t\) for the rest of the construction. +\end{proposition} + +\begin{proof} +Use the uniform optimizer bound \([0,U]\) from +\Cref{i:limiting-transform} with \(K=W_1\). All shifted points +\(x=(\beta,h+\beta Au)\) lie in the fixed compact \(K'\) from that +proof. Choose one \(\sigma>0\) such that +\[ + K''=\{x+(s,0):x\in K',\ \abs{s}\le\sigma\}\subset\Omega, +\] +and set \(E'_t=\sup_{K''}\abs{q^{(t)}-q_*}\to0\). +For every \((e,d)\in\partial q^{(t)}(x)\), its two supporting +inequalities at \(x+(\sigma,0)\) and \(x-(\sigma,0)\), together +with the independence of \(q_*\) from \(\beta\), imply +\begin{equation} + -\frac{2E'_t}{\sigma}\le e\le\frac{2E'_t}{\sigma}. + \label{i:energy-support-error} +\end{equation} +These are fixed steps on an enlarged compact, so the estimate also +covers outer parameters on the boundary of \(W_1\). + +At a shifted point corresponding to an active \(u\), +\Cref{k:active-parabola} gives \(d=u\) for every underlying support. +By \Cref{k:subdifferential}, the actual active generators of +\(\partial P_1^t(\lambda)\) are precisely the vectors in the union +\[ + G_t(\lambda)= + \bigcup_{u\ \mathrm{active}} + \left\{(e+Au^2/2,u):(e,u)\in + \partial q^{(t)}(\beta,h+\beta Au)\right\}, + \qquad + \partial P_1^t(\lambda)=\conv G_t(\lambda). +\] +For a nearest \(d_i\in D_*\), every such generator \(w\) +satisfies +\[ + \norm{w-v_i} + \le\frac{2E'_t}{\sigma} + +\left(1+\frac A2(U+4p)\right)\abs{u-d_i}. +\] +Equations \eqref{i:active-density-error} and +\eqref{i:energy-support-error} show that this tends uniformly to zero +over all outer parameters, all active optimizers, and all underlying +supports. Thus, for all sufficiently large \(t\), +\[ + G_t(\lambda)\subset + \partial P_1^t(\lambda)\cap + \bigcup_i\clB{v_i}{\tau/2} + \qquad(\lambda\in W_1). +\] +Taking convex hulls proves one inclusion in \eqref{i:initial-hull}; +the other follows from convexity of the subdifferential. Finally, +\eqref{i:initial-pressure-error} follows from +\Cref{i:limiting-transform}. All earlier size conditions are lower +bounds on \(t\), so they can be imposed at the same time. +\end{proof} diff --git a/preprints/A-radial-continuum-phase-transition-with-algebraic-decay-September-24-2026/build/sections/introduction.tex b/preprints/A-radial-continuum-phase-transition-with-algebraic-decay-September-24-2026/build/sections/introduction.tex new file mode 100644 index 0000000000000000000000000000000000000000..5afbf814beddf72acba8096c48d2a4aebfefe321 --- /dev/null +++ b/preprints/A-radial-continuum-phase-transition-with-algebraic-decay-September-24-2026/build/sections/introduction.tex @@ -0,0 +1,230 @@ +\section{Introduction}\label{sec:introduction} + +A phase transition is a singularity that appears in the thermodynamic +limit of a particle system. We construct one for a radial pair +interaction with a bounded continuous core and an explicit integrable +power-law bound on its tail. + +We consider identical classical particles in three-dimensional Euclidean space. +For a radial pair potential \(\phi\), a cube \(\Lambda_L=[0,L]^3\), and \(N\ge0\), set +\begin{equation}\label{eq:partition} + U^\phi(x_1,\ldots,x_N)=\sum_{1\le i0\). +A potential is \emph{stable} if \(U^\phi\ge-BN\) for a constant \(B\) +and every finite configuration. + +Let \(p\) denote the packing constant +\[ + p=\lim_{L\to\infty}\frac{M_L}{L^3}, + \qquad + M_L=\max\bigl\{\#X:X\subset[0,L]^3,\ + \abs{x-y}\ge1\ \text{for distinct }x,y\in X\bigr\}. +\] +The existence and positivity of \(p\) are proved in \Cref{f:packing}; +its numerical value is not needed. + +\begin{theorem}\label{thm:main} +There is a bounded continuous function +\(\phi:[0,\infty)\to\R\) and finite constants \(B,C>0\) such that +\[ + \sum_{i0\). +Its finite one-sided derivatives at \(\beta_c\) satisfy +\[ + f'_-(\beta_c,\rho)>f'_+(\beta_c,\rho). +\] +In particular, \(f(\,\cdot\,,\rho)\) is not differentiable at \(\beta_c\). +\end{theorem} + +\subsection{Context and the role of the tail} + +The existence of continuum phase transitions belongs to the program +highlighted by Simon's problems in mathematical physics +\citep{Simon1984}. We address the explicit interaction class and +canonical singularity in \Cref{thm:main}. + +Rigorous continuum coexistence results precede the present construction. +Ruelle proved a transition for a symmetric two-species classical system +\citep{Ruelle1971}. Lebowitz, Mazel, and Presutti established a +liquid--vapor transition for a continuum model with an attractive +pair term and a repulsive four-body term +\citep{LebowitzMazelPresutti1998,LebowitzMazelPresutti1999}. +Dereudre and Renaud-Chan establish density coexistence for a class of +saturated continuum interactions +\citep[Theorems~1--2]{DereudreRenaudChan2026}. +Their radial-kernel example integrates over the union of particle +neighborhoods, giving a many-body energy +\citep[Section~2.3, Example~3]{DereudreRenaudChan2026}. +The species or many-body structures are part of these models; the +problem considered here uses a single species and a sum of radial pair +interactions. + +Long-range weak attractions provide a classical route from microscopic +particle interactions to a nonconvex mean-field free energy. The +Lebowitz--Penrose theory identifies the limiting free energy through +convexification of a reference free energy with a quadratic attraction +term \citep{LebowitzPenrose1966}. Gerardi, Marchioro, Olivieri, and +Presutti extended this framework from hard-core references to +superstable interactions \citep{GerardiEtAl1973}. In the present problem, the limit of +an interaction as its range tends to infinity is only an intermediate +object: the required singularity must belong to one fixed pair +potential. Moreover, a pointwise power bound constrains how the masses +of its successive attractions may depend on their ranges. + +The companion article \citep[Theorem~1.1]{OpenAICompanion2026} constructs +a radial potential with a divergent repulsive core, an integrable +\(o(r^{-3})\) tail, and a common transition inverse temperature for an +open interval of densities. Both articles use small clusters and +separated supports of a pressure in two variables. Here we prove the +bounded-core construction and the quantitative range estimate needed +for a fixed power margin. All proofs required for the present theorem +are included below. The two articles concern different potentials, +each with its own density interval and common critical inverse temperature. + +A recent theorem of He, Jauslin, Lebowitz, and Peled proves a +liquid--vapor transition for a finite-range modified Kac pair +interaction \citep[Equation~(1.13), Theorems~1.3 and~1.5]{HeJauslinLebowitzPeled2026}. +Their interaction is defined using a fixed partition into boxes, and +the coexistence theorem is formulated at a fixed inverse temperature. +Our physical interaction is a radial function of particle separation; +boxes enter only the estimates used to analyze its integrals. The +conclusion here is a strict inverse-temperature derivative jump of the +canonical free energy at one common critical point throughout an open +interval of densities. It is an existence result for +the constructed infinite-range interaction, with no identification of +its phases as conventional fluid phases. + +The estimates also have several classical ancestors. The quadratic +occupation control is part of the superstability framework developed by Ruelle \citep{Ruelle1970}. +For our particular core, the needed bound follows from the weighted +graph maximization argument of Motzkin and Straus +\citep[Theorem~1]{MotzkinStraus1965}; we reproduce the argument and its +geometric application. Supporting slopes and partial convex duality +are used in their standard finite-dimensional sense +\citep[Theorem~11.1 and Proposition~11.3]{RockafellarWets1998}. +The local occupation estimates, the quantitative attraction comparison, +and the persistence argument are proved at their precise strengths in +this paper. + +\subsection{Proof strategy} + +There are three tasks: construct a stable family of bounded potentials, +compare a long-range attraction with its ideal density transform, and +preserve a sufficiently large set of pressure supports while the +attraction ranges grow. We describe how their outputs fit together. + +\paragraph{A bounded reference family.} +A parameter \(t>1\) fixes the microscopic core; it is chosen once and is +independent of \(\beta\). Below distance one, the core has height +\(t^2\) outside a narrow shell at distances comparable to \(\ee^{-t}\); +inside the shell it has a zero-energy plateau. It vanishes beyond a +fixed radius. Four particles can form a tetrahedron with all six pair +distances in the zero plateau, whereas five cannot lie in the shell. This gives a quadratic lower bound +in the particle counts of small cubes. The bound survives the addition +of nonnegative radial attractions with a prescribed total mass. +\Cref{f:section,f:local-section} use it to establish the canonical and +grand-canonical limits, uniform occupation moments, and continuity of +pressure under a small change of attraction mass. + +For a potential \(\varphi\) in this family and a density \(d\ge0\), +write +\[ + H_\varphi(\beta,d) + =-\lim_{L\to\infty}\frac{1}{tL^3} + \log Z^\varphi_{L,\lfloor dL^3\rfloor}(\beta), + \qquad + Q_\varphi(\beta,h) + =\sup_{d\ge0}\{hd-H_\varphi(\beta,d)\}. +\] +Thus \(H_\varphi=\beta f_\varphi/t\), and \(h\) is the field dual to +density. The pressure \(Q_\varphi\) is convex jointly in +\((\beta,h)\). A supporting slope \((e,d)\) at \((\beta_0,h_0)\) +means that +\[ + Q_\varphi(\beta,h)\ge Q_\varphi(\beta_0,h_0) + +e(\beta-\beta_0)+d(h-h_0) +\] +for every \(\beta>0\) and \(h\in\R\). +Two such slopes with the same density \(d=\rho\) and distinct first +coordinates give two distinct upper supporting slopes of +\(H_\varphi(\,\cdot\,,\rho)\). This is the route to the derivative +jump, proved explicitly in \Cref{m:canonical-corner}. + +\paragraph{An attraction with controlled error.} +For a pressure \(Q\), an ideal attraction of mass \(a>0\) acts by +\begin{equation}\label{eq:intro-transform} + (T_aQ)(\beta,h) + =\sup_{u\in\R} + \left\{Q(\beta,h+\beta au)-\frac{\beta a}{2}u^2\right\}. +\end{equation} +Square completion identifies its maximizing shifts \(u\) with +maximizing densities after addition of the quadratic attraction term. +\Cref{k:section-transform} proves attainment, uniform bounds on these +shifts, and an exact formula for all supporting slopes of \(T_aQ\). + +Fix a nonnegative radial \(C^1\) kernel \(w\), supported in the unit +ball and of integral one, and put \(w_R(x)=R^{-3}w(x/R)\). +If the reference interaction \(\varphi_0\) has range at most \(R_0\), +then \Cref{k:range} compares the physical attraction \(-ta w_R\) +with the ideal transform: +\[ + \bigl\|Q_{\varphi_0-ta w_R}-T_aQ_{\varphi_0}\bigr\|_{\mathcal D} + \le C_{\mathcal D,t}\sqrt{R_0/R} +\] +on each compact parameter set \(\mathcal D\). The constant is +uniform over earlier reference interactions and over positive masses +\(a\) tending to zero. Boxes of side comparable to +\(\sqrt{R_0R}\) balance the cost of deleting reference interactions +across boxes against the variation of the new kernel inside them. +For the lower bound, independent reference grand ensembles are placed +in these boxes. Completing a square avoids any need to identify their +mean density with the maximizing shift, so the comparison remains valid +at coexistence without selecting a phase. + +\paragraph{Three slope groups and one fixed potential.} +As \(t\) grows, the bare-core pressure tends to +\(\max\{0,ph,4ph-9p\}\). Its branches come from vacuum, single +particles, and four-particle clusters, respectively. A first ideal +attraction turns these into three affine branches with noncollinear +slopes in the \((e,d)\) plane. \Cref{i:section} proves that, for one +large finite \(t\), the actual generating supports remain near these +three slopes throughout a parameter neighborhood. + +Uniform convergence of convex pressures by itself need not preserve +a corner. The additional control comes from the following property +of a maximizing shift in \eqref{eq:intro-transform}: at its shifted +parameter \((\beta,h+\beta au)\), the reference pressure has a +parabolic upper bound in the field variable +(\Cref{k:active-parabola}). In \Cref{m:one-step} this bound confines +nearby supports to one density band and hence to one of the separated +slope groups. An error \(\delta\) in pressure then costs +\(O(\delta/a)\) in the slopes. + +After the first attraction, take \(R_{n+1}=R_n^2\) and +\(a_n=R_n^{-1/16}\). The range errors are \(O(R_n^{-1/4})\), and +their ratios to the next masses are \(O(R_n^{-1/8})\). All relevant +losses are summable. \Cref{sec:iteration} consequently preserves a +common two-dimensional ball of pressure supports, passes it to the +limit, and uses horizontal slices near density \(5p/3\) to obtain the +canonical singularity at a common inverse temperature throughout a +density interval. The same schedule gives +\(\sum_n a_nR_n^{1/32}<\infty\), which supplies the pointwise tail +bound. The core, all attraction parameters, and the density interval +are fixed before the limiting critical parameter is selected. diff --git a/preprints/A-radial-continuum-phase-transition-with-algebraic-decay-September-24-2026/build/sections/iteration.tex b/preprints/A-radial-continuum-phase-transition-with-algebraic-decay-September-24-2026/build/sections/iteration.tex new file mode 100644 index 0000000000000000000000000000000000000000..b7af1ac23d29e4311f661a80c9afdae42960d0eb --- /dev/null +++ b/preprints/A-radial-continuum-phase-transition-with-algebraic-decay-September-24-2026/build/sections/iteration.tex @@ -0,0 +1,478 @@ +\section{The multiscale potential and the canonical singularity} +\label{sec:iteration} + +Fix \(t\) as in \Cref{i:initial-separation}. We will preserve its +three groups of generating supports while replacing ideal attractions +by physical ones. The essential step is quantitative: a pressure error +\(\delta\), followed by an ideal attraction of mass \(a\), enlarges +the slope groups by \(O(\delta/a+a)\) and costs \(O(a)\) in the +parameter domain. We first prove this step, then choose one sequence +of masses and ranges for which both losses are summable. + +\subsection{Preservation under one attraction} + +Let \(q\) be a physical pressure and \(P\) a convex approximation +whose subdifferential is generated near the three initial slopes. +In the iteration, \(P\) will be the preceding ideal transform and +\(q\) its finite-range realization. We ask how closely the generating +supports of \(T_aq\) remain in those three groups when +\(\norm{P-q}_{W_1}\le\delta\). At each active shifted point, the +parabolic bound first isolates one group in a neighborhood of radius +proportional to \(a\). Secant inequalities in that neighborhood then +transfer the support bound from \(P\) to \(q\). + +Recall the parameter balls +\(W(r)=\clB{\lambda_*}{r}\), \(W_1=W(1/4)\), +the slope vectors \(v_i=(Ad_i^2/2,d_i)\), and the constants +\(\tau,\gamma>0\) from \Cref{i:geometry}. +By \Cref{k:transform-properties}, choose \(U_*>0\) bounding all active +shifts for every transform \(T_aQ\) of a family pressure, with \(00\), and then \(l>0\), sufficiently small that +\begin{equation}\label{eq:band-constants} + \frac{k}{2\beta_{\min}}+\frac{2C_*l}{k}<\frac p{16}. +\end{equation} + +Set +\[ + S_* = \beta_{\max}U_*+k+l+1. +\] +Here \(\beta_{\max}U_*a\) bounds the displacement to an active +shifted point, \(ka\) will be a field increment, and \(la\) the +radius on which all supports will be controlled. All these constants +depend only on the fixed \(t\), the family, and the parameter compacts. + +\begin{lemma}[One-step preservation of generating supports] +\label{m:one-step} +Let \(q\) be a pressure from the family \eqref{f:family}, and let +\(P\) be a finite convex function on \(\Omega\) that is +\(C_*\)-Lipschitz on \(W_1\). Suppose +\[ + \begin{gathered} + 02\), define +\begin{equation}\label{eq:mass-range} + R_n=R_1^{2^{n-1}},\qquad + a_1=A,\qquad a_n=R_n^{-1/16}\quad(n\ge2). +\end{equation} +Let +\begin{equation}\label{eq:truncated-models} + \varphi_n=\psi_t-t\sum_{j=1}^n a_jw_{R_j}, + \qquad q_n=Q_{\varphi_n}, + \qquad P_n=T_{a_n}q_{n-1}\quad(n\ge1), +\end{equation} +where \(\varphi_0=\psi_t\) and \(q_0=q^{(t)}\). +Thus \(P_1=T_Aq^{(t)}\) is the pressure used in +\Cref{i:initial-separation}. + +Choose \(R_1\) large enough initially that +\(\sum_{n\ge2}a_n<1\). +Every truncated potential then belongs to the stable family +\eqref{f:family}. +By \Cref{k:range}, for all sufficiently large \(R_1\), +\begin{equation}\label{eq:approximation-errors} + \norm{q_n-P_n}_{W_1}\le\delta_n, + \qquad \delta_n=C_3R_n^{-1/4}\quad(n\ge1), +\end{equation} +with one constant \(C_3\) independent of \(n\) and \(R_1\). +Indeed, the reference range is at most two for \(n=1\), giving the +stronger estimate \(O(R_1^{-1/2})\). +For \(n\ge2\), it is at most \(R_{n-1}=\sqrt{R_n}\). +The ratio threshold in \Cref{k:range} is uniform in the reference, so it +suffices to enlarge the same \(R_1\). + +The crucial relations are +\begin{equation}\label{eq:summable-errors} + a_{n+1}=R_n^{-1/8},\qquad + \frac{\delta_n}{a_{n+1}}=C_3R_n^{-1/8}, + \qquad + \sum_{n\ge1}\left(a_{n+1}+\delta_n+ + \frac{\delta_n}{a_{n+1}}\right) + \longrightarrow0\quad(R_1\to\infty). +\end{equation} +For example, for every \(q>0\), +\[ + \sum_{n\ge1}R_n^{-q} + =\sum_{n\ge1}(R_1^{-q})^{2^{n-1}} + \le \frac{R_1^{-q}}{1-R_1^{-q}}. +\] +This explicit bound will meet all the conditions below. + +By \Cref{k:transform-properties}, fix \(C_0>0\), independently of +\(R_1\), such that +\begin{equation}\label{eq:small-mass-increment} + 0\le T_aQ-Q\le C_0a\qquad\text{on }W_1 +\end{equation} +for all family pressures and \(0 + \frac{\gamma}{8}-\frac{2\gamma}{32} + =\frac{\gamma}{16}>0. +\end{align*} +Thus a minimizer \(\lambda_n\) is interior, and +\(g\in\partial P_n(\lambda_n)\). +By \Cref{m:propagation}, this subdifferential is generated by its +intersections with the three balls \(\clB{v_i}{r_n}\). +The vector \(g\) lies outside the convex hull of any two of the larger +balls \(\clB{v_i}{\tau}\). +Each of the three intersections must therefore be nonempty. +Choose one actual support from each. +Their triangle contains \(\clB{g}{\gamma}\) by \Cref{i:geometry}, and +their convex hull is contained in \(\partial P_n(\lambda_n)\). +\end{proof} + +\subsection{The final potential and its decay} + +Define one potential, now with all parameters fixed, by +\begin{equation}\label{eq:final-potential} + \Phi(x)=\phi(\abs{x}) + =\psi_t(\abs{x})-t\sum_{n\ge1}a_nw_{R_n}(x). +\end{equation} + +\begin{proposition}[Stability and algebraic decay]\label{m:tail} +The function \(\phi\) is bounded and continuous on \([0,\infty)\), +is stable with \(B=t^2/2\), and satisfies +\(\abs{\phi(r)}\le Cr^{-3-1/32}\) for all \(r\ge1\), where +\begin{equation}\label{eq:tail-constant} + C=2^{3+1/32}t^2+t\norm{w}_{\infty} + \left(A R_1^{1/32} + +\sum_{n\ge2}R_1^{-2^{n-1}/32}\right)<\infty. +\end{equation} +\end{proposition} + +\begin{proof} +Since \(\sum_na_nn}a_j\longrightarrow0. +\] +Choose a subsequence of \(\lambda_n\) converging to +\(\lambda_c=(\beta_c,h_c)\in W(1/8)\). +In particular, \(\beta_c\in[7/8,9/8]\). +For each \(v\in\clB{g}{\gamma}\) and \(\lambda\in W_1\), +\eqref{eq:common-pressure-ball} gives +\[ + P_n(\lambda)\ge P_n(\lambda_n)+v\cdot(\lambda-\lambda_n). +\] +Uniform convergence and continuity pass this inequality to the limit. +It is a supporting inequality for \(Q_\Phi\) on a neighborhood of +\(\lambda_c\). +Such a local support extends globally on the convex domain \(\Omega\): +a violation at another point would, by convexity along the joining +segment, give a violation arbitrarily close to \(\lambda_c\). +Consequently, +\begin{equation}\label{eq:limiting-support-ball} + \clB{g}{\gamma}\subset\partial Q_\Phi(\beta_c,h_c). +\end{equation} + +We state explicitly the duality implication used to finish the proof. +It is the supporting-hyperplane form of Fenchel duality +\citep[Theorem~11.1 and Proposition~11.3]{RockafellarWets1998}, applied +only in the field coordinate and with the inverse temperature retained +as a parameter. + +\begin{lemma}[From pressure supports to canonical slopes]\label{m:canonical-corner} +Suppose that \(H(\beta,d)\) and \(Q(\beta,h)\) have the duality and +finiteness properties of \Cref{f:thermodynamics}; assume also, as there, +that \(H(\,\cdot\,,d)\) is concave on \((0,\infty)\) for every \(d>0\). +If +\[ + (s_1,\rho),(s_2,\rho)\in\partial Q(\beta_c,h_c), + \qquad \rho>0,\quad s_10\), and set +\[ + \eta=\min\{\gamma/2,\rho_0/2\},\qquad + I=(\rho_0-\eta,\rho_0+\eta)\subset(0,\infty). +\] +For every \(\rho\in I\), the two points +\[ + (g_1-\gamma/2,\rho),\qquad(g_1+\gamma/2,\rho) +\] +lie in \(\clB{g}{\gamma}\), since their squared distances from \(g\) +are \((\gamma/2)^2+(\rho-\rho_0)^2<\gamma^2/2\). +By \eqref{eq:limiting-support-ball}, both support the same pressure at +the same \((\beta_c,h_c)\), independently of \(\rho\). +Lemma~\ref{m:canonical-corner} therefore gives finite one-sided +canonical slopes whose difference is at least \(\gamma\) for every +\(\rho\in I\). +The all-sequence canonical limit of \Cref{f:thermodynamics} applies +in particular to \(N_L=\lfloor\rho L^3\rfloor\) for every \(\beta>0\). +The specified free energy equals +\[ + f(\beta,\rho)=\frac{t}{\beta}H_\Phi(\beta,\rho). +\] +For each \(\rho\in I\), it is finite for every \(\beta>0\). +The finite concave function \(H_\Phi(\,\cdot\,,\rho)\) is continuous +at \(\beta_c\); hence the common derivative term from \(1/\beta\) +cancels in the difference, giving +\[ + f'_-(\beta_c,\rho)-f'_+(\beta_c,\rho) + =\frac{t}{\beta_c} + \bigl(H'_{\Phi,-}(\beta_c,\rho) + -H'_{\Phi,+}(\beta_c,\rho)\bigr) + \ge\frac{t\gamma}{\beta_c}>0. +\] +The potential and the density interval are fixed before the limiting +critical parameter is selected, so this one \(\beta_c\) works throughout +\(I\). +\end{proof} + diff --git a/preprints/A-radial-continuum-phase-transition-with-algebraic-decay-September-24-2026/build/sections/local_bounds.tex b/preprints/A-radial-continuum-phase-transition-with-algebraic-decay-September-24-2026/build/sections/local_bounds.tex new file mode 100644 index 0000000000000000000000000000000000000000..0a6c972a7e80736aea7d13c01bd4bd93b9a89da8 --- /dev/null +++ b/preprints/A-radial-continuum-phase-transition-with-algebraic-decay-September-24-2026/build/sections/local_bounds.tex @@ -0,0 +1,245 @@ +\section{Occupation bounds and perturbation estimates} +\label{f:local-section} + +The next estimates turn the quadratic stability bound into uniform +control of local particle numbers and changes of pressure. They play +the role of the occupation bounds in the theory of superstable +interactions \citep{Ruelle1970}. We prove them for the precise family +of deletions and interpolations used in the range comparison. + +From this section onward the occupation estimates are used with $t$ +fixed. Their constants may depend on $t$ and on a compact set of +$(\beta,h)$, but never on the number or sizes of the attraction ranges. +The finite volume systems in \eqref{f:modified-energy} are included; +in particular, deleting interactions between larger boxes will not +require a new thermodynamic limit theorem. + +If a bounded region meets $k$ small cells, its grand partition +function has the exact decomposition +\begin{equation}\label{f:cell-measure} + \Xi^v(\beta,h) + =\sum_{n_1,\ldots,n_k\ge0} + \frac{\ee^{th\sum_i n_i}}{\prod_i n_i!} + \int_{\prod_i C_i^{n_i}}\exp(-\beta U^v(X))\,\dd X. +\end{equation} +Indeed, the $N!/\prod_i n_i!$ assignments of $N=\sum_i n_i$ labels +to disjoint cells cancel the global $1/N!$. This identity also gives +the conditional grand distribution within one cell after the outside +configuration has been specified. + +\begin{proposition}[Uniform occupation moments]\label{f:moments} +Fix $t$ satisfying \eqref{f:t-requirements} and a compact +$\mathcal D\subset\Omega$. There are constants +$C_{\exp},C_1<\infty$, depending only on $t$ and $\mathcal D$, such +that every grand canonical measure for an interaction +\eqref{f:modified-energy} satisfies +\begin{equation}\label{f:moment-estimates} + \sup_i\mathbb E\exp(\beta D_0n_i^2/4)\le C_{\exp}, + \qquad \sup_{i,j}\mathbb E[n_i n_j]\le C_1 + \qquad ((\beta,h)\in\mathcal D). +\end{equation} +The bounds hold for every bounded region with finitely many cropped +cells, and uniformly for all permitted deletions and interpolations. +\end{proposition} + +\begin{proof} +If the region meets no small cell, its grand measure is the vacuum and +the assertion is immediate. Otherwise, let the number of cells be positive. +Write $\alpha=D_0-M/2$, $u_0=\beta D_0/4$, and enclose +$\mathcal D$ in $\beta\in[\beta_-,\beta_+]$, $\abs{h}\le H$. +We first check, before taking any maximum, that +$F_i:=\mathbb E\exp(u_0n_i^2)$ is finite in each fixed volume. +By \eqref{f:cell-stability}, insertion of this exponential into +\eqref{f:cell-measure} bounds its numerator by the product of $k$ +one-variable series with exponents +\[ + -\beta(\alpha-D_0/4)n_i^2 + -\beta\alpha\sum_{j\ne i}n_j^2 + +(th+\beta B_0)\sum_j n_j. +\] +Each spatial integral contributes at most +$\prod_j(b^3)^{n_j}$, with the factors $1/n_j!$ already in +\eqref{f:cell-measure}. The quadratic coefficients are positive in +magnitude because +\[ + \alpha-D_0/4=3D_0/4-M/2\ge11D_0/16>0. +\] +Thus all these series are finite, and the denominator $\Xi^v$ is at +least one. In particular, +$F_{\max}:=\max_iF_i<\infty$. To bound this maximum independently of +the number of cells, we shall prove +\[ + F_{\max}\le C_2 F_{\max}^{\eta_0}, + \qquad \eta_0=4M^2/D_0^2\le1/16<1, +\] +where $C_2$ depends only on $t$ and $\mathcal D$. +The ratio of attraction strength to core repulsion will therefore +make the dependence on neighboring moments sublinear. + +Condition on the configuration $Y$ outside cell $i$. For $n$ +particles inside that cell, the part of the energy involving at least +one inside particle is bounded below by +\begin{equation}\label{f:conditional-energy} + (D_0-J_{ii}/2)n^2-B_0n-nG_i(Y), + \qquad G_i(Y)=\sum_{j\ne i}J_{ij}n_j. +\end{equation} +This uses the complete core estimate inside the cell, discards +nonnegative core terms across cells, and bounds all attractions by +their cell suprema. Dividing the conditional weights by the +outside-only Boltzmann factor leaves a denominator containing the +empty-cell term one. Hence the conditional exponential moment is +at most its unnormalized numerator. The inequality +\[ + nG_i\le D_0n^2/4+G_i^2/D_0 +\] +and the factor $\exp(u_0n^2)$ leave the quadratic coefficient +$D_0/2-J_{ii}/2\ge7D_0/16$. Consequently +\begin{equation}\label{f:conditional-moment} + \mathbb E[\exp(u_0n_i^2)\mid Y] + \le C_2\exp(\beta G_i(Y)^2/D_0), +\end{equation} +where the finite constant +\[ + C_2=\sum_{n\ge0}\frac{b^{3n}}{n!} + \exp\left(-\frac{7\beta_-D_0}{16}n^2 + +(tH+\beta_+B_0)n\right) +\] +depends only on $t$ and $\mathcal D$ and is at least one. + +To close this estimate, Cauchy--Schwarz with the weights $J_{ij}$ +and the row bound give +\[ + G_i^2\le M\sum_{j\ne i}J_{ij}n_j^2. +\] +Set $\theta_j=J_{ij}/M$ for $j\ne i$ and +$\theta_\varnothing=1-\sum_{j\ne i}\theta_j\ge0$. +With the exponent $\eta_0$ chosen above, +$\beta G_i^2/D_0\le\eta_0u_0\sum_{j\ne i}\theta_jn_j^2$. +Apply convexity of the exponential with the additional zero argument +of weight $\theta_\varnothing$, and then the concavity of +$x\mapsto x^{\eta_0}$, to obtain +\[ + \begin{split} + \mathbb E\exp(\beta G_i^2/D_0) + &\le\theta_\varnothing+ + \sum_{j\ne i}\theta_j\mathbb E\exp(\eta_0u_0n_j^2)\\ + &\le\theta_\varnothing+\sum_{j\ne i}\theta_jF_j^{\eta_0} + \le F_{\max}^{\eta_0}, + \end{split} +\] +where the last inequality uses $F_{\max}\ge1$. +Taking expectations in \eqref{f:conditional-moment} and then the +maximum over cells yields +$F_{\max}\le C_2F_{\max}^{\eta_0}$, and therefore +\[ + F_{\max}\le C_2^{1/(1-\eta_0)}=:C_{\exp}. +\] +This bound is independent of the number of cells. Since +$\exp(u_0n_i^2)\ge u_0n_i^2$ and +$u_0\ge\beta_-D_0/4$, the second moments are uniformly bounded. +Cauchy--Schwarz gives the mixed moment bound in +\eqref{f:moment-estimates}. + +Cropping a cell only decreases its volume and the relevant suprema. +Every deletion or interpolation allowed in \eqref{f:modified-energy} +retains the full intra-cell core and the same row bound. All steps +above therefore apply without change to those systems. +\end{proof} + +\begin{corollary}[Absolute perturbation bound]\label{f:perturbation} +In a fixed bounded region, let $v_\theta=(1-\theta)v_0+\theta v_1$, +$0\le\theta\le1$, be bounded symmetric pair interactions satisfying +the hypotheses of \Cref{f:moments} with the same $t$ and $M$. +Let $J^{\mathrm{diff}}$ be a symmetric nonnegative matrix such that +\[ + \abs{v_1(x,y)-v_0(x,y)}\le J^{\mathrm{diff}}_{ij} + \quad (x\in C_i,\ y\in C_j). +\] +Then, uniformly for $(\beta,h)\in\mathcal D$, +\begin{equation}\label{f:interpolation-bound} + \abs{\log\Xi^{v_1}(\beta,h)-\log\Xi^{v_0}(\beta,h)} + \le\frac{\beta C_1}{2}\sum_{i,j}J^{\mathrm{diff}}_{ij}. +\end{equation} +Thus signed deletions are controlled by the absolute interaction +difference, with no cancellation assumption. +\end{corollary} + +\begin{proof} +Differentiation in $\theta$ is justified directly in finite volume. +The empty region is immediate. Put $\alpha=D_0-M/2>0$. +If there are $k\ge1$ cells and the region has volume $V$, the common +energy bound gives +$U^{v_\theta}\ge\alpha N^2/k-B_0N$, while boundedness of the pair +difference gives $\abs{\partial_\theta U^{v_\theta}}\le C N^2$. +Uniformly in $\theta$ and on $\mathcal D$, the differentiated grand +series is therefore dominated by a constant times +\begin{equation}\label{f:derivative-domination} + \sum_{N\ge0}(1+N^2)\frac{V^N}{N!} + \exp\left(-\beta_-\alpha N^2/k + +(tH+\beta_+B_0)N\right)<\infty. +\end{equation} +The volume and the constant in this domination are fixed only for +the differentiation; the resulting estimate below is uniform in +volume. Hence +\[ + \frac{\dd}{\dd\theta}\log\Xi^{v_\theta} + =-\beta\mathbb E_\theta\partial_\theta U^{v_\theta}. +\] +For every configuration, +\[ + \abs{\partial_\theta U^{v_\theta}} + \le\frac12\sum_{i,j}J^{\mathrm{diff}}_{ij}n_i n_j, +\] +where the diagonal uses $n_i(n_i-1)\le n_i^2$. +Apply \Cref{f:moments} and integrate over $\theta\in[0,1]$. +\end{proof} + +\begin{corollary}[Pressure control by attraction mass] +\label{f:tail-pressure} +For a finite or countably infinite member of \eqref{f:family}, let +\[ + \varphi_k=\psi_t-t\sum_{j\le k}\alpha_jw_{R_j}, + \qquad m_k=\sum_{j>k}\alpha_j. +\] +For every compact $\mathcal D\subset\Omega$ there is a constant +$C_{\mathcal D,t}<\infty$, independent of the ranges, their number, +and $k$, such that +\begin{equation}\label{f:tail-pressure-bound} + 0\le Q_\varphi(\beta,h)-Q_{\varphi_k}(\beta,h) + \le C_{\mathcal D,t}m_k + \qquad ((\beta,h)\in\mathcal D). +\end{equation} +The same upper bound holds for the difference of normalized finite +volume log grand partition functions in every cube of side $L\ge1$. +\end{corollary} + +\begin{proof} +Interpolate the whole tail at once: +\[ + \varphi_{k,\theta} + =\varphi_k-\theta t\sum_{j>k}\alpha_jw_{R_j}, + \qquad 0\le\theta\le1. +\] +Even for infinitely many terms, the tail converges uniformly and is +a bounded continuous pair interaction, of supremum norm at most +$t\norm{w}_\infty m_k$. Thus the derivative exists for each finite +configuration and the domination \eqref{f:derivative-domination} +justifies differentiation directly, including the infinite endpoint. +All intermediate attractions obey the original mass bound. + +By the proof of \eqref{f:row-bound}, the difference matrix can be +chosen with +\[ + \sup_i\sum_jJ^{\mathrm{diff}}_{ij}\le c_btm_k. +\] +There are at most $216L^3$ cells. Dividing +\eqref{f:interpolation-bound} by $tL^3$ gives the claimed finite +volume upper bound, for instance with +$C_{\mathcal D,t}=108\beta_+C_1c_b$. Its lower bound follows because +adding a nonnegative attraction decreases the energy. + +\Cref{f:thermodynamics} already supplies thermodynamic pressure +limits for both endpoints, including the infinite sum. Passing to +those limits proves \eqref{f:tail-pressure-bound}; convergence of +truncation pressures is a consequence, not an input, of this estimate. +\end{proof} diff --git a/preprints/A-radial-continuum-phase-transition-with-algebraic-decay-September-24-2026/build/sections/range.tex b/preprints/A-radial-continuum-phase-transition-with-algebraic-decay-September-24-2026/build/sections/range.tex new file mode 100644 index 0000000000000000000000000000000000000000..e4d44451d4850fabba4dd4d29968bce7f203b194 --- /dev/null +++ b/preprints/A-radial-continuum-phase-transition-with-algebraic-decay-September-24-2026/build/sections/range.tex @@ -0,0 +1,392 @@ +\section{A uniform quantitative range estimate} +\label{k:section-range} + +We now compare the ideal transform with a physical attraction of finite +range. The estimate is uniform over all preceding attraction ranges and +over arbitrarily small positive added masses. This uniformity will allow +the error and the next mass to be chosen by a single explicit schedule. +The quadratic density term is the one underlying the Lebowitz--Penrose +limit \citep{LebowitzPenrose1966,GerardiEtAl1973}; our purpose here is a quantitative +estimate with constants uniform across the complete reference family. + +\begin{theorem}[Quantitative approximation by a long-range attraction] +\label{k:range} +Fix an admissible $t$ and a compact set $\mathcal D\subset\Omega$. +There are constants $C_{\mathcal D,t}<\infty$ and +$K_{\mathcal D,t}<\infty$ with the following property. Let $R_0\ge2$, +and let +\[ + \varphi_0=\psi_t-t\sum_j\alpha_jw_{R_j}, + \qquad 1\le R_j\le R_0, + \qquad \alpha_j\ge0, +\] +where the sum may be empty, finite, or countably infinite. If +$a>0$, $\sum_j\alpha_j+a\le A+1$, and $R/R_0\ge K_{\mathcal D,t}$, +then +\begin{equation} + \sup_{\lambda\in\mathcal D} + \abs{Q_{\varphi_0-ta w_R}(\lambda)-T_aQ_{\varphi_0}(\lambda)} + \le C_{\mathcal D,t}\sqrt{\frac{R_0}{R}}. + \label{k:range-estimate} +\end{equation} +Both constants are independent of $R_0$, the number and values of the +reference ranges, the reference masses, and the positive mass $a$. +\end{theorem} + +Use boxes of an intermediate side $s$, with $R_0\le s\le R$. +Deleting the reference interaction between different boxes costs +$O(R_0/s)$ in pressure. The suprema and infima of the new kernel on +pairs of boxes have row sums which, after multiplication by $s^3$, +approximate its unit integral within $O(s/R)$. Density bounds will give +pressure errors of the same order. The suprema will give a product upper +bound; the infima will give a lower bound from independent reference +ensembles in the boxes. Balancing these two errors explains +the choice $s\asymp\sqrt{R_0R}$. We begin with the deletion estimate, +which must include both the positive and negative parts of the reference +interaction. + +\begin{lemma}[Grand pressure in a finite box] +\label{k:box-pressure} +Let $\varphi_0$ and $R_0$ be as in \Cref{k:range}, allowing +$\sum_j\alpha_j\le A+1$. Write +\[ + Q^s_{\varphi_0}(\lambda) + =\frac1{ts^3}\log\Xi_s^{\varphi_0}(\lambda). +\] +For every compact $\mathcal K\subset\Omega$ and every $s\ge R_0$ with +$s/b\in\N$, +\begin{equation} + \sup_{\lambda\in\mathcal K} + \abs{Q^s_{\varphi_0}(\lambda)-Q_{\varphi_0}(\lambda)} + \le C_{\mathcal K,t}\frac{R_0}{s}. + \label{k:box-pressure-bound} +\end{equation} +More generally, for the same $s$, partition $\Lambda_{ms}$ into $m^3$ +boxes of side $s$, aligned with the small-cell grid. In the potential +$\varphi_0-ta w_R$, where $a\ge0$, $R\ge1$, and +$\sum_j\alpha_j+a\le A+1$, delete only reference interactions between +distinct boxes. If $\Xi^{\mathrm{del}}_{ms}$ is the resulting grand +partition function, then +\begin{equation} + \sup_{\lambda\in\mathcal K} + \frac{\abs{\log\Xi_{ms}^{\varphi_0-ta w_R}(\lambda) + -\log\Xi^{\mathrm{del}}_{ms}(\lambda)}}{t(ms)^3} + \le C_{\mathcal K,t}\frac{R_0}{s}. + \label{k:deletion} +\end{equation} +The constants have the uniformity asserted in \Cref{k:range} and are +also independent of $m,s,R$. +\end{lemma} + +\begin{proof} +Put $S_0=t\sum_j\alpha_jw_{R_j}$ and $S_{\mathrm{new}}=ta w_R$. +The series for $S_0$ converges uniformly, is supported in the ball of +radius $R_0$, and has the row bound \eqref{f:row-bound}, including when +there are countably many reference terms. +Multiply the reference interaction on pairs belonging to distinct large +boxes by a parameter $\theta\in[0,1]$. Every intermediate interaction +retains all intra-small-cell core terms, and its attractive part is +bounded by $S_0+S_{\mathrm{new}}$. This has small-cell row sum at most +$M$. Therefore \Cref{f:moments,f:perturbation} apply throughout this +interpolation, whether or not the intermediate interaction is translation +invariant. + +The absolute difference is bounded on every pair by +\[ + (\psi_t+S_0)(x-y) + \boldsymbol 1\{x,y\text{ belong to distinct large boxes}\}. +\] +Its row sum over small cells is at most +$C_b t^2+c_b t(A+1)$: the core is bounded by $t^2$ and supported within +distance two, and the attraction has the common row bound already proved. +This estimate uses an absolute majorant and thus remains valid when +$\varphi_0$ changes sign on a pair of cells. + +A nonzero difference row must belong to a small cell within distance +$R_0+\sqrt3 b$ of the boundary of its large box. Since $R_0\ge2$ and +$s\ge R_0$, there are at most $C_b s^2R_0$ such small cells per large +box. The bound follows either by subtracting the volume of the inner +cube when the strip is thin, or from $s^3\le C s^2R_0$ when the strip +fills a fixed fraction of the box. Thus the sum of all difference +majorants is at most +\[ + C_b m^3s^2R_0\bigl(C_b t^2+c_b t(A+1)\bigr). +\] +The finite-volume interpolation bound of \Cref{f:perturbation}, divided +by $t(ms)^3$, proves \eqref{k:deletion}. + +For $a=0$ the deleted system consists of independent reference boxes, +and its grand partition function is exactly +\begin{equation} + \Xi^{\mathrm{del}}_{ms} + =\bigl(\Xi_s^{\varphi_0}\bigr)^{m^3}. + \label{k:factorization} +\end{equation} +For completeness, with prescribed box counts $(n_z)$ of total $N$, the +$N!/\prod_z n_z!$ assignments of labels to boxes cancel the global +$1/N!$ and leave $\prod_z1/n_z!$. Summing the nonnegative product +integrals over all $(n_z)$ gives \eqref{k:factorization}, including +the zero-count terms. Its normalized logarithm equals +$Q^s_{\varphi_0}$ for every $m$. Letting $m\to\infty$ in +\eqref{k:deletion} and using only the pressure limit of the full +reference proves \eqref{k:box-pressure-bound}. +\end{proof} + +The next estimate uses the global $C^1$ bound of $w$. Radial monotonicity +and strict positivity on its support are not required. + +\begin{lemma}[Upper and lower kernel row sums] +\label{k:kernel-rows} +Let $B_z$ be the box of side $s$ centered at $z$, where $z$ ranges +over a translated lattice $s\mathbb Z^3$, and suppose $0D_{\mathrm{cut}}V}B_n\le1 + \qquad(V\ge V_0). + \label{k:grand-tail} +\end{equation} +Both $D_{\mathrm{cut}}$ and $V_0$ depend only on $\mathcal D,t$. +They are fixed before any local occupation constants: the tail lemma +uses only the canonical quadratic bound, even with this auxiliary +weight. + +Let $U$ be the common active-shift bound on $\mathcal D$ from +\Cref{k:transform-properties}. Choose one compact rectangle +$\mathcal D'\subset\Omega$ containing $\mathcal D$ and every point +\begin{equation} + (\beta,h+\beta av),\qquad + (\beta,h)\in\mathcal D,\quad + 0\le a\le A+1,\quad 0\le v\le\max\{U,D_{\mathrm{cut}}\}. + \label{k:shifted-compact} +\end{equation} +Only now fix the moment and deletion constants of +\Cref{f:moments,k:box-pressure} on $\mathcal D'$. All these constants +are independent of the eventual box side and of the reference. + +Put $q=R/R_0$ and choose +\begin{equation} + s=b\left\lfloor\frac{\sqrt{R_0R}}b\right\rfloor. + \label{k:box-side} +\end{equation} +A single lower bound on $q$, uniform for $R_0\ge2$, ensures +\begin{equation} + R_0\le s\le R,\qquad + \frac12R_0\sqrt q\le s\le R_0\sqrt q,\qquad + s^3\ge V_0,\qquad + \eta=C_w\frac sR\le1. + \label{k:ratio-conditions} +\end{equation} +Partition $\Lambda_L$, $L=ms$, into $m^3$ boxes $B_z$, and delete +reference interactions between different boxes as in +\Cref{k:box-pressure}. The new attraction remains intact. All limits +in $m$ below hold with $s,R,R_0$, the reference, and the parameters fixed. + +\medskip\noindent\emph{Upper bound.} +Let $N_z$ be the particle count in $B_z$, and write +$W_R=\sum_{i0$, define +\begin{equation} + T_aQ(\beta,h) + =\sup_{u\in\R}\left\{ + Q(\beta,h+\beta au)-\frac{\beta a}{2}u^2\right\}. + \label{k:transform} +\end{equation} +An \emph{active shift} at $\lambda=(\beta,h)$ is a value of $u$ attaining +this supremum. Its shifted parameter is +$x_u=(\beta,h+\beta au)$. +\end{definition} + +\begin{proposition}[Attainment and uniform bounds] +\label{k:transform-properties} +Suppose the size conditions on $t$ in \Cref{f:stability} hold, including +$ct>4(A+1)$. For every $01$ and $\lambda$ +in a fixed compact. Thus the maximizing-density bound is also uniform +in sufficiently large $t$. + +If $d$ maximizes \eqref{k:density-transform}, setting $u=d$ in +\eqref{k:square-completion} makes $u$ active. Conversely, fix an active +$u$. The density supremum defining $Q(\beta,h+\beta au)$ is attained: +its fixed shifted field and the canonical quadratic bound give coercivity. +For an attaining density $d$, \eqref{k:square-completion} can equal +$T_aQ(\beta,h)$ only if $d$ maximizes \eqref{k:density-transform} and +$(u-d)^2=0$. This proves the claimed exact identity of maximizing sets +and also attainment in \eqref{k:transform}. + +For fixed $u$, the branch +\[ + F_u(\beta,h)=Q(\beta,h+\beta au)-\beta au^2/2 +\] +is convex: its parameter map is invertible and linear, and its last +term is linear. The supremum of these branches is convex and is finite +by \eqref{k:coercivity}. If $d$ is a maximizing density, pressure +duality and the nonnegativity of the added quadratic give +\[ + Q(\beta,h)\le T_aQ(\beta,h) + \le Q(\beta,h)+\frac{\beta a}{2}d^2. +\] +This proves \eqref{k:small-mass}, with +$C_{\mathcal D}=\tfrac12\sup_{\mathcal D}\beta\, U^2$. +The canonical bound and pressure duality bound $Q$ uniformly on every +compact at fixed $t$; \eqref{k:small-mass} does the same for $T_aQ$. +Convexity then gives the asserted local Lipschitz bounds. +\end{proof} + +We recall the elementary convex facts used below. For a finite convex +function $F$ on $\Omega$, write +\[ + \partial F(\lambda)= + \{v\in\R^2:F(y)\ge F(\lambda)+\ip{v}{y-\lambda} + \text{ for all }y\in\Omega\}. +\] +Such a function is continuous and locally Lipschitz. Indeed, convexity +first bounds it above on a simplex by its values at the vertices; +reflection about an interior point supplies a lower bound on a smaller +neighborhood, and one-dimensional secant inequalities give a Lipschitz +bound on a still smaller neighborhood. The directional derivative +\[ + F'(\lambda;z)=\lim_{r\downarrow0} + \frac{F(\lambda+rz)-F(\lambda)}r +\] +is finite and sublinear in $z$. A linear functional dominated by it +exists by linear separation (equivalently, the finite-dimensional +Hahn--Banach theorem). The secant inequalities show that its vector is +a member of $\partial F(\lambda)$. Thus this set is nonempty. It is +convex and closed by its defining inequalities, and locally bounded by +the local Lipschitz bound, hence compact. Those same inequalities and +continuity show that its graph is closed. In particular, uniform bounds +for convex functions on a larger compact neighborhood give uniform bounds +for all their supporting slopes on a smaller compact neighborhood. +We write a pressure support as $(e,d)$, with coordinates dual to +$(\beta,h)$: $d$ is the density coordinate and $e$ the +inverse-temperature slope coordinate. + +\begin{lemma}[Parabolic bound at an active point] +\label{k:active-parabola} +Let $a>0$ be as in \Cref{k:transform-properties}, and let $u$ be active +at $\lambda=(\beta,h)$. Then for every $r\in\R$, +\begin{equation} + Q(x_u+(0,r))\le Q(x_u)+ur+\frac{r^2}{2\beta a}. + \label{k:parabola} +\end{equation} +Every member of $\partial Q(x_u)$ has second coordinate $u$. +\end{lemma} + +\begin{proof} +Compare the active shift $u$ with $u+r/(\beta a)$ in +\eqref{k:transform}. Expanding the two quadratic penalties proves +\eqref{k:parabola}. If $(e,d)\in\partial Q(x_u)$, the supporting +inequality gives +\[ + dr\le Q(x_u+(0,r))-Q(x_u) + \le ur+\frac{r^2}{2\beta a}. +\] +Divide separately for $r>0$ and $r<0$ and let $r\to0$ to obtain $d=u$. +\end{proof} + +The subdifferential of a supremum of convex functions is governed by +its active branches. General forms of this principle appear in +\citet[p.~864]{HantouteLopezZalinescu2008}. Here the compactness of +active shifts and the parabolic bound give the following exact formula; +we prove both inclusions for this setting, including nonsmooth pressures. + +\begin{proposition}[Exact subdifferential formula] +\label{k:subdifferential} +For each fixed $00$, let $\lambda_r=\lambda+rz$, +choose an active shift $u_r$ there, and choose +$v_r\in\partial F_{u_r}(\lambda_r)$. The branch support at the endpoint +gives +\begin{equation} + \ip{v_0}{z} + \le\frac{T_aQ(\lambda_r)-T_aQ(\lambda)}r + \le\frac{F_{u_r}(\lambda_r)-F_{u_r}(\lambda)}r + \le\ip{v_r}{z}. + \label{k:endpoint-support} +\end{equation} +All $u_r$ are uniformly bounded near $\lambda$; their shifted points +stay in one compact subset of $\Omega$, and the underlying subgradients +of $Q$ are bounded there. Along a sequence $r\downarrow0$, pass to +convergent subsequences of $u_r$ and these subgradients. Continuity of +$Q$ and $T_aQ$ makes the limiting shift active at $\lambda$, and the +closed subgradient graph makes the limiting $v_r$ a member of +$G(\lambda)$. Thus \eqref{k:endpoint-support} implies +\[ + \ip{v_0}{z}\le\max_{v\in G(\lambda)}\ip{v}{z}. +\] +This holds in every direction. Separation from the compact convex set +$\conv G(\lambda)$ proves $v_0\in\conv G(\lambda)$. +\end{proof} + +All compactness arguments in \Cref{k:subdifferential} concern a fixed +positive $a$. The bounds needed when $a$ decreases to zero are supplied +separately by \Cref{k:transform-properties}: active shifts, shifted +parameter compacts, and local Lipschitz constants are uniform. At +$a=0$ the shift representation would have every $u\in\R$ active; no +compactness assertion for that degenerate representation is needed. diff --git a/preprints/A-radial-continuum-phase-transition-with-algebraic-decay-September-24-2026/paper.pdf b/preprints/A-radial-continuum-phase-transition-with-algebraic-decay-September-24-2026/paper.pdf new file mode 100644 index 0000000000000000000000000000000000000000..ec2712db17bbe29cba2d49c25a3eab8c501d675a --- /dev/null +++ b/preprints/A-radial-continuum-phase-transition-with-algebraic-decay-September-24-2026/paper.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:d53cafa68c322a072098e8a3cb2e3b7e059b4239ba95466d089632b8ce368587 +size 536600 diff --git a/preprints/A-rational-obstruction-to-strong-chromatic-splitting-at-height-three-September-25-2026/README.md b/preprints/A-rational-obstruction-to-strong-chromatic-splitting-at-height-three-September-25-2026/README.md new file mode 100644 index 0000000000000000000000000000000000000000..06f4e7b7df2650f6bef067b4576be8fdcde8e292 --- /dev/null +++ b/preprints/A-rational-obstruction-to-strong-chromatic-splitting-at-height-three-September-25-2026/README.md @@ -0,0 +1,17 @@ +# [A rational obstruction to strong chromatic splitting at height three](paper.pdf) + +**Author:** OpenAI + +**Date:** September 25, 2026 + +## Citation + +```bibtex +@misc{OAI:A-rational-obstruction-to-strong-chromatic-splitting-at-height-three-September-25-2026, + author = {{OpenAI}}, + title = {{A rational obstruction to strong chromatic splitting at height three}}, + howpublished = {OpenAI Math Release preprint + \href{https://github.com/openai/math/blob/main/preprints/A-rational-obstruction-to-strong-chromatic-splitting-at-height-three-September-25-2026/paper.pdf}{OAI:A-rational-obstruction-to-strong-chromatic-splitting-at-height-three-September-25-2026}}, + year = {2026} +} +``` diff --git a/preprints/A-rational-obstruction-to-strong-chromatic-splitting-at-height-three-September-25-2026/SUPPORT-COVERAGE.md b/preprints/A-rational-obstruction-to-strong-chromatic-splitting-at-height-three-September-25-2026/SUPPORT-COVERAGE.md new file mode 100644 index 0000000000000000000000000000000000000000..558d743abdcb3d24b06cb60070b9712630894321 --- /dev/null +++ b/preprints/A-rational-obstruction-to-strong-chromatic-splitting-at-height-three-September-25-2026/SUPPORT-COVERAGE.md @@ -0,0 +1,17 @@ +# Support coverage + +The manuscript contains the full argument for Theorem 1.1 and Corollary 1.2, including the integral coefficient conventions, the completed-theory application, the geometric comparison, Witt transfer, and ordinary finite-amplitude descent. The mathematical references in the paper supply the explicitly cited external inputs. The files below support only the stated portions of that argument. + +## Exact-arithmetic Chevalley--Eilenberg calculation + +`support/r-ce/ce_certificate.py` constructs the trivial-coefficient Chevalley--Eilenberg matrices \(d_0,\ldots,d_3\) for \(\mathfrak{sl}_2\) and \(\mathfrak{sl}_3\) from matrix commutators. It uses exact integers and rational `Fraction` arithmetic. Its complete generated arrays and transcript are `support/r-ce/ce_certificate.json` and `support/r-ce/ce_certificate.txt`. + +The checked identities are the characteristic-zero arithmetic used in Appendix A.1: the matrix shapes, rational ranks \((0,3,0,0)\) and \((0,8,20,35)\), the displayed nonzero minors, the adjacent products \(d_1d_0=0\), \(d_2d_1=0\), and \(d_3d_2=0\), closure of the trace cocycle, and its value \(2\) on the specified \(\mathfrak{sl}_2\) triple. The program does not calculate an augmented-image rank for the trace. Appendix A.1 proves non-boundary separately by restricting to the rank-two block. + +This support does not check characteristic-\(p\) ranks, the building proof in Appendix A.2, the continuous group/Lie comparison, the geometric and Witt arguments, ordinary descent, or the main theorem. It supplies no computation for another paper in this family. `BUILDING.md` gives the guarded replay command and the required Python assertion setting. + +The active guarded trio is in `support/r-ce/`. The separate directory `support/historical/r-ce/` preserves the earlier CE script, data, and transcript byte for byte, with the bounded arithmetic scope stated above. + +Exact artifact and source-locator identities are listed in [SUPPORT-COVERAGE.json](SUPPORT-COVERAGE.json). + +No Lean source or Lean verification is supplied for this article. The arithmetic support is not an independent review of the final manuscript. diff --git a/preprints/A-rational-obstruction-to-strong-chromatic-splitting-at-height-three-September-25-2026/build/main.tex b/preprints/A-rational-obstruction-to-strong-chromatic-splitting-at-height-three-September-25-2026/build/main.tex new file mode 100644 index 0000000000000000000000000000000000000000..65324c62fb728f11dd65b8fdd74b0627a52d8d3c --- /dev/null +++ b/preprints/A-rational-obstruction-to-strong-chromatic-splitting-at-height-three-September-25-2026/build/main.tex @@ -0,0 +1,33 @@ +\documentclass[11pt]{article} +\input{preamble} +\title{A rational obstruction to strong chromatic splitting\\at height three} +\author{OpenAI} +\date{September 25, 2026} +\begin{document} +\maketitle +\begin{abstract} +For every prime \(p\ge5\), the canonical map +\(L_0L_{K(3)}S\to L_0L_{K(2)}L_{K(3)}S\) for the sphere spectrum \(S\) +is nonzero on \(\pi_{-3}\). Consequently, the height-three strong chromatic +splitting formula is false in this range, even as an equivalence of +underlying \(E(2)\)-local spectra without specified summand maps. +\end{abstract} +\tableofcontents +\input{sections/01-introduction} +\input{sections/02-cohomology} +\input{sections/03-completed-tower} +\input{sections/04-modifications} +\input{sections/05-degree-three} +\input{sections/06-witt-transfer} +\input{sections/07-detection} +\input{sections/wedge-obstruction} +\input{sections/08-elementary-background} +\input{sections/09-frobenius} +\input{sections/10-oriented-projection} +\begingroup +\small +\raggedright +\bibliographystyle{plain} +\bibliography{references} +\endgroup +\end{document} diff --git a/preprints/A-rational-obstruction-to-strong-chromatic-splitting-at-height-three-September-25-2026/build/preamble.tex b/preprints/A-rational-obstruction-to-strong-chromatic-splitting-at-height-three-September-25-2026/build/preamble.tex new file mode 100644 index 0000000000000000000000000000000000000000..48362d8db912739ff0bfb075ec104e70994bed99 --- /dev/null +++ b/preprints/A-rational-obstruction-to-strong-chromatic-splitting-at-height-three-September-25-2026/build/preamble.tex @@ -0,0 +1,67 @@ +\usepackage[T1]{fontenc} +\usepackage[utf8]{inputenc} +\usepackage{lmodern} +\input{glyphtounicode} +\pdfgentounicode=1 +\usepackage[a4paper,margin=29mm,headheight=15pt]{geometry} +\usepackage{amsmath,amssymb,amsthm,mathtools,mathrsfs,bm} +\usepackage{microtype} +\usepackage{enumitem,booktabs,array,graphicx,needspace} +\usepackage{tikz-cd} +\usepackage{xcolor} +\usepackage{xurl} +\ifdefined\pdfinfoomitdate\pdfinfoomitdate=1\fi +\ifdefined\pdftrailerid\pdftrailerid{}\fi +\ifdefined\pdfsuppressptexinfo\pdfsuppressptexinfo=15\fi +\definecolor{linkblue}{RGB}{28,63,99} +\usepackage[colorlinks=true,linkcolor=linkblue,citecolor=linkblue,urlcolor=linkblue, + pdfencoding=auto,psdextra]{hyperref} +\usepackage[capitalise,noabbrev,nameinlink]{cleveref} +\numberwithin{equation}{section} +\newtheorem{theorem}{Theorem}[section] +\newtheorem{proposition}[theorem]{Proposition} +\newtheorem{lemma}[theorem]{Lemma} +\newtheorem{corollary}[theorem]{Corollary} +\theoremstyle{definition} +\newtheorem{definition}[theorem]{Definition} +\newtheorem{convention}[theorem]{Convention} +\theoremstyle{remark} +\newtheorem{remark}[theorem]{Remark} +\makeatletter +\newcommand{\reserveStatementSpace}{% + \if@nobreak\else\Needspace{6\baselineskip}\fi} +\makeatother +\AddToHook{env/theorem/before}{\reserveStatementSpace} +\AddToHook{env/proposition/before}{\reserveStatementSpace} +\AddToHook{env/lemma/before}{\reserveStatementSpace} +\AddToHook{env/corollary/before}{\reserveStatementSpace} +\AddToHook{env/definition/before}{\reserveStatementSpace} +\AddToHook{env/convention/before}{\reserveStatementSpace} +\AddToHook{env/remark/before}{\reserveStatementSpace} +\newcommand{\Q}{\mathbb Q} +\newcommand{\Z}{\mathbb Z} +\newcommand{\F}{\mathbb F} +\newcommand{\C}{\mathbb C} +\newcommand{\cO}{\mathcal O} +\newcommand{\cts}{\mathrm{cts}} +\DeclareMathOperator{\GL}{GL} +\DeclareMathOperator{\Lie}{Lie} +\DeclareMathOperator{\Gal}{Gal} +\DeclareMathOperator{\Hom}{Hom} +\DeclareMathOperator{\Aut}{Aut} +\DeclareMathOperator{\End}{End} +\DeclareMathOperator{\Spec}{Spec} +\DeclareMathOperator{\Spd}{Spd} +\DeclareMathOperator{\tr}{tr} +\DeclareMathOperator{\rank}{rank} +\DeclareMathOperator{\id}{id} +\DeclareMathOperator{\RHom}{RHom} +\newcommand{\Rlim}{R\!\varprojlim} +\setlist{topsep=4pt,itemsep=2pt,parsep=0pt} +\setlength{\emergencystretch}{2em} +\allowdisplaybreaks[1] +\setcounter{tocdepth}{1} +\hypersetup{pdftitle={A rational obstruction to strong chromatic splitting at height three}, + pdfsubject={Chromatic splitting, rational homotopy, and Fargues-Fontaine modifications}, + pdfauthor={OpenAI}} +\input{unicode-mappings} diff --git a/preprints/A-rational-obstruction-to-strong-chromatic-splitting-at-height-three-September-25-2026/build/references.bib b/preprints/A-rational-obstruction-to-strong-chromatic-splitting-at-height-three-September-25-2026/build/references.bib new file mode 100644 index 0000000000000000000000000000000000000000..eb40bcbc04e037b2d4c27a649c13d7dc7eb5ee85 --- /dev/null +++ b/preprints/A-rational-obstruction-to-strong-chromatic-splitting-at-height-three-September-25-2026/build/references.bib @@ -0,0 +1,311 @@ +@misc{BMR2026, + author = {Barthel, Tobias and Mann, Lucas and Ray, Rin and Schlank, Tomer M. and Senger, Andrew and Weinstein, Jared and Zhou, Xinyu}, + title = {On the chromatic splitting conjecture in coheight 1}, + year = {2026}, + note = {\href{https://math.bu.edu/people/jsweinst/ChromaticSplitting.pdf}{Preprint dated August 26, 2026}}, + url = {https://math.bu.edu/people/jsweinst/ChromaticSplitting.pdf} +} + +@misc{Clausen2025, + author = {Clausen, Dustin}, + title = {Duality and linearization for {$p$}-adic {Lie} groups}, + year = {2025}, + note = {\href{https://arxiv.org/abs/2506.18174v1}{arXiv:2506.18174v1}}, + eprint = {2506.18174}, + archivePrefix = {arXiv}, + url = {https://arxiv.org/abs/2506.18174v1} +} + +@misc{LurieEllipticII2018, + author = {Lurie, Jacob}, + title = {{Elliptic Cohomology II: Orientations}}, + year = {2018}, + note = {\href{https://www.math.ias.edu/~lurie/papers/Elliptic-II.pdf}{Author manuscript dated April 26, 2018}}, + url = {https://www.math.ias.edu/~lurie/papers/Elliptic-II.pdf} +} + +@misc{BB2019, + author = {Barthel, Tobias and Beaudry, Agn{\`e}s}, + title = {Chromatic structures in stable homotopy theory}, + year = {2019}, + note = {\href{https://arxiv.org/abs/1901.09004v2}{arXiv:1901.09004v2}, April 29, 2019}, + eprint = {1901.09004}, + archivePrefix = {arXiv}, + url = {https://arxiv.org/abs/1901.09004v2} +} + +@misc{BSSW2025, + author = {Barthel, Tobias and Schlank, Tomer M. and Stapleton, Nathaniel and Weinstein, Jared}, + title = {On the rationalization of the {$K(n)$}-local sphere}, + year = {2025}, + note = {\href{https://arxiv.org/abs/2402.00960v2}{arXiv:2402.00960v2}, September 9, 2025}, + eprint = {2402.00960}, + archivePrefix = {arXiv}, + url = {https://arxiv.org/abs/2402.00960v2} +} + +@article{CDN2021, + author = {Colmez, Pierre and Dospinescu, Gabriel and Nizio{\l}, Wies{\l}awa}, + title = {Integral {$p$}-adic {\'e}tale cohomology of {Drinfeld} symmetric spaces}, + journal = {Duke Mathematical Journal}, + volume = {170}, + number = {3}, + pages = {575--613}, + year = {2021}, + doi = {10.1215/00127094-2020-0084}, + note = {Cited preprint: \href{https://arxiv.org/abs/1905.11495v1}{arXiv:1905.11495v1}, May 27, 2019}, + url = {https://arxiv.org/abs/1905.11495v1} +} + +@misc{FS, + author = {Fargues, Laurent and Scholze, Peter}, + title = {Geometrization of the local {Langlands} correspondence}, + year = {2024}, + note = {\href{https://arxiv.org/abs/2102.13459v4}{arXiv:2102.13459v4}, November 27, 2024}, + eprint = {2102.13459}, + archivePrefix = {arXiv}, + url = {https://arxiv.org/abs/2102.13459v4} +} + +@incollection{Faltings2002, + author = {Faltings, Gerd}, + title = {A relation between two moduli spaces studied by {V. G. Drinfeld}}, + booktitle = {Algebraic number theory and algebraic geometry}, + series = {Contemporary Mathematics}, + volume = {300}, + pages = {115--129}, + publisher = {American Mathematical Society}, + year = {2002}, + note = {\href{https://www.sas.rochester.edu/mth/sites/doug-ravenel/otherpapers/faltings-97.pdf}{14-page author manuscript}, Introduction, p.~1}, + url = {https://www.sas.rochester.edu/mth/sites/doug-ravenel/otherpapers/faltings-97.pdf} +} + +@article{SW2013, + author = {Scholze, Peter and Weinstein, Jared}, + title = {Moduli of {$p$}-divisible groups}, + journal = {Cambridge Journal of Mathematics}, + volume = {1}, + number = {2}, + pages = {145--237}, + year = {2013}, + note = {Locators refer to the \href{https://www.math.uni-bonn.de/people/scholze/Moduli.pdf}{62-page author preprint}}, + url = {https://www.math.uni-bonn.de/people/scholze/Moduli.pdf} +} + +@misc{ALM2026, + author = {Ansch{\"u}tz, Johannes and {Le Bras}, Arthur-C{\'e}sar and Mann, Lucas}, + title = {A 6-functor formalism for solid quasi-coherent sheaves on the {Fargues--Fontaine} curve}, + year = {2026}, + note = {\href{https://arxiv.org/abs/2412.20968v2}{arXiv:2412.20968v2}, submitted September 8, 2026; manuscript dated September 10, 2026}, + eprint = {2412.20968}, + archivePrefix = {arXiv}, + url = {https://arxiv.org/abs/2412.20968v2} +} + +@misc{ScholzeDiamonds, + author = {Scholze, Peter}, + title = {{\'E}tale cohomology of diamonds}, + year = {2026}, + note = {\href{https://arxiv.org/abs/1709.07343v4}{arXiv:1709.07343v4}, submitted April 14, 2026; manuscript dated April 15, 2026}, + eprint = {1709.07343}, + archivePrefix = {arXiv}, + url = {https://arxiv.org/abs/1709.07343v4} +} + +@misc{Tang2026, + author = {Tang, Jiacheng}, + title = {Profinite and solid cohomology}, + year = {2026}, + note = {\href{https://arxiv.org/abs/2410.08933v2}{arXiv:2410.08933v2}, January 27, 2026}, + eprint = {2410.08933}, + archivePrefix = {arXiv}, + url = {https://arxiv.org/abs/2410.08933v2} +} + +@misc{Anschutz2020, + author = {Ansch{\"u}tz, Johannes}, + title = {Solid group cohomology}, + year = {2020}, + note = {\href{https://janschuetz.perso.math.cnrs.fr/skripte/homology_profinite.pdf}{La Tourette notes}, September 2020. Written by Ansch{\"u}tz, reread by Arthur-C{\'e}sar Le Bras; talk given by Le Bras}, + url = {https://janschuetz.perso.math.cnrs.fr/skripte/homology_profinite.pdf} +} + +@misc{ScholzeAnalyticGeometry, + author = {Scholze, Peter}, + title = {Lectures on Analytic Geometry}, + year = {2019}, + note = {\href{https://www.math.uni-bonn.de/people/scholze/Analytic.pdf}{Bonn course notes}, winter term 2019/20; all results joint with Dustin Clausen. Locators refer to the 110-page author PDF accessed September 27, 2026}, + url = {https://www.math.uni-bonn.de/people/scholze/Analytic.pdf} +} + +@article{Lazard1965, + author = {Lazard, Michel}, + title = {Groupes analytiques {$p$}-adiques}, + journal = {Publications Math{\'e}matiques de l'IH{\'E}S}, + volume = {26}, + pages = {5--219}, + year = {1965}, + doi = {10.1007/BF02684303}, + note = {\href{https://numdam.org/articles/10.1007/BF02684303/}{DOI: 10.1007/BF02684303}}, + url = {https://numdam.org/articles/10.1007/BF02684303/} +} + +@article{HKN2011, + author = {Huber, Annette and Kings, Guido and Naumann, Niko}, + title = {Some complements to the {Lazard} isomorphism}, + journal = {Compositio Mathematica}, + volume = {147}, + number = {1}, + pages = {235--262}, + year = {2011}, + doi = {10.1112/S0010437X10004884}, + note = {\href{https://doi.org/10.1112/S0010437X10004884}{Published version}. Statement numbering refers to \href{https://arxiv.org/abs/0904.3863v3}{arXiv:0904.3863v3}, March 24, 2010}, + url = {https://doi.org/10.1112/S0010437X10004884} +} + +@article{ChevalleyEilenberg1948, + author = {Chevalley, Claude and Eilenberg, Samuel}, + title = {Cohomology theory of {Lie} groups and {Lie} algebras}, + journal = {Transactions of the American Mathematical Society}, + volume = {63}, + number = {1}, + pages = {85--124}, + year = {1948}, + doi = {10.1090/S0002-9947-1948-0024908-8}, + note = {\href{https://www.sas.rochester.edu/mth/sites/doug-ravenel/otherpapers/chev-eilen.pdf}{Published text}; DOI: 10.1090/S0002-9947-1948-0024908-8}, + url = {https://www.sas.rochester.edu/mth/sites/doug-ravenel/otherpapers/chev-eilen.pdf} +} + +@article{Koszul1950, + author = {Koszul, Jean-Louis}, + title = {Homologie et cohomologie des alg{\`e}bres de {Lie}}, + journal = {Bulletin de la Soci{\'e}t{\'e} Math{\'e}matique de France}, + volume = {78}, + pages = {65--127}, + year = {1950}, + doi = {10.24033/bsmf.1410}, + note = {\href{https://numdam.org/articles/10.24033/bsmf.1410/}{DOI: 10.24033/bsmf.1410}}, + url = {https://numdam.org/articles/10.24033/bsmf.1410/} +} + +@article{HS1999, + author = {Hovey, Mark and Strickland, Neil P.}, + title = {Morava {$K$}-theories and localisation}, + journal = {Memoirs of the American Mathematical Society}, + volume = {139}, + number = {666}, + pages = {viii+100}, + year = {1999}, + doi = {10.1090/memo/0666}, + note = {\href{https://www.sas.rochester.edu/mth/sites/doug-ravenel/otherpapers/kn.pdf}{Author manuscript}; DOI: 10.1090/memo/0666}, + url = {https://www.sas.rochester.edu/mth/sites/doug-ravenel/otherpapers/kn.pdf} +} + +@misc{Bosco2023, + author = {Bosco, Guido}, + title = {On the {$p$}-adic pro-{\'e}tale cohomology of {Drinfeld} symmetric spaces}, + year = {2023}, + note = {\href{https://arxiv.org/abs/2110.10683v2}{arXiv:2110.10683v2}, June 9, 2023}, + eprint = {2110.10683}, + archivePrefix = {arXiv}, + url = {https://arxiv.org/abs/2110.10683v2} +} + +@incollection{Hovey1995, + author = {Hovey, Mark}, + title = {Bousfield localization functors and {Hopkins}' chromatic splitting conjecture}, + booktitle = {The {\v C}ech Centennial}, + series = {Contemporary Mathematics}, + volume = {181}, + pages = {225--250}, + publisher = {American Mathematical Society}, + year = {1995}, + note = {\href{https://www.sas.rochester.edu/mth/sites/doug-ravenel/otherpapers/chromatic-splitting.pdf}{Author manuscript}, July 1993}, + url = {https://www.sas.rochester.edu/mth/sites/doug-ravenel/otherpapers/chromatic-splitting.pdf} +} + +@article{Beaudry2017, + author = {Beaudry, Agn{\`e}s}, + title = {The chromatic splitting conjecture at {$n=p=2$}}, + journal = {Geometry \& Topology}, + volume = {21}, + number = {6}, + pages = {3213--3230}, + year = {2017}, + doi = {10.2140/gt.2017.21.3213}, + note = {\href{https://msp.org/gt/2017/21-6/gt-v21-n6-p02-p.pdf}{Published text}; DOI: 10.2140/gt.2017.21.3213}, + url = {https://msp.org/gt/2017/21-6/gt-v21-n6-p02-p.pdf} +} + +@article{Behrens2012, + author = {Behrens, Mark}, + title = {The homotopy groups of {$S_{E(2)}$} at {$p\geq 5$} revisited}, + journal = {Advances in Mathematics}, + volume = {230}, + number = {2}, + pages = {458--492}, + year = {2012}, + doi = {10.1016/j.aim.2012.02.023}, + note = {Cited locators refer to \href{https://arxiv.org/abs/1002.0807v4}{arXiv:1002.0807v4}, March 23, 2012}, + eprint = {1002.0807}, + archivePrefix = {arXiv}, + url = {https://arxiv.org/abs/1002.0807v4} +} + +@article{ShimomuraYabe1995, + author = {Shimomura, Katsumi and Yabe, Atsuko}, + title = {The homotopy groups {$\pi_*(L_2S^0)$}}, + journal = {Topology}, + volume = {34}, + number = {2}, + pages = {261--289}, + year = {1995}, + note = {\href{https://www.sas.rochester.edu/mth/sites/doug-ravenel/otherpapers/shimomura-yabe.pdf}{Published text}}, + url = {https://www.sas.rochester.edu/mth/sites/doug-ravenel/otherpapers/shimomura-yabe.pdf} +} + +@article{GHM2014, + author = {Goerss, Paul G. and Henn, Hans-Werner and Mahowald, Mark}, + title = {The rational homotopy of the {$K(2)$}-local sphere and the chromatic splitting conjecture for the prime 3 and level 2}, + journal = {Documenta Mathematica}, + volume = {19}, + pages = {1271--1290}, + year = {2014}, + note = {\href{https://ems.press/content/serial-article-files/26260}{Published text}; Theorem 5.11 and its proof are on page 1288}, + url = {https://ems.press/content/serial-article-files/26260} +} + +@article{BGH2022, + author = {Beaudry, Agn{\`e}s and Goerss, Paul G. and Henn, Hans-Werner}, + title = {Chromatic splitting for the {$K(2)$}-local sphere at {$p=2$}}, + journal = {Geometry \& Topology}, + volume = {26}, + number = {1}, + pages = {377--476}, + year = {2022}, + doi = {10.2140/gt.2022.26.377}, + note = {Cited locators refer to \href{https://arxiv.org/abs/1712.08182v4}{arXiv:1712.08182v4}, submitted October 6, 2020; manuscript dated October 8, 2020}, + eprint = {1712.08182}, + archivePrefix = {arXiv}, + url = {https://arxiv.org/abs/1712.08182v4} +} + +@misc{LRZ2024, + author = {Li, Shizhang and Reinecke, Emanuel and Zavyalov, Bogdan}, + title = {Relative {Poincar\'e} duality in nonarchimedean geometry}, + year = {2024}, + note = {\href{https://arxiv.org/abs/2410.08200v1}{arXiv:2410.08200v1}}, + url = {https://arxiv.org/abs/2410.08200v1} +} + +@article{Venjakob2002, + author = {Venjakob, Otmar}, + title = {On the structure theory of the {Iwasawa} algebra of a {$p$}-adic {Lie} group}, + journal = {Journal of the European Mathematical Society}, + volume = {4}, + pages = {271--311}, + year = {2002}, + doi = {10.1007/s100970100038}, + note = {\href{https://ems.press/content/serial-article-files/31498}{DOI: 10.1007/s100970100038}}, + url = {https://ems.press/content/serial-article-files/31498} +} diff --git a/preprints/A-rational-obstruction-to-strong-chromatic-splitting-at-height-three-September-25-2026/build/sections/01-introduction.tex b/preprints/A-rational-obstruction-to-strong-chromatic-splitting-at-height-three-September-25-2026/build/sections/01-introduction.tex new file mode 100644 index 0000000000000000000000000000000000000000..aced45dd24ccd61fdd34a59f0f807274c3bdfadd --- /dev/null +++ b/preprints/A-rational-obstruction-to-strong-chromatic-splitting-at-height-three-September-25-2026/build/sections/01-introduction.tex @@ -0,0 +1,263 @@ +\section{Introduction} +\label{sec:introduction} + +Let \(S\) be the sphere spectrum, fix a prime \(p\), and write +\(S_p^\wedge\) for its \(p\)-completion. We use \(L_i\) for localization +at Johnson--Wilson theory \(E(i)\) and \(L_{K(i)}\) for localization at +Morava \(K\)-theory of height \(i\), all at \(p\). In particular, \(L_0\) +is rationalization. Chromatic splitting concerns the lower-height +overlap \(L_{n-1}L_{K(n)}S\) in the fracture square for \(L_nS\). +Hopkins' strong formulation, recorded by Hovey +\cite[Conjecture~4.2]{Hovey1995}, prescribes an assembly of this overlap +from localized spheres. + +At height three, the proposed underlying spectrum is +\begin{equation}\label{eq:proposed-wedge} +\begin{aligned} + \mathcal W_3={}&L_2\bigl(S_p^\wedge\vee\Sigma^{-1}S_p^\wedge\bigr)\\ + &{}\vee L_1\bigl(\Sigma^{-3}S_p^\wedge\vee\Sigma^{-4}S_p^\wedge\bigr)\\ + &{}\vee L_0\bigl(\Sigma^{-5}S_p^\wedge\vee\Sigma^{-6}S_p^\wedge + \vee\Sigma^{-8}S_p^\wedge\vee\Sigma^{-9}S_p^\wedge\bigr). +\end{aligned} +\end{equation} +The strong prediction is an equivalence +\(L_2L_{K(3)}S\simeq\mathcal W_3\), with specified maps, including the +canonical unit on the first summand. The revised formulation of +Barthel--Beaudry retains this odd-prime prediction +\cite[Conjecture~6.3 and Remark~6.4]{BB2019}. We disprove even the +underlying equivalence of \(E(2)\)-local spectra. + +The obstruction is a natural map. For every spectrum \(X\), the +\(K(2)\)-localization unit induces +\[ + \tau_X:L_0X\longrightarrow L_0L_{K(2)}X. +\] + +\begin{theorem}\label{thm:main} +For every prime \(p\geq5\), the canonical map +\begin{equation}\label{eq:main-map} + \tau_{L_{K(3)}S}:L_0L_{K(3)}S\longrightarrow + L_0L_{K(2)}L_{K(3)}S +\end{equation} +is nonzero on \(\pi_{-3}\). +\end{theorem} + +\begin{corollary}\label{cor:strong} +For every prime \(p\geq5\), there is no equivalence of \(E(2)\)-local +spectra \(L_2L_{K(3)}S\simeq\mathcal W_3\), with \(\mathcal W_3\) as in +\eqref{eq:proposed-wedge}. Thus the height-three strong chromatic +splitting formula is false in this range, whether or not an equivalence +is required to preserve its specified unit summand. +\end{corollary} + +\subsection{Why the canonical map matters} + +The rational homotopy groups of the \(K(n)\)-local sphere are known +for every positive height \(n\) and every prime. Barthel--Schlank--Stapleton--Weinstein +identify their algebra with an exterior algebra over \(\Q_p\), with +primitive generators in degrees \(1-2i\), \(1\leq i\leq n\) +\cite[Theorem~A]{BSSW2025}. At height three the resulting degrees are +\[ + 0,-1,-3,-4,-5,-6,-8,-9, +\] +exactly the shifts appearing in \eqref{eq:proposed-wedge}. Thus the +rational degree pattern does not distinguish the proposed wedge from +the actual overlap. + +The natural transformation \(\tau\) does distinguish them. In +\(\mathcal W_3\), a degree-\(-3\) class can occur only in the +lower-height part, which is \(K(2)\)-acyclic. The two top-height +summands have no rational homotopy in that degree. It follows that +\(\tau_{\mathcal W_3}\) vanishes on \(\pi_{-3}\). The formal argument +in \cref{sec:wedge-obstruction} also compares the maps for +\(L_{K(3)}S\) and \(L_2L_{K(3)}S\). Naturality makes the conclusion +valid for any hypothetical equivalence, with no assumption on its +individual summand maps. + +At height two, Hovey records Hopkins' splitting +for \(p>3\), based on Shimomura--Yabe's calculations +\cite[\S4]{Hovey1995}, \cite[Theorem~2.4]{ShimomuraYabe1995}. Behrens later reorganized those calculations +and displayed the predicted overlap homotopy groups +\cite[Remark~7.8]{Behrens2012}. Goerss--Henn--Mahowald proved the +splitting at \(p=3\) \cite[Theorem~5.11]{GHM2014}. At \(p=2\), +Beaudry disproved the original formula \cite[Theorem~1.4]{Beaudry2017}, +while Beaudry--Goerss--Henn subsequently proved a refined splitting +with additional Moore-spectrum summands and retained the weak unit +splitting \cite[Theorem~1.1.6]{BGH2022}. + +The canonical map already controls the height-two argument of +Goerss--Henn--Mahowald. At \(p=3\), their proof identifies the rational +map from the \(K(2)\)-local sphere to its \(K(1)\)-localization with +projection onto the degrees zero and minus one, followed by canonical +localization; the fracture square then reconstructs the splitting +\cite[proof of Theorem~5.11, p.~1288]{GHM2014}. +Theorem~\ref{thm:main} finds a nonzero degree-minus-three component at +height three, which obstructs \(\mathcal W_3\). A finite filtration +with these cofibers may retain nonzero attaching maps that a wedge +would discard. The existence of a retraction of the canonical unit +in weak chromatic splitting remains a separate question. + +\subsection{A detecting spectrum} + +The proof of Theorem~\ref{thm:main} has two tasks: construct a map +from \(L_{K(3)}S\) to an ordinary \(K(2)\)-local spectrum, and prove +that this map preserves a nonzero rational degree-\(-3\) class. Locality +then forces the map to factor through the canonical localization unit. +The class is detected in degree-three stabilizer cohomology; the main +argument compares its height-three and height-two realizations. + +Fix \(p\geq5\), put \(k=\F_{p^6}\), and choose \(C=\C_p\) with +compatible constant-field data. For \(n=2,3\), let \(\Gamma_n\) be the +height-\(n\) Honda formal group over \(k\), and put +\[ + E_n=E(k,\Gamma_n),\qquad + P_n=\Aut_k(\Gamma_n)=\cO_{D_n}^{\times}. +\] +Here \(D_n\) is the division algebra over \(\Q_p\) of invariant +\(1/n\), and \(P_n\) is the nonextended stabilizer. Set +\[ + G=P_3\times P_2,\qquad + \Delta=\Gal(k/\F_p),\qquad G_3=P_3\rtimes\Delta. +\] +The field \(k\) contains the endomorphism fields of both Honda groups. + +On the height-exactly-two stratum \(k((u_2))\) of the height-three +deformation space, the Gross--Torii extension identifies the connected +height-two group with \(\Gamma_2\). Its valued-field completion +\(\widehat L\) carries the two framing actions, hence an action of +\(G\). The solid Lubin--Tate construction of +Barthel--Mann--Ray--Schlank--Senger--Weinstein--Zhou \cite{BMR2026} +gives the connected completed theory +\[ + \widehat B^\square=E^\square(\widehat L,(\Gamma_2)_{\widehat L}), + \qquad \mathcal D=\bigl((\widehat B^\square)^{hG}\bigr)(*). +\] +The superscript \(\square\) records the solid structure and continuous +action; \((*)\) is evaluation in ordinary spectra. Fixed points are +formed before this evaluation. + +Section~\ref{sec:completed-tower} constructs equivariant maps +\(E_3^\square\to\widehat B^\square\leftarrow E_2^\square\). +The first requires a section on oriented deformations, obtained by +adjoining the marked \'etale factor to the connected deformation. +The second is the canonical connected base-change map. Restriction +from \(G_3\) to \(P_3\), inflation along \(G\to P_3\), and the first +map define \(g:L_{K(3)}S\to\mathcal D\). Every profinite evaluation +of \(\widehat B^\square\) is \(K(2)\)-local, so its bar totalization +\(\mathcal D\) is too. Thus +\begin{equation}\label{eq:overview-detector} + L_{K(3)}S\xrightarrow{\eta}L_{K(2)}L_{K(3)}S + \xrightarrow{\overline g}\mathcal D, + \qquad g=\overline g\eta. +\end{equation} +The source identification and these actual maps are established in +\cref{prop:completed-theory,lem:local-detector}. + +Write \(N_3:P_3\to\Z_p^\times\) for reduced norm and +\(P_3^1=N_3^{-1}(\mu_{p-1})\). The completed comparison +\[ + E_2^\square\xrightarrow{\sim}(\widehat B^\square)^{hP_3^1} +\] +reduces the detector's coefficients to height two. We prove the +individual-coefficient comparison as well as this instance of +\cite[Theorem~5.2.6]{BMR2026}. The corresponding equivalence for the +uncompleted half sphere remains a separate conjecture +\cite[Conjectures~1.1.1 and~5.1.13]{BMR2026}, with a conditional +coheight-one consequence in \cite[Proposition~5.1.17]{BMR2026}. +The completed target in \eqref{eq:overview-detector} suffices here. + +\subsection{Transporting the primitive} + +The starting classes are nonzero elements +\(e_{P_n}\in H^3_{\cts}(P_n,\Q_p)\), for \(n=2,3\). +Under rational Lie comparison they are represented, up to nonzero +scalars, by +\[ + (U,V,W)\longmapsto\tr_{\mathrm{red}}\bigl(U[V,W]\bigr). +\] +This is the classical cocycle attached to an invariant bilinear form +\cite[\S21]{ChevalleyEilenberg1948}, \cite[\S11]{Koszul1950}. +Section~\ref{sec:cohomology} constructs these classes and their linear +counterparts \(e_{J_n}\) for +\[ + J_n=\{g\in\GL_n(\Q_p):|\det g|_p=1\}. +\] +The trace class restricts nontrivially from rank three to rank two; +the same compatibility holds for the relevant parabolic subgroup, +with determinant lattices retained. The low-rank calculation and the building argument are given +in \cref{app:background}. + +To compare division-algebra and linear classes, we use a line +modification of a form of the rank-\(n\), degree-one Fargues--Fontaine +bundle \(\mathcal E_n^0=\cO(1/n)\), retaining a determinant lattice. +The stack \(\mathcal M_n\) of such modifications has two presentations: +\[ + \mathcal M_n\simeq[\mathbb P_C^{n-1,\diamondsuit}/P_n] + \simeq[\Omega_C^{n-1,\diamondsuit}/J_n]. +\] +Here projective space parametrizes quotient lines of the untilt fiber, +and Drinfeld space \(\Omega_C^{n-1}\) is the complement of the +\(\Q_p\)-rational hyperplanes. The relative bundle correspondence and +basic-stratum theorems of Fargues--Scholze provide the torsors in +families \cite[Theorem~II.2.19, Corollary~II.2.20, and +Theorems~III.2.4 and~III.4.5]{FS}. Keeping a lattice in the determinant +local system gives the structure group \(J_n\). These presentations +follow Faltings's Lubin--Tate/Drinfeld duality +\cite[Introduction]{Faltings2002}, in the perfectoid form of +Scholze--Weinstein with its exchanged period maps +\cite[Theorem~7.2.3]{SW2013}. Section~\ref{sec:modifications} +constructs the determinant-lattice comparison used here. + +For the geometric cohomology below, write \(H^i_{\mathrm b}\) for +integral cohomology followed by inversion of \(p\). Integral cohomology +is the derived inverse limit of finite \(p\)-power coefficient complexes, +as specified in Section~\ref{sec:cohomology}. +The two presentations have complementary roles. The projective bundle +formula singles out a one-dimensional Galois-invariant line in +\(H^3_{\mathrm b}(\mathcal M_n)\), containing the nonzero image of \(e_{P_n}\). +Using the Galois weights in the integral Drinfeld cohomology theorem +of Colmez--Dospinescu--Nizioł \cite[Theorems~1.1 and~5.1]{CDN2021}, +we prove that the image of \(e_{J_n}\) is also nonzero and belongs +to that line. Thus the two +classes are proportional. + +The completed tower carries a surjection from the rank-three bundle +to the rank-two bundle. A common line modification turns this into +an exact sequence of rational local systems of ranks \(1,3,2\). +Parabolic restriction and projective-bundle injectivity then give +\[ + e_{P_3}|_{Y_C}=a\,e_{P_2}|_{Y_C},\qquad a\in\Q_p^\times, + \quad + Y_C=[\Spd\widehat L/G]\times_{\Spd k}\Spd C. +\] +Section~\ref{sec:degree-three} proves this relation. Both classes on +\(Y_C\) could still vanish, so nonvanishing requires a further step. + +Section~\ref{sec:witt-transfer} first reflects the relation from +\(Y_C\) to \(Y=[\Spd\widehat L/G]\) with Witt coefficients. The +maps from constants become equivalences after \(P_3^1\)-descent; +a bounded finite free complex proves injectivity under the change +of constants \(W(k)\to W(C^\flat)\). The continuous additive +retraction onto Witt constants of Barthel--Schlank--Stapleton--Weinstein +\cite[Proposition~2.5.1 and Corollary~2.5.9]{BSSW2025} then proves +nonvanishing of the height-two class in deformation coefficients. +If \(c_n\) is the image of \(e_{P_n}\) under projection from \(G\) +and the coefficient unit, the result is +\[ + c_3=a\,c_2\ne0 + \quad\text{in }H^3(G,\pi_0\widehat B^\square)[1/p]. +\] + +Finally, Section~\ref{sec:detection} uses the descent spectral +sequences for the actual map \(g\). A finite cohomological bound +produces a finite integral filtration in each ordinary homotopy +degree. After rationalization, central scalar automorphisms kill +all coefficient degrees \(t\ne0\). With \(s\) denoting group-cohomological +degree, the nonzero coefficient map in bidegree \((s,t)=(3,0)\) therefore survives to degree \(-3\) ordinary +homotopy. The factorization \eqref{eq:overview-detector} proves the +main theorem, and Section~\ref{sec:wedge-obstruction} gives its +wedge consequence. + +Continuous invariants throughout the proof retain their solid +coefficient objects, and inversion of \(p\) follows integral cohomology. +This order matters for the noncompact spaces and groups that occur here. diff --git a/preprints/A-rational-obstruction-to-strong-chromatic-splitting-at-height-three-September-25-2026/build/sections/02-cohomology.tex b/preprints/A-rational-obstruction-to-strong-chromatic-splitting-at-height-three-September-25-2026/build/sections/02-cohomology.tex new file mode 100644 index 0000000000000000000000000000000000000000..b37da1b84bad7db1d434e42058f46ca566844b01 --- /dev/null +++ b/preprints/A-rational-obstruction-to-strong-chromatic-splitting-at-height-three-September-25-2026/build/sections/02-cohomology.tex @@ -0,0 +1,658 @@ +\section{Integral cohomology and degree-three group classes} +\label{sec:cohomology} + +Our goal is to construct the degree-three classes that will be transported +through the two-height tower, together with the exact restriction maps +between their linear realizations. We first fix the integral coefficient +convention and prove the finite-resolution statement that will also +control the later Galois-weight and ordinary-descent arguments. + +\subsection{Coefficients and continuous descent} + +Put $\Lambda_m=\Z/p^m$. For a small v-stack $X$ occurring below, put +\[ + R\Gamma_{\mathrm{int}}(X) + =\Rlim_m R\Gamma_v(X,\Lambda_m),\qquad + H^i_{\mathrm{int}}(X)=H^iR\Gamma_{\mathrm{int}}(X),\qquad + H^i_{\mathrm b}(X)=H^i_{\mathrm{int}}(X)[1/p]. +\] +For a locally profinite group $G$, define separately in $D(\mathrm{Ab})$ +\begin{equation}\label{eq:integral-group-cochains} +\begin{gathered} + R\Gamma_{\mathrm{int}}(G)=\Rlim_m C_{\cts}^{\bullet}(G,\Lambda_m), + \\ + H^i_{\mathrm{int}}(G)=H^iR\Gamma_{\mathrm{int}}(G), + \qquad H^i_{\mathrm b}(G)=H^i_{\mathrm{int}}(G)[1/p]. +\end{gathered} +\end{equation} +Here $C_{\cts}^{\bullet}$ is the inhomogeneous continuous cochain complex, +with trivial discrete coefficients. The standard resolution in the +condensed classifying topos $BG_{\mathrm{pro\acute et}}$, the topos of +condensed sets with $G$-action, gives the natural comparison +\[ + C_{\cts}^{\bullet}(G,\Lambda_m)\simeq + \RHom_{\Lambda_m[\underline G]\text{-}\mathrm{Cond}} + (\underline{\Lambda_m},\underline{\Lambda_m}). +\] +This is the external derived Hom; its continuous-cochain calculation for +these solid topological coefficients is the standard-resolution proof in +\cite[\S2, pp.~5--7, Lemmas~2.1--2.2]{Anschutz2020}. +We next construct the maps from this group complex to geometric quotients. + +For a condensed $G$-complex $K$, write $R\Gamma(G,K)$ for its +\emph{internal} derived condensed invariants, and $R\Gamma(G,K)(*)$ +for their point value, which is the external derived Hom +\cite[\S2, p.~8]{Anschutz2020}. Topological coefficients in this +notation are implicitly condensed. Reduction on finite cochains is +degreewise surjective: a function lifts by composing with a section of +the finite coefficient reduction. Compatible finite continuous +functions are exactly continuous $\Z_p$-valued functions. Hence +\begin{equation}\label{eq:group-cochain-bridge} + R\Gamma_{\mathrm{int}}(G)\simeq C_{\cts}^{\bullet}(G,\Z_p) + \simeq R\Gamma(G,\underline{\Z_p})(*). +\end{equation} +For the last comparison, $\Z_p$ with its $p$-adic topology is solid by +\cite[\S1, pp.~4--5]{Anschutz2020}, so the same standard-resolution +proof applies. We do not identify the internal invariant object with +the discrete condensation of the external cochain complex for a +noncompact group; see \cite[Lemma~2.5 and Remark~2.6]{Anschutz2020}. +The comparisons in \eqref{eq:group-cochain-bridge} are natural for +continuous group homomorphisms. +For compact $G$, continuous functions $G^r\to\Q_p$ are bounded, so +\[ + C_{\cts}(G^r,\Z_p)[1/p]=C_{\cts}(G^r,\Q_p). +\] +Consequently $H^*_{\mathrm b}(G)$ agrees with ordinary continuous +$\Q_p$-cohomology for compact $G$. We use this last identification +only in that case. + +Tate twists are taken at each finite level before taking the derived +limit. All maps of coefficients and all descent maps are formed +integrally. In particular, $H^i_{\mathrm b}(X)$ is not defined as +$H^i_v(X,\widehat\Z_p[1/p])$. Moving inversion of $p$ before +cohomology can change the result for a noncompact space or group. + +For an analytic adic space in this paper, its \'etale site agrees with +the \'etale site of its diamond +\cite[Lemma~15.6]{ScholzeDiamonds}. On a locally spatial diamond, +bounded-below \'etale complexes have the same derived cohomology on +the diamond \'etale, quasi-pro-\'etale, and v-sites +\cite[Proposition~14.10]{ScholzeDiamonds}. We apply these statements +to finite constant coefficients, one level at a time. The integral +analytic pro-\'etale comparison used for Drinfeld space is the +separate comparison in \cite[\S5]{CDN2021}. Quotient stacks are +handled by descent for their covers. Only after the finite-level +comparisons do we take $\Rlim_m$. + +Here is the map from group classes to geometric quotients. For a small +v-stack base $B$, put $B_BG=[B/\underline G]$, where $G$ acts trivially +on $B$. A $G$-space $X$ over $B$ gives a map $[X/G]\to B_BG$; a +$G$-torsor $T\to Z$ over $B$ gives its classifying map $Z\to B_BG$. +At each finite level define +\begin{equation}\label{eq:finite-constant-cochains} +\begin{split} + u_{X,G,m}:C_{\cts}^{\bullet}(G,\Lambda_m) + &\longrightarrow + \operatorname{Tot}_{a}R\Gamma_v + (X\times\underline{G^a},\Lambda_m)\\ + &\simeq R\Gamma_v([X/G],\Lambda_m). +\end{split} +\end{equation} +In degree $a$, pull a locally constant function on $G^a$ back along +the projection from $X\times\underline{G^a}$. Its finitely many +clopen fibers define a section of the finite constant sheaf, followed +by the canonical degree-zero truncation map to derived sections. +These maps commute with +the faces and degeneracies of the action groupoid. The last +equivalence is its \v{C}ech descent. The case $X=B$ defines the +universal map into $B_BG$, and the general map is its pullback along +$[X/G]\to B_BG$. Likewise the map obtained from the \v{C}ech nerve +of $T\to Z$ is the pullback along its classifying map. + +If $\phi:G'\to G$ is a continuous homomorphism and $f:X'\to X$ is +$\phi$-equivariant over a base map, this construction gives a +commutative square +\begin{equation}\label{eq:constant-cochain-naturality} +\begin{tikzcd}[column sep=large] + C_{\cts}^{\bullet}(G,\Lambda_m) + \arrow[r,"u_{X,G,m}"] \arrow[d,"\phi^*"'] + &R\Gamma_v([X/G],\Lambda_m)\arrow[d,"{[f/\phi]^*}"]\\ + C_{\cts}^{\bullet}(G',\Lambda_m) + \arrow[r,"u_{X',G',m}"'] + &R\Gamma_v([X'/G'],\Lambda_m). +\end{tikzcd} +\end{equation} +It commutes already in each finite cosimplicial degree. The same +naturality holds for coefficient reduction, maps of finite coefficient +sheaves, and automorphisms of the base. Taking $\Rlim_m$ defines +$u_{X,G,\mathrm{int}}:R\Gamma_{\mathrm{int}}(G)\to +R\Gamma_{\mathrm{int}}([X/G])$; denote the induced maps on cohomology +after $[1/p]$ by $u_{X,G,\mathrm b}$. No acyclicity of $B$ or $X$ +is used in this construction. + +There is one restricted case in which the universal map is an +equivalence. Let $B_C=\Spd C$ for the algebraically closed perfectoid +field $C=\C_p$, and abbreviate $B_CG=[B_C/\underline G]$. +For every profinite set $S$, the actual constants map +is an equivalence +\begin{equation}\label{eq:geometric-point-constants} + C_{\cts}(S,\Lambda_m)[0]\xrightarrow{\ \sim\ } + R\Gamma_v(B_C\times\underline S,\Lambda_m). +\end{equation} +For the empty set both sides are zero. Otherwise write $S=\varprojlim_i S_i$ +over its finite quotients, including the one-point quotient. The example +after \cite[Definition~7.8]{ScholzeDiamonds} constructs the affinoid +pro-\'etale limit $\operatorname{Spa}(C,\cO_C)\times S$ of the finite +unions indexed by $S_i$; the associated-diamond construction and +\cite[Example~11.12 and Lemma~15.2]{ScholzeDiamonds} identify its +diamond with $B_C\times\underline S$. +The point $\operatorname{Spa}(C,\cO_C)$ is strictly totally disconnected +by \cite[Definition~7.15 and Proposition~7.16]{ScholzeDiamonds}, so its +\'etale covers split and finite constant coefficients have no higher +cohomology there. The preceding \cite[Lemma~15.6]{ScholzeDiamonds} +comparison transfers this assertion to the finite unions of their +diamonds. The continuity statement +\cite[Proposition~14.9]{ScholzeDiamonds} computes the \'etale +cohomology on the limit as the filtered colimit of that on the finite +unions. Its degree-zero group is +$\varinjlim_i C(S_i,\Lambda_m)=C_{\cts}(S,\Lambda_m)$ and its higher +groups vanish. Proposition~14.10 of the same source, valid for the +coefficient ring $\Lambda_m$, transfers this calculation to the v-site. +Every comparison is induced by the constants map, and is natural in +$S$ and in automorphisms of $C$. + +For compact $G$, apply \eqref{eq:geometric-point-constants} to +$S=G^a$ in every degree of \eqref{eq:finite-constant-cochains}. +It follows that $u_{B_C,G,m}$ is an equivalence. The same argument +applies to the finite \'etale fiber of $(\Z/p^m)(t)$ over $C$, +where $p$ is invertible, retaining its arithmetic action. The +compatible Tate line is finite free of rank one, so its tensor factor +commutes with the cochain terms and their derived coefficient limit. +Thus the integral comparison for +$B_CG$ is also compatible with Tate twists and arithmetic Galois +actions, without a choice of basis for the Tate fiber. This +geometric-point calculation is used below for the compact groups +$P_n$; it is not asserted over $\Spd k$. + +For descent with geometric coefficients it is useful to retain +profinite parameters. Denote by $\mathscr C_{X,m}$ the condensed +derived complex whose derived value on a profinite set $S$ is +\[ + \mathscr C_{X,m}(S)=R\Gamma_v(X\times\underline S,\Lambda_m), + \qquad \mathscr C_X=\Rlim_m\mathscr C_{X,m}. +\] +Descent for profinite covers gives these condensed complexes. The value +of $\mathscr C_X$ at the point is $R\Gamma_{\mathrm{int}}(X)$. +A group acting on $X$ acts continuously +on this complex. The coefficient complexes used below are +\emph{derived solid}: they lie in the derived category of solid +condensed modules. Here is a concrete check of this condition. +On an affinoid perfectoid cover, the characteristic-$p$ structure +sheaf $\cO^\flat$ has no higher cohomology, and its values with +profinite parameters are continuous-function modules. Their complete +linear topology expresses them as limits of discrete modules, so +they are solid. Perfectoid hypercover descent computes the general +$\cO^\flat$ complex by limits of these objects; this is also the +construction in \cite[Theorem~3.2.1(iii) and Remark~3.2.2]{BMR2026}. +The Artin--Schreier +sequence gives the $\F_p$ complex, the coefficient exact sequences +give the $\Z/p^m$ complexes, and $\Rlim_m$ gives $\mathscr C_X$. +Derived solidity is preserved by these limits and extensions +\cite[Theorem~4.4(ii)]{Tang2026}. This argument applies with the +group action and with all the profinite parameters retained. It +also gives derived $p$-completeness of $\mathscr C_X$. + +For compact $G$, the condensed standard resolution with its derived +parameter values retained gives +\begin{equation}\label{eq:geometric-group-descent} + R\Gamma(G,\mathscr C_X)(*)\simeq + \Rlim_m\operatorname{Tot}_{a}R\Gamma_v + (X\times\underline{G^a},\Lambda_m) + \simeq R\Gamma_{\mathrm{int}}([X/G]). +\end{equation} +Indeed all $G^a$ are profinite. In the standard resolution, the +degree-$a$ derived Hom is precisely the derived parameter value on +$G^a$; this is the formal condensed construction in +\cite[\S2, pp.~5--6]{Anschutz2020}, before any acyclicity simplification. +The parameterized complex is bounded below, and the derived +totalization computes its point-valued invariants. The second +comparison is quotient \v{C}ech descent at each finite level; the +derived limit and totalization commute because both are limits. +This is the geometric descent comparison used here. On the same bar +terms, using \cite[Lemma~2.2]{Anschutz2020} for the source parameter +value, the finite unit $\underline{\Lambda_m}\to\mathscr C_{X,m}$ +is exactly the map in \eqref{eq:finite-constant-cochains}. The complexes +$\mathscr C_{X,m}$ and $\mathscr C_X$ are concentrated in cohomological +degrees at least zero, so their units factor through their degree-zero +cohomology by the truncation triangle. Thus the derived-invariants map +agrees with $u_{X,G,\mathrm{int}}$ after the coefficient limit, and its +factorization through degree zero gives the constants bottom-row edge map. + +The next lemma makes these descent groups computable from a finite +complex. Its final assertion will let us use arithmetic weights known +on ordinary cohomology groups while retaining the solid coefficients. + +\begin{lemma}[Finite resolutions and scalar actions] +\label{lem:solid-resolutions} +Let $G$ be a compact $p$-adic analytic group without elements of order +$p$, and let $\Lambda_G=\Z_p[[G]]$. There is a finite resolution +$P_\bullet\to\Z_p$ by finitely generated projective profinite +$\Lambda_G$-modules. Its length can be taken to be $\dim G$. +For any derived-solid $\Z_p$-complex $K$ with a continuous $G$-action, +the point-valued derived invariants are computed by +\begin{equation} +\label{eq:finite-resolution} + R\Gamma(G,K)(*)\simeq + \Hom_{\Lambda_G}(P_\bullet,K). +\end{equation} +Here the right side means the finite total complex of morphisms in +solid modules. In particular, invariants of a module in the heart +have cohomology only in degrees $0,\ldots,\dim G$. + +Suppose $K$ is bounded below and an operator $\gamma$ commutes with +$G$. If $\gamma$ acts by $\lambda_j\in\Q_p$ on $H^j(K)(*)[1/p]$, +then it acts by the same scalar on every term in the $j$th row of +the rationalized descent spectral sequence +\[ + H^s\bigl(G,H^j(K)\bigr)(*)[1/p] + \ \Longrightarrow\ H^{s+j}\bigl(R\Gamma(G,K)(*)\bigr)[1/p]. +\] +For $K=\mathscr C_X$, the abutment is $H^{s+j}_{\mathrm b}([X/G])$. +\end{lemma} + +\begin{proof} +First construct the resolution in profinite modules. A compact $p$-adic analytic group has a noetherian completed group +ring; if it has no $p$-torsion, then +$\operatorname{pd}_{\Lambda_G}\Z_p=\operatorname{cd}_p(G)=\dim G$. +Here projective dimension is in profinite modules. +These results for compact groups, including the agreement with +abstract finitely generated modules, are recorded in +\cite[\S\S1.1--1.2, pp.~275--276]{Venjakob2002}. +The noetherianity input for analytic pro-$p$ groups is +\cite[V.2.2.4]{Lazard1965}; the compact-group statement uses the +finite-index passage recorded by Venjakob. +Choose successive finite free presentations. Noetherianity keeps +their kernels finitely generated, and the projective-dimension +bound makes the last syzygy projective. Their natural compact +topologies give the asserted profinite resolution. + +To apply the resolution to the geometric coefficient complexes, we +pass from continuous actions to solid modules over the completed group +ring. Write +$A=\underline{\Z_p}[\underline G]$ for the free condensed group +ring. A continuous action on a complex means an object of +$D(A\text{-}\mathrm{Cond})$ with derived-solid underlying complex. +By \cite[Lemma~1.3]{Anschutz2020}, the group-ring analytic +structure has underived solidification; its normalized ring is +\[ + A^\square\simeq\underline{\Z_p[[G]]} + =\underline{\varprojlim_{m,V}(\Z/p^m)[G/V]}, +\] +where $V$ runs through open normal subgroups. The ring +identification, including multiplication, is +\cite[\S3.6, preceding Proposition~3.21]{Tang2026}. +Analyticity gives fully faithful embeddings of the \emph{unbounded} +derived category +of solid $G$-modules into both $D(A\text{-}\mathrm{Cond})$ and +$D(A^\square\text{-}\mathrm{Cond})$, as stated in +\cite[\S1, pp.~3--4]{Anschutz2020}. +The solid heart is characterized by solidity of the underlying +condensed module. The essential-image criterion of Clausen--Scholze's +analytic theory extends this characterization to unbounded complexes: +membership is detected on cohomology objects +\cite[Proposition~12.4 and Remark~12.5]{ScholzeAnalyticGeometry}. +Thus its objects are precisely the complexes with derived-solid +underlying coefficients. Consequently the external derived Hom +complexes satisfy +\[ + R\Gamma(G,K)(*) + \simeq\RHom_{A\text{-}\mathrm{Cond}}(\underline{\Z_p},K) + \simeq\RHom_{G\text{-}\mathrm{Solid}}(\underline{\Z_p},K). +\] +The first identification is the external derived Hom for the condensed +classifying topos $BG_{\mathrm{pro\acute et}}$ +\cite[\S2, pp.~5 and~8]{Anschutz2020}; the last category is that of +solid $\Z_p[[G]]$-modules by +\cite[Proposition~3.21]{Tang2026}. +They apply to the given complex $K$ itself, including when its +coefficients are neither profinite nor derived $p$-complete. + +The profinite resolution stays exact and projective in solid +modules \cite[Theorems~3.2 and~3.14(i)]{Tang2026}. +Exactness can be seen by testing on extremally +disconnected profinite sets: maps from such a set lift through a +surjection of profinite spaces. Each $P_i$ is a retract of +$\Lambda_G^{r_i}$, and +\[ + \Hom_{\Lambda_G}(\Lambda_G,M)=M(*). +\] +Evaluation at the point is exact, since the point is extremally +disconnected. This proves projectivity against an arbitrary solid +coefficient module $M$, even when $M$ is not profinite. A bounded +projective resolution computes derived Hom against any complex, +which proves the displayed calculation and the amplitude assertion. + +Finally, $\Hom_{\Lambda_G}(P_i,H^j(K))$ is a retract of a finite +sum of $H^j(K)(*)$. The retract commutes with $\gamma$, because +$\gamma$ commutes with the $G$-action. After inverting $p$, the +entire finite complex computing the $j$th row therefore has scalar +action $\lambda_j$. Its cohomology has the same action. +This uses a scalar identity on point values only; it does not +deduce an identity of condensed sheaves from their point values. +Boundedness of the resolution gives a finite filtration in each +total degree, so exact rationalization preserves the spectral +sequence and its abutment. +\end{proof} + +\subsection{Compact groups and the affine building} + +Put +\[ + J_n=\{g\in\GL_n(\Q_p):|\det g|_p=1\}\qquad(n=2,3). +\] +Let $D_n$ denote the division algebra of degree $n$ whose maximal +order has unit group $P_n$. We need only the following low-degree +part of the rational Lie algebra calculation. + +\begin{proposition}[The group classes] +\label{prop:group-classes} +For $G=P_n$ or $J_n$, where $n=2,3$, +\[ + H^1_{\mathrm b}(G)=\Q_p, + \qquad H^2_{\mathrm b}(G)=0, + \qquad H^3_{\mathrm b}(G)=\Q_p. +\] +For $n=2,3$ and every compact open subgroup $K\subset J_n$, +restriction gives an isomorphism in every degree, with the compact +comparison identifying its target: +\[ + H^*_{\mathrm b}(J_n)\xrightarrow{\mathrm{res}}H^*_{\mathrm b}(K) + \cong H^*(\mathfrak{gl}_n(\Q_p),\Q_p). +\] +In degree three the block inclusion $J_2\hookrightarrow J_3$ +induces a nonzero map. Fix arbitrary nonzero classes +\[ + e_{P_n}\in H^3_{\mathrm b}(P_n),\qquad + e_{J_n}\in H^3_{\mathrm b}(J_n). +\] +\end{proposition} + +\begin{proof} +For an inclusion \(K_1\subset K_2\) of compact open subgroups of +\(\GL_n(\Q_p)\), choose a \(K_2\)-stable lattice \(\Lambda\) and then +\(r\) large enough that +\[ + U=1+p^r\End_{\Z_p}(\Lambda)\subset K_1. +\] +This is a uniform subgroup normal in \(K_2\). In the division-algebra +case one uses a sufficiently deep congruence subgroup of +\(\cO_{D_n}^{\times}\), which is normal in the relevant compact group. +For these small groups, Lazard's comparison is +\cite[Theorem~3.3.3 and Remark~3.3.4]{HKN2011} with trivial +\(\Z_p\)-coefficients, followed by inversion of \(p\). +Its coefficient naturality, rational agreement, and group naturality are +\cite[Theorem~3.1.1(1)--(3)]{HKN2011}, with numbering in the arXiv +revision specified in the bibliography. Equip the uniform groups with +their lower \(p\)-series. At these odd primes the proof of +Theorem~3.3.3 gives \(e=1\), and the associated graded target is generated +in degree one, as required by the group-naturality clause. Inclusions and +the block homomorphisms preserve these filtrations. The common subgroup +\(U\) therefore identifies restriction with the identity on the same +ambient Lie algebra cohomology. + +Here is why passing back to a compact group is valid even for an index +divisible by \(p\). For \(U\triangleleft K\), the augmentation and norm +of the finite permutation module \(\Z[K/U]\) induce, by continuous +Shapiro, the identities +\[ + \operatorname{cor}\operatorname{res}=[K:U],\qquad + \operatorname{res}\operatorname{cor} + =\sum_{q\in K/U}q^* +\] +at each finite coefficient level, compatibly with reduction in \(m\). +Shapiro here is the derived induction--restriction adjunction: induction +for an open subgroup is a direct sum on underlying condensed modules and +is exact by (AB4) \cite[Theorems~2.2 and~2.4]{Tang2026}; restriction is +also exact. Their adjunction therefore descends directly to derived +categories. Its naturality identifies the +permutation maps with the actual restrictions and conjugations in the +finite continuous bar complexes. Apply \(\Rlim_m\) to these identities +before taking cohomology and inverting \(p\). Only then divide by +\([K:U]\). This gives +\(H^*_{\mathrm b}(K)\xrightarrow{\sim}H^*_{\mathrm b}(U)^{K/U}\) +by restriction, with the compact +continuous-cochain interpretation of \eqref{eq:group-cochain-bridge}. + +The finite quotient acts trivially in every Lie degree. The +Chevalley--Eilenberg complex is a finite-dimensional algebraic +\(\GL_n\)-representation after a splitting-field extension. Cartan's +formula \(\mathcal L_X=d\iota_X+\iota_Xd\) makes the derivative of its +action on cohomology zero. In characteristic zero, an algebraic action +of the connected group \(\GL_n\) with zero derivative is trivial. +Thus all inner automorphisms act trivially, in every degree; the same +conclusion descends for the division algebra. This proves the compact +restriction identification without an all-degree numerical calculation. + +After a finite splitting-field extension, either Lie algebra is +$\mathfrak{gl}_n$. The finite Chevalley--Eilenberg calculation in +\Cref{app:background} gives its asserted groups in degrees one through +three, with degree-three representative +\[ + (X,Y,Z)\longmapsto\tr\bigl(X[Y,Z]\bigr), +\] +or its reduced-trace version for $D_n$. +This is the invariant-form cocycle of +\cite[\S21]{ChevalleyEilenberg1948} and \cite[\S11]{Koszul1950}; +the appendix records the normalization and restriction used here. +These representatives are defined over $\Q_p$ and invariant under +inner automorphisms, so the assertions descend from the splitting +field. Restriction of the displayed form to the upper-left or +lower-right $2\times2$ block is the corresponding form for +$\mathfrak{gl}_2$, whose class is nonzero. This proves the compact +calculations and their compatibility. + +To pass to $J_n$, let $\mathcal B_n$ be the reduced affine building +of $\GL_n(\Q_p)$. It is contractible of dimension $n-1$. +The group $J_n$ preserves vertex types, its face stabilizers +$K_\sigma$ are compact open, and a chamber $\Delta$ is a +fundamental domain, including its faces. A proof of these building +properties by lattice norms is included in \Cref{app:background}. +Write its bounded augmented cellular resolution as +\[ + P_r=\bigoplus_{\substack{\sigma\subset\Delta\\\dim\sigma=r}} + \Z[\underline{J_n/K_\sigma}]. +\] +It is exact also as a complex of condensed modules. Indeed, on a +profinite parameter set a continuous section of a discrete cellular +chain group has finite image. Contractibility lifts each of the finitely +many cycles in that image; choosing those lifts on its clopen fibers +gives a continuous lift. The same exact derived induction--restriction +adjunction therefore supplies continuous Shapiro for each summand. +At coefficient level \(\Lambda_m\), applying derived Hom gives a finite +filtration with graded terms +\(\bigoplus_{\dim\sigma=r}C_{\cts}^{\bullet}(K_\sigma,\Lambda_m)[-r]\) +in \(D(\mathrm{Ab})\). +A cellular boundary is the literal signed projection of the corresponding +coset modules, so naturality of that adjunction makes each face map the +signed restriction between stabilizers, with no index multiplier. +All of these statements are compatible with coefficient reduction. + +Take \(\Rlim_m\) of this finite filtration. There are finitely many +orbit terms in each of finitely many cellular degrees, so this step uses +only finite sums and cones in the cellular direction. Taking cohomology of +the resulting finite filtration gives the integral spectral sequence +\[ + E_1^{r,s}= + \bigoplus_{\substack{\sigma\subset\Delta\\\dim\sigma=r}} + H^s_{\mathrm{int}}(K_\sigma) + \ \Longrightarrow\ H^{r+s}_{\mathrm{int}}(J_n), +\] +with precisely those signed restriction faces. It does not interchange +an infinite group-cochain totalization with the derived inverse limit. +Only after this construction do we invert \(p\); exact rationalization +preserves the finite filtration and its abutment. + +Under the compact comparison, every restriction between face +stabilizers is the identity on the same Lie algebra cohomology; +any change of representatives contributes an inner conjugation, +which also acts trivially. Thus each rationalized row is the +cellular cochain complex of the simplex $\Delta$ with constant +coefficients $H^s(\mathfrak{gl}_n,\Q_p)$. +It has cohomology only in cellular degree zero. +The resulting edge map is restriction to a vertex stabilizer. +Restriction further to compact open subgroups, using intersections +and the compact comparison, proves the asserted canonical +identifications. The block restriction can now be tested on +compact open subgroups, where it is the trace calculation above. +\end{proof} + +\subsection{The parabolic restriction} + +The relevant subgroup remembers a lattice only in each determinant +line. In a basis adapted to a line it is +\[ + D=\left\{ + \begin{pmatrix}a&v\\0&A\end{pmatrix}: + a\in\Z_p^\times,\quad A\in J_2,\quad + v\in\Q_p^{1\times2}\right\}\subset J_3. +\] +Write $\rho_3:D\hookrightarrow J_3$ for the inclusion and +$\rho_2:D\to J_2$ for the quotient action. + +\begin{lemma}[Unipotent acyclicity and restriction] +\label{lem:parabolic} +Inflation induces an isomorphism +\[ + H^*_{\mathrm b}(\Z_p^\times\times J_2) + \xrightarrow{\ \sim\ }H^*_{\mathrm b}(D). +\] +There is a scalar $b\in\Q_p^\times$ such that +\[ + \rho_3^*e_{J_3}=b\,\rho_2^*e_{J_2}. +\] +\end{lemma} + +\begin{proof} +The kernel of $D\to\Z_p^\times\times J_2$ is +$N=\Q_p^2$, with its additive topology. +Fix $m$ and exhaust it by the compact open subgroups +$N_r=p^{-r}\Z_p^2$ for $r\geq0$. Their finite-coefficient +cohomology, computed by the two-variable Koszul resolution, is +\[ + H^q(N_r,\Z/p^m)=\bigwedge^q(\Z/p^m)^2. +\] +The standard bases identify restriction from $N_{r+1}$ to $N_r$ +with multiplication by $p^q$ in degree $q$: restriction multiplies +each degree-one homomorphism by $p$, and is compatible with cup +products. For $q>0$ this inverse system is pro-zero; for $q=0$ +it is constant with identity transition maps. + +This calculation computes the cohomology of $N$ without a missing +derived-limit term. Indeed a continuous cochain on $N$ is exactly +a compatible family of cochains on the $N_r$. +Restriction of cochains is surjective, since $N_r^a$ is clopen in +$N_{r+1}^a$ and a function extends by zero on its complement. +Thus the ordinary inverse limit of the cochain complexes realizes +their derived inverse limit. The Milnor exact sequence has +$\varprojlim^1_r=0$ in every degree: this holds both for a constant +tower with identity maps and for a pro-zero tower. It follows that +the constant map is an equivalence +\[ + \Lambda_m\xrightarrow{\ \sim\ }C_{\cts}^{\bullet}(N,\Lambda_m). +\] +The same proof works with coefficients +$C_{\cts}(S,\Z/p^m)$ for every profinite parameter space $S$. +The finite Koszul complex gives finite sums of this module, the +positive-degree restrictions are again $p^q$, and extension by +zero is continuous on $N_r^a\times S$. +The condensed standard resolution evaluated on each such parameter +identifies this calculation with the internal equivalence +\[ + \underline{\Lambda_m}\xrightarrow{\ \sim\ } + R\Gamma(N,\underline{\Lambda_m}). +\] +Thus it proves an equivalence of coefficient objects for extension +descent. The equivalence is the map of constants, so it is equivariant +under the Levi action. + +Now take $\Rlim_m$. In degree zero the parameter modules have +surjective transition maps, by lifting locally constant values, +and their limit is $C_{\cts}(S,\Z_p)$. +Hence the internal coefficient limit satisfies +\[ + \underline{\Z_p}\xrightarrow{\ \sim\ } + \Rlim_m R\Gamma(N,\underline{\Lambda_m}). +\] +At each finite level, internal descent for the split extension identifies +$R\Gamma(D,\underline{\Lambda_m})$ with +$R\Gamma(\Z_p^\times\times J_2, +R\Gamma(N,\underline{\Lambda_m}))$. +The Levi-equivariant constant equivalence therefore induces inflation. +After point evaluation, the standard-resolution comparison identifies +this map with finite continuous-cochain inflation. Taking $\Rlim_m$ +and using \eqref{eq:integral-group-cochains} gives the claimed +isomorphism integrally before rationalization. + +Finally, the compact resolution for $\Z_p^\times$ and the finite +building resolution for $J_2$ give the usual rational product +calculation. Since $H^*_{\mathrm b}(\Z_p^\times)$ is exterior +on one degree-one generator and $H^2_{\mathrm b}(J_2)=0$, +degree three of the Levi is exactly $H^3_{\mathrm b}(J_2)$. +Thus $\rho_3^*e_{J_3}$ is a scalar multiple of +$\rho_2^*e_{J_2}$. Restricting to the section +$A\mapsto\operatorname{diag}(1,A)$ shows that this scalar is +nonzero, by \cref{prop:group-classes}. +\end{proof} + +The vanishing just proved is specific to our order of operations. +For example, continuous homomorphisms $\Q_p\to\Q_p$ give a +nonzero $H^1_{\cts}(\Q_p,\Q_p)$, whereas the preceding integral +calculation gives $H^1_{\mathrm b}(\Q_p)=0$. + +\subsection{The integral Drinfeld theorem} + +Let $\Omega_C^d$ be the base change to $C$ of Drinfeld's symmetric +space over $\Q_p$, the complement of the $\Q_p$-rational +hyperplanes in $\mathbf P^d$. The following precise input supplies +the weights used in \cref{prop:primitive-comparison}. + +\begin{theorem}[Colmez--Dospinescu--Nizio\l] +\label{thm:drinfeld} +For $0\leq j\leq d$ there are compatible +$\GL_{d+1}(\Q_p)\times\Gal(\overline\Q_p/\Q_p)$-equivariant +isomorphisms of groups +\[ + H^j_{\mathrm{int}}(\Omega_C^d)(j) + \simeq\operatorname{Sp}_j(\Z_p)^*, + \qquad + H^j_{\mathrm{\acute et}}(\Omega_C^d,\F_p(j)) + \simeq\operatorname{Sp}_j(\F_p)^*. +\] +Here $\operatorname{Sp}_j$ is the generalized Steinberg +representation of \cite[\S4.1]{CDN2021}, the star denotes its +continuous coefficient-linear dual with weak topology, and +the right sides have trivial Galois action. These groups vanish +for $j>d$, and $H^0_{\mathrm{int}}(\Omega_C^d)=\Z_p$ by constants. +In particular, on the untwisted integral group in degree $j$, +an arithmetic Galois element $\gamma$ acts by +$\chi_{\mathrm{cyc}}(\gamma)^{-j}$. +\end{theorem} + +The group calculation is \cite[Theorem~1.1]{CDN2021}, with its integral +cohomology convention. The comparison with +$H^j_{\mathrm{pro\acute et}}(\Omega_C^d,\widehat\Z_p(j))$ +is made explicitly in the proof in \cite[\S5]{CDN2021}; finite +coefficient site comparison and $\Rlim_m$ identify it with the +convention above. Thus the scalar assertion holds on the +integral point groups themselves. To identify the stated degree-zero +map, choose a $C$-point of $\Omega_C^d$ by taking $d+1$ coordinates +linearly independent over $\Q_p$. By +\eqref{eq:geometric-point-constants}, evaluation there splits the +constants unit at each finite level and after $\Rlim_m$. The cited +calculation makes $H^0_{\mathrm{int}}(\Omega_C^d)$ a free rank-one +$\Z_p$-module, so this split inclusion from $\Z_p$ is an isomorphism. +This identifies the actual constants map locally from the degree-zero +calculation. Applying +\cref{lem:solid-resolutions} to a compact open uniform subgroup +of $J_{d+1}$ propagates it to the group-cohomology rows, without +any assumption about cohomology of a rationalized sheaf. diff --git a/preprints/A-rational-obstruction-to-strong-chromatic-splitting-at-height-three-September-25-2026/build/sections/03-completed-tower.tex b/preprints/A-rational-obstruction-to-strong-chromatic-splitting-at-height-three-September-25-2026/build/sections/03-completed-tower.tex new file mode 100644 index 0000000000000000000000000000000000000000..44c140d36ecd927b56cbe243972f1eb9ce1f9815 --- /dev/null +++ b/preprints/A-rational-obstruction-to-strong-chromatic-splitting-at-height-three-September-25-2026/build/sections/03-completed-tower.tex @@ -0,0 +1,744 @@ +\section{The completed coheight-one tower} +\label{sec:completed-tower} + +We now construct the geometric and spectral comparison objects used by +the detector. The spectral output is a pair of equivariant maps from the +two Morava theories to a completed height-two theory. We also need descent for each +coefficient module and ordinary \(K(2)\)-locality of every profinite +evaluation. These properties allow the later coefficient calculation to +detect the canonical localization map. The inputs +from Barthel--Mann--Ray--Schlank--Senger--Weinstein--Zhou include the +characteristic-\(p\) tower, its cohomology calculations, and the tensor +and completion constructions +in the version dated 26 August 2026 \cite{BMR2026}. We first establish +the connected target and its coefficient descent. +We then construct the map from height three by splitting a completed +connected--\'etale group. The detailed oriented deformation argument is +given in \cref{app:oriented-projection}. + +\subsection{The characteristic-\(p\) tower} + +Recall that $k=\F_{p^6}$ and that the height-$n$ Honda groups $\Gamma_n$ are over $k$, +for $n=2,3$. All their geometric endomorphisms are defined over $k$. +Let $D_n$ be the division algebra of invariant $1/n$ over $\Q_p$, with +the convention identifying +\[ + P_n=\Aut_k(\Gamma_n)=\cO_{D_n}^{\times},\qquad + N_n:P_n\longrightarrow\Z_p^{\times} +\] +with its reduced norm. Thus $P_n$ is the nonextended stabilizer. Set +\[ + P_n^1=N_n^{-1}(\mu_{p-1}),\qquad + \overline Z=P_3/P_3^1\simeq\Z_p,\qquad + G=P_3\times P_2,\qquad H=P_2\times\overline Z. +\] +The identification of $\overline Z$ with $\Z_p$ will not require a choice +of generator. We write $\mathcal E_n^0=\cO(1/n)$ for the corresponding +rank-$n$, degree-one Fargues--Fontaine bundle. A $P_n$-structured form +$\mathcal E_n$ means a form whose torsor of frames has been reduced from +$D_n^{\times}$ to $\cO_{D_n}^{\times}$. Equivalently, its determinant +framing is specified up to $\Z_p^{\times}$; this is determinant lattice +data, not a lattice in a rank-$n$ rational local system. + +The height-exactly-two stratum of the height-three deformation space is +the Laurent field +\[ + K=k((u_2)),\qquad p=u_1=0,\quad u_2\ne0. +\] +Let $L/K$ be the Gross--Torii extension classifying the isomorphisms +$(\Gamma_2)_L\simeq(\Gamma^{\mathrm{un}}_{3,L})^{\circ}$, and let +$\widehat L$ be its valued-field completion. The two change-of-framing +actions give the continuous $G$-action on $\widehat L$. Fix compatible +determinant identifications for the two Honda isocrystals, and hence an +identification +$\iota:\det\mathcal E_3^0\simeq\det\mathcal E_2^0$. +These choices exist over $k$: the determinant identifications in +\cite[\S2.7]{BMR2026} require $\F_{p^2}\subset k$. + +\begin{proposition}[The completed tower]\label{prop:tower} +The field $\widehat L$ is perfectoid of characteristic $p$, and is +isomorphic, as a valued field over $k$, to the completed perfection of a +Laurent-series field: +\[ + \widehat L\simeq + \left(\bigcup_{r\ge0}k((z^{1/p^r}))\right)^{\wedge}. +\] +There is a $G$-equivariant identification of v-sheaves over $\Spd k$ +\[ + \Spd\widehat L\simeq + \operatorname{Surj}(\mathcal E_3^0,\mathcal E_2^0). +\] +Consequently +\[ + Y=[\Spd\widehat L/G] +\] +classifies surjections $\mathcal E_3\twoheadrightarrow\mathcal E_2$ +between $P_n$-structured forms. In particular $Y$ carries the universal +surjection used below. + +Let $S_3=\ker N_3$ and $T_3=P_3/S_3=\Z_p^{\times}$. Then +\[ + [\Spd\widehat L/S_3] + \simeq\operatorname{BC}(-1/2)^* +\] +as v-sheaves, with the following residual $P_2\times T_3$-action. +On extension classes in $\operatorname{Ext}^1(\mathcal E_2^0,\cO)$, +write $\lambda_*$ for pushout by $\lambda$ on $\cO$ and $b^*$ for +pullback by $b$ on $\mathcal E_2^0$. For the framing action +$f\mapsto bfg^{-1}$ and $a=N_3(g)$, the action is +\[ + (b,a)\cdot\xi + =\bigl(aN_2(b)^{-1}\bigr)_*(b^{-1})^*\xi. +\] +In particular, the $T_3$-action is pushout by $a$ on the kernel. +Only this restriction of the action on the extension chart is used below. +\end{proposition} + +\begin{proof} +The imported result is \cite[Proposition 2.7.1(1)--(5)]{BMR2026}, whose +hypotheses are $h\ge2$, a perfect field containing +$\F_{p^{h(h-1)}}$, and the coheight-one Gross--Torii extension with the +specified determinant identifications. Here $h=3$ and +$k=\F_{p^6}$ satisfy those hypotheses exactly. + +We recall why its formulation gives all surjections and the indicated +structure groups. The infinite-level two-tower theorem +\cite[Theorem 2.6.3]{BMR2026}, followed by forgetting the étale framing, +identifies $\Spd\widehat L$ with exact sequences +\[ + 0\longrightarrow\cO\longrightarrow\mathcal E_3^0 + \longrightarrow\mathcal E_2^0\longrightarrow0 +\] +whose determinant identification is $\iota$. Forgetting that framing is +the $\Z_p^{\times}$-torsor in the proof of +\cite[Proposition 2.7.1(2)]{BMR2026}. Given a surjection, its kernel is +the line +$\det\mathcal E_3^0\otimes(\det\mathcal E_2^0)^{-1}$; the fixed +$\iota$ trivializes this line and recovers the sequence. Thus no +additional condition is imposed on the surjection. Descent along the +two framing torsors gives exactly the asserted description of $Y$, also +in families. + +Quotienting by $S_3$ instead allows the source bundle to vary while +retaining its determinant trivialization. It therefore classifies +extensions +\[ + 0\longrightarrow\cO\longrightarrow\mathcal E + \longrightarrow\mathcal E_2^0\longrightarrow0 +\] +with middle term of slope $1/3$. By +\cite[Proposition 2.7.1(5)]{BMR2026}, these are precisely the extensions +that are nonsplit at every geometric point, which form +$\operatorname{BC}(-1/2)^*$. + +To specify the joint action, let $i_f:\cO\to\mathcal E_3^0$ be the +kernel injection normalized by $\iota$. If $f'=bfg^{-1}$, the +determinant isomorphisms for the two exact sequences give +\[ + g i_f=i_{f'}\,N_3(g)N_2(b)^{-1}. +\] +Indeed, $g$ acts on the source determinant by $N_3(g)$ and $b$ acts on +the target determinant by $N_2(b)$. The induced diagram of extensions +therefore has kernel map $N_3(g)N_2(b)^{-1}$ and quotient map $b$. +Functoriality of extensions by pushout and pullback proves the displayed +$P_2\times T_3$-action. This calculation is an identity of bundle maps +on every test object and commutes with base change. For $b=1$, it is +pushout by $a=N_3(g)$. On the extension group, $a_*=(aI_2)^*$, +so this is the action by inverse pullback of the central scalar $a^{-1}I_2$. +The underlying quotient and this $T_3$ restriction +are the parts of \cite[Proposition 2.7.1(5)]{BMR2026} used below; the +joint action includes the target determinant factor just computed. +The field assertions are parts (1) and (4) of the same proposition. +\end{proof} + +\subsection{Acyclicity of constants} + +Choose $C=\C_p$ and a map $\Spd C\to\Spd k$, and write $Y_C$ for +base change over $\Spd k$. Put +\[ + \widetilde Y=\Spd\widehat L,\qquad + \widetilde Y_C=\widetilde Y\times_{\Spd k}\Spd C. +\] +The quotient torsors $\widetilde Y\to Y$ and $\widetilde Y_C\to Y_C$ +give classifying maps +\[ + Y\longrightarrow B_kG=[\Spd k/\underline G],\qquad + Y_C\longrightarrow B_CG=[\Spd C/\underline G]. +\] +These are relative classifying stacks; the nonextended actions fix +their respective bases. If $q_n:G\to P_n$ is projection, the classes +denoted below by $e_{P_n}|_Y$ and $e_{P_n}|_{Y_C}$ mean respectively +$u_{\widetilde Y,G,\mathrm b}(q_n^*e_{P_n})$ and +$u_{\widetilde Y_C,G,\mathrm b}(q_n^*e_{P_n})$. +Their base-change compatibility is +\eqref{eq:constant-cochain-naturality} for +$\widetilde Y_C\to\widetilde Y$, already at each finite level. Introduce +\[ + X=[\Spd\widehat L/P_3^1],\qquad + X_C=X\times_{\Spd k}\Spd C; + \quad Y=[X/H],\quad Y_C=[X_C/H]. +\] +The notation $\cO^{\flat}$ denotes the characteristic-$p$ structure +sheaf. Cohomology with these coefficients is derived solid cohomology; +on the group quotients used here, quotient descent computes it as +continuous derived invariants of the parameterized geometric coefficient +complex. The comparisons +with integral constant coefficients will be made in +\Cref{prop:witt-comparison}. + +\begin{proposition}[Acyclicity of the maps of constants] +\label{prop:constant-acyclicity} +The actual maps of constants induce equivalences +\begin{align} + k&\xrightarrow{\ \sim\ }R\Gamma(P_3^1,\widehat L), + \label{eq:finite-constant-map}\\ + C^{\flat}&\xrightarrow{\ \sim\ } + R\Gamma_v(X_C,\cO^{\flat}). + \label{eq:geometric-constant-map} +\end{align} +These are equivalences of derived solid algebras with residual $H$-action. +The $H$-action on the left sides is trivial. The maps are natural for +the change of constants $k\to C^{\flat}$ and for the arithmetic actions +preserving the chosen base data. +\end{proposition} + +\begin{proof} +Apply the underlying cohomology calculation of +\cite[Proposition~3.6.9]{BMR2026} to the quotient in \Cref{prop:tower}, +retaining only its $T_3$-action. The map of constants and the vanishing +in the other degrees give a fiber sequence of derived solid $k$-modules +with $T_3$-action +\[ + k\longrightarrow R\Gamma(S_3,\widehat L) + \longrightarrow k_{\chi_T}[-3]. +\] +Here $\chi_T$ denotes the actual character on the line in degree three. +This is the restriction of the argument combining quotient descent and cohomology in +\cite[Theorem~3.6.10]{BMR2026} that is needed here. For $b=1$, the action +formula in \Cref{prop:tower} is pushout by $a\in T_3$ on the kernel. +On the extension group this equals pullback by $aI_2$, or the +action by inverse pullback of $a^{-1}I_2$. The reduced norm of $aI_2$ +in $D_2^\times$ is $a^2$. Hence the subgroup +\[ + P_3^1/S_3=\mu_{p-1}\subset T_3 +\] +acts on the last term by $\alpha\mapsto\alpha^{2}$ or +$\alpha\mapsto\alpha^{-2}$; either convention gives the same vanishing +used below. The displayed exponent is also checked in +\cite[Lemma 5.2.5]{BMR2026}. This is a subgroup of the norm quotient; +it must not be replaced by the central $\mu_{p-1}$ in $P_3$. +Since $p\ge5$, this character is nontrivial. The averaging idempotent +for the finite group $\mu_{p-1}$ is defined in characteristic $p$, so its +derived invariants are exact and annihilate $k_{\chi_T}$. +Taking these invariants proves that the underlying map in +\eqref{eq:finite-constant-map} is an equivalence. The map itself is +$H$-equivariant: it is induced by constants from the original $G$-action +on $\widehat L$, followed by $P_3^1$-descent, and $G$ fixes $k$. +The forgetful functor from $H$-equivariant objects detects equivalences, +so this gives the required equivalence with its full residual action. +No splitting of the fiber sequence or joint character formula is needed. + +For completeness, the finite-field hypothesis in this import is +substantive. Its proof uses the solid Frobenius comparison +\cite[Lemma 3.6.3 and Corollary 3.6.8]{BMR2026} for +$\bigl(\bigcup_{r\ge0}k((z^{1/p^r}))\bigr)^{\wedge}$ +over a \emph{finite} field $k$. +The geometric base change of this point is a punctured perfectoid ball; +the proof solves coefficient Frobenius minus identity on its convergent functions +and then requires surjectivity in solid modules. For this precise +function ring, \Cref{lem:continuous-as-section} supplies a continuous +section and hence proves surjectivity on every profinite parameter space. +This justifies the solid enhancement directly, without applying a +linear open-mapping theorem to Frobenius minus identity. The resulting natural +comparison commutes with the $S_3$-action. Thus +\Cref{prop:tower} verifies the required hypothesis before finite-field +descent is applied. + +For the geometric statement, take a quadratic extension $F/\Q_p$ that +is a maximal subfield of $D_2$. Proposition 3.1.2 of \cite{BMR2026} +identifies +\[ + \bigl(\operatorname{BC}(-1/2)^*\bigr)_C + \simeq[\mathcal H^1_{F,C}/F], +\] +where $\mathcal H^1_{F,C}$ is the one-dimensional Drinfeld upper +half-plane for $F$ and the additive group $F$ acts by translations. +This identification is equivariant for the central $\Q_p^{\times}$. +It is distinct from the rank-$n$ Drinfeld space +$\Omega_C^{n-1}$ used in the modification argument. +Proposition 3.5.6 of \cite{BMR2026} computes the solid +$\cO^{\flat}$-cohomology of this quotient: it is $C^{\flat}$ in degree +zero, a character line in degree $[F:\Q_p]+1=3$, and zero otherwise. +By the $b=1$ action formula in \Cref{prop:tower}, this norm-quotient +$\mu_{p-1}$ acts by kernel scalars. Expressing those as central +pullbacks on the target gives reduced-norm character +$\alpha\mapsto\alpha^2$ or its inverse. Both are nontrivial, so the +top cohomology has no $\mu_{p-1}$-invariants; the choice of action or +dual convention does not affect the conclusion. +The degree-zero identification is the map of constants. Applying the +same averaging argument proves \eqref{eq:geometric-constant-map}. + +Only central equivariance of the Drinfeld description was needed to +prove this vanishing. The constant map on $X_C$ itself has the full +residual $H$-equivariance, and an equivariant map is an equivalence when +its underlying map is. This proves the full assertion, without assuming +a $P_2$-equivariant splitting or a larger equivariance of that chart. +Finally, the actions of $P_2$ and $\overline Z$ are over the fixed base, +so they fix $k$ and $C^{\flat}$ pointwise. All arrows used above are +induced by structure maps and group descent; their stated naturality +follows. +\end{proof} + +\subsection{Completed Lubin--Tate theory and coefficient descent} + +Write $E_n=E(k,\Gamma_n)$ for ordinary Morava $E$-theory and +$E_n^{\square}=E^{\square}(k^{\delta},\Gamma_n)$ for its solid +enhancement, where $k^{\delta}$ is the discrete solid ring. For the +valued field $\widehat L$, its solid ring of parameters is +\[ + \widehat L(T)=C_{\cts}(T,\widehat L) +\] +on profinite sets $T$. Define +\[ + \widehat B^{\square} + =E^{\square}(\widehat L,(\Gamma_2)_{\widehat L}). +\] +Here $\pi_t$ of a solid spectrum is its solid homotopy module; $(*)$ +will denote evaluation in ordinary spectra at the one-point set. +In this section and in \cref{sec:witt-transfer,sec:detection}, +$H^s(A,M)$ denotes the point-valued continuous cohomology +$H^s((R\Gamma(A,M))(*))$ for a solid coefficient module $M$. + +\begin{proposition}[The completed theory and its coefficients] +\label{prop:completed-theory} +There are continuous $G$-equivariant maps of solid $E_\infty$-rings +\begin{equation}\label{eq:completed-map} + E_3^{\square}\longrightarrow\widehat B^{\square} + \longleftarrow E_2^{\square}, +\end{equation} +where $G$ acts through $P_3$ on the first source and through $P_2$ on +the second. The right arrow is canonical connected base change. +The target is even periodic, with +\[ + \pi_0\widehat B^{\square}(*)\simeq W(\widehat L)[[u'_1]]. +\] +Invariantly, its coefficients are complete base changes of the +height-two deformation ring and its invertible even coefficient lines. +The natural map $W(\widehat L)\to\pi_0\widehat B^{\square}$ is +$G$-equivariant. Every profinite evaluation +$\widehat B^{\square}(T)$ is an ordinary $K(2)$-local spectrum. + +The second map in \eqref{eq:completed-map} induces an $H$-equivariant +equivalence +\[ + E_2^{\square}\xrightarrow{\ \sim\ } + (\widehat B^{\square})^{hP_3^1}, +\] +where $\overline Z$ acts trivially on the source. Moreover, for every +$t\in\Z$ it induces an equivalence on the individual coefficient +complexes +\[ + \pi_tE_2^{\square}\xrightarrow{\ \sim\ } + R\Gamma(P_3^1,\pi_t\widehat B^{\square}). +\] +In particular, for $s\ge0$ and all $t$ there are natural isomorphisms +\begin{equation}\label{eq:coefficient-descent} + H^s(H,\pi_tE_2^{\square}) + \xrightarrow{\ \sim\ } + H^s(G,\pi_t\widehat B^{\square}). +\end{equation} +These comparisons preserve the coefficient maps from Witt constants. + +Finally, put $\Delta=\Gal(k/\F_p)$ and +$G_n=P_n\rtimes\Delta$. The ordinary stabilizer-descent identification +is +\[ + \bigl((E_n^{\square})^{hG_n}\bigr)(*)\simeq L_{K(n)}S, + \qquad n=2,3, +\] +compatibly with the canonical maps to $E_n$. +\end{proposition} + +The following subsections prove the proposition. Its coefficient +statements concern the connected +base-change map from \(E_2\). We prove them independently of the map +from \(E_3\); that map will be attached to the same endpoint by the +oriented splitting construction below. + +\subsection{The connected endpoint and its coefficients} + +For the Honda pairs over \(k\), the deformation ideals are +\((p,u_1,u_2)\) at height three and \((p,u'_1)\) at height two. +They are finite regular sequences in the respective deformation rings. +The connected completed pair uses the Noetherian pair +\((k,\Gamma_2)\) as a witness: every +\(C_{\cts}(T,\widehat L)\) is perfect, since inverse Frobenius is +continuous, and flat over \(k\). Thus the endpoint formulas of +\cite[Proposition~4.5.5]{BMR2026} apply through this fixed witness. +The finite Koszul argument below explains the reduction through +hypersheafification and totalization in these instances. + +The pointwise coefficient formula and the coefficient-line base change +follow from \cite[Lemma 4.2.12 and Proposition 4.5.5]{BMR2026}, applied to +the constant connected height-two group. The formal parameter $u'_1$ +is a coordinate inherited from a deformation coordinate for $E_2$; no +assertion that $P_2$ fixes this parameter is intended. Witt vectors and +power series carry their limit structures on profinite parameters. +Functoriality of the perfect-residue-ring deformation construction gives +the asserted map of Witt constants. The same pointwise description, +together with \cite[Proposition 4.4.2]{BMR2026}, proves $K(2)$-locality +of every evaluation. + +The same formulas identify the condensed connected theory +\(E^{\mathrm{cond}}(\widehat L,(\Gamma_2)_{\widehat L})\) +with \(\widehat B^\square\). For a profinite set \(T\), its evaluation is +\[ + E\bigl(C_{\cts}(T,\widehat L), + (\Gamma_2)_{C_{\cts}(T,\widehat L)}\bigr). +\] +The canonical connected +base-pair map defines +\[ + c^\square:E_2^\square\longrightarrow\widehat B^\square. +\] +It is continuous \(G\)-equivariant by functoriality of the connected +base-pair construction: \(P_3\) acts through the base and \(P_2\) +changes the connected framing. In particular \(G\) acts on its +source through \(P_2\). The family naturality is checked in +\cref{app:oriented-projection}, before constructing the oriented section. +The coefficient and locality statements above apply to this endpoint +without using the full completed intermediate. + +We give the coefficient argument separately from the spectral one. +Proposition~4.3.12 of \cite{BMR2026} states the following descent result: +if $(R,\Gamma)$ is a Noetherian Luriean pair and +$S^{-1}\to S^{\bullet}$ is an augmented cosimplicial diagram of flat, +relatively perfect $R$-algebras which is a limit diagram in derived +commutative rings, then both +\[ + E(S^{-1},\Gamma)\longrightarrow + \operatorname{Tot} E(S^{\bullet},\Gamma) + \quad\text{and}\quad + \pi_{2j}E(S^{-1},\Gamma)\longrightarrow + \operatorname{Tot}\pi_{2j}E(S^{\bullet},\Gamma) +\] +are equivalences; the latter totalization is in $D(\Z)$. +We use this statement for fixed witnesses with finite regular deformation +ideals, and give the finite Koszul justification for their reduction +through totalization. +For any one witness $(R_0,\Gamma_0)$, put +$A_0=\pi_0E(R_0,\Gamma_0)$ and $I_0=\ker(A_0\to R_0)$. +Assume the indicated generators of $I_0$ form a finite regular sequence. +Thus $R_0=A_0/I_0$ has a finite free Koszul resolution and is a perfect +$A_0$-module. Each profinite hypercover diagram for that endpoint is +$A_0$-linear by functoriality from its fixed witness. For the final bar diagram this is +$A_0=W(k)[[u'_1]]$ and $I_0=(p,u'_1)$; its maps are $A_0$-linear +because $P_3^1$ acts trivially on the $E_2$ source. +Derived tensor with $R_0$ therefore commutes with all limits in +$D(A_0)$, including each of these totalizations. + +By \cite[Lemma~4.2.12]{BMR2026}, the coefficient objects are derived +$I_0$-complete and their derived reductions are the characteristic-$p$ +terms in the augmented diagram. The comparison cone is again derived +complete, since complete objects are closed under limits and cofibers. +Its reduction modulo $I_0$ vanishes by the characteristic-$p$ limit +diagram. If $I_0=(x_1,\ldots,x_c)$, each quotient by +$(x_1^m,\ldots,x_c^m)$ has a finite filtration with subquotients finite +free over $R_0$. The cone therefore vanishes modulo all these quotients. +Their ideals are cofinal with powers of $I_0$, so derived completeness +implies that the cone vanishes. Tensoring with an invertible even +coefficient line also preserves limits. This proves the required +individual-coefficient comparison for every even degree at every +endpoint; the odd coefficient modules vanish. The argument uses the +perfect Koszul quotient for the tensor--totalization interchange. + +\paragraph{From coefficient descent to spectral descent.} +These coefficient comparisons also give the spectral comparisons in +ordinary spectra. They give descent for each Eilenberg--Mac Lane +layer, because the forgetful functor from $H\Z$-modules preserves limits; +exactness of totalization then gives descent for every finite Postnikov +interval. For an augmented cosimplicial spectrum +$Z^{-1}\to Z^\bullet$, fix $b$ and $q\leq b$. The canonical fiber sequence +\[ + \tau_{[q-1,b]}Z\longrightarrow\tau_{\leq b}Z + \longrightarrow\tau_{\leq q-2}Z +\] +has a $(q-2)$-coconnective last term, and totalization preserves this +bound. Hence the first arrow induces an isomorphism on $\pi_q$ both on +the augmentation object and after totalization. The finite-interval +comparison therefore proves the degree-$q$ comparison for +$\tau_{\leq b}Z$. Degrees above $b$ vanish on both sides. +Thus $\tau_{\leq b}Z^{-1}\to\operatorname{Tot}\tau_{\leq b}Z^\bullet$ +is an equivalence for every $b$. Taking the limit over $b$, using +Postnikov completeness of ordinary spectra and commutation of limits, +proves the spectral comparison without a bounded-below assumption. + +\paragraph{Descent along the completed tower.} +Apply the descent result to \((R,\Gamma)=(k,\Gamma_2)\) and the +\(P_3^1\)-bar diagram of \(\widehat L\), augmented by the constants +\(k\); its map of theories is \(c^\square\). To justify the application +in the solid category, evaluate first on an extremally +disconnected profinite parameter set $T$. Such evaluation is exact on +condensed modules and preserves limits; moreover, the forgetful functor +from derived commutative algebras creates limits and detects equivalences. +Thus \eqref{eq:finite-constant-map} identifies +\[ + C_{\cts}(T,k)\ \simeq\ + \operatorname{Tot} + C_{\cts}\bigl((P_3^1)^{\bullet}\times T,\widehat L\bigr) +\] +in derived commutative rings. All displayed rings are perfect and +flat over $k$, hence relatively perfect. The restricted descent argument +therefore applies to this diagram. The endpoint formulas of +Proposition 4.5.5 and the cotensor comparison of Remark 4.5.6 in +\cite{BMR2026}, with the witness verification above, identify its +spectral and coefficient terms with the evaluations of the required +solid objects and their profinite cotensors. Extremally +disconnected parameter sets detect equivalences; thus this proves both +the spectral and individual-coefficient assertions for \(c^\square\). +Odd coefficient modules vanish on both sides. This also proves the +specialization of \cite[Theorem 5.2.6]{BMR2026}, whose numerical +hypothesis is exactly $p-1\nmid2$. +Every comparison is induced by the same augmented diagram and its +commuting actions. It therefore retains the residual \(H\)-action and +the maps from Witt constants; \(\overline Z\) acts trivially on the +\(E_2\) source. + +\subsection{The map from height three} + +The connected theory now has the coefficient descent and locality needed +for detection. To map the height-three sphere into it, we start with the +tensor construction and then pass from the full completed group to its +connected summand. The direction of this last map requires a section of +the deformation functor, rather than the forgetful transformation alone. + +The uncompleted tensor maps are the starting point. Examples~4.6.2--4.6.3 of +\cite{BMR2026} use the pointwise constructions of Theorems~4.4.6 and~4.4.8, +followed by Construction~4.5.2, to give +\[ + E_3^{\square}\longrightarrow L_{K(2)}^{\square}E_3^{\square} + \longrightarrow B^{\square}\longleftarrow E_2^{\square}, +\] +where +\[ + B^{\square} + =L_{K(2)}^{\square} + (E_3^{\square}\otimes_{SW(k)}^{\square}E_2^{\square}) + \simeq E^{\square}(L,\Gamma^{\mathrm{un}}_{3,L}). +\] +Here \(SW(k)\) is the spherical Witt-vector ring and +\(L_{K(2)}^{\square}\) is localization in solid spectra. These maps have +the projection actions, as in \cite[(5.2.2)--(5.2.3)]{BMR2026}. +The tensor construction uses the constant pair \((k,\Gamma_2)\), whose +base is Noetherian and perfect and whose group is connected of height +two, as required in \cite[Construction~4.4.7]{BMR2026}. The fixed-witness +descent argument above identifies the required hypersheafifications once +the following regular quotient is checked. + +For the localized $K$-pair and its uncompleted ind-\'{e}tale $L/K$ +extension, the witness has classical deformation ring +\[ + A_K=\bigl(W(k)[[u_1,u_2]][u_2^{-1}]\bigr)^{\wedge}_{(p,u_1)}, + \qquad A_K/(p,u_1)=K. +\] +The sequence $(p,u_1)$ remains regular under localization and this +Noetherian completion. The localized witness is Noetherian Luriean +by \cite[Proposition~4.4.4 and Remark~4.2.2]{BMR2026}; the displayed +$p$-basis below also verifies that $K$ is $F$-finite. The profinite witness hypotheses are +\cite[Propositions~4.5.9, 4.5.10, and 4.5.12, Definition~4.5.11, +and Examples~4.6.1--4.6.3]{BMR2026}. +Concretely, $K(T)=C_{\cts}(T,K)$ is flat over the field $K$ and +relatively perfect: the decomposition +$K=\bigoplus_{j=0}^{p-1}u_2^jK^p$ has continuous coefficient-extraction +maps and hence applies to continuous functions on $T$. +Compactness of $T$ supplies a uniform denominator for inversion of +$u_2$. This verifies the local-field instance directly; no commutation +of arbitrary localization with totalization is required. +The ind-\'{e}tale $L$-extension retains the same witness as in +Proposition~4.5.12. + +\begin{lemma}[The completed oriented projection] +\label{lem:completed-oriented-map} +Put \(F=\widehat L(*)\), regarded here as the ordinary completed field. +For a nonempty profinite set \(T\), put \(R_T=C_{\cts}(T,F)\) and let +\(\Gamma_T=\Gamma^{\mathrm{un}}_{3,R_T}\) be the full completed +\(p\)-divisible group. Let \(C_T\) be the base change of the connected +term in the uniquely split sequence over \(F\). It is identified with +\((\Gamma_2)_{R_T}\) by the fixed tautological Gross--Torii formal +isomorphism. There is a continuous adic \(E_\infty\)-map +\[ + r_T:E(R_T,\Gamma_T)\longrightarrow E(R_T,C_T). +\] +The maps are natural for profinite pullbacks and every continuous +\(G=P_3\times P_2\) family of pair maps. With the terminal value at the +empty parameter set, pointwise application and hypersheafification give +a continuous \(G\)-equivariant map of condensed \(E_\infty\)-rings +\[ + r^{\mathrm{cond}}: + E^{\mathrm{cond}}(\widehat L,\Gamma^{\mathrm{un}}_{3,\widehat L}) + \longrightarrow E^{\mathrm{cond}}(\widehat L,C_{\widehat L}). +\] + +Let \(b_T:E_2\to E(R_T,\Gamma_T)\) be the pointwise ordinary +\(E_2\) tensor-factor map followed by completion, and let +\(c_T:E_2\to E(R_T,C_T)\) be the canonical connected map induced by +\(k\to R_T\). Then \(r_Tb_T\simeq c_T\), naturally for the same +parameters and pair families. This homotopy preserves the connected +residue tag and orientation, and hence its coefficient and even-line maps. +After condensation and solid localization, it gives +\[ + r^{\mathrm{cond}}b^\square\simeq c^\square +\] +as continuous \(G\)-equivariant maps of underlying solid spectra, where +\(b^\square\) is the tensor-factor map followed by completion and +\(c^\square\) is the canonical connected base-change map. +\end{lemma} + +\begin{proof}[Construction; full proof in \cref{app:oriented-projection}] +Fix a parameter set \(T\). Write \(\mathscr D_\Gamma^{\mathrm{or}}\) +and \(\mathscr D_C^{\mathrm{or}}\) for oriented deformations with the +specified residue tags of the full and connected reference groups. +The connected--\'{e}tale sequence is uniquely split over the perfect +field \(F\). Let \(Q_T\) be the base change of its \'{e}tale quotient, +so \(\Gamma_T=C_T\oplus Q_T\). An oriented deformation \(C_A\) +over a complete adic test ring \(A\) has a +contractible space of lifts of the tagged \'{e}tale quotient \(Q_T\). +With such a marked lift \(Q_A\), send it to +\(C_A\oplus Q_A\), retaining its connected orientation. This constructs +\(\mathfrak s\) in the left diagram: +\begin{equation}\label{eq:oriented-section-diagram} +\begin{tikzcd}[column sep=large] + \mathscr D_C^{\mathrm{or}} + \arrow[r,bend left=18,"\mathfrak s"] + &\mathscr D_\Gamma^{\mathrm{or}} + \arrow[l,bend left=18,"\mathfrak f"] +\end{tikzcd} +\qquad +\begin{tikzcd}[column sep=large] + E(R_T,C_T)\arrow[r,bend left=18,"i_T"] + &E(R_T,\Gamma_T)\arrow[l,bend left=18,"r_T"] . +\end{tikzcd} +\end{equation} +Here \(\mathfrak f\) takes the connected deformation, and +\(\mathfrak f\mathfrak s\simeq\mathrm{id}\). Corepresentability +reverses the arrows and gives \(r_Ti_T\simeq\mathrm{id}\). +In particular the required map is \(r_T\), supplied by the split +section. The forgetful map alone supplies \(i_T\) in the other direction. + +The tensor-factor identity keeps track of which connected map results. +The tensor construction compares the two oriented formal groups through +their common Quillen formal group. On the split deformation its comparison +is the inverse of the connected inclusion. The resulting connected tag +and orientation are therefore exactly those defining \(c_T\), so +\(r_Tb_T\simeq c_T\). The appendix verifies this cancellation on the +actual tags and proves its compatibility with adic refinement and every +continuous group-family map. It then upgrades the identity to the +continuous solid maps in the statement. This identifies the right-hand +tensor composite with the canonical map used in coefficient descent. +\end{proof} + +Completion of the pair diagrams followed by the constructed map gives +\[ + B^\square\longrightarrow + E^{\mathrm{cond}}(\widehat L,\Gamma^{\mathrm{un}}_{3,\widehat L}) + \xrightarrow{r^{\mathrm{cond}}}\widehat B^\square. +\] +The completion map is \cite[Example~4.6.4]{BMR2026}; the full completed +pair uses the perfect-field witness verified in +\cref{app:oriented-projection}. The composite from \(E_3^\square\) +is therefore an actual continuous equivariant ring map. Take this +composite and \(c^\square\) as the arrows in \eqref{eq:completed-map}. +The later detection argument uses this unital equivariant map from +\(E_3^\square\) and the coefficient descent for \(c^\square\). +The identity \(r^{\mathrm{cond}}b^\square\simeq c^\square\) also +identifies the right arrow with the \(E_2\) tensor composite. +This construction uses no coefficient formula +or locality assertion for the nonconnected intermediate, and does not +assert that \(K\to\widehat L\) is relatively perfect. + +\subsection{The ordinary sphere source} + +It remains to identify the source and the sphere unit after evaluation. +We first spell out the effect of hypersheafification at the point. Let \(a_{\mathrm{hyp}}\) be +hypersheafification on profinite sets with the finite jointly surjective +topology. Every covering sieve of the one-point set is maximal: a +nonempty member of a covering family has a point and hence a section. +Consequently associated-sheaf formation for abelian presheaves leaves +their point value unchanged. The unit +\(P\to a_{\mathrm{hyp}}P\) for a spectrum-valued presheaf is an +isomorphism on associated homotopy sheaves. Evaluating these sheaves at +the point therefore gives an isomorphism on every ordinary homotopy +group, and hence +\[ + P(*)\xrightarrow{\sim}(a_{\mathrm{hyp}}P)(*). +\] +This proof applies to the accessible presheaves on full profinite sets +used here and is natural in the unit. It is the full-profinite version of +\cite[Lemma~13.5 and Remark~13.6]{Clausen2025}; the discrete spectra are +solid by Example~13.4(1) there. + +Definition~4.5.7 of \cite{BMR2026} computes condensed localization by +pointwise ordinary localization followed by hypersheafification, and its +restriction to solid spectra is solid localization. Discrete condensation +is the hypersheafification of the constant presheaf, so +\(V^\delta(*)\simeq V\). Thus, naturally in an ordinary spectrum \(V\) +and its maps, +\[ + \bigl(L_{K(n)}^\square(V^\delta)\bigr)(*)\simeq L_{K(n)}V, +\] +with the evaluated localization unit equal to the ordinary unit. +In particular the input \(V=S\) is the ordinary sphere before any +completion. Corollary~4.6.8 of \cite{BMR2026} constructs +\(L_{K(n)}^\square S^\delta\to E_n^\square\) by applying this functor +to the ordinary map \(L_nS\to E_n\). The natural point comparison +identifies it with +\[ + L_{K(n)}L_nS\longrightarrow L_{K(n)}E_n. +\] +Since \(S\to L_nS\) is a \(K(n)\)-equivalence and \(E_n\) is +\(K(n)\)-local, this is precisely the factor of the ordinary sphere +unit \(S\to E_n\) through \(L_{K(n)}S\). Lemma~4.5.8 and the endpoint +formula of Proposition~4.5.5 in that paper identify the target at the +point naturally with \(E_n\): the discrete-adjunction map used there is +adjoint at the point to the identity of \(E_n\). + +Proposition~4.6.9 of \cite{BMR2026} identifies the \v Cech augmentation +of that same map with +\(L_{K(n)}^\square S^\delta\xrightarrow{\sim}(E_n^\square)^{hG_n}\). +Its composite to \(E_n^\square\) is the map just identified. Evaluation +at the point preserves limits, so this gives the asserted ordinary +fixed-point equivalence with its canonical map to \(E_n\). +The coefficient-field scope is also exact: Example~4.6.1 there allows an +algebraic field containing the endomorphism field. Here +\(k=\F_{p^6}\) is finite and contains both required fields. The Honda +groups descend to \(\F_p\), and all geometric endomorphisms are +\(k\)-rational. The descent identifications from the \(\F_p\)-models split +the pair-automorphism sequences, giving exactly +\(G_n=P_n\rtimes\Gal(k/\F_p)\). Thus the cited solid descent theorem +applies to this enlarged finite field, and the point comparison above +identifies its source and map for the uncompleted sphere \(S\). + +\begin{proof}[Proof of \cref{prop:completed-theory}] +The connected endpoint calculation gives the coefficients and ordinary +locality of every evaluation. The tensor construction and the oriented +section supply the left equivariant ring map; the right map is the +canonical connected base change already constructed. +The constants calculation and finite Koszul +argument prove both fixed-point descent and descent for each coefficient +module, with their residual actions and maps from Witt constants. +Taking derived \(H\)-invariants in the extension +\(P_3^1\to G\to H\) gives \eqref{eq:coefficient-descent}; +\(\overline Z\) acts trivially on the \(E_2\) source. +The preceding point-evaluation argument proves the ordinary +stabilizer-descent identification with its sphere unit. +\end{proof} + +\begin{remark} +The completion in \eqref{eq:completed-map} is essential to the statements +proved here. The uncompleted half-sphere equivalence is +\cite[Conjectures 1.1.1 and 5.1.13]{BMR2026}; +\cite[Proposition 5.1.17]{BMR2026} proves an implication from that +conjecture to coheight-one splitting. The completed theorem above +supplies an actual target for detection and does not assert an +equivalence from the uncompleted fixed points to that target. +\end{remark} diff --git a/preprints/A-rational-obstruction-to-strong-chromatic-splitting-at-height-three-September-25-2026/build/sections/04-modifications.tex b/preprints/A-rational-obstruction-to-strong-chromatic-splitting-at-height-three-September-25-2026/build/sections/04-modifications.tex new file mode 100644 index 0000000000000000000000000000000000000000..62fe49a3465f97c378ebc471490f61db6d76e4ba --- /dev/null +++ b/preprints/A-rational-obstruction-to-strong-chromatic-splitting-at-height-three-September-25-2026/build/sections/04-modifications.tex @@ -0,0 +1,351 @@ +\section{Two realizations of a modification}\label{sec:modifications} + +We compare two descriptions of a line modification of a rank-$n$, +degree-one bundle. Framing the original bundle gives a projective +space with a division-algebra action; framing its slope-zero +modification gives a Drinfeld space with a linear-group action. +The common quotient will carry both degree-three classes. We then +apply the same modification to the universal rank-three-to-rank-two +surjection, obtaining the exact sequence that relates the two heights. + +We work over $\Spd C$. For a perfectoid test object $S$ over this +base, write $\mathcal X_S^{\mathrm{FF}}$ for its relative +Fargues--Fontaine curve, with untilt divisor +$i_\infty:\infty\hookrightarrow\mathcal X_S^{\mathrm{FF}}$. +At the fixed geometric point determined by $C$, write +$\mathcal X_C^{\mathrm{FF}}$ for the corresponding curve. +For a locally profinite group $A$, write +$B_CA=[\Spd C/\underline A]$ for its relative classifying stack; +all classifying maps in this section are over this base. +Write $\mathcal E_n^0=\cO(1/n)$, of rank $n$ and degree one. +A \emph{determinant lattice} of a slope-zero rank-$n$ bundle means a +$\Z_p$-lattice in its rank-one determinant local system. +Its frame group is +\[ + J_n=\{g\in\GL_n(\Q_p):|\det(g)|_p=1\}. +\] +This is the structure that will relate the two heights. + +We use the following precise forms of the relative bundle theorems. +Let $\bar k$ be an algebraic closure of $k$. Since $C^\flat$ is +algebraically closed, fix an extension $\bar k\hookrightarrow C^\flat$ +of the embedding $k\hookrightarrow C^\flat$ underlying the chosen map +$\Spd C\to\Spd k$, and hence a compatible map +$\Spd C\to\Spd\bar k$. Geometrically semistable bundles of slope zero +correspond to pro-\'etale $\Q_p$-local systems, by +\cite[Theorem II.2.19 and Corollary II.2.20]{FS}. +Pulling the geometrically trivial and basic strata of +\cite[Theorems III.2.4 and III.4.5]{FS} back along this map gives the +relative classifying descriptions $B_C\GL_n(\Q_p)$ and +$B_C\Aut(\mathcal E_n^0)$, respectively. +Thus their sheaves of frames are actual torsors in families. +The later arithmetic $G_F$-descent is established by the local +construction below. We will apply the bundle statements after proving +the required slope conditions. Geometric points alone are not being +used to identify arbitrary stacks. + +\subsection{The lower and upper modifications} + +\begin{lemma}\label{lem:lower-modification} +Let $\mathcal E_n$ be a form of $\mathcal E_n^0$, and let +$q:i_\infty^*\mathcal E_n\twoheadrightarrow\mathcal L$ be a quotient +line. The elementary lower modification +\[ + 0\longrightarrow\mathcal V_n\longrightarrow\mathcal E_n + \longrightarrow i_{\infty *}\mathcal L\longrightarrow0 +\] +is geometrically semistable of slope zero. It therefore defines a +rank-$n$ rational local system. +\end{lemma} + +\begin{proof} +The kernel is a vector bundle: locally at the Cartier divisor, the +quotient is a coordinate modulo a local equation of the divisor, and +the kernel has a basis obtained by multiplying that coordinate by the +local equation. At a geometric point, +$\rank(\mathcal V_n)=n$ and $\deg(\mathcal V_n)=0$. +If it had a positive-slope Harder--Narasimhan piece, its maximal +destabilizing subbundle $\mathcal F$ would have rank $b\frac1n + =\mu(\mathcal E_n). +\] +One may saturate its image in $\mathcal E_n$ without decreasing degree. +Hence $\mathcal V_n$ is semistable of degree zero at every geometric +point. The relative slope-zero equivalence stated above now applies. +\end{proof} + +Conversely, an elementary upper modification of $\cO^n$ at $\infty$ +is determined by a line of modification directions. Locally at a +geometric point, if $t$ is a uniformizer at $\infty$ and +$L_0=(B_{\mathrm{dR}}^+)^n$, its lattice is +\[ + L_\ell=L_0+B_{\mathrm{dR}}^+t^{-1}\widetilde\ell + \subset t^{-1}L_0 , +\] +where $\widetilde\ell$ lifts the direction line $\ell$. +Intrinsically the direction lies in +$(\cO(\infty)/\cO)^n$; tensoring every coordinate by the same +one-dimensional space identifies its projectivization with +$\mathbb P^{n-1}_C$. + +\begin{lemma}\label{lem:drinfeld-locus} +The upper modification $\mathcal E_\ell$ is semistable of slope $1/n$ +if and only if $\ell$ lies in no proper $\Q_p$-rational subspace. +Consequently its semistable locus is +\[ + \Omega_C^{n-1} + =\mathbb P_C^{n-1}\setminus + \bigcup_{\substack{W\subset\Q_p^n\\ \dim W=n-1}}\mathbb P(W_C). +\] +On this locus $\mathcal E_\ell$ is a form of $\mathcal E_n^0$. +\end{lemma} + +\begin{proof} +If $\ell\subset W_C$ for a proper rational subspace $W$, modify +$W\otimes_{\Q_p}\cO$ along $\ell$. The resulting rank-$\dim W$, +degree-one bundle embeds as a subbundle of $\mathcal E_\ell$; the +quotient is the unchanged trivial bundle associated to $\Q_p^n/W$. +Its slope is greater than $1/n$, so $\mathcal E_\ell$ is not semistable. + +For the converse, take a saturated destabilizing subbundle +$\mathcal F\subset\mathcal E_\ell$ of rank $bn-1$; +the finite Hom calculation below then makes every ordinary +point-valued $E_2^{s,j}$ in that row zero. No vanishing assertion for +the entire condensed module $H^j(K_n)$ is needed. + +Choose the finite projective resolution +$P_\bullet\to\Z_p$ over $\Lambda=\Z_p[[U]]$ from +\cref{lem:solid-resolutions}. The unit +$\underline{\Z_p}\to K_n$ factors through $H^0(K_n)[0]$, since +$K_n$ is concentrated in cohomological degrees at least zero. +The degree-zero calculation and local constants argument following +\cref{thm:drinfeld} show that this unit is an isomorphism on point +values. Each $\Hom_\Lambda(P_i,-)$ is a retract of a finite sum of +point values, and the equivariant unit commutes with the idempotent +defining that retract. It follows that the unit gives an isomorphism +of the integral bottom-row computing complexes +\[ + \Hom_\Lambda(P_\bullet,\underline{\Z_p}) + \xrightarrow{\ \sim\ } + \Hom_\Lambda(P_\bullet,H^0(K_n)). +\] +After $[1/p]$, \eqref{eq:group-cochain-bridge} and +\eqref{eq:geometric-group-descent} identify the restricted group +class with the constants bottom-row edge class induced by +$u_{\Omega_C^{n-1},U,\mathrm b}$. This uses only the point-value +isomorphism, not an identification of the entire condensed module. + +For a fixed $j$, the rationalized row is +computed by +\[ + \Hom_\Lambda(P_\bullet,H^j(K_n))[1/p]. +\] +Every term is a direct summand of a finite sum of +$H^j(K_n)(*)[1/p]$. By the integral Drinfeld theorem, +\cite[Theorem 1.1]{CDN2021}, a Galois element $\gamma$ acts on the latter +by the scalar $\chi_{\mathrm{cyc}}(\gamma)^{-j}$. +It commutes with $\Lambda$ and with the idempotents defining the +projective summands, so it has the same scalar action on the entire +row and on its cohomology. No identification of a solid module by +its point value is involved. + +The constants row $j=0$ has trivial Galois action. +Choose $\gamma\in G_F$ with $\chi_{\mathrm{cyc}}(\gamma)$ of infinite +order. The only possible incoming differentials to bidegree $(3,0)$ +are +\[ + d_2:E_2^{1,1}\longrightarrow E_2^{3,0}, + \qquad + d_3:E_3^{0,2}\longrightarrow E_3^{3,0}. +\] +The second source is absent when $n=2$. +After inversion of $p$, the respective source scalars are +$\chi_{\mathrm{cyc}}(\gamma)^{-1}$ and +$\chi_{\mathrm{cyc}}(\gamma)^{-2}$, whereas the target scalar is $1$. +Every differential commutes with $\gamma$, so both are zero. +There are no outgoing differentials from the bottom row. +Hence the nonzero bottom-row image of the restriction of $e_{J_n}$ +survives in $H^3_{\mathrm b}([\Omega_C^{n-1}/U])$. +Its abutment image is +$u_{\Omega_C^{n-1},U,\mathrm b}(\operatorname{res}^{J_n}_U e_{J_n})$ +by the unit calculation +above. Naturality \eqref{eq:constant-cochain-naturality} for +$U\hookrightarrow J_n$ identifies it with the pullback of +$u_{\Omega_C^{n-1},J_n,\mathrm b}(e_{J_n})$. +Thus the class on $[\Omega_C^{n-1}/J_n]=\mathcal M_n$ is nonzero. + +Finally, that class is $G_F$-invariant. Its classifying map to +$B_CJ_n$ is equivariant for the action on the $\Spd C$ base, while +$G_F$ fixes $J_n$ and the untwisted finite coefficients. Naturality +of $u_{B_C,J_n,m}$ for this base action passes through the derived +coefficient limit and rationalization. It therefore sends the +invariant group class to an invariant geometric class. +Equation~\eqref{eq:invariant-line} therefore places both nonzero +classes on the same one-dimensional $\Q_p$-line, proving +\cref{eq:primitive-comparison}. +\end{proof} + +\subsection{Transport along the rank-one filtration} + +\begin{theorem}\label{thm:geometric-transport} +There is a scalar $a\in\Q_p^\times$ such that the classes from the two +stabilizer factors satisfy +\begin{equation}\label{eq:geometric-transport} + e_{P_3}|_{Y_C}=a\,e_{P_2}|_{Y_C} + \quad\text{in }H^3_{\mathrm b}(Y_C). +\end{equation} +\end{theorem} + +\begin{proof} +By \cref{lem:parabolic}, for the inclusion $\rho_3:D\to J_3$ and +projection $\rho_2:D\to J_2$ there is a scalar $b\in\Q_p^\times$ +with +\[ + \rho_3^*e_{J_3}=b\,\rho_2^*e_{J_2} + \quad\text{in }H^3_{\mathrm b}(D). +\] +Apply $u_{B_C,D,\mathrm b}$ to this group-cohomology equality and +pull it back along $d:\mathcal T\to B_CD$. +The exact filtration in \cref{prop:filtered-modification} identifies +the two $J_n$-classifying maps with the composites induced by +$D\to J_n$, and \eqref{eq:constant-cochain-naturality} identifies +their pullbacks with these same group classes. The $P_n$-framing +torsors in that proposition give the first and last equalities below. +Consequently +\[ + \begin{aligned} + \pi^*(e_{P_3}|_{Y_C}) + &=m_3^*(e_{P_3}|_{\mathcal M_3})\\ + &=\kappa_3\,d^*\rho_3^*e_{J_3}\\ + &=\kappa_3b\,d^*\rho_2^*e_{J_2}\\ + &=\frac{\kappa_3b}{\kappa_2}\, + m_2^*(e_{P_2}|_{\mathcal M_2})\\ + &=\frac{\kappa_3b}{\kappa_2}\,\pi^*(e_{P_2}|_{Y_C}). + \end{aligned} +\] +All three scalars are nonzero. The projective morphism +$\pi:\mathcal T\to Y_C$ satisfies +\cref{lem:projective-bundle}, so $\pi^*$ is injective. +Taking $a=\kappa_3b/\kappa_2$ proves the theorem. +\end{proof} + +This transports the primitive through a filtration of rational local +systems. The equality itself does not assert nonvanishing on $Y_C$. +The next section tests its coefficient image using the completed +tower and proves the required nonvanishing there. diff --git a/preprints/A-rational-obstruction-to-strong-chromatic-splitting-at-height-three-September-25-2026/build/sections/06-witt-transfer.tex b/preprints/A-rational-obstruction-to-strong-chromatic-splitting-at-height-three-September-25-2026/build/sections/06-witt-transfer.tex new file mode 100644 index 0000000000000000000000000000000000000000..22c7a5d018cd34b1d18eaf471a8220d6701c347f --- /dev/null +++ b/preprints/A-rational-obstruction-to-strong-chromatic-splitting-at-height-three-September-25-2026/build/sections/06-witt-transfer.tex @@ -0,0 +1,337 @@ +\section{Transfer through Witt coefficients} +\label{sec:witt-transfer} + +Witt cohomology supplies an injective base-change map with which to +reflect the relation of \cref{thm:geometric-transport} from $Y_C$ to +$Y$. We first prove this injectivity by reducing to trivial Witt +coefficients for a compact group. We then carry the relation to +deformation coefficients and prove nonvanishing there using the +rank-two theory and the retraction onto its Witt constants. + +Recall the notation +\[ + G=P_3\times P_2,\qquad + X=[\Spd\widehat L/P_3^1],\qquad + H=P_2\times\overline Z, + \quad \overline Z=P_3/P_3^1. +\] +Thus $Y=X/H$, with the residual action supplied by quotient +descent. For any of these stacks $Z$, write +\[ + R\Gamma_v(Z,W(\cO^\flat)) + :=\Rlim_m R\Gamma_v(Z,W_m(\cO^\flat)), + \qquad + H_W^i(Z):=H^i\bigl((R\Gamma_v(Z,W(\cO^\flat)))(*)\bigr). +\] +The derived global sections are solid complexes; \((*)\) is the exact, +limit-preserving point evaluation, so \(H_W^i(Z)\) is an ordinary abelian +group. This is an integral definition. Inverting $p$ below always takes +place after this derived limit and after cohomology. + +\subsection{Witt constants and base change} + +In \cite[Remark~3.6.12(2)]{BMR2026}, Barthel and coauthors propose +studying mixed-characteristic Witt cohomology on the unquotiented +punctured Banach--Colmez space. Here the finite norm-unit quotient +$P_3^1/\ker N_3\simeq\mu_{p-1}$ has already removed the higher +characteristic-$p$ contribution in \cref{prop:constant-acyclicity}. +That constants equivalence lifts through the finite Witt levels, +giving the following more restricted comparison. + +\begin{proposition}[Witt comparison]\label{prop:witt-comparison} +The natural constant maps give the commutative square of integral +cohomology groups +\begin{equation}\label{eq:witt-square} +\begin{tikzcd}[column sep=large,row sep=large] + H^i(H,W(k)) \arrow[r,"\sim"] \arrow[d] + & H_W^i(Y) \arrow[d] + \\ + H^i(H,W(C^\flat)) \arrow[r,"\sim"] + & H_W^i(Y_C). +\end{tikzcd} +\end{equation} +The actions of $H$ on the two left-hand coefficient rings are trivial. +The right vertical map is injective after inverting $p$, for every $i$. +Moreover, quotient descent gives a natural identification +\begin{equation}\label{eq:witt-group-identification} + R\Gamma_v(Y,W(\cO^\flat)) + \simeq R\Gamma(G,W(\widehat L)). +\end{equation} +\end{proposition} + +\begin{proof} +For a characteristic-$p$ ring $R$, restriction of Witt components and +Verschiebung give an exact sequence of additive groups +\begin{equation}\label{eq:witt-devissage} + 0\longrightarrow R\xrightarrow{\ V^{m-1}\ }W_m(R) + \longrightarrow W_{m-1}(R)\longrightarrow0 + \qquad(m\geq2). +\end{equation} +The sequence is natural for ring maps and automorphisms. We use it as +a sequence of additive coefficient objects; no assertion of linearity +over $R$ is needed. It also gives an exact sequence of the associated +solid modules and of the corresponding sheaves on the v-site. + +\Cref{prop:constant-acyclicity} identifies the actual constant maps +\[ + k\longrightarrow R\Gamma(P_3^1,\widehat L),\qquad + C^\flat\longrightarrow R\Gamma_v(X_C,\cO^\flat) +\] +as equivalences with their residual $H$-actions. As proved there, +the characteristic-$p$ vanishing uses only the central scalar +restrictions of \cite[Propositions~3.5.6 and~3.6.9]{BMR2026}, identified +with $T_3$ by the determinant calculation in \cref{prop:tower}, and finite +norm-unit +invariants as in \cite[Lemma~5.2.5]{BMR2026}. Full $H$-equivariance +comes from the actual maps of constants and quotient descent. +Apply \eqref{eq:witt-devissage} to the source and target of each map. +Induction on $m$, using the resulting morphisms of exact triangles, +gives equivalences +\begin{align*} + W_m(k)&\xrightarrow{\sim} + R\Gamma(P_3^1,W_m(\widehat L)),\\ + W_m(C^\flat)&\xrightarrow{\sim} + R\Gamma_v(X_C,W_m(\cO^\flat)). +\end{align*} +Every map used in this induction is the map induced by constants. +In particular the equivalences are $H$-equivariant, including the +$\overline Z$-action; $H$ fixes the constant rings. + +Derived global sections and derived invariants preserve limits. +Passing to $\Rlim_m$ therefore gives the same equivalences with full +Witt coefficients. Here the ordinary Witt module is also the derived +limit of its truncations: in Witt coordinates, restriction has the +continuous set-theoretic section obtained by appending a zero. +Consequently restriction is surjective on continuous functions from +every profinite parameter space. The truncation towers have no higher +derived-limit term in the solid coefficient category. The argument +applies to $k$, $C^\flat$, and $\widehat L$ with their respective +topologies. Taking residual $H$-invariants gives the horizontal +isomorphisms of \eqref{eq:witt-square}. Naturality of the constant +maps under $k\to C^\flat$ gives its commutativity. + +For \eqref{eq:witt-group-identification}, recall +$\widetilde Y=\Spd\widehat L$ from \cref{sec:completed-tower}. +This affinoid perfectoid space is acyclic for $\cO^\flat$, also with +profinite parameters. Applying \eqref{eq:witt-devissage} gives, for +every profinite set $S$, the finite equivalence induced by sections +\[ + \alpha_{S,m}:C_{\cts}(S,W_m(\widehat L))[0] + \xrightarrow{\ \sim\ } + R\Gamma_v(\widetilde Y\times\underline S,W_m(\cO^\flat)). +\] +Here Witt coordinates identify $W_m(C_{\cts}(S,\widehat L))$ with +$C_{\cts}(S,W_m(\widehat L))$. The equivalence is natural in $S$ +and in the $G$-action. Write $u^a_{\widetilde Y,G,m}$ for the +degree-$a$ map of \eqref{eq:finite-constant-cochains}. The actual +finite coefficient units give the commutative square +\begin{equation}\label{eq:finite-witt-constant-square} +\begin{tikzcd}[column sep=large] + C_{\cts}(G^a,\Lambda_m)[0] + \arrow[r,"u^a_{\widetilde Y,G,m}"] \arrow[d,"\iota_m"'] + &R\Gamma_v(\widetilde Y\times\underline{G^a},\Lambda_m) + \arrow[d,"\omega_m"]\\ + C_{\cts}(G^a,W_m(\widehat L))[0] + \arrow[r,"\alpha_{G^a,m}","\sim"'] + &R\Gamma_v(\widetilde Y\times\underline{G^a},W_m(\cO^\flat)). +\end{tikzcd} +\end{equation} +The left map applies $\Lambda_m=W_m(\F_p)\to W_m(\widehat L)$ +pointwise; the right map is induced by the corresponding sheaf unit. +Both composites are the same locally constant Witt section. These +squares commute with the bar faces, including the action face, and +with restriction in $m$. The same calculation with an extra +profinite parameter retains the solid structure. The condensed +standard resolution, quotient descent, and then $\Rlim_m$ therefore +give \eqref{eq:witt-group-identification}. The full Witt module is +the derived limit of its truncations as shown above. Under point +evaluation of this identification, the Witt image of +$u_{\widetilde Y,G,\mathrm{int}}$ is exactly the coefficient map +induced by $\Z_p\to W(\widehat L)$ and +\eqref{eq:group-cochain-bridge}. + +It remains to prove injectivity, since an equivalence of the horizontal +groups alone would not imply it. The finite-resolution statement of +\cref{lem:solid-resolutions} applies to $H$. Indeed an element of +order $p$ in $P_2$ would have minimal polynomial of degree $p-1>2$ +over $\Q_p$, whereas the reduced characteristic polynomial in $D_2$ +has degree two. Thus $P_2$, and hence $H=P_2\times\Z_p$, has no +$p$-torsion. +Choose a bounded resolution $P_\bullet\to\Z_p$ by finitely generated +projective $\Z_p[[H]]$-modules. For a trivial solid coefficient module +$M$, the point-valued cochain complex is +\[ + \Hom_{\Z_p[[H]]}(P_\bullet,M) + =\Hom_{\Z_p}(Q_\bullet,M), + \qquad + Q_\bullet=\Z_p\otimes_{\Z_p[[H]]}P_\bullet. +\] +Every $Q_j$ is finite free over $\Z_p$. Thus this complex is obtained +from a fixed bounded finite free $\Z_p$-complex by scalar extension +to the underlying module of $M$; its differentials are the same +$\Z_p$-linear matrices. This description is natural in $M$ and is +valid for the valued Witt topology as well as for the $p$-adic one, +by \cref{lem:solid-resolutions}. + +Take $A=W(k)$ and $B=W(C^\flat)$. The map $A\to B$ is injective. +Since $A$ is a discrete valuation ring with uniformizer $p$ and $B$ +is $p$-torsion-free, $B$ is flat over $A$. Applying the finite +complex just described gives +\begin{equation}\label{eq:witt-scalar-extension} + H^i(H,A)[1/p]\otimes_{A[1/p]}B[1/p] + \xrightarrow{\sim}H^i(H,B)[1/p]. +\end{equation} +The field extension $A[1/p]\hookrightarrow B[1/p]$ is faithfully +flat, so the natural map from the first cohomology group to its +scalar extension is injective. The square now proves the assertion. +All infinite limits preceded this finite-complex argument. +\end{proof} + +\subsection{Reflecting the geometric relation} + +Via \eqref{eq:group-cochain-bridge}, let $c_n^W$ denote the image of +$e_{P_n}$ under projection from $G$ and the coefficient map +$\Z_p\to W(\widehat L)$: +\[ + c_n^W\in H^3(G,W(\widehat L))[1/p]\qquad(n=2,3). +\] +Write $c_n$ for its further image under the natural equivariant map +$W(\widehat L)\to\pi_0\widehat B^{\square}$ of +\cref{prop:completed-theory}. + +\begin{proposition}[Transport to deformation coefficients] +\label{prop:coefficient-transport} +For the nonzero scalar $a\in\Q_p^\times$ of +\cref{thm:geometric-transport}, one has +\begin{equation}\label{eq:witt-relation} + c_3^W=a\,c_2^W + \quad\hbox{in }H^3(G,W(\widehat L))[1/p], +\end{equation} +and hence +\begin{equation}\label{eq:deformation-relation} + c_3=a\,c_2 + \quad\hbox{in }H^3(G,\pi_0\widehat B^{\square})[1/p]. +\end{equation} +Under \eqref{eq:coefficient-descent}, the preimage of $c_2$ in +\[ + H^3(H,\pi_0E_2^{\square})[1/p] +\] +is obtained by inflating $e_{P_2}$ and then including constants. +\end{proposition} + +\begin{proof} +Let $\omega_{Z,m}$ be the map on cohomology complexes induced by +$\Lambda_m=W_m(\F_p)\to W_m(\cO^\flat)$ for $Z=Y$ or $Y_C$. +These maps are compatible with restriction in $m$. Their derived +inverse limit gives a map +\[ + H^3_{\mathrm b}(Z)\longrightarrow H_W^3(Z)[1/p]. +\] +Let $j:Y_C\to Y$ denote base change, and put +\[ + \beta_m=\omega_{Y,m}u_{\widetilde Y,G,m},\qquad + \beta_{C,m}=\omega_{Y_C,m}u_{\widetilde Y_C,G,m}. +\] +The finite constant maps and the finite Witt units give +\begin{equation}\label{eq:finite-witt-base-change} +\begin{tikzcd}[column sep=large] + C_{\cts}^{\bullet}(G,\Lambda_m) + \arrow[r,"\beta_m"] \arrow[d,equal] + &R\Gamma_v(Y,W_m(\cO^\flat))\arrow[d,"j^*"]\\ + C_{\cts}^{\bullet}(G,\Lambda_m) + \arrow[r,"\beta_{C,m}"'] + &R\Gamma_v(Y_C,W_m(\cO^\flat)). +\end{tikzcd} +\end{equation} +This square commutes in each cosimplicial degree before totalization, +and is compatible with $q_n:G\to P_n$ and reduction in $m$. +Take the derived coefficient limit, cohomology, and then invert $p$. +The classifying-pullback convention and +\eqref{eq:constant-cochain-naturality} give +$j^*(e_{P_n}|_Y)=e_{P_n}|_{Y_C}$. Moreover, +\eqref{eq:finite-witt-constant-square} identifies the Witt image of +$e_{P_n}|_Y$ with $c_n^W$ under +\eqref{eq:witt-group-identification}. Thus the geometric relation says +that $c_3^W-a c_2^W$ maps to zero in $H_W^3(Y_C)[1/p]$. +The right vertical map of \eqref{eq:witt-square} is injective after +inverting $p$, so this difference is already zero on $Y$. This proves +\eqref{eq:witt-relation}. Applying the coefficient map +$W(\widehat L)\to\pi_0\widehat B^{\square}$ then proves +\eqref{eq:deformation-relation}. + +Finally, \eqref{eq:coefficient-descent} is induced by the actual map +$E_2^{\square}\to\widehat B^{\square}$, by first taking +$P_3^1$-invariants and then $H$-invariants. Its compatibility with +constants is part of \cref{prop:completed-theory}, whose finite Koszul +argument proves the required coefficient instance of the descent +construction in \cite[Proposition~4.6.12]{BMR2026}. +The action of $\overline Z$ on the source is trivial. Thus +inflation from $P_2$ followed by this map is exactly the coefficient +map defining $c_2$. This proves the last assertion while retaining +the residual action. +\end{proof} + +\subsection{Nonvanishing in the deformation ring} + +The reflected relation reduces nonvanishing to $c_2$. Coefficient +descent identifies this class with constants in the rank-two theory; +the additive retraction onto Witt constants detects it there. + +\begin{lemma}[Nonvanishing of constants]\label{lem:constant-nonvanishing} +The class $c_2$ is nonzero. Consequently $c_3$ is nonzero as well. +\end{lemma} + +\begin{proof} +Set $A_k=\pi_0E_2^{\square}(*)$. The class corresponding to $c_2$ +under \eqref{eq:coefficient-descent} restricts along $P_2\hookrightarrow H$ +to the coefficient image of $e_{P_2}$ in $H^3(P_2,A_k)[1/p]$. +It suffices to prove that this image is nonzero. + +Choose an algebraic closure $\overline k$ of $k$ and let +$A_{\overline k}$ be the deformation ring of the same height-two +formal group after this base extension. Deformation theory gives +a natural continuous $P_2$-equivariant map +\[ + A_k\longrightarrow A_{\overline k}. +\] +In compatible coordinates these rings are $W(k)[[u'_1]]$ and +$W(\overline k)[[u'_1]]$, with their $(p,u'_1)$-adic topologies; +equivariance follows from deformation base change, without requiring +the coordinate to be fixed. Since $P_2$ is nonextended, it fixes +$W(\overline k)$ pointwise. + +The splitting theorem of Barthel--Schlank--Stapleton--Weinstein +\cite[Proposition~2.5.1 and Corollary~2.5.9]{BSSW2025} gives a +continuous additive equivariant retraction +\[ + r:A_{\overline k}\longrightarrow W(\overline k) +\] +of the inclusion of Witt constants. Restrict its equivariance to $P_2$. +Additivity and continuity make it $\Z_p$-linear. After condensation +it is a morphism of solid $P_2$-modules, and hence +induces a retraction on the continuous derived invariants used here. + +The composite of the coefficient maps +\[ + \Z_p\longrightarrow A_k\longrightarrow A_{\overline k} + \xrightarrow{r}W(\overline k) +\] +is the usual inclusion of constants. Applying the finite free +complex argument from \cref{prop:witt-comparison} with $P_2$ in +place of $H$ gives +\[ + H^3(P_2,W(\overline k))[1/p] + \cong H^3_{\mathrm b}(P_2) + \otimes_{\Q_p}W(\overline k)[1/p]. +\] +The image of $e_{P_2}$ here is $e_{P_2}\otimes1$, which is nonzero +by \cref{prop:group-classes}. If its image in +$H^3(P_2,A_k)[1/p]$ vanished, the displayed composite would also +send it to zero, a contradiction. Restriction to $P_2$ therefore +proves $c_2\ne0$, and \eqref{eq:deformation-relation} with $a\ne0$ +proves $c_3\ne0$. +\end{proof} + +We have thus identified the image of the degree-three group class +under the completed coefficient map. The remaining step is to +detect it in the homotopy of an ordinary $K(2)$-local spectrum. diff --git a/preprints/A-rational-obstruction-to-strong-chromatic-splitting-at-height-three-September-25-2026/build/sections/07-detection.tex b/preprints/A-rational-obstruction-to-strong-chromatic-splitting-at-height-three-September-25-2026/build/sections/07-detection.tex new file mode 100644 index 0000000000000000000000000000000000000000..065c051d57c0d9a56a74708ae2bef0b5e1208409 --- /dev/null +++ b/preprints/A-rational-obstruction-to-strong-chromatic-splitting-at-height-three-September-25-2026/build/sections/07-detection.tex @@ -0,0 +1,302 @@ +\section{Detection on ordinary spectra} +\label{sec:detection} + +We now turn the coefficient relation into a statement about the natural +localization map. Keep $G=P_3\times P_2$ and +$H=P_2\times\overline Z$, and put +\[ + \Delta=\Gal(k/\F_p),\qquad + G_3=P_3\rtimes\Delta,\qquad + \mathcal D=\bigl((\widehat B^{\square})^{hG}\bigr)(*). +\] +Thus $\Delta$ has order six. The symbol $\mathcal D$ denotes an ordinary spectrum: +the continuous fixed points are formed in solid spectra and then evaluated +at the point. + +\begin{lemma}[The ordinary local detector]\label{lem:local-detector} +There is a natural map of ordinary spectra +\begin{equation}\label{eq:ordinary-detector} + g:L_{K(3)}S\longrightarrow\mathcal D. +\end{equation} +The spectrum $\mathcal D$ is $K(2)$-local, so $g$ factors through the canonical +localization unit: +\begin{equation}\label{eq:detector-factorization} + L_{K(3)}S\xrightarrow{\eta}L_{K(2)}L_{K(3)}S + \xrightarrow{\overline g}\mathcal D. +\end{equation} +\end{lemma} + +\begin{proof} +Use the ordinary descent identification in +\cref{prop:completed-theory} to form the composite +\begin{equation}\label{eq:detector-construction} +\begin{split} + L_{K(3)}S + &\simeq\bigl((E_3^{\square})^{hG_3}\bigr)(*) + \longrightarrow\bigl((E_3^{\square})^{hP_3}\bigr)(*)\\ + &\longrightarrow\bigl((E_3^{\square})^{hG}\bigr)(*) + \longrightarrow\bigl((\widehat B^{\square})^{hG}\bigr)(*). +\end{split} +\end{equation} +The first arrow restricts the group action, the second is inflation along +$G\to P_3$, and the last is induced by the equivariant map +\eqref{eq:completed-map}. In the middle term with group $G$, the action on +$E_3^{\square}$ is through $P_3$. Restriction and inflation are the +natural maps of continuous fixed points. The source equivalence and its ordinary unit are the point-valued +form of \cite[Proposition~4.6.9]{BMR2026} proved in +\cref{prop:completed-theory}; the last map uses the local split-section +completed map of \cref{lem:completed-oriented-map} and that proposition. + +For every profinite set $T$, the evaluation +$\widehat B^{\square}(T)$ is the ordinary height-two Lubin--Tate theory +described in \cref{prop:completed-theory}, with the Noetherian witness +verified there; hence it is $K(2)$-local +\cite[Proposition~4.4.2]{BMR2026}. +The bar construction gives +\[ + \mathcal D\simeq\operatorname{Tot}\bigl(\widehat B^{\square}(G^{\bullet})\bigr). +\] +Ordinary local spectra are closed under limits, so $\mathcal D$ is +$K(2)$-local. This also follows from the +pointwise criterion for solid locality +\cite[Definition~4.5.7 and the following paragraph]{BMR2026}. +The universal property of ordinary $K(2)$-localization now gives +\eqref{eq:detector-factorization}. +\end{proof} + +\subsection{Convergence before rationalization} + +We use integral continuous cohomology of the solid homotopy modules in +\cref{lem:solid-resolutions}. For such a module $M$, recall that +$H^s(A,M)$ denotes the point evaluation of $R^s\Gamma(A,M)$. +In particular, $[1/p]$ below is applied to this integral group after it +has been computed. For $M=\Z_p$, the natural comparison +\eqref{eq:group-cochain-bridge} identifies these groups with the +finite continuous-cochain derived-limit convention of +\eqref{eq:integral-group-cochains}. + +The compact groups needed here have finite cohomological dimension in +this coefficient category. Indeed, if the degree-$n$ division algebra +defining $P_n$ contained an element of order $p$, it would contain the +field $\Q_p(\zeta_p)$: the nontrivial element has the irreducible +cyclotomic minimal polynomial of degree $p-1$. Its reduced +characteristic polynomial has degree $n$ and annihilates it. +Since $p-1>n$ for $n=2,3$ and $p\geq5$, this is impossible. +Thus neither $P_2$ nor $P_3$ has $p$-torsion. The same holds for +$G$ and $H$. An order-$p$ element of $G_3$ would project trivially to +$\Delta$, since $p\nmid6$, and hence would lie in $P_3$. Thus +\cref{lem:solid-resolutions} gives a uniform finite cohomological bound +for each of these groups. We only need the existence of a common bound +$d$, independent of the homotopy degree of the coefficient module. + +\begin{lemma}[Finite-amplitude descent]\label{lem:descent-convergence} +For the continuous actions used in \eqref{eq:detector-construction}, +there are natural, strongly convergent spectral sequences +\begin{equation}\label{eq:integral-descent-ss} + E_2^{s,t}=H^s(A,\pi_tX) + \ \Longrightarrow\ \pi_{t-s}\bigl((X^{hA})(*)\bigr), + \qquad 0\leq s\leq d. +\end{equation} +They give, after exact rationalization, natural strongly convergent +spectral sequences with $E_2$-terms $H^s(A,\pi_tX)[1/p]$ and abutments +$\pi_{t-s}((X^{hA})(*))[1/p]$. +This statement applies to the unbounded even-periodic theories here. +\end{lemma} + +\begin{proof} +Let $F(X)=(X^{hA})(*)$. This is an exact functor preserving limits and +upper homotopy bounds. For the latter assertion, coconnective objects +remain coconnective under evaluation and limits. Although $X$ need not +be a module spectrum over $H\Z_p$, each Postnikov layer $HM$ is one. +The forgetful functor from module spectra preserves limits, so its +continuous fixed points agree with the derived module invariants. +On this Eilenberg--Mac Lane object, the finite +resolution in \cref{lem:solid-resolutions} gives +\[ + \pi_{-s}F(HM)=H^s(A,M),\qquad + F(HM)\in\mathrm{Sp}_{[-d,0]}. +\] +The solid spectra are used with the hypersheaf convention of +\cite[Section~4.5, especially Proposition~4.5.5]{BMR2026}. +Truncations remain solid, since they retain or kill each solid homotopy +module. Evaluation on an extremally disconnected profinite set is +$t$-exact: homotopy sheafification evaluated on this projective object +agrees with pointwise homotopy. It therefore commutes with truncation. +These evaluations detect equivalences, so Postnikov completeness of +ordinary spectra gives +$X\simeq\varprojlim_n\tau_{\leq n}X$. + +Put $T_n=F(\tau_{\leq n}X)$. Exactness identifies the successive fiber as +\[ + \operatorname{fib}(T_{n+1}\longrightarrow T_n) + \simeq F\bigl(\Sigma^{n+1}H\pi_{n+1}X\bigr) + \in\mathrm{Sp}_{[n+1-d,n+1]}. +\] +Consequently $\pi_qT_{n+1}\to\pi_qT_n$ is an isomorphism for +$n\geq q+d$; the analogous tower in degree $q+1$ is also eventually +constant. The Milnor exact sequence therefore has zero derived-limit +term, and gives +\begin{equation}\label{eq:finite-postnikov-window} + \pi_qF(X)\cong\pi_qF(\tau_{\leq q+d}X). +\end{equation} +The infinitely negative tail causes no further issue: since +$F(\tau_{\leq q-1}X)$ is concentrated in degrees at most $q-1$, the +fiber sequence for this lower cutoff gives +\[ + \pi_qF(\tau_{[q,q+d]}X) + \xrightarrow{\ \sim\ }\pi_qF(\tau_{\leq q+d}X). +\] +Hence each abutment degree is computed by a finite Postnikov interval. + +The exact couple of this Postnikov construction has the $E_2$-term in +\eqref{eq:integral-descent-ss}. Its filtration in degree $q$ is +\[ + \operatorname{Fil}^{s}\pi_qF(X) + =\ker\!\left(\pi_qF(X)\longrightarrow + \pi_qF(\tau_{\leq q+s-1}X)\right), + \quad 0\leq s\leq d+1. +\] +It starts with the whole group, ends in zero by +\eqref{eq:finite-postnikov-window}, and has associated graded +$E_\infty^{s,q+s}$. This proves the asserted strong convergence and its +naturality. Tensoring the exact couple and these finite filtrations with +$\Q$ is exact. It therefore gives the rational spectral sequence and +its stated abutment; no exchange of rationalization with an infinite +totalization is involved. The groups here are $\Z_p$-modules, so this +tensor operation agrees with $[1/p]$. +No uniform torsion exponent as $t$ varies is required: each fixed +homotopy degree has the finite filtration just constructed. +\end{proof} + +\subsection{The surviving coefficient row} + +\begin{lemma}[Central weights]\label{lem:central-weights} +For $t\ne0$ and every $s$, one has +\[ + H^s(G_3,\pi_tE_3^{\square})[1/p]=0, + \qquad + H^s(G,\pi_t\widehat B^{\square})[1/p]=0. +\] +\end{lemma} + +\begin{proof} +Odd homotopy modules vanish. For $t=2m$, scalar automorphisms +$a\in1+p\Z_p$ lift canonically to scalar automorphisms of the universal +deformation. They fix its deformation ring, and act by $a^m$ on its +$m$th invariant-differential line $\pi_{2m}E_n^{\square}$. +This is the central character used in +\cite[Lemma~2.6.1]{BSSW2025}; replacing the periodicity convention by its +dual replaces $m$ by $-m$ and has the same consequence. + +Choose $b=1+p$. It is central in $G_3$, since the finite Galois action +fixes these scalars. A central group element acting on a coefficient +module induces the identity on its group cohomology. This follows +already from the bar resolution: its coefficient action is chain +homotopic to the action by conjugation, which is the identity for a +central element. Consequently $(b^m-1)$ annihilates +$H^s(G_3,\pi_{2m}E_3^{\square})$. For $m\ne0$ this is a nonzero +$p$-adic scalar, and is invertible after $p$ is inverted. This proves +the first vanishing. + +For the target, use the natural coefficient descent +\eqref{eq:coefficient-descent} to identify its cohomology with +$H^s(H,\pi_tE_2^{\square})$. The factor $\overline Z$ acts trivially +on these coefficients, and the central scalars in $P_2$ act by the same +character $b^m$. The preceding argument applies to $H$, and proves +the second vanishing. +\end{proof} + +\begin{proposition}\label{prop:nonzero-detector} +The ordinary map $g$ of \eqref{eq:ordinary-detector} is nonzero on +rational homotopy in degree $-3$: +\[ + \pi_{-3}L_0g:\pi_{-3}L_0L_{K(3)}S + \longrightarrow\pi_{-3}L_0\mathcal D + \quad\text{is nonzero}. +\] +\end{proposition} + +\begin{proof} +First extend the source primitive to the extended stabilizer. +By \cref{prop:group-classes}, $e_{P_3}$ is represented under rational +Lie comparison by a nonzero multiple of +\[ + (X,Y,Z)\longmapsto\tr_{\mathrm{red}}\bigl(X[Y,Z]\bigr). +\] +In the standard presentation of the degree-three division algebra, +take its maximal unramified subfield $\Q_{p^3}$ and an element $\Pi$ +such that +\[ + \Pi x=\sigma(x)\Pi\quad(x\in\Q_{p^3}),\qquad \Pi^3=p, +\] +where $\sigma$ is arithmetic Frobenius of the unramified cubic +extension $\Q_{p^3}/\Q_p$, inducing $z\mapsto z^p$ on the residue +field. Its extension fixing $\Pi$ equals conjugation +$y\mapsto\Pi y\Pi^{-1}$, as is checked on the generators +$\Q_{p^3}$ and $\Pi$. It therefore preserves the reduced-trace form. +Its action on $P_3$ factors through the order-three quotient of +$\Delta$; the order-two kernel also fixes the $\Z_p$-coefficients. +Naturality of Lie comparison therefore makes $e_{P_3}$ invariant under +$\Delta$. Since $6$ is a unit in $\Z_p$, finite-group averaging and +extension descent give +\[ + H^3(G_3,\Z_p)[1/p] + \xrightarrow{\ \sim\ } + \bigl(H^3(P_3,\Z_p)[1/p]\bigr)^\Delta. +\] +Let $\widetilde e_3$ be the class restricting to $e_{P_3}$, and let +$z_3$ be its image under the unit-coefficient map +$\Z_p\to\pi_0E_3^{\square}$. + +Under \eqref{eq:group-cochain-bridge}, restriction from $G_3$ to $P_3$ +and inflation along $G\to P_3$ on the constant classes are the +derived limits of their finite continuous-cochain pullbacks. These +are the group maps used in \eqref{eq:constant-cochain-naturality}. +All coefficient maps in the following square are the maps induced by +\eqref{eq:detector-construction}: +\[ +\begin{tikzcd}[column sep=large] + H^3(G_3,\Z_p)[1/p] + \arrow[r] \arrow[d,"{\mathrm{res},\,\mathrm{inf}}"'] + & H^3(G_3,\pi_0E_3^{\square})[1/p]\arrow[d]\\ + H^3(G,\Z_p)[1/p]\arrow[r] + & H^3(G,\pi_0\widehat B^{\square})[1/p]. +\end{tikzcd} +\] +On bar cochains, the right vertical map is restriction, pullback along +$G\to P_3$, and the coefficient map of +$E_3^{\square}\to\widehat B^{\square}$, in that order. +Its composite with the source coefficient unit is the target unit, +which is also $\Z_p\to W(\widehat L)\to\pi_0\widehat B^{\square}$. +Here the first map is the integral limit of the finite Witt units in +\eqref{eq:finite-witt-constant-square}; the map from full Witt vectors +to deformation coefficients is applied only after that limit. +The source descent identification used here is the canonical one in +\cref{prop:completed-theory}. +In particular the image of $z_3$ is exactly the class called $c_3$ in +\cref{prop:coefficient-transport}: it is obtained from $e_{P_3}$ by +projection and the actual constant-coefficient inclusion. That +proposition and \cref{lem:constant-nonvanishing} give +\[ + c_3=a\,c_2\ne0,\qquad a\in\Q_p^{\times}. +\] +Thus the map between the rational descent $E_2$-terms is nonzero in +bidegree $(s,t)=(3,0)$. + +Apply \cref{lem:descent-convergence} to the source and target of $g$. +By \cref{lem:central-weights}, only $t=0$ survives after rationalization. +The differential has bidegree $(r,r-1)$, so every rational differential +vanishes. For total homotopy degree $-3$, the finite filtration has +just one possibly nonzero graded piece, namely $(s,t)=(3,0)$. +The nonzero map on this piece is therefore a nonzero map on the +abutments. These abutments are the ordinary rational homotopy groups +asserted in the proposition. +\end{proof} + +\begin{proof}[Proof of \cref{thm:main}] +Rationalize \eqref{eq:detector-factorization}. If the canonical map +$L_0L_{K(3)}S\to L_0L_{K(2)}L_{K(3)}S$ were zero on $\pi_{-3}$, +its composite $L_0g$ would also be zero there. This contradicts +\cref{prop:nonzero-detector}. Every step above applies to each prime +$p\geq5$, proving the theorem. +\end{proof} diff --git a/preprints/A-rational-obstruction-to-strong-chromatic-splitting-at-height-three-September-25-2026/build/sections/08-elementary-background.tex b/preprints/A-rational-obstruction-to-strong-chromatic-splitting-at-height-three-September-25-2026/build/sections/08-elementary-background.tex new file mode 100644 index 0000000000000000000000000000000000000000..2e1b836800bccb7916b028b763c488f49ea0b166 --- /dev/null +++ b/preprints/A-rational-obstruction-to-strong-chromatic-splitting-at-height-three-September-25-2026/build/sections/08-elementary-background.tex @@ -0,0 +1,174 @@ +\appendix +\section{The low-rank cohomology and the building} +\label{app:background} + +We give the two elementary ingredients in the proof of +\Cref{prop:group-classes}. The calculations are over characteristic +zero; no reduction modulo $p$ of the matrices below is used. +The degree-three trace class is the classical cocycle associated to an +invariant symmetric bilinear form +\cite[\S21]{ChevalleyEilenberg1948}, \cite[\S11]{Koszul1950}. +We give the low-rank calculation and block restriction in the +normalization used in this paper. + +\subsection{A finite Chevalley--Eilenberg calculation} +\label{app:ce} + +For $\mathfrak{sl}_n$, order the matrix units $E_{ij}$, $i\ne j$, +lexicographically, followed by $H_i=E_{ii}-E_{nn}$ for $1\le i0$. +Metric convergence of normalized norms is equivalent to uniform +convergence of these ratios. Near any fixed norm they are uniformly +bounded, so for sufficiently small $t$ the displayed maximum is +identically zero. This also proves continuity at $t=0$ and hence +contractibility. + +Finally, the type of $[g\cO^n]$ is $v_p(\det g)$ modulo $n$. +Each chamber has every type once and $J_n$ preserves types. +Start a target chamber at its unique type-zero vertex. Scaling its +lattice by an integral power of $p$ makes its determinant valuation zero; +an adapted basis for its residue flag then carries the standard chamber +to it by an element of $J_n$. Thus $J_n$ is chamber-transitive. +A vertex stabilizer in $J_n$ actually preserves its lattice, since +$gL=aL$ implies $0=v_p(\det g)=n v_p(a)$. It is consequently a +conjugate of $\GL_n(\Z_p)$, compact open. Face stabilizers are finite +intersections of these groups and fix their vertices pointwise because +types are distinct. The closed chamber is therefore a fundamental +domain, including faces, with no inversions. These are exactly the +properties used by the finite cellular resolution in \cref{sec:cohomology}. diff --git a/preprints/A-rational-obstruction-to-strong-chromatic-splitting-at-height-three-September-25-2026/build/sections/09-frobenius.tex b/preprints/A-rational-obstruction-to-strong-chromatic-splitting-at-height-three-September-25-2026/build/sections/09-frobenius.tex new file mode 100644 index 0000000000000000000000000000000000000000..da1268f4b6898ef175a21d2c0ad424c89a664b0e --- /dev/null +++ b/preprints/A-rational-obstruction-to-strong-chromatic-splitting-at-height-three-September-25-2026/build/sections/09-frobenius.tex @@ -0,0 +1,81 @@ +\section{A continuous section for the Frobenius comparison} +\label{app:frobenius} + +The solid Frobenius comparison used in +\Cref{prop:constant-acyclicity} requires surjectivity on continuous +families, not only on point values. We supply the topological step for +its precise function ring. This avoids applying a linear open-mapping +theorem to Frobenius minus identity, which is not linear over the +algebraically closed coefficient field. + +Let $C$ be an algebraically closed complete nonarchimedean field of +characteristic $p$, containing $k=\F_q$, and let +\[ + R=H^0(\widetilde{\mathbb B}^{\,*}_C,\cO) + =\left\{\sum_{\alpha\in\Z[1/p]}c_\alpha t^\alpha: + \text{the series converges for every }0<|t|<1\right\}. +\] +Its Fr\'echet topology is given by the Gauss seminorms +$\|f\|_r=\sup_\alpha |c_\alpha|r^\alpha$, $0100$ both approximate lists are zero. +The data at zero and the constant terms are, exactly, +\begin{equation}\label{cert:fixed-inputs} + (\widetilde c_{1,0},\widetilde d_{1,0},C_1) + =(1,.44,-.013),\qquad + (\widetilde c_{2,0},\widetilde d_{2,0},C_2) + =(0,-.368,.017). +\end{equation} +The zero-node data are included in all evaluations of the tabulated functions. +They are excluded when forming a column of the operator $S$. +Direct integer addition gives +\begin{equation}\label{cert:table-norms} +\begin{aligned} + \sum_{n\ge0}(|\widetilde c_{1,n}|+|\widetilde d_{1,n}|) + &=2.2516666247<2.29,\\ + \sum_{n\ge0}(|\widetilde c_{2,n}|+|\widetilde d_{2,n}|) + &=.8927477314<2.29,\\ + \sum_{n>40}(|\widetilde c_{1,n}|+|\widetilde d_{1,n}|) + &=.0000197471<.00002,\\ + \sum_{n>40}(|\widetilde c_{2,n}|+|\widetilde d_{2,n}|) + &=.0000161623<.00002. +\end{aligned} +\end{equation} +Here and below an index $n\ge0$ in a coefficient sum ranges over +$\{0\}\cup\mathcal N$. + +\begingroup +\small +\setlength{\tabcolsep}{7pt} +\begin{longtable}{r r r r r} +\caption{Rational coefficient approximations. Each of the four coefficient columns is +scaled by $10^{10}$.}\label{tab:coefficients}\\ +\toprule +$n$ & $10^{10}\widetilde c_{1,n}$ & $10^{10}\widetilde d_{1,n}$ + & $10^{10}\widetilde c_{2,n}$ & $10^{10}\widetilde d_{2,n}$\\ +\midrule +\endfirsthead +\multicolumn{5}{c}{Table~\thetable\ (continued)}\\ +\toprule +$n$ & $10^{10}\widetilde c_{1,n}$ & $10^{10}\widetilde d_{1,n}$ + & $10^{10}\widetilde c_{2,n}$ & $10^{10}\widetilde d_{2,n}$\\ +\midrule +\endhead +\midrule +\multicolumn{5}{r}{Continued on the next page}\\ +\endfoot +\bottomrule +\endlastfoot +\input{data/coefficients.tex} +\end{longtable} +\endgroup + +Write $J=\{1,3,4\}$ and $E=I_f\setminus J$. Matrix rows and columns on a +node set always list all $c$ entries first and all $d$ entries second, with +nodes increasing within each group. In particular the order on $J$ is +$(c_1,c_3,c_4,d_1,d_3,d_4)$. The two rational matrices used to certify the +small inverses are +\begin{equation}\label{cert:Wplus} + W_+=10^{-4}\begin{pmatrix} + \input{data/W_plus.tex} + \end{pmatrix}, +\end{equation} +\begin{equation}\label{cert:Wminus} + W_-=10^{-4}\begin{pmatrix} + \input{data/W_minus.tex} + \end{pmatrix}. +\end{equation} +These matrices are exact inputs. The $W$-based inverse argument below +uses their products and column sums; it does not require a computed +matrix inverse. We describe the supplied program's additional inverse +checks in Section~\ref{cert:evaluation}. + +\subsection{Moments and the integration rule}\label{cert:quadrature} + +For a finite input $p=PR$ supported on nodes at most $100$, put +\begin{equation}\label{cert:moments} + U_{p,l}(m,y)=\frac{y^l p^{(l)}(m)}{l!},\qquad + X_{p,l}(m,y)=\frac{y^l K_p^{(l)}(m)}{l!}. +\end{equation} +The domain for the quadrature error estimate is stated explicitly in +Lemma~\ref{cert:moment-error}. + +We use the first Fej\'er quadrature rule, whose cosine-sum weights are +described in \cite[Section~2, formula~(2.3)]{Waldvogel2006}. Our weights +are half the mass-two weights there, normalized to average over an interval; +for even $N=256$ the terminal cosine term in that formula vanishes. +We prove the exactness, positivity and error bound needed here. +Set $N=256$, and for $0\le v0,\qquad \sum_v o_v=1. +\] +Thus the six pieces together have total weight one. + +For clarity, the complete density on piece $j$ used in this appendix is +\begin{equation}\label{cert:density} + f_j(t)=2\pi\sum_n e^{-i\pi tn} + \left\{-\pi c_n\sum_{l=j}^6(t_l-t)P_l e^{i\pi t_ln} + +i d_n\sum_{l=j}^6 P_l e^{i\pi t_ln}\right\}. +\end{equation} +It includes node $0$ when that coefficient is part of the input. The +folded atoms have masses $e_jCP_j$ at $t_j$, where $e_0=1$ and +$e_j=2$ for $j>0$. The factors multiplying this density and these masses +for $U$ and $X$, respectively, are +\begin{equation}\label{cert:analytic-factors} + e^{i\pi tm}\frac{(i\pi ty)^l}{l!},\qquad + \lambda(t)e^{i\pi z(t)m}\frac{(i\pi z(t)y)^l}{l!}. +\end{equation} +The real part is taken after summation. + +\begin{lemma}[Quadrature moment error]\label{cert:moment-error} +Let $p=PR$ have a real coefficient list supported in $\{0\}\cup I_f$, +with $\|(c,d)\|_1\le3$, and let $C\in\RR$ satisfy $|C|\le1$. +Let $m,y\in\RR$ and $l\in\{0,\ldots,30\}$ satisfy +$0\le m\le100$ and $|y|\le3/2$. +The quadrature rule~\eqref{cert:rule}, with exact evaluations at its nodes +and at the atoms, has error less than $10^{-22}$ in each moment +$U_{p,l}(m,y)$ and $X_{p,l}(m,y)$. +\end{lemma} + +This domain contains all basis inputs and the two tabulated inputs used below. + +\begin{proof} +Put $\rho=1/6$, $c_j=(2j-1)/12$, $a_j=c_j+ih$, and +$\Delta_j=|a_j|^2-\rho^2$. These last quantities increase from +$\Delta_1=167/1200$ to $\Delta_6=389/400$, so the disk +$|t-c_j|\le\rho$ does not contain the pole $-ih$. All factors in +\eqref{cert:density} and~\eqref{cert:analytic-factors} are analytic there. +The reciprocal-disk calculation is exact: for $\zeta=1/(t+ih)$, +\[ + |1-a_j\zeta|^2-\rho^2|\zeta|^2 + =\Delta_j\left|\zeta-\frac{\overline a_j}{\Delta_j}\right|^2 + -\frac{\rho^2}{\Delta_j}. +\] +Thus inversion maps the disk for $t+ih$ onto the disk with center +$(c_j-ih)/\Delta_j$ and radius $\rho/\Delta_j$. In particular, +\[ + |\zeta|\le\frac{|a_j|+\rho}{\Delta_j} + =\frac1{|a_j|-\rho}. +\] +The right side decreases with $c_j$, and $|a_1|=\sqrt{601}/60$. +Consequently the following are analytic bounds on every disk: +\[ +\begin{aligned} + |t|&\le\frac{13}{12}<1.1,\\ + |\lambda(t)|&\le\frac{120}{\sqrt3(\sqrt{601}-10)}<4.774<5,\\ + |z(t)|&\le\frac{80}{\sqrt{601}-10}+\frac25<5.912<6.2. +\end{aligned} +\] +For the two finer comparisons, it suffices to square the rational lower +bounds $1.732^2=2.999824<3$ and $24.515^2=600.985225<601$, and then use +\[ +\begin{gathered} + 4.774(1.732)(24.515-10)=120.01826452>120,\\ + (5.912-.4)(24.515-10)=80.00668>80. +\end{gathered} +\] +The image disk for $z=-B\zeta-ih$ has center +$-B(c_j-ih)/\Delta_j-ih$ and radius $B\rho/\Delta_j$. Its lowest +imaginary part is therefore $B(h-\rho)/\Delta_j-h$. Since $h>\rho$, +this decreases with $\Delta_j$, giving +\begin{equation}\label{cert:disk-imaginary} + \operatorname{Im}z(t)\ge + B\frac{h-1/6}{c_j^2+h^2-(1/6)^2}-h + \ge-\frac{1402}{17505}>-.081. +\end{equation} +Since $\sum_{l=1}^6|P_l|=10$ and $|t_l-t|\le13/12$ for $l\ge j$, +the analytic continuation of~\eqref{cert:density} on this disk is bounded +by +\[ + 3\max\{2\pi^2\cdot10\cdot13/12,\,2\pi\cdot10\} + e^{100\pi/6}<1000e^{100\pi/6}. +\] +Use $|a^l/l!|\le e^{|a|}$ for the divided powers. Including all factors, +the direct and transformed integrands are bounded, respectively, by +\[ +\begin{aligned} + 1000e^{200\pi/6+1.5\pi\cdot1.1}&<5.38\cdot10^{50},\\ + 5000e^{100\pi/6+100\pi\cdot.081+1.5\pi\cdot6.2} + &<1.51\cdot10^{50}. +\end{aligned} +\] +We may therefore use the common bound $M=10^{53}$. +The real interval has half-length $1/12$, half the disk radius. +Cauchy's coefficient estimate bounds the tail of the Taylor polynomial +of degree $255$ by $M\sum_{k=256}^{\infty}2^{-k}=M2^{-255}$. +Exactness, positivity and total mass one for the integral and quadrature +rule give discrepancy at most +\begin{equation}\label{cert:quadrature-error} + 2^{-254}10^{53}<3.455\cdot10^{-24}<10^{-22}. +\end{equation} +Atoms are evaluated separately and incur no quadrature error. +\end{proof} + +\subsection{Interval evaluation}\label{cert:evaluation} + +For a finite input $p$ in the real finite-input domain of +Lemma~\ref{cert:moment-error} and a node row $m\in I_f$, form the two real numbers +\begin{equation}\label{cert:pair} + \mathcal S_m(p)= + \left(-\frac{X_{p,0}(m,1)}{Q_m},\; + \frac{D_mX_{p,0}(m,1)-X_{p,1}(m,1)}{Q_m}\right). +\end{equation} +With input $c_n=1$ or $d_n=1$ and all other data zero, these are the +entries of the corresponding column of $S$ at row node $m$. + +Let $D=S_{J,J}$ and $U=S_{J,E}$. + +For the Bernstein certificates, use exactly the triples $(i,m,y)$ and +orders $\nu$ specified in Lemma~\ref{signs:bernstein}; in particular $y$ +is signed and no leftward interval at $m=1$ is used for index $1$. +Express its power coefficients in terms of the moments as +\begin{equation}\label{cert:power-from-moments} + T_r=\sigma_i y^{-\nu} + \left(U_{\widetilde p_i,r+\nu}(m,y) + +X_{\widetilde p_{3-i},r+\nu}(m,y)\right). +\end{equation} +The signed scaling is exact. Definition~\eqref{cert:moments} gives +$U_{p,l}(m,y)=y^lU_{p,l}(m,1)$ and the same identity for $X$. +Therefore, for positive or negative $y$, +\[ + T_r=\sigma_i y^r\left(U_{\widetilde p_i,r+\nu}(m,1) + +X_{\widetilde p_{3-i},r+\nu}(m,1)\right). +\] +This is the identity used when the implementation caches unscaled moments +at $y=1$; it does not discard the orientation of a half-gap. + +Equations~\eqref{cert:rule}, \eqref{cert:density}, +\eqref{cert:analytic-factors}, \eqref{cert:pair}, and +\eqref{cert:power-from-moments} already specify all +sums in the finite certificates below. The following pseudocode records their +algebraic organization. A measure is a list of pairs $(t,g)$ in the order +used by the supplied interval implementation: first the $6\cdot256$ +quadrature pairs $(\tau_{jv},o_v f_j(\tau_{jv})/6)$, with +$j=1,\ldots,6$ outermost and $v=0,\ldots,255$ innermost, and then the seven +atom pairs $(t_j,e_jCP_j)$ for $j=0,\ldots,6$. The atom masses are zero for +a basis input. The pseudocode is not a byte-for-byte execution trace. +The supplied source fixes the detailed arithmetic and summation order for +an actual interval run, including the factoring and caching described below. + +\noindent +\begin{minipage}{\linewidth} +\begin{lstlisting}[basicstyle=\ttfamily\footnotesize] +Moments(measure, m, y, L): + U[0..L] = 0; X[0..L] = 0 + for (t,g) in measure: + lam = i/(b*(t+i*h)) + z = -B/(t+i*h)-i*h + u = g*exp(i*pi*t*m) + v = g*lam*exp(i*pi*z*m) + for r = 0..L: + U[r] += real(u); X[r] += real(v) + u *= (i*pi*t*y)/(r+1) + v *= (i*pi*z*y)/(r+1) + return U,X + +Pair(measure,m): + U,X = Moments(measure,m,1,1) + return -X[0]/Q[m], (D[m]*X[0]-X[1])/Q[m] + +for basis column c[n] or d[n], n in I_f: + measure = Measure(basis column; all other data zero) + column = stack Pair(measure,m), m in J + if n in J: + also form rows m in E intersect [7,16] + test column sums of W[eta] and the defect/product matrices + +for i in {1,2}, m in I_f: + c,d = Pair(Measure(full table input 3-i),m) + test abs(table_c[i,m]-c) + test abs(table_d[i,m]+indicator(i==1 and m==1)-d) + +for (i,m,y,nu) in the prescribed half-gap cases: + U,X = Moments for the two full table inputs, through L=30 + T[r] = sigma[i]*(U[i,r+nu]+X[3-i,r+nu])/y**nu + for k = 0..28: + Bk = T[0]; a = 1 + for r = 1..k: + a *= (k+1-r)/(29-r) + Bk += a*T[r] + test the appropriate lower bound for Bk +\end{lstlisting} +\end{minipage} + +Here \texttt{stack} means all first entries followed by all second entries. +The recurrence for $a$ equals $\binom{k}{r}/\binom{28}{r}$. +\texttt{Measure} denotes this finite list and introduces no additional +approximation. The fixed data at zero are used only for the two full +table inputs. Reusing a measure means reusing its whole finite list. + +For the accompanying certified evaluation, every real or complex quantity +in these sums is enclosed using interval arithmetic. The pair calculation +forms the moments at $y=1$. For the Bernstein calculation, the +implementation also forms and caches the unscaled moments at $y=1$, then +uses the signed $y^r$ identity following~\eqref{cert:power-from-moments}. +Each unscaled real moment enclosure is enlarged by +$[-10^{-22},10^{-22}]$, as justified by Lemma~\ref{cert:moment-error}. +Consequently the analytic-remainder contribution to the enclosure radius +of a Bernstein coefficient is bounded by +\[ +\begin{aligned} + 2\cdot10^{-22}\sum_{r=0}^k + \frac{\binom{k}{r}}{\binom{28}{r}}|y|^r + &\le2\cdot10^{-22}\sum_{r=0}^{28}(3/2)^r\\ + &=4\cdot10^{-22}\bigl((3/2)^{29}-1\bigr)<5.2\cdot10^{-17}. +\end{aligned} +\] +The last inequality is an exact rational comparison. It is not correct +to regard $10^{-22}$ as an unamplified per-Bernstein error. +Arb rounding and dependency effects can further widen the balls; all of +them are propagated by the subsequent products, absolute values, column +sums, and Bernstein combinations. An upper bound is accepted only when +the complete enclosure lies strictly below the asserted rational bound, +and a lower bound only when it lies strictly above. This procedure uses +enclosures of the exact integrals, not a quadrature error inferred from +observed numerical agreement. + +The ball-arithmetic inclusion methodology is described in +\cite{Johansson2017}. The supplied implementation uses Arb through +python-flint at $192$ bits of precision; the citation does not by itself +certify this local environment, its call domains, or a run. +The entry point \path{verification/verify_all.py} runs two verifiers in +the same directory, \path{interval_certificate.py} and +\path{interval_bounds.py}. Their output records, \path{finite.json} and +\path{scalar.json}, and the aggregate record \path{summary.json} are +placed in its \path{results/} subdirectory. +The first verifier checks the exact bounds in~\eqref{cert:matrix-table}, +\eqref{cert:exterior-block}, \eqref{cert:residual-bounds}, and +\eqref{cert:bernstein-table}, as well as all 2436 individual Bernstein +inequalities in Lemma~\ref{signs:bernstein}. It also asserts +four finite brackets: the two defect brackets, the exterior bracket, and +the bracket for $(2,16,-1/2,28)$. All its comparisons and extrema use +complete enclosures. + +The mathematical $W$ argument in the next subsection uses only products +and column sums. The supplied program also calls a certified inverse +for $I-\eta D$ and tests its inverse and cross-product norms as +additional checks; these calls are not needed for the $W$ deduction. + +\subsection{Matrix and residual certificates}\label{cert:matrix-bounds} + +The following are strict upper bounds +for the indicated column norms. The last three columns use only +columns whose node lies in the stated range. They refer to the exact +operator entries in~\eqref{cert:pair}. +\begin{equation}\label{cert:matrix-table} +\begin{array}{c|cc|ccc} + \eta& + \|W_\eta\|& + \|I-W_\eta(I-\eta D)\|& + \|W_\eta U_{7:9}\|& + \|W_\eta U_{12:16}\|& + \|W_\eta U_{19:100}\|\\ \hline + +1&30.84&.001&3.54&.44&.25\\ + -1&1.83&.001&.73&.32&.17 +\end{array} +\end{equation} +The unmultiplied exterior block also satisfies +\begin{equation}\label{cert:exterior-block} + \|S_{E\cap[7,16],J}\|<.088. +\end{equation} +The certified enclosure calculation described in +Section~\ref{cert:evaluation} proves these inequalities. +For example, the two defect norms are enclosed respectively in +\[ +\begin{gathered} + [.00025422624910652544,\ .00025422624910652546],\\ + [.00042288165102553355,\ .00042288165102553357], +\end{gathered} +\] +and the exterior block norm is enclosed in +$[.08688748182289641591,\ .08688748182289641592]$. +These intervals are stated with rational endpoints rounded outwards. + +The three printed product ranges partition $E$, so $\|W_\eta U\|$ is the +maximum of the three grouped product norms. The exact bounds in +\eqref{cert:matrix-table} and~\eqref{cert:exterior-block} give the primary +matrix conclusion directly. Put $A_\eta=I-\eta D$. +Since $\|I-W_\eta A_\eta\|<.001$, the finite square +matrix $W_\eta A_\eta$ is invertible by a geometric series, hence so +is $A_\eta$, and +\[ + A_\eta^{-1}=(W_\eta A_\eta)^{-1}W_\eta. +\] +It follows from~\eqref{cert:matrix-table} that +\[ + \|A_\eta^{-1}\|<\frac{30.84}{1-.001}<32,\qquad + \|A_\eta^{-1}U\|<\frac{3.54}{1-.001}<3.7. +\] +Equation~\eqref{cert:exterior-block} gives the remaining bound $<.09$ +in Lemma~\ref{interpolation:finite}. + +For the two full tabulated inputs, including~\eqref{cert:fixed-inputs}, +the residual certificate is +\begin{equation}\label{cert:residual-bounds} +\begin{aligned} + \left|\widetilde c_{i,m}+\frac{X_{\widetilde p_{3-i},0}(m,1)}{Q_m}\right| + &<3\cdot10^{-10},\\ + \left|\widetilde d_{i,m}+\mathbf1_{\{i=1,\ m=1\}} + +\frac{X_{\widetilde p_{3-i},1}(m,1) + -D_mX_{\widetilde p_{3-i},0}(m,1)}{Q_m}\right| + &<3\cdot10^{-10} + \qquad(i=1,2,\ m\in I_f). +\end{aligned} +\end{equation} +The primary evaluation in Section~\ref{cert:evaluation} proves +\eqref{cert:residual-bounds} for these exact residuals. Since +$3\cdot10^{-10}<10^{-9}$, these inequalities, the preceding primary +matrix conclusion, and~\eqref{cert:table-norms} prove every finite +assertion of Lemma~\ref{interpolation:finite}. + +\subsection{Bernstein coefficient certificates}\label{cert:bernstein-bounds} + +\begin{samepage} +Let $B_{i,m,y,k}$ denote the sum $B_k$ in +\eqref{signs:bernstein-bound}. Each entry below is a strict lower bound +for every such sum in its row, for all $0\le k\le28$ and both half-gap +directions when applicable. +\begin{equation}\label{cert:bernstein-table} +\begin{array}{c|cc} + m& i=1&i=2\\ \hline + 0&\text{---}&.762\\ + 1&.314&.191\\ + 3,4&.028&.024\\ + 7,9&.114&.112\\ + 12,13,15,16&.0105&.0094\\ + 19,21&.123&.113\\ + 24,25,27,28&.0111&.0113\\ + 31,33&.121&.126\\ + 36,37,39,40&.0107&.0115 +\end{array} +\end{equation} +\end{samepage} +These bounds concern the exact derivatives of the tabulated functions, +not derivatives rounded to satisfy interpolation identities. +For example, the coefficient at $(i,m,y,k)=(2,16,-1/2,28)$ is enclosed in +\[ + [.00951875955680,\ .00951875955684]. +\] +The primary evaluation in Section~\ref{cert:evaluation} proves the grouped +bounds~\eqref{cert:bernstein-table} for the exact derivatives of the tabulated +functions. Each printed lower bound is stronger than its corresponding +threshold in~\eqref{signs:bernstein-bound}. Therefore these exact grouped +bounds imply Lemma~\ref{signs:bernstein}. + +\subsection{Scalar envelope and sign bounds}\label{cert:scalar-bounds} + +The remaining finite substitutions involve elementary real expressions. +For the endpoint and density envelopes in +\eqref{interpolation:piece-envelopes}, it suffices to test the finitely +many endpoints and residues specified there. The endpoint formulas +for $|\lambda|$, $|z|$, and $\operatorname{Im}z$ use rational numbers and +square roots. The density expressions additionally use sine and cosine +at rational multiples of $\pi$. The endpoint tests are certified by +the same elementary enclosures used above; their monotonicity and +convexity extensions to whole intervals were proved in +Section~\ref{sec:interpolation}. + +For the row envelope in~\eqref{interpolation:row-envelope}, certified +outward bounds are +\begin{equation}\label{cert:row-substitutions} +\begin{array}{c|c} + \text{row cut }x&\text{upper bound for the finite envelope sum}\\ \hline + 0&27.618723\\ + 4&.269094\\ + 16&.014122\\ + 100&4.617010\cdot10^{-10} +\end{array} +\end{equation} +The atom-column analogue at cut $100$ is less than +$3.131186\cdot10^{-8}$. These bounds imply all the row-tail constants +used in Section~\ref{sec:interpolation}. + +For the sign proof, the following outward bounds record the relevant +substitutions. The expressions themselves, and the reason they bound +whole intervals or tails, are given at the cited equations. +\begin{center} +\begin{tabular}{p{.65\linewidth}r} +\toprule +Expression & Certified bound\\ +\midrule +\eqref{signs:rational-lower-sum}, index $1$ & $>.01206834$\\ +\eqref{signs:rational-lower-sum}, index $2$ & $>.01656468$\\ +$\sum_{j=1}^6 M_j(\pi t_j)^2/6$ & $<76.043$\\ +Right side of \eqref{signs:tail-second-derivative} +before its final comparison & $<.005303$\\ +\eqref{signs:perturbation-envelope} divided by $\nu!$, +$(s_*,\nu)=(0,0)$ & $<.003189$\\ +Same expression, $(s_*,\nu)=(1/2,1)$ & $<.003736$\\ +Same expression, $(s_*,\nu)=(1/2,2)$ & $<.009146$\\ +Same expression, $(s_*,\nu)=(2,2)$ & $<.001796$\\ +$2.29\sum_{j=1}^6M_j/6+.017\sum_{j=0}^6m_j^*$ & $<72.669$\\ +\bottomrule +\end{tabular} +\end{center} +In the fourth row the expression means the explicit sum with +$.00005\cdot77$, not the derivative supremum to its left. +For the five cases of~\eqref{signs:taylor-representatives}, the +corresponding bounds for $\mathcal E(m,Y,\nu)$ in +\eqref{signs:taylor-bound} are +\begin{equation}\label{cert:taylor-substitutions} +\begin{array}{c|c} + (m,Y,\nu)&\text{upper bound}\\ \hline + (0,1/2,0)&1.578\cdot10^{-10}\\ + (1,1,1)&2.007\cdot10^{-5}\\ + (1,1,2)&5.735\cdot10^{-6}\\ + (3,1,2)&9.581\cdot10^{-11}\\ + (4,3/2,2)&6.133\cdot10^{-7} +\end{array} +\end{equation} +The first two rational-part bounds can be obtained by exact fraction +addition alone. All the other displayed scalar bounds follow by +substitution and outward enclosure of the explicit exponential, +trigonometric and rational expressions. The scalar verifier records +22 named strict comparisons: the four row substitutions, the atom tail, +the nine sign-table rows, the five Taylor representatives, and the three +disk-integrand and quadrature comparisons. It also checks the endpoint +and residue envelopes, the coarser row-tail comparisons, the finite-inverse +substitution giving a bound below $90$, +and the distance substitution using $91$ and $2540$. The derivation of +these whole-space coefficients from the Schur estimates remains the +analytic argument in Section~\ref{sec:interpolation}. +Its disk assertions use the coarser bounds $5$ and $6.2$ for +$|\lambda|$ and $|z|$; the finer $4.774$ and $5.912$ bounds were proved +analytically above and are not literal scalar-program assertions. +At the six gap midpoints the program evaluates the product defining $P$ +and checks that each quotient is greater than $.68$; it does not assert +the symbolic radical identities for those values. No sampling of an +unknown function between interpolation nodes is involved. The +logarithmic concavity argument and exact midpoint values in +Lemma~\ref{signs:P-lower-bound} supply the global lower bound for $P$. diff --git a/preprints/A-sharp-Fourier-certificate-for-planar-circle-packing-September-23-2026/build/sections/fourier.tex b/preprints/A-sharp-Fourier-certificate-for-planar-circle-packing-September-23-2026/build/sections/fourier.tex new file mode 100644 index 0000000000000000000000000000000000000000..cf9bd076db908f6d31a120c3d17894c166a15a73 --- /dev/null +++ b/preprints/A-sharp-Fourier-certificate-for-planar-circle-packing-September-23-2026/build/sections/fourier.tex @@ -0,0 +1,401 @@ +\section{Gaussian cardinal functions}\label{sec:fourier} + +The construction uses a family of entire functions whose values and first +derivatives can be prescribed at a periodic set of nodes. Multiplication by +a fixed Gaussian turns these functions into radial Schwartz functions. Their +Fourier transforms have strictly stronger Gaussian decay, which will make the +coupled interpolation equations tractable. + +Set +\begin{equation}\label{fourier:parameters} + b=\frac{\sqrt3}{2},\qquad h=\frac25,\qquad B=b^{-2}=\frac43, + \qquad s=b\abs{x}^2. +\end{equation} +The same coordinate, \(s=b\abs{\xi}^2\), is used on the Fourier side. +To see the lattice geometry behind this choice, let \(\Lambda\) have basis +\begin{equation}\label{fourier:lattice} + u=\frac{1}{\sqrt b}(1,0),\qquad + v=\frac{1}{\sqrt b}(1/2,b). +\end{equation} +This oriented basis has determinant one. If \(\mathcal R\) is +counterclockwise rotation through \(90\) degrees, its dual basis is +\(-\mathcal R v,\mathcal R u\). Hence +\(\Lambda^*=\mathcal R\Lambda\): the lattice and its dual have the same +shell radii. For \(a=ju+kv\), with \(j,k\in\ZZ\), the squared radius in +the common coordinate is +\begin{equation}\label{fourier:shell-coordinate} + s_a:=b\abs{a}^2=j^2+jk+k^2. +\end{equation} + +We impose interpolation conditions at the positive nodes of a periodic set: +\begin{equation}\label{fourier:nodes} + A=\{0,1,3,4,7,9\},\qquad + \mathcal N=(12\ZZ+A)\cap(0,\infty). +\end{equation} +The set \(\mathcal N\) contains every positive value of \(j^2+jk+k^2\). +Indeed, +\(j^2+jk+k^2\equiv(j-k)^2\pmod3\), so it is never \(2\pmod3\). +Modulo \(4\), it is \(0\) when \(j,k\) are both even and is odd otherwise. +Combining these restrictions gives precisely the six allowed +residues in \(A\). The inclusion is strict: \(15\in\mathcal N\), but +\(j^2+jk+k^2=15\) would imply +\((2j+k)^2\equiv2k^2\pmod5\). Since \(2\) is not a square modulo \(5\), +this forces \(5\mid k\) and then \(5\mid j\), which would make \(25\) +divide \(15\). Thus we require zeros at some nodes that are not lattice +shell coordinates. +This deliberate enlargement uses a periodic residue pattern with an +elementary trigonometric vanishing function. Covering planar shell values by +periodic sets also occurs in Sardari's Fourier interpolation construction +\cite[Section~6.1]{Sardari2021}; the Gaussian transformation and the +coupled inversion needed here will be proved directly. + +\subsection{The cardinal family} + +The values and first derivatives will be encoded by double-pole and +simple-pole terms multiplied by a common vanishing factor. This is the +same cardinal pattern that occurs in Gaussian Beurling--Selberg +interpolation \cite[formulas~(1.2)--(1.3)]{CarneiroLittmannVaaler2013}. +The nodes and simultaneous Fourier constraints here are different. + +For \(-6\le j\le6\), put \(t_j=j/6\), and define +\begin{equation}\label{fourier:polynomial} + P(s)=\prod_{a\in A}\left(2\sin\frac{\pi(s-a)}{12}\right)^2 + =\sum_{j=-6}^{6}P_j e^{i\pi t_js}. +\end{equation} +Here \(P_{-j}=\overline{P_j}\), and +\begin{equation}\label{fourier:coefficients} + (P_0,P_1,\ldots,P_6) + =\left(5,-1+2ib,1+2ib,-2,-\frac12+ib,-1-2ib,1\right). +\end{equation} +For example, with \(w=e^{i\pi s/6}\) and \(\rho=e^{i\pi/6}\), each factor is +\[ + \left(2\sin\frac{\pi(s-a)}{12}\right)^2 + =-\rho^{-a}w^{-1}(w-\rho^a)^2. +\] +The six minus signs cancel, and \(\rho^{-\sum_{a\in A}a}=\rho^{-24}=1\). +Thus the product in \eqref{fourier:polynomial} equals +\[ + w^{-6}\prod_{a\in A}(w-\rho^a)^2. +\] +Expanding this Laurent polynomial gives +\eqref{fourier:coefficients}. + +Every zero \(n\in12\ZZ+A\) of \(P\) is double. Define \(Q_n,D_n\) by +\begin{equation}\label{fourier:node-expansion} + P(n+u)=Q_nu^2\bigl(1+D_nu+O(u^2)\bigr). +\end{equation} +If \(a\in A\) is the residue of \(n\) modulo \(12\), then +\begin{equation}\label{fourier:node-data} + \begin{split} + Q_n&=\left(\frac\pi6\right)^2 + \prod_{\substack{a'\in A\\a'\ne a}} + \left(2\sin\frac{\pi(a-a')}{12}\right)^2,\\ + D_n&=\frac\pi6\sum_{\substack{a'\in A\\a'\ne a}} + \cot\frac{\pi(a-a')}{12}. + \end{split} +\end{equation} +Thus \(Q_n,D_n\) are periodic in \(n\), and their exact values are +\[ +\begin{array}{c|c|c} + a&Q_n&D_n\\\hline + 0&\pi^2/3&-7\pi\sqrt3/18\\ + 1&(2\pi^2/3)(2-\sqrt3)&\pi(3+\sqrt3)/18\\ + 3&(2\pi^2/3)(2-\sqrt3)&-\pi(3+\sqrt3)/18\\ + 4&\pi^2/3&7\pi\sqrt3/18\\ + 7&(2\pi^2/3)(2+\sqrt3)&-\pi(3-\sqrt3)/18\\ + 9&(2\pi^2/3)(2+\sqrt3)&\pi(3-\sqrt3)/18 +\end{array} +\] +The expansion follows by isolating the vanishing sine factor; logarithmic +differentiation of the remaining factors gives \(D_n\). The table then uses +the elementary trigonometric values at multiples of \(\pi/12\). +In particular, +\begin{equation}\label{fourier:node-bounds} + Q_n>\frac74=1.75,\qquad \abs{D_n}<\frac{53}{25}=2.12. +\end{equation} +To check these inequalities with rational bounds, use +\(157/50<\pi<22/7\) and \(\sqrt3<26/15\). The minimum of \(Q_n\) is +greater than \(197192/112500>7/4\), and the maximum of \(\abs{D_n}\) is +less than \(286/135<53/25\). + +Let \(C\in\RR\), and let \((c_n,d_n)_{n\in\{0\}\cup\mathcal N}\) be a +real coefficient list with +\begin{equation}\label{fourier:list-norm} + \norm{(c,d)}_1 + :=\sum_{n\in\{0\}\cup\mathcal N}(\abs{c_n}+\abs{d_n})<\infty. +\end{equation} +Away from the nodes, define +\begin{equation}\label{fourier:cardinal} + p(s)=P(s)R(s),\qquad + R(s)=C+\sum_{n\in\{0\}\cup\mathcal N} + \left(\frac{c_n}{(s-n)^2}+\frac{d_n}{s-n}\right). +\end{equation} +We next show that the apparent singularities are removable, with uniform +control of the resulting entire function. + +\begin{lemma}[Finite-measure representation]\label{fourier:measure-lemma} +The function in \eqref{fourier:cardinal} extends to an entire function with +the representation +\begin{equation}\label{fourier:measure} + p(s)=\int_{-1}^{1}e^{i\pi ts}\,\dd\mu(t),\qquad s\in\CC, +\end{equation} +where \(\mu\) is a finite complex measure, supported in \([-1,1]\), with +\(\mu(-E)=\overline{\mu(E)}\) for Borel sets \(E\). +Its total variation satisfies +\begin{equation}\label{fourier:variation} + \norm{\mu}_{\mathrm{TV}} + \le25\abs C+\frac{65\pi^2}{18}\sum_n\abs{c_n} + +\frac{32\pi}{3}\sum_n\abs{d_n}. +\end{equation} +The atoms of \(\mu\) have mass \(CP_j\) at \(t_j\), for \(-6\le j\le6\). +For real-axis evaluations, the remaining measure can be folded onto +\([0,1]\): on \(t_{j-1}0\qquad(-1\le t\le1). +\end{equation} +Consequently every derivative of \(K_p(s)\) decays exponentially as +\(s\to+\infty\). +\end{proposition} + +\begin{proof} +By \eqref{fourier:measure}, +\[ + q_p(x)=\int_{-1}^{1}e^{-\pi b(h-it)\abs{x}^2}\,\dd\mu(t). +\] +Every spatial derivative of the integrand is a polynomial in \(x\), whose +coefficients are bounded uniformly for \(t\in[-1,1]\), times a Gaussian of +modulus \(e^{-\pi bh\abs{x}^2}\). The finite variation of \(\mu\) +justifies differentiation under the integral and bounds every Schwartz +seminorm. Reality follows from the real values of \(p\) on the real axis, +and radiality from the dependence on \(\abs{x}^2\). + +For \(\operatorname{Re}a>0\), the two-dimensional Gaussian transform in +our Fourier convention is +\[ + \widehat{e^{-\pi a\abs{x}^2}}(\xi) + =a^{-1}e^{-\pi\abs{\xi}^2/a}. +\] +This follows from the one-dimensional Gaussian integral by taking products +and analytically continuing in \(a\) to the right half-plane. +Apply this with \(a=b(h-it)\). Fubini's theorem applies because the +integrands have a common integrable Gaussian majorant. We have +\[ + a^{-1}=\lambda(t),\qquad + -\frac{\pi\abs{\xi}^2}{a}+\pi hs=i\pi z(t)s, +\] +which gives \eqref{fourier:kernel} and the asserted Fourier transform. +The curve \(z([-1,1])\) is compact, so \(K_p\) is entire by differentiation +under its defining integral. Also, +\[ + \lambda(-t)=\overline{\lambda(t)},\qquad + z(-t)=-\overline{z(t)}. +\] +Thus the same folded measure and real-part convention as in +Lemma~\ref{fourier:measure-lemma} apply to \(K_p\) and its real-axis +derivatives. + +Finally, the imaginary part in \eqref{fourier:damping} is smallest at +\(\abs t=1\), where its value is \(26/435\). Differentiating +\eqref{fourier:kernel} gives, for \(s\ge0\), +\[ + \abs{K_p^{(k)}(s)} + \le\norm{\mu}_{\mathrm{TV}} + \max_{\abs t\le1}\bigl(\abs{\lambda(t)}\, + (\pi\abs{z(t)})^k\bigr) + e^{-26\pi s/435}. +\] +This proves the decay assertion. +\end{proof} + +No cardinal representation of \(K_p\) is assumed. Only its entireness, its values and +derivatives, and the estimates obtained from \eqref{fourier:kernel} will be +used in the interpolation equations. In particular, the measure in that +formula is supported on the real interval \([-1,1]\); later complex-disk +estimates concern specified finite integrands, not an analytic continuation +in the integration variable of an arbitrary \(\ell^1\) density. + +\begin{remark}[A real-radius Hankel representation]\label{fourier:hankel} +For real \(s\ge0\), the same transform has the representation +\begin{equation}\label{fourier:hankel-formula} + K_p(s)=e^{\pi hs}\frac{\pi}{b}\int_0^\infty + e^{-\pi hu}p(u) + J_0\!\left(\frac{2\pi\sqrt{us}}{b}\right)\,\dd u, +\end{equation} +where +\(J_0(v)=(2\pi)^{-1}\int_0^{2\pi}e^{-iv\cos\theta}\,\dd\theta\) +for real \(v\). To prove this, use polar coordinates in the Fourier +integral for \(q_p\), perform the angular integral, and put \(u=b r^2\). +The resulting radial integral is +\(\widehat q_p(\xi)=(\pi/b)\int_0^\infty e^{-\pi hu}p(u) +J_0(2\pi\sqrt{us}/b)\,\dd u\), with \(s=b\abs{\xi}^2\). +Absolute convergence follows from \(\abs{p(u)}\le\norm{\mu}_{\mathrm{TV}}\) +and \(\abs{J_0(v)}\le1\). Multiplication by \(e^{\pi hs}\) gives +\eqref{fourier:hankel-formula}. This is an identity on the nonnegative real +axis; it is not used as a numerical sign certificate below. +\end{remark} + +\subsection{The coupled interpolation conditions} + +We use two cardinal functions \(p_1,p_2\), with coefficients +\(c_{i,n},d_{i,n},C_i\) for function index \(i\in\{1,2\}\). Their +coefficients at positive nodes will be determined later; fix +\begin{equation}\label{fourier:fixed} + (c_{1,0},d_{1,0},C_1)=(1,0.44,-0.013),\qquad + (c_{2,0},d_{2,0},C_2)=(0,-0.368,0.017). +\end{equation} +All terminating decimals in the construction denote exact rational numbers. +Write \(K_i=K_{p_i}\) and \(H_i=p_i+K_{3-i}\). The associated pair is +\begin{equation}\label{fourier:pair} + F(x)=e^{-\pi hs}H_1(s),\qquad + \widehat F(\xi)=e^{-\pi hs}H_2(s), + \quad s=b\abs{x}^2\ \text{or}\ b\abs{\xi}^2, +\end{equation} +respectively. Indeed, \(F=q_{p_1}+\widehat{q_{p_2}}\), and Fourier +inversion gives \(\widehat F=\widehat{q_{p_1}}+q_{p_2}\), since the +functions are even. Thus both functions in \eqref{fourier:pair} are real, +radial, and Schwartz for every pair of admissible coefficient lists. + +The required interpolation conditions are +\begin{equation}\label{fourier:interpolation} + \begin{split} + H_1(n)&=H_2(n)=0,\\ + H_2'(n)&=0,\qquad H_1'(n)=-Q_n\mathbf1_{\{n=1\}} + \qquad(n\in\mathcal N). + \end{split} +\end{equation} +Here primes denote derivatives with respect to \(s\). +They give zeros of order at least two for both functions at every positive +node except that \(H_1\) has a prescribed simple zero at \(1\). + +\begin{lemma}[Coefficient form of the interpolation]\label{fourier:equivalence} +For two real \(\ell^1\) coefficient lists, conditions +\eqref{fourier:interpolation} are equivalent to +\begin{equation}\label{fourier:coefficient-equations} + \begin{split} + c_{i,n}&=-\frac{K_{3-i}(n)}{Q_n},\\ + d_{i,n}&=\frac{D_nK_{3-i}(n)-K_{3-i}'(n)}{Q_n} + -\mathbf1_{\{i=1,\ n=1\}} + \qquad(i=1,2,\ n\in\mathcal N). + \end{split} +\end{equation} +\end{lemma} + +\begin{proof} +By \eqref{fourier:node-values}, +\[ + H_i(n)=Q_nc_{i,n}+K_{3-i}(n),\qquad + H_i'(n)=Q_n(d_{i,n}+D_nc_{i,n})+K_{3-i}'(n). +\] +The first equation in \eqref{fourier:coefficient-equations} is therefore +equivalent to \(H_i(n)=0\). Substituting it into the second displayed +identity gives the derivative condition in +\eqref{fourier:interpolation} exactly when the second coefficient equation +holds. Division by \(Q_n\) is legitimate by +\eqref{fourier:node-bounds}. +\end{proof} diff --git a/preprints/A-sharp-Fourier-certificate-for-planar-circle-packing-September-23-2026/build/sections/independent-certification.tex b/preprints/A-sharp-Fourier-certificate-for-planar-circle-packing-September-23-2026/build/sections/independent-certification.tex new file mode 100644 index 0000000000000000000000000000000000000000..911beac05a27d6f1e8a9d7a779ed6166132fb5c2 --- /dev/null +++ b/preprints/A-sharp-Fourier-certificate-for-planar-circle-packing-September-23-2026/build/sections/independent-certification.tex @@ -0,0 +1,863 @@ +\section{Alternative certification methods}\label{ind:section} + +The primary certificate does not use the methods in this appendix. +We develop several ways to assess or reproduce its finite quantities. +The first gives a fixed rational evaluation of every $N=256$ moment and +proves a uniform error theorem. A separate exact-rational +program checks the first $6\times6$ block with an $N=64$ rule. +Rational-center Taylor integration gives a different, arbitrarily refinable +enclosure method, and the final proposition gives a local formula for the +direct cardinal derivatives. + +All terminating decimals denote exact rational numbers. The full +$N=256$ rational recipe and the Taylor method specify computations and +their error bounds without asserting unreported finite values. The +$N=64$ program has the narrower scope stated with its proof. + +\subsection{Rational exponential primitives}\label{cert:rational-primitives} + +The fixed rational procedures below use rational arithmetic alone, with a +complex number represented by a pair of rationals. Put +\[ + \begin{aligned} + p_*&=3.141592653589793238462643383279502884,\\ + b_*&=.866025403784438646763723170752936183, + \end{aligned} +\] +and retain $B=4/3$ and $h=2/5$ exactly. Both $|p_*-\pi|$ and $|b_*-b|$ +are less than $10^{-36}$. For example, the identity +$\pi/4=4\arctan(1/5)-\arctan(1/239)$ and alternating series through +power $57$ certify the first assertion; comparing $(b_*\pm10^{-36})^2$ +with $3/4$ certifies the second. +Define +\[ + [w]_{40}=10^{-40}\lfloor10^{40}w\rfloor +\] +for real $w$, and apply it componentwise to complex numbers. Define an +approximation $e_*(u)$ to $e^{i\pi u}$ by the following finite recursion: +\begin{equation}\label{cert:estar} + v_0=\left[\sum_{q=0}^{32}\frac{(ip_*u/1024)^q}{q!}\right]_{40}, + \qquad v_{a+1}=[v_a^2]_{40}\ (0\le a<10),\qquad e_*(u)=v_{10}. +\end{equation} +Real and imaginary parts of $e_*(u)$ give the cosine and sine +replacements when $u$ is real. + +\begin{lemma}[Rounded exponential]\label{cert:rounded-exponential} +For every $u\in\mathbb Q(i)$ satisfying +$\Re(ip_*u)\le0$ and $|p_*u|\le1000$, the recursion +\eqref{cert:estar} satisfies +\[ + |e_*(u)-e^{ip_*u}|<3.542\cdot10^{-34}<10^{-32}. +\] +\end{lemma} + +\begin{proof} +The reduced argument has modulus less than one, so the Taylor remainder +in the first step of~\eqref{cert:estar} is less than $3/33!$. +A componentwise floor contributes less than $2\cdot10^{-40}$ in complex +absolute value. Each exact exponential subsequently squared has modulus +at most one. Thus the recurrence +\[ + e_0=3/33!+2\cdot10^{-40},\qquad + e_{a+1}=2e_a+e_a^2+2\cdot10^{-40} +\] +bounds the accumulated error. Exact rational evaluation gives +$e_{10}<3.542\cdot10^{-34}$, as claimed. +\end{proof} + +\paragraph{Positive exponential upper bounds.} +For $x\ge0$ and an integer $n\ge0$ with $x9\cdot10^{-34}. +\end{aligned} +\] +Here $\sin x\ge2x/\pi$ on $[0,\pi/2]$ follows from concavity. +It follows that every real rounded node $\tau_{jv}^{\#}$ remains strictly +inside $(t_{j-1},t_j)$. The density is therefore evaluated on its stated +piece, and all rounded nodes and exact atoms lie in $[0,1]$. +The remaining real exponential calls in the density and direct factors +now have $|u|\le100$; the residue arguments for $Q_n,D_n$ have $|u|<1$. +They too satisfy the call hypotheses. + +For real $t\in[0,1]$, the exact identities +\[ +\begin{gathered} + |\lambda(t)|\le\frac5{\sqrt3}<3,\\ + |z(t)|^2=h^2+\frac{B^2-2Bh^2}{t^2+h^2}\le\left(\frac{44}{15}\right)^2,\\ + \operatorname{Im}z(t)=h\left(\frac B{t^2+h^2}-1\right) + \ge\frac{26}{435}>.0597 +\end{gathered} +\] +use $B^2-2Bh^2=304/225>0$. They apply in particular at every rounded +node before the floor in~\eqref{cert:lambda-z-star}. That floor lowers the +imaginary part by less than $10^{-40}$ and changes the complex value by +less than $2\cdot10^{-40}$. Thus +\[ + \operatorname{Im}z_*\ge\frac{26}{435}-10^{-40}>0, + \qquad |z_*|<\frac{44}{15}+2\cdot10^{-40}<3. +\] +For the exact transformed arguments, $100\pi|z(t)|<922$; for the rounded +ones, $|p_*z_*m|<100(3.142)(3)<1000$ and +$\Re(ip_*z_*m)=-p_*m\operatorname{Im}z_*\le0$. +The exponential estimate already proved therefore applies to every +transformed call as well. + +For a quadrature node write $t=\tau_{jv}$ and $t^{\#}=\tau_{jv}^{\#}$; +at an atom put $t=t^{\#}=t_j$. Along the intervening real segment, +$|t+ih|\ge h$ and $b,b_*>.86$. The derivative bounds for the reciprocal +factors and the change of $b$ give +\[ +\begin{aligned} + |\lambda_* -\lambda(t)| + &<8(9\cdot10^{-34})+4\cdot10^{-36}+2\cdot10^{-40}<10^{-29},\\ + |z_*-z(t)| + &<\frac{25}{3}(9\cdot10^{-34})+2\cdot10^{-40}<10^{-29}. +\end{aligned} +\] +Indeed $1/(b_*h^2)<8$, $B/h^2=25/3$, and the change of $1/b$ contributes +less than $10^{-36}/(.86^2h)<4\cdot10^{-36}$. At atoms the node-error +terms vanish. Both transformed exponential arguments lie in the closed +left half-plane after multiplication by $i$. The integral first-difference +formula there gives, for example, +\[ + |e_*(z_*m)-e^{i\pi z(t)m}| + <10^{-32}+100(3.142\cdot10^{-29}+3\cdot10^{-36})<4\cdot10^{-27}. +\] +The direct exponential-factor error is smaller. The arguments of the +divided powers differ by less than +$1.5(3.142\cdot10^{-29}+3\cdot10^{-36})<5\cdot10^{-29}$. +The divided powers and their derivatives are bounded along the connecting +segment by $e^{1.5\cdot3.142\cdot3}<1.5\cdot10^6$; this also gives the +stated bound $e^{1.5\pi\cdot3}<1.5\cdot10^6$. One rational check of this +exponential bound uses $1.5(3.142)(3)/16<.884$ and +\[ +\begin{gathered} + \sum_{q=0}^{8}\frac{.884^q}{q!} + +\frac{.884^9}{9!}\frac1{1-.884/10}<2.421,\\ + 2.421^{16}<1.5\cdot10^6. +\end{gathered} +\] +The first expression majorizes the exponential series because every +successive ratio in its tail is at most $.884/10$. +The divided-power errors are therefore less than +$7.5\cdot10^{-23}<1.5\cdot10^{-22}$. +Using $|\lambda|<3$, modulus at most one for the exact exponentials, and +modulus less than two for their approximations, the transformed-factor +error is at most +\[ + 2(1.5\cdot10^6)10^{-29} + +3(1.5\cdot10^6)(4\cdot10^{-27}) + +3(1.5\cdot10^{-22})<3\cdot10^{-20}. +\] +The same bound applies to the direct factor. + +For completeness, the density estimate can be obtained without relying on +cancellation. Its real exponential factors, including evaluation at a +rounded node, have error less than +$3.542\cdot10^{-34}+100(3.142(9\cdot10^{-34})+10^{-36})<3\cdot10^{-31}$. +Let $A_n,B_n$ denote the inner sums in~\eqref{cert:density} with and without +the factor $t_l-t$, respectively, and put $\#$ on their replacements. +On the common piece, $|t_l-t|,|t_l-t^{\#}|\le1$ for $l\ge j$. +The displayed list for $P_l$ gives +$\sum_{l=1}^6|P_l^{\#}-P_l|<7\cdot10^{-36}$ and +$\sum_{l=1}^6|P_l^{\#}|<11$. Hence $|A_n^{\#}|,|B_n^{\#}|<22$ and +\[ +\begin{aligned} + |A_n^{\#}-A_n|,\ |B_n^{\#}-B_n| + &<22(9\cdot10^{-34})+2(7\cdot10^{-36})\\ + &\quad+10(3\cdot10^{-31})<4\cdot10^{-30}. +\end{aligned} +\] +The bracket in~\eqref{cert:density} has approximate magnitude at most +$70(|c_n|+|d_n|)$ and error less than +$1.3\cdot10^{-29}(|c_n|+|d_n|)$. Expanding the outer product once and +using list norm at most three gives the pointwise bound +\[ +\begin{aligned} + |f_j^{\#}(t^{\#})-f_j(t)| + &<2(10^{-36})(3)(2)(70)\\ + &\quad+2(3.142)(3)\bigl(70(3\cdot10^{-31})+1.3\cdot10^{-29}\bigr)\\ + &<7\cdot10^{-28}<10^{-22}. +\end{aligned} +\] + +On the real segment $|f_j|<1000$, and each atom has magnitude at most six +and error less than $10^{-28}$. The latter error also follows directly +from the displayed $P_l$ list. The exact transformed factor has magnitude +below $3\cdot1.5\cdot10^6$, so every approximate factor has magnitude +below $4.6\cdot10^6$. The exact quadrature weights have total mass one, +and the sum of the absolute weight errors over all six pieces is less than +$10^{-29}$. Separating weight, density, factor, and atom errors, and then +adding the analytic quadrature error, bounds either moment error by +\[ +\begin{split} + &10^{-29}(1001)(4.6\cdot10^6)+10^{-22}(4.6\cdot10^6) + +1000(3\cdot10^{-20})\\ + &\quad+7\bigl(10^{-28}(4.6\cdot10^6)+6(3\cdot10^{-20})\bigr) + +10^{-22}<5\cdot10^{-16}<10^{-14}. +\end{split} +\] + +Finally, for the residue arguments the sine and cosine replacement errors +are less than $3.542\cdot10^{-34}+10^{-36}<4\cdot10^{-34}$. +Every nonzero exact sine has magnitude at least $\sin(\pi/12)>1/4$, and +its replacement has magnitude greater than $1/5$. Each squared factor +$(2\sin)^2$ in $Q_n$ and its replacement is less than five in magnitude, +and their difference is less than $5\cdot10^{-33}$. Telescoping the five +factors and allowing for $(p_*/6)^2-(\pi/6)^2$ gives a $Q_n$ error less +than $2\cdot10^{-29}<10^{-25}$. For a cotangent, division by the two sine +lower bounds gives an error less than +$5(4\cdot10^{-34})+20(4\cdot10^{-34})=10^{-32}$. +Summing five terms and allowing for the prefactor gives a $D_n$ error less +than $6\cdot10^{-32}<10^{-25}$, as required. +\end{proof} + +\subsection{Propagation to finite certificates}\label{cert:propagation} + +We record stability estimates before applying any numerical margin. +The first controls the two operator coordinates under errors in the moments +and node constants. The second controls Bernstein sums from errors in +the signed-$y$ moments. + +To propagate a moment radius, abbreviate $X_\ell=X_{p,\ell}(m,1)$ and +write $\mathcal S_{m,c},\mathcal S_{m,d}$ for the two components of +\eqref{cert:pair}, with hats for their rational approximations. +Suppose $|X_0|,|X_1|\le H$, +$|D_m|\le d$, and rational approximations have errors at most +$\epsilon,\epsilon,\eta_Q,\eta_D$ in $X_0,X_1,Q_m,D_m$, respectively. +If both exact and approximate $Q_m$ exceed $q>0$, expanding the numerator +and then the reciprocal in \eqref{cert:pair} gives +\begin{equation}\label{ind:pair-stability} + \max\{|\widehat{\mathcal S}_{m,c}-\mathcal S_{m,c}|, + |\widehat{\mathcal S}_{m,d}-\mathcal S_{m,d}|\} + \le \frac{(d+1+\eta_D)\epsilon+H\eta_D}{q} + +\frac{(d+1)H\eta_Q}{q^2}. +\end{equation} +For example, the perturbed second numerator differs by at most +$(d+\eta_D)\epsilon+H\eta_D+\epsilon$, and the exact numerator is bounded +by $(d+1)H$; this proves the inequality, which also dominates the first +entry's error. The Taylor budgets and the separated and common-kernel +first-block estimates below use $q=7/4$ and $d=2.12$. The separate-rounding +comparison uses $q=7/4$ and $d=4.3$. +The rational lower bound $Q_m>197192/112500$ proved after +\eqref{fourier:node-bounds} exceeds $7/4$ by $1268/450000$, so +$\eta_Q\le10^{-25}$ also ensures that the approximate $Q_m$ exceeds $q$. + +\paragraph{Analytic pair-error allowance.} +For an input $p$ in the rational finite-input domain of +Lemma~\ref{cert:rational-moment-error} and a row $m\in I_f$, define the rational +version $\mathcal S_m^\#$ by using the quantities with superscript $\#$ +throughout~\eqref{cert:pair}. We record the division estimate explicitly +for both basis and full-table inputs. By +\eqref{interpolation:piece-envelopes} and the atom masses there, the total +variation of the folded measure for any input in the real finite-input +domain of Lemma~\ref{cert:moment-error} is at most +\[ + 3\sum_{j=1}^6\frac{M_j}{6}+\sum_{j=0}^6m_j^*=119.642<133. +\] +On the real segment, $|\lambda|,|z|<3$ and +$|e^{i\pi z m}|\le1$ for $m\ge0$. Thus, using $\pi<22/7$, +\[ + |X_{p,0}(m,1)|<133(3)<400,\qquad + |X_{p,1}(m,1)|<133(3)(3)(22/7)=3762<4000. +\] +For this calculation abbreviate $X_j=X_{p,j}(m,1)$, $Q=Q_m$, and $D=D_m$. +Write $\varepsilon=10^{-14}$ and $\delta=10^{-25}$. By +Lemma~\ref{cert:rational-moment-error}, the errors in $X_0,X_1$ are less than +$\varepsilon$ and those in $Q,D$ are less than $\delta$. +Equation~\eqref{fourier:node-bounds} gives $Q>1.75$, $|D|<2.12$, and hence +$Q^{\#}>1.7$. For $\mathcal A=DX_0-X_1$ and +$\mathcal A^{\#}=D^{\#}X_0^{\#}-X_1^{\#}$, +\[ + |\mathcal A|<4848,\qquad + |\mathcal A^{\#}-\mathcal A|<2.12\varepsilon+(400+\varepsilon)\delta+\varepsilon + <3.121\cdot10^{-14}. +\] +The two quotient differences are consequently bounded by +\[ +\begin{aligned} + \left|\frac{X_0^{\#}}{Q^{\#}}-\frac{X_0}{Q}\right| + &<\frac{10^{-14}}{1.7}+\frac{400\cdot10^{-25}}{1.75(1.7)} + <6\cdot10^{-15},\\ + \left|\frac{\mathcal A^{\#}}{Q^{\#}}-\frac{\mathcal A}{Q}\right| + &<\frac{3.121\cdot10^{-14}}{1.7} + +\frac{4848\cdot10^{-25}}{1.75(1.7)} + <1.84\cdot10^{-14}<2\cdot10^{-14}. +\end{aligned} +\] +In particular, the uniform error in each entry of $\mathcal S_m^{\#}$ is +strictly less than $10^{-12}$; we retain this looser allowance below. + +\subsubsection{Bernstein multipliers}\label{ind:bernstein} +For every signed half-gap in Lemma~\ref{signs:bernstein}, $y\ne0$ and +$|y|^{-\nu}\le4$. Directly from \eqref{cert:moments}, +\[ + y^{-\nu}\bigl(U_{p,r+\nu}(m,y)+X_{q,r+\nu}(m,y)\bigr) + =y^r\bigl(U_{p,r+\nu}(m,1)+X_{q,r+\nu}(m,1)\bigr). +\] +This identity also holds for negative $y$ and gives exactly the +power coefficients of \eqref{cert:power-from-moments}; it does not replace +their derivatives by interpolated jets. Two moment errors less than +$10^{-14}$ give a $T_r$ error less than $8\cdot10^{-14}$. +For fixed $0\le k\le28$, the rational recurrence $a_0=1$ and +$a_r=a_{r-1}(k+1-r)/(29-r)$ satisfies +\begin{equation}\label{ind:bernstein-multiplier} + a_r=\prod_{j=1}^r\frac{k+1-j}{29-j} + =\frac{k!}{(k-r)!}\frac{(28-r)!}{28!} + =\frac{\binom{k}{r}}{\binom{28}{r}}\qquad(0\le r\le k). +\end{equation} +Each factor is in $[0,1]$. There are +$\sum_{k=0}^{28}k=28\cdot29/2=406$ nonconstant multipliers, in addition +to $a_0=1$ for each $k$. Every Bernstein sum has at most $29$ terms, +so its error is strictly less than +$29(8\cdot10^{-14})=2.32\cdot10^{-12}$. + +\input{sections/rational-check} + +\subsection{Comparison of first-block majorants}\label{ind:small} + +The preceding $N=64$ proof separates the value and derivative integrands. +Two coarser arguments instead use one majorant for both. They concern the +same basis inputs $n\in J$, row nodes $m\in J$, $y=1$, and orders zero and +one, with no atoms. The table compares their disk and node bounds with +the resulting entry errors: +\begingroup +\small +\[ +\begin{array}{c|c|c|c} + \text{route}&\text{disk majorant}&|D_m|&\text{entry error}\\ \hline + \text{separated}&M_0<24028,\ M_1<468015&<2.12&<7\cdot10^{-14}\\ + \text{common kernel}&M=10^6&<2.12&<4.1\cdot10^{-13}\\ + \text{separate rounding}&M=750000&<4.3&<5.23\cdot10^{-13} +\end{array} +\] +\endgroup +The separated route uses the explicit density bound $D_0$ from +Section~\ref{cert:rational-small}. The two common routes use density and +exponential bounds $(1800,2.8)$ and $(2500,3)$, respectively. + +\subsubsection{A common kernel majorant}\label{ind:small-kernel} + +The rational comparisons +\begin{equation}\label{ind:small-exponentials} + \mathcal E_{32}(44/21)<42/5,\qquad + \mathcal E_{32}(891/875)<14/5 +\end{equation} +follow by clearing positive denominators in~\eqref{ind:positive-exponential}; +both arguments are nonnegative and less than $34$. Since +$4\pi/6<44/21$ and $4\pi(.081)<891/875$, the two basis densities on the +complex disks satisfy +\[ +\begin{aligned} + |u_n(t)|&<2(22/7)^2(65/6)(42/5)=12584/7<1800,\\ + |v_n(t)|&<2(22/7)10(42/5)=3696/7<1800. +\end{aligned} +\] +The bounds $|\lambda|<5$, $|e^{i\pi zm}|<2.8$, and +$\pi|z|<6.2\pi<20$ therefore bound both transformed integrands by +$1800(5)(2.8)(20)=504000<10^6$. The positive degree-63 rule and the +half-radius Cauchy estimate give, for each moment, +\begin{equation}\label{ind:small-kernel-quadrature} + |X_\ell-X_{\ell,\mathrm{quad}}|\le10^6 2^{-62}<2.17\cdot10^{-13}. +\end{equation} +The factor two here compares the integral and quadrature averages; a direct +Taylor average has only one tail, as in Lemma~\ref{ind:taylor-average}. + +The rounding proof in Section~\ref{cert:rational-small} applies to these +same inputs and gives +finite moment-rounding error below $10^{-26}$, hence below $10^{-14}$. +It also gives $|X_0|,|X_1|<10^4$, $|\delta Q|<3\cdot10^{-35}$, +$|\delta D|<4\cdot10^{-36}$, and both denominators above $7/4$. +Thus the total moment allowance is $2.27\cdot10^{-13}$. In +\eqref{ind:pair-stability}, the additional $Q,D$ contribution is less than +\[ + \frac{10^4(4\cdot10^{-36})+(4\cdot10^{-36})(2.27\cdot10^{-13})}{7/4} + +\frac{3.12\cdot10^4(3\cdot10^{-35})}{(7/4)^2}<4\cdot10^{-31}. +\] +Consequently +\begin{equation}\label{ind:small-kernel-entry} + \frac{3.12(2.27\cdot10^{-13})}{1.75}+4\cdot10^{-31} + <4.1\cdot10^{-13}<10^{-12}. +\end{equation} + +\subsubsection{A separate rounding majorant}\label{ind:small-rounding} + +The weaker bounds $e^{4\pi/6}<9$ and $e^{4\pi(.081)}<3$ also follow +from~\eqref{ind:small-exponentials}. They bound either basis density by +$2500$; for the larger density, +$2(22/7)^2(65/6)9<2500$. With the kernel bound $5$ and derivative +factor $6.2\pi<20$, this gives the distinct common majorant +$M=2500(5)(3)(20)=750000$. Its moment quadrature radius is +\begin{equation}\label{ind:small-rounding-quadrature} + \varepsilon_q=750000\,2^{-62} + <1.626303258728257\cdot10^{-13}. +\end{equation} +Using $|D_m|<4.3$ and $Q_m>7/4$, the pair quadrature error is less than +$(5.3/(7/4))\varepsilon_q<4.926\cdot10^{-13}$. +The same finite rounding allowance below $10^{-14}$ applies because the +schedule and inputs have not changed. Substitution of the same polynomial +$Q,D$ errors in~\eqref{ind:pair-stability}, now with $d=4.3$, bounds their +additional contribution by $6\cdot10^{-31}$. Therefore +\begin{equation}\label{ind:small-rounding-entry} + \frac{5.3}{7/4}\bigl(750000\,2^{-62}+10^{-14}\bigr) + +6\cdot10^{-31}<5.23\cdot10^{-13}<10^{-12}. +\end{equation} + +All three routes support the program's $10^{-12}$ entry radius. They do +not extend its coverage: the public rational algorithm still checks only +the 36 first-block entries and the two inverse certificates, not the +residual or Bernstein tables. + +\subsection{Rational-center Taylor integration}\label{ind:taylor} + +For a finite input in the real moment domain of Lemma~\ref{cert:moment-error}, let +$F^U_{j,\ell}$ and $F^X_{j,\ell}$ be the density \eqref{cert:density} +multiplied by the respective factors in \eqref{cert:analytic-factors}. +The input is supported on nodes at most $100$, has coefficient-list norm at +most $3$ and $|C|\le1$, and the parameters satisfy +$0\le m\le100$, $|y|\le3/2$, and $\ell\in\{0,\ldots,30\}$. The coefficients, +$C,m,y$ are rational when a rational enclosure is requested. The six +centers and the two radii are +\begin{equation}\label{ind:centers} + c_j=\frac{2j-1}{12},\qquad R=\frac16,\qquad a=\frac1{12} + \quad(1\le j\le6). +\end{equation} +Each $F^U_{j,\ell}$ and $F^X_{j,\ell}$ is analytic on a neighborhood of +$|t-c_j|\le R$: the input list is finite and the only rational-kernel pole, +$-ih$, is outside that disk. This analytic assertion is made for finite +inputs. An arbitrary $\ell^1$ list need not give a complex-frequency +density analytic on these disks. + +\begin{lemma}[Taylor averages with rational coefficient radii] +\label{ind:taylor-average} +Suppose $F(c_j+x)=\sum_{k\ge0}A_kx^k$ is analytic on a neighborhood of +$|x|\le R$ and $|F|\le M$ there. Let $D\ge0$ be an integer. For +$0\le k\le D$, suppose +$\widehat A_k\in\mathbb Q(i)$ and $e_k\in\mathbb Q_{\ge0}$ satisfy +$|A_k-\widehat A_k|\le e_k$. Then +\begin{align}\label{ind:taylor-average-error} + \left|6\int_{t_{j-1}}^{t_j}F(t)\,\dd t- + \sum_{2r\le D}\widehat A_{2r}\frac{a^{2r}}{2r+1}\right| + \le M2^{-D}+\sum_{2r\le D}e_{2r}\frac{a^{2r}}{2r+1}. +\end{align} +If the final sum of radii is at most $\delta$ for each of the six pieces, +their combined integral has error at most $M2^{-D}+\delta$. Discrete +atoms are not included in this estimate and are to be enclosed separately. +\end{lemma} + +\begin{proof} +Since $2a=1/6$, the expression $6\int F$ is the interval average. The +average of $x^k$ on $[-a,a]$ is zero for odd $k$ and is $a^k/(k+1)$ for +even $k$. Thus the degree-$D$ Taylor polynomial has the exact average +\begin{equation}\label{ind:even-average} + \sum_{2r\le D} A_{2r}\frac{a^{2r}}{2r+1}. +\end{equation} +Cauchy's estimate gives $|A_k|\le M R^{-k}$. Bounding the average of the +absolute tail by its supremum on $[-a,a]$ gives the conservative bound +\[ + M\sum_{k>D}(a/R)^k=M\sum_{k>D}2^{-k}=M2^{-D}. +\] +Replacing the even coefficients adds the displayed weighted sum of their +radii. Finally, each of the six integrals is $1/6$ times its average, so +the six factors $1/6$ sum to one. Taking a real part cannot increase the +error. Atoms have no integral tail of this kind. +\end{proof} + +Here are explicit rational operations for obtaining the coefficient +enclosures in this lemma. Write +$\mathcal B(a,r)=\{w\in\mathbb C:|w-a|\le r\}$, where +$a\in\mathbb Q(i)$ and $r\in\mathbb Q_{\ge0}$, and put +$\langle a\rangle_1=|\Re a|+|\Im a|$. The following enclosure operations +use rational centers and radii: +\begin{align}\label{ind:ball-operations} + \mathcal B(a,r)+\mathcal B(b,s)&\subseteq\mathcal B(a+b,r+s),\nonumber\\ + \mathcal B(a,r)\mathcal B(b,s)&\subseteq + \mathcal B\bigl(ab,\langle a\rangle_1s+\langle b\rangle_1r+rs\bigr),\nonumber\\ + \mathcal B(a,r)^{-1}&\subseteq\mathcal B(a^{-1},r/\ell_0^2) + \quad\text{if }0<\ell_0\le |a|-r. +\end{align} +In the last line $a^{-1}\in\mathbb Q(i)$, and the condition on a rational +$\ell_0$ can be checked by squaring rational quantities. The bound follows +from $|w^{-1}-a^{-1}|=|w-a|/(|w||a|)$. These operations allow repeated +use of the same enclosure; possible dependence between its occurrences +does not invalidate inclusion. + +Constant exponentials also admit arbitrarily small rational radii. +Use the rational bounds $\mathcal T_n,\mathcal E_n$ from +\eqref{ind:positive-exponential}. If +$w\in\mathcal B(w_0,r)$, choose rational $q\ge|w_0|$, for example +$q=\langle w_0\rangle_1$, and nonnegative integers $n,s$ +with $q4/5$. Taking real and imaginary parts of the constant exponential +enclosures also encloses cosine and sine with the same radius. The nonzero +sine denominators in \eqref{fourier:node-data} have modulus greater than +$1/4$. Consequently sufficiently refined enclosures certify fixed +positive lower bounds for the moduli of all denominators used above or in +computing $Q_m,D_m$, and separately a positive lower bound for $Q_m$ at +pair division. For each fixed $D$ there are finitely many operations in +\eqref{ind:convolution}--\eqref{ind:divided-power-recurrence}. Their +radius formulas tend to zero when their input radii tend to zero, while +\eqref{ind:constant-exponential} has radius tending to zero by refining +its input and increasing its two truncation orders. Thus one may refine +the rational inputs and exponential orders, recompute the displayed +rational radii, and stop when any prescribed finite collection of positive +coefficient and atom budgets holds. The convergence just proved ensures +that this rational stopping test terminates. This is a computability +statement for each specified finite set of inputs, not a claim that any +of its moment sums have been evaluated. + +We next make sufficient budgets explicit. Put +$h_D=\lfloor D/2\rfloor+1$. For rational $\delta>0$, the conditions +\begin{equation}\label{ind:coefficient-budget} + e_{j,2r}\le \frac{\delta(2r+1)12^{2r}}{h_D} + \quad(0\le2r\le D) +\end{equation} +on each piece and moment imply that the weighted coefficient-radius sum in +\eqref{ind:taylor-average-error} is at most $\delta$. + +We now combine these integration budgets with the pair estimate +\eqref{ind:pair-stability} and the Bernstein estimate from +Section~\ref{ind:bernstein}. + +\begin{proposition}[Two conditional Taylor budgets]\label{ind:taylor-budgets} +The following choices give sufficient rational error enclosures for the +same finite quantities used in Appendix~\ref{sec:certificates}. + +For a basis input at $n\in J=\{1,3,4\}$, a row $m\in J$, $y=1$, and +$\ell=0,1$, take $D=63$, $h_D=32$, $M=10^6$, and $\delta=10^{-14}$ in +\eqref{ind:coefficient-budget}. There are no atoms. If $Q_m,D_m$ have +rational errors at most $10^{-25}$, either entry of \eqref{cert:pair} +has error less than $2.13\cdot10^{-13}<10^{-12}$. + +For the full rational finite-input domain at the start of this subsection, +take $D=255$, $h_D=128$, $M=10^{53}$, and $\delta=4\cdot10^{-15}$ in +\eqref{ind:coefficient-budget}, and require each of the seven atom-product +radii in \eqref{ind:atom-radii} to be at most $4\cdot10^{-15}/7$. +Each $U$ and $X$ moment then has error less than $10^{-14}$. For node +rows $m\in I_f$, with rational $Q_m,D_m$ errors at most $10^{-25}$, either +entry of \eqref{cert:pair} has error less than $1.8\cdot10^{-14}$. +\end{proposition} + +\begin{proof} +The first-block majorant $10^6$ is proved in +Section~\ref{ind:small-kernel}. Lemma~\ref{ind:taylor-average} gives +\[ + \epsilon\le10^6 2^{-63}+10^{-14}<1.19\cdot10^{-13}. +\] +On the real interval a basis density has modulus below $214$, +$|\lambda|<3$, and $\pi|z|<10$; thus the two exact moments have modulus +below $10^4$. Substitution of $H=10^4$, $q=7/4$, $d=2.12$, and +$\eta_Q=\eta_D=10^{-25}$ in \eqref{ind:pair-stability} gives +\[ + \frac{(3.12+10^{-25})1.19\cdot10^{-13}+10^4 10^{-25}}{7/4} + +\frac{3.12\cdot10^4 10^{-25}}{(7/4)^2}<2.13\cdot10^{-13}. +\] +The pointwise majorant $10^{53}$ for the full domain was proved in +Lemma~\ref{cert:moment-error}. In the second case the total moment radius +is at most +\[ + 10^{53}2^{-255}+8\cdot10^{-15} + <1.728\cdot10^{-24}+8\cdot10^{-15}<10^{-14}. +\] +For the value and first-derivative moments in \eqref{cert:pair}, the real +density is bounded by $1000$, the sum of seven atom moduli by $42$, and +the two factors by $3$ and $30$. Hence both exact moments are bounded +by $H=4\cdot10^4$. The right side of \eqref{ind:pair-stability} is then +at most +\[ + \frac{(3.12+10^{-25})10^{-14}+4\cdot10^4 10^{-25}}{7/4} + +\frac{3.12\cdot4\cdot10^4 10^{-25}}{(7/4)^2}<1.8\cdot10^{-14}. +\] +All inequalities here are comparisons of the displayed rational bounds. +The refinement argument above proves that their coefficient and atom +budgets can be met for any prescribed finite set of these moments. +\end{proof} + +For a matrix with $r$ rows and entry error at most $\varepsilon$, the +induced column-norm error is at most $r\varepsilon$, and multiplication +by $W_\eta$ adds the factor $\|W_\eta\|_1$. In particular $r=6$ for the +$J$ row block and $r=12$ for the rows on $E\cap[7,16]$. In a tabulated +residual, the other summands in \eqref{cert:residual-bounds} are exact +rationals, so its error is just the corresponding pair error. Finally, +for the signed half-gaps of Lemma~\ref{signs:bernstein}, the two moment +errors below $10^{-14}$ give a power-coefficient error below +$8\cdot10^{-14}$ and a Bernstein-coefficient error below +$2.32\cdot10^{-12}$, as proved in Section~\ref{ind:bernstein}. +These bounds specify the margins required in a Taylor-based comparison; +they do not themselves assert the matrix, residual, or Bernstein values +printed in Appendix~\ref{sec:certificates}. + +\subsection{Direct node-Taylor representation}\label{ind:node-taylor} + +There is also a local Taylor representation for the direct cardinal +function that does not integrate its spectral density. It applies to +the full $\ell^1$ class of \eqref{fourier:cardinal}. Write +$\mathcal N_0=\{0\}\cup\mathcal N$ and, for $m\in\mathcal N_0$, put +\begin{equation}\label{ind:node-radius} + \rho_m=\min_{\substack{n\in\mathcal N_0\\n\ne m}}|m-n|. +\end{equation} +The consecutive gaps in one period of $\mathcal N_0$ are +$1,2,1,3,2,3$. Thus the minimum exists and is positive: +$\rho_m=1$ for $m=0$ and for residues $0,1,3,4$, while $\rho_m=2$ +for residues $7,9$. This is a guaranteed radius to the nearest other +node; it may be smaller than the analytic radius for a particular list +whose corresponding coefficients vanish. + +\begin{proposition}\label{ind:node-taylor-proposition} +Let $p=PR$ have a real $\ell^1$ coefficient list as in +\eqref{fourier:list-norm}. Define +\begin{equation}\label{ind:node-P-coefficients} + A_k(m)=\frac{P^{(k)}(m)}{k!} + =\frac1{k!}\sum_{v=-6}^6P_v(i\pi t_v)^k e^{i\pi t_vm}. +\end{equation} +Then $A_0(m)=A_1(m)=0$. The regular series has coefficients +\begin{equation}\label{ind:node-regular-coefficients} + r_j(m)=C\mathbf1_{\{j=0\}}+ + (-1)^j\sum_{\substack{n\in\mathcal N_0\\n\ne m}} + \left(\frac{(j+1)c_n}{(m-n)^{j+2}} + +\frac{d_n}{(m-n)^{j+1}}\right), +\end{equation} +and converges for $|u|<\rho_m$. For $0<|u|<\rho_m$, the Laurent +expansion is +\[ + R(m+u)=\frac{c_m}{u^2}+\frac{d_m}{u} + +\sum_{j=0}^{\infty}r_j(m)u^j. +\] +For every integer $\ell\ge0$ the direct derivative is therefore +\begin{equation}\label{ind:node-derivatives} + \frac{p^{(\ell)}(m)}{\ell!} + =c_mA_{\ell+2}(m)+d_mA_{\ell+1}(m) + +\sum_{k=2}^{\ell}A_k(m)r_{\ell-k}(m), +\end{equation} +where the last sum is empty for $\ell=0,1$. +\end{proposition} + +\begin{proof} +For $n\ne m$, the geometric series for $(m-n+u)^{-1}$ and its derivative +give +\[ + \frac1{m-n+u}=\sum_{j\ge0}\frac{(-1)^ju^j}{(m-n)^{j+1}},\qquad + \frac1{(m-n+u)^2}=\sum_{j\ge0} + \frac{(-1)^j(j+1)u^j}{(m-n)^{j+2}}, +\] +whenever $|u|<|m-n|$. To justify summing these series over the infinite +list, fix $0\le r<\rho_m$ and set $q=r/\rho_m<1$. Since +$|m-n|\ge\rho_m$ for every other node, the sum of absolute values on +$|u|\le r$ is at most +\[ + \frac{\sum_{n\ne m}|c_n|}{\rho_m^2} + \sum_{j\ge0}(j+1)q^j + +\frac{\sum_{n\ne m}|d_n|}{\rho_m}\sum_{j\ge0}q^j + = + \frac{\sum_{n\ne m}|c_n|}{\rho_m^2(1-q)^2} + +\frac{\sum_{n\ne m}|d_n|}{\rho_m(1-q)}<\infty. +\] +This proves absolute uniform convergence on every smaller closed disk, +permits exchanging the node and Taylor sums, and gives +\eqref{ind:node-regular-coefficients}. It also proves precisely why the +nearest-other-node distance is a valid common radius. + +The finite Fourier expansion \eqref{fourier:polynomial} gives +\eqref{ind:node-P-coefficients}, and the double zero of $P$ gives +$A_0=A_1=0$. Multiply its entire Taylor series by the convergent Laurent +series for $R$. The two principal terms are canceled by the double zero; +collecting the coefficient of $u^\ell$ gives +\eqref{ind:node-derivatives}. In particular $A_2=Q_m$ and $A_3=Q_mD_m$, +so the cases $\ell=0,1$ reproduce \eqref{fourier:node-values}. +\end{proof} + +For finite rational input, the constants in this direct formula can be +enclosed by the rational operations above. For an arbitrary $\ell^1$ +input the formula is an exact convergence statement; a computable error +for truncating its node sum additionally requires a bound for that list's +tail. It represents the direct function $p$ only. The transformed +function $K_p$ still has the kernel representation +\eqref{fourier:kernel}, and this local identity neither places $K_p$ in +the cardinal class nor replaces the exact-derivative certificates for the +tabulated functions. diff --git a/preprints/A-sharp-Fourier-certificate-for-planar-circle-packing-September-23-2026/build/sections/interpolation.tex b/preprints/A-sharp-Fourier-certificate-for-planar-circle-packing-September-23-2026/build/sections/interpolation.tex new file mode 100644 index 0000000000000000000000000000000000000000..236b2530f5f016819110a07ae5f8fa1a1e7a554d --- /dev/null +++ b/preprints/A-sharp-Fourier-certificate-for-planar-circle-packing-September-23-2026/build/sections/interpolation.tex @@ -0,0 +1,401 @@ +\section{Solving the interpolation equations}\label{sec:interpolation} + +The equations \eqref{fourier:coefficient-equations} couple two coefficient +lists through the transformed part of the construction. We solve them on +$\ell^1$, using the strict damping of the Fourier kernel to control every +row beyond a fixed finite set. + +Let +\[ + X=\ell^1(\mathcal N;\RR^2),\qquad + \norm{x}_1=\sum_{n\in\mathcal N}\bigl(\abs{c_n}+\abs{d_n}\bigr) + \quad\text{for }x=((c_n,d_n))_{n\in\mathcal N}. +\] +When writing matrices, we order all $c$ coordinates first and all $d$ +coordinates second, with increasing nodes within each group. All operator +norms below are the induced $\ell^1$ norms, equivalently the maximum +absolute column sum for a finite matrix. A subscript specifying a set of +nodes includes both coordinates at each node; a star denotes all positive +nodes. + +For $x\in X$, let $p_x$ be the function \eqref{fourier:cardinal} with these +positive-node coefficients and with $C=c_0=d_0=0$. Define the real linear +operator $S$ by +\begin{equation}\label{interpolation:S} + (Sx)_{c,n}=-\frac{K_{p_x}(n)}{Q_n},\qquad + (Sx)_{d,n}=\frac{D_nK_{p_x}(n)-K_{p_x}'(n)}{Q_n}. +\end{equation} +The bounds proved below show, in particular, that $S$ is a bounded operator +on $X$. + +We use two successive decompositions of this operator. Set +\begin{equation}\label{interpolation:node-blocks} + J=\{1,3,4\},\qquad E=\mathcal N\cap[7,100],\qquad + I_f=J\cup E=\mathcal N\cap[1,100],\qquad D=S_{J,J}. +\end{equation} +A finite inverse handles the coordinates on $J$. A first Schur complement +then incorporates $E$, using separate bounds for its rows through $16$ +and beyond $16$; a second incorporates the infinite tail beyond $100$. +The tail bounds apply to output rows uniformly over all input columns. +The first output rows require a separate, larger bound even for columns +indexed by large nodes. + +\subsection{Uniform envelopes for the Fourier terms} + +We use the following rational upper and lower bounds. As throughout the +construction, every finite decimal denotes its exact rational value. +\begin{table}[htbp] +\centering +\begin{tabular}{c|rrr|rr} +$j$ & $b_j^*$ & $a_j^*$ & $g_j$ & $M_j$ & $\beta_j$\\ \hline +0 & 2.887 & 2.934 & 2.933 & --- & ---\\ +1 & 2.665 & 2.713 & 2.440 & 56 & 0\\ +2 & 2.218 & 2.269 & 1.567 & 48 & 4.8\\ +3 & 1.804 & 1.859 & .900 & 35 & 3.17\\ +4 & 1.486 & 1.548 & .482 & 25 & 1.946\\ +5 & 1.250 & 1.320 & .224 & 19 & 1.217\\ +6 & 1.073 & 1.151 & .0597 & 6.284 & .792 +\end{tabular} +\caption{Envelopes for the folded Fourier integrals.} +\label{interpolation:envelopes} +\end{table} + +At each endpoint $t_j=j/6$, +\[ + \abs{\lambda(t_j)}\le b_j^*,\qquad + \abs{z(t_j)}\le a_j^*,\qquad + \operatorname{Im}z(t_j)\ge g_j. +\] +On the interval $t_{j-1}\le t\le t_j$, the corresponding bounds are +\begin{equation}\label{interpolation:piece-envelopes} + \abs{\lambda(t)}\le b_{j-1}^*,\quad + \abs{z(t)}\le a_{j-1}^*,\quad + \operatorname{Im}z(t)\ge g_j+\beta_j(t_j-t),\quad + \abs{u_n(t)},\abs{v_n(t)}\le M_j. +\end{equation} +The density bounds hold uniformly for +$n\in\{0\}\cup\mathcal N$. + +Here are direct ways to verify these estimates. The identities +\[ + \abs{\lambda(t)}=\frac{1}{b\sqrt{t^2+h^2}},\qquad + \abs{z(t)}^2=h^2+\frac{B^2-2Bh^2}{t^2+h^2},\qquad + \operatorname{Im}z(t)=h\left(\frac{B}{t^2+h^2}-1\right) +\] +show that the first two quantities decrease on $[0,1]$. The nonnegative +quantity $2hBt/(t^2+h^2)^2$, which is minus the derivative of +$\operatorname{Im}z$, has its minimum on each of the six intervals at an +endpoint. Its only interior critical point on the positive axis is a +maximum. These observations give the first three bounds in +\eqref{interpolation:piece-envelopes} by endpoint substitution. +For the densities, $e^{i\pi t_l n}$ depends only on $n$ modulo $12$, +whereas $\abs{e^{-i\pi tn}}=1$ on the real interval. The norm of an +affine function of $t$ is convex. Thus an upper bound for both folded +densities on the $j$th interval is +\begin{equation}\label{interpolation:density-check} + 2\max_{a\in A}\max\left\{ + \pi\abs{\sum_{l=j}^6P_le^{i\pi t_la}},\quad + \pi^2\max_{u\in\{t_{j-1},t_j\}} + \abs{\sum_{l=j}^6(t_l-u)P_le^{i\pi t_la}} + \right\}. +\end{equation} +Substitution gives a value less than the displayed $M_j$ in every case. +For the constant input $C=1$, the absolute folded atom masses are +\[ + (m_0^*,\ldots,m_6^*)=(5,4,4,4,2,4,2). +\] + +For $s\ge0$, set +\begin{equation}\label{interpolation:L} + \begin{aligned} + L_j(s)&=b_{j-1}^*e^{-\pi g_js} + \min\left\{\frac16,\frac{1}{\pi\beta_js}\right\} + &&(1\le j\le6),\\ + L_j^P(s)&=m_j^*b_j^*e^{-\pi g_js} + &&(0\le j\le6). + \end{aligned} +\end{equation} +When $\beta_js=0$, the minimum in the first line is understood to be +$1/6$. Integrating the exponential envelope in +\eqref{interpolation:piece-envelopes} over a piece of length $1/6$ gives +its factor $L_j(s)$. Differentiating the kernel contributes +$(i\pi z(t))^\ell$. Consequently, for every integer $\ell\ge0$ and every +input $p$ of the form \eqref{fourier:cardinal}, +\begin{equation}\label{interpolation:moment-bound} + \abs{K_p^{(\ell)}(s)} + \le w\sum_{j=1}^6 M_jL_j(s)(\pi a_{j-1}^*)^\ell + +\abs C\sum_{j=0}^6L_j^P(s)(\pi a_j^*)^\ell, + \qquad s\ge0, +\end{equation} +where $w=\sum_{n\in\{0\}\cup\mathcal N}(\abs{c_n}+\abs{d_n})$. +The argument applies first to finite lists and then to all such lists by +the finite-measure representation and absolute summability. + +Both $Q_n$ and $D_n$ are $12$-periodic on the nodes. Moreover, +\[ + L_j(n+12k)\le e^{-12k\pi g_j}L_j(n)\qquad(k\ge0), +\] +because the minimum factor in \eqref{interpolation:L} is nonincreasing. +For every real cutoff $x\ge0$, the value and derivative terms in +\eqref{interpolation:S}, together with +\eqref{interpolation:moment-bound}, therefore imply +\begin{equation}\label{interpolation:row-envelope} + \norm{S_{>x,*}} + \le + \sum_{n\in\mathcal N\cap(x,x+12]} + \sum_{j=1}^6 + \frac{1+\abs{D_n}+\pi a_{j-1}^*}{Q_n}\, + \frac{M_jL_j(n)}{1-e^{-12\pi g_j}}. +\end{equation} +Here $>x$ means a strict inequality for the row node. This is a bound +uniform over all input columns. It applies as well to a single input coefficient +$c_0$ or $d_0$. + +For completeness, this estimate also justifies the operator definition on +the whole space. Apply the bound first to one positive-node basis column, +then sum with the absolute coefficients of a finitely supported list. +The case $x=0$ bounds the resulting output norm by a constant times the +input norm. Finite lists are dense in $X$, so this map has a unique bounded +extension to $X$. The basis measures converge in total variation by +Lemma~\ref{fourier:measure-lemma}; hence $K_{p_x}(n)$ and $K_{p_x}'(n)$ +converge at each fixed node as the lists are truncated. The extended map +therefore has exactly the coordinates in \eqref{interpolation:S}, not merely +the coordinates of an auxiliary matrix limit. + +For $x=0,4,16,100$, respectively, the right side of +\eqref{interpolation:row-envelope} is less than +\[ + 27.7,\qquad .272,\qquad .0145,\qquad 4.7\cdot10^{-10}. +\] +We shall use the slightly weaker bounds +\begin{equation}\label{interpolation:tails} + \norm{S_{>0,*}}<28,\qquad + \norm{S_{>4,*}}<.28,\qquad + \norm{S_{>16,*}}<.015,\qquad + \norm{S_{>100,*}}<5\cdot10^{-10}. +\end{equation} +For the input $C=1$, the analogue of +\eqref{interpolation:row-envelope} replaces $M_jL_j$ by $L_j^P$, +uses $a_j^*$ in place of $a_{j-1}^*$, and sums over $0\le j\le6$. +For rows greater than $100$, it gives the bound +\begin{equation}\label{interpolation:atom-tail} + \sum_{n\in\mathcal N,\ n>100} + \left(\frac{\abs{K_P(n)}}{Q_n} + +\frac{\abs{D_nK_P(n)-K_P'(n)}}{Q_n}\right) + <3.2\cdot10^{-8}. +\end{equation} +The scalar substitutions in these envelope estimates use only the explicit +constants and the node data. Section~\ref{cert:scalar-bounds} records the +finite scalar bounds and the precise coverage of their supplied verifier. + +\begin{remark}[Compactness of the interpolation operator] +\label{interpolation:compactness} +Let $P_M$ keep the output coordinates with node at most $M$. Then $P_MS$ +has finite rank. The right side of \eqref{interpolation:row-envelope} +tends to zero as $x\to\infty$: there are only six nodes in any period, +the periodic $Q_n,D_n$ are uniformly bounded with $Q_n$ bounded away from +zero, and every $L_j(n)$ contains the factor $e^{-\pi g_jn}$ with $g_j>0$. +Thus $\norm{S-P_MS}\to0$. The operator $S$ is a norm limit of finite-rank +operators and is compact on $X$. This property describes the tail +structure; the explicit inverse below is obtained independently from the +quantitative block estimates. +\end{remark} + +\subsection{The finite certificates} + +There are $51$ nodes in $I_f$ and hence $102$ scalar coordinates. +The approximate lists $\widetilde x_1,\widetilde x_2$ are given by +Table~\ref{tab:coefficients}, with the indicated factor $10^{-10}$, +and are zero at positive nodes beyond $100$. Append the fixed data +\eqref{fourier:fixed} to obtain $\widetilde p_i$, and write +$\widetilde K_i=K_{\widetilde p_i}$. + +\begin{lemma}[Finite certificates]\label{interpolation:finite} +For each $\eta\in\{1,-1\}$, the matrix $I-\eta D$ is invertible, and +\begin{equation}\label{interpolation:finite-matrices} + \norm{(I-\eta D)^{-1}}<32,\qquad + \norm{(I-\eta D)^{-1}S_{J,E}}<3.7,\qquad + \norm{S_{E\cap[7,16],J}}<.09. +\end{equation} +For $i=1,2$ and $n\in I_f$, the table satisfies +\begin{equation}\label{interpolation:finite-residuals} + \begin{split} + \abs{\widetilde c_{i,n}+\frac{\widetilde K_{3-i}(n)}{Q_n}} + &<10^{-9},\\ + \abs{\widetilde d_{i,n}+\mathbf1_{\{i=1,\ n=1\}} + +\frac{\widetilde K_{3-i}'(n)-D_n\widetilde K_{3-i}(n)}{Q_n}} + &<10^{-9}. + \end{split} +\end{equation} +For each $i$, the norm of the tabulated list, including node zero, is less +than $2.29$, and its norm on nodes greater than $40$ is less than $.00002$. +\end{lemma} + +\begin{proof} +The primary interval calculation in Appendix~\ref{sec:certificates} proves +the exact bounds~\eqref{cert:matrix-table}, \eqref{cert:exterior-block}, and +\eqref{cert:residual-bounds}, using the quadrature assertion of +Lemma~\ref{cert:moment-error} for the analytic moment remainder. +The elementary argument in Section~\ref{cert:matrix-bounds} gives the +matrix conclusions, and~\eqref{cert:table-norms} gives the list norms. +\end{proof} + +\subsection{Inversion on the whole coefficient space} + +We first prove an inverse bound that keeps finite and tail residuals +separate. + +\begin{lemma}\label{interpolation:inverse} +For each $\eta\in\{1,-1\}$, $I-\eta S$ has a bounded inverse on $X$. +If $r=r_f+r_t$, with $r_f$ supported on $I_f$ and $r_t$ on +$\mathcal N\cap(100,\infty)$, then +\begin{equation}\label{interpolation:inverse-bound} + \norm{(I-\eta S)^{-1}r}_1 + \le91\norm{r_f}_1+2540\norm{r_t}_1. +\end{equation} +\end{lemma} + +\begin{proof} +Fix $\eta$ and first restrict to the finite block $I_f$. Put +\[ + V=(I-\eta D)^{-1},\qquad L=S_{E,J},\qquad U=S_{J,E}. +\] +The first block of tail estimates and the finite certificates give +\[ + \norm L + \le\norm{S_{E\cap[7,16],J}}+\norm{S_{E\cap(16,100],J}} + <.09+.015<.11. +\] +Eliminating the $J$ coordinates from +$(I-\eta S_{I_f,I_f})x_f=r_f$ leaves on $E$ the operator +\[ + I-\eta S_{E,E}-LVU. +\] +Its difference from the identity has norm less than +\[ + \norm{S_{E,E}}+\norm L\norm{VU} + <.28+.11\cdot3.7=.687<1. +\] +It is therefore invertible by a Neumann series. Solving this block and +substituting into the $J$ equations yields an inverse $V_f$ for the whole +finite compression, with +\begin{equation}\label{interpolation:finite-inverse-bound} + \begin{split} + \norm{V_f} + &\le\max\left\{ + 32+\frac{(1+3.7)\cdot.11\cdot32}{1-.687},\quad + \frac{1+3.7}{1-.687}\right\}\\ + &<90. + \end{split} +\end{equation} +Indeed, the two entries in the maximum bound the output norm arising +from a unit input on $J$ and on $E$, respectively. + +Now write $X=X_f\oplus X_t$, where $X_t$ is the infinite tail space. +The full equation $(I-\eta S)x=r$ is equivalent, after elimination of +$x_f$, to +\begin{equation}\label{interpolation:tail-schur} + \bigl(I-\eta S_{t,t}-S_{t,f}V_fS_{f,t}\bigr)x_t + =r_t+\eta S_{t,f}V_fr_f. +\end{equation} +Put $\delta=5\cdot10^{-10}$. By +\eqref{interpolation:tails}, +\[ + \norm{S_{t,t}}<\delta,\qquad + \norm{S_{t,f}}<\delta,\qquad + \norm{S_{f,t}}<28. +\] +The operator subtracted from the identity in +\eqref{interpolation:tail-schur} thus has norm less than +\[ + \delta(1+90\cdot28)=2521\delta<1. +\] +Its Neumann series converges in the bounded operators on $X_t$. +Together with the finite inverse this proves invertibility on the full +space $X$, without any passage to a limit of finite-section solutions. +Quantitatively, +\[ + \norm{x_t}_1 + \le\frac{\norm{r_t}_1+90\delta\norm{r_f}_1}{1-2521\delta}, + \qquad + \norm{x_f}_1\le90\norm{r_f}_1+2520\norm{x_t}_1. +\] +Adding these inequalities gives coefficients less than $91$ and $2540$ +for $\norm{r_f}_1$ and $\norm{r_t}_1$, respectively, which proves +\eqref{interpolation:inverse-bound}. +\end{proof} + +\begin{proposition}[Exact interpolation and coefficient enclosure] +\label{interpolation:solution} +There are unique real lists $x_1,x_2\in X$ satisfying +\eqref{fourier:coefficient-equations} with the fixed data +\eqref{fourier:fixed}. For each $i=1,2$, +\begin{equation}\label{interpolation:coefficient-error} + \sum_{n\in\mathcal N} + \bigl(\abs{c_{i,n}-\widetilde c_{i,n}} + +\abs{d_{i,n}-\widetilde d_{i,n}}\bigr)<.00003. +\end{equation} +The norm of each exact list, including its node-zero coefficients, is +less than $2.3$. The corresponding functions satisfy all the +interpolation conditions \eqref{fourier:interpolation}. +\end{proposition} + +The uniqueness here is within the specified coefficient space, with the +fixed node-zero data and constants. Triangular-shell values and first +derivatives do not determine an arbitrary radial Schwartz function and +its Fourier transform uniquely \cite[Theorem~1.7]{Sardari2021}. + +\begin{proof} +Let $p_i^0$ denote \eqref{fourier:cardinal} with the fixed values +$(C_i,c_{i,0},d_{i,0})$ and all positive-node coefficients zero. Let +$e\in X$ have its $d_1$ coordinate equal to one and all other coordinates +zero. Define $g_i\in X$ by +\[ + (g_i)_{c,n}=-\frac{K_{p_{3-i}^0}(n)}{Q_n},\qquad + (g_i)_{d,n}= + \frac{D_nK_{p_{3-i}^0}(n)-K_{p_{3-i}^0}'(n)}{Q_n} + -\mathbf1_{\{i=1\}}e_{d,n}. +\] +The estimates for the node-zero and constant columns show that these +are summable lists. The equations to solve are precisely +\[ + x_1-Sx_2=g_1,\qquad x_2-Sx_1=g_2. +\] +Their sum is an equation for $I-S$, and their difference is an equation +for $I+S$. Lemma~\ref{interpolation:inverse} gives unique real solutions +and hence unique $x_1,x_2$. + +It remains to locate these solutions relative to the table. Set +\[ + \rho_i=\widetilde x_i-S\widetilde x_{3-i}-g_i. +\] +There are $102$ scalar coordinates in $I_f$, so +\eqref{interpolation:finite-residuals} gives +\[ + \norm{(\rho_i)_f}_1<1.1\cdot10^{-7}=:R_f. +\] +On the tail, both the tabulated positive-node list and the vector $e$ +vanish. The density-column estimate, now including node zero, and +\eqref{interpolation:atom-tail} imply +\[ + \norm{(\rho_i)_t}_1 + <2.29\cdot5\cdot10^{-10}+.017\cdot3.2\cdot10^{-8}=:R_t. +\] +Here $2.29$ bounds the whole tabulated coefficient list, and $.017$ +bounds both fixed constants in absolute value. + +The errors $x_i-\widetilde x_i$ are recovered by taking half the sum +and half the difference of the solutions with residuals +$-(\rho_1+\rho_2)$ and $-(\rho_1-\rho_2)$. Applying +\eqref{interpolation:inverse-bound} to both gives, for each list, +\[ + \norm{x_i-\widetilde x_i}_1 + \le2(91R_f+2540R_t) + =.00002860012<.00003. +\] +This proves \eqref{interpolation:coefficient-error}. Since the fixed +node-zero coefficients agree, the total exact coefficient norm is +less than $2.29+.00003<2.3$. Finally, +\eqref{fourier:coefficient-equations} is equivalent to +\eqref{fourier:interpolation} by Lemma~\ref{fourier:equivalence}. +\end{proof} diff --git a/preprints/A-sharp-Fourier-certificate-for-planar-circle-packing-September-23-2026/build/sections/introduction.tex b/preprints/A-sharp-Fourier-certificate-for-planar-circle-packing-September-23-2026/build/sections/introduction.tex new file mode 100644 index 0000000000000000000000000000000000000000..9292ad7549152501aab97208b24a8be0a3f89391 --- /dev/null +++ b/preprints/A-sharp-Fourier-certificate-for-planar-circle-packing-September-23-2026/build/sections/introduction.tex @@ -0,0 +1,193 @@ +\section{Introduction} +\label{sec:introduction} + +The planar circle-packing problem asks what fraction of the plane can be +occupied by congruent disks with disjoint interiors. Scale their radius +to $1/2$, so their centers form a set $\mathcal C\subset\RR^2$ with +$|x-y|\ge1$ for distinct centers. If $B_R$ is the disk of radius $R$ +centered at the origin, we use the upper covered-area density +\[ + \overline\delta(\mathcal C) + =\limsup_{R\to\infty} + \frac{\vol\bigl(B_R\cap\bigcup_{c\in\mathcal C}B(c,1/2)\bigr)} + {\vol B_R}. +\] +The triangular lattice attains $\pi/(2\sqrt3)$, and Thue's theorem gives +the matching upper bound. Hales presents an elementary geometric proof +based on an idea of Rogers~\cite[pp.~442--443]{Hales2000}. + +The Fourier-analytic approach of Cohn and Elkies~\cite{CohnElkies2003} +asks whether the same optimum can be certified by one function with +compatible signs before and after Fourier transformation. The density +theorem alone does not supply such a function. Our result is a sharp +Fourier certificate for the classical optimum. + +Throughout, the Fourier transform is +\[ + \widehat f(\xi)=\int_{\RR^2}f(x)e^{-2\pi i x\cdot\xi}\,\dd x. +\] +For real radial integrable functions it is real and radial. A Schwartz +function is a smooth function whose derivatives decay faster than every +inverse power of the radius; this regularity is stronger than the +admissibility conditions in the Cohn--Elkies theorem. + +\begin{theorem}[Sharp Fourier certificate]\label{intro:main} +There is a real radial Schwartz function $f:\RR^2\to\RR$ such that +\begin{equation}\label{intro:conditions} + \widehat f(0)=1,\qquad f(0)=\frac{2}{\sqrt3},\qquad + \widehat f(\xi)\ge0\quad(\xi\in\RR^2),\qquad + f(x)\le0\quad(|x|\ge1). +\end{equation} +\end{theorem} + +Theorem~\ref{intro:main} gives an affirmative answer to the +two-dimensional case of Cohn and Elkies's Conjecture~7.3 +\cite{CohnElkies2003}, concerning sharp auxiliary functions for their +linear programming bound. Their Definition~2.2 and Theorem~3.1 apply to +this Schwartz function and to arbitrary packings, not only periodic ones. +Their opposite Fourier phase gives the same transform for these even +functions. +For disks of radius $1/2$ they give +\[ + \vol B(0,1/2)\,\frac{f(0)}{\widehat f(0)} + =\frac{\pi}{4}\frac{2}{\sqrt3} + =\frac{\pi}{2\sqrt3}. +\] +This is the density of the lattice generated by $(1,0)$ and +$(1/2,\sqrt3/2)$. Thus the theorem identifies an extremizer for this +analytic bound. To put it in the equal-origin formulation of +Cohn and Elkies's Theorem~3.2, which their Conjecture~7.3 invokes, set +\[ + g(x)=\frac{\sqrt3}{2}\, + f\bigl((\sqrt3/2)^{1/2}x\bigr). +\] +The two-dimensional scaling rule gives $g(0)=\widehat g(0)=1$, +$\widehat g\ge0$, and $g(x)\le0$ for +$|x|\ge(2/\sqrt3)^{1/2}$, exactly the requested planar witness. +The sign estimates also determine the zeros in their required domains. +For $f$, every zero with $|x|\ge1$ has integral squared radius. This +restriction recovers a classical equality consequence: among periodic +packings, equality in the density bound forces the triangular lattice +(Corollary~\ref{normalization:periodic-uniqueness}). The equal-origin +witness $g$ has zeros outside the shells of the triangular lattice of +covolume one, so it does not meet the additional zero-set requirement of +Cohn and Elkies's Conjecture~8.1. Section~\ref{sec:fourier} gives an explicit +extra node, and Section~\ref{sec:normalization} records the zero sets in +their respective sign domains and final radial coordinates. + +\subsection{Fourier certificates and interpolation} + +Cohn and Elkies formulated their sharpness conjecture in dimensions +$2$, $8$, and $24$~\cite[Conjecture~7.3]{CohnElkies2003}. +Their finite Laguerre--Gaussian approximations already combine prescribed +double roots, numerical searches for the remaining parameters, and rigorous +sign checks by rational arithmetic and Sturm root counts +\cite[Section~7, pp.~704--705]{CohnElkies2003}. Exact attainment of the +conjectured sharp bound remained unresolved in that approach. +Viazovska constructed the required function in dimension +$8$~\cite[Theorems~3 and~4]{Viazovska2017}, and Cohn, Kumar, Miller, +Radchenko, and Viazovska constructed its dimension-$24$ counterpart +\cite[Section~4]{CKMRV2017}. +Their constructions use modular forms to control a function and its +Fourier transform simultaneously. +Cohn and Elkies report that Gorbachev independently obtained essentially +the same linear programming bound \cite[p.~694]{CohnElkies2003}. + +Interpolation gives a broader setting for these constructions. +Radchenko and Viazovska recovered even Schwartz functions on the real +line from their values and Fourier values at the square roots of +nonnegative integers~\cite[Theorem~1]{RadchenkoViazovska2019}. +Cohn, Kumar, Miller, Radchenko, and Viazovska developed interpolation +using values and first derivatives in dimensions $8$ and $24$ and used +it to prove universal optimality of the $E_8$ and Leech lattices +\cite[Theorems~1.4, 1.7, and~1.9]{CKMRV2022}. + +The planar shell data have a different feature. +Cohn, Kumar, Miller, Radchenko, and Viazovska conjectured that values +and first derivatives of a radial Schwartz function and its Fourier +transform on the nonzero shells of the triangular lattice normalized to +covolume one do not determine +the function uniquely~\cite[Conjecture~7.5]{CKMRV2022}. +Sardari proved this nonuniqueness by constructing infinitely many +linearly independent radial Schwartz functions with double zeros, +before and after Fourier transformation, on those shells +\cite[Theorem~1.7]{Sardari2021}. +Thus prescribing the shell data alone does not single out a planar +certificate. We solve a specified interpolation problem within an +explicit family of Gaussian-damped cardinal functions and establish +the signs separately. + +A 2026 bachelor's thesis abstract by Zhitniaia~\cite{Zhitniaia2026} +announces an integral representation of a magic function for the +hexagonal lattice using weakly holomorphic modular forms and theta +functions for $\Gamma_0(12)$. Our comparison is limited to the +institutional abstract, which does not specify the complete sign, +normalization, and regularity statement. The precise relation of that +claim to Theorem~\ref{intro:main} remains to be determined from the +full thesis. + +\subsection{The construction} + +For a sharp Fourier bound on a lattice, Poisson summation forces +the auxiliary function and its Fourier transform to vanish on the +nonzero shells of the lattice and its dual, respectively +\cite[Section~5]{CohnElkies2003}. +After normalizing the triangular lattice to covolume one, it is +isometric to its dual, so the two sets of shell radii coincide. +This normalization makes the coupled Fourier equations symmetric. +At the end we dilate the resulting function to put the first +physical-space shell at the exclusion radius one in +Theorem~\ref{intro:main}. +We cover their squared radii, in a fixed scaled coordinate, by a +periodic set of interpolation nodes. +Periodic supersets of the values of the triangular quadratic form +also occur in Sardari's construction +\cite[Section~6.1]{Sardari2021}; here a fixed periodic set permits +an explicit trigonometric cardinal family. + +An interior zero of a differentiable function with a fixed sign must +also have zero first derivative. We therefore impose zeros of order +at least two on the Fourier side and at every physical-space shell +beyond the first. The first physical-space shell is the boundary of +the required sign region, so a simple zero is possible there; its +prescribed negative derivative will fix the normalization. + +A trigonometric polynomial vanishes doubly on the chosen set. +Dividing it by linear and quadratic factors produces functions +that prescribe values and first derivatives at individual nodes. +This Hermite-cardinal pattern has a precedent in the Gaussian +Beurling--Selberg constructions of Carneiro, Littmann, and Vaaler +\cite[formulas~(1.2)--(1.3)]{CarneiroLittmannVaaler2013}; the Fourier +coupling and sign estimates below are proved for the present family. +Multiplication by a fixed Gaussian gives radial Schwartz functions. +Their Fourier transforms admit integral representations with +additional Gaussian decay. We use that decay to solve the coupled +interpolation equations: a finite matrix handles the first nodes, +and explicit estimates control the remaining infinite system. The inversion +acts on the full unweighted $\ell^1$ coefficient space. Its uniqueness +concerns the prescribed coefficient lists within this chosen family. + +The resulting functions have exact zeros, but the coefficient +lists used for computation are finite approximations. +An explicit per-list $\ell^1$ error bound connects these approximations +to the exact solution. Near a zero, a raw error divided by its vanishing +power would be singular. We instead subtract the lower Taylor terms of +the error before division; positive Bernstein coefficients then certify +the signs on complete half-gaps. A separate sine-product barrier and +second-derivative estimate control the entire unbounded region. +Poisson summation first equates the two origin values. +Differentiating its dilation identity then determines their common +positive value from the prescribed first-shell derivative, so the +final normalization is exact. + +Section~\ref{sec:fourier} develops the cardinal functions and their +Fourier transforms. Section~\ref{sec:interpolation} solves the +interpolation equations, and Section~\ref{sec:signs} proves the +global signs and identifies their zeros. Section~\ref{sec:normalization} +establishes the exact normalization, performs the final dilation, and +derives the packing consequences. Appendix~\ref{sec:certificates} gives +the finite data and the primary interval certificate, including the +analytic integration error that connects its finite sums to exact moments. +Appendix~\ref{ind:section} gathers the rational alternatives and +independent checks, with their precise mathematical and computational +scopes. diff --git a/preprints/A-sharp-Fourier-certificate-for-planar-circle-packing-September-23-2026/build/sections/normalization.tex b/preprints/A-sharp-Fourier-certificate-for-planar-circle-packing-September-23-2026/build/sections/normalization.tex new file mode 100644 index 0000000000000000000000000000000000000000..c559a6037b296055211ffc486eeb412b1c5f740f --- /dev/null +++ b/preprints/A-sharp-Fourier-certificate-for-planar-circle-packing-September-23-2026/build/sections/normalization.tex @@ -0,0 +1,311 @@ +\section{Poisson normalization} +\label{sec:normalization} + +The prescribed first-shell derivative determines the remaining scalar +normalization. Poisson summation first equates the values at the origin; +its differentiated dilation identity then proves that their common value is +positive, without estimating that value numerically. + +Let the exact coefficient lists be those of +Proposition~\ref{interpolation:solution}, and put +\[ + G_i(s)=e^{-\pi hs}H_i(s),\qquad F(x)=G_1(b|x|^2). +\] +Section~\ref{sec:fourier} proves that $F$ is real, radial, and Schwartz, +with $\widehat F(\xi)=G_2(b|\xi|^2)$. The interpolation conditions give +\begin{equation}\label{normalization:jets} + \begin{split} + G_1(n)&=G_2(n)=0,\\ + G_2'(n)&=0,\qquad + G_1'(n)=-Q_1e^{-\pi h}\mathbf1_{\{n=1\}} + \qquad(n\in\mathcal N). + \end{split} +\end{equation} +Indeed, $G_i'(n)=e^{-\pi hn}H_i'(n)$ because $H_i(n)=0$. + +\subsection{The dilation identity} + +Use the lattice $\Lambda$ of covolume one and the shell coordinate $s_a$ from +\eqref{fourier:lattice}--\eqref{fourier:shell-coordinate}. Every positive +$s_a$ lies in $\mathcal N$. Exactly six lattice vectors have +$s_a=1$: from +$4(j^2+jk+k^2)=(2j+k)^2+3k^2=4$, one obtains +\[ + (j,k)=(1,0),\ (-1,0),\ (0,1),\ (0,-1),\ (1,-1),\ (-1,1). +\] + +As shown in Section~\ref{sec:fourier}, $\Lambda^*=\mathcal R\Lambda$. +Apply Poisson summation on $\Lambda$ to +$F_t(x)=F(\sqrt t\,x)$, whose Fourier transform is +$t^{-1}\widehat F(\xi/\sqrt t)$. Radiality makes the sums over +$\Lambda^*$ and $\Lambda$ equal, and gives, for every $t>0$, +\begin{equation}\label{normalization:poisson} + \sum_{a\in\Lambda}G_1(ts_a) + =\frac1t\sum_{a\in\Lambda}G_2(s_a/t). +\end{equation} + +We will differentiate this identity twice. Here is the needed convergence +justification. On a compact subinterval of $(0,\infty)$, each of the first +two $t$ derivatives of $F(\sqrt t\,x)$ is a finite sum of spatial derivatives +of $F$ multiplied by polynomials in $x$ and bounded functions of $t$. +The same statement holds for $t^{-1}\widehat F(x/\sqrt t)$. +The Schwartz bounds therefore imply, for every $M$, a common bound +$C_M(1+|x|)^{-M}$ for these functions and their first two $t$ derivatives. +The number of points of a fixed planar lattice in a radius-$R$ disk is +$O(R^2)$, so $\sum_{a\in\Lambda}(1+|a|)^{-M}$ converges for $M>2$. +Uniform convergence on compact $t$ intervals now permits both termwise +differentiations of \eqref{normalization:poisson}. + +\begin{lemma}[Exact origin value]\label{normalization:origin-lemma} +The Fourier pair constructed above satisfies +\begin{equation}\label{normalization:origin} + F(0)=\widehat F(0)=6Q_1e^{-\pi h}>0. +\end{equation} +\end{lemma} + +\begin{proof} +At $t=1$, all nonzero summands in \eqref{normalization:poisson} vanish by +\eqref{normalization:jets}. Hence $G_1(0)=G_2(0)$. +The derivative of its left side at $1$ is +\[ + \sum_{a\in\Lambda}s_aG_1'(s_a)=-6Q_1e^{-\pi h}, +\] +because only the six vectors on the first shell contribute. On the right, +\[ + \left.\frac{\dd}{\dd t}\left(t^{-1}G_2(s/t)\right)\right|_{t=1} + =-G_2(s)-sG_2'(s). +\] +For $s=s_a>0$, both terms vanish by the exact jets, while $s_a=0$ contributes +$-G_2(0)$. Equality of the derivatives gives +$G_2(0)=6Q_1e^{-\pi h}$. The positivity follows from $Q_1>0$ in +\eqref{fourier:node-bounds}. This calculation uses the Fourier pairing and +the exact interpolation conditions, but not the global sign estimates. +\end{proof} + +\subsection{The final dilation} + +\begin{proof}[Proof of Theorem~\ref{intro:main}] +Define +\begin{equation}\label{normalization:scaling} + f(x)=\frac{F(x/\sqrt b)}{bF(0)},\qquad + \widehat f(\xi)=\frac{\widehat F(\sqrt b\,\xi)}{F(0)}. +\end{equation} +The second identity is the two-dimensional Fourier scaling rule. Since +$F(0)>0$, the dilation and normalization preserve real-valuedness, +radiality, and Schwartz regularity. They give +$\widehat f(0)=1$ and $f(0)=1/b=2/\sqrt3$. +By Proposition~\ref{signs:global} and the positive factor $e^{-\pi hs}$, +$G_2(s)=e^{-\pi hs}H_2(s)\ge0$ for $s\ge0$ and +$G_1(s)=e^{-\pi hs}H_1(s)\le0$ for $s\ge1$. +Thus $\widehat f\ge0$ everywhere. If $|x|\ge1$, +then $b|x/\sqrt b|^2=|x|^2\ge1$, so $f(x)\le0$. These are exactly the +four assertions of \eqref{intro:conditions}. +\end{proof} + +The scaling also records the prescribed contact data in the final radial +coordinates. Write $f_{\mathrm{rad}}(r)=f(r,0)$ and +$\widehat f_{\mathrm{rad}}(\rho)=\widehat f(\rho,0)$ for $r,\rho\ge0$. Then +\[ + f_{\mathrm{rad}}(r)=\frac{G_1(r^2)}{bF(0)},\qquad + \widehat f_{\mathrm{rad}}(\rho)=\frac{G_2(b^2\rho^2)}{F(0)}. +\] +Proposition~\ref{signs:global} therefore gives the exact zero sets in the +respective sign domains: +\begin{equation}\label{normalization:zero-sets} + \begin{aligned} + \{r^2:r\ge1,\ f_{\mathrm{rad}}(r)=0\}&=\mathcal N,\\ + \{\rho^2:\rho\ge0,\ \widehat f_{\mathrm{rad}}(\rho)=0\} + &=b^{-2}\mathcal N=\frac43\mathcal N. + \end{aligned} +\end{equation} +The physical zero at $r=1$ is simple, every other listed physical zero is +exactly double, and every listed Fourier zero is exactly double. Indeed, +the Gaussian and scalar factors do not vanish, while the changes of +variable $s=r^2$ and $s=b^2\rho^2$ have nonzero derivatives at these positive +radii, so they preserve the orders established for $H_1,H_2$. +At the exclusion radius, the radial derivative is +\begin{equation}\label{normalization:radial-slope} + f_{\mathrm{rad}}'(1) + =\frac{2G_1'(1)}{bF(0)}=-\frac{1}{3b}=-\frac{2}{3\sqrt3}. +\end{equation} + +\subsection{The packing consequence} + +The relation between the signs and packing density is particularly visible +for a periodic packing. Suppose the center set is +$\bigcup_{j=1}^N(L+t_j)$, where $L$ has covolume $V$, the $t_j$ are distinct +modulo $L$, and distinct centers have distance at least one. Absolute +convergence follows from the Schwartz bounds. Every summand below except +the $N$ diagonal terms at the origin is nonpositive, so +\[ + \mathcal S:=\sum_{j,k=1}^N\sum_{v\in L}f(v+t_j-t_k)\le Nf(0). +\] +Poisson summation on each translated lattice gives +\[ + \mathcal S=\frac1V\sum_{w\in L^*}\widehat f(w) + \left|\sum_{j=1}^N e^{2\pi i w\cdot t_j}\right|^2 + \ge\frac{N^2}{V}\widehat f(0), +\] +because every Fourier summand is nonnegative. It follows that +$N/V\le f(0)/\widehat f(0)=2/\sqrt3$. The disks have area $\pi/4$, hence +their covered-area density is at most $\pi/(2\sqrt3)$. + +For an arbitrary, not necessarily periodic, packing, the same implication +is Theorem~3.1 of Cohn and Elkies~\cite{CohnElkies2003}. Its admissibility +hypotheses hold because both $f$ and $\widehat f$ are Schwartz. +For disks of radius $1/2$, the Cohn--Elkies center-density normalization is +one quarter of the center-number density in their upper-density convention. +Multiplying it by the unit-disk area $\pi$ therefore gives the corresponding +covered-area density, equivalently $\pi/4$ times that center-number density. +Their upper density takes a supremum over observation centers, so it bounds +the fixed-origin limsup used in the introduction. +For the fixed-origin convention and separated centers, the +covered area inside $B_R$ lies between $(\pi/4)$ times the number of centers +in $B_{R-1/2}$ and $(\pi/4)$ times the number in $B_{R+1/2}$; division by +$\pi R^2$ and passage to the limsup matches the convention in the +introduction. The lattice generated by $(1,0)$ and $(1/2,b)$ has one center +per area $b$ and attains this value. This is the classical packing-density +theorem recovered from the new Fourier certificate, not a new density claim. + +The restriction of the physical zero set to integer squared radii also +recovers the classical periodic equality case. We use the even integral +lattice argument of Cohn and Elkies~\cite[Section~8]{CohnElkies2003}. + +\begin{corollary}[Periodic equality case]\label{normalization:periodic-uniqueness} +If a periodic packing of disks of radius $1/2$ has covered-area density +$\pi/(2\sqrt3)$, then its center set is the image of +$\ZZ(1,0)+\ZZ(1/2,b)$ under a Euclidean isometry. +\end{corollary} + +\begin{proof} +Write the center set as +$\mathcal C=\bigcup_{j=1}^N(L+t_j)$, with the notation of the periodic +packing argument above. Equality of its density with $\pi/(2\sqrt3)$ gives +$N/V=f(0)/\widehat f(0)$. Thus both inequalities in the Poisson comparison +are equalities. Absolute convergence permits us to write the defect in +the physical inequality as +\[ + 0=Nf(0)-\mathcal S + =\sum_{\substack{1\le j,k\le N,\ v\in L\\ + v\ne0\ \text{or}\ j\ne k}} + -f(v+t_j-t_k). +\] +Every summand is nonnegative, so every one is zero. Any difference of +distinct centers occurs in this sum, and \eqref{normalization:zero-sets} +therefore gives +\[ + |x-y|^2\in\mathcal N\subset\ZZ + \qquad(x,y\in\mathcal C,\ x\ne y). +\] +Only the physical zero set is used here. + +Translate one center to the origin and dilate by $\sqrt2$. Put +\[ + L_0=\sqrt2\,L,\qquad + \mathcal C_0=\sqrt2\,(\mathcal C-t_1),\qquad + \Gamma=\operatorname{span}_{\ZZ}\mathcal C_0. +\] +Here $\Gamma$ consists of finite integer linear combinations of the +scaled centers, and $L_0\subset\mathcal C_0$ because the first coset becomes +$L_0$. All squared distances in $\mathcal C_0$ are even integers. Since +$0\in\mathcal C_0$, polarization gives +\[ + \langle x,y\rangle + =\frac{|x|^2+|y|^2-|x-y|^2}{2}\in\ZZ + \qquad(x,y\in\mathcal C_0). +\] +Integer linear combinations preserve these integral pairings, and their +squared norms remain even because the cross terms carry a factor of two. +Thus every pairing in $\Gamma$ is integral and every squared norm in +$\Gamma$ is even. + +In particular, pairing with a basis of $L_0$ shows +$L_0\subset\Gamma\subset L_0^*$. Both $L_0$ and $L_0^*$ are lattices of +rank two, so $L_0^*/L_0$ is finite. Hence $\Gamma$ is a finite union of +$L_0$ cosets and is itself a lattice of rank two. Put +$d:=[\Gamma:L_0]$. A basis of $\Gamma$ has a Gram matrix +\[ + \begin{pmatrix}a&m\\m&c\end{pmatrix}, + \qquad a,c\in2\ZZ_{>0},\quad m\in\ZZ. +\] +Its determinant $ac-m^2=\operatorname{covol}(\Gamma)^2$ is a positive +integer congruent to $0$ or $3$ modulo $4$. It is therefore at least $3$, +and $\operatorname{covol}(\Gamma)\ge\sqrt3$. +On the other hand, $\mathcal C_0$ consists of $N$ distinct $L_0$ cosets, +all among the $d$ cosets of $\Gamma$. Since +$\operatorname{covol}(L_0)=2V$, we obtain +\[ + \sqrt3\le\operatorname{covol}(\Gamma) + =\frac{2V}{d}\le\frac{2V}{N}=\sqrt3. +\] +Equality holds throughout, so $d=N$ and the two sets of cosets coincide: +$\mathcal C_0=\Gamma$. + +It remains to identify this lattice of Gram determinant $3$. Choose a +shortest nonzero vector $w_1\in\Gamma$ and extend it to a lattice basis +$w_1,w_2$; this is possible because a shortest vector is primitive. +Put $a=|w_1|^2$, and replace $w_2$ by an integer translate along $w_1$ +so that $m=\langle w_1,w_2\rangle$ satisfies $|m|\le a/2$. +With $c=|w_2|^2\ge a$ and the even integer $a\ge2$, +\[ + 3=ac-m^2\ge\frac34a^2. +\] +Hence $a=2$. The equation $2c-m^2=3$, with $c$ even and $|m|\le1$, +forces $c=2$ and $m=\pm1$. This is the Gram matrix of the triangular +lattice with shortest length $\sqrt2$. Scaling back by $1/\sqrt2$ proves +the assertion. +\end{proof} + +\subsection{A second-derivative consistency identity} + +The same dilation identity contains an additional relation among the exact +curvatures. It is not needed for the construction or its sign proof. +For $n\in\mathcal N$, let +$r_\Lambda(n)=\#\{a\in\Lambda:s_a=n\}$, allowing the value zero for nodes +that are not lattice radii; in particular, $r_\Lambda(1)=6$. + +\begin{proposition}[Second differentiated Poisson identity] +\label{normalization:second-poisson} +The exact pair satisfies the absolutely convergent identity +\begin{equation}\label{normalization:curvature-identity} + \sum_{n\in\mathcal N}r_\Lambda(n)n^2 + \bigl(G_1''(n)-G_2''(n)\bigr)=2G_2(0). +\end{equation} +Using the global signs, it follows that +\begin{equation}\label{normalization:curvature-inequality} + H_1''(1)-H_2''(1)\ge2(1-\pi h)Q_1. +\end{equation} +\end{proposition} + +\begin{proof} +The convergence argument above permits a second differentiation of +\eqref{normalization:poisson}. At $t=1$, +\[ + \left.\frac{\dd^2}{\dd t^2}\left(t^{-1}G_2(s/t)\right)\right|_{t=1} + =2G_2(s)+4sG_2'(s)+s^2G_2''(s). +\] +On the left the corresponding term is $s^2G_1''(s)$. At every positive +node, the first two terms on the right vanish by +\eqref{normalization:jets}; at the origin they contribute $2G_2(0)$. +Grouping the remaining absolutely convergent lattice sums by their squared +radii proves \eqref{normalization:curvature-identity}. + +For a positive node $n>1$, $G_1$ has a local maximum equal to zero and +$G_2$ has a local minimum equal to zero, by Proposition~\ref{signs:global} +and $e^{-\pi hs}>0$. +Therefore $G_1''(n)\le0\le G_2''(n)$. All terms in +\eqref{normalization:curvature-identity} beyond the first shell are +nonpositive. Since $r_\Lambda(1)=6$ and $G_2(0)=6Q_1e^{-\pi h}$, +\[ + G_1''(1)-G_2''(1)\ge2Q_1e^{-\pi h}. +\] +Finally, differentiating $G_i=e^{-\pi hs}H_i$ twice and inserting the +exact value and slope at $1$ gives +\[ + G_1''(1)-G_2''(1) + =e^{-\pi h}\bigl(H_1''(1)-H_2''(1)+2\pi hQ_1\bigr). +\] +This yields \eqref{normalization:curvature-inequality}. Unlike the origin +calculation, this last inequality is conditional on the global signs that +were proved in Section~\ref{sec:signs}. +\end{proof} diff --git a/preprints/A-sharp-Fourier-certificate-for-planar-circle-packing-September-23-2026/build/sections/rational-check.tex b/preprints/A-sharp-Fourier-certificate-for-planar-circle-packing-September-23-2026/build/sections/rational-check.tex new file mode 100644 index 0000000000000000000000000000000000000000..134fb988b80be3321c355e226224aefbe4e7ad24 --- /dev/null +++ b/preprints/A-sharp-Fourier-certificate-for-planar-circle-packing-September-23-2026/build/sections/rational-check.tex @@ -0,0 +1,173 @@ +\subsection{An independent rational check of the first block} +\label{cert:rational-small} + +The optional program \path{verification/check_sjj_rational.py} checks +the $6\times6$ block on $J=\{1,3,4\}$ by exact rational arithmetic. +It uses the cosine rule \eqref{cert:rule} with $N=64$ and the sum over +$1\le a\le31$, and otherwise the rounded exponential, kernels and +basis densities specified above. There are no atom terms for these +basis inputs. The following error bound makes this smaller check +independent of the full Arb calculation. + +\begin{lemma}\label{cert:rational-small-error} +For each basis input $c_n=1$ or $d_n=1$, $n\in J$, with all other data +zero, and each output node $m\in J$, the rational program computes +both entries of $\mathcal S_m(p)$ with absolute error less than +$10^{-12}$. +\end{lemma} + +\begin{proof} +We separate quadrature error from rounding in the finite sum. Use the +same complex disks of radius $1/6$ as in +Lemma~\ref{cert:moment-error}. The basis density on any disk is bounded by +\[ + D_0=2\pi^2\cdot10\cdot\frac{13}{12}e^{4\pi/6}. +\] +This also bounds a simple-pole basis density. The bounds +$|\lambda|<5$, $|z|<6.2$ and $\operatorname{Im}z>-.081$ proved there +show that the transformed value and first-derivative integrands are +bounded respectively by +\[ + M_0=5D_0e^{4\pi\cdot.081}<24028, + \qquad M_1=6.2\pi M_0<468015. +\] +Use the rational upper bound $e^x\le\mathcal E_n(x)$ from +\eqref{ind:positive-exponential}. Put +$\overline\pi=p_*+10^{-36}>\pi$. Here +$x=743\overline\pi/750$ lies in $[0,4)$, so the formula applies with $n=40$. +Using $M_0=(325/3)\pi^2e^{743\pi/750}$, the exact rational comparisons +\[ + \frac{325}{3}\overline\pi^2 \mathcal E_{40}(743\overline\pi/750)<24028, + \qquad + \frac{31}{5}\overline\pi\, + \frac{325}{3}\overline\pi^2 \mathcal E_{40}(743\overline\pi/750)<468015 +\] +prove the displayed bounds for $M_0,M_1$. Each comparison is a fraction +inequality after clearing positive denominators; no rounded decimal +evaluation of an exponential is used. +Discrete cosine orthogonality now gives exactness through degree $63$. +The weights sum to one and each is at least $1/(64\cdot63)>0$ by +the same telescoping identity. Cauchy's estimate on the half-radius +real segments bounds the degree-$63$ Taylor tails by $M_i2^{-63}$. +The integral and quadrature have total mass one, after combining all +six pieces. Thus their discrepancies in the two moments are at most +\[ + \epsilon_0=2^{-62}M_0<5.211\cdot10^{-15},\qquad + \epsilon_1=2^{-62}M_1<1.015\cdot10^{-13}. +\] +Using $Q_m>1.75$ and $|D_m|<2.12$, the quadrature error in either +entry of \eqref{cert:pair} is consequently less than +\begin{equation}\label{cert:rational-small-quadrature} + \frac{2.12\epsilon_0+\epsilon_1}{1.75}<6.432\cdot10^{-14}. +\end{equation} + +For completeness, two changes in the rational implementation require +care in reusing the rounding estimates. First, for a real argument $u$ +the program evaluates $e_*(r)$ with $r=u\bmod2\in[0,2)$. +Periodicity is used for the exact target: +$e^{i\pi r}=e^{i\pi u}$. The approximation $p_*$ is not treated as +giving an exactly periodic exponential. Since $r$ is real and +$|p_*r|<2(3.142)<1000$, Lemma~\ref{cert:rounded-exponential}, followed by +$|e^{ip_*r}-e^{i\pi r}|\le2\cdot10^{-36}$, gives an absolute error +less than $2\cdot10^{-32}$ for each such replacement. When $u$ +contains an approximate quadrature node, first compare at that +approximate argument and then use the first-difference bound for the +exact exponential. This avoids any assumption of continuity of the +modulo operation. + +Second, with $N=64$, the sum of the absolute cosine coefficients in +one weight is less than one. Including the final floor, the weight +error is therefore at most +\[ + \frac{2\cdot10^{-32}}N+10^{-40} + <\frac{10^{-29}}N. +\] +Summing the stronger first expression, rather than the relaxed bound on +the right, gives the total weight error +\begin{equation}\label{cert:rational-small-weight-total} + N\left(\frac{2\cdot10^{-32}}N+10^{-40}\right) + =2\cdot10^{-32}+64\cdot10^{-40}<3\cdot10^{-32}. +\end{equation} +The node error is less than $2\cdot10^{-32}/12<2\cdot10^{-33}$. +The exact nodes are at distance at least $1/98304$ from the endpoints +of their own interval $[(j-1)/6,j/6]$, since +$1-\cos(\pi/128)\ge2/128^2$. Thus the approximate nodes remain in their +own intervals, in particular in $[0,1]$. Their transformed frequencies +have positive imaginary part even after the floor, because +$\operatorname{Im}z\ge26/435$ there. The real-segment estimates in the +proof of Lemma~\ref{cert:rational-moment-error} also give $|z_*|<3$. +Thus every transformed call has its argument in the closed left +half-plane and satisfies $|p_*z_*m|<3.142(3)(4)<1000$. + +Here is a direct propagation bound for these smaller inputs. The +real-segment estimates are $|\lambda|<3$, $|z|<3$, and basis-density +modulus less than $214$. For one piece put +\[ + A_n=\sum_{l=j}^6P_le^{i\pi t_ln},\qquad + B_n=\sum_{l=j}^6t_lP_le^{i\pi t_ln}. +\] +Their moduli are at most ten. Replacing $b$ and the real exponentials +changes each sum by less than $3\cdot10^{-31}$; including the node +movement changes $B_n-tA_n$ by less than $7\cdot10^{-31}$. +The phase $e^{-i\pi tn}$ changes by less than $5\cdot10^{-32}$, +since $n\le4$. Substitution in \eqref{fourier:folded} therefore gives +a density error less than $3\cdot10^{-29}$. +The derivatives of the rational kernels, using $b,b_*>4/5$ and +$|t+ih|\ge2/5$, give errors less than $2\cdot10^{-32}$ in both +$\lambda$ and $z$, including their floors. The first-difference +formula for exponentials in the closed left half-plane, with $m\le4$, +then bounds the errors in +\[ + g_0(t)=\lambda(t)e^{i\pi mz(t)},\qquad + g_1(t)=g_0(t)i\pi z(t) +\] +by $2\cdot10^{-30}$ and $3\cdot10^{-29}$, respectively. +The exact factors have moduli below $3$ and $36$; their approximations +have moduli below $4$ and $37$. Product differences bound each +derivative-integrand error by +\[ + 3\cdot10^{-29}\cdot37+214\cdot3\cdot10^{-29} + <8\cdot10^{-27}; +\] +the value-integrand bound is smaller. By +\eqref{cert:rational-small-weight-total}, the total absolute weight error +is below $3\cdot10^{-32}$, and the approximate integrands have modulus +below $10^4$. Hence each rounded moment differs from its exact finite +quadrature moment by less than $10^{-26}$. + +Finally, for residues $1,3,4$, the exact polynomial forms are +\[ +\begin{gathered} + Q_1=Q_3=\frac{4\pi^2}{3}(1-b),\qquad Q_4=\frac{\pi^2}{3},\\ + D_1=\frac{\pi(3+2b)}{18},\qquad + D_3=-\frac{\pi(3+2b)}{18},\qquad D_4=\frac{7\pi b}{9}. +\end{gathered} +\] +The program replaces $\pi,b$ by $p_*,b_*$ in these expressions. +Using $|p_*^2-\pi^2|<6.3\cdot10^{-36}$, $\pi^2<10$, and $b,b_*<1$ gives +\[ +\begin{aligned} + |Q_m^\#-Q_m| + &<\max\{(4/3)(6.3+10),\,6.3/3\}10^{-36}<3\cdot10^{-35},\\ + |D_m^\#-D_m| + &<\max\{(5+44/7)/18,\,29/9\}10^{-36}<4\cdot10^{-36}. +\end{aligned} +\] +The exact rational lower bound used +after~\eqref{ind:pair-stability} therefore shows that both denominators +$Q_m,Q_m^\#$ exceed $7/4$. The moments have modulus below $10^4$, so the +additional division and $D_m$ errors are less than $4\cdot10^{-31}$. +Together with \eqref{cert:rational-small-quadrature}, these bounds +give an error less than $7\cdot10^{-14}<10^{-12}$ in either matrix +entry, as asserted. +\end{proof} + +If every entry of this $6\times6$ matrix has error below +$\varepsilon=10^{-12}$, its induced column-norm error is below +$6\varepsilon$. The program accordingly enlarges the computed +defect norm by $\|W_\eta\|_1\,6\varepsilon$, then verifies that the +result is below $.003$ and that the resulting inverse bound is below +$32$. This is the same perturbation-of-identity argument as in +Section~\ref{cert:matrix-bounds}. It checks both small inverse +certificates without asserting that the full $N=256$ rational +prescription has been executed. diff --git a/preprints/A-sharp-Fourier-certificate-for-planar-circle-packing-September-23-2026/build/sections/signs.tex b/preprints/A-sharp-Fourier-certificate-for-planar-circle-packing-September-23-2026/build/sections/signs.tex new file mode 100644 index 0000000000000000000000000000000000000000..8c36f33c60059da9e2defafe40156c8331932665 --- /dev/null +++ b/preprints/A-sharp-Fourier-certificate-for-planar-circle-packing-September-23-2026/build/sections/signs.tex @@ -0,0 +1,437 @@ +\section{Global signs}\label{sec:signs} + +We now prove the global signs for the exact functions supplied by +Proposition~\ref{interpolation:solution}. The strict bounds will also +identify their zeros within the respective sign domains. Write +\[ + \sigma_1=-1,\qquad \sigma_2=1,\qquad + \delta=3\cdot10^{-5}. +\] +The difference between each exact coefficient list and its tabulated list has +norm less than $\delta$. Their fixed coefficients at zero and their constants +$C_i$ agree. Tildes will always denote the functions constructed from the +finite tabulated lists, so that +$\widetilde H_i=\widetilde p_i+K_{\widetilde p_{3-i}}$. +We use the spectral envelopes of Section~\ref{sec:interpolation} throughout. + +\subsection{Taylor quotients on half-gaps} + +Set $\mathcal N_0=\{0\}\cup\mathcal N$. For each +$m\in\mathcal N_0\cap[0,40]$, let $m^+$ be its successor in +$\mathcal N_0$ and include +\[ + y=\frac{m^+-m}{2}. +\] +If $m>0$, also include $y=(m^--m)/2$, where $m^-$ is its predecessor. +Thus $y$ is signed, and the associated half-gap is +\[ + I_{m,y}=\{m+yv:0\le v\le1\}. +\] +For function index $2$ we use every such interval, with order +$\nu=0$ at $m=0$ and $\nu=2$ otherwise. For function index $1$ we use only +those intervals contained in $[1,\infty)$, with $\nu=1$ at $m=1$ and +$\nu=2$ otherwise. In particular, the only odd order occurs on the +rightward half-gap from $1$. +There are $22$ centers from $0$ through $40$. They give $43$ half-gaps for +index $2$ and, after removing the two intervals not contained in $[1,\infty)$, $41$ for index +$1$, for a total of $84$ certified intervals. + +Fix one of the specified triples $(i,m,y)$. For $s=m+u$ in its interval, +we will prove positivity of +\[ + \frac{\sigma_i H_i(m+u)}{u^\nu}. +\] +Here $u^0=1$, and for $\nu>0$ the quotient at $u=0$ is interpreted by +continuity. The interpolation conditions make the derivatives of $H_i$ +of orders below $\nu$ vanish at $m$, so this extension exists. Positivity +of the quotient gives the required sign: every order is even except for +the rightward half-gap from $1$, where $u\ge0$. +Figure~\ref{fig:half-gaps} shows the node-to-midpoint intervals on which +these quotients are studied. +\input{figures/half-gaps} + +The tabulated function $\widetilde H_i$ approximates $H_i$ but need not +reproduce its exact zero and slope at a node. We therefore compare the +exact quotient with the Taylor series of $\widetilde H_i$ after deleting +the terms of orders below $\nu$. Put $\Delta_i=H_i-\widetilde H_i$. +Since $\widetilde H_i$ is entire and the lower derivatives of $H_i$ vanish, +this Taylor series gives +\begin{align}\label{signs:deleted-jets} + &\frac{\sigma_i H_i(m+u)}{u^\nu} + -\sigma_i\sum_{r=0}^{\infty} + \frac{\widetilde H_i^{(r+\nu)}(m)}{(r+\nu)!}u^r + \nonumber\\ + &\hspace{1cm}= + \frac{\sigma_i}{u^\nu} + \left(\Delta_i(m+u)- + \sum_{j=0}^{\nu-1}\frac{\Delta_i^{(j)}(m)}{j!}u^j\right). +\end{align} +For $\nu=0$, the sum on the second line is empty and the right-hand side is +simply $\sigma_i\Delta_i(m+u)$. For $\nu>0$, Taylor's integral remainder +gives the exact identity +\[ + \frac{\Delta_i(m+u)-\sum_{j=0}^{\nu-1} + \Delta_i^{(j)}(m)u^j/j!}{u^\nu} + =\frac{1}{(\nu-1)!}\int_0^1(1-t)^{\nu-1} + \Delta_i^{(\nu)}(m+tu)\,\dd t. +\] +It holds for either sign of $u$ and extends continuously to $u=0$. +Since $m+tu$ stays in the same half-gap, the absolute value of the +right-hand side of \eqref{signs:deleted-jets} is at most +\begin{equation}\label{signs:coefficient-error} + \frac{1}{\nu!}\sup_{s\in I_{m,y}} + \abs{\Delta_i^{(\nu)}(s)}. +\end{equation} +The same bound applies at $u=0$ by continuity and for either orientation +of the half-gap. Deleting the lower Taylor terms is essential here: +dividing the raw table error by $u^\nu$ could be singular at the node. + +After writing $u=yv$, the first $29$ terms of the tabulated series in +\eqref{signs:deleted-jets} form a polynomial of degree at most $28$ in +$v$. We bound it on $[0,1]$ using the standard range enclosure by Bernstein +coefficients \cite[Section~3]{TitiGarloff2019}, retaining its elementary +proof below. + +\begin{lemma}[Finite Bernstein certificate]\label{signs:bernstein} +For every specified triple $(i,m,y)$, put +\begin{equation}\label{signs:power-coefficients} + T_r=\sigma_i y^r + \frac{\widetilde H_i^{(r+\nu)}(m)}{(r+\nu)!}, + \qquad 0\le r\le28. +\end{equation} +Then, for $0\le k\le28$, +\begin{equation}\label{signs:bernstein-bound} + B_k:=\sum_{r=0}^k + \frac{\binom{k}{r}}{\binom{28}{r}}T_r + >\begin{cases} + 0.74,&m=0,\\ + 0.18,&m=1,\\ + 0.02,&m=3,4,\\ + 0.009,&4\begin{cases} + 0.735,&m=0,\\ + 0.168,&m=1,\\ + 0.017,&m=3,4,\\ + 0.006,&40.011\qquad(s\ge s_0,\ i=1,2). +\end{equation} +For the tabulated coefficients put +$u_n^*=\sigma_i\widetilde c_{i,n}$, +$v_n^*=\sigma_i\widetilde d_{i,n}$, and $[w]_-:=\min\{w,0\}$. +Using +\[ + \frac{1}{s-n}=\frac1s+\frac{n}{s(s-n)} +\] +and discarding positive terms gives the following lower bound for +$\sigma_i\widetilde R_{i,0}(s)$, uniformly for $s\ge s_0$: +\begin{equation}\label{signs:rational-lower-sum} + \sigma_i C_i+ + \frac{[\sum_{n\le40}v_n^*]_-}{s_0} + +\sum_{n\le40} + \left(\frac{[u_n^*]_-}{(s_0-n)^2} + +\frac{[v_n^*]_-n}{s_0(s_0-n)}\right), +\end{equation} +where both sums include $n=0$. Each negative reciprocal term is smallest +at $s=s_0$. The exact rational sums in +\eqref{signs:rational-lower-sum} exceed $0.012$ for index $1$ and $0.016$ +for index $2$. To pass to the exact rational part, use $s-n\ge1.5>1$ and +the coefficient enclosure to obtain +\[ + \begin{split} + \abs{R_{i,0}(s)-\widetilde R_{i,0}(s)} + &\le\sum_{n\le40}\left( + \frac{\abs{c_{i,n}-\widetilde c_{i,n}}}{(s-n)^2} + +\frac{\abs{d_{i,n}-\widetilde d_{i,n}}}{s-n}\right)\\ + &<\frac{\delta}{s_0-40}=0.00002. + \end{split} +\] +The finite lower sums therefore imply the asserted exact bound +\eqref{signs:rational-tail}. + +The coefficient norm of the tabulated part with $n>40$ is less than +$0.00002$ for either function. Hence the exact coefficient norm for +$p_{i,1}$ is less than $0.00005$. The derivative envelopes imply +\begin{align}\label{signs:tail-second-derivative} + \sup_{s\ge s_0}\abs{E_i''(s)} + \le {}&0.00005\cdot77 + +2.3\sum_{j=1}^6 M_jL_j(s_0)(\pi a_{j-1}^*)^2 + \nonumber\\ + &+0.017\sum_{j=0}^6 L_j^P(s_0)(\pi a_j^*)^2 + <0.006. +\end{align} +For the direct part we used +$\sum_{j=1}^6 M_j(\pi t_j)^2/6<77$; for the transformed part we used the +all-node coefficient norm less than $2.3$ from +Proposition~\ref{interpolation:solution}. + +It remains to give the quadratic lower bound for $P$. +\begin{lemma}\label{signs:P-lower-bound} +If $m\in12\ZZ+A$ is a nearest zero of $P$ to a real number $s$, then +\[ + P(s)\ge0.68(s-m)^2. +\] +\end{lemma} + +\begin{proof} +On either half-gap from $m$, consider $P(m+u)/u^2$, with value $Q_m$ at +$u=0$. Put $a=\pi/12$. For a sine factor whose residue is $a'\ne m$ +modulo $12$, the second derivative of its logarithm is +$-2a^2\csc^2(a(m+u-a'))<0$. These factors stay nonzero on the half-gap. +The factor vanishing at $m$, after division by $u^2$, has the same +logarithmic concavity because +\[ + \frac{\dd^2}{\dd u^2}\log\abs{\frac{\sin(au)}{u}} + =\frac1{u^2}-\frac{a^2}{\sin^2(au)}\le0. +\] +Here $\abs{\sin(au)}\le\abs{au}$, and the logarithm has a smooth limiting +value at $u=0$. Squaring the divided sine only multiplies this second +derivative by two. +The quotient is therefore logarithmically concave on each half-gap, +including its continuous value at the center. Its minimum is bounded +below by the smaller of its endpoint values. At the center +$Q_m>1.75$. At the gap midpoints, direct substitution gives the values +listed below; periodicity reduces the calculation to one period. +To evaluate them, put $\theta=\pi(s-2)/12$ and $c=\cos(2\theta)$. +The identity $4\sin v\sin w=2(\cos(v-w)-\cos(v+w))$ shows that the paired +unsquared factors for residues $(0,4)$, $(1,3)$, and $(7,9)$ are, +respectively, +\[ + 1-2c,\qquad \sqrt3-2c,\qquad \sqrt3+2c. +\] +Consequently +\[ + P(s)=(4c^2-3)^2(1-2c)^2, + \qquad c=\cos\frac{\pi(s-2)}6, +\] +which evaluates every entry directly. +\begin{center} +\begin{tabular}{c|c} +Gap between consecutive zeros & Midpoint quotient $P(s)/(s-m)^2$\\ +\hline +$[0,1],\ [3,4]$ & $12-8\sqrt2$\\ +$[1,3]$ & $1$\\ +$[7,9]$ & $9$\\ +$[4,7],\ [9,12]$ & $\dfrac89(3+\sqrt2+\sqrt3+\sqrt6)$ +\end{tabular} +\end{center} +The distances to the two ends of a gap agree at its midpoint, so this +table supplies the endpoint value for both half-gaps. The smallest value +is $12-8\sqrt2>0.68$: indeed $(283/200)^2>2$, so +$\sqrt2<283/200$ and $12-8(283/200)=0.68$. The other three values are +at least $1$. This proves the claim. +\end{proof} + +For any $s\ge41.5$, choose a nearest node $m\ge43$; at $s=41.5$, choose +$43$ rather than the tied node $40$. Both $H_i$ and $P R_{i,0}$ vanish +to order at least two at $m$, by \eqref{fourier:interpolation} and the absence of a +pole of $R_{i,0}$ there. Thus +$E_i=H_i-P R_{i,0}$ vanishes to order at least two at $m$. The segment between +$s$ and $m$ lies in $[41.5,\infty)$, and +\eqref{signs:tail-second-derivative} gives +\[ + \abs{E_i(s)}\le\frac{0.006}{2}(s-m)^2. +\] +Using \eqref{signs:rational-tail} and +Lemma~\ref{signs:P-lower-bound}, we obtain on the entire unbounded region +\begin{equation}\label{signs:tail-margin} + \sigma_iH_i(s) + \ge\left(0.011\cdot0.68-\frac{0.006}{2}\right)(s-m)^2 + =0.00448(s-m)^2\ge0. +\end{equation} + +\begin{proposition}[Global signs and zeros]\label{signs:global} +The exact functions have the zero sets +\begin{equation}\label{signs:zero-sets} + \{s\ge1:H_1(s)=0\}=\mathcal N,\qquad + \{s\ge0:H_2(s)=0\}=\mathcal N, +\end{equation} +and satisfy the strict inequalities +\[ + H_1(s)<0\quad(s\ge1,\ s\notin\mathcal N),\qquad + H_2(s)>0\quad(s\ge0,\ s\notin\mathcal N). +\] +The zero of $H_1$ at $1$ is simple. Its zeros at $n\in\mathcal N$ with +$n>1$, and the zeros of $H_2$ at every $n\in\mathcal N$, have order exactly +two as zeros of the entire functions of $s$. +Consequently $F(x)\le0$ when $b\abs{x}^2\ge1$, and +$\widehat F(\xi)\ge0$ for every $\xi\in\RR^2$. +\end{proposition} + +\begin{proof} +At every node $n\in\mathcal N$, the exact interpolation conditions give +$H_1(n)=H_2(n)=0$. Away from the nodes, the quotient bounds +\eqref{signs:finite-quotient} are strict on the finite region, and +\eqref{signs:tail-margin} is strict on the remaining region. This proves +\eqref{signs:zero-sets} and the asserted strict signs. + +The interpolation slope $H_1'(1)=-Q_1\ne0$ makes its zero at $1$ simple. +For $H_2$ at every node in $\mathcal N\cap[1,40]$, and for $H_1$ at those +nodes greater than $1$, the chosen order is $\nu=2$. +The endpoint value in \eqref{signs:finite-quotient} is then +$\sigma_iH_i''(m)/2>0$, so the zero has order exactly two. +For a node $m\in\mathcal N$ with $m\ge43$, take $u$ sufficiently small that +$m$ remains the nearest node to $m+u$ and $m+u\ge41.5$. The exact value and first derivative +at $m$ vanish, so \eqref{signs:tail-margin} yields +\[ + \frac{\sigma_iH_i''(m)}2 + =\lim_{u\to0}\frac{\sigma_iH_i(m+u)}{u^2} + \ge0.00448>0. +\] +These zeros also have order exactly two. Finally, the factors +$e^{-\pi h s}$ in $F$ and $\widehat F$ are positive, which gives their +non-strict sign inequalities. +\end{proof} diff --git a/preprints/A-sharp-Fourier-certificate-for-planar-circle-packing-September-23-2026/paper.pdf b/preprints/A-sharp-Fourier-certificate-for-planar-circle-packing-September-23-2026/paper.pdf new file mode 100644 index 0000000000000000000000000000000000000000..55d2b483df0ba641692f6f4eaef4c55dfe8acfdc --- /dev/null +++ b/preprints/A-sharp-Fourier-certificate-for-planar-circle-packing-September-23-2026/paper.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:3a532659c4aa989d4d9cca9d33810940b61c835597bb0b1b50c16d3487001329 +size 540259 diff --git a/preprints/A-sharp-Fourier-certificate-for-planar-circle-packing-September-23-2026/verification/README.md b/preprints/A-sharp-Fourier-certificate-for-planar-circle-packing-September-23-2026/verification/README.md new file mode 100644 index 0000000000000000000000000000000000000000..199208d1dd7eca50a6418c16df52a624783e2f29 --- /dev/null +++ b/preprints/A-sharp-Fourier-certificate-for-planar-circle-packing-September-23-2026/verification/README.md @@ -0,0 +1,16 @@ +# Packing certificate verification + +Requires Python 3 and `python-flint==0.9.0`. Run with assertions enabled; do not use `-O`, `-OO`, or `PYTHONOPTIMIZE`. + +From the article directory: + +```sh +python3 -m pip install -r verification/requirements.txt +python3 -B verification/verify_all.py +``` + +Keep `verification/data/certificate-inputs.json`, the three TSV tables in `verification/data/`, and the corresponding TeX tables in `build/data/` together. The verifier checks their identities and writes `finite.json`, `scalar.json`, and `summary.json` under `verification/results/`. + +Success is a zero exit with `"status": "pass"` in the printed summary. A failed run may leave older or partial result files; those files do not establish success for that run. + +The main command checks the finite matrix, residual, Bernstein, and scalar comparisons. The analytic quadrature, continuum, and infinite-tail arguments remain in the article. diff --git a/preprints/A-sharp-Fourier-certificate-for-planar-circle-packing-September-23-2026/verification/certificate_data.py b/preprints/A-sharp-Fourier-certificate-for-planar-circle-packing-September-23-2026/verification/certificate_data.py new file mode 100644 index 0000000000000000000000000000000000000000..1f1ecbbfdbc9fc610cbff2c96fc3617c2551bbfb --- /dev/null +++ b/preprints/A-sharp-Fourier-certificate-for-planar-circle-packing-September-23-2026/verification/certificate_data.py @@ -0,0 +1,98 @@ +"""Validate the public certificate manifest and all exact displayed tables.""" + +from pathlib import Path +import hashlib +import json +import platform + +VERIFICATION_DIR = Path(__file__).resolve().parent +WRITEUP_DIR = VERIFICATION_DIR.parent +EXPECTED_SHA = "f967c6559e568142f1a9c174f54e7087bfa7934dc7f38792084db6351c5676c4" + + +def sha256(path): + return hashlib.sha256(Path(path).read_bytes()).hexdigest() + + +def _integer_rows(text): + return [[int(value) for value in line.split()] + for line in text.strip().splitlines() if line.strip()] + + +def _tex_integer_rows(text): + rows = [] + for line in text.strip().splitlines(): + line = line.strip() + if not line: + continue + assert line.endswith("\\\\"), "TeX data row lacks row terminator" + rows.append([int(value.strip()) for value in line[:-2].split("&")]) + return rows + + +def load_data(): + """Return tables after exact comparison with the public manifest. + + The fixed digest identifies this mathematical certificate's complete + input specification. Independently parsed TSV and TeX tables must agree + as integer arrays. No internal submission or historical review is needed. + """ + if not __debug__: + raise RuntimeError("Run without -O: certificate assertions are required") + source_path = VERIFICATION_DIR / "data" / "certificate-inputs.json" + source_sha256 = sha256(source_path) + assert source_sha256 == EXPECTED_SHA, "Certificate manifest hash mismatch" + manifest = json.loads(source_path.read_text()) + assert manifest["schema"] == "planar-fourier-packing-certificate-inputs-v1" + tables = manifest["tables"] + paths = {} + parsed = {} + for name in ("coefficients", "W_plus", "W_minus"): + table = tables[name] + expected_scale = 10**10 if name == "coefficients" else 10**4 + assert table["scale_denominator"] == expected_scale + tsv_path = VERIFICATION_DIR / "data" / (name + ".tsv") + tex_path = WRITEUP_DIR / "build" / "data" / (name + ".tex") + rows = _integer_rows(tsv_path.read_text()) + assert rows == table["rows"], "TSV differs from public manifest: " + name + assert _tex_integer_rows(tex_path.read_text()) == rows, "TeX differs from TSV: " + name + for path in (tsv_path, tex_path): + assert sha256(path) == manifest["files_sha256"][path.name], "Data hash mismatch: " + path.name + paths[path.name] = path + parsed[name] = rows + rows = parsed["coefficients"] + assert len(rows) == 51 and all(len(row) == 5 for row in rows) + for name in ("W_plus", "W_minus"): + assert len(parsed[name]) == 6 and all(len(row) == 6 for row in parsed[name]) + nodes = [row[0] for row in rows] + assert nodes == [n for n in range(1, 101) if n % 12 in {0, 1, 3, 4, 7, 9}] + return { + "source_path": source_path, + "source_sha256": source_sha256, + "data_sha256": {name: sha256(path) for name, path in paths.items()}, + "rows": rows, + "nodes": nodes, + "W_plus": parsed["W_plus"], + "W_minus": parsed["W_minus"], + "data_identities_match": True, + } + + +def provenance(data, script_path): + return { + "source_sha256": data["source_sha256"], + "source_kind": "public certificate input manifest", + "data_sha256": data["data_sha256"], + "data_identities_match": data["data_identities_match"], + "python_version": platform.python_version(), + "certificate_file": Path(script_path).name, + "certificate_sha256": sha256(script_path), + } + + +def write_result(name, result): + output_dir = VERIFICATION_DIR / "results" + output_dir.mkdir(exist_ok=True) + path = output_dir / name + path.write_text(json.dumps(result, indent=2) + "\n") + return path diff --git a/preprints/A-sharp-Fourier-certificate-for-planar-circle-packing-September-23-2026/verification/check_sjj_rational.py b/preprints/A-sharp-Fourier-certificate-for-planar-circle-packing-September-23-2026/verification/check_sjj_rational.py new file mode 100644 index 0000000000000000000000000000000000000000..01c6aa7a7d0a6eba2761a5527076edcb5612793c --- /dev/null +++ b/preprints/A-sharp-Fourier-certificate-for-planar-circle-packing-September-23-2026/verification/check_sjj_rational.py @@ -0,0 +1,236 @@ +#!/usr/bin/env python3 +"""Exact rational finite quadrature for S[{1,3,4},{1,3,4}]. + +All finite arithmetic uses fractions.Fraction. The analytic quadrature and +transcendental error proof gives abs(S_exact - S_finite) < 10^-12 per entry. +The N=64 certificate is independent of the full Arb finite certificate. +Its complete entry-error estimate is proved in the article appendix. +""" + +from fractions import Fraction as F +from functools import lru_cache +import json +import time + +from certificate_data import load_data, provenance, write_result + + +SCALE = 10**40 +PI = F("3.141592653589793238462643383279502884") +BETA = F(".866025403784438646763723170752936183") +H = F(2, 5) +B = F(4, 3) +N = 64 +J = [1, 3, 4] +ENTRY_ERROR = F(1, 10**12) + + +class C: + """Complex numbers with exact rational real and imaginary components.""" + + __slots__ = ("r", "i") + + def __init__(self, r=0, i=0): + self.r = F(r) + self.i = F(i) + + def __add__(self, other): + other = to_c(other) + return C(self.r + other.r, self.i + other.i) + + __radd__ = __add__ + + def __neg__(self): + return C(-self.r, -self.i) + + def __sub__(self, other): + return self + -to_c(other) + + def __rsub__(self, other): + return to_c(other) + -self + + def __mul__(self, other): + if not isinstance(other, C): + return C(self.r * other, self.i * other) + return C(self.r * other.r - self.i * other.i, + self.r * other.i + self.i * other.r) + + __rmul__ = __mul__ + + def __truediv__(self, other): + if not isinstance(other, C): + return C(self.r / other, self.i / other) + denominator = other.r * other.r + other.i * other.i + return C((self.r * other.r + self.i * other.i) / denominator, + (self.i * other.r - self.r * other.i) / denominator) + + def __rtruediv__(self, other): + return to_c(other) / self + + +def to_c(value): + return value if isinstance(value, C) else C(value) + + +def floor_rational(value): + return F((value * SCALE).__floor__(), SCALE) + + +def floor_complex(value): + return C(floor_rational(value.r), floor_rational(value.i)) + + +I = C(0, 1) + + +def estar(u): + x = I * PI * to_c(u) / 1024 + term = C(1) + value = C(1) + for q in range(1, 33): + term = term * x / q + value = value + term + value = floor_complex(value) + for _ in range(10): + value = floor_complex(value * value) + return value + + +@lru_cache(None) +def ereal(u): + # The target exp(i*pi*u) is 2-periodic. The reduced rational evaluation + # has direct error below 1e-32 and keeps rational inputs short. + return estar(u % 2) + + +def atan_bracket(d): + upper = sum((F((-1)**k, (2*k + 1) * d**(2*k + 1)) + for k in range(29)), F(0)) + return upper - F(1, 59 * d**59), upper + + +def decimal_out(value, places=22, upper=False): + scaled = value * 10**places + integer = scaled.__ceil__() if upper else scaled.__floor__() + sign = "-" if integer < 0 else "" + digits = str(abs(integer)).rjust(places + 1, "0") + return sign + digits[:-places] + "." + digits[-places:] + + +def interval(value, radius): + return [decimal_out(value - radius), + decimal_out(value + radius, upper=True)] + + +def run_certificate(): + """Verify both rational inverse certificates and persist their enclosures.""" + start = time.monotonic() + data = load_data() + # Retain both exact transcendental-constant bracket checks. + epsilon = F(1, 10**36) + al, au = atan_bracket(5) + bl, bu = atan_bracket(239) + assert PI - epsilon < 16*al - 4*bu <= 16*au - 4*bl < PI + epsilon + assert (BETA - epsilon)**2 < F(3, 4) < (BETA + epsilon)**2 + + weights = [] + points = [] + for v in range(N): + alpha = F(2*v + 1, 2*N) + weights.append(floor_rational( + (1 - sum((2 * ereal(2*a*alpha).r / F(4*a*a - 1) + for a in range(1, N//2)), F(0))) / N)) + points.append(ereal(alpha).r) + print("weights complete", time.monotonic() - start, flush=True) + + polynomial = [C(5), C(-1, 2*BETA), C(1, 2*BETA), C(-2), + C(-F(1, 2), BETA), C(-1, -2*BETA), C(1)] + q_values = {1: 2*PI*PI*(2 - 2*BETA)/3, + 3: 2*PI*PI*(2 - 2*BETA)/3, 4: PI*PI/3} + d_values = {1: PI*(2*BETA + 3)/18, + 3: -PI*(2*BETA + 3)/18, 4: 7*BETA*PI/9} + matrix = [[F(0) for _ in range(6)] for _ in range(6)] + + for j in range(1, 7): + tails = {} + for n in J: + a = C() + b0 = C() + for ell in range(j, 7): + u = polynomial[ell] * ereal(F(ell*n, 6)) + a = a + u + b0 = b0 + u * F(ell, 6) + tails[n] = (a, b0) + for v in range(N): + t = (2*j - 1 + points[v]) / 12 + tt = C(t, H) + lam = floor_complex(I / (BETA*tt)) + z = floor_complex(-B/tt - I*H) + densities = [] + for n in J: + phase = ereal(-t*n) + a, b0 = tails[n] + densities.append((phase*(-2*PI*PI)*(b0 - t*a), + phase*(2*PI)*I*a)) + for ri, m in enumerate(J): + g0 = lam * estar(z*m) + g1 = g0 * I * PI * z + for ci in range(6): + density = densities[ci % 3][ci // 3] + x0 = (density*g0).r * weights[v] / 6 + x1 = (density*g1).r * weights[v] / 6 + matrix[ri][ci] += -x0 / q_values[m] + matrix[ri + 3][ci] += (d_values[m]*x0 - x1) / q_values[m] + print("piece", j, "complete", time.monotonic() - start, flush=True) + + result = { + **provenance(data, __file__), + "status": "pass", + "arithmetic": "exact fractions.Fraction", + "floor_decimal_places": 40, + "constant_absolute_error_bound": "1e-36", + "exponential_taylor_degree": 32, + "exponential_squarings": 10, + "entry_error_bound": "1e-12", + "quadrature_N": N, + "nodes": J, + "order": ["c1", "c3", "c4", "d1", "d3", "d4"], + "selected_d1_d1": interval(matrix[3][3], ENTRY_ERROR), + "matrix_enclosures": [[interval(x, ENTRY_ERROR) for x in row] + for row in matrix], + "W_checks": {}, + } + for sign, integers in [(1, data["W_plus"]), (-1, data["W_minus"])]: + w = [[F(x, 10000) for x in row] for row in integers] + wnorm = max(sum(abs(w[i][j]) for i in range(6)) for j in range(6)) + error = [[F(i == j) - sum( + w[i][k] * (F(k == j) - sign*matrix[k][j]) for k in range(6)) + for j in range(6)] for i in range(6)] + residual = max(sum(abs(error[i][j]) for i in range(6)) + for j in range(6)) + # ||W * (S_exact-S_finite)||_1 <= ||W||_1 * 6*entry_error. + radius = wnorm * 6 * ENTRY_ERROR + residual_upper = residual + radius + assert residual_upper < F(3, 1000), (sign, "residual norm") + inverse_upper = wnorm / (1 - residual_upper) + assert inverse_upper < 32, (sign, "inverse norm") + result["W_checks"][str(sign)] = { + "W_column_norm": decimal_out(wnorm, upper=True), + "residual_norm_enclosure": interval(residual, radius), + "inverse_norm_upper": decimal_out(inverse_upper, upper=True), + } + result["elapsed_seconds"] = time.monotonic() - start + write_result("sjj-rational.json", result) + print(json.dumps({"status": result["status"], + "selected": result["selected_d1_d1"], + "W_checks": result["W_checks"], + "seconds": result["elapsed_seconds"]}, indent=2), flush=True) + return result + + +def main(): + run_certificate() + + +if __name__ == "__main__": + main() diff --git a/preprints/A-sharp-Fourier-certificate-for-planar-circle-packing-September-23-2026/verification/data/W_minus.tsv b/preprints/A-sharp-Fourier-certificate-for-planar-circle-packing-September-23-2026/verification/data/W_minus.tsv new file mode 100644 index 0000000000000000000000000000000000000000..c39d9e66853317e7a60a13d7295a7ba22f243e9b --- /dev/null +++ b/preprints/A-sharp-Fourier-certificate-for-planar-circle-packing-September-23-2026/verification/data/W_minus.tsv @@ -0,0 +1,6 @@ +13265 -1358 -385 -381 815 -164 +616 10426 686 -484 182 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"2dc6a2a19f0abdb0e0ca3f57261d44938a7c9f1ae8697dc816d7911875038577", + "coefficients.tsv": "b62c6b3ea62ab8d7906214f009b23b232c16602cb6ae0487ae8fa362c0ad4ced" + } +} diff --git a/preprints/A-sharp-Fourier-certificate-for-planar-circle-packing-September-23-2026/verification/data/coefficients.tsv b/preprints/A-sharp-Fourier-certificate-for-planar-circle-packing-September-23-2026/verification/data/coefficients.tsv new file mode 100644 index 0000000000000000000000000000000000000000..9ca3bdaf95a790c9d3ecbd1160974dd65c24573f --- /dev/null +++ b/preprints/A-sharp-Fourier-certificate-for-planar-circle-packing-September-23-2026/verification/data/coefficients.tsv @@ -0,0 +1,51 @@ +1 -670689128 -6398628799 -250018302 4107732082 +3 93524468 564897270 -186215974 -65204793 +4 -64146977 121133118 82882809 -327822296 +7 -1358193 -16329465 1979670 21637038 +9 -2932910 -940875 3636920 2145806 +12 3646952 -34846939 -6005382 34369042 +13 3183411 63030995 -574123 -75237042 +15 11390977 27049982 -11312377 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+96 -2 -4 1 3 +97 2 -6 -2 4 +99 0 -6 0 5 +100 0 2 0 -2 diff --git a/preprints/A-sharp-Fourier-certificate-for-planar-circle-packing-September-23-2026/verification/interval_bounds.py b/preprints/A-sharp-Fourier-certificate-for-planar-circle-packing-September-23-2026/verification/interval_bounds.py new file mode 100644 index 0000000000000000000000000000000000000000..3ff31d3d4bd58057a970ca4884062726f338473c --- /dev/null +++ b/preprints/A-sharp-Fourier-certificate-for-planar-circle-packing-September-23-2026/verification/interval_bounds.py @@ -0,0 +1,215 @@ +#!/usr/bin/env python3 +"""Arb scalar envelopes, tail bounds, Taylor errors, and quadrature error. + +Call run_certificate() after constructing the finite certificate context. +Importing this module does not run a certificate. +""" +import json +from math import factorial +import flint +from flint import arb, ctx +from certificate_data import provenance, write_result + + +def run_certificate(c): + a=lambda x:arb(str(x)) + printed_scalar_bounds=[] + + def printed_upper(name,value,bound): + assert valuea(bound),(name,str(value),bound) + printed_scalar_bounds.append({'name':name,'enclosure':str(value),'lower':bound}) + + pi,b,h,B=c.pi,c.b,c.h,c.B + bs=list(map(a,['2.887','2.665','2.218','1.804','1.486','1.250','1.073'])) + aa=list(map(a,['2.934','2.713','2.269','1.859','1.548','1.320','1.151'])) + gg=list(map(a,['2.933','2.440','1.567','.900','.482','.224','.0597'])) + mm=list(map(a,[56,48,35,25,19,'6.284'])) + bet=list(map(a,[0,'4.8','3.17','1.946','1.217','.792'])) + atoms=list(map(a,[5,4,4,4,2,4,2])) + + endpoint_checks=[] + for j,t in enumerate(c.ts): + lam=1/(b*(t*t+h*h).sqrt()) + z=(h*h+(B*B-2*B*h*h)/(t*t+h*h)).sqrt() + im=h*(B/(t*t+h*h)-1) + assert lamgg[j] + endpoint_checks.append({'j':j,'lambda_abs':str(lam),'z_abs':str(z),'Im_z':str(im)}) + for j in range(1,7): + for t in (c.ts[j-1],c.ts[j]): + deriv=2*h*B*t/(t*t+h*h)**2 + assert deriv>=bet[j-1] + for n in c.A: + phase=[c.P[l]*c.exp_pi(c.ts[l]*n) for l in range(j,7)] + ss=sum(phase) + assert 2*pi*abs(ss)a('1.75') and abs(c.D[n])0 and bet[j-1]>0: + factor=factor.min(1/(pi*bet[j-1]*s)) + out.append(bs[j-1]*(-pi*gg[j]*s).exp()*factor) + return out + + + def laps(s): + return [atoms[j]*bs[j]*(-pi*gg[j]*s).exp() for j in range(7)] + + + tail_bounds=[] + for cut,bound in [(0,'27.7'),(4,'.272'),(16,'.0145'),(100,'4.7e-10')]: + firsts=[n for n in range(cut+1,cut+13) if n%12 in c.A] + total=arb(0) + for n in firsts: + factors=ls(n) + for j in range(1,7): + total+=(1+abs(c.D[n%12])+pi*aa[j-1])/c.Q[n%12]*mm[j-1]*factors[j-1]/(1-(-12*pi*gg[j]).exp()) + assert total40) + for i in range(2)] + assert exact_norms[0]==22516666247 + assert exact_norms[1]==8927477314 + assert exact_tail_norms[0]==197471 + assert exact_tail_norms[1]==161623 + + # Coefficient norms and the exact-solution distance asserted in the text. + norms=[sum(abs(cc)+abs(dd) for cc,dd in tab.values()) for tab in c.tab] + tail_norms=[sum(abs(cc)+abs(dd) for n,(cc,dd) in tab.items() if n>40) for tab in c.tab] + assert all(xa('.012' if i==0 else '.016') + printed_lower('rational tail index '+str(i+1),lower,('.01206834','.01656468')[i]) + assert lower-a('.00003')/(s0-40)>a('.011') + r_lowers.append(str(lower)) + + direct_second=sum(mm[j-1]*(pi*c.ts[j])**2/6 for j in range(1,7)) + assert direct_second<77 + printed_upper('direct second derivative envelope',direct_second,'76.043') + tail_second=a('.00005')*77+a('2.3')*sum(mm[j-1]*ls(s0)[j-1]*(pi*aa[j-1])**2 for j in range(1,7))+a('.017')*sum(laps(s0)[j]*(pi*aa[j])**2 for j in range(7)) + assert tail_seconda('.68') + midpoint_values.append({'left':left,'right':right,'quotient':str(quotient)}) + + # Approximate-to-exact quotient errors. + origin_error=a('.00003')*sum(mm[j-1]*(arb(1)/6+ls(0)[j-1]) for j in range(1,7)) + assert origin_error-a('.081') + assert 1/(b*(h-rho))<5 + assert B/(h-rho)+h arb(str(bound)), (description, str(x), bound) + + + printed_brackets=[] + + def assert_printed_bracket(name,value,lower,upper): + assert_lower(value,lower,name) + assert_upper(value,upper,name) + record={'name':name,'enclosure':enclose(value),'lower':lower,'upper':upper} + printed_brackets.append(record) + return record + + def exp_pi(t): + return (I*pi*t).exp() + + + Q, D = {}, {} + for a in A: + Q[a] = (pi/6)**2 + D[a] = arb(0) + for d in A: + if a == d: + continue + x = pi*(arb(a-d)/12) + Q[a] *= (2*x.sin())**2 + D[a] += pi/6*x.cos()/x.sin() + + + def make_grid(): + N = 256 + weights, xs = [], [] + for v in range(N): + alpha = arb(2*v+1)/(2*N) + weights.append((1-2*sum((2*a*pi*alpha).cos()/arb(4*a*a-1) + for a in range(1,128)))/N) + xs.append((pi*alpha).cos()) + assert all(w > 0 for w in weights) + assert sum(weights).contains(1) + grid = [] + for j in range(1,7): + for x,w in zip(xs, weights): + t = (2*j-1+x)/12 + z = -B/(acb(t)+I*h)-I*h + lam = I/(b*(acb(t)+I*h)) + grid.append((j,t,w/6,z,lam)) + for j,t in enumerate(ts): + z = -B/(acb(t)+I*h)-I*h + lam = I/(b*(acb(t)+I*h)) + grid.append((0,t,arb(0),z,lam)) + return grid + + + def basis_measure(n,kind): + sums={} + for j in range(1,7): + phases=[P[l]*exp_pi(ts[l]*(n % 12)) for l in range(j,7)] + aa=sum(phases) + bb=sum(ts[l]*phases[l-j] for l in range(j,7)) + sums[j]=(aa,bb) + result=[] + for j,t,w,z,lam in grid: + if j==0: + result.append(acb(0)) + continue + aa,bb=sums[j] + phase=exp_pi(-t*n) + density=-2*pi*pi*phase*(bb-t*aa) if kind==0 else 2*I*pi*phase*aa + result.append(w*density) + return result + + + def table_measure(index): + result=[acb(0) for _ in grid] + for n,(c,d) in tab[index].items(): + for kind,coeff in enumerate((c,d)): + if coeff==0: + continue + basis=basis_measure(n,kind) + result=[a+coeff*x for a,x in zip(result,basis)] + for j in range(7): + result[-7+j]=(1 if j==0 else 2)*C[index]*P[j] + return result + + + factor_cache={} + + + def k_factors(m): + if m not in factor_cache: + f0=[lam*exp_pi(z*m) for _,t,w,z,lam in grid] + f1=[v*I*pi*z for v,(_,t,w,z,lam) in zip(f0,grid)] + factor_cache[m]=(f0,f1) + return factor_cache[m] + + + def pair(measure,m): + f0,f1=k_factors(m) + x0=sum(v*f for v,f in zip(measure,f0)).real+ERR + x1=sum(v*f for v,f in zip(measure,f1)).real+ERR + q,d=Q[m % 12],D[m % 12] + return (-x0/q,(d*x0-x1)/q) + + + def moments(measure,m,L): + us=[acb(0) for _ in range(L+1)] + ks=[acb(0) for _ in range(L+1)] + for mass,(_,t,w,z,lam) in zip(measure,grid): + u=mass*exp_pi(t*m) + k=mass*lam*exp_pi(z*m) + pu=I*pi*t + pk=I*pi*z + for ell in range(L+1): + us[ell]+=u + ks[ell]+=k + u=u*pu/(ell+1) + k=k*pk/(ell+1) + return ([v.real+ERR for v in us],[v.real+ERR for v in ks]) + + + def norm1(mat): + return max_ball([sum(abs(mat[i,j]) for i in range(mat.nrows())) + for j in range(mat.ncols())]) + + + start=time.time() + grid=make_grid() + measures=[table_measure(0),table_measure(1)] + print("Built rigorous quadrature grid and both tabulated measures",flush=True) + + # Matrix columns in precisely the specified all-c, then all-d order. + J=[1,3,4] + E=[n for n in nodes if n not in J] + cols=[(n,k) for k in range(2) for n in nodes] + SJ=[] + SJlow=[] + for n,k in cols: + mu=basis_measure(n,k) + v=[pair(mu,m) for m in J] + SJ.append([x[0] for x in v]+[x[1] for x in v]) + if n in J: + low=[pair(mu,m) for m in E if m<=16] + SJlow.append([x[0] for x in low]+[x[1] for x in low]) + jcols=[i for i,(n,k) in enumerate(cols) if n in J] + ecols=[i for i,(n,k) in enumerate(cols) if n in E] + dmat=arb_mat([[SJ[j][i] for j in jcols] for i in range(6)]) + umat=arb_mat([[SJ[j][i] for j in ecols] for i in range(6)]) + ident=arb_mat([[int(i==j) for j in range(6)] for i in range(6)]) + wp=arb_mat([[arb(x)/10000 for x in row] for row in source_data["W_plus"]]) + wm=arb_mat([[arb(x)/10000 for x in row] for row in source_data["W_minus"]]) + matrix_results=[] + for sign,w in [(1,wp),(-1,wm)]: + inv=(ident-sign*dmat).inv() + invnorm=norm1(inv);cross=norm1(inv*umat) + defect=norm1(ident-w*(ident-sign*dmat)) + assert_upper(invnorm,32,"inverse norm") + assert_upper(cross,3.7,"cross norm") + assert_upper(defect,.001,"W inverse defect") + defect_bracket={1:('.00025422624910652544','.00025422624910652546'), + -1:('.00042288165102553355','.00042288165102553357')}[sign] + assert_printed_bracket('W defect for sign '+str(sign),defect,*defect_bracket) + wnorm=norm1(w) + assert_upper(wnorm,'30.84' if sign==1 else '1.83',"W column norm") + grouped_cross=[] + group_bounds=('3.54','.44','.25') if sign==1 else ('.73','.32','.17') + for (lo,hi),bound in zip(((7,9),(12,16),(19,100)),group_bounds): + indices=[j for j,column in enumerate(ecols) if lo<=cols[column][0]<=hi] + block=arb_mat([[umat[i,j] for j in indices] for i in range(6)]) + value=norm1(w*block) + assert_upper(value,bound,('W grouped cross',sign,lo,hi)) + grouped_cross.append({'node_range':[lo,hi],'norm':enclose(value),'upper_bound':bound}) + matrix_results.append({'sign':sign,'inverse_norm':enclose(invnorm),'cross_norm':enclose(cross),'W_defect':enclose(defect),'W_norm':enclose(wnorm),'W_grouped_cross':grouped_cross}) + low_norm=max_ball([sum(abs(x) for x in v) for v in SJlow]) + assert_upper(low_norm,.088,"low exterior to J") + assert_printed_bracket('low exterior norm',low_norm, + '.08688748182289641591','.08688748182289641592') + print("Certified matrix and exterior-block bounds",flush=True) + + residuals=[] + for i in range(2): + for m in nodes: + cc,dd=pair(measures[1-i],m) + c,d=tab[i][m] + r0=abs(c-cc);r1=abs(d+int(i==0 and m==1)-dd) + assert_upper(r0,"3e-10","c residual") + assert_upper(r1,"3e-10","d residual") + residuals.append({'function':i+1,'node':m,'c':enclose(r0),'d':enclose(r1)}) + print("Certified tabulated residual bounds",flush=True) + + centers=[0]+[n for n in nodes if n<=40] + all_moments={} + for m in centers: + all_moments[m]=[moments(measures[i],m,30) for i in range(2)] + print("Computed interval Taylor coefficients at every sign-test center",flush=True) + + sign_results=[] + bernstein_example=None + sign_minima=[] + full_nodes=[0]+nodes + for m in centers: + q=full_nodes.index(m) + ys=[arb(full_nodes[q+1]-m)/2] + if q>0: + ys.append(arb(full_nodes[q-1]-m)/2) + for y in ys: + for i in range(2): + if i==0 and (m==0 or (m==1 and y<0)): + continue + nu=0 if m==0 else (1 if i==0 and m==1 else 2) + sig=-1 if i==0 else 1 + direct=all_moments[m][i][0] + transformed=all_moments[m][1-i][1] + coeffs=[sig*y**r*(direct[r+nu]+transformed[r+nu]) for r in range(29)] + bern=[sum(arb(comb(k,r))/comb(28,r)*coeffs[r] for r in range(k+1)) for k in range(29)] + bound=.74 if m==0 else (.18 if m==1 else (.02 if m in (3,4) else .009)) + for k,value in enumerate(bern): + assert_lower(value,bound,('Bernstein',i+1,m,str(y),k)) + if i==1 and m==16 and y==-arb(1)/2: + record=assert_printed_bracket('Bernstein (2,16,-1/2,28)',bern[28], + '.00951875955680','.00951875955684') + bernstein_example={'function':2,'node':16,'y':'-0.5','k':28,**record} + minimum=-max_ball([-value for value in bern]) + sign_minima.append((i+1,m,minimum)) + sign_results.append({'function':i+1,'node':m,'y':enclose(y),'minimum':enclose(minimum),'required_lower_bound':bound}) + assert bernstein_example is not None + assert len(printed_brackets)==4 + print("Certified individual Bernstein coefficient bounds",flush=True) + + # Exact counterparts of every grouped lower bound in the source table. + group_specs=[([0],None,'.762'),([1],'.314','.191'),([3,4],'.028','.024'), + ([7,9],'.114','.112'),([12,13,15,16],'.0105','.0094'), + ([19,21],'.123','.113'),([24,25,27,28],'.0111','.0113'), + ([31,33],'.121','.126'),([36,37,39,40],'.0107','.0115')] + grouped_bernstein=[] + for group,bound1,bound2 in group_specs: + for i,bound in enumerate((bound1,bound2),1): + if bound is None: + continue + values=[value for function,node,value in sign_minima if function==i and node in group] + value=-max_ball([-v for v in values]) + assert_lower(value,bound,('grouped Bernstein',i,group)) + grouped_bernstein.append({'function':i,'nodes':group,'minimum':enclose(value),'lower_bound':bound}) + + result={'source_sha256':EXPECTED_SHA,'status':'pass','method':'Arb balls for the exact cosine quadrature, enlarged by 1e-22 per moment for the analytic quadrature remainder','python_flint_version':flint.__version__,'precision_bits':ctx.prec,'elapsed_seconds':time.time()-start,'matrix_bounds':matrix_results,'low_exterior_norm':enclose(low_norm),'residuals':residuals,'Bernstein_intervals':sign_results,'quadrature_moment_error_added':'1e-22'} + # Explicit coverage assertions prevent an accidentally shortened run passing. + assert nodes == [n for n in range(1,101) if n % 12 in A] + assert len(cols)==102 and len(jcols)==6 and len(ecols)==96 + assert dmat.nrows()==dmat.ncols()==6 + assert umat.nrows()==6 and umat.ncols()==96 + assert len(SJlow)==6 and all(len(column)==12 for column in SJlow) + assert len(residuals)==102 + assert {(r["function"],r["node"]) for r in residuals} == {(i,m) for i in (1,2) for m in nodes} + assert len(centers)==22 and len(sign_results)==84 + assert sum(r["function"]==1 for r in sign_results)==41 + assert sum(r["function"]==2 for r in sign_results)==43 + assert len({(r["function"],r["node"],r["y"]) for r in sign_results})==84 + result["coverage"]={"finite_nodes":51,"matrix_signs":2,"residual_pairs":102, + "sign_centers":22,"half_gap_intervals":84, + "function_1_intervals":41,"function_2_intervals":43, + "Bernstein_inequalities":84*29} + result['grouped_Bernstein_bounds']=grouped_bernstein + result['printed_brackets']=printed_brackets + result['Bernstein_example']=bernstein_example + result.update(provenance(source_data, __file__)) + write_result("finite.json", result) + print(json.dumps({'status':'pass','elapsed_seconds':result['elapsed_seconds'],'matrix_bounds':matrix_results,'residual_pairs':len(residuals),'sign_intervals':len(sign_results)},indent=2),flush=True) + + return SimpleNamespace(pi=pi,b=b,h=h,B=B,A=A,P=P,ts=ts,Q=Q,D=D, + tab=tab,C=C,exp_pi=exp_pi,ROOT=ROOT,SOURCE=SOURCE, + EXPECTED_SHA=EXPECTED_SHA,source_data=source_data, + result=result,dmat=dmat,umat=umat) + + +if __name__ == "__main__": + run_certificate() diff --git a/preprints/A-sharp-Fourier-certificate-for-planar-circle-packing-September-23-2026/verification/requirements.txt b/preprints/A-sharp-Fourier-certificate-for-planar-circle-packing-September-23-2026/verification/requirements.txt new file mode 100644 index 0000000000000000000000000000000000000000..388b3abf3e5e746b922e1bd1ed888165685786b0 --- /dev/null +++ b/preprints/A-sharp-Fourier-certificate-for-planar-circle-packing-September-23-2026/verification/requirements.txt @@ -0,0 +1 @@ +python-flint==0.9.0 diff --git a/preprints/A-sharp-Fourier-certificate-for-planar-circle-packing-September-23-2026/verification/verify_all.py b/preprints/A-sharp-Fourier-certificate-for-planar-circle-packing-September-23-2026/verification/verify_all.py new file mode 100644 index 0000000000000000000000000000000000000000..b96f1531d4853dc2d34351431571b833480e58b8 --- /dev/null +++ b/preprints/A-sharp-Fourier-certificate-for-planar-circle-packing-September-23-2026/verification/verify_all.py @@ -0,0 +1,38 @@ +#!/usr/bin/env python3 +"""Run the finite and scalar Arb certificates, then write their results.""" + +import json + +from certificate_data import VERIFICATION_DIR, sha256, write_result +from interval_certificate import run_certificate as run_finite +from interval_bounds import run_certificate as run_scalar + + +def main(): + finite = run_finite() + scalar = run_scalar(finite) + assert finite.result["status"] == scalar["status"] == "pass" + assert finite.result["source_sha256"] == scalar["source_sha256"] + assert finite.result["data_sha256"] == scalar["data_sha256"] + result = { + "status": "pass", + "source_sha256": finite.result["source_sha256"], + "data_sha256": finite.result["data_sha256"], + "data_identities_match": finite.result["data_identities_match"], + "python_version": finite.result["python_version"], + "python_flint_version": finite.result["python_flint_version"], + "precision_bits": finite.result["precision_bits"], + "coverage": finite.result["coverage"], + "certificate_sources_sha256": { + path.name: sha256(path) + for path in sorted(VERIFICATION_DIR.glob("*.py")) + }, + "result_files": ["finite.json", "scalar.json"], + } + write_result("summary.json", result) + print(json.dumps(result, indent=2), flush=True) + return result + + +if __name__ == "__main__": + main() diff --git a/preprints/A-sharp-entropy-bound-and-the-simplex-inequality-for-isotropic-constants-October-5-2026/README.md b/preprints/A-sharp-entropy-bound-and-the-simplex-inequality-for-isotropic-constants-October-5-2026/README.md new file mode 100644 index 0000000000000000000000000000000000000000..38e26a1c648cebd8bcf4e6ed5c020e71e58d605f --- /dev/null +++ b/preprints/A-sharp-entropy-bound-and-the-simplex-inequality-for-isotropic-constants-October-5-2026/README.md @@ -0,0 +1,30 @@ +# [A sharp entropy bound and the simplex inequality for isotropic constants](isotropic-simplex.pdf) + +**Author:** OpenAI + +**Date:** October 5, 2026 + +## Citation + +```bibtex +@misc{OAI:A-sharp-entropy-bound-and-the-simplex-inequality-for-isotropic-constants-October-5-2026, + author = {{OpenAI}}, + title = {{A sharp entropy bound and the simplex inequality for isotropic constants}}, + howpublished = {OpenAI Math Release preprint + \href{https://github.com/openai/math/blob/main/preprints/A-sharp-entropy-bound-and-the-simplex-inequality-for-isotropic-constants-October-5-2026/isotropic-simplex.pdf}{OAI:A-sharp-entropy-bound-and-the-simplex-inequality-for-isotropic-constants-October-5-2026}}, + year = {2026} +} +``` + +## Verification + +The scalar certificate checker uses only Python 3's standard library and +exact rational arithmetic. From this paper's root directory, run: + +```sh +python3 verification/verify_scalar.py +``` + +A successful run exits with status zero and ends with `ALL CHECKS PASSED`. +It checks the rational bounds used in the scalar estimates; the analytic +inputs are proved in the paper's appendix. diff --git a/preprints/A-sharp-entropy-bound-and-the-simplex-inequality-for-isotropic-constants-October-5-2026/build/manuscript/approximation.tex b/preprints/A-sharp-entropy-bound-and-the-simplex-inequality-for-isotropic-constants-October-5-2026/build/manuscript/approximation.tex new file mode 100644 index 0000000000000000000000000000000000000000..15e690d68312fd52f23f41db03e43a040c5d3303 --- /dev/null +++ b/preprints/A-sharp-entropy-bound-and-the-simplex-inequality-for-isotropic-constants-October-5-2026/build/manuscript/approximation.tex @@ -0,0 +1,206 @@ +\section{Approximation of log-concave densities}\label{geo:section} + +The smooth estimate of Theorem~\ref{mat:entropy-smooth} requires a +potential with positive, bounded Hessian. We remove those restrictions +while preserving the entropy, covariance, and first moments. The same +approximation will let us study equality by applying the strict +remainder estimate to smooth densities whose entropy gaps tend to zero. + +\subsection{Convex approximation with entropy convergence} + +We use extended-valued convex potentials: $V:\R^m\to(-\infty,+\infty]$ is +proper if it is finite somewhere and never takes the value $-\infty$. +The convention $e^{-\infty}=0$ allows bounded supports. +The elementary tail estimate below supplies a common integrable bound for +the approximation, including its entropy integrand. + +\begin{lemma}\label{geo:coercivity} +Let $V:\R^m\to(-\infty,+\infty]$ be proper, lower semicontinuous, and convex, +with +\[ + 0<\int_{\R^m}e^{-V(x)}\,\dd x<\infty. +\] +Then there are $a>0$ and $b\geq0$ such that +\begin{equation}\label{geo:linear-coercivity} + V(x)\geq a|x|-b\qquad(x\in\R^m). +\end{equation} +In particular, $e^{-V}$ has moments of every order, and +$\int |V|e^{-V}<\infty$, where $|V|e^{-V}$ is interpreted as zero when +$V=+\infty$. +\end{lemma} + +\begin{proof} +The convex set $\{V<\infty\}$ has positive measure and hence nonempty +interior: otherwise its affine hull would be a proper affine subspace. +Choose a simplex with all vertices in this set and with nonempty interior. +Convexity bounds $V$ above by the largest of its values at the vertices +throughout the simplex. Consequently, some ball $B(x_0,r)$ lies in a +sublevel set $\{V\leq M\}$, where $M\geq V(x_0)$. + +The closed convex set $A=\{V\leq M+1\}$ has finite volume, since +\[ + |A|\leq e^{M+1}\int e^{-V}. +\] +It is bounded. Indeed, the convex hull of $B(x_0,r)$ and a point $z\in A$ +contains a cone with base an $(m-1)$-dimensional disk of radius $r$ through +$x_0$, perpendicular to $z-x_0$, and height $|z-x_0|$. Its volume tends to +infinity with $|z-x_0|$. In dimension one the same assertion follows from +the length of the interval joining $x_0$ to $z$. + +Choose $R>r$ so large that $A\subset B(x_0,R)$. If +$d=|x-x_0|\geq R$ and $V(x)<\infty$, put +$y=x_0+(R/d)(x-x_0)$. Then $y\notin A$, and convexity gives +\[ + M+1V(x_0)+d/R$. The inequality is automatic if $V(x)=+\infty$. +On the compact ball $\overline B(x_0,R)$, lower semicontinuity and properness +give a finite lower bound: a sequence with values tending to $-\infty$ +would have a convergent subsequence and violate lower semicontinuity. +Combining these two bounds proves \eqref{geo:linear-coercivity}, after +enlarging $b$. + +The moment assertion follows from $e^{-V(x)}\leq e^b e^{-a|x|}$. +For the entropy assertion, the elementary bound +\begin{equation}\label{geo:entropy-domination} + |u|e^{-u}\leq C_b e^{-u/2}\qquad(u\geq-b) +\end{equation} +holds because $|u|e^{-u/2}$ is bounded on $[-b,\infty)$. +Substitute $u=V(x)$ and use \eqref{geo:linear-coercivity}. +\end{proof} + +\begin{lemma}\label{geo:approximation} +Let $f=e^{-V}$ be a log-concave probability density on $\R^m$, represented +by a proper lower semicontinuous convex potential $V$. There are smooth +convex potentials $V_j$ and probability densities +\[ + f_j=Z_j^{-1}e^{-V_j},\qquad Z_j=\int_{\R^m}e^{-V_j}, +\] +such that +\begin{equation}\label{geo:hessian-bounds} + \frac2j\Id\preceq D^2V_j\preceq\left(j+\frac2j\right)\Id + \qquad(j\geq1), +\end{equation} +and +\begin{equation}\label{geo:approximation-limits} + Z_j\longrightarrow1,\qquad \norm{f_j-f}_{L^1}\longrightarrow0,\qquad + h(f_j)\longrightarrow h(f),\qquad + \Cov(f_j)\longrightarrow\Cov(f). +\end{equation} +The covariance limit is entrywise, the means of $f_j$ converge to the mean +of $f$, and all these quantities are finite. +\end{lemma} + +\begin{proof} +Fix $a,b$ from Lemma~\ref{geo:coercivity}. For each positive integer $j$, +define the Moreau envelope \cite[Section~7]{Moreau1965} +\begin{equation}\label{geo:moreau} + Q_j(x)=\inf_{y\in\R^m}\left\{V(y)+\frac j2|x-y|^2\right\}. +\end{equation} +The expression in braces is lower semicontinuous, strictly convex on its +effective domain, and tends to infinity as $|y|\to\infty$. It therefore +has a unique minimizer $p_j(x)$, and $Q_j$ is finite everywhere. Moreover, +\begin{equation}\label{geo:envelope-lower} + Q_j(x)\geq a|x|-b-\frac{a^2}{2j}. +\end{equation} +Indeed, $V(y)\geq a|x|-a|x-y|-b$, and minimizing +$jr^2/2-ar$ over $r\geq0$ proves the bound. + +For completeness, the needed regularity of $Q_j$ follows directly from +the minimization. The optimality condition is +$j(x-p_j(x))\in\partial V(p_j(x))$, where $\partial V$ denotes the convex +subdifferential. To obtain it, compare the minimum at $p=p_j(x)$ with +$p+s(z-p)$, use convexity to bound $V(p+s(z-p))$ above, divide by $s>0$, +and let $s\downarrow0$. The resulting inequality is +$V(z)\geq V(p)+j\langle x-p,z-p\rangle$. +Adding the two subgradient inequalities at $p_j(x)$ and +$p_j(x')$ gives +\[ + \langle x-x',p_j(x)-p_j(x')\rangle + \geq |p_j(x)-p_j(x')|^2. +\] +Consequently both $p_j$ and $x\mapsto x-p_j(x)$ are $1$-Lipschitz. +Evaluating the infimum at $p_j(x)$ and, in the reverse direction, at +$p_j(x+h)$ shows that +\[ + \nabla Q_j(x)=j(x-p_j(x)). +\] +For example, the first comparison gives an upper error $j|h|^2/2$ +after subtracting $j\langle x-p_j(x),h\rangle$, and the second gives +a lower error of order $j|h|^2$ by the Lipschitz estimate just proved. +Thus $Q_j$ is continuously differentiable with $j$-Lipschitz gradient. +It is convex, since the function minimized in \eqref{geo:moreau} is jointly +convex in $(x,y)$. + +We next check convergence before smoothing. The functions $Q_j$ increase +with $j$, and $Q_j(x)\leq V(x)$ when $V(x)$ is finite. If $Q_j(x)$ stays +bounded above as $j\to\infty$, the identity at the minimizer and $V\geq-b$ +imply +\[ + |x-p_j(x)|^2\leq\frac{2(Q_j(x)+b)}j\longrightarrow0. +\] +Lower semicontinuity then gives +$V(x)\leq\liminf_j V(p_j(x))\leq\lim_jQ_j(x)$. +This proves +\begin{equation}\label{geo:envelope-limit} + Q_j(x)\longrightarrow V(x)\quad\hbox{for every }x, +\end{equation} +including the value $+\infty$. + +Choose a nonnegative even smooth function $\eta$ supported in the unit ball +with integral one, and set $\eta_\varepsilon(x)=\varepsilon^{-m} +\eta(x/\varepsilon)$. Define +\begin{equation}\label{geo:smooth-potentials} + V_j=Q_j*\eta_{1/j}+\frac{|x|^2}j. +\end{equation} +Convolution preserves convexity and the Lipschitz bound on the gradient; +hence \eqref{geo:hessian-bounds} holds. Evenness gives +$\int z\eta(z)\,\dd z=0$, so Jensen's inequality and +\eqref{geo:envelope-lower} yield the common bound +\begin{equation}\label{geo:common-tail} + V_j(x)\geq Q_j(x)\geq a|x|-B, + \qquad B=b+\frac{a^2}2. +\end{equation} + +Smoothing at this scale does not affect the limit. The $j$-Lipschitz +bound on the gradient gives +\[ + Q_j(x-z)\leq Q_j(x)-\langle\nabla Q_j(x),z\rangle+\frac j2|z|^2. +\] +Integrating against $\eta_{1/j}$ cancels the linear term, so +\[ + 0\leq (Q_j*\eta_{1/j})(x)-Q_j(x)\leq\frac1{2j} + \qquad(x\in\R^m). +\] +Together with \eqref{geo:envelope-limit}, this proves +$V_j(x)\to V(x)$ everywhere. + +Now \eqref{geo:common-tail} dominates $(1+|x|^2)e^{-V_j(x)}$ by an +integrable function independent of $j$. Applying +\eqref{geo:entropy-domination} with $B$ in place of $b$ also gives +\[ + |V_j(x)|e^{-V_j(x)}\leq C e^{-a|x|/2}. +\] +At points where $V=+\infty$, both $e^{-V_j}$ and $V_je^{-V_j}$ tend to +zero. Dominated convergence therefore proves convergence of the +normalizing constants, first moments, second moments, and the integrals +$\int V_je^{-V_j}$. Finally, +\[ + h(f_j)=\frac1{Z_j}\int V_je^{-V_j}+\log Z_j, +\] +which proves all of \eqref{geo:approximation-limits}. +\end{proof} + +\begin{proof}[Proof of the inequality in Theorem~\ref{intro:entropy}] +Apply Theorem~\ref{mat:entropy-smooth} to the normalized potentials +$V_j+\log Z_j$ of Lemma~\ref{geo:approximation}. Their Hessians satisfy +\eqref{geo:hessian-bounds}, so +\[ + h(f_j)\geq m+\frac12\log\det\Cov(f_j). +\] +The covariance of $f$ is positive definite: zero variance in a nonzero +direction would concentrate this Lebesgue density on an affine hyperplane. +Thus \eqref{geo:approximation-limits} permits passage to the limit in both +sides and proves the inequality. +\end{proof} + diff --git a/preprints/A-sharp-entropy-bound-and-the-simplex-inequality-for-isotropic-constants-October-5-2026/build/manuscript/certificate.tex b/preprints/A-sharp-entropy-bound-and-the-simplex-inequality-for-isotropic-constants-October-5-2026/build/manuscript/certificate.tex new file mode 100644 index 0000000000000000000000000000000000000000..acc2100dd82b5dca62038e4b55ea866fa64e221a --- /dev/null +++ b/preprints/A-sharp-entropy-bound-and-the-simplex-inequality-for-isotropic-constants-October-5-2026/build/manuscript/certificate.tex @@ -0,0 +1,612 @@ +\section{Verification of the scalar estimates}\label{cert:verification} + +This appendix proves the Gaussian estimates used in the transport and scalar +arguments. All terminating decimals in this appendix denote exact rational +numbers. The finite certificates below concern entire intervals: their +validity follows from coefficient bounds and polynomial positivity, with no +sampling assumption. +In the negative tail, $k$ and its first three derivatives are small and +can be approximated by Gaussian excess moments. In the positive tail, +$k\sim a/2$, so we subtract this leading term and control derivatives of +the remaining logarithmic correction. On the compact interval, we +propagate the differential equations in their stable directions---backward +from $q(10)$ and forward from $k(-6)$---and then certify polynomial +positivity by exact arithmetic. + +\begin{proof}[Proof of Lemma~\ref{sc:bounds} and Lemma~\ref{tr:scalar-basic}] +We use the definitions of $q,k,d,b,l$ from the transport section and of +$p,h_0,\mathsf D,\mathsf L,\mathsf E,K_0,J_0,A_*,\eta_*,\nu_*$ +from the scalar section. The differential identities needed below are +\begin{equation}\label{cert:identities} + q'=qd,\qquad k'=b=1-kd,\qquad + l=k-q+ab,\qquad l'=2b-qd+al. +\end{equation} +In particular, +\begin{equation}\label{cert:derived} + \begin{split} + p&=b+kl-l^2/\alpha,\\ + \mathsf E&=p+b(.75-b)-l^2/\alpha,\\ + h_0&=k(b-.5)-2(b-.28)l/\alpha,\\ + \frac{d\log M}{d\log q} + &=\frac{2(1-qd)}{d^2} + -\frac{2bl}{d(1-b^2)},\qquad M=d^{-2}(1-b^2). + \end{split} +\end{equation} +We verify the estimates separately on $a\le-6$, $a\ge10$, and +$[-6,10]$. The central calculation is also specified in +\texttt{verification/verify\_scalar.py}; it uses rational arithmetic only. + +\subsection*{The negative tail} +Put $r=-a\ge6$, $u=r^{-2}$, and +\[ + I_j(r)=\int_0^\infty v^j e^{-rv-v^2/2}\,dv\qquad(j\ge0). +\] +Writing $\varphi$ for the standard Gaussian density, the lower Gaussian tail +is $P=\Phi(-r)=\varphi(r)I_0(r)$. Thus +\[ + q=\frac{\varphi(r)}{1-P},\qquad + k=\frac{(1-P)(-\log(1-P))}{\varphi(r)}. +\] +Integration by parts gives +\[ + I_1=1-rI_0,\qquad I_2=I_0-rI_1,\qquad I_3=2I_1-rI_2. +\] +The inequality $0\le1-(1-P)(-\log(1-P))/P\le2P$ gives +$|k-I_0|\le2P/r$. Since $P\le\varphi(r)/r$ and +$\varphi(6)<7\cdot10^{-9}$, we have $q\le1.01\varphi(r)$. +Applying \eqref{cert:identities} successively therefore gives +\[ + |b-I_1|\le4\varphi(r)/r,\quad + |l-I_2|\le6\varphi(r),\quad + |l'-I_3|\le9r\varphi(r). +\] +Each Gaussian error decreases faster than its required inverse power for +$r\ge6$: indeed $r^N\varphi(r)$ is decreasing there for $N\le7$. +Checking at $r=6$ yields, with $g_0=k,g_1=b,g_2=l,g_3=l'$, +\begin{equation}\label{cert:left-errors} + |g_j-I_j|\le .01u\frac{j!}{r^{j+1}}\qquad(0\le j\le3). +\end{equation} +The elementary bounds $1-v^2/2\le e^{-v^2/2}\le1$ also give +\begin{equation}\label{cert:integral-errors} + 1-\frac{(j+1)(j+2)}2u + \le \frac{I_j}{j!/r^{j+1}}\le1. +\end{equation} +For $\zeta=1+.01u$, consequences we shall use are +\[ + \begin{array}{c} + 00$. Moreover, +\[ + p\ge u(1-3.01u)-4\zeta^2u^3/\alpha + \ge u(1-4.54u)\ge\frac1{7+r^2}. +\] +The last two inequalities follow by inserting $u\le1/36$; after dividing +by positive factors they reduce to +$3.01+4\zeta^2u/\alpha<4.54$ and +$(1-4.54u)(1+7u)\ge1$. +The two summands of $h_0$ in \eqref{cert:derived} have opposite signs +and magnitudes at most $.501/r$ and $.43/r$. Hence +\[ + h_0^2\le .501^2u<.67^2u(1-4.54u)\le .67^2p. +\] +This establishes the required bounds on $p,h_0$ in this tail. + +For clarity, the remaining substitutions can be made with the following +ratios, avoiding cancellation in $p$: +\begin{equation}\label{cert:left-ratios} + \frac{kl}{b}\le\frac{2u\zeta^2}{1-3.01u},\qquad + \frac{l^2}{\alpha b}\le\frac{4u^2\zeta^2}{\alpha(1-3.01u)}, + \qquad .93<\frac bp<1.05. +\end{equation} +They imply the $O$ row of the coefficient table in Lemma~\ref{sc:bounds}. +For example, $\mathsf D\le D_*(7+r^2)<.295(7+r^2)$, +$\mathsf L\le2\zeta/(\alpha r^3)<.60$, +\[ + .370. +\] +This proves the local matrix bound on $a\le-6$. + +\subsection*{The positive tail} +For $r=a\ge10$ put $u=r^{-2}$ and +$\theta=\log(r\sqrt{2\pi})$. Gaussian tail integration gives +\[ + F(u)=\int_0^\infty e^{-v-uv^2/2}\,dv,\qquad + \Phi(-r)=\frac{\varphi(r)}r F(u),\qquad + k=\frac r2+\frac{C(u,\theta)}r, +\] +where +\begin{equation}\label{cert:C} + C=F\theta+G,\qquad + G=\frac{F-1}{2u}-F\log F. +\end{equation} +Let $\mathcal U=u\partial_u$, with $\theta$ held fixed, and let +$\mathcal D=r\,d/dr=\partial_\theta-2\mathcal U$. Define +\[ + B=(1-\mathcal D)C,\qquad J=(2-\mathcal D)B,\qquad + R=(3-\mathcal D)J. +\] +Here $B,J,R$ are auxiliary scalar functions used only in this appendix. +Differentiating $k$ gives +\begin{equation}\label{cert:right-derivatives} + b=.5-uB,\qquad l=J/r^3,\qquad l'=-R/r^4. +\end{equation} +We first establish uniform bounds for these three derivatives: +\begin{equation}\label{cert:right-errors} + |B-(\theta-1.5)|<.04,\qquad + |J-(2\theta-4)|<.13,\qquad + |R-(6\theta-14)|<.75. +\end{equation} + +Differentiation under the integral gives, for $0\le u\le.01$, +\[ + 1-u\le F\le1,\qquad |F'|\le1,\qquad + |F''|\le6,\qquad |F'''|\le90,\qquad |F^{(4)}|\le2520. +\] +Indeed the $j$th derivative is bounded in absolute value by +$(2j)!/2^j$. To control the apparent quotient in $G$, use +$(F(u)-1)/u=\int_0^1F'(tu)\,dt$. Since $F\ge.99$ and +$|1+\log F|\le1$, differentiating \eqref{cert:C} yields +\[ + |G''|\le15+6+1/.99<23,\qquad + |G'''|\le315+90+18/.99+1/.99^2<435. +\] +The values $F(0)=1$, $F'(0)=-1$, $G(0)=-1/2$, and $G'(0)=5/2$ +therefore give +\[ + C=\theta-.5+(2.5-\theta)u+R_0,\qquad + g=11.5+3\theta,\qquad c_0=435+90\theta, +\] +with +\begin{equation}\label{cert:Euler-remainder} + |\mathcal U^jR_0|\le2^jg u^2+c_0u^3\mathbf1_{\{j=3\}} + \qquad(0\le j\le3). +\end{equation} +The same bounds apply to $\partial_\theta R_0$ with $g=3,c_0=90$; +all higher $\theta$ derivatives vanish. For example, +$\mathcal U^2R_0=uR_0'+u^2R_0''$ and +$\mathcal U^3R_0=uR_0'+3u^2R_0''+u^3R_0'''$, so these statements +follow directly from Taylor's theorem and the derivative bounds just given. + +Expanding the three operators in \eqref{cert:right-derivatives}, the +linear-in-$u$ errors are respectively +\[ + (8.5-3\theta)u,\qquad (37-12\theta)u,\qquad + (197-60\theta)u. +\] +The errors contributed by $R_0$ are bounded respectively by +\begin{equation}\label{cert:operator-errors} + (5g+3)u^2,\qquad(30g+33)u^2,\qquad + (210g+321+8c_0u)u^2. +\end{equation} +For example the second operator is +$2-3\mathcal D+\mathcal D^2$ and the third is +$6-11\mathcal D+6\mathcal D^2-\mathcal D^3$; substituting +$\mathcal D=\partial_\theta-2\mathcal U$ into +\eqref{cert:Euler-remainder} gives exactly the displayed constants. + +Here is a rational way to check uniformity on the unbounded interval. +Write $x=\log(r/10)\ge0$, so $u=.01e^{-2x}$ and +$\theta=\theta_0+x$, where $3.221<\theta_0<3.222$. +The terms in \eqref{cert:operator-errors} decrease with $x$. For the +linear-in-$u$ terms, maximize a linear function times $e^{-2x}$ at +its endpoint or stationary point, using $e^{-t}\le1/(1+t)$ for $t\ge0$. +More explicitly, set +\[ + \begin{aligned} + g_+&=11.5+3(3.222),& c_+&=435+90(3.222),\\ + a_B&=3(3.222)-8.5,& a_J&=12(3.222)-37,\\ + a_R&=197-60(3.222). + \end{aligned} +\] +Upper bounds for the total errors are +\[ + \begin{array}{c|l|c} + &\text{rational upper bound}&\text{a larger rational}\\\hline + B&\displaystyle\frac{.015}{2-2a_B/3} + +\frac{5g_++3}{10^4}&.024\\[2mm] + J&\displaystyle\frac{.06}{2-a_J/6} + +\frac{30g_++33}{10^4}&.102\\[2mm] + R&\displaystyle\max\left\{\frac{197-60(3.221)}{100}, + \frac{.3}{2+a_R/30}\right\} + +\frac{210g_++321}{10^4}+\frac{8c_+}{10^6}&.624 + \end{array} +\] +These prove \eqref{cert:right-errors}. The bound on $\theta_0$ itself +can be checked with the logarithm series +\[ + \log y=2\sum_{j=0}^{N}\frac{z^{2j+1}}{2j+1}+\mathcal R_N,\qquad + z=\frac{y-1}{y+1},\qquad + |\mathcal R_N|\le + \frac{2|z|^{2N+3}}{(2N+3)(1-z^2)}, +\] +applied to $\theta_0=\tfrac12\log(200\pi)$. For a completely +specified enclosure, use Machin's identity +$\pi=16\arctan(1/5)-4\arctan(1/239)$ and the alternating arctangent +sums through degrees $47$ and $49$. In the logarithm series first divide +the argument by a power of two to place it in $[1,2)$, and take $N=35$ +for that argument and for $\log2$. These rational operations give +$3.221523626198<\theta_0<3.221523626199$. + +Combining \eqref{cert:right-errors} with the preceding Taylor bounds +on $C$ gives the following estimates needed in the lemmas: +\begin{equation}\label{cert:right-consequences} + 1.72\theta-4.13>0$, +$R-1.7J>2.6\theta-8.171>0$, and $3J-R>.86$. +For the bounds involving $k$ and $p$, we have +$C\ge .99\theta-.5>0$ by \eqref{cert:C}, whereas its Taylor expansion gives +\[ + C-(\theta-.5)\le + u\{2.5-\theta+.01(11.5+3\theta)\}<0. +\] +Thus $00$. The Gaussian excess above $r$ +is $V/r$, so +\[ + rd=\mathbb EV,\qquad 1-qd=r^{-2}\operatorname{Var}V. +\] +Taylor's lower bound in the numerator and $F\le1$ give +$\mathbb EV\ge1-3u$ and $\mathbb EV^2\ge2(1-6u)$. +Monotone weighting of the exponential density gives +$\mathbb EV\le1$. Hence $d\ge.97/r$, +$\operatorname{Var}V\ge.88$, and +\[ + \chi\le\frac{J}{.97r^2}<.027<.097. +\] +Equation \eqref{cert:derived} now implies +\[ + \frac{d\log M}{d\log q} + \ge2(.88)-\frac{.027}{1-.5^2}>.68. +\] +This verifies the logarithmic slope required by Lemma~\ref{tr:scalar-basic}. + +For the matrix inequality, use $\mathsf E\le p+.482(.75-.482)$ +in \eqref{cert:N}. The first part of $N_{11}$ increases with $b$ +on $[.482,.5]$, and $(p+v)^2/p$ increases with $p$ when +$00. +\] +This finishes the positive tail. + +\subsection*{Polynomial enclosures on the central interval} +We next provide all data and rules for the finite certificate on $[-6,10]$. +On each row of Table~\ref{cert:centers}, write $a=w+rx$, $-1\le x\le1$. +The corresponding closed intervals are consecutive and cover $[-6,10]$. +The tabulated constants are rational seeds; we do not assume that they are +Gaussian function values. The differential residuals and the two analytic +anchors $q(10)$ and $k(-6)$ will certify the entire piecewise polynomial +family, including all its initial coefficients. +Construct polynomials $P,\widehat k$ of degree $14$ from their tabulated constant +terms by the recurrences +\begin{equation}\label{cert:recurrence} + D=P-a,\qquad + P_{i+1}=\frac r{i+1}(PD)_i,\qquad + \widehat k_{i+1}=\frac r{i+1}(1-\widehat k D)_i\quad(0\le i<14). +\end{equation} +The subscript denotes a coefficient, so the constant $1$ contributes only +when $i=0$. At each step only coefficients already computed occur on the +right. The letters $P,\widehat k,D$ denote polynomial approximations to $q,k,d$, +respectively, throughout this central calculation. + +\begin{table}[htbp] +\centering +\caption{Rational input data for the central polynomial certificate.} +\label{cert:centers} +\begin{tabular}{r r l l} +\hline +$w$&$r$&$P_0$&$\widehat k_0$\\\hline +$-5.25$&$.75$&$.0000004128471303$&$.1842076703035094$\\ +$-3.75$&$.75$&$.0003526268606772$&$.2507500253088710$\\ +$-2.5$&$.5$&$.0176378254869167$&$.3531628936050266$\\ +$-1.5$&$.5$&$.1387897504588508$&$.4981884857033017$\\ +$-.5$&$.5$&$.5091604338370335$&$.7246172144765372$\\ +$.25$&$.25$&$.9635539794164037$&$.9475979387933716$\\ +$1.25$&$.75$&$1.7288166273310539$&$1.3000948982566192$\\ +$3$&$1$&$3.2830986549304364$&$2.0126493036044195$\\ +$5$&$1$&$5.1865039671258417$&$2.9046537878842420$\\ +$7$&$1$&$7.1375456132265036$&$3.8366560415481663$\\ +$9$&$1$&$9.1085231050028685$&$4.7898159350739631$\\\hline +\end{tabular} +\end{table} + +For a polynomial $A$, let $[A]$ denote the sum of the absolute values of +its coefficients. Direct use of \eqref{cert:recurrence} gives +\begin{equation}\label{cert:residuals} + r[(PD)_{\deg\ge14}]< + \begin{cases}2\cdot10^{-9},&w=-3.75,\\7\cdot10^{-11},&\text{otherwise},\end{cases} + \qquad r[(\widehat k D)_{\deg\ge14}]<2\cdot10^{-10}. +\end{equation} +For each adjacent pair, the values of both $P$ and $\widehat k$ at their common +endpoint differ by less than $2\cdot10^{-10}$. Also +\begin{equation}\label{cert:additional-poly} + D>.09,\qquad [\widehat k]<\begin{cases}2,&w\le.25,\\5.6,&w>.25, + \end{cases} +\end{equation} +and the first-row value $\widehat k(-1)$ differs from $.16237766$ by less than +$3\cdot10^{-9}$, while the last-row value $P(1)$ differs from +$10.098093234$ by less than $2\cdot10^{-10}$. +These are rational coefficient checks. For example the first residual is +bounded by +\[ + r\sum_{i=14}^{28}\left|\sum_{j=0}^{14}P_jD_{i-j}\right|, +\] +with missing coefficients taken as zero. The bound on $D$ follows from +the quartic lower-bound rule \eqref{cert:Bernstein} below, after charging +discarded coefficients to the remainder. + +Here are analytic anchors for those coefficient checks. For any $r>0$, +Taylor's formula under the integral gives alternating upper and lower +bounds for $I_0(r)$ by +\begin{equation}\label{cert:Mills} + S_N(r)=\sum_{i=0}^{N}\frac{(-1)^i(2i-1)!!}{r^{2i+1}}, + \qquad S_{2j+1}(r)\le I_0(r)\le S_{2j}(r), +\end{equation} +where $(-1)!!=1$. At $r=10$, use $S_{11},S_{12}$ and $q(10)=1/I_0(10)$; +at $r=6$, use $S_{13},S_{14}$ and +$|k(-6)-I_0(6)|\le2\varphi(6)/36$. Exact substitution gives +\[ + |q(10)-10.098093234|<4\cdot10^{-10},\qquad + |k(-6)-.16237766|<4\cdot10^{-8}. +\] +The rough bound $\varphi(6)<7\cdot10^{-9}$ used here follows, for example, +from $\sqrt{2\pi}>12/5$ and +$e^{18}>\sum_{i=0}^{59}18^i/i!$: division gives +$(5/12)/\sum_{i=0}^{59}18^i/i!<7\cdot10^{-9}$. + +These anchors and residuals imply the following +uniform, rather than just endpoint, errors: +\begin{equation}\label{cert:global-errors} + |q-P|\le e_q=10^{-8},\qquad |k-\widehat k|\le e_k=8\cdot10^{-7}. +\end{equation} +Indeed the equation for $q-P$ has multiplier $r(q+P-a)>0$, so backward +integration from $a=10$ is stable. Its absolute error increases by at most +twice each bound in \eqref{cert:residuals} and by each joining error. +The resulting bound is +\[ + 6\cdot10^{-10}+2(2\cdot10^{-9}+10\cdot7\cdot10^{-11}) + +10\cdot2\cdot10^{-10}=8\cdot10^{-9}4,\\ + &.67^2p-\{k(b-.5)-2(b-.28)l/\alpha\}^2;\\[1mm] + \mathcal G_2:\quad& l'-z_{\min}l,\ z_{\max}l-l',\ + H-.8l,\ 1.77l-H,\ \chi_{\max}d-l,\\ + &2(1-qd)(1-b^2)-2bld-m_0d^2(1-b^2);\\[1mm] + \mathcal G_3:\quad& p-D_*/\mathsf D_{\max}\quad + \text{(omitted for bin }O\text{)},\\ + &D_*\mathsf E-K_{\min}p,\ K_{\max}p-D_*\mathsf E,\ + J_{\max}p-D_*f_1;\\[1mm] + \mathcal G_4:\quad& hp-D_*\mathsf E^2,\\ + &p(hv-o^2)-D_*(\mathsf E^2v-2\mathsf E\,o\,f_1+h f_1 f_1). +\end{align*} +Products are evaluated from left to right, with repeated multiplication +for powers. In the last expression the product in the middle is +$2\mathsf E\,o\,f_1$. The bounds below are the minimum over all members +of a group, so positivity proves every inequality listed in it. + +\begin{table}[htbp] +\centering +\caption{Lower bounds for the four polynomial groups, in units of $10^{-3}$.} +\label{cert:group-bounds} +\begin{tabular}{r r r r r} +\hline +$w$&$\mathcal G_1$&$\mathcal G_2$&$\mathcal G_3$&$\mathcal G_4$\\\hline +$-5.25$&$7$&$1$&$1$&$1$\\ +$-3.75$&$1$&$1$&$2$&$2$\\ +$-2.5$&$2$&$2$&$2$&$5$\\ +$-1.5$&$2$&$2$&$2$&$8$\\ +$-.5$&$2$&$3$&$5$&$14$\\ +$.25$&$2$&$2$&$8$&$11$\\ +$1.25$&$2$&$.8$&$8$&$20$\\ +$3$&$2$&$.8$&$8$&$7$\\ +$5$&$1$&$.8$&$10$&$15$\\ +$7$&$3$&$.6$&$11$&$11$\\ +$9$&$2$&$.35$&$12$&$9$\\\hline +\end{tabular} +\end{table} + +Table~\ref{cert:group-bounds} is obtained by the rational operations +\eqref{cert:recurrence}, \eqref{cert:aux-polys}, +\eqref{cert:aux-errors}, \eqref{cert:product-error}, and +\eqref{cert:Bernstein}, with no additional function evaluations. +The accompanying checker reproduces all $44$ bounds, all $22$ +differential residual bounds, the $20$ joining bounds, the anchor checks, +and the coefficient norm bounds used to obtain +\eqref{cert:global-errors}. The same endpoint enclosures give +$q(-1)>.287$, $q(-2)<.056$, $q(-3)<.005$, and $q(4)>4$, as used +in the secant argument. The logarithm series above also certifies +\[ + \log\frac{q(-1)}{q(-2)}>1.62,\quad + \log\frac{q(-1)}{q(-3)}>4,\quad + \log\frac{q(4)}{q(-2)}>4,\quad + \log\frac{q(4)}{q(-3)}>6.5. +\] + +We explain why these finite checks finish the proof. Group $\mathcal G_1$ +gives $00$, the bounds for $p,h_0$, and the bounds for +$b,\mathsf L$. Its strict positive margins give $p>0$. +Group $\mathcal G_2$ gives the bounds for $z,h_*,\chi$ after division by +$l$ or $d$, and the logarithmic slope bound by division by +$d^2(1-b^2)$. Group $\mathcal G_3$ gives the remaining coefficient bounds +after division by $p$; the $O$ bound for $\mathsf D$ instead follows from +$D_*<.295$ and the lower bound on $p$. +Finally +\[ + Q-N-\operatorname{diag}(\ell_1,.75) + =\begin{pmatrix}h&o\\o&v\end{pmatrix} + -\frac{D_*}{p}\binom{\mathsf E}{f_1} + \begin{pmatrix}\mathsf E&f_1\end{pmatrix}. +\] +Its leading minor and determinant, multiplied by $p$, are exactly the +two expressions in $\mathcal G_4$. Their positivity proves the matrix +inequality. Continuity supplies every bin endpoint, including the stronger +of the two adjacent logarithmic slope bounds. Together with the two tails, +this proves both lemmas on the whole real line. +\end{proof} diff --git a/preprints/A-sharp-entropy-bound-and-the-simplex-inequality-for-isotropic-constants-October-5-2026/build/manuscript/geometry.tex b/preprints/A-sharp-entropy-bound-and-the-simplex-inequality-for-isotropic-constants-October-5-2026/build/manuscript/geometry.tex new file mode 100644 index 0000000000000000000000000000000000000000..3bc85ba4c720f3b40f2607fd5714827d0232a72d --- /dev/null +++ b/preprints/A-sharp-entropy-bound-and-the-simplex-inequality-for-isotropic-constants-October-5-2026/build/manuscript/geometry.tex @@ -0,0 +1,169 @@ +\section{The simplex inequality and its equality cases}\label{sec:geometry} + +We transfer Theorem~\ref{intro:entropy} to convex bodies using the exponential +density on a cone. This construction appears in +\cite[Proposition~2(b)]{FradeliziMeyer2008}; its entropy reduction is +\cite[Lemma~5.2 and Proposition~5.2]{FradeliziMarinSola2026}. +The moment formulas also appear in \cite[Lemma~2.5]{Klartag2018}. +The calculation identifies equality in the body inequality with entropy +equality in one higher dimension. The support of the resulting +exponential product then determines the body. + +\begin{proof}[Proof of Theorem~\ref{intro:simplex}] +Let $K\subset\R^n$ be a convex body. Translation preserves both $L_K$ +and the property of being a simplex, so assume that $K$ has centroid +zero. Write $\Sigma_K=\Cov(X)$ for $X$ uniform on $K$, and define +\begin{equation}\label{eq:body-cone} + \mathcal C_K=\{(tx,t):x\in K,\ t\geq0\}\subset\R^{n+1}. +\end{equation} +Convexity of $K$ makes $\mathcal C_K$ convex. It is closed because $K$ +is compact: a convergent sequence $(t_jx_j,t_j)$ with $t_j\to0$ has +$t_jx_j\to0$, and when the limiting height is positive one can divide +by that height. The cone has nonempty interior, and its only point +at height zero is the origin. + +Define the log-concave density +\begin{equation}\label{eq:body-cone-density} + f_K(y,t)=\frac{e^{-t}}{n!\,|K|}\, + \mathbf 1_{\mathcal C_K}(y,t). +\end{equation} +The change of variables $y=tx$, with Jacobian $t^n$ for $t>0$, gives +\[ + \int_{\mathcal C_K}e^{-t}\,\dd y\,\dd t + =|K|\int_0^\infty t^ne^{-t}\,\dd t=n!\,|K|, +\] +so $f_K$ is a probability density. The same change of variables shows +that it is the density of $(TX,T)$, where $T$ is independent of $X$ +and has density $t^ne^{-t}/n!$ on $(0,\infty)$. Integration by parts +therefore gives +\[ + \E T=n+1,\qquad + \E T^2=(n+1)(n+2),\qquad + \Var(T)=n+1. +\] +Since $\E X=0$, independence makes the cross-covariance of $TX$ and +$T$ vanish. Hence +\begin{align} + h(f_K)&=n+1+\log(n!\,|K|),\label{eq:body-cone-entropy}\\ + \Cov(f_K)&= + \begin{pmatrix} + (n+1)(n+2)\Sigma_K&0\\ + 0&n+1 + \end{pmatrix},\label{eq:body-cone-covariance}\\ + \det\Cov(f_K)&=(n+1)^{n+1}(n+2)^n\det\Sigma_K. + \label{eq:body-cone-determinant} +\end{align} +Applying the entropy inequality of Theorem~\ref{intro:entropy} in +dimension $n+1$ yields +\begin{equation}\label{eq:body-cone-bound} + (n!)^2|K|^2\geq + (n+1)^{n+1}(n+2)^n\det\Sigma_K. +\end{equation} +This is precisely +\[ + L_K\leq + \frac{(n!)^{1/n}}{(n+1)^{(n+1)/(2n)}\sqrt{n+2}}. +\] +Every step from the entropy inequality to \eqref{eq:body-cone-bound} +is reversible. Thus equality in this bound holds if and only if +$f_K$ has equality in Theorem~\ref{intro:entropy}. + +Suppose equality holds. By that theorem, the law with density $f_K$ +is an invertible affine image of the product of $n+1$ mean-one +exponential laws. The closed support of $f_K$ is exactly +$\mathcal C_K$: it has positive density on the interior of the cone, +and this interior is dense in the cone. Taking closed supports gives +\begin{equation}\label{eq:body-cone-affine-orthant} + \mathcal C_K=b+M[0,\infty)^{n+1} +\end{equation} +for an invertible matrix $M$ and a vector $b$. + +The origin is the unique extreme point of $\mathcal C_K$. Indeed, +every nonzero $z\in\mathcal C_K$ is the midpoint of $0$ and $2z$, +whereas $0=(u+v)/2$ with $u,v\in\mathcal C_K$ forces the +nonnegative heights of $u$ and $v$ to vanish, and hence $u=v=0$. +The orthant also has the origin as its unique extreme point. +Invertible affine maps preserve extreme points, so +\eqref{eq:body-cone-affine-orthant} forces $b=0$. + +Choose positive multiples $w_0,\ldots,w_n$ of the respective columns +of $M$, and let $a_i$ be the last coordinate of $w_i$. Each $w_i$ is a nonzero point of +$\mathcal C_K$, so $a_i>0$. Put $p_i=w_i/a_i$. The section at +height one is +\begin{equation}\label{eq:body-cone-section} + K\times\{1\} + =\left\{\sum_{i=0}^n c_iw_i: + c_i\geq0,\ \sum_{i=0}^n c_i a_i=1\right\} + =\operatorname{conv}\{p_0,\ldots,p_n\}. +\end{equation} +The vectors $p_i$ are linearly independent, since the vectors $w_i$ +are. They are consequently affinely independent in the height-one +hyperplane. Thus $K$ is a simplex. Figure~\ref{fig:cone-section} +illustrates the normalization of the generating rays in this step. + +Conversely, let $K=\operatorname{conv}\{v_0,\ldots,v_n\}$ be a +simplex, and let $M$ have columns $(v_i,1)$. These columns are +linearly independent and +\[ + M[0,\infty)^{n+1}=\mathcal C_K, + \qquad |\det M|=n!\,|K|. +\] +For the determinant identity, subtract the first column from the +others and expand along the last row; the remaining determinant is +$n!$ times the volume of $K$, up to sign. If $E_0,\ldots,E_n$ are +independent mean-one exponential variables, the density of +$M(E_0,\ldots,E_n)^{\mathsf T}$ at $(y,t)\in\mathcal C_K$ is +\[ + \frac{1}{|\det M|}\exp\left(-\sum_{i=0}^n + (M^{-1}(y,t))_i\right) + =\frac{e^{-t}}{n!\,|K|}, +\] +because the last row of $M$ consists of ones. This is exactly +$f_K$, up to immaterial boundary values. Theorem~\ref{intro:entropy} +therefore gives equality in \eqref{eq:body-cone-bound}. +\end{proof} + +For $n=1$, every convex body is an interval, hence a simplex, and the +constant is $1/\sqrt{12}$. Indeed, an interval of length $\ell$ has +variance $\ell^2/12$, in agreement with the formula above. + +\begin{figure}[t] +\centering +\begin{tikzpicture}[scale=1.25, + x={(1cm,0cm)},y={(0.45cm,0.35cm)},z={(0cm,1.5cm)}, + line cap=round,line join=round,font=\small] + \coordinate (O) at (0,0,0); + \coordinate (P0) at (-1,-0.65,1); + \coordinate (P1) at (1,-0.65,1); + \coordinate (P2) at (0,1,1); + \coordinate (W0) at (-1.4,-0.91,1.4); + \fill[gray!8] (-1.4,-1,1)--(1.4,-1,1)--(1.4,1.3,1) + --(-1.4,1.3,1)--cycle; + \draw[gray!55] (-1.4,-1,1)--(1.4,-1,1)--(1.4,1.3,1) + --(-1.4,1.3,1)--cycle; + \node[anchor=west] at (1.4,1.3,1) {$t=1$}; + \draw[gray!70] (O)--(P2); + \fill[blue!12] (P0)--(P1)--(P2)--cycle; + \draw[blue!65!black,thick] (P0)--(P1)--(P2)--cycle; + \draw[thick] (O)--(P0) (O)--(P1); + \draw[thick,dashed,->] (P0)--(-1.65,-1.0725,1.65); + \draw[thick,dashed,->] (P1)--(1.5,-0.975,1.5); + \draw[thick,dashed,->] (P2)--(0,1.5,1.5); + \fill (W0) circle[radius=1.3pt] + node[above right] {$w_0$} node[left] {$t=a_0>1$}; + \fill (O) circle[radius=1.3pt] node[below] {$0$}; + \fill (P0) circle[radius=1.3pt] node[left] {$p_0$}; + \fill (P1) circle[radius=1.3pt] node[right] {$p_1$}; + \fill (P2) circle[radius=1.3pt] node[above left] {$p_2$}; + \node at (0,0,1) {$K\times\{1\}$}; + \node[align=left,anchor=west] at (2.3,-1,0.65) + {$p_i=w_i/a_i$\\$a_i=\text{height of }w_i>0$}; +\end{tikzpicture} +\caption{The height-one section of a cone generated by three independent +rays. Dividing each generator $w_i$ by its positive height $a_i$ +produces the vertices $p_i$ of the section; one generator above the +section is shown explicitly. Formula +\eqref{eq:body-cone-section} gives the same correspondence in every +dimension.} +\label{fig:cone-section} +\end{figure} diff --git a/preprints/A-sharp-entropy-bound-and-the-simplex-inequality-for-isotropic-constants-October-5-2026/build/manuscript/introduction.tex b/preprints/A-sharp-entropy-bound-and-the-simplex-inequality-for-isotropic-constants-October-5-2026/build/manuscript/introduction.tex new file mode 100644 index 0000000000000000000000000000000000000000..8dcf5266e427bc40f61062ba2b7ea77660cf052d --- /dev/null +++ b/preprints/A-sharp-entropy-bound-and-the-simplex-inequality-for-isotropic-constants-October-5-2026/build/manuscript/introduction.tex @@ -0,0 +1,192 @@ +\section{Introduction} + +The isotropic constant compares the covariance of a uniform distribution +with the volume of its support. For a convex body $K\subset\R^n$, +meaning a compact convex set with nonempty interior, write $|K|$ for its +Lebesgue volume, $b_K=|K|^{-1}\int_Kx\,\dd x$ for its centroid, and +\[ + \Sigma_K=\frac1{|K|}\int_K(x-b_K)(x-b_K)^{\mathsf T}\,\dd x. +\] +Its isotropic constant is +\begin{equation}\label{intro:isotropic-constant} + L_K=\left(\frac{\det\Sigma_K}{|K|^2}\right)^{1/(2n)}. +\end{equation} +This quantity is unchanged by invertible affine maps. The slicing +problem, arising in Bourgain's work on convex bodies +\cite{Bourgain1986}, asks whether it admits an upper bound independent +of dimension. Klartag proved an $O(n^{1/4})$ bound +\cite{Klartag2006}. Subsequent progress used stochastic localization, +introduced by Eldan \cite{Eldan2013}, leading through subpolynomial +and polylogarithmic estimates \cite{Chen2021,KlartagLehec2022,Klartag2023} +to Klartag and Lehec's dimension-independent bound using Guan's +covariance estimate \cite{Guan2024,KlartagLehec2025}. +The sharp form, also called the strong isotropic constant conjecture, +asks, in each dimension, whether a simplex maximizes $L_K$. +The distinction is substantial: a universal bound does not +identify the extremal value or an extremizing body. + +In the plane, Campi, Colesanti, and Gronchi proved the simplex bound, +and Saroglou established uniqueness of the maximizing body +\cite{CampiColesantiGronchi1999,Saroglou2010}. +In higher dimensions, geometric variation arguments have imposed +restrictions on possible maximizers. Rademacher proved that a +simplicial polytope maximizing the isotropic constant among all convex +bodies must be a simplex \cite{Rademacher2016}. More recent restrictions on local +maximizers in terms of decomposability appear in \cite{Kipp2026}. +We prove the sharp inequality for arbitrary convex bodies and classify +all equality cases. + +\begin{theorem}[The simplex inequality]\label{intro:simplex} +For every integer $n\ge1$, every convex body $K\subset\R^n$ satisfies +the following bound, with equality if and only if $K$ is a simplex: +\begin{equation}\label{intro:simplex-bound} + L_K\le + \frac{(n!)^{1/n}}{(n+1)^{(n+1)/(2n)}\sqrt{n+2}}. +\end{equation} +\end{theorem} + +The theorem also has a volume-product consequence. For a convex body +$K$ with $0$ in its interior, define its polar by +$K^\circ=\{y\in\R^n:\langle x,y\rangle\le1\text{ for every }x\in K\}$. +Using exponential measures on convex cones, Klartag proved that the +sharp simplex bound implies +\begin{equation}\label{intro:mahler} + |K|\,|K^\circ|\ge\frac{(n+1)^{n+1}}{(n!)^2}, +\end{equation} +the nonsymmetric Mahler inequality +\cite[Corollary~1.2]{Klartag2018}. Thus Theorem~\ref{intro:simplex} +also gives this inequality in every dimension. + +The proof proceeds through differential entropy. For a probability +density $f$ on $\R^m$, define, when the integrals are finite, +\[ + h(f)=-\int_{\R^m}f(x)\log f(x)\,\dd x, + \qquad + \Cov(f)=\int_{\R^m}(x-\bar x)(x-\bar x)^{\mathsf T}f(x)\,\dd x, + \quad \bar x=\int_{\R^m}xf(x)\,\dd x. +\] +All logarithms are natural, and $0\log0=0$. A log-concave density can +be represented, up to a null set, as $f=e^{-V}$ for a proper lower +semicontinuous convex function $V:\R^m\to(-\infty,+\infty]$. +The value $+\infty$ permits a density to vanish. Such a probability +density has finite entropy and moments, and its covariance is positive +definite; the needed tail bound is proved in Lemma~\ref{geo:coercivity}. + +\begin{theorem}[The sharp entropy bound]\label{intro:entropy} +Let $m\ge1$ and let $f$ be a log-concave probability density on $\R^m$. +Then +\begin{equation}\label{intro:entropy-bound} + h(f)\ge m+\frac12\log\det\Cov(f). +\end{equation} +Equality holds if and only if $f$ is, almost everywhere, the density +of $M(E_1,\ldots,E_m)^{\mathsf T}+b$, where $M$ is an invertible real +$m\times m$ matrix, $b\in\R^m$, and the $E_i$ are independent random +variables with density $e^{-u}\mathbf1_{\{u\ge0\}}$. +\end{theorem} + +Sharpness follows from a product of $m$ independent exponential variables +of mean one: its entropy is $m$ and its covariance is the identity. +Both sides of \eqref{intro:entropy-bound} increase by $\log|\det A|$ +under an invertible affine change $x\mapsto Ax+b$. Thus the theorem +asserts that a log-concave density with identity covariance has entropy +at least $m$. + +Bobkov and Madiman developed the relation between entropy bounds for +log-concave measures and the slicing problem \cite{BobkovMadiman2011}. +Fradelizi and Mar\'in Sola formulated the sharp entropy statement +\eqref{intro:entropy-bound} and proved its equivalence, across dimensions, +with the simplex bound \cite[Conjecture~3(iii) and Proposition~5.2]{FradeliziMarinSola2026}. +In dimension one, Melbourne, Nayar, and Roberto proved the sharp bound, +attained by a one-sided exponential distribution +\cite[Theorem~1.1]{MelbourneNayarRoberto2026}. +Theorem~\ref{intro:entropy} establishes the inequality and identifies +all equality cases in every dimension. +The cone reduction used here follows the preceding geometric and +entropic work; we include it both to derive the dimensional constant +and to transfer the equality classification to convex bodies. + +\subsection{Ideas of the proof} + +We first assume that $f=e^{-V}$ is smooth and that the Hessian of $V$ is +bounded above and below by positive multiples of the identity. +Let $\nabla\phi$ transport standard Gaussian measure to $f$, and put +$J=D^2\phi$. Brenier's theorem and Caffarelli's regularity and +contraction theory supply this positive definite matrix field and its +uniform bounds +\cite{Brenier1991,McCann1995,Caffarelli2000,CorderoErausquinFigalli2019}. +The scalar model is the transport from a Gaussian variable to a +mean-one exponential variable, +\[ + t(a)=-\log\Phi(-a),\qquad q(a)=t'(a), +\] +where $\Phi$ is the standard Gaussian distribution function. Since $q$ +is an increasing bijection onto $(0,\infty)$, spectral calculus defines +the symmetric matrix field $A=q^{-1}(J)$. + +Two features of this parametrization drive the proof. First, an affine +change of the target can be chosen so that $\E A=0$, where expectation +is Gaussian. This nonlinear normalization is proved by a homotopy and +a compactness argument; ordinary covariance normalization does not +give the required identity. Second, differentiating the transport +equation and using convexity of $V$ gives an energy estimate for $A$. +The estimate retains a nonnegative term involving pairs of distinct +eigenvalues of $A$. Discarding that term would lose the control needed +for matrix functions whose derivatives do not commute. + +The zero mean permits an orthogonal decomposition +\[ + A(x)=\sum_{r=1}^m x_rX_r+Y(x), + \qquad X_r=\E[x_rA(x)], +\] +where $Y$ has Gaussian chaos degrees at least two. The first term lies +in the equality space of the Gaussian Poincar\'e inequality and hence +has no energy surplus. We instead use its constant matrices $X_r$ to +choose a supporting plane for the covariance log determinant. +The higher-degree part supplies a strict spectral gap. A covariance +identity involving the resolvent of the Ornstein--Uhlenbeck operator +then expresses the signed deficit +$m+\tfrac12\log\det\Cov(f)-h(f)$ in terms of these two parts. + +The resulting estimate has both noncommuting matrix terms and scalar +divided differences. A commutator square controls the former. +One scalar inequality absorbs the cost of the supporting plane and +the terms without derivatives. A second controls the divided +differences, charging their off-diagonal error to the +eigenvalue-separation term retained earlier. +After this cancellation, the entropy deficit is bounded above by a +quadratic expression that is nonpositive on every chaos of degree at +least two. Retaining strict slack also controls $Y$ and forces +$\sum_rX_r^2$ towards the identity when the entropy gap tends to zero. All numerical constants in the scalar inequalities are +exact rationals. Their verification uses polynomial bounds on a compact +interval and separate analytic estimates on both tails. + +For equality, it is essential that this remainder applies to nearly +sharp densities. The expected extremizers have restricted support and +affine potentials, so equality cannot be studied solely within the +smooth, uniformly convex class. We approximate an arbitrary equality +density while preserving entropy and covariance. The remainder then +forces the reparametrized Jacobians to converge in Gaussian $L^2$ to +a linear field $A(x)=\sum_rx_rX_r$, with $\sum_rX_r^2=I$. +Since $q$ is Lipschitz in Frobenius norm, the original Jacobians +converge to $q(A)$, and Gaussian Poincar\'e gives convergence of the +centered maps. + +The local compatibility of this limiting Jacobian is decisive. +The constant and linear terms of its mixed-derivative identities +force the symmetric matrices $X_r$ to commute. Simultaneous +orthogonal diagonalization then gives independent one-dimensional +exponential transports. Only the nonvanishing of $q'(0)$ and $q''(0)$ +is needed for the commutation argument, isolating a potentially +reusable algebraic step. No uniform Hessian bounds on the smooth +approximants are required. Their affine normalizations are controlled +only after the limiting law and its covariance have been identified. + +Section~\ref{tr:section} constructs the normalization and proves the +transport energy estimate. Section~\ref{mat:section} decomposes the +entropy deficit and proves the smooth case using two scalar +propositions. Section~\ref{sc:section} proves those propositions; +Appendix~\ref{cert:verification} supplies the quantitative one-variable +bounds. Section~\ref{geo:section} removes the smoothness assumptions. +Section~\ref{sec:rigidity} classifies equality in the entropy bound. +Section~\ref{sec:geometry} applies that classification to the exponential +density on the cone over $K$, completing Theorem~\ref{intro:simplex}. diff --git a/preprints/A-sharp-entropy-bound-and-the-simplex-inequality-for-isotropic-constants-October-5-2026/build/manuscript/main.tex b/preprints/A-sharp-entropy-bound-and-the-simplex-inequality-for-isotropic-constants-October-5-2026/build/manuscript/main.tex new file mode 100644 index 0000000000000000000000000000000000000000..208e79bad81ad827bd6da83e44332fd397714940 --- /dev/null +++ b/preprints/A-sharp-entropy-bound-and-the-simplex-inequality-for-isotropic-constants-October-5-2026/build/manuscript/main.tex @@ -0,0 +1,64 @@ +\documentclass[11pt]{article} +\usepackage[T1]{fontenc} +\usepackage{lmodern} +\usepackage[margin=1in]{geometry} +\usepackage{amsmath,amssymb,amsthm,mathtools} +\usepackage{booktabs,array,longtable} +\usepackage{tikz} +\usepackage{microtype} +\usepackage{xcolor} +\usepackage[numbers,sort&compress]{natbib} +\usepackage[colorlinks=true,linkcolor=blue!55!black,citecolor=blue!55!black,urlcolor=blue!55!black]{hyperref} +\hypersetup{pdftitle={A sharp entropy bound and the simplex inequality for isotropic constants},pdfauthor={OpenAI}} +\newcommand{\R}{\mathbb R} +\newcommand{\N}{\mathbb N} +\newcommand{\Dgap}{\delta} +\newcommand{\mathscr}[1]{\mathcal{#1}} +\newcommand{\E}{\mathbb E} +\DeclareMathOperator{\tr}{tr} +\DeclareMathOperator{\Cov}{Cov} +\DeclareMathOperator{\Var}{Var} +\DeclareMathOperator{\Sym}{Sym} +\newcommand{\Id}{I} +\newcommand{\dd}{\,\mathrm d} +\newcommand{\norm}[1]{\lVert #1\rVert} +\newcommand{\abs}[1]{\lvert #1\rvert} +\newcommand{\ip}[2]{\langle #1,#2\rangle} +\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} +\allowdisplaybreaks[1] + +\title{A sharp entropy bound and the simplex inequality\protect\\ for isotropic constants} +\author{OpenAI} +\date{October 5, 2026} + +\begin{document} +\maketitle +\begin{abstract} +We prove the strong isotropic constant conjecture: in each dimension, +simplices are the unique maximizers of the isotropic constant among +convex bodies. We also prove the sharp entropy bound +$h(f)\ge m+\tfrac12\log\det\Cov(f)$ for every log-concave probability +density on $\R^m$, with equality precisely for invertible affine +images of products of one-sided exponential laws. +\end{abstract} + +\input{introduction} +\input{transport} +\input{matrix} +\input{scalar} +\input{approximation} +\input{rigidity} +\input{geometry} +\appendix +\input{certificate} +\bibliographystyle{plainnat} +\bibliography{references} +\end{document} diff --git a/preprints/A-sharp-entropy-bound-and-the-simplex-inequality-for-isotropic-constants-October-5-2026/build/manuscript/matrix.tex b/preprints/A-sharp-entropy-bound-and-the-simplex-inequality-for-isotropic-constants-October-5-2026/build/manuscript/matrix.tex new file mode 100644 index 0000000000000000000000000000000000000000..8fcbac092ff2151d68dbdcad17f2e7c7bcb55119 --- /dev/null +++ b/preprints/A-sharp-entropy-bound-and-the-simplex-inequality-for-isotropic-constants-October-5-2026/build/manuscript/matrix.tex @@ -0,0 +1,585 @@ +\section{Gaussian decomposition of the entropy deficit} +\label{mat:section} + +The transport estimate of Proposition~\ref{tr:surplus} controls the +Gaussian energy of the reparametrized Jacobian. Its degree-one part, +however, has no Poincar\'e surplus. We use that part to choose a matrix +supporting plane for the log determinant. The remaining Gaussian +chaoses then supply the coercivity needed to prove the entropy bound. +We retain a strict remainder that will also identify the equality cases. +Throughout this section, $\E$ denotes integration against the standard +Gaussian measure $\gamma_m$. + +Define a scalar weight on $[0,\infty)$ by +\begin{equation}\label{mat:rigidity-weight} + \omega(\lambda)= + \frac{3(\lambda-1)^2}{37+3\lambda} + \left(\frac3{20}+\frac{49}{100}\frac{37}{37+3\lambda}\right) + \min\left\{\frac{13}{100},\frac{32}{25\sqrt\lambda}\right\}, +\end{equation} +where the minimum is $13/100$ at $\lambda=0$. +This weight is continuous and nonnegative, vanishes only at $1$, and +satisfies $\omega(\lambda)/\sqrt\lambda\to24/125$ as +$\lambda\to\infty$. + +\begin{theorem}[The smooth entropy bound with a remainder] +\label{mat:entropy-smooth}\label{prop:slack} +Let $m\ge1$ be an integer and let $\mu=e^{-V}\,\dd y$ be a +probability measure on $\R^m$, where +$V\in C^\infty(\R^m)$ and +$c_-\Id\le D^2V\le c_+\Id$ for constants $00$, +\begin{equation}\label{mat:dual-bound} + \tr(SW)\le\|u\|_S^2+\|v\|_S^2. +\end{equation} +\end{lemma} + +\begin{proof} +Put $F=\nabla\phi$. For two degree-$d$ components of $F$, Gaussian +integration by parts gives +\[ + \sum_\ell\E[(\partial_\ell F_{i,d})(\partial_\ell F_{j,d})] + =d\,\E[F_{i,d}F_{j,d}]. +\] +Differentiation lowers the degree by one, and $R$ therefore cancels the +factor $d$. Summing over $d\ge1$ proves +$C_{ij}=\sum_\ell\E[J_{i\ell}R J_{j\ell}]$. Symmetry of $J$ gives +the first matrix identity in \eqref{mat:covariance-completion}. +Expanding the row Gram matrix of $q-(\Id+N)k$ gives cross terms +$-2\E(qk)=-2\E t$. The remaining term is +\[ + \E\bigl[k(\Id+N)k\bigr] + =\E\left(k^2+\sum_r K_r^2\right), +\] +by entrywise integration by parts. This proves the second identity. +The calculation is valid by the $H^1$--$H^{-1}$ pairing, even if $Nk$ +is not square-integrable. + +For a test field $F$, the pairing with $u-\partial^*v$ is +$\langle u,F\rangle-\langle v,\partial F\rangle$. Cauchy--Schwarz +bounds it by +$(\|u\|^2+\|v\|^2)^{1/2}\|F\|_{H^1}$. Taking the dual norm proves +\eqref{mat:dual-bound} for $S=\Id$, and applying this argument to +$S^{1/2}U$ proves the general case. +\end{proof} + +Separate the degree-one part of $A$ by writing +\begin{equation}\label{mat:chaos} + A=\sum_{r=1}^m x_rX_r+Y,\qquad + X_r=\E(x_rA),\qquad B=\sum_rX_r^2. +\end{equation} +The matrices $X_r$ are constant and symmetric; $Y$ contains only chaoses +of degree at least two. We shall use +\begin{equation}\label{mat:gradient-fields} + Z_r=\partial_rA=X_r+G_r,\qquad + G_r=\partial_rY,\qquad + g_r=\partial_rN^{-1}Y, + \qquad V_0=\|G\|^2-\|Y\|^2. +\end{equation} +Thus $\partial^*g=Y$. All three derivative fields are symmetric. + +For comparison, the coordinatewise Gaussian-to-exponential transport +satisfies +\[ + A(x)=\operatorname{diag}(x_1,\ldots,x_m),\qquad B=\Id. +\] +We therefore choose the supporting plane for $\log\det C$ as a +regularized inverse of $B$ centered at this value. Fix +\begin{equation}\label{mat:dual-constants} + \alpha=\frac{3}{40},\qquad s=\frac1{1-\alpha}, +\end{equation} +and set +\begin{equation}\label{mat:dual-matrices} + S=(\Id/s+\alpha B)^{-1},\qquad T=S-\Id, + \qquad H=-T(B-\Id). +\end{equation} +These constant matrices commute with one another, though generally not +with functions of $A$. Directly from their definition, +\begin{equation}\label{mat:dual-relations} + 00$ for every +real $a$. To reserve part of the negative $H$ term for the cost of +$\log\det S$, fix +\begin{equation}\label{mat:completion-constants} + \epsilon=\frac3{20},\qquad\kappa=\frac{49}{100}, +\end{equation} +and put +\[ + \rho=\epsilon\Id+\kappa S/s. +\] +Then $\Id-\rho>0$. A weighted completion of the square gives +\begin{align} + \tau\left[T\left(Yh_0+\sum_rX_rM_r\right)\right] + &\le\tau\bigl(H(\Id-\rho)p\bigr) + +\frac{\|M p^{-1/2}\|^2}{4(1-\epsilon)} + +\frac{\alpha s\|Yh_0p^{-1/2}\|^2} + {4(1-\epsilon-\kappa)}. + \label{mat:weighted-completion} +\end{align} +To verify that the order of the weights is valid, write +$Q_0=H(\Id-\rho)$ and $F=Yh_0+\sum_rX_rM_r$. +Apply $\langle a,b\rangle\le\|a\|^2+\frac14\|b\|^2$ to +$Q_0^{1/2}p^{1/2}$ and $Q_0^{-1/2}TFp^{-1/2}$. +The resulting second square has left weight bounded above by +\[ + T^2Q_0^{-1}\le\alpha S(\Id-\rho)^{-1}, +\] +with equality on the range of $H$. Here inverses are interpreted as +pseudoinverses on the common nullspace of $Q_0$ and $T$. +Since +\[ + (\Id-\rho)S^{-1} + =\frac{1-\epsilon-\kappa}{s}\Id + +(1-\epsilon)\alpha XX^t, +\] +the same block argument as in \eqref{mat:block-bound} proves +\eqref{mat:weighted-completion}. Thus this step does not require $p$ +to commute with $S$. + +\subsection{Two scalar estimates and the final chaos bound} + +The following estimate combines the cost $\tr(T-\log S)$ with the +negative $H$ term left by \eqref{mat:weighted-completion} and the +terms quadratic in $Y$. A second estimate will control the derivative +residuals, including their dependence on $Z$, by $\mathcal C$ and the +higher-chaos energy. The proof of the first estimate is given in +Section~\ref{sc:absorption-proof}. Its strict remainder controls the +eigenvalues of $B$ through the weight \eqref{mat:rigidity-weight}. + +\begin{proposition}[Absorption with a first-chaos remainder] +\label{mat:scalar-absorption} +Let $A\in L^2(\gamma_m;\operatorname{Sym}_m)$ satisfy $\E A=0$, and +define $X_r,Y,B$ by \eqref{mat:chaos} and $S,T,H$ by +\eqref{mat:dual-matrices}. With the constants +\eqref{mat:dual-constants}, \eqref{mat:constants}, and +\eqref{mat:completion-constants}, the functions \eqref{mat:scalar-functions}, and +$\rho=\epsilon\Id+\kappa S/s$, one has +\begin{equation}\label{mat:absorption-inequality} + \begin{split} + &\tr(T-\log S)-\tau(H\rho p) + +s\tau\left[Y^2\left((b-\beta)^2+ + \frac{\alpha h_0^2}{4(1-\epsilon-\kappa)p}\right)\right]\\ + &\hspace{15mm}\le\frac{69}{500}\|Y\|^2 + -\frac1{1000}\sum_{i=1}^m\omega(\lambda_i), + \end{split} +\end{equation} +where $\lambda_1,\ldots,\lambda_m$ are the eigenvalues of $B$. +All scalar factors in the expected traces are evaluated at $A$. +\end{proposition} + +The remaining derivative terms are compared entrywise in an eigenbasis +of $A$. This comparison retains a multiple of the eigenvalue-separation +surplus $\mathcal C$ rather than discarding it. + +\begin{proposition}[Two-eigenvalue estimate]\label{mat:scalar-pair} +Let $a_i,a_j\in\R$ and $G,g,Z\in\R$. Set +\[ + \delta=\bigl|\log(q(a_i)/q(a_j))\bigr|,\qquad + r_{ij}=\frac{1+m_0(\min(a_i,a_j))}{16} + \left(1+\frac{\delta^2}{30}+\frac{\delta^4}{1680}\right), +\] +where $m_0$ is the six-interval lower bound in \eqref{tr:bins}. Define the endpoint residuals +\[ + \begin{array}{lll} + D_j=Gb_j+Z\Delta k_{ij},&L_j=Gl_j+Z\Delta b_{ij},& + M_j=Ge_j+Z\psi_{ij}+cgb_j,\\ + D_i=Gb_i+Z\Delta k_{ji},&L_i=Gl_i+Z\Delta b_{ji},& + M_i=Ge_i+Z\psi_{ji}+cgb_i. + \end{array} +\] +Let +$Q(u,v)=\frac{43}{100}u^2+\frac{26}{100}uv+\frac{68}{25}v^2$. +Then +\begin{align} + &\operatorname{avg}_{v=i,j}\left[ + s(D_v-\beta g)^2+\left(1+\frac s8\right)(D_v-\gamma g)^2 + +\gamma^2(s-1)g^2+\frac{L_v^2}{\alpha} + +\frac{M_v^2}{4(1-\epsilon)p_v}\right]\notag\\ + &\hspace{15mm}\le Q(G,g-G/2)+r_{ij}\delta^2Z^2. + \label{mat:pair-inequality} +\end{align} +Here $\operatorname{avg}_{v=i,j}$ is the arithmetic mean of the two +endpoint expressions, counting both also when $i=j$. Moreover +$p_v=p(a_v)>0$, and all divided differences extend continuously to +$a_i=a_j$. +\end{proposition} + +The proof of Proposition~\ref{mat:scalar-pair} occupies +Section~\ref{sc:pair-proof}. We now show that these two scalar estimates +close the matrix argument. + +\begin{proof}[Proof of Theorem~\ref{mat:entropy-smooth}] +Apply Lemma~\ref{mat:expansion} in +\eqref{mat:deficit-reduction}. Bound its last line by +Lemma~\ref{mat:commutator}, its first square by +\eqref{mat:block-bound}, and its linear $T$ term by +\eqref{mat:weighted-completion}. The $H$ terms combine to +$-\tau(H\rho p)$. Since $Y=Y^t$ and $h_0,p,b$ are commuting functions +of $A$, the two terms quadratic in $Y$ have exactly the form appearing +in \eqref{mat:absorption-inequality}, by cyclicity of the trace; +no commutation with $Y$ is required. Set +$\Omega_B=\sum_{i=1}^m\omega(\lambda_i)$. Consequently, +\begin{align} + \mathscr D\le {}&-V_0-\mathcal C+\frac{69}{500}\|Y\|^2 + -\frac{\Omega_B}{1000} + +\|D-\beta g\|_S^2 + +\left(1+\frac s8\right)\|D-\gamma g\|^2 + +\frac1\alpha\|L\|^2\nonumber\\ + &\hspace{22mm}+\gamma^2\tau\left(T\sum_rg_r^2\right) + +\frac{\|Mp^{-1/2}\|^2}{4(1-\epsilon)}. + \label{mat:pre-pair} +\end{align} +Use $S\le s\Id$ and $T\le(s-1)\Id$. In an eigenbasis of $A$, the +entries of $G_r,g_r,Z_r$ are symmetric in $i,j$. Pairing the entries +$(i,j)$ and $(j,i)$ therefore gives precisely the endpoint average in +\eqref{mat:pair-inequality}; right multiplication by $p^{-1/2}$ gives +the denominator $p_j$ for entry $(i,j)$. Proposition~\ref{mat:scalar-pair} +now bounds the last five terms of \eqref{mat:pre-pair} by +\[ + \E\sum_{i,j,r}Q\bigl((G_r)_{ij},(g_r)_{ij}-(G_r)_{ij}/2\bigr) + +\mathcal C. +\] +The separation surplus is the same $\mathcal C$ as in +Proposition~\ref{tr:surplus}: its coefficient at $(i,j)$ is independent +of the derivative index $r$, so summing over $r$ is unchanged by rotating +that index. All changes of eigenbasis are pointwise; no eigenvectors are +differentiated. The separation surplus cancels. +Because $Q$ has constant coefficients, +its summed quadratic form is invariant under orthonormal changes of +basis. We may therefore return to fixed coordinates and apply the +Gaussian chaos decomposition. + +For a degree-$d$ component of $Y$, where $d\ge2$, the associated +components of $g$ and $G$ satisfy $g=G/d$, and +$\|G\|^2=d\|Y\|^2$. Different degrees remain orthogonal after +differentiation. The required bound follows, degree by degree, from +\begin{equation}\label{mat:final-polynomial} + Q(1,u-1/2)+\frac7{50}u\le1-u, + \qquad 0\le u\le\frac12. +\end{equation} +Indeed, the right side minus the left side factors as +$\frac1{50}(1-2u)(1+68u)\ge0$. Set $u=1/d$, multiply by +$\|G\|^2$, and sum over the chaoses; the sums converge because +$Y\in H^1(\gamma_m)$. The summed $Q$ term is at most +$V_0-\frac7{50}\|Y\|^2$. Substitution into +\eqref{mat:pre-pair} therefore gives +\[ + \mathscr D\le-\frac1{500}\|Y\|^2-\frac{\Omega_B}{1000}, +\] +which is \eqref{mat:strict-remainder}. +\end{proof} diff --git a/preprints/A-sharp-entropy-bound-and-the-simplex-inequality-for-isotropic-constants-October-5-2026/build/manuscript/references.bib b/preprints/A-sharp-entropy-bound-and-the-simplex-inequality-for-isotropic-constants-October-5-2026/build/manuscript/references.bib new file mode 100644 index 0000000000000000000000000000000000000000..6f8513597d97eaf4f84820a18b86c43af14a30cc --- /dev/null +++ b/preprints/A-sharp-entropy-bound-and-the-simplex-inequality-for-isotropic-constants-October-5-2026/build/manuscript/references.bib @@ -0,0 +1,243 @@ +@article{Bourgain1986, + author = {Bourgain, Jean}, + title = {On high dimensional maximal functions associated to convex bodies}, + journal = {American Journal of Mathematics}, + volume = {108}, number = {6}, pages = {1467--1476}, year = {1986}, + doi = 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Theorem~\ref{mat:entropy-smooth} controls the nonlinear part of the normalized +matrix field and the sum of squares of its linear coefficients. We now +show that this control determines every equality case. The structural +step is local: the first two nonconstant terms of $q(A(x))$ force the +coefficients of a linear symmetric matrix field $A$ to commute whenever +$q(A)$ is a Jacobian. + +\subsection{Compatibility of a nonlinear Jacobian} + +\begin{lemma}[Rigidity of a linear matrix field]\label{lem:linear-jacobian} +Let $m\ge1$, and let $X_1,\ldots,X_m$ be real symmetric $m\times m$ matrices satisfying +\[ + \sum_{r=1}^m X_r^2=I, + \qquad A(x)=\sum_{r=1}^m x_rX_r. +\] +Suppose $F\in W^{1,2}_{\mathrm{loc}}(\R^m;\R^m)$ has weak Jacobian +$DF=q(A)$ almost everywhere. Then there are an orthogonal matrix $U$, +signs $\sigma_1,\ldots,\sigma_m\in\{-1,1\}$, and $w\in\R^m$ such that +\begin{equation}\label{eq:rigid-map} + U^{\mathsf T}F(Uy) + =\bigl(\sigma_i t(\sigma_i y_i)\bigr)_{i=1}^m+w + \qquad\text{for almost every }y\in\R^m. +\end{equation} +In particular, $F_\#\gamma_m$ is an affine image of the product of $m$ +mean-one exponential laws, and its covariance matrix is $I$. +\end{lemma} + +\begin{proof} +Write $e_1,\ldots,e_m$ for the standard basis and $J=q(A)$. +Commutation of distributional derivatives of $F$ gives +\begin{equation}\label{eq:jacobian-compatibility} + (\partial_rJ)e_s=(\partial_sJ)e_r + \qquad(1\le r,s\le m). +\end{equation} +The scalar function $q$ is real analytic near zero. Since $A(0)=0$, its +convergent power series, interpreted by matrix multiplication, gives +\[ + J(x)=q(0)I+q'(0)A(x)+\tfrac12q''(0)A(x)^2+O(|x|^3) +\] +near zero; the differentiated remainder is $O(|x|^2)$. +The identity $q'(a)=q(a)(q(a)-a)$ yields +\[ + q(0)=\sqrt{\frac2\pi},\qquad + q'(0)=\frac2\pi>0,\qquad + q''(0)=\sqrt{\frac2\pi}\left(\frac4\pi-1\right)>0. +\] +Thus \eqref{eq:jacobian-compatibility}, which is a pointwise identity +between analytic functions near zero, has constant term +\begin{equation}\label{eq:coefficient-symmetry} + X_re_s=X_se_r. +\end{equation} +Equivalently, $A(x)z=A(z)x$ for all $x,z\in\R^m$; in particular, +$A(x)e_s=X_sx$. Its degree-one term is +\[ + (X_rA(x)+A(x)X_r)e_s + =(X_sA(x)+A(x)X_s)e_r. +\] +The terms with leftmost factor $A(x)$ cancel by +\eqref{eq:coefficient-symmetry}. The remaining equality is +$(X_rX_s-X_sX_r)x=0$ near zero, and hence +\begin{equation}\label{eq:coefficients-commute} + X_rX_s=X_sX_r\qquad(1\le r,s\le m). +\end{equation} + +Commuting real symmetric matrices admit a common orthonormal eigenbasis. +Choose the orthogonal matrix $U$ with those basis vectors as columns, +and rotate both the input and the output. Then +\[ + \widetilde A(y)=U^{\mathsf T}A(Uy)U + =\sum_{i=1}^m y_i\widetilde X_i, + \qquad + \widetilde X_i=U^{\mathsf T} + \left(\sum_{r=1}^m U_{ri}X_r\right)U +\] +has diagonal coefficients. Orthogonality of $U$ gives +$\sum_i\widetilde X_i^2=I$, and the identity $A(x)z=A(z)x$ gives +$\widetilde X_i e_j=\widetilde X_j e_i$. If $i\ne j$, these two vectors +are multiples of different basis vectors, so both are zero. +Consequently, only the $i$th diagonal entry of $\widetilde X_i$ can be +nonzero. The sum-of-squares identity makes that entry a sign: +\[ + \widetilde X_i=\sigma_i e_ie_i^{\mathsf T}, + \qquad + \widetilde A(y)=\operatorname{diag}(\sigma_1y_1,\ldots,\sigma_my_m). +\] +It follows that +\[ + D_y\bigl(U^{\mathsf T}F(Uy)\bigr) + =\operatorname{diag}\bigl(q(\sigma_1y_1),\ldots,q(\sigma_my_m)\bigr). +\] +Since $t'=q$ and $\sigma_i^2=1$, the right side is also the Jacobian of +$y\mapsto(\sigma_i t(\sigma_i y_i))_{i=1}^m$. +Their difference has zero weak gradient on the connected set $\R^m$ +and is therefore almost everywhere constant. This proves +\eqref{eq:rigid-map}. + +For independent standard Gaussian coordinates $G_i$, the variables +$t(\sigma_iG_i)$ are independent mean-one exponentials. Their variances +are one, and the signs and the orthogonal matrix preserve covariance +$I$. This proves the final assertion. +\end{proof} + +\subsection{Compactness and the proof of the equality classification} + +We first record the functional-calculus estimate that turns convergence +of the normalized fields into convergence of transport Jacobians. The +Frobenius norm is essential to this elementary argument. + +\begin{lemma}\label{lem:spectral-lipschitz} +For real symmetric matrices $M,N\in\R^{m\times m}$, +\begin{equation}\label{eq:spectral-lipschitz} + |q(M)-q(N)|_{\mathrm F}\le |M-N|_{\mathrm F}. +\end{equation} +\end{lemma} + +\begin{proof} +Let $(u_i)$ and $(v_k)$ be orthonormal eigenbases of $M$ and $N$, with +eigenvalues $\lambda_i$ and $\mu_k$. Then +\[ + u_i^{\mathsf T}(q(M)-q(N))v_k + =(q(\lambda_i)-q(\mu_k))\,u_i^{\mathsf T}v_k, +\] +whereas the corresponding entry of $M-N$ is +$(\lambda_i-\mu_k)u_i^{\mathsf T}v_k$. The scalar bound $00$, it consequently has a positive infimum on +$\{\lambda\ge0:|\lambda-1|\ge\varepsilon\}$. Thus every eigenvalue +of $B_j$ tends to $1$, and $B_j\to I$ in operator norm. + +Since $\sum_r|X_{j,r}|_{\mathrm F}^2=\tr B_j$, the coefficients form +a bounded sequence in a finite-dimensional space. Pass to a subsequence +such that $X_{j,r}\to X_r$ for every $r$. Each $X_r$ is symmetric, +and Gaussian orthogonality together with \eqref{eq:vanishing-slack} +gives +\begin{equation}\label{eq:linear-field-limit} + \sum_{r=1}^mX_r^2=I,\qquad + A_j\longrightarrow A(x):=\sum_{r=1}^m x_rX_r + \quad\text{in }L^2(\gamma_m). +\end{equation} +Lemma~\ref{lem:spectral-lipschitz} now gives +\begin{equation}\label{eq:jacobian-limit} + J_j=q(A_j)\longrightarrow J:=q(A) + \quad\text{in }L^2(\gamma_m). +\end{equation} + +To obtain a limiting map, remove the irrelevant translations: +$\widehat F_j=F_j-\E F_j$. Each $\widehat F_j$ lies in +$H^1(\gamma_m;\R^m)$: its law has finite second moments and, for this +fixed $j$, its Jacobian has the bounds \eqref{tr:Jacobian-bounds} +applied to the transformed target. Gaussian Poincar\'e, applied +componentwise to the centered difference, gives +\begin{equation}\label{eq:map-cauchy} + \norm{\widehat F_j-\widehat F_k}_{L^2(\gamma_m)}^2 + \le \norm{J_j-J_k}_{L^2(\gamma_m)}^2. +\end{equation} +Consequently $\widehat F_j$ converges strongly in Gaussian $H^1$ to a +centered map $\widehat F$ with weak Jacobian $D\widehat F=J$. +Indeed, \eqref{eq:map-cauchy} and \eqref{eq:jacobian-limit} give +convergence of both maps and first derivatives; passage to weak +derivatives is also valid on every bounded set, where the Gaussian +density has a positive lower bound. In particular, +$\widehat F\in W^{1,2}_{\mathrm{loc}}(\R^m;\R^m)$. +Lemma~\ref{lem:linear-jacobian} applies to \eqref{eq:linear-field-limit} +and shows that $\widehat F_\#\gamma_m$ is an affine exponential product +with covariance $I$. + +It remains to return from the normalized laws to $f$. +By Cauchy--Schwarz, the strong $L^2(\gamma_m)$ convergence of the +centered maps gives +\begin{equation}\label{eq:normalized-covariance-limit} + D_j:=\Cov((\widehat F_j)_\#\gamma_m) + =P_jC_jP_j^{\mathsf T}\longrightarrow I. +\end{equation} +We can now bound the normalizing matrices uniformly, without any +uniform Hessian bounds on the approximating potentials. +Choose positive constants $c,C_0,d,D$ such that, for all sufficiently +large $j$, +\[ + cI\le C_j\le C_0I,\qquad dI\le D_j\le DI. +\] +Congruence by $P_j$ then yields +\begin{equation}\label{eq:normalization-compactness} + \frac d{C_0}I\le P_jP_j^{\mathsf T}\le\frac Dc I. +\end{equation} +Hence $P_j$ and $P_j^{-1}$ are bounded. Pass to a further subsequence +with $P_j\to P$; the lower bound in +\eqref{eq:normalization-compactness} makes $P$ invertible. + +The law of $\widehat F_j$ is the image of $f_j(x)\,\dd x$ under +$x\mapsto P_j(x-a_j)$. For a bounded continuous function $\psi$, +\begin{align*} + &\left|\int\psi(P_j(x-a_j))f_j(x)\,\dd x + -\int\psi(P(x-a))f(x)\,\dd x\right|\\ + &\quad\le\norm{\psi}_{\infty}\norm{f_j-f}_{L^1} + +\int\left|\psi(P_j(x-a_j))-\psi(P(x-a))\right|f(x)\,\dd x + \longrightarrow0. +\end{align*} +The first term tends to zero by \eqref{eq:equality-approximation}; +the second does so by dominated convergence. Strong convergence of +$\widehat F_j$ also identifies the weak limit of these laws as +$\widehat F_\#\gamma_m$. It follows that the image of $f(x)\,\dd x$ +under $x\mapsto P(x-a)$ is the affine exponential product already +determined above. Since $P$ is invertible, $f$ itself has the form +asserted in Theorem~\ref{intro:entropy}, up to equality almost everywhere. + +Conversely, the density of a product of $m$ independent mean-one +exponentials is log-concave, has entropy $m$, and has covariance $I$. +Every invertible affine image remains log-concave and has the same +entropy gap by affine invariance. Thus every density in the stated +class satisfies $\Dgap=0$. +\end{proof} diff --git a/preprints/A-sharp-entropy-bound-and-the-simplex-inequality-for-isotropic-constants-October-5-2026/build/manuscript/scalar.tex b/preprints/A-sharp-entropy-bound-and-the-simplex-inequality-for-isotropic-constants-October-5-2026/build/manuscript/scalar.tex new file mode 100644 index 0000000000000000000000000000000000000000..f9d3954eba740a9325aa445bd863066d8f92cccc --- /dev/null +++ b/preprints/A-sharp-entropy-bound-and-the-simplex-inequality-for-isotropic-constants-October-5-2026/build/manuscript/scalar.tex @@ -0,0 +1,486 @@ +\section{The two scalar estimates}\label{sc:scalar}\label{sc:section} + +The matrix argument has reduced the entropy inequality to two estimates. +Proposition~\ref{mat:scalar-absorption} controls the part depending on the +first Gaussian chaos; Proposition~\ref{mat:scalar-pair} controls the remaining +quadratic form, one pair of eigenvalues at a time. We prove them here. +The functions $q,d,k,b,l$ and the six intervals +$O,P_1,R_1,U_1,V_1,T_1$ are those of the transport section, and the constants +$\alpha,s,\beta,c,\gamma,\epsilon,\kappa$ and functions $p,e,h_0$ are those +of the matrix section. All finite decimals below denote exact rational +numbers. + +We first state the one-variable bounds used in both proofs. Their verification +is given in Appendix~\ref{cert:verification}; in particular, none of the +bounds in this section is inferred from numerical sampling. For the +quadratic form in Proposition~\ref{mat:scalar-pair}, set +\begin{equation}\label{sc:auxiliary-functions} + \begin{gathered} + D_*=\frac{1}{4(1-\epsilon)},\qquad + \mathsf D=\frac{D_*}{p},\quad \mathsf L=\frac{l}{\alpha},\quad + \mathsf E=e+\gamma b + =p+b(.75-b)-\frac{l^2}{\alpha},\\ + K_0=\mathsf D\mathsf E,\qquad J_0=\mathsf D cb,\\ + A_*=s+1+s/8,\quad + \eta_*=s\beta+(1+s/8)\gamma,\quad + \nu_*=s\beta^2+(9s/8)\gamma^2. + \end{gathered} +\end{equation} +The symbols $\mathsf D,\mathsf L,\mathsf E$ denote scalar functions, to +be distinguished from the matrix residuals $D_r,L_r,M_r$ of the preceding +section. + +\begin{lemma}[One-variable bounds]\label{sc:bounds} +For every $a\in\R$, +\begin{equation}\label{sc:p-h-bounds} + \begin{gathered} + p(a)>0,\qquad |h_0(a)|\le .67\sqrt{p(a)},\\ + p(a)\ge\begin{cases}(7+a^2)^{-1},&a\le0,\\ .25,&a\ge0,\end{cases} + \qquad p(a)\ge .46\quad(a\ge4). + \end{gathered} +\end{equation} +The following interval bounds hold: +\begin{equation}\label{sc:interval-table} +\begin{array}{c|cccccc} + &b&\mathsf L\le&\mathsf D\le&K_0&J_0\le&\ell_1\\\hline + O &[0,.09]&.60&.295(7+a^2)&[.37,.56]&.72&.13\\ + P_1 &[.083,.15]&1.01&4.3&[.29,.46]&.72&.13\\ + R_1 &[.14,.23]&1.19&3.00&[.29,.371]&.68&.13\\ + U_1 &[.22,.342]&1.19&1.72&[.29,.398]&.61&.123\\ + V_1 &[.337,.45]&.83&.87&[.31,.4]&.50&.060\\ + T_1 &[.445,.5]&.20&.640&[.32,.4]&.48&.022 +\end{array} +\end{equation} +Here $\ell_1$ is the positive number in the last column, and +\begin{equation}\label{sc:local-surplus} + Q- + \begin{pmatrix} + A_*b^2-\eta_*b+\nu_*/4+l^2/\alpha&\nu_*/2-\eta_*b\\ + \nu_*/2-\eta_*b&\nu_* + \end{pmatrix} + -\mathsf D + \begin{pmatrix}\mathsf E\\cb\end{pmatrix} + \begin{pmatrix}\mathsf E&cb\end{pmatrix} + \succeq\begin{pmatrix}\ell_1&0\\0&.75\end{pmatrix}. +\end{equation} +Here $Q=\left(\begin{smallmatrix}.43&.13\\.13&2.72\end{smallmatrix}\right)$, +as in Proposition~\ref{mat:scalar-pair}. +Finally, writing +\begin{equation}\label{sc:derivative-functions} + z=\frac{l'}{l},\qquad h_*=1+3b+kz,\qquad \chi=\frac{l}{d}, +\end{equation} +we have +\begin{equation}\label{sc:derivative-table} +\begin{array}{c|ccc} + &z&h_*&\chi\le\\\hline + a\le-1&[-.08,.70]&[.8,1.77]&.070\\ + a\ge-1&[-.475,0]&[.8,1.77]&.097 +\end{array} +\end{equation} +At an endpoint shared by two intervals, either applicable row may be used. +\end{lemma} + +The first line of the lemma supplies a lower bound for the mean of $p$ from +only a second moment. The matrix surplus in~\eqref{sc:local-surplus} +then provides room to absorb the secant errors in the second proposition. +We treat these two uses separately. + +\subsection{Absorption using a second moment}\label{sc:absorption-proof} + +\begin{proof}[Proof of Proposition~\ref{mat:scalar-absorption}] +By $0.689$. +We also have +\begin{equation}\label{sc:rho-loss} + H\rho d_u\le .027. +\end{equation} +To check this, note that $\rho\le.64$ and that $H\le\lambda$ for $\lambda\ge1$, +whereas $H\le\alpha s$ for $\lambda\le1$. In the former case, +\[ + H\rho d_u\le + \frac{.64\cdot1.28^2\cdot.13}{16(.46-.13)}<.02582; +\] +in the latter, $\alpha s\cdot.64\cdot .13^3/[16(.46-.13)]<.027$. + +For completeness, we next verify the elementary logarithmic bound needed +to compare $T-\log S$ with the positive term in~\eqref{sc:p-mean}. +Set +\[ + F_0(S)=\frac{S(S-1-\log S)}{(1-S)^2},\qquad F_0(1)=\tfrac12. +\] +For $00). +\end{equation} +Here is a direct verification, including the point $S=1$ by continuity. +For $x>0$, let +\[ + J(x)=\frac{(.478+1.4x^2)(1-x^2)^2}{x} + -(x^2-1-2\log x). +\] +Then +\[ + x^2J'(x)=P(x):=-.478+2x+.444x^2-2x^3-6.966x^4+7x^6. +\] +Descartes' rule of signs gives at most three positive roots of $P$, +counting multiplicity. The signs at $.24,.26,.6$, together with +$P(1)=0$ and $P'(1)>0$, give exactly three: one in $(.24,.26)$, one in +$(.26,.6)$, and $1$. Thus the only local minima of $J$ occur at the +first root and at $1$. We have $J(1)=0$ and $J(.25)>.14$; the latter +follows already from $\log2<.694$. On $[.24,.26]$, +$|P(.25)|<.01$ and $|P'|<4$, hence $|J'|<2$. +The first minimum is therefore positive. Also $J(x)\to+\infty$ at +both ends of $(0,\infty)$. This proves $J\ge0$, which is equivalent to +\eqref{sc:F0-sqrt} away from $x=1$. + +Using $\alpha\lambda=1/S-1/s$, inequality~\eqref{sc:F0-sqrt} gives +\[ + \frac{\alpha F_0(S)\sqrt\lambda}{\rho} + \le \sqrt\alpha\sqrt{1-S/s}\, + \frac{.478+1.4S}{.15+.45325S} + \le \frac{\sqrt{.075}\cdot .478}{.15}<.88. +\] +The last ratio decreases in $S$ because +$1.4\cdot.15<.478\cdot.45325$. Finally, +$.089<.688\cdot.13$ and $.88<.688\cdot1.28$ prove the last inequality +in~\eqref{sc:log-bound}. + +For $\lambda\ne1$ we have +\[ + \frac{T-\log S}{H\rho}=\frac{\alpha F_0(S)}{\rho}. +\] +Combining~\eqref{sc:p-mean}--\eqref{sc:log-bound}, and treating $\lambda=1$ +by continuity, gives +\[ + T-\log S-H\rho\int p\,d\nu_v + \le -.001H\rho u+.027r. +\] +The definitions give $H\rho u=\omega(\lambda)$. Sum over an orthonormal +eigenbasis of the matrix $B$, and add~\eqref{sc:y-cost}. +This summation evaluates $\tau(H\rho p(A))$ without requiring $p(A)$ +to commute with $B$. Likewise, \eqref{sc:y-cost} may be integrated +against the positive semidefinite matrix $Y^2$ without a commutation +assumption. Since $\sum_v r=\norm{Y}^2$, the result is +\[ + (.027+.111)\norm{Y}^2-.001\sum_i\omega(\lambda_i) + =\frac{69}{500}\norm{Y}^2-\frac1{1000}\sum_i\omega(\lambda_i), +\] +as asserted. +\end{proof} + +\subsection{A quadratic estimate for two endpoints}\label{sc:pair-proof} + +\begin{proof}[Proof of Proposition~\ref{mat:scalar-pair}] +Fix two endpoints $a_-\le a_+$ and write +\[ + \delta=\log q(a_+)-\log q(a_-),\qquad m_*=m_0(a_-). +\] +When the endpoints coincide, all secant increments vanish, and +\eqref{sc:local-surplus} immediately proves the assertion. +We henceforth suppose $\delta>0$. + +We first isolate the cost of the secant increments at one endpoint $a_j$. +Use coordinates $(G,h)$ with $h=g-G/2$. With $Z=0$, the left side of +the endpoint quadratic form in Proposition~\ref{mat:scalar-pair} is +exactly the two matrices subtracted from $Q$ in~\eqref{sc:local-surplus}. +For an increment vector $(x,y,w)$, define +\begin{align} + \mathcal N_j(x,y,w)^2={}&A_*x^2+y^2/\alpha+\mathsf D_jw^2\notag\\ + &+\frac{((A_*b_j-\eta_*/2)x+\mathsf L_jy+K_{0j}w)^2}{\ell_{1j}} + +\frac{(-\eta_*x+J_{0j}w)^2}{.75}. + \label{sc:increment-norm} +\end{align} +Completing squares against the surplus +$\ell_{1j}G^2+.75h^2$ in~\eqref{sc:local-surplus} shows that the full +endpoint form is bounded above by +\begin{equation}\label{sc:one-endpoint} + Q(G,h)+Z^2\mathcal N_j(\Delta k,\Delta b,\psi_j)^2. +\end{equation} +Indeed, the two coefficients of $2GZ$ and $2hZ$ are precisely the +numerators of the two squared terms in~\eqref{sc:increment-norm}. +The first line of that formula is the coefficient of $Z^2$ before +completion of squares. + +The next step estimates these increment norms by integrating derivatives +between the endpoints. Recall that a secant slope is the uniform average +of the derivative over $[a_-,a_+]$. With $\xi=\log q(a)$, differentiation +therefore gives the vector +\begin{equation}\label{sc:derivative-vector} + v_j(a)=\chi(a)(1,z(a),W_j(a)),\qquad + W_j(a)=h_*(a)-2b_j-2\mathsf L_jz(a). +\end{equation} +More explicitly, $(\Delta k,\Delta b,\psi_j)$ is the uniform average in +$a$ of $\int_{\log q(a_j)}^{\log q(a)}v_j(q^{-1}(e^\xi))\,d\xi$. +This follows from $b'=l$, $l'=zl$, $d(\log q)/da=d$, and +$f''=l(1+3b+kz)=lh_*$. +Put +\[ + F_j(a)=\mathcal N_j(v_j(a))^2. +\] + +We record the bounds furnished by Lemma~\ref{sc:bounds}. If the endpoint +$a_j$ belongs to $R_1,U_1,V_1,T_1$, respectively, then +\begin{equation}\label{sc:F-simple} + F_j(a)\le .295,\quad .23,\quad .23,\quad .39 + \qquad\hbox{for all }a. +\end{equation} +For $a\le-1$, the $R_1$ bound improves to $.20$. +For $a_j\in P_1$, the bounds are $.42^2$ when $a\le-1$ and $.61^2$ +when $a\ge-1$. If $a_j\in O$, write $t_j=-a_j$; then +\begin{equation}\label{sc:F-negative} + \frac{F_j(a)}{\chi(a)^2}\le + \begin{cases} + 20+1.04(7+t_j^2),&a\le-1,\\ + 12+1.63(7+t_j^2),&a\ge-1. + \end{cases} +\end{equation} +For clarity, the complete finite arithmetic behind these bounds is given +by the following table. Its three entries bound, in order, +\[ + |W_j|,\qquad + |A_*b_j-\eta_*/2+\mathsf L_jz+K_{0j}W_j|,\qquad + |-\eta_*+J_{0j}W_j|. +\] +\begin{equation}\label{sc:substitution-table} +\begin{array}{c|cc} + &a\le-1&a\ge-1\\\hline + O&(1.867,.52,1.314)&(2.34,.55,1.155)\\ + P_1&(1.766,.49,1.813)&(2.564,.433,1.155)\\ + R_1&(1.681,.664,2.057)&(2.621,.420,1.155)\\ + U_1&(1.55,.847,2.101)&(2.461,.615,1.155)\\ + V_1&(1.263,.912,1.787)&(1.885,.770,1.205)\\ + T_1&(.913,.869,1.386)&(1.071,.840,1.251) +\end{array} +\end{equation} +To verify it, substitute the intervals in +\eqref{sc:interval-table} and~\eqref{sc:derivative-table}; each expression +is affine in each of its variables separately, so its extrema occur at +corners of the corresponding box. For the middle expression, first write +it as +\[ + (A_*-2K_{0j})b_j-\eta_*/2+K_{0j}h_*+ + (1-2K_{0j})\mathsf L_jz. +\] +Substituting these three bounds into~\eqref{sc:increment-norm}, together +with the bounds for $\mathsf D_j,\ell_{1j},z,\chi$, proves +\eqref{sc:F-simple}--\eqref{sc:F-negative}. All these are finite rational +comparisons; the exact verifier accompanying Appendix~\ref{cert:verification} +also checks them. + +It remains to compare the integrated derivative cost with the logarithmic +separation surplus. Let $u_-$ be the mean of +$(\log q(a)-\log q(a_-))/\delta$ under the uniform probability measure +on $[a_-,a_+]$, and put $u_+=1-u_-$. In the $\xi$ coordinate this measure +has density proportional to $D_0=1/d$. The excess-variance bound in +Lemma~\ref{tr:q-properties} gives +\[ + 0\le \frac{d\log D_0}{d\xi}=\frac{1-qd}{d^2}\le1. +\] +The density $D_0$ is increasing, whereas $D_0e^{-\xi}$ is decreasing. +The covariance of an increasing function with an increasing function is +nonnegative, and with a decreasing function is nonpositive. Applying this +observation to the coordinate $\xi$ compares its mean first with constant +density and then with density proportional to $e^\xi$. The latter comparison +gives +$u_-\le\tfrac12+\tfrac12\coth(\delta/2)-1/\delta$, and hence +\begin{equation}\label{sc:mean-distance} + \tfrac12\le u_-\le\tfrac12+\delta/12, + \qquad u_-^2+u_+^2\le\tfrac12(1+\delta^2/36). +\end{equation} +For the upper bound one uses $\coth x-1/x\le x/3$, obtained by comparing +the power series of $x\cosh x$ and $(1+x^2/3)\sinh x$. + +Suppose $P_-$ and $P_+$ have the following property: on every partial +$\xi$-interval starting at the indicated endpoint, the mean of +$\mathcal N_j(v_j)$ is at most $\sqrt{P_j}$. Pointwise bounds for $F_j$ +are one way to ensure this property. Minkowski's inequality, followed by +the uniform average in $a$, gives +\[ + \mathcal N_-(\Delta k,\Delta b,\psi_-)\le\sqrt{P_-}\,\delta u_-, + \qquad + \mathcal N_+(\Delta k,\Delta b,\psi_+)\le\sqrt{P_+}\,\delta u_+. +\] +Since $u_-\ge u_+\ge0$, +\[ + P_-u_-^2+P_+u_+^2 + \le\max\left(P_-,\frac{P_-+P_+}{2}\right)(u_-^2+u_+^2). +\] +Averaging~\eqref{sc:one-endpoint} and applying +\eqref{sc:mean-distance} therefore proves the desired estimate whenever +\begin{equation}\label{sc:threshold} + \max\left(P_-,\frac{P_-+P_+}{2}\right)\le + \Theta_{m_*}(\delta):= + \frac{1+m_*}{4}\, + \frac{1+\delta^2/30+\delta^4/1680}{1+\delta^2/36}. +\end{equation} +Indeed, the resulting coefficient of $Z^2$ is at most +$\delta^2(1+\delta^2/36)\Theta_{m_*}(\delta)/4 +=r_{-+}\delta^2$. + +We verify~\eqref{sc:threshold} for all endpoint positions. The function +$\Theta_m$ increases in $\delta\ge0$: differentiation with respect to +$y=\delta^2$ leaves a positive numerator +$1/180+2y/1680+y^2/(1680\cdot36)$. +We will also use the following observation about partial intervals: +if a smaller bound holds on an initial segment of logarithmic length $L$, +then its fraction in a partial interval of length $h\le\delta$ is at least +$\min(1,L/h)\ge L/\delta$, provided $\delta\ge L$. +If $a_-\in R_1$, take $P_-\le.295$ and $P_+\le.39$; +then $(P_-+P_+)/2\le.3425<\Theta_{.38}(0)=.345$. +For $a_-\in U_1$ or $V_1$, use $P_-\le.23$, $P_+\le.39$ and +$\Theta_{.68}(0)=.42$. For $a_-\in T_1$, both bounds are $.39<.42$. + +Now suppose $a_-\in P_1$. If $a_+\le-1$, both endpoint costs are at +most $.20<\Theta_{.19}(0)$, using the $R_1$ row at $a_+=-1$. +Otherwise, the part of the logarithmic +interval below $-1$ has length at least $1.62$. Hence every partial +interval from the lower endpoint has mean norm at most +\[ + .61-\frac{(.61-.42)1.62}{\delta},\qquad + P_-=(.61-.3078/\delta)^2. +\] +The length assertion follows from $q(-1)>.287$, $q(-2)<.056$. +If the upper endpoint lies in $T_1$, then $\delta>4$ because $q(4)>4$. +For $\delta\le4$, therefore, $P_+\le.23$ and +$P_-\le.284143<\Theta_{.19}(0)=.2975$. +For $4\le\delta\le5$, use $P_-<.300787$, $P_+\le.39$, and +\[ + \max(.300787,(.300787+.39)/2)<.345394 + <\Theta_{.19}(4). +\] +For $\delta\ge5$, use $P_-\le.61^2=.3721$ and +\[ + \max(.3721,(.3721+.39)/2)=.38105<\Theta_{.19}(5). +\] +This covers $P_1$. + +Finally suppose $a_-\in O$. For any endpoint $a_j\in O$ and any +argument $a\le-1$ between the endpoints, +\[ + t_j^2\le a^2+2\delta,\qquad |a|\chi(a)\le .09. +\] +The first follows by integrating $d\ge-a$ on the negative half-line; +the second follows from $l\le .075\cdot1.19<.09$ and $d\ge|a|$. +Using $\chi\le.07$ in~\eqref{sc:F-negative} gives +\begin{equation}\label{sc:O-low-cost} + F_j(a)\le .144+.011\delta. +\end{equation} +If the whole interval lies at or below $-1$, take +$P_-=.144+.011\delta$ and +$P_+\le\max(P_-,.20)$; the latter bound also covers upper endpoints +in $P_1$ and $R_1$. +For $0\le\delta\le4$, both are below $.25$. For $\delta\ge4$, use +\begin{equation}\label{sc:theta-tail} + \Theta_0(\delta)\ge .25+.00260\delta^2. +\end{equation} +This follows on clearing denominators and using $\delta^2\ge16$. +The quadratic $.106-.011\delta+.00260\delta^2$ is strictly positive, +so~\eqref{sc:threshold} follows in this case too. + +If $a_+>-1$, the first part of the logarithmic interval has length at +least $4$, since $q(-3)<.005$ and $q(-1)>.287$. +For $a\ge-1$, the same integration gives $t_-^2\le1+2\delta$; +using $\chi\le.097$ in~\eqref{sc:F-negative} yields +$F_-(a)\le .236+.031\delta$. +On every partial interval from the lower endpoint, Jensen's inequality +therefore permits the uniform bound +\[ + P_-=.236+.031\delta-\frac{4}{\delta}(.092+.020\delta) + =.156+.031\delta-.368/\delta. +\] +For partial intervals contained below $-1$, the same bound follows from +\eqref{sc:O-low-cost}, because $\delta\ge4$. +By~\eqref{sc:theta-tail}, +\[ + \Theta_0(\delta)-P_- + \ge .094-.031\delta+.00260\delta^2+.368/\delta>0; +\] +the quadratic has negative discriminant and positive leading coefficient. +If $a_+\notin T_1$, then $P_+\le.23<\Theta_0(\delta)$. +If $a_+\in T_1$, then $\delta>6.5$ from $q(4)>4$ and $q(-3)<.005$, +and +$P_+\le.39<\Theta_0(6.5)<\Theta_0(\delta)$. +This proves~\eqref{sc:threshold} in every case and completes the proof. +\end{proof} diff --git a/preprints/A-sharp-entropy-bound-and-the-simplex-inequality-for-isotropic-constants-October-5-2026/build/manuscript/transport.tex b/preprints/A-sharp-entropy-bound-and-the-simplex-inequality-for-isotropic-constants-October-5-2026/build/manuscript/transport.tex new file mode 100644 index 0000000000000000000000000000000000000000..64bbbfb36dd49fccc75381fa7eb00e56ee47a627 --- /dev/null +++ b/preprints/A-sharp-entropy-bound-and-the-simplex-inequality-for-isotropic-constants-October-5-2026/build/manuscript/transport.tex @@ -0,0 +1,552 @@ +\section{Gaussian transport and a matrix surplus}\label{tr:section} + +We first work with a probability density $e^{-V}$ on $\mathbb R^m$ satisfying +\begin{equation}\label{tr:smooth-assumptions} + V\in C^\infty(\mathbb R^m),\qquad + 00,\qquad 0a$, and write +$\mathbb E_a$ for expectation under this law. Gaussian integration +by parts gives +\[ + \mathbb E_a X_a=d(a),\qquad + \operatorname{Var}(X_a)=1-q(a)d(a). +\] +Since $X_a>0$ almost surely, $d>0$ and hence $q'=qd>0$. +Its strictly positive variance gives $q'<1$. + +For completeness, this variance is at most the square of its mean. +The survival function +\[ + S_a(u)=\frac{\Phi(-a-u)}{\Phi(-a)},\qquad u\ge0, +\] +satisfies $S_a(u+v)\le S_a(u)S_a(v)$, since its hazard rate $q(a+u)$ +is increasing. Integrating over $u,v\ge0$ yields +\[ + \frac12\mathbb E_a X_a^2 + =\int_0^\infty uS_a(u)\,du + \le\left(\int_0^\infty S_a(u)\,du\right)^2=d(a)^2. +\] +Consequently $1-qd=\operatorname{Var}(X_a)\le d^2$. Differentiating +$q'=qd$ now gives +$q''=q(d^2+qd-1)\ge0$. Finally, $q(a)\to0$ as $a\to-\infty$, +whereas $q(a)>a$ for $a>0$. These limits prove surjectivity. +\end{proof} + +We shall also need two positivity bounds and a quantitative growth estimate. +Their proof, including +the rational bounds on the compact interval and estimates on both tails, +is given in Appendix~\ref{cert:verification}. +Put +\begin{equation}\label{tr:M-definition} + D_0(a)=\frac1{d(a)},\qquad M(a)=D_0(a)^2(1-b(a)^2). +\end{equation} +A lower bound for the growth of $M$ in $\log q$ will let us compare +uniform averages in $q$ with uniform averages in $\log q$ in the +surplus estimate. +Define a nondecreasing step function $m_0$ by the following table. At a +finite endpoint we take the value in the interval to its right. +\begin{equation}\label{tr:bins} +\begin{array}{c|rrrrrr} + &O&P_1&R_1&U_1&V_1&T_1\\\hline + a&(-\infty,-3)&[-3,-2)&[-2,-1)&[-1,\tfrac12)&[\tfrac12,4)&[4,\infty)\\ + m_0(a)&0&.19&.38&.68&.68&.68 +\end{array} +\end{equation} +All terminating decimals in this paper denote exact rational numbers. + +\begin{lemma}[Scalar bounds]\label{tr:scalar-basic} +The functions in \eqref{tr:scalar-functions} satisfy, on $\mathbb R$, +\begin{equation}\label{tr:scalar-basic-equations} + 00,\qquad + \frac{d\log M}{d\log q} + :=\frac{M'}{Md}\ge m_0. +\end{equation} +\end{lemma} + +For a real symmetric matrix, scalar functions will always be interpreted +by spectral functional calculus. If $f$ is differentiable, its divided +difference is +\begin{equation}\label{tr:divided-difference} + f[u,v]=\begin{cases}(f(u)-f(v))/(u-v),&u\ne v,\\f'(u),&u=v. + \end{cases} +\end{equation} +The classical Daleckii--Krein formula +\cite{DaleckiiKrein1965} states that, in an eigenbasis of a symmetric +matrix $A$, the derivative of $f(A)$ in a symmetric direction $Z$ has +entries $f[a_i,a_j]Z_{ij}$; see also +\cite[Section~2.4]{Noferini2017}. This formula +will allow us to keep the off-diagonal entries of matrix derivatives. + +\subsection{Transport and the choice of coordinates} + +Brenier's theorem gives a convex potential $\phi$ such that +$(\nabla\phi)_\#\gamma_m=\mu$; its gradient is unique up to null sets +\cite{Brenier1991,McCann1995}. For $m\ge2$ we use the full-space +regularity theorem of Cordero-Erausquin and Figalli +\cite[Theorem~1.1]{CorderoErausquinFigalli2019}, which extends +Caffarelli's interior theory \cite{Caffarelli1992} to the unbounded +domains considered here. It applies with both domains equal to $\R^m$, because both densities +are smooth and locally bounded away from zero. For $m=1$, the same +smoothness follows directly by expressing monotone transport through +the two distribution functions. Thus $\phi$ is smooth in every dimension. +Caffarelli's contraction theorem \cite{Caffarelli2000} and its inverse +bound then imply that the Jacobian +\begin{equation}\label{tr:Jacobian} + J=D^2\phi +\end{equation} +satisfies +\begin{equation}\label{tr:Jacobian-bounds} + c_+^{-1/2}I\le J\le c_-^{-1/2}I. +\end{equation} +For the lower bound, apply the contraction theorem to the inverse +transport. We use its quantitative form: a source potential with +Hessian at most $aI$ and a target potential with Hessian at least $bI$ +give a transport Lipschitz constant at most $\sqrt{a/b}$; see also +\cite[Theorem~1]{ChewiPooladian2023}. + +Define the symmetric matrix field +\begin{equation}\label{tr:A-definition} + A=q^{-1}(J). +\end{equation} +The bounds \eqref{tr:Jacobian-bounds} place the spectrum of $A$ in a +compact interval. Our preferred coordinates are characterized by +$\mathbb EA=0$. They need not be the usual isotropic coordinates. + +\begin{proposition}[Affine normalization]\label{tr:normalization} +Let $\mu=e^{-V}dy$ satisfy \eqref{tr:smooth-assumptions}. There exists a +positive definite symmetric matrix $P$ such that the Brenier map from +$\gamma_m$ to $P_\#\mu$ has Jacobian $J_P$ satisfying +\[ + \mathbb E q^{-1}(J_P)=0. +\] +The transformed density again satisfies +\eqref{tr:smooth-assumptions}, with possibly different positive constants. +\end{proposition} +\begin{proof} +We use a homotopy from the Gaussian and show that its zeros stay in a +compact subset of the positive definite cone. For $0\le\theta\le1$, +define +\[ + W_\theta(y)=(1-\theta)|y|^2/2+\theta V(y),\qquad + d\mu_\theta=Z_\theta^{-1}e^{-W_\theta(y)}\,dy, +\] +where $Z_\theta$ is the normalizing integral. +There are constants $a_-,a_+>0$, independent of $\theta$, with +$a_-I\le D^2W_\theta\le a_+I$. These densities have uniformly +Gaussian tails. Their covariance matrices $C_\theta$ depend +continuously on $\theta$, are positive definite, and therefore satisfy +\begin{equation}\label{tr:homotopy-covariance} + c_0I\le C_\theta\le C_0I +\end{equation} +for fixed $c_0,C_0>0$. + +For a positive definite symmetric $P$, let $J_{\theta,P}$ be the +Jacobian of the transport to $P_\#\mu_\theta$. Define a map with +values in the vector space of symmetric matrices by +\[ + F(\theta,P)=\mathbb E q^{-1}(J_{\theta,P}). +\] +Contraction in both directions gives +\begin{equation}\label{tr:homotopy-spectral} + \frac{\lambda_{\min}(P)}{\sqrt{a_+}}I + \le J_{\theta,P}\le\frac{\|P\|_{\mathrm{op}}}{\sqrt{a_-}}I. +\end{equation} + +We first verify continuity of $F$. Restrict $P$ to any compact subset +of the positive definite cone. The transports have common Lipschitz +constants by \eqref{tr:homotopy-spectral}; their values at zero are +bounded because their means and second moments are bounded. Every +convergent sequence $(\theta_j,P_j)$ thus has locally uniformly +convergent subsequences of transport maps. Their limits push forward +$\gamma_m$ to the limiting target and remain gradients of convex +functions, so Brenier uniqueness identifies the limit. Write $J_j,J$ +for the corresponding Jacobians. They converge weak-* locally and +obey common positive spectral bounds. + +The change-of-variables equation expresses $\log\det J_j$ in terms +of the target potential composed with its transport map. It therefore +converges locally uniformly to $\log\det J$. On a fixed compact +spectral interval, strict concavity of $\log\det$ gives a constant +$c>0$ such that +\[ + \log\det J_j\le\log\det J+ + \operatorname{tr}\bigl(J^{-1}(J_j-J)\bigr)-c|J_j-J|_{\mathrm F}^2. +\] +Integration on any ball, followed by weak-* convergence, proves +$J_j\to J$ in local $L^2$. Functional calculus then gives local +$L^2$ convergence of $q^{-1}(J_j)$. Their common spectral bounds +control the Gaussian tails, proving continuity of $F$. + +We next bound all zeros, uniformly in $\theta$. Suppose +$\mathbb EA=0$, where $A=q^{-1}(J_{\theta,P})$. Convexity of $q$ +and Jensen's inequality on the expected spectral measure of each unit +vector give +\[ + \mathbb E J_{\theta,P}\ge q(0)I. +\] +Gaussian integration by parts gives +$\mathbb E(x_r\partial_i\phi)=(\mathbb E J_{\theta,P})_{ir}$. +Orthogonal projection of the centered components of $\nabla\phi$ +onto the span of $x_1,\ldots,x_m$ consequently yields +\[ + P C_\theta P\ge(\mathbb E J_{\theta,P})^2\ge q(0)^2I. +\] +Together with \eqref{tr:homotopy-covariance}, this bounds $P^{-1}$ +uniformly. The lower estimate in \eqref{tr:homotopy-spectral} now +gives a uniform lower bound $A\ge a_*I$, where we may take $a_*\le0$. +Since $q'\le1$, +\[ + q(a)\le q(a_*)+a-a_*\quad(a\ge a_*), + \qquad + \mathbb E\operatorname{tr}J_{\theta,P} + \le m\bigl(q(a_*)-a_*\bigr). +\] +On the other hand, Gaussian Poincar\'e and +\eqref{tr:homotopy-spectral} imply +\[ + c_0\|P\|_{\mathrm{op}}^2 + \le\operatorname{tr}(P C_\theta P) + \le\mathbb E\operatorname{tr}J_{\theta,P}^2 + \le\frac{\|P\|_{\mathrm{op}}}{\sqrt{a_-}} + \mathbb E\operatorname{tr}J_{\theta,P}. +\] +This also bounds $P$ uniformly. + +Choose a bounded open set in the space of symmetric matrices whose +closure lies in the positive definite cone and whose interior contains +every zero just bounded. The map $F(\theta,\cdot)$ has no zeros on its +boundary. At $\theta=0$, the target is $N(0,P^2)$, so +$J_{0,P}=P$ and $F(0,P)=q^{-1}(P)$. Its unique zero is $q(0)I$; +its derivative there is multiplication by the positive scalar +$1/q'(0)$. Its Brouwer degree is therefore nonzero. Homotopy +invariance of degree gives a zero at $\theta=1$, as required. +\end{proof} + +The quantity +$m+\tfrac12\log\det\operatorname{Cov}(\mu)-h(\mu)$ is unchanged +by an invertible affine transformation. Indeed, a linear map $P$ +adds $\log|\det P|$ to entropy and $2\log|\det P|$ to the covariance +log determinant. We may therefore impose the normalization from +Proposition~\ref{tr:normalization}. From now on $V,J,A,C$ refer to +this transformed density and transport, and +\begin{equation}\label{tr:mean-zero} + \mathbb EA=0. +\end{equation} + +\subsection{The transport equation and its energy} + +We shall differentiate the Jacobian and integrate the resulting equation. +The twice-differentiated change-of-variables equation is part of +Caffarelli's transport method \cite[Section~5]{Caffarelli2000}. +The next lemma establishes the finite energy needed to do this without +additional decay assumptions on higher derivatives of $V$. +For a matrix field $F$, set +\[ + |F|_{\mathrm F}^2=\operatorname{tr}(F^{\mathsf T}F),\qquad + \|F\|^2=\mathbb E|F|_{\mathrm F}^2,\qquad + \|\partial F\|^2=\sum_{r=1}^m\|\partial_rF\|^2. +\] +The same convention sums over any displayed derivative index of a +collection of matrix fields. + +\begin{lemma}[Jacobian equation and finite energy]\label{tr:energy} +Under \eqref{tr:smooth-assumptions}, put $J_r=\partial_rJ$ and +\[ + \mathcal L u=\operatorname{tr}(J^{-1}D^2u) + -\nabla V(\nabla\phi)\cdot\nabla u. +\] +Then +\begin{equation}\label{tr:Jacobian-equation} + \mathcal L J_{ij} + =-\delta_{ij}+(JD^2V(\nabla\phi)J)_{ij} + +\operatorname{tr}(J^{-1}J_iJ^{-1}J_j). +\end{equation} +Moreover $\|\partial J\|<\infty$. The matrix fields $A$ and $k(A)$ +belong to Gaussian $H^1$, and the integrations by parts below with +smooth matrix functions of $J$ are justified by compact cutoffs. +\end{lemma} +\begin{proof} +The density equation is +\begin{equation}\label{tr:density-equation} + \log\det J=V(\nabla\phi)-|x|^2/2-(m/2)\log(2\pi). +\end{equation} +Differentiating twice gives \eqref{tr:Jacobian-equation}. The divergence +of the cofactor matrix of a Hessian vanishes. Applying this identity +to $J$, and then differentiating \eqref{tr:density-equation} once, +shows that +\[ + \mathcal L u=e^{|x|^2/2}\operatorname{div} + \bigl(e^{-|x|^2/2}J^{-1}\nabla u\bigr). +\] +Thus, for smooth $u$ and compactly supported smooth $v$, +\begin{equation}\label{tr:divergence-form} + \mathbb E(v\mathcal Lu) + =-\mathbb E\langle\nabla v,J^{-1}\nabla u\rangle. +\end{equation} + +Choose $0<\lambda\le\Lambda<\infty$ such that +$\lambda I\le J\le\Lambda I$. Taking the trace in +\eqref{tr:Jacobian-equation}, and using $D^2V\ge0$, gives +\[ + \mathcal L(\operatorname{tr}J) + \ge-m+\Lambda^{-2}\sum_r|J_r|_{\mathrm F}^2. +\] +For a compactly supported cutoff $0\le\chi\le1$, integration by parts +and $|\nabla\operatorname{tr}J|\le\sqrt m\,|\partial J|$ imply +\begin{align*} + \Lambda^{-2}\mathbb E\chi^2|\partial J|^2 + &\le m+2\lambda^{-1}\sqrt m\, + \mathbb E\bigl(\chi|\nabla\chi|\,|\partial J|\bigr)\\ + &\le m+\tfrac12\Lambda^{-2}\mathbb E\chi^2|\partial J|^2 + +2m\Lambda^2\lambda^{-2}\mathbb E|\nabla\chi|^2. +\end{align*} +Let $\chi$ tend to one through standard radial cutoffs with gradients +bounded by a constant divided by their radii. Fatou's lemma proves +$\|\partial J\|<\infty$. + +All eigenvalues of $J$ lie in $[\lambda,\Lambda]$. The Fr\'echet +derivatives of $q^{-1}$ and of $k\circ q^{-1}$ are bounded on that +spectral interval, so $A,k(A)\in H^1(\gamma_m)$. The same applies to +any smooth scalar function on this interval. For such a matrix +multiplier, insert a cutoff into \eqref{tr:divergence-form} and let it +tend to one. The terms containing its gradient vanish by +Cauchy--Schwarz and the energy bound; the remaining differentiated +terms are bounded by a constant times $1+|\partial J|^2$. +This justifies the asserted integrations by parts. +\end{proof} + +\subsection{Keeping the eigenvalue separation} + +Set $Z_r=\partial_r A$. At each point diagonalize $A$, and write +$a_i$ for its eigenvalues and $\lambda_i=q(a_i)$ for those of $J$. +For real $a,b$, define +\begin{equation}\label{tr:surplus-weight} + \begin{gathered} + \delta(a,b)=|\log q(a)-\log q(b)|,\qquad + m_*(a,b)=m_0(\min(a,b)),\\ + r(a,b)=\frac{1+m_*(a,b)}{16} + \left(1+\frac{\delta(a,b)^2}{30} + +\frac{\delta(a,b)^4}{1680}\right). + \end{gathered} +\end{equation} +We use the abbreviations $\delta_{ij}=\delta(a_i,a_j)$ and +$r_{ij}=r(a_i,a_j)$. The quantity +\begin{equation}\label{tr:surplus-definition} + \mathcal C=\mathbb E\sum_{i,j,r} + r_{ij}\delta_{ij}^2(Z_r)_{ij}^2 +\end{equation} +is independent of the choice of orthonormal basis within repeated +eigenspaces. In particular, the factor $\delta_{ij}^2$ vanishes when +$a_i=a_j$. + +\begin{proposition}[Matrix surplus]\label{tr:surplus} +For the transport satisfying \eqref{tr:smooth-assumptions}, with the +notation above, +\begin{equation}\label{tr:surplus-inequality} + \mathbb E\operatorname{tr}k(A)^2 + \ge\|\partial A\|^2-\|\partial k(A)\|^2+\mathcal C. +\end{equation} +\end{proposition} +\begin{proof} +The proof has two parts. Symmetry of the third derivatives of $\phi$ +first replaces the transport equation by averages of the scalar +function $M$. Comparing arithmetic and logarithmic averages then +produces the term $\mathcal C$. + +The identity $qk=t$ will match the entropy term in the covariance +completion of Section~\ref{mat:section}; we first estimate the derivative +cost of $k(A)$. Put $F(\lambda)=k(q^{-1}(\lambda))^2$. Multiply +\eqref{tr:Jacobian-equation} by $F(J)$ in trace and integrate by parts. +Lemma~\ref{tr:energy} justifies this operation. At a fixed point +choose an orthonormal eigenbasis of $J$ and rotate all three indices +of the symmetric tensor +$\Theta_{ijr}=\partial_rJ_{ij}=\partial_i\partial_j\partial_r\phi$ +into that basis. The quadratic term on the right of the Jacobian +equation has trace contraction +\[ + \sum_i F(\lambda_i)\operatorname{tr}(J^{-1}J_iJ^{-1}J_i) + =\sum_{i,j,r}\frac{F(\lambda_i)}{\lambda_j\lambda_r}\Theta_{ijr}^2, +\] +where we used the symmetry of $\Theta$. The term obtained by integration +by parts is +\[ + \sum_r\lambda_r^{-1}\operatorname{tr}\bigl((\partial_rF(J))J_r\bigr) + =\sum_{i,j,r}\frac{F[\lambda_i,\lambda_j]}{\lambda_r}\Theta_{ijr}^2. +\] +The remaining term is +$\operatorname{tr}(F(J)JD^2VJ)\ge0$. +Discarding it gives +\begin{equation}\label{tr:initial-multiplier} + \mathbb E\operatorname{tr}k(A)^2 + \ge\mathbb E\sum_{i,j,r}\Theta_{ijr}^2 + \left(\frac{F(\lambda_i)}{\lambda_j\lambda_r} + +\frac{F[\lambda_i,\lambda_j]}{\lambda_r}\right). +\end{equation} +Only pointwise changes of basis are made here; no eigenvectors are +differentiated. + +Regard $M$ as a function of $\lambda=q(a)$. The identities +\eqref{tr:scalar-identities} give +\begin{equation}\label{tr:lambdaF} + \frac{d}{d\lambda}(\lambda F(\lambda)) + =k^2+\frac{2kb}{d} + =\frac{1-b^2}{d^2}=M. +\end{equation} +For two positive numbers $u,v$, let $M^{\mathrm{ar}}_{uv}$ be the +uniform average of $M$ on the interval between $u$ and $v$, with its +continuous interpretation when $u=v$. With the weight +$\Theta_{ijr}^2/(\lambda_i\lambda_j\lambda_r)$, symmetrization in +$i,j$ changes the coefficient in \eqref{tr:initial-multiplier} to +\[ + \frac{\lambda_i+\lambda_j}{2}\, + M^{\mathrm{ar}}_{\lambda_i\lambda_j}. +\] +In the full sum we can replace this coefficient by +$\lambda_rM^{\mathrm{ar}}_{\lambda_i\lambda_j}$, decreasing its +value. To see this, order three eigenvalues as $u\le v\le w$. +Lemma~\ref{tr:scalar-basic} makes $M$ increasing, so +\[ + M^{\mathrm{ar}}_{uv}\le M^{\mathrm{ar}}_{uw} + \le M^{\mathrm{ar}}_{vw}. +\] +The ordered coefficients $u+v-2w$, $u+w-2v$, and $v+w-2u$ sum to +zero. Their scalar product with these three ordered averages is +nonnegative, by the rearrangement inequality. Full symmetry of +$\Theta$ permits precisely this averaging over permutations of the +three indices. We have proved +\begin{equation}\label{tr:arithmetic-energy} + \mathbb E\operatorname{tr}k(A)^2 + \ge\mathbb E\sum_{i,j,r} + \frac{\Theta_{ijr}^2}{\lambda_i\lambda_j\lambda_r}\, + \lambda_r M^{\mathrm{ar}}_{\lambda_i\lambda_j}. +\end{equation} + +We now fix a pair of eigenvalues. Bars in the rest of the proof denote +uniform averages in $\log\lambda$ over their interval. Set +\[ + x=\delta_{ij}/2,\qquad w_0=\frac{x}{\sinh x},\qquad m_*=m_*(a_i,a_j), +\] +with $w_0=1$ at $x=0$. Since $da/d\log\lambda=D_0$, +the divided-difference formula shows that the coefficients of +$(Z_r)_{ij}^2$ and $(\partial_r k(A))_{ij}^2$, respectively, when +written with the same weight as in \eqref{tr:arithmetic-energy}, are +\begin{equation}\label{tr:derivative-coefficients} + \lambda_r w_0^2\overline{D_0}^{\,2},\qquad + \lambda_r w_0^2\overline{bD_0}^{\,2}. +\end{equation} +Both $D_0(1+b)$ and $D_0(1-b)=k$ are increasing: $d'=qd-1<0$, +and $b'=l>0$, $k'=b>0$. The covariance inequality for two increasing +functions on an interval therefore yields +\begin{equation}\label{tr:logarithmic-chebyshev} + \overline M\ge + M_1:=\overline{D_0}^{\,2}-\overline{bD_0}^{\,2} + \ge\tfrac34\overline{D_0}^{\,2}. +\end{equation} + +We also need a quantitative comparison of the two averaging measures. +On the logarithmic interval, \eqref{tr:scalar-basic-equations} says +that $M(\lambda)\lambda^{-m_*}$ is increasing. Exponential weighting +and the same covariance inequality give +\begin{equation}\label{tr:average-comparison} + \frac{M^{\mathrm{ar}}_{\lambda_i\lambda_j}}{\overline M} + \ge\frac{\operatorname{shc}((1+m_*)x)} + {\operatorname{shc}(x)\operatorname{shc}(m_*x)}, + \qquad \operatorname{shc}(x)=\frac{\sinh x}{x}. +\end{equation} +Indeed, after centering the logarithmic interval, the left side is +$\mathbb E_0(e^sM)/[\mathbb E_0e^s\,\mathbb E_0M]$, where +$\mathbb E_0$ is its uniform average. Under the measure proportional +to $e^{m_*s}$, the increasing factor $Me^{-m_*s}$ can only increase +the mean of $e^s$. Taking $M=e^{m_*s}$ gives the displayed ratio. + +Here is a convenient explicit lower bound for this ratio. Define +$A_0(x)=x\coth x-1$, with $A_0(0)=0$. The identity +\[ + x\coth x+\frac{x^2}{\sinh^2x}\ge2 +\] +implies both $A_0(x)\ge1-w_0^2$ and that $A_0(x)/x^2$ is +nonincreasing for $x>0$. To verify the identity, clear the denominator +and expand +$\tfrac{x}{2}\sinh(2x)+x^2-\cosh(2x)+1$ in powers of $x$; +the coefficients vanish through degree four and are nonnegative +thereafter. Since $0\le m_*\le1$, we obtain +$A_0(m_*x)\ge m_*^2A_0(x)$. The addition formula for $\sinh$ now gives +\begin{align} + \frac{\operatorname{shc}((1+m_*)x)} + {\operatorname{shc}(x)\operatorname{shc}(m_*x)} + &=1+\frac{m_*A_0(x)+A_0(m_*x)}{1+m_*}\notag\\ + &\ge1+m_*(1-w_0^2). + \label{tr:hyperbolic-comparison} +\end{align} +Every formula here extends continuously to $m_*=0$ or $x=0$. + +Combining \eqref{tr:logarithmic-chebyshev}--\eqref{tr:hyperbolic-comparison}, +the amount by which the coefficient in +\eqref{tr:arithmetic-energy} exceeds the difference of the two +coefficients in \eqref{tr:derivative-coefficients} is at least +\[ + \lambda_r(1+m_*)(1-w_0^2)M_1. +\] +Relative to the coefficient of $(Z_r)_{ij}^2$, this is at least +\[ + \frac34(1+m_*)(w_0^{-2}-1) + \ge\frac{1+m_*}{16} + \left(\delta_{ij}^2+\frac{\delta_{ij}^4}{30} + +\frac{\delta_{ij}^6}{1680}\right). +\] +The last inequality is the Taylor expansion of +$(\sinh x/x)^2$, whose omitted coefficients are positive. +Substituting in \eqref{tr:arithmetic-energy} and summing proves +\eqref{tr:surplus-inequality}. +\end{proof} + +We have obtained two inputs for the entropy argument: the matrix field +$A$ has no constant Gaussian component, and convexity supplies +\eqref{tr:surplus-inequality}. The next section combines these facts +with a Gaussian covariance identity and separates the first chaos of +$A$ from its higher chaoses. diff --git a/preprints/A-sharp-entropy-bound-and-the-simplex-inequality-for-isotropic-constants-October-5-2026/isotropic-simplex.pdf b/preprints/A-sharp-entropy-bound-and-the-simplex-inequality-for-isotropic-constants-October-5-2026/isotropic-simplex.pdf new file mode 100644 index 0000000000000000000000000000000000000000..6fd924789df8720547eddd4fb0c7836e603da4be --- /dev/null +++ b/preprints/A-sharp-entropy-bound-and-the-simplex-inequality-for-isotropic-constants-October-5-2026/isotropic-simplex.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:b6cb0115cd04d4eb0c24db7e600ba8dd9519848a92e87fa8a661f435e41e95a0 +size 567145 diff --git a/preprints/A-sharp-entropy-bound-and-the-simplex-inequality-for-isotropic-constants-October-5-2026/verification/verify_scalar.py b/preprints/A-sharp-entropy-bound-and-the-simplex-inequality-for-isotropic-constants-October-5-2026/verification/verify_scalar.py new file mode 100644 index 0000000000000000000000000000000000000000..d90ced29edee5e652cb8c3c94a3c84e490ddb9e5 --- /dev/null +++ b/preprints/A-sharp-entropy-bound-and-the-simplex-inequality-for-isotropic-constants-October-5-2026/verification/verify_scalar.py @@ -0,0 +1,512 @@ +#!/usr/bin/env python3 +"""Reproduce the scalar certificate using only Python's standard library. + +Run: python3 verification/verify_scalar.py + +All input decimals, polynomial operations, enclosures, and comparisons are exact +fractions. Printed decimals are rounded OUTWARDS; no float is used. No sampling, +numerical integration, external data, or imported certificate module is used. + +Mathematical inputs, proved in the accompanying scalar appendix, are the Mills +alternating-integral bounds, the Gaussian ODEs, their contractive error comparison, +the quartic Bernstein enclosure, the elementary integral tail estimates, and the +multi-affine corner principle. This program checks their rational hypotheses and +consequences; it is not a formal proof assistant for those calculus statements. +""" + +from fractions import Fraction as F +from itertools import product +from math import comb, factorial + + +def q(value): + """Reject binary floating-point input, including in future table edits.""" + if isinstance(value, float): + raise TypeError("Use a decimal string, integer, or Fraction") + return F(value) + + +def require(condition, message): + # Deliberately not assert: verification also works under python -O. + if not condition: + raise ArithmeticError(message) + + +def down(value, places=12): + n = value.numerator * 10**places // value.denominator + sign = "-" if n < 0 else "" + n = abs(n) + return f"{sign}{n // 10**places}.{n % 10**places:0{places}d}" + + +def up(value, places=12): + n = -((-value.numerator * 10**places) // value.denominator) + sign = "-" if n < 0 else "" + n = abs(n) + return f"{sign}{n // 10**places}.{n % 10**places:0{places}d}" + + +def parse_rows(data): + return [[None if x == "-" else q(x) for x in line.split()] + for line in data.strip().splitlines()] + + +ROWS = parse_rows(""" +-5.25 .75 .0000004128471303 .1842076703035094 +-3.75 .75 .0003526268606772 .2507500253088710 +-2.5 .5 .0176378254869167 .3531628936050266 +-1.5 .5 .1387897504588508 .4981884857033017 +-.5 .5 .5091604338370335 .7246172144765372 +.25 .25 .9635539794164037 .9475979387933716 +1.25 .75 1.7288166273310539 1.3000948982566192 +3 1 3.2830986549304364 2.0126493036044195 +5 1 5.1865039671258417 2.9046537878842420 +7 1 7.1375456132265036 3.8366560415481663 +9 1 9.1085231050028685 4.7898159350739631 +""") + +# b_min, b_max, L_max, D_max, K_min, K_max, J_max, L_1, m_0. +BINS = parse_rows(""" +0 .09 .60 - .37 .56 .72 .13 0 +.083 .15 1.01 4.3 .29 .46 .72 .13 .19 +.14 .23 1.19 3 .29 .371 .68 .13 .38 +.22 .342 1.19 1.72 .29 .398 .61 .123 .68 +.337 .45 .83 .87 .31 .4 .50 .060 .68 +.445 .5 .20 .640 .32 .4 .48 .022 .68 +""") +CLAIMED = [[v / 1000 for v in row] for row in parse_rows(""" +7 1 1 1 +1 1 2 2 +2 2 2 5 +2 2 2 8 +2 3 5 14 +2 2 8 11 +2 .8 8 20 +2 .8 8 7 +1 .8 10 15 +3 .6 11 11 +2 .35 12 9 +""")] + +ALPHA, BETA, S, GAMMA, DS = F(3, 40), F(7, 25), F(40, 37), F(3, 4), F(5, 17) +ASTAR = S + 1 + S / 8 +MSTAR = S * BETA + (1 + S / 8) * GAMMA +MPRIME = S * BETA**2 + (9 * S / 8) * GAMMA**2 +EQ, EK = F(1, 10**8), F(8, 10**7) + + +def add(a, b): + out = [F(0)] * max(len(a), len(b)) + for i, x in enumerate(a): + out[i] += x + for i, x in enumerate(b): + out[i] += x + return out + + +def scale(a, c): + return [x * q(c) for x in a] + + +def sub(a, b): + return add(a, scale(b, -1)) + + +def mul(a, b): + out = [F(0)] * (len(a) + len(b) - 1) + for i, x in enumerate(a): + for j, y in enumerate(b): + out[i + j] += x * y + return out + + +def norm(a): + return sum(map(abs, a), F(0)) + + +def endpoint(a, x): + return sum((v * x**i for i, v in enumerate(a)), F(0)) + + +class Quartic: + """Pair (polynomial, uniform error) on [-1,1], as in the appendix.""" + + def __init__(self, f, error=0): + f = f if isinstance(f, list) else [q(f)] + self.f = (f + [F(0)] * 5)[:5] + self.error = q(error) + norm(f[5:]) + + def __add__(self, other): + other = as_quartic(other) + return Quartic(add(self.f, other.f), self.error + other.error) + + __radd__ = __add__ + + def __neg__(self): + return Quartic(scale(self.f, -1), self.error) + + def __sub__(self, other): + return self + -as_quartic(other) + + def __rsub__(self, other): + return as_quartic(other) + -self + + def __mul__(self, other): + other = as_quartic(other) + error = (norm(self.f) * other.error + norm(other.f) * self.error + + self.error * other.error) + return Quartic(mul(self.f, other.f), error) + + __rmul__ = __mul__ + + def __truediv__(self, constant): + return self * (1 / q(constant)) + + def __pow__(self, n): + result = Quartic(1) + for _ in range(n): + result = result * self + return result + + def lower(self): + return min(sum((F(comb(i, j), comb(4, j)) * eta**j * self.f[j] + for j in range(i + 1)), F(0)) + for i in range(5) for eta in (-1, 1)) - self.error + + +def as_quartic(value): + return value if isinstance(value, Quartic) else Quartic(value) + + +def polynomials(row): + w, r, p0, k0 = row + a = [w, r] + p, k = [F(0)] * 15, [F(0)] * 15 + p[0], k[0] = p0, k0 + for i in range(14): + d = sub(p, a) + p[i + 1] = r * mul(p, d)[i] / (i + 1) + k[i + 1] = r * ((1 if i == 0 else 0) - mul(k, d)[i]) / (i + 1) + d = sub(p, a) + b = sub([F(1)], mul(k, d)) + j = add(sub(k, p), mul(a, b)) + y = add(sub(scale(b, 2), mul(p, d)), mul(a, j)) + eb = norm(k) * EQ + EK * (EQ + norm(d)) + ej = EQ + EK + norm(a) * eb + ey = 2 * eb + (norm(p) + norm(d) + EQ) * EQ + norm(a) * ej + # Construct FULL b,j,y before the quartic truncation, as specified. + approximants = [Quartic(p, EQ), Quartic(k, EK), Quartic(d, EQ), + Quartic(b, eb), Quartic(j, ej), Quartic(y, ey), Quartic(a)] + return approximants, (p, k, d) + + +def groups(row, approximants): + w = row[0] + idx = (0 if w < -3 else 1 if w < -2 else 2 if w < -1 + else 3 if w < F(1, 2) else 4 if w < 4 else 5) + bmin, bmax, lmax, dmax, kmin, kmax, jmax, l1, m0 = BINS[idx] + zmin, zmax, chimax = map(q, ("-.08", ".7", ".07") if w < -1 + else ("-.475", "0", ".097")) + qq, k, d, b, l, lp, a = approximants + p = b + k * l - l * l / ALPHA + e = p + b * (q(".75") - b) - l * l / ALPHA + hp = l + 3 * b * l + k * lp + g1 = [b - bmin, bmax - b, l, ALPHA * lmax - l, + p * (7 + a * a) - 1 if w < 0 else p - q(".46" if idx == 5 else ".25"), + q(".67")**2 * p - (k * (b - q(".5")) - (b - q(".28")) * l * (2 / ALPHA))**2] + g2 = [lp - zmin * l, zmax * l - lp, hp - q(".8") * l, + q("1.77") * l - hp, chimax * d - l, + 2 * (1 - qq * d) * (1 - b * b) - 2 * b * l * d - m0 * d * d * (1 - b * b)] + g3 = [DS * e - kmin * p, kmax * p - DS * e, jmax * p - DS * q("1.5") * b] + if dmax is not None: + g3.append(p - DS / dmax) + f1 = q("1.5") * b + h = q(".43") - l1 - ASTAR * b * b + MSTAR * b - MPRIME / 4 - l * l / ALPHA + v = q("1.97") - MPRIME + o = q(".13") - MPRIME / 2 + MSTAR * b + g4 = [h * p - DS * e * e, + p * (h * v - o * o) - DS * (e * e * v - 2 * e * o * f1 + h * f1 * f1)] + return [g1, g2, g3, g4] + + +def atan_interval(x, n=24): + """Alternating atan series, 0= 2: + x /= 2 + shifts += 1 + + def unit(v): + z = (v - 1) / (v + 1) + lo = 2 * sum((z**(2*i + 1) / (2*i + 1) for i in range(n)), F(0)) + hi = lo + 2 * z**(2*n + 1) / ((2*n + 1) * (1 - z*z)) + return lo, hi + + lo, hi = unit(x) + l2, h2 = unit(F(2)) + return lo + shifts * l2, hi + shifts * h2 + + +def mills(r, n): + double_factorial, total = 1, F(0) + for i in range(n + 1): + if i: + double_factorial *= 2 * i - 1 + total += F((-1)**i * double_factorial, r**(2*i + 1)) + return total + + +def anchors(): + a5, a239 = atan_interval(F(1, 5)), atan_interval(F(1, 239)) + pi_lo, pi_hi = 16 * a5[0] - 4 * a239[1], 16 * a5[1] - 4 * a239[0] + require(F(3) < pi_lo < pi_hi < F(22, 7), "Machin pi interval") + # theta0=log(10 sqrt(2 pi))=log(200 pi)/2. + theta_lo = log_interval(200 * pi_lo)[0] / 2 + theta_hi = log_interval(200 * pi_hi)[1] / 2 + require(q("3.221") < theta_lo < theta_hi < q("3.222"), "theta0 interval") + print("theta0 enclosure:", down(theta_lo), "< theta0 <", up(theta_hi)) + # sqrt(2pi)>12/5 and e^18 exceeds this positive Taylor sum. + exp18_lo = sum((F(18**i, factorial(i)) for i in range(60)), F(0)) + require(2*pi_lo > F(12, 5)**2, "sqrt(2pi)>12/5") + require(F(5, 12) / exp18_lo < F(7, 10**9), "f6<7e-9") + q_lo, q_hi = 1 / mills(10, 12), 1 / mills(10, 11) + q_error = max(abs(q("10.098093234") - v) for v in (q_lo, q_hi)) + require(q_error < F(4, 10**10), "q(10) Mills anchor") + # |k(-6)-I0(6)| <= 2 f6/36 < 7/(18*10^9). + k_lo = mills(6, 13) - F(7, 18 * 10**9) + k_hi = mills(6, 14) + F(7, 18 * 10**9) + k_error = max(abs(q(".16237766") - v) for v in (k_lo, k_hi)) + require(k_error < F(4, 10**8), "k(-6) Mills anchor") + print("Mills anchor absolute errors <=", up(q_error, 15), "and", up(k_error, 15)) + print("Machin pi, logarithm series, Gaussian-density bound, Mills anchors: PASS") + + +def central(): + previous = None + residual_q = residual_k = jumps_q = jumps_k = forcing_k = F(0) + min_d = F(100) + points = {} + print("Central quartic group lower bounds (rounded down):") + for ri, row in enumerate(ROWS): + w, r = row[:2] + approximants, (p, k, d) = polynomials(row) + rp, rk = r * norm(mul(p, d)[14:]), r * norm(mul(k, d)[14:]) + require(rp < (F(2, 10**9) if ri == 1 else F(7, 10**11)), "P residual") + require(rk < F(2, 10**10), "Q residual") + residual_q += 2 * rp + residual_k += 2 * rk + forcing_k += 2 * r * EQ * norm(k) + d_lower = Quartic(d).lower() + require(d_lower > q(".09"), "D polynomial positive") + min_d = min(min_d, d_lower) + require(norm(k) < (2 if w <= F(1, 4) else q("5.6")), "Q coefficient norm") + if previous is not None: + old_row, old_p, old_k = previous + require(old_row[0] + old_row[1] == w - r, "contiguous intervals") + jq = abs(endpoint(old_p, 1) - endpoint(p, -1)) + jk = abs(endpoint(old_k, 1) - endpoint(k, -1)) + require(jq < F(2, 10**10) and jk < F(2, 10**10), "joining errors") + jumps_q += jq + jumps_k += jk + else: + require(w - r == -6, "left endpoint") + k_poly_anchor = abs(endpoint(k, -1) - q(".16237766")) + require(k_poly_anchor < F(3, 10**9), "left Q anchor") + previous = row, p, k + for side in (-1, 1): + points[w + r * side] = endpoint(p, side) + lows = [min(v.lower() for v in group) for group in groups(row, approximants)] + require(all(v >= c for v, c in zip(lows, CLAIMED[ri])), f"groups at w={w}") + print(f" {str(w):>5}:", " ".join(down(v) for v in lows)) + require(ROWS[-1][0] + ROWS[-1][1] == 10, "right endpoint") + q_poly_anchor = abs(endpoint(p, 1) - q("10.098093234")) + require(q_poly_anchor < F(2, 10**10), "right P anchor") + q_budget = F(4, 10**10) + q_poly_anchor + jumps_q + residual_q + k_budget = F(4, 10**8) + k_poly_anchor + jumps_k + residual_k + forcing_k + require(q_budget < EQ and k_budget < EK, "global ODE error budgets") + print("44 central group bounds, 22 residuals, 20 joins, 11 D bounds, 11 Q norms: PASS") + print("D polynomial uniform lower bound >=", down(min_d)) + print("Endpoint polynomial anchor errors <=", up(q_poly_anchor, 15), "and", up(k_poly_anchor, 15)) + print("Integrated k forcing <=", up(forcing_k, 15)) + print("Global ODE error budgets <=", up(q_budget, 15), "< 1e-8;", up(k_budget, 15), "< 8e-7") + require(points[-1] - EQ > q(".287"), "q(-1)>.287") + require(points[-2] + EQ < q(".056"), "q(-2)<.056") + require(points[-3] + EQ < q(".005"), "q(-3)<.005") + require(points[4] - EQ > 4, "q(4)>4") + for ratio, bound in [(F(287, 56), q("1.62")), (F(4)/q(".056"), F(4)), + (F(287, 5), F(4)), (F(4)/q(".005"), q("6.5"))]: + require(log_interval(ratio)[0] > bound, "secant log length") + print("Four q endpoint bounds and four secant logarithmic lengths: PASS") + + +def left_tail(): + u = F(1, 36) + eta = 1 + u / 100 + b_over_u_lo = 1 - q("3.01") * u + kl_over_b = 2 * u * eta**2 / b_over_u_lo + l2_over_alpha_b = 4 * u**2 * eta**2 / (ALPHA * b_over_u_lo) + require(eta * u < q(".028"), "left b upper") + require(2 * eta / (216 * ALPHA) < q(".60"), "left L upper") + require(1 / (1 + kl_over_b) > q(".93"), "left b/p lower") + require(1 / (1 - l2_over_alpha_b) < q("1.05"), "left b/p upper") + require(q("3.01") + 4 * u * eta**2 / ALPHA < q("4.54"), "left p lower") + require((1 - q("4.54") * u) * (1 + 7 * u) > 1, "left p inverse quadratic") + require(eta / 2 < q(".501"), "left h0 first magnitude") + require(4 * BETA * eta * u / ALPHA < q(".43"), "left h0 second magnitude") + require(q(".501")**2 < q(".67")**2 * (1 - q("4.54") * u), "left h0 bound") + require(DS * (1 + q(".93") * (q(".75") - q(".028") - l2_over_alpha_b)) > q(".37"), "left K lower") + require(DS * (1 + q("1.05") * q(".75")) < q(".56"), "left K upper") + require(DS * q("1.5") * q("1.05") < q(".72"), "left J upper") + require(DS < q(".295"), "left D upper coefficient") + require(3 * F(1, 6) * eta / (1 - q("6.01") * u) < q(".61"), "left z upper") + require(1 + 3 * q(".028") + eta * q(".61") / 6 < q("1.77"), "left hstar upper") + require(2 * eta * u*u < q(".07"), "left chi upper") + # All f(r)*r^n used here decrease for r>=6 (n<=7). Check at r=6. + f6 = F(7, 10**9) + gaussian_errors = [2*f6/36, 4*f6/6, 6*f6, 54*f6] + for j, err in enumerate(gaussian_errors): + require(err < F(1, 100) * u * F(factorial(j), 6**(j+1)), "left Gaussian correction") + qd = q("1.01") * f6 * (6 + q("1.01") * f6) + bld = q(".028") * (2 * eta / 216) * (6 + q("1.01") * f6) + require(2 * (1-qd) * (1-q(".028")**2) - 2*bld > 0, "left log M derivative") + require(q(".028") * (1 + kl_over_b + q(".75")) < q(".05072"), "left E upper") + sub11 = ASTAR*q(".028")**2 + MPRIME/4 + (2*eta/216)**2/ALPHA + q(".56")*q(".05072") + sub22 = MPRIME + q(".72") * q("1.5") * q(".028") + require(sub11 < q(".224") and sub22 < q(".802"), "left matrix diagonals") + require(q("1.5") * q(".56") < MSTAR, "left offdiagonal correction nonpositive") + require(MPRIME/2 - MSTAR*q(".028") > q(".34") and MPRIME/2 < q(".386"), "left matrix offdiagonal") + margin = (q(".43")-q(".224")-q(".13")) * (q("2.72")-q(".802")-q(".75")) - (q(".13")-q(".386"))**2 + require(margin > 0, "left matrix determinant") + print("Left-tail Gaussian corrections and sufficient scalar/matrix margins: PASS") + print("Left-tail determinant margin =", down(margin)) + + +def right_tail(): + # H=(F1-1)/(2u)=1/2 integral_0^1 F1'(tu)dt. + # For F1 log F1, use .99<=F1<=1 and 0<=1+log F1<=1. + g2 = 15 + 6 + F(100, 99) + g3 = 315 + 90 + 18*F(100, 99) + F(100, 99)**2 + require(g2 < 23 and g3 < 435, "G1 derivative bounds") + print("G1 derivative upper bounds <=", up(g2), "< 23;", up(g3), "< 435") + tlo, thi = q("3.221"), q("3.222") + g0, c0 = q("11.5") + 3*thi, 435 + 90*thi + ab, aj, ce = 3*thi-q("8.5"), 12*thi-37, 197-60*thi + # x=log(r/10), u=.01 exp(-2x), theta=theta0+x. For linear + # times exponential, locate its sole critical point then use + # exp(-t)<=1/(1+t). The higher-order remainder terms decrease. + require(0 < ab < F(3, 2) and 0 < aj < 6 and ce > 0, "tail critical point ranges") + require(15 < 4*(5*g0+3) and 90 < 4*(30*g0+33) + and 630 < 4*(210*g0+321) and 90 < 6*c0, "decreasing tail remainders") + errb = q(".015") / (2 - 2*ab/3) + (5*g0+3)/10000 + errj = q(".06") / (2-aj/6) + (30*g0+33)/10000 + erre = max((197-60*tlo)/100, q(".3") / (2+ce/30)) + (210*g0+321)/10000 + 8*c0/1000000 + require(errb < q(".04") and errj < q(".13") and erre < q(".75"), "right error bounds") + print("Right-tail B,J,E error upper bounds <=", up(errb), up(errj), up(erre)) + # These derivative-sign checks extend the following endpoint bounds to + # every x>=0, where theta=theta0+x and u=.01 exp(-2x). + require(1 < 2*(tlo-q("1.46")), "decreasing u B upper") + require(2 < 3*(2*tlo-q("3.87")), "decreasing J/r^3 upper") + require(1 < 2*(tlo-q(".475")), "decreasing u C upper") + require(1/(tlo-q(".475")) + 2/(2*tlo-q("3.87")) < 2, + "decreasing u C J upper") + require(1 < tlo-q("1.46"), "decreasing B/r upper") + require((thi-q("1.46"))/10 < q(".18"), "right B/r upper") + # C1=theta-.5+(2.5-theta)u+R0, |R0|<=g0 u^2. + # The upper correction is negative since both coefficients in + # 2.5-theta+.01*(11.5+3theta) decrease with theta. + require(q("2.5")-tlo+(q("11.5")+3*tlo)/100 < 0, "right C1 upper") + require(q(".99")*tlo-q(".5")-(q("11.5")+3*thi)/10000 > 0, + "right C1 positive") + require((6*tlo-q("14.75"))/(2*tlo-q("3.87")) > q("1.7"), "right E/J lower") + require(-q("13.25") < -3*q("4.13"), "right E/J upper 3") + require(q(".5")-(thi-q("1.46"))/100 > q(".482"), "right b lower") + require((2*thi-q("3.87"))/1000 < q(".0026"), "right l upper") + require(q(".5")+(thi-q(".475"))/100 < q(".528"), "right k/r upper") + require((thi-q(".475"))*(2*thi-q("3.87"))/100 < q(".075"), "right u C1 J upper") + require(q(".5")-q(".605")/100-q(".0026")**2/ALPHA > q(".493"), "right p lower") + require(-q(".395")+q(".075") < 0, "right p upper .5") + require(q(".528")*q(".18")+2*q(".22")*q(".0026")/ALPHA < q(".14"), "right h0") + require(q(".14")**2 < q(".67")**2*q(".493"), "right h0 over sqrt p") + require(DS*(1+(q(".125")-q(".0026")**2/ALPHA)/q(".5")) > q(".367"), "right K lower") + require(DS*(1+q(".482")*(q(".75")-q(".482"))/q(".493")) < q(".4"), "right K upper") + require(DS/q(".493") < q(".640") and DS*q(".75")/q(".493") < q(".48"), "right D,J upper") + require(1+3*q(".482")-3*q(".528") > q(".8"), "right hstar lower") + require(1+3*q(".5")-q("1.7")*q(".5") < q("1.77"), "right hstar upper") + require(F(3, 10) < q(".475"), "right z lower") + require((2*thi-q("3.87"))/(100*q(".97")) < q(".027"), "right chi upper") + require(2*q(".88")-q(".027")/(1-q(".5")**2) > q(".68"), "right log M derivative") + extra = q(".482")*(q(".75")-q(".482")) + sub11 = ASTAR/4-MSTAR/2+MPRIME/4+q(".0026")**2/ALPHA+DS*(q(".5")+extra)**2/q(".5") + sub22 = MPRIME+DS*(q("1.5")*q(".5"))**2/q(".493") + require(sub11 < q(".403") and sub22 < q("1.12"), "right matrix diagonals") + require(MPRIME/2+(-MSTAR+q("1.5")*q(".367"))*q(".5") > q(".080"), "right matrix offdiagonal lower") + require(MPRIME/2+(-MSTAR+q("1.5")*q(".4"))*q(".482") < q(".13"), "right matrix offdiagonal upper") + margin = (q(".43")-q(".403")-q(".022"))*(q("2.72")-q("1.12")-q(".75"))-q(".05")**2 + require(margin > 0, "right matrix determinant") + print("Right-tail sufficient scalar/matrix margins: PASS") + print("Right-tail determinant margin =", down(margin)) + + +SECANTS = parse_rows(""" +1.867 .52 1.314 2.34 .55 1.155 +1.766 .49 1.813 2.564 .433 1.155 +1.681 .664 2.057 2.621 .420 1.155 +1.55 .847 2.101 2.461 .615 1.155 +1.263 .912 1.787 1.885 .770 1.205 +.913 .869 1.386 1.071 .840 1.251 +""") + + +def secants(): + limits = [None, [q(".42")**2, q(".61")**2], [q(".20"), q(".295")], + [q(".23")]*2, [q(".23")]*2, [q(".39")]*2] + for ri, (name, row, triples) in enumerate(zip(("O", "P1", "R1", "U1", "V1", "T1"), BINS, SECANTS)): + bmin, bmax, lmax, dmax, kmin, kmax, jmax, l1, _ = row + bounds = [] + for zi, (zm, zM) in enumerate(((q("-.08"), q(".7")), (q("-.475"), F(0)))): + claimed = triples[3*zi:3*zi+3] + maxima = [F(0)]*3 + for b, l, k, j, h, z in product((bmin, bmax), (F(0), lmax), (kmin, kmax), + (F(0), jmax), (q(".8"), q("1.77")), (zm, zM)): + w = h-2*b-2*l*z + vals = [abs(w), abs(ASTAR*b-MSTAR/2+l*z+k*w), abs(-MSTAR+j*w)] + maxima = list(map(max, zip(maxima, vals))) + require(all(x <= y for x, y in zip(maxima, claimed)), f"secant corners {name},{zi}") + w, a, m = claimed + chi = (q(".07"), q(".097"))[zi] + cst = ASTAR+max(abs(zm), abs(zM))**2/ALPHA+a*a/l1+m*m/q(".75") + if dmax is not None: + bound = chi*chi*(cst+dmax*w*w) + require(bound < limits[ri][zi], f"F_j bound {name},{zi}") + bounds.append(up(bound)) + else: + require(cst < (20, 12)[zi], "O constant term") + require(q(".295")*w*w < (q("1.04"), q("1.63"))[zi], "O quadratic term") + bounds.append(up(cst)+" + "+up(q(".295")*w*w)+" (7+t0^2)") + quantity = "F_j/chi^2" if dmax is None else "F_j" + print("Secant", name, quantity, "upper bounds:", "; ".join(bounds), "PASS") + print("36 corner-derived constants and all 12 secant F_j bounds: PASS") + + +def main(): + print("Exact scalar certificate (standard-library Fraction arithmetic)") + anchors() + central() + left_tail() + right_tail() + secants() + print("ALL CHECKS PASSED. Decimal displays are outward rounded; all tests are rational.") + + +if __name__ == "__main__": + main() diff --git a/preprints/A-single-lattice-covering-bound-of-order-n-log-n-September-23-2026/README.md b/preprints/A-single-lattice-covering-bound-of-order-n-log-n-September-23-2026/README.md new file mode 100644 index 0000000000000000000000000000000000000000..f11ec5236e6859cdca3b2f7b0ae9e5c7ece4b487 --- /dev/null +++ b/preprints/A-single-lattice-covering-bound-of-order-n-log-n-September-23-2026/README.md @@ -0,0 +1,17 @@ +# [A single-lattice covering bound of order n log n](paper.pdf) + +**Author:** OpenAI + +**Date:** September 23, 2026 + +## Citation + +```bibtex +@misc{OAI:A-single-lattice-covering-bound-of-order-n-log-n-September-23-2026, + author = {{OpenAI}}, + title = {{A single-lattice covering bound of order $n\log n$}}, + howpublished = {OpenAI Math Release preprint + \href{https://github.com/openai/math/blob/main/preprints/A-single-lattice-covering-bound-of-order-n-log-n-September-23-2026/paper.pdf}{OAI:A-single-lattice-covering-bound-of-order-n-log-n-September-23-2026}}, + year = {2026} +} +``` diff --git a/preprints/A-single-lattice-covering-bound-of-order-n-log-n-September-23-2026/build/blocks.tex b/preprints/A-single-lattice-covering-bound-of-order-n-log-n-September-23-2026/build/blocks.tex new file mode 100644 index 0000000000000000000000000000000000000000..9257d4bdf8e44389566b175320ad684c60498689 --- /dev/null +++ b/preprints/A-single-lattice-covering-bound-of-order-n-log-n-September-23-2026/build/blocks.tex @@ -0,0 +1,482 @@ +\section{Folded Gaussian blocks} +\label{sec:blocks} + +We next turn the uniform Boolean sampler into a distribution adapted to +Gaussian weights. The folded Gaussian and biased-bit description follow +the setup of Li and Liu~\cite[Section~3]{LL} and +\cite[Section~2]{LLC}; all estimates required for the hierarchy are +proved here. Throughout this section all constants, including the +lower threshold on the integer block size $b$, are absolute. For a finite +set or group, an expectation with an unqualified subscript denotes the +uniform average. + +Each point of a unit cube has two adjacent integer corrections in each +coordinate, zero and one. We attach a Gaussian weight to every resulting +bit vector. Summing these weights folds the Gaussian onto the cube; their +normalized values form a product law on the bits. We retain cube locations +where this sum has its typical size and no single bit vector carries too +much weight. After discretizing the cube, a line subgroup lets us choose +at most one usable location in each coset while retaining almost all +of the mean weight. +The preimage of that subgroup is one lattice. The next section will use +these block lattices together. + +\subsection{The folded distribution and its grid} + +Set +\begin{equation} + h_b=\sqrt{\frac94\log b},\qquad + f_b(t)=h_b\gamma_1(h_bt)+h_b\gamma_1(h_b(t-1)),\qquad + F_b(t)=\prod_{j=1}^b f_b(t_j), + \label{eq:folded-density} +\end{equation} +where $0\le t\le1$ in the definition of $f_b$ and +$t\in[0,1]^b$ in that of $F_b$. At $t\in[0,1]^b$, let $P_{b,t}$ be the +product probability distribution on $\{0,1\}^b$ whose $j$th coordinate has +probabilities +\[ + P_{b,t}(\eps_j=e) + =\frac{h_b\gamma_1(h_b(t_j-e))}{f_b(t_j)},\qquad e\in\{0,1\}. +\] +Thus the following identity holds for every bit vector: +\begin{equation} + h_b^b\gamma_b\bigl(h_b(t-\eps)\bigr) + =F_b(t)P_{b,t}(\eps). + \label{eq:folded-identity} +\end{equation} +Write +\[ + a_b=\int_0^1 f_b(t)\,dt,\qquad + \mu_b=\frac1{a_b}\int_0^1f_b(t)\log f_b(t)\,dt. +\] + +\begin{lemma}[Folded Gaussian grid] +\label{lem:folded-grid} +For every sufficiently large $b$, there is a prime $p_b$ satisfying +\begin{equation} + b\mu_b-4b^{0.56}\le\log p_b\le b\mu_b-3b^{0.56}. + \label{eq:block-prime} +\end{equation} +Let $G_b=\F_{p_b}^b$, and associate with $z\in G_b$ the anchor +$t_z=z/p_b\in[0,1)^b$, using the representatives +$0,\ldots,p_b-1$ in every coordinate. Define the eligible set +$E_b\subset G_b$ by the two conditions +\begin{equation} + \left|\log F_b(t_z)-b\mu_b\right|\le b^{0.56},\qquad + \#\left\{j:\left|t_{z,j}-\frac12\right|\le h_b^{-2}\right\} + \ge b^{0.70}. + \label{eq:block-eligible} +\end{equation} +Then +\begin{equation} + \E_{z\in G_b}F_b(t_z)=1+O(b^{-1/20}),\qquad + \E_{z\in G_b}F_b(t_z)\ind_{E_b}(z)=1+O(b^{-1/20}). + \label{eq:block-mass} +\end{equation} +Moreover, $\mu_b=\log h_b+O(1)$. Put +$\eta_b=bh_b^2/p_b$. If $t$ belongs to the grid cell +$Q_z=t_z+[0,p_b^{-1})^b$, then, for every $\eps\in\{0,1\}^b$, +\begin{equation} + \begin{split} + e^{-\eta_b}F_b(t_z)&\le F_b(t)\le e^{\eta_b}F_b(t_z),\\ + e^{-\eta_b}\frac{F_b(t_z)}{p_b}P_{b,t_z}(\eps) + &\le\frac{h_b^b}{p_b}\gamma_b\bigl(h_b(t-\eps)\bigr) + \le e^{\eta_b}\frac{F_b(t_z)}{p_b}P_{b,t_z}(\eps). + \end{split} + \label{eq:block-rounding} +\end{equation} +There is an absolute $c>0$ such that, for every eligible $z$, +every $t\in Q_z$, and every $\eps\in\{0,1\}^b$, +\begin{equation} + \frac{F_b(t_z)}{p_b}P_{b,t_z}(\eps)\le e^{-c b^{0.70}}, + \qquad + \frac{h_b^b}{p_b}\gamma_b\bigl(h_b(t-\eps)\bigr) + \le e^{-c b^{0.70}}. + \label{eq:block-atom} +\end{equation} +\end{lemma} + +\begin{proof} +First, if $X$ is a standard normal random variable, then +$a_b=\Prob(|X|\le h_b)$. The map +\[ + X\longmapsto + \begin{cases} + X/h_b,&X\ge0,\\ + 1+X/h_b,&X<0 + \end{cases} +\] +pushes its conditional law on $[-h_b,h_b]$ to the probability density +$f_b/a_b$ on $[0,1]$. At the resulting point $t$, one summand in +$f_b(t)$ equals $h_b\gamma_1(X)$, whereas +$f_b(t)\le2h_b/\sqrt{2\pi}$ everywhere. Consequently +\[ + \log h_b-\frac12\log(2\pi) + -\frac12\E[X^2\mid |X|\le h_b] + \le\mu_b\le\log h_b+\log\frac2{\sqrt{2\pi}}. +\] +The conditional second moment is bounded by an absolute constant, proving +$\mu_b=\log h_b+O(1)$. The elementary Gaussian tail estimate also gives +\begin{equation} + 1-a_b\le C h_b^{-1}e^{-h_b^2/2} + =C h_b^{-1}b^{-9/8},\qquad + \int_{[0,1]^b}F_b(t)\,dt=a_b^b=1-O(b^{-1/8}). + \label{eq:folded-total-mass} +\end{equation} + +Here is an elementary way to choose the prime, avoiding any quantitative +prime-gap input. Since $\mu_b\ge\log h_b-C\to\infty$, +$T=b\mu_b-3b^{0.56}$ is positive for large $b$. Set +$l=\lfloor e^T/2\rfloor$. For a prime $q$, the factorial valuation formula +gives +\[ + v_q\binom{2l}{l} + =\sum_{k\ge1}\left(\left\lfloor\frac{2l}{q^k}\right\rfloor + -2\left\lfloor\frac l{q^k}\right\rfloor\right) + \le\lfloor\log_q(2l)\rfloor. +\] +Thus the power of each prime in the binomial coefficient is at most $2l$. +If all its prime divisors were at most $l/\log(2l)$, their number would be +at most $l/\log(2l)$, and hence +$\log\binom{2l}{l}\le l$. This contradicts +$\binom{2l}{l}\ge4^l/(2l+1)>e^l$ for large $l$. +There is therefore a prime divisor $p_b$ with +\[ + \frac l{\log(2l)} + \frac12b^{0.56}\right) + \le C b^{-0.12}(\log b)^2=O(b^{-1/20}). +\] +For one coordinate, on the interval +$|t-1/2|\le(2h_b^2)^{-1}$ we have +$f_b(t)\ge c h_b e^{-h_b^2/8}$, with an absolute $c>0$. +Its probability under $f_b/a_b$ is therefore at least +$c h_b^{-1}b^{-9/32}$. The number $N$ of coordinates in this interval is +binomial with mean +\[ + \E N\ge c h_b^{-1}b^{23/32}\ge2b^{0.70} +\] +for sufficiently large $b$, since $23/32>0.70$. Its variance is at most +its mean, so +\[ + \Prob(N0$ and increasing +the absolute threshold. Equation~\eqref{eq:block-rounding} proves the +same assertion at all points of the cell, again decreasing $c$ if needed. +\end{proof} + +The exponents are chosen to leave room between four different estimates: +\[ + \frac12<0.56,\qquad + \frac{0.56}{0.9}<0.65<0.69<0.70<\frac{23}{32}. +\] +The information window exceeds the square-root fluctuation scale; the +previous block's weight cap fits the sampler; and the number of balanced +coordinates exceeds the number of fair bits needed. The first block will +have size about $(\log\log n)^2$, so $2(0.70)>1$ also makes its atom +bound stronger than the final logarithmic loss. None of these values +is intended to be optimal. + +\subsection{Usable anchors and sparse line maxima} + +The next lemma has two roles. Most of the folded Gaussian mass is kept on +anchors whose biased bits sample a prescribed preceding block uniformly +in its base point. A suitable line in the new grid then permits this mass +to be represented by one selected anchor in each coset. + +\begin{lemma}[Usable anchors] +\label{lem:usable} +There are absolute constants $A,B>0$ with the following property. +Let $b\ge B$ be an integer, and use the objects in +Lemma~\ref{lem:folded-grid}. Suppose that $H$ is a finite abelian group, +$\log|H|\le b^4$, and $g_0:H\to[0,\infty)$ satisfies +\[ + \frac12\le\bar g_0:=\E_{x\in H}g_0(x)\le2,\qquad + g_0(x)\le\exp(5a^{0.56})\quad(x\in H) +\] +for some $1\le a\le b^{1/0.9}$. Then one can choose shifts +$w_1,\ldots,w_b\in H$, a usable set $U_b\subset E_b$, and a +one-dimensional linear subspace $C_b\subset G_b$ such that, writing +$W\eps=\sum_{j=1}^b\eps_jw_j$, +\begin{equation} + \sum_{\eps\in\{0,1\}^b}P_{b,t_z}(\eps)g_0(x-W\eps) + \ge(1-A b^{-1/100})\bar g_0 + \qquad(z\in U_b,\ x\in H). + \label{eq:block-biased-sampler} +\end{equation} +Define +\begin{equation} + v_b(z)=\frac{F_b(t_z)}{p_b}\ind_{U_b}(z),\qquad + g_b(x)=\max_{c\in C_b}v_b(x-c),\qquad + \bar g_b=\E_{x\in G_b}g_b(x). + \label{eq:block-line-weight} +\end{equation} +These choices satisfy +\begin{equation} + |\bar g_b-1|\le A b^{-1/100},\qquad + 0\le g_b(x)\le\exp(5b^{0.56})\quad(x\in G_b),\qquad + \left|\E_z F_b(t_z)\ind_{U_b}(z)-1\right| + \le A b^{-1/20}. + \label{eq:block-usable-bounds} +\end{equation} +There is also a base version with no $H$, $g_0$, or $W$: take $U_b=E_b$ +and choose $C_b$ so that the conclusions concerning $g_b$ in +Equation~\eqref{eq:block-usable-bounds} hold. +\end{lemma} + +\begin{proof} +We first select $W$ and $U_b$ when preceding data are given. +For large $b$, the bound on $a$ implies +\[ + 5a^{0.56}\le5b^{0.56/0.9}\le b^{0.65}, +\] +because $0.56/0.9<0.65$. Thus $g_0$ satisfies the cap in +Lemma~\ref{lem:sampler}. Choose the columns $w_j$ independently and +uniformly from $H$. + +Fix an eligible anchor $z$. A bit whose probability of being $1$ is $q$ +has the following exact mixture representation: with probability +$2\min(q,1-q)$ use a fair bit, and otherwise use its more likely value. +Make the mixture choices independently at the $b$ coordinates. Write +$S\subset\{1,\ldots,b\}$ for the coordinates chosen to be fair, and +$d_j$ for the deterministic value at $j\notin S$. +Conditional on these choices, the resulting vector has fair independent +bits on $S$ and the values $d_j$ elsewhere; averaging these conditional +laws recovers $P_{b,t_z}$ exactly. + +By Equation~\eqref{eq:block-bit-odds}, at least $b^{0.70}$ coordinates +are selected into $S$ with probability at least $2\alpha$, where +$\alpha=(1+e)^{-1}$. Apply Chebyshev's inequality to the number selected +among these coordinates. Its mean is at least $2\alpha b^{0.70}$, +its variance is at most its mean, and $b^{0.69}$ is at most half its mean +for sufficiently large $b$. Hence +\begin{equation} + \Prob\bigl(|S|0$, choose $c\in C_b$ attaining its maximum and let +$z=x-c\in U_b$, again using standard representatives. Then +\begin{equation} + \lambda=\frac{k-z}{p_b}\in L_b,\qquad + \tau=t-\lambda=\frac{z+\{p_b t\}}{p_b}\in Q_z,\qquad + g_b(x)=\frac{F_b(t_z)}{p_b}. + \label{eq:block-cell-identity} +\end{equation} +For all $\eps\in\{0,1\}^b$, the point $\lambda+\eps$ lies in $L_b$, +its residual $t-\lambda-\eps=\tau-\eps$ has coordinates in $[-1,1]$, +and +\begin{equation} + \frac{h_b^b}{p_b}\gamma_b\bigl(h_b(t-\lambda-\eps)\bigr) + \ge e^{-\eta_b}g_b(x)P_{b,t_z}(\eps). + \label{eq:block-cell-lower} +\end{equation} +The corresponding individual weight is at most $e^{-c b^{0.70}}$. +\end{lemma} + +\begin{proof} +The integer vector $k-z$ reduces to $c$ modulo $p_b$, proving +$\lambda\in L_b$. The remaining identities in +Equation~\eqref{eq:block-cell-identity} follow from +$p_b t=k+\{p_b t\}$ and the chosen maximizing anchor. Adding an integer +bit vector preserves $L_b$. Since $\tau\in[0,1)^b$, the residual +coordinate bound follows. Apply +Equations~\eqref{eq:block-rounding} and~\eqref{eq:block-atom} at +$\tau\in Q_z$ to obtain the stated lower and upper estimates. +\end{proof} + +In the hierarchy below, the preceding group is $H=G_a$ with +$a\le b^{1/0.9}$. Its size satisfies the remaining hypothesis of +Lemma~\ref{lem:usable}, because +\[ + \log|G_a|=a\log p_a=O(a^2\log\log a) + \le O(b^{2/0.9}\log\log b)\le b^4 +\] +for sufficiently large $b$. Both this bound and the cap comparison in +the proof have strict exponent margins. Thus the same absolute cutoff +works at every level, independently of the number of blocks. diff --git a/preprints/A-single-lattice-covering-bound-of-order-n-log-n-September-23-2026/build/figures/suffix-tree.tex b/preprints/A-single-lattice-covering-bound-of-order-n-log-n-September-23-2026/build/figures/suffix-tree.tex new file mode 100644 index 0000000000000000000000000000000000000000..65dd86e492afd23d085906e566256a7c70f51043 --- /dev/null +++ b/preprints/A-single-lattice-covering-bound-of-order-n-log-n-September-23-2026/build/figures/suffix-tree.tex @@ -0,0 +1,20 @@ +\begin{tikzpicture}[x=1cm,y=1cm, + every node/.style={font=\small}, + edge/.style={-{Latex[length=1.6mm]},thin}] +\node (root) at (0,0) {$P$}; +\node (left) at (-2.7,-1.2) {$P_v$}; +\node (right) at (2.7,-1.2) {$P_{v+1}$}; +\draw[edge] (root) -- node[above left] {$l[q]=v$} (left); +\draw[edge] (root) -- node[above right] {$l[q]=v+1$} (right); +\node (la) at (-4,-2.5) {$P_{v,a}$}; +\node (lb) at (-1.4,-2.5) {$P_{v,a+1}$}; +\node (ra) at (1.4,-2.5) {$P_{v+1,b}$}; +\node (rb) at (4,-2.5) {$P_{v+1,b+1}$}; +\draw[edge] (left) -- node[above left] {$l[q-1]=a$} (la); +\draw[edge] (left) -- node[above right] {$a+1$} (lb); +\draw[edge] (right) -- node[above left] {$b$} (ra); +\draw[edge] (right) -- node[above right] {$b+1$} (rb); +\foreach \name in {la,lb,ra,rb} + \draw[densely dotted] (\name.south) -- ++(0,-0.45); +\node at (0,-3.4) {Continue through the earlier coordinates.}; +\end{tikzpicture} diff --git a/preprints/A-single-lattice-covering-bound-of-order-n-log-n-September-23-2026/build/final.tex b/preprints/A-single-lattice-covering-bound-of-order-n-log-n-September-23-2026/build/final.tex new file mode 100644 index 0000000000000000000000000000000000000000..e9cc3a74a426d9bff0287cbe6cc9f2d88614c0c6 --- /dev/null +++ b/preprints/A-single-lattice-covering-bound-of-order-n-log-n-September-23-2026/build/final.tex @@ -0,0 +1,303 @@ +\section{One shear and exact coverage}\label{sec:final} + +We extend the simultaneous Boolean-shift argument of Li and Liu +\cite[Theorem~4.3]{LL} to patterns whose branch bases may depend on later +coordinates. Unit separation between siblings still allows one linear +shear to treat every pattern in a finite family. Throughout this section +all measures on a lattice torus are normalized Haar measures. + +\subsection{A simultaneous shear for finitely many patterns} + +A labeled pattern in $\R^q$ is a finite nonempty set of points, with one +label assigned to each point. Call it \emph{binary by suffixes} if, +after fixing any suffix of coordinates, the next coordinate has at most +two possible values, and two such values always differ by $1$. +For $q=0$ a pattern is a single label. Such a pattern has at most $2^q$ +points. Deleting points preserves the suffix rule; the result is again a +pattern whenever it is nonempty. Taking a nonempty slice at the last +coordinate and then deleting that coordinate also preserves the rule. + +Figure~\ref{fig:suffix-tree} shows why a fixed global binary cube is not +required. + +\begin{lemma}[Simultaneous shear]\label{lem:shear} +Let $\mathcal T=\R^m/\Lambda_h$, and let $U_a\subset\mathcal T$ be +measurable sets with $|U_a|\le\eta_a$, where $\eta_a>0$. +For $0\le q\le D$, let $\mathcal P_q$ be finite families of labeled +patterns in $\R^q$, binary by suffixes. Suppose they are closed under +last-coordinate slicing. Write +\[ + Q=\sum_{q=0}^D|\mathcal P_q|,\qquad L_*>\max(1,Q). +\] +There are real vectors $Z_1,\ldots,Z_D\in\R^m$ such that, for every +$P\in\mathcal P_q$, +\begin{equation}\label{eq:shear-bound} + \left|\bigcap_{l\in P} + \left(U_{a(l)}+ + \pi_{\Lambda_h}\left(\sum_{j=1}^q l[j]Z_j\right)\right)\right| + \le L_*^{|P|-1}\prod_{l\in P}\eta_{a(l)}. +\end{equation} +Here $l[j]$ denotes the $j$th scalar coordinate of $l$. +\end{lemma} + +\begin{proof} +The assertion in dimension zero is the assumed bound on each $U_a$. +Suppose $Z_1,\ldots,Z_{q-1}$ have been fixed so that all preceding +estimates hold. A pattern with just one value of $l[q]$ preserves its +bound under the resulting common translation. + +Otherwise the two last-coordinate values are $v$ and $v+1$. +Let the two slices have cardinalities $k_1$ and $k_2$, and let $A$ and +$B$ be their intersection sets. Choose $Z_q$ uniformly in a fixed +fundamental parallelepiped of $\Lambda_h$. Translation invariance gives +\[ + |(A+\pi(vZ_q))\cap(B+\pi((v+1)Z_q))| + =|A\cap(B+\pi(Z_q))|. +\] +This identity is valid even when $v$ is not an integer: the products +$vZ_q$ and $(v+1)Z_q$ are formed in $\R^m$ before projection. +Only the relative shift must be Haar uniform. By Fubini, +\[ + \E_{Z_q}|A\cap(B+\pi(Z_q))|=|A||B| + \le L_*^{k_1+k_2-2}\prod_{l\in P}\eta_{a(l)}. +\] +Markov's inequality bounds the probability of violating +Equation~\eqref{eq:shear-bound} by $1/L_*$. If either slice has +measure zero the intersection has measure zero almost surely, +which gives the same conclusion. A union bound over +$\mathcal P_q$ has total probability less than one, so one choice of +$Z_q$ works for all its patterns. Induction proves the result. +\end{proof} + +\begin{figure}[htbp] +\centering +\input{figures/suffix-tree} +\caption{The suffix rule used by the shear induction. The bases $a$ and +$b$ may differ because the later coordinate has already been fixed. +Only the separation of siblings is prescribed. Removing leaves, or an +entire subtree, preserves the rule.} +\label{fig:suffix-tree} +\end{figure} + +\subsection{Removing a small hole} + +The next argument is the classical completion-by-dilation method of +Rogers~\cite{Rog59}. We use the small-dilation form also given by +Fejes T\'oth~\cite[Lemma~2]{FT09}, and include its proof for the +single-lattice setting. + +\begin{lemma}[Completion]\label{lem:completion} +Let $K\subset\R^n$ be a convex body and $\Gamma$ a full-rank lattice. +If its hole proportion $\delta=1-|\pi_\Gamma(K)|$ satisfies +\[ + \delta<(1-\delta)k^{-n} +\] +for a positive integer $k$, then $(1+1/k)K+\Gamma=\R^n$. +\end{lemma} + +\begin{proof} +Multiplication by $k$ on $\R^n/\Gamma$ maps $\pi_\Gamma(K/k)$ +onto $\pi_\Gamma(K)$. Its image has measure at most $k^n$ times +the measure of the original set: partition a fundamental +parallelepiped into $k^n$ cells on which the map is injective. +Consequently +\[ + |\pi_\Gamma(K/k)|\ge k^{-n}(1-\delta)>\delta. +\] +For every point $x$ of the torus, the sets $\pi_\Gamma(K)$ and +$x-\pi_\Gamma(K/k)$ have measures summing to more than one and +therefore intersect. Thus +$\pi_\Gamma(K)+\pi_\Gamma(K/k)$ is the whole torus. +Finally, convexity gives $K+K/k=(1+1/k)K$. Neither symmetry nor +the inclusion $0\in K$ is required. +\end{proof} + +\subsection{Proof of the covering theorem} + +\begin{proof}[Proof of Theorem~\ref{thm:main}] +All constants in the vertical construction, including its cutoff $B$, +are fixed first. For sufficiently large $n$, put $s=\log\log n$ and +use Proposition~\ref{prop:vertical} with $b_1=\lceil s^2\rceil$. +Retain its notation $L,M,D,D_v,b_i,h_i$, and set +\[ + R^2=\sum_{i=1}^{r+1}b_i h_i^2,\qquad m=n-D. +\] +Here $D=O(s^2+1)$ and $R$ is bounded by a fixed polynomial in $s$. +After passing to the affine coordinates of Lemma~\ref{lem:sections}, +there are at most $(Cm^2)^D$ horizontal bodies $J_a$ such that every +$v$ with $\|v\|\le R$ has a label satisfying +\begin{equation}\label{eq:final-sections} + J_a\subset K_v,\qquad + c_{\rm sec}\vol_n(K)\gamma_D(v) + \le\vol_m(J_a)\le + C_{\rm sec}\vol_n(K)\gamma_D(v). +\end{equation} + +Choose an absolute constant $C_*$ later and put +\begin{equation}\label{eq:target-density} + \rho=C_*n\log n,\qquad + D_h=\frac{\vol_n(K)}{\rho D_v},\qquad + u_a=\frac{\vol_m(J_a)}{D_h}. +\end{equation} +For any vertical target $y$, use its pattern from +Proposition~\ref{prop:vertical} and label each raw point $l$ by a +body contained in the section at $y-Ml$. The residual bound permits +Equation~\eqref{eq:final-sections}. With the chosen horizontal covolume, +the normalized section volume is comparable to the corresponding +normalized Gaussian weight multiplied by $\rho$: +\[ + c_{\rm sec}\rho D_v\gamma_D(y-Ml) + \le u_{a(l)}\le + C_{\rm sec}\rho D_v\gamma_D(y-Ml). +\] +Thus the total weight supplies the sum of the horizontal loads, while +the atom bound controls each load separately. These estimates give +\begin{equation}\label{eq:final-loads} + \sum_l u_{a(l)}\ge c_3\rho,\qquad + \max_l u_{a(l)}\le U_n:=C\rho\exp(-c b_1^{0.70}), +\end{equation} +where $c_3=c_{\rm sec}\kappa>0$ is absolute. + +Three scale comparisons are needed. First, +\begin{equation}\label{eq:load-upper-check} + \frac{U_n}{m} + \le C'\exp\bigl(s-c's^{1.4}\bigr)\longrightarrow0. +\end{equation} +Second, every pattern has at most $2^D$ points, so deleting those with +$u_{a(l)}<\sqrt m$ loses at most +\begin{equation}\label{eq:load-discard-check} + 2^D\sqrt m=n^{1/2+o(1)}=o(\rho). +\end{equation} +Third, the logarithm of the number of labels is +$O(D\log m)=o(\sqrt m)$. These statements follow from +$D=O(s^2+1)$ and $\log n=e^s$; all their constants are independent +of $K$. In particular the retained pattern is nonempty and has load +at least $c_3\rho/2$ for large $n$. + +Restrict the label list to $\sqrt m\le u_a\le U_n$. +Lemma~\ref{lem:horizontal} gives one lattice $\Lambda_h$ of covolume +$D_h$ for which the hole sets +\[ + U_a=(\R^m/\Lambda_h)\setminus\pi_{\Lambda_h}(J_a) +\] +satisfy $|U_a|\le e^{-u_a/2}$ simultaneously. We next select one +shear that works for every retained pattern, not a separate shear +for each target. + +\paragraph{A finite family for all targets.} +Initially restrict to $y=Mz$ with $z\in[0,1)^D$. This is enough +because $L$ contains $\Z^D$; after the final lattice is defined, +integer raw translations will reduce any target to this region. +The finite-alphabet conclusion of Proposition~\ref{prop:vertical} +places each scalar raw coordinate in $[-Q_{\rm raw},Q_{\rm raw}]$, where +$Q_{\rm raw}=1+(D+2)^D$. In block $i\le r$ it is a multiple of $1/p_i$; +in the terminal block it is an integer. The logarithmic alphabet size +is therefore at most +\[ + O(D\log(D+2)+\max_i\log p_i)=O(s^{c_0}) +\] +for some fixed absolute exponent $c_0$. + +Order scalar coordinates by increasing block index, and take +$\mathcal P_q$ to be \emph{all} labeled patterns on the first $q$ +coordinate alphabets that are binary by suffixes, for $0\le q\le D$. +This includes every retained full pattern: the base in a block depends +only on later blocks, and deleting points cannot introduce a third +value at any suffix. The families are closed under slicing. + +Let $\mathcal A_j$ be the alphabet of scalar coordinate $j$, and let +$N_{\rm lab}$ be the number of retained section labels. A labeled point +consists of a coordinate tuple and one such label, so +$N=N_{\rm lab}\prod_{j=1}^D|\mathcal A_j|$ bounds their number in +each of these coordinate spaces. Consequently +\[ + \log(N+1)=O(D\log n+s^{c_1}),\qquad + Q:=\sum_{q=0}^D|\mathcal P_q| + \le (D+1)(N+1)^{2^D}. +\] +For example, the last bound follows by encoding a set of at most +$2^D$ points as a list of that length, padded with a dummy symbol. +Take $L_*=2(Q+1)$. Then +\begin{equation}\label{eq:pattern-cost} + 2^D\log L_* + \le C4^D(D\log n+s^{c_1})+C2^D\log(D+2) + =o(n\log n). +\end{equation} +Indeed $4^D$ times any fixed polynomial in $s$ is +$\exp(O(s^2))=o(n)$. + +\paragraph{The covering lattice.} +Apply Lemma~\ref{lem:shear} with $\eta_a=e^{-u_a/2}$ and set +$Zl=\sum_{j=1}^D l[j]Z_j$. Define +\begin{equation}\label{eq:full-lattice} + \Gamma=\{(h+Zl,Ml):h\in\Lambda_h,\ l\in L\}. +\end{equation} +It is a full-rank lattice, being the image of the product lattice +$\Lambda_h\times L$ under an invertible block triangular linear +map. Its covolume is exactly $D_hD_v$, so +$\vol_n(K)/\det\Gamma=\rho$. + +For a fixed $y$ and a retained labeled pattern, each $J_{a(l)}$ +lies in the section $K_{y-Ml}$. The uncovered horizontal points of +$K+\Gamma$, viewed in $\mathcal T=\R^m/\Lambda_h$, therefore lie in +\[ + \bigcap_l\bigl(U_{a(l)}+\pi_{\Lambda_h}(Zl)\bigr). +\] +For $y\in M[0,1)^D$, Lemma~\ref{lem:shear} and the retained load give +the uniform bound +\begin{equation}\label{eq:uniform-holes} + \text{horizontal hole fraction at }y + \le\exp\left(2^D\log L_*-\frac{c_3\rho}{4}\right). +\end{equation} +For other $y$, reduce $M^{-1}y$ modulo $\Z^D$ and translate by the +corresponding point $(Zk,Mk)\in\Gamma$. This only translates the +horizontal section on its torus, so the same bound holds. + +Let $F_h,F_v$ be fundamental parallelepipeds of $\Lambda_h$ and +$ML$. Their product is a measurable fundamental domain for +$\Gamma$: first reduce the vertical coordinate using a point +$(Zl,Ml)$, and then reduce the horizontal coordinate using +$\Lambda_h$. The union $K+\Gamma$ is closed, since translates of +the compact set $K$ by a lattice are locally finite. Its complement +is therefore measurable, and Fubini integrates +Equation~\eqref{eq:uniform-holes} over $F_v$. No measurable selection +of auxiliary patterns is needed. Equations~\eqref{eq:target-density} +and~\eqref{eq:pattern-cost} yield +\[ + \delta:=1-|\pi_\Gamma(K)| + \le\exp\left(o(n\log n)-\frac{c_3C_*}{4}n\log n\right) + \le n^{-2n} +\] +once $C_*$ is a sufficiently large absolute constant and then $n$ +is sufficiently large. Since $n^{-2n}<(1-n^{-2n})n^{-n}$ for +$n\ge2$, Lemma~\ref{lem:completion} with $k=n$ gives +$(1+1/n)K+\Gamma=\R^n$. Thus +\[ + \Lambda=(1+1/n)^{-1}\Gamma +\] +is a single covering lattice for $K$, of density +\[ + \frac{\vol_n(K)}{\det\Lambda} + =(1+1/n)^n\rho\le eC_*n\log n. +\] + +\paragraph{Affine invariance and the remaining dimensions.} +Translations of a body do not change the covering property, and +invertible linear maps applied to both body and lattice preserve +density. Hence the affine normalization can be undone. + +There remains only a fixed finite set of dimensions. In each such +dimension choose a maximum-volume simplex with vertices in $K$, +which exists by compactness and has positive volume. Put its vertices +at $0,e_1,\ldots,e_n$ by an affine map. Replacing the $j$th nonzero +vertex by any $x\in K$ changes the determinant to $x_j$, so maximality +gives $|x_j|\le1$. Thus +\[ + [0,1/n]^n\subset K\subset[-1,1]^n. +\] +The lattice $n^{-1}\Z^n$ covers by translates of the smaller cube +and therefore of $K$, with density at most $(2n)^n$. +The maximum of $(2n)^n/(n\log n)$ over the remaining finite set +can be absorbed in one absolute constant. This proves the theorem +for every $n\ge2$. +\end{proof} diff --git a/preprints/A-single-lattice-covering-bound-of-order-n-log-n-September-23-2026/build/hierarchy.tex b/preprints/A-single-lattice-covering-bound-of-order-n-log-n-September-23-2026/build/hierarchy.tex new file mode 100644 index 0000000000000000000000000000000000000000..bb47ce0129daee1099f7f4985e02d6e8013e9607 --- /dev/null +++ b/preprints/A-single-lattice-covering-bound-of-order-n-log-n-September-23-2026/build/hierarchy.tex @@ -0,0 +1,341 @@ +\section{A hierarchy of branching lattice points} +\label{sec:hierarchy} + +The preceding block estimates produce a function whose mean is close to one, +although the function may vanish at many targets. We now arrange blocks in +decreasing sizes. The bits in one block sample the mean of the preceding +block; a final block of bounded size supplies a starting point. Keeping all +the intermediate branches is what preserves their total weight. +The data are chosen from the largest block to the smallest: each new +block samples the function already fixed in its predecessor. For a given +target, the lattice points are selected in the reverse order, starting +with the terminal block and then choosing the earlier blocks conditional +on the suffix already selected. + +\subsection{The block sizes and the lattice} + +Fix a sufficiently large absolute integer $B$, whose requirements will be +specified below. Starting with an integer $b_1\ge B$, form a finite list by +putting +\[ + b_{i+1}=\lceil b_i^{0.9}\rceil +\] +whenever this number is at least $B$, and otherwise stopping. Let $r$ be the +last index. Increasing $B$ ensures that +\begin{equation} + b_{i+1}\le \tfrac12 b_i,\qquad + B\le b_r1$ satisfies +\begin{equation} + \sum_{\epsilon\in\{0,1\}^{b_i}} P_{i,t_z}(\epsilon) + g_{i-1}(x-W_{i-1}\epsilon) + \ge (1-A b_i^{-1/100})\bar g_{i-1} + \quad (x\in G_{i-1}). + \label{eq:hierarchy-biased-sampler} +\end{equation} +All constants in these estimates are independent of the number of blocks. + +For $i\le r$, define the raw coordinate lattice +\begin{equation} + L_i=\Z^{b_i}+p_i^{-1}C_i. + \label{eq:hierarchy-raw-lattice} +\end{equation} +Here the right side means the union of the integer translates of the +standard representatives divided by $p_i$; it is independent of the choice +of representatives. It is a lattice of determinant $p_i^{-1}$. Indeed, +choose a generator $c$ of $C_i$ with its $j$th coordinate equal to $1$. +The vector $c/p_i$ and the vectors $e_k$ for $k\ne j$ are a basis: +they generate $e_j=p_i(c/p_i)-\sum_{k\ne j}c_k e_k$ and all of $L_i$, +and their determinant has absolute value $1/p_i$. + +Adjoin a terminal block of size +\[ + T=b_{r+1}=b_r\lceil\log_2 p_r\rceil, + \qquad h_{r+1}=1,\qquad L_{r+1}=\Z^T. +\] +Its size is bounded by a constant depending only on the fixed cutoff $B$. +Define $W_r$, of size $b_r$ by $T$, by using the columns +$2^k e_j\pmod {p_r}$ for each $1\le j\le b_r$ and +$0\le k<\lceil\log_2 p_r\rceil$. Consequently +\begin{equation} + \{W_r\epsilon:\epsilon\in\{0,1\}^T\}=G_r. + \label{eq:hierarchy-terminal-exhaustion} +\end{equation} +Use standard integer representatives for every $W_i$, and put +\[ + A_i=p_i^{-1}W_i\quad (1\le i\le r),\qquad + L=\prod_{i=1}^{r+1}L_i,\qquad D=\sum_{i=1}^{r+1}b_i. +\] +Each entry of $A_i$ belongs to $[0,1)$ and is an integer multiple of +$1/p_i$. On raw block coordinates $\ell=(\ell_1,\ldots,\ell_{r+1})$, +define the invertible block upper triangular map $M$ by +\begin{equation} + (M\ell)_i=h_i(\ell_i+A_i\ell_{i+1})\quad (i\le r), + \qquad (M\ell)_{r+1}=\ell_{r+1}. + \label{eq:hierarchy-map} +\end{equation} +In particular, $\Lambda_v=ML$ is a single full-rank lattice, with +\begin{equation} + D_v=\det\Lambda_v=\prod_{i=1}^r\frac{h_i^{b_i}}{p_i}. + \label{eq:hierarchy-determinant} +\end{equation} +The fractional off-diagonal entries do not change this conclusion: $M$ is +one invertible linear map applied to the product lattice $L$. + +\begin{proposition}[Weighted vertical patterns] +\label{prop:vertical} +There are absolute constants $B,C,c,\kappa>0$ such that, for every integer +$b_1\ge B$, the construction above has +\begin{equation} + D\le C(b_1+1),\qquad + R^2:=\sum_{i=1}^{r+1}b_i h_i^2 + \le C\bigl(b_1\log(b_1+1)+1\bigr). + \label{eq:hierarchy-dimension-radius} +\end{equation} +For every $y\in\R^D$ there is a finite nonempty pattern +$\mathcal P_y\subset L$ such that +\begin{align} + \|(y-M\ell)_i\|_\infty&\le h_i + \qquad(\ell\in\mathcal P_y,\ 1\le i\le r+1), + \label{eq:hierarchy-residual}\\ + \sum_{\ell\in\mathcal P_y}D_v\gamma_D(y-M\ell)&\ge\kappa, + \label{eq:hierarchy-total}\\ + \max_{\ell\in\mathcal P_y}D_v\gamma_D(y-M\ell) + &\le \exp(-c b_1^{0.70}). + \label{eq:hierarchy-cap} +\end{align} +The terminal block is constant on the pattern. For each ordinary block +$i$ and each fixed suffix $(\ell_{i+1},\ldots,\ell_{r+1})$, its retained +values belong to $\lambda_i+\{0,1\}^{b_i}$ for one +$\lambda_i\in L_i$ depending on that suffix. In particular +$|\mathcal P_y|\le 2^D$. + +Order the scalar coordinates by increasing block index and denote them by +$\ell[1],\ldots,\ell[D]$. Conditional on any fixed scalar suffix +$\ell[j+1],\ldots,\ell[D]$, the possible next values $\ell[j]$ are at most +two; if there are two, they differ by $1$. This property is preserved upon +deleting any subset of the pattern. + +If $y\in M[0,1)^D$, all raw coordinates can in addition be restricted to +the fixed finite alphabets +\begin{equation} + \begin{aligned} + \ell_i&\in\bigl(p_i^{-1}\Z\cap[-Q_{\rm raw},Q_{\rm raw}]\bigr)^{b_i} + &&(i\le r),\\ + \ell_{r+1}&\in\bigl(\Z\cap[-Q_{\rm raw},Q_{\rm raw}]\bigr)^T, + &&Q_{\rm raw}=1+(D+2)^D. + \end{aligned} + \label{eq:hierarchy-alphabets} +\end{equation} +For an ordinary block the logarithm of each scalar alphabet size is +$O(D\log(D+2)+b_i\log b_i)$; the terminal version omits the second term. +All implicit constants are absolute. +\end{proposition} + +\begin{proof} +The size and radius bounds follow from Equation~\eqref{eq:hierarchy-sizes}, +the formula $h_i^2=(9/4)\log b_i$, and the absolute bound on $T$. We prove +the weight estimates by an induction that allows an empty intermediate +pattern. + +\paragraph{Truncated patterns.} +Retain just the ordinary blocks $1,\ldots,i$, omitting the interaction of +block $i$ with its successor. Write $M^{[i]}$ for this truncated map, +$d_i=\sum_{j\le i}b_j$, and +$D_v^{[i]}=\prod_{j\le i}h_j^{b_j}/p_j$. We claim that for every +$y=(y_1,\ldots,y_i)\in\R^{d_i}$ there is a pattern with total weight at +least +\begin{equation} + \sum_{\ell\in\mathcal P^{[i]}_y} + D_v^{[i]}\gamma_{d_i}(y-M^{[i]}\ell) + \ge a_i g_i(x_i),\qquad + x_i=\lfloor p_i y_i/h_i\rfloor\pmod {p_i}, + \label{eq:hierarchy-induction} +\end{equation} +where +\begin{equation} + a_i\ge\prod_{j=1}^i(1-C_0 b_j^{-1/100}) + \label{eq:hierarchy-product} +\end{equation} +for an absolute $C_0$. The residual and block-branching properties in the +proposition are part of this induction. Empty patterns are permitted when +$g_i(x_i)=0$. + +Fix the lexicographic order on the standard representatives of each +finite group; when a maximum has ties, choose the least maximizing anchor. +Suppose $g_i(x_i)>0$. Choose a usable anchor $z\in x_i-C_i$ realizing +the maximum in Equation~\eqref{eq:hierarchy-functions}. By +Lemma~\ref{lem:block-cell} there is $\lambda_i\in L_i$ such that +\[ + t=y_i/h_i-\lambda_i\in t_z+[0,1/p_i)^{b_i}. +\] +Consider all choices $\ell_i=\lambda_i+\epsilon$, with +$\epsilon\in\{0,1\}^{b_i}$. Their individual block weights satisfy +\begin{equation} + \frac{h_i^{b_i}}{p_i}\gamma_{b_i}(h_i(t-\epsilon)) + \ge e^{-\delta_i}\frac{F_i(t_z)}{p_i}P_{i,t_z}(\epsilon), + \qquad \delta_i=\frac{b_i h_i^2}{p_i}, + \label{eq:hierarchy-rounded-weight} +\end{equation} +by Equation~\eqref{eq:block-rounding}. Since each coordinate of $t$ is in +$[0,1)$, the corresponding residual has absolute value at most $h_i$. +For $i=1$, summing Equation~\eqref{eq:hierarchy-rounded-weight} over all +bits proves Equation~\eqref{eq:hierarchy-induction} with $a_1=e^{-\delta_1}$. + +For $i>1$, attach the preceding induction pattern to each bit choice, +using the modified target whose last block is +\[ + \widetilde y_{i-1} + =y_{i-1}-h_{i-1}A_{i-1}(\lambda_i+\epsilon), +\] +and whose earlier blocks are unchanged. This is allowed even when its +preceding pattern is empty. The new preceding grid index is exactly +$x-W_{i-1}\epsilon$, where +\begin{equation} + x=\left\lfloor p_{i-1}y_{i-1}/h_{i-1} + -W_{i-1}\lambda_i\right\rfloor\pmod {p_{i-1}}. + \label{eq:hierarchy-grid-shift} +\end{equation} +Indeed, the vector $W_{i-1}\epsilon$ is integral and hence commutes with +the coordinate floor as a translation. The fixed vector +$W_{i-1}\lambda_i$ need not be integral; in particular, no compatibility +between the adjacent primes is required. + +Patterns attached to different bit vectors are disjoint because their +last blocks differ. Since $F_i(t_z)/p_i=g_i(x_i)$, multiplying their +weights by Equation~\eqref{eq:hierarchy-rounded-weight} and summing over +every bit vector gives total weight at least +\[ + a_{i-1}e^{-\delta_i}g_i(x_i) + \sum_{\epsilon\in\{0,1\}^{b_i}} + P_{i,t_z}(\epsilon)g_{i-1}(x-W_{i-1}\epsilon). +\] +Equation~\eqref{eq:hierarchy-biased-sampler} recovers the preceding mean +from this sum. Thus we may take +\[ + \begin{aligned} + a_i&=a_{i-1}e^{-\delta_i}(1-A b_i^{-1/100})\bar g_{i-1}\\ + &\ge a_{i-1}e^{-\delta_i} + (1-A b_i^{-1/100})(1-A b_{i-1}^{-1/100}), + \end{aligned} +\] +where the second line uses the mean bound in +Equation~\eqref{eq:hierarchy-functions}. The prime estimates imply that +$\delta_i$ decays faster than any fixed negative power of $b_i$. As +$b_{i-1}\ge b_i$, increasing one absolute constant $C_0$ bounds the new +multiplier below by $1-C_0b_i^{-1/100}$ and yields +Equation~\eqref{eq:hierarchy-product}, including its base case. This step +uses the weighted sum over all bits, not a choice of one successful bit. + +\paragraph{Summable losses.} +Set $a=1/100$. Equation~\eqref{eq:hierarchy-sizes} gives the explicit bound +\begin{equation} + \sum_{j=1}^r b_j^{-a} + \le b_r^{-a}\sum_{k=0}^{r-1}2^{-ak} + \le \frac{B^{-a}}{1-2^{-a}}. + \label{eq:hierarchy-summability} +\end{equation} +Choose $B$ so that $C_0B^{-a}\le1/2$. Using +$\log(1-u)\ge-2u$ for $0\le u\le1/2$, the products in +Equation~\eqref{eq:hierarchy-product} are bounded below by the positive +absolute constant +\[ + a_*:=\exp\left(-\frac{2C_0B^{-a}}{1-2^{-a}}\right). +\] +This lower bound is independent of $r$ and $b_1$. + +\paragraph{The terminal correction.} +For the full target $y$, choose a terminal raw point of the form +$\ell_{r+1}=\lfloor y_{r+1}\rfloor+\epsilon$. +After accounting for its integer part, varying $\epsilon$ shifts the last +ordinary grid index by $-W_r\epsilon$. By +Equation~\eqref{eq:hierarchy-terminal-exhaustion} we can place that index at +any point of $G_r$. Since $\bar g_r\ge1/2$, choose it at a point with +$g_r\ge1/2$. Apply Equation~\eqref{eq:hierarchy-induction} to the first $r$ +blocks with this fixed terminal contribution subtracted from $y_r$. +Every coordinate of the terminal residual has absolute value at most one, +and therefore its Gaussian factor is at least $(2\pi e)^{-T/2}$. +If $T_*(B)$ is an upper bound for all possible terminal dimensions, the +resulting full pattern has total weight at least +\[ + \kappa:=\tfrac12 a_*(2\pi e)^{-T_*(B)/2}>0. +\] +The cutoff $B$ is now fixed, so $\kappa$ is absolute. This proves +Equation~\eqref{eq:hierarchy-total} and, in particular, nonemptiness. + +\paragraph{Individual weights and branching.} +Every retained point comes from a usable anchor in every ordinary block. +The atom estimate in Equation~\eqref{eq:block-atom}, followed by the upper +rounding estimate in Equation~\eqref{eq:block-rounding}, bounds its +individual normalized block weight by +\[ + e^{\delta_i}\exp(-c_0 b_i^{0.70}) + \le \exp(-c b_i^{0.70}), +\] +after increasing $B$ and decreasing the absolute constant $c>0$. +The terminal Gaussian factor is at most $(2\pi)^{-T/2}\le1$. +Multiplication over all blocks proves Equation~\eqref{eq:hierarchy-cap}. +The preceding construction already proves all residual bounds. + +Different bit vectors at a recursive step have different last blocks. +Thus a fixed suffix of later blocks identifies a unique recursive branch. +On that branch the construction chose one base $\lambda_i$ before +enumerating its bits and retained a subset of its Boolean translate. +Distinct branches are distinct raw lattice points. Once a scalar suffix is fixed, +the next coordinate can therefore only be the corresponding coordinate +of $\lambda_i$ or that coordinate plus one. Further deletion of points +cannot destroy this assertion. This proves the stated scalar suffix +rule and the cardinality bound. + +\paragraph{Finite alphabets.} +Finally suppose $y=Mz$ with $z\in[0,1)^D$, and put $e_i=z_i-\ell_i$. +Equation~\eqref{eq:hierarchy-residual} and the entry bounds on $A_i$ imply +\[ + \|e_{r+1}\|_\infty\le1,\qquad + \|e_i\|_\infty\le1+b_{i+1}\|e_{i+1}\|_\infty + \le1+D\|e_{i+1}\|_\infty. +\] +Backward iteration bounds all coordinates of $\ell$ by +$1+(D+2)^D$. Also $L_i\subset p_i^{-1}\Z^{b_i}$ and the terminal +lattice is integral. These facts give Equation~\eqref{eq:hierarchy-alphabets}. +Each ordinary scalar alphabet has at most $2Q_{\rm raw}p_i+1$ elements; +$\log p_i=O(b_i\log b_i)$ gives its asserted logarithmic bound. +\end{proof} + +For the covering construction we take +$b_1=\lceil(\log\log n)^2\rceil$ once $n$ is sufficiently large. +Proposition~\ref{prop:vertical} then has +$D=O((\log\log n)^2+1)$, and both $D$ and $R$ satisfy the polynomial +bounds required in Lemma~\ref{lem:sections}. diff --git a/preprints/A-single-lattice-covering-bound-of-order-n-log-n-September-23-2026/build/main.tex b/preprints/A-single-lattice-covering-bound-of-order-n-log-n-September-23-2026/build/main.tex new file mode 100644 index 0000000000000000000000000000000000000000..f85aeb1072c0db55f7482adadae5ea4392d0581a --- /dev/null +++ b/preprints/A-single-lattice-covering-bound-of-order-n-log-n-September-23-2026/build/main.tex @@ -0,0 +1,85 @@ +\pdftrailerid{} +\documentclass[11pt]{article} +\usepackage[T1]{fontenc} +\usepackage{lmodern} +\ifdefined\pdfglyphtounicode + \pdfgentounicode=1 + \pdfglyphtounicode{tfm:lmex10/parenleftbig}{0028} + \pdfglyphtounicode{tfm:lmex10/parenrightbig}{0029} + \pdfglyphtounicode{tfm:lmex10/vextendsingle}{23D0} + \pdfglyphtounicode{tfm:lmex10/parenleftBig}{0028} + \pdfglyphtounicode{tfm:lmex10/parenrightBig}{0029} + \pdfglyphtounicode{tfm:lmex10/parenleftbigg}{0028} + \pdfglyphtounicode{tfm:lmex10/parenrightbigg}{0029} + \pdfglyphtounicode{tfm:lmex10/floorleftbigg}{230A} + \pdfglyphtounicode{tfm:lmex10/floorrightbigg}{230B} + \pdfglyphtounicode{tfm:lmex10/braceleftbigg}{007B} + \pdfglyphtounicode{tfm:lmex10/bracerightbigg}{007D} + \pdfglyphtounicode{tfm:lmex10/parenleftBigg}{0028} + \pdfglyphtounicode{tfm:lmex10/parenrightBigg}{0029} + \pdfglyphtounicode{tfm:lmex10/braceleftBigg}{007B} + \pdfglyphtounicode{tfm:lmex10/parenlefttp}{239B} + \pdfglyphtounicode{tfm:lmex10/parenrighttp}{239E} + \pdfglyphtounicode{tfm:lmex10/parenleftbt}{239D} + \pdfglyphtounicode{tfm:lmex10/parenrightbt}{23A0} + \pdfglyphtounicode{tfm:lmex10/summationtext}{2211} + \pdfglyphtounicode{tfm:lmex10/producttext}{220F} + \pdfglyphtounicode{tfm:lmex10/summationdisplay}{2211} + \pdfglyphtounicode{tfm:lmex10/productdisplay}{220F} + \pdfglyphtounicode{tfm:lmex10/integraldisplay}{222B} + \pdfglyphtounicode{tfm:lmex10/intersectiondisplay}{22C2} + \pdfglyphtounicode{tfm:lmex10/tildewide}{02DC} + \pdfglyphtounicode{tfm:lmex10/bracketleftBig}{005B} + \pdfglyphtounicode{tfm:lmex10/bracketrightBig}{005D} + \pdfglyphtounicode{tfm:lmex10/radicalbigg}{221A} + \pdfglyphtounicode{tfm:lmsy10/mapsto}{007C} +\fi +\usepackage[margin=1.05in]{geometry} +\usepackage{amsmath,amssymb,amsthm,mathtools} +\usepackage{microtype} +\usepackage{enumitem} +\usepackage{flafter} +\usepackage{tikz} +\usetikzlibrary{arrows.meta,positioning} +\usepackage{xcolor} +\usepackage[colorlinks=true,linkcolor=blue!45!black,citecolor=blue!45!black, + urlcolor=blue!45!black,pdfauthor={OpenAI}, + pdftitle={A single-lattice covering bound of order n log n}]{hyperref} +\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{\Z}{\mathbb Z} +\newcommand{\F}{\mathbb F} +\newcommand{\E}{\mathbb E} +\newcommand{\Prob}{\mathbb P} +\newcommand{\vol}{\operatorname{vol}} +\newcommand{\ind}{\mathbf 1} +\newcommand{\eps}{\varepsilon} +\DeclareMathOperator{\dist}{dist} +\setlist{itemsep=3pt,topsep=5pt} +\setlength{\emergencystretch}{2em} +\title{A single-lattice covering bound\\of order \texorpdfstring{$n\log n$}{n log n}} +\author{OpenAI} +\date{September 23, 2026} +\begin{document} +\maketitle +\begin{abstract} +Every convex body in $\R^n$, $n\ge2$, admits a covering by translates +along one full-rank lattice with density at most $Cn\log n$, for an absolute +constant $C$. No symmetry or boundary regularity is assumed. +\end{abstract} +\input{overview} +\input{preliminaries} +\input{sampler} +\input{blocks} +\input{hierarchy} +\input{final} +\input{references} +\end{document} diff --git a/preprints/A-single-lattice-covering-bound-of-order-n-log-n-September-23-2026/build/overview.tex b/preprints/A-single-lattice-covering-bound-of-order-n-log-n-September-23-2026/build/overview.tex new file mode 100644 index 0000000000000000000000000000000000000000..1d9db20232cd783d88aa1370311a09e1f1186a48 --- /dev/null +++ b/preprints/A-single-lattice-covering-bound-of-order-n-log-n-September-23-2026/build/overview.tex @@ -0,0 +1,126 @@ +\section{Introduction}\label{sec:introduction} + +For a convex body $K\subset\R^n$, its lattice covering density is +\[ + \theta_L(K)=\inf\left\{ + \frac{\vol_n(K)}{\det\Lambda}:\Lambda\text{ is a full-rank lattice and } + K+\Lambda=\R^n\right\}. +\] +Here a convex body is compact with nonempty interior, and +$\det\Lambda$ denotes lattice covolume. The density is the average number +of translates covering a point in a fundamental cell: integrating the +covering multiplicity over that cell gives $\vol_n(K)$. Thus density +measures the overlap needed to cover space when the centers must form +one lattice. We prove the following bound. + +\begin{theorem}\label{thm:main} +There is an absolute constant $C>0$ such that, for every integer +$n\ge2$ and every convex body $K\subset\R^n$, one full-rank lattice +$\Lambda$ satisfies +\[ + K+\Lambda=\R^n,\qquad + \frac{\vol_n(K)}{\det\Lambda}\le Cn\log n. +\] +No symmetry or boundary regularity is assumed. +\end{theorem} + +\paragraph{History and significance.} +Rogers proved that every convex body in dimension $n\ge3$ admits a +translative covering of density at most +$n\log n+n\log\log n+5n$~\cite{Rog57}. +The centers in that construction may be a finite union of lattice +cosets. Fejes T\'oth later obtained the same leading density using only +$O(\log n)$ cosets of one lattice~\cite{FT09}. +Requiring the centers to form one lattice is a stronger +constraint, and a translative bound does not establish +Theorem~\ref{thm:main}. +For general lattice coverings, Rogers obtained the bound +$n^{\log_2\log n+O(1)}$~\cite{Rog59}. Ordentlich, Regev, and Weiss +replaced it by the universal quadratic bound +$\theta_L(K)=O(n^2)$~\cite[Theorem~1.1]{ORW}. Li and Liu subsequently proved +\[ + \theta_L(K)\le Cn\log n\,(\log\log n)^{10/3+o(1)} +\] +uniformly over convex bodies~\cite[Theorem~1.2]{LL}. +Their binary-correction companion improves the exponent of +$\log\log n$ to +$\frac23\log_2(2\pi e)+o(1)$~\cite[Theorem~1.1]{LLC}. +Theorem~\ref{thm:main} removes this remaining iterated-logarithmic loss +and gives the constant-factor $n\log n$ bound. +In every sufficiently large dimension, Li and Liu also give centrally +symmetric bodies with lattice covering density at least +$cn\log n$~\cite[Theorem~1.3]{LL}. Together these +results determine the supremum of lattice covering density over bodies +in dimension $n$, up to absolute constant factors, as $n\log n$. +This statement concerns the supremum over bodies, rather than the covering +density of every individual body. A translative lower bound must exclude +all center configurations and is a separate question; the present theorem +and its proof concern the single-lattice upper bound. + +\paragraph{The new mechanism.} +The proof develops the horizontal--vertical framework and binary-correction +ideas of Li and Liu~\cite{LL,LLC}. Write a point as $(x,y)$, where $x$ +is the horizontal coordinate and the vertical coordinate $y$ has much +smaller dimension. Translating vertically lets several horizontal sections +of the same body contribute to covering a given height. The pointwise +Gaussian marginal theorem of Eldan and Klartag~\cite{EK} controls the +volumes of these sections after an affine normalization. The lattice +mean-hole method of Rogers and Schmidt~\cite{Rog58,Sch58}, in the +form of Ordentlich--Regev--Weiss~\cite[Theorem~2.3]{ORW}, then provides +one horizontal lattice with an exponentially small uncovered fraction for +every section on a finite list. + +The remaining task is to distribute these horizontal contributions by one +linear shear of a vertical lattice. A binary correction offers two adjacent +integer choices in each vertical coordinate. In Li and Liu's two-stage +construction, one auxiliary correction is chosen for an entire weighted +Boolean cube~\cite[Theorem~6.2]{LLC}. Here the corrections themselves +form a hierarchy: we retain their weighted branches through all +intermediate blocks. + +The total Gaussian mass of the retained family, multiplied by the +vertical lattice covolume, stays above an absolute constant, +while every individual atom becomes very small. These two properties +play different roles: the total mass supplies enough horizontal load, +and the atom bound keeps every section within the range of the mean-hole +estimate. The corrections are arranged in successively smaller blocks, +with sizes approximately $b,b^{0.9},b^{0.9^2},\ldots$, stopped at a +fixed absolute cutoff. Each block loses only $O(b^{-1/100})$ of the +mass. The resulting losses are summable even when the number of blocks +grows. + +Two auxiliary statements isolate features of potential independent use. +The finite-group sampler in Lemma~\ref{lem:sampler} converts an +almost-everywhere average into a uniform lower average using random +subset sums; its cost depends on $\log|H|$, rather than $|H|$. +Lemma~\ref{lem:shear} extends the simultaneous Boolean-shift argument +of Li and Liu~\cite[Theorem~4.3]{LL} to finitely many binary-suffix +patterns with one linear shear. The values of the branch bases can depend +on later choices; only the unit separation of two siblings is needed. +This preserves the single-lattice structure throughout. + +\paragraph{Organization.} +Section~\ref{sec:preliminaries} derives the precise section and horizontal +lattice estimates from two established results, stating their hypotheses +explicitly. Section~\ref{sec:sampler} proves the uniform finite-group +sampler. Sections~\ref{sec:blocks} and~\ref{sec:hierarchy} construct +the folded Gaussian blocks and their weighted branching hierarchy. +Section~\ref{sec:final} chooses the common shear and removes the final +holes by a dilation of factor $1+1/n$. All new probabilistic and +geometric arguments are proved below; no estimate on a single successful +branch substitutes for a bound on their total weight. + +\paragraph{Notation and constants.} +Euclidean volume is written $\vol_d$. Expectations over finite groups +use uniform probability; $|A|$ denotes cardinality for a finite set. +On a lattice torus, $|A|$ instead denotes normalized Haar measure. +The standard Gaussian density is +\[ + \gamma_d(y)=(2\pi)^{-d/2}\exp(-\|y\|^2/2). +\] +All logarithms are natural except $\log_2$. The constants $c,C$ may +change from line to line and are absolute. Asymptotic statements concern +sufficiently large dimensions or block sizes; their thresholds never +depend on the convex body. The particular decimal exponents below are +chosen for strict inequalities, not optimization. A fixed cutoff may be +large, but it is chosen before the dimension and remains absolute. diff --git a/preprints/A-single-lattice-covering-bound-of-order-n-log-n-September-23-2026/build/preliminaries.tex b/preprints/A-single-lattice-covering-bound-of-order-n-log-n-September-23-2026/build/preliminaries.tex new file mode 100644 index 0000000000000000000000000000000000000000..bd9ac137959cb151f7d00ba34190a6acc93c1876 --- /dev/null +++ b/preprints/A-single-lattice-covering-bound-of-order-n-log-n-September-23-2026/build/preliminaries.tex @@ -0,0 +1,205 @@ +\section{Geometric and horizontal preparations} +\label{sec:preliminaries} + +We use two established results: the pointwise Gaussian marginal theorem +of Eldan and Klartag, and the Rogers--Schmidt mean-hole estimate. From +the first we derive a finite list of bodies contained in nearby horizontal +sections; from the second we obtain one horizontal lattice that works for +the whole list. Neither result requires symmetry. This section develops the finite-section +and common-lattice preparation of Li and Liu~\cite[Proposition~4.1 +and Lemma~4.2]{LL}, with the precise bounds used in our construction. + +\subsection{A finite list of horizontal sections} + +Theorem~1 of Eldan and Klartag~\cite{EK} has the following consequence. +There are absolute constants $\alpha,\beta,\gamma,C_0>0$ such that, for +all sufficiently large $n$, every isotropic random vector $X\in\R^n$ +with a log-concave density has the following property: for each integer +$1\le D\le n^\alpha$, there is a $D$-dimensional subspace $E$ whose +orthogonal marginal has density $f_E$ satisfying +\begin{equation} + \left|\frac{f_E(y)}{\gamma_D(y)}-1\right| + \le C_0 n^{-\gamma} + \qquad (y\in E,\ \|y\|\le n^\beta). + \label{eq:gaussian-marginal-input} +\end{equation} +Here isotropic means $\E X=0$ and $\E XX^{\mathsf T}=I_n$; orthonormal +coordinates identify $E$ with $\R^D$. For a uniform law on a convex +body, we use the canonical marginal density given by the section integral. +We only use the +lower bound in Equation~\eqref{eq:gaussian-marginal-input}. + +\begin{lemma}[Section labels] +\label{lem:sections} +There are absolute constants $c,C>0$ with the following property. +Fix $M\ge1$. For all sufficiently large $n$, with threshold depending +only on $M$, suppose that $D$ is a positive integer and +\[ + 1\le D,R\le M(1+\log\log n)^M,\qquad m=n-D. +\] +Every convex body in $\R^n$ has an invertible affine image +$K\subset\R^m\times\R^D$ and a list of at most $(Cm^2)^D$ convex +bodies $J_a\subset\R^m$ such that, writing $K_y=\{x:(x,y)\in K\}$, +for every $\|y\|\le R$ some label $a$ satisfies +\begin{equation} + J_a\subset K_y, + \qquad + c\,\vol_n(K)\gamma_D(y) + \le \vol_m(J_a) + \le C\,\vol_n(K)\gamma_D(y). + \label{eq:section-labels} +\end{equation} +\end{lemma} + +\begin{proof} +An affine transformation makes the uniform law on the given body +isotropic: its covariance matrix is positive definite because the body +has nonempty interior. The uniform density is log-concave. Put +$S=2DR+2$. Uniformly over the displayed range of $D,R$, for sufficiently +large $n$ we have +\[ + D\le n^\alpha,\qquad S0$, where $B_m$ is the unit ball. Continuity of volume under +decreasing compact neighborhoods shows that $y\mapsto\vol_m(K_y)$ is +upper semicontinuous. Its almost-everywhere Gaussian lower bound therefore +extends to every $\|y\|\le S0$ such that every Borel set +$J\subset\R^m$ of volume $u\le\eta_m$ satisfies +\begin{equation} + \left|\int h_{\Lambda}(J)\,d\mu_m(\Lambda)-e^{-u}\right| + \le C_{\mathrm R}e^{-\eta_m}. + \label{eq:mean-hole-input} +\end{equation} + +\begin{lemma}[Common horizontal lattice] +\label{lem:horizontal} +Let $J_1,\ldots,J_N$ be convex bodies in $\R^m$, let $D_h>0$, and set +$u_a=\vol_m(J_a)/D_h$. If +\begin{equation} + \max_a u_a\le\eta_m, + \qquad + (1+C_{\mathrm R})\sum_{a=1}^N e^{-u_a/2}<1, + \label{eq:horizontal-criterion} +\end{equation} +there is a lattice $\Lambda_h$ of determinant $D_h$ satisfying +\begin{equation} + h_{\Lambda_h}(J_a)\le e^{-u_a/2} + \qquad (1\le a\le N). + \label{eq:horizontal-holes} +\end{equation} +In particular, this conclusion holds for all sufficiently large $m$ if +\[ + \min_a u_a\ge\sqrt m,\qquad + \max_a u_a=o(m),\qquad + \log N=o(\sqrt m). +\] +The dimension threshold in this last formulation depends only on the +three numerical bounds, not on the shapes of the bodies. +\end{lemma} + +\begin{proof} +Replace every $J_a$ by $J'_a=D_h^{-1/m}J_a$, so that +$\vol_m(J'_a)=u_a$. Equation~\eqref{eq:mean-hole-input} gives +\[ + \E_{\Lambda\sim\mu_m}h_{\Lambda}(J'_a) + \le e^{-u_a}+C_{\mathrm R}e^{-\eta_m} + \le(1+C_{\mathrm R})e^{-u_a}. +\] +Markov's inequality bounds the probability that +$h_{\Lambda}(J'_a)>e^{-u_a/2}$ by +$(1+C_{\mathrm R})e^{-u_a/2}$. The sum of these probabilities is less +than one by Equation~\eqref{eq:horizontal-criterion}. Thus a single +unimodular lattice $\Lambda$ satisfies all the desired inequalities. +Set $\Lambda_h=D_h^{1/m}\Lambda$; scaling identifies the normalized +tori, preserves the hole proportions, and gives determinant $D_h$. + +Finally, $\eta_m/m\to\tfrac14\log(27/16)>0$, so +$\max_a u_a=o(m)$ ensures the first condition in +Equation~\eqref{eq:horizontal-criterion}. The other two numerical +bounds give +\[ + (1+C_{\mathrm R})\sum_{a=1}^N e^{-u_a/2} + \le(1+C_{\mathrm R})\exp\{\log N-\tfrac12\sqrt m\}=o(1), +\] +which ensures the second. +\end{proof} diff --git a/preprints/A-single-lattice-covering-bound-of-order-n-log-n-September-23-2026/build/references.tex b/preprints/A-single-lattice-covering-bound-of-order-n-log-n-September-23-2026/build/references.tex new file mode 100644 index 0000000000000000000000000000000000000000..0a5df6ac02e0f657ccb198c464f1e8e2e5eed0ef --- /dev/null +++ b/preprints/A-single-lattice-covering-bound-of-order-n-log-n-September-23-2026/build/references.tex @@ -0,0 +1,68 @@ +\begingroup +\small +\raggedright +\pdfbookmark[1]{References}{references} +\begin{thebibliography}{9} + +\bibitem{EK} +Ronen Eldan and Bo'az Klartag, +\emph{Pointwise estimates for marginals of convex bodies}, +Journal of Functional Analysis \textbf{254} (2008), no.~8, 2275--2293. +\href{https://doi.org/10.1016/j.jfa.2007.08.014}{doi:10.1016/j.jfa.2007.08.014}. +\href{https://arxiv.org/abs/0708.2513v1}{arXiv:0708.2513v1}. + +\bibitem{FT09} +G\'abor Fejes T\'oth, +\emph{A note on covering by convex bodies}, +Canadian Mathematical Bulletin \textbf{52} (2009), no.~3, 361--365. +\href{https://doi.org/10.4153/CMB-2009-039-x}{doi:10.4153/CMB-2009-039-x}. + +\bibitem{LL} +Heng Li and Xizhi Liu, +\emph{Nearly sharp bounds for lattice coverings by convex bodies}, +preprint, version~2, 10~August 2026. +\href{https://arxiv.org/abs/2607.28429v2}{arXiv:2607.28429v2}. + +\bibitem{LLC} +Heng Li and Xizhi Liu, +\emph{Two-stage binary correction for nearly linear lattice coverings}, +unpolished manuscript, 2026. +\href{https://github.com/xliu2022/xliu2022.github.io/blob/4b360838cb034784293bb6ba2128483fcec6e3ce/Unpolished_Manuscripts/Two-Stage\%20Binary\%20Correction\%20for\%20Nearly\%20Linear\%20Lattice\%20Coverings.pdf}{Author-hosted manuscript}, +accessed 23~September 2026. + +\bibitem{ORW} +Or Ordentlich, Oded Regev, and Barak Weiss, +\emph{New bounds on the density of lattice coverings}, +Journal of the American Mathematical Society \textbf{35} (2022), no.~1, +295--308. +\href{https://doi.org/10.1090/jams/984}{doi:10.1090/jams/984}. +\href{https://arxiv.org/abs/2006.00340v1}{arXiv:2006.00340v1}. + +\bibitem{Rog57} +C.~A. Rogers, +\emph{A note on coverings}, +Mathematika \textbf{4} (1957), no.~1, 1--6. +\href{https://doi.org/10.1112/S0025579300001030}{doi:10.1112/S0025579300001030}. + + +\bibitem{Rog58} +C.~A. Rogers, +\emph{Lattice coverings of space: The Minkowski--Hlawka theorem}, +Proceedings of the London Mathematical Society (3) \textbf{8} (1958), +447--465. +\href{https://doi.org/10.1112/plms/s3-8.3.447}{doi:10.1112/plms/s3-8.3.447}. + +\bibitem{Rog59} +C.~A. Rogers, +\emph{Lattice coverings of space}, +Mathematika \textbf{6} (1959), no.~1, 33--39. +\href{https://doi.org/10.1112/S002557930000190X}{doi:10.1112/S002557930000190X}. + +\bibitem{Sch58} +Wolfgang M. Schmidt, +\emph{The measure of the set of admissible lattices}, +Proceedings of the American Mathematical Society \textbf{9} (1958), +390--403. +\href{https://doi.org/10.2307/2032994}{doi:10.2307/2032994}. +\end{thebibliography} +\endgroup diff --git a/preprints/A-single-lattice-covering-bound-of-order-n-log-n-September-23-2026/build/sampler.tex b/preprints/A-single-lattice-covering-bound-of-order-n-log-n-September-23-2026/build/sampler.tex new file mode 100644 index 0000000000000000000000000000000000000000..11b812581ad26544ea4975262629f2489a0ae959 --- /dev/null +++ b/preprints/A-single-lattice-covering-bound-of-order-n-log-n-September-23-2026/build/sampler.tex @@ -0,0 +1,198 @@ +\section{A uniform Boolean sampler} +\label{sec:sampler} + +We next construct an averaging operator from random subset sums. The +exponential-potential argument develops the finite-group binary correction +of Li and Liu~\cite[Theorem~4.1]{LLC}; we prove the weighted estimate with +random shifts used below, including all required quantitative bounds. A second +moment estimate alone gives a good approximation at most base points. A +second group of random shifts upgrades this to a lower bound at \emph{every} +base point. This distinction will allow a later block to sample an earlier +one even when its base point depends on other lattice choices. + +\begin{lemma}[Uniform Boolean sampling] +\label{lem:sampler} +There is an absolute constant $b_0$ with the following property. Let +$b\ge b_0$, let $H$ be a finite abelian group written additively, and suppose +that +\[ + \log |H|\le b^4, + \qquad 0\le g\le \exp(b^{0.65}), + \qquad \bar g:=\E_{x\in H}g(x)\in[1/2,2]. +\] +Let $s$ be an integer satisfying $b^{0.69}\le s\le b$, and choose +$w_1,\ldots,w_s$ independently and uniformly in $H$. With +$c_*=(\log 2)/8$, the probability that +\begin{equation} +\label{eq:sampler-conclusion} + 2^{-s}\sum_{\eps\in\{0,1\}^s} + g\left(x-\sum_{i=1}^s\eps_iw_i\right) + \ge (1-3b^{-1/10})\bar g + \qquad\text{for every }x\in H +\end{equation} +is at least +\begin{equation} +\label{eq:sampler-probability} + 1-\exp(-c_*b^{0.69}) + -b\exp\left(-\frac{c_*}{2}b^{0.59}\right). +\end{equation} +In particular, the failure probability is at most $b^{-1/2}$ after +increasing $b_0$. +\end{lemma} + +\begin{proof} +Put $\delta=b^{-1/10}$, $k=\lfloor s/2\rfloor$, and $j_*=s-k$. +We use the first $k$ shifts to define +\[ + A(x)=2^{-k}\sum_{\eps\in\{0,1\}^k} + g\left(x-\sum_{i=1}^k\eps_iw_i\right). +\] +Temporarily take $x$ independently uniform in $H$. For two distinct bit +vectors $\eps,\eps'$, their arguments in this average are independent +uniform points. To see this, choose an index at which the bits differ and +condition on all other shifts. The map from $x$ and that remaining shift +to the two arguments is a bijection of $H\times H$: subtracting the +arguments determines the shift with coefficient $1$ or $-1$, and then +determines $x$. This works in every finite abelian group, with no +restriction on its order. Consequently, +\begin{equation} +\label{eq:sampler-second-moment} + \E_{w_1,\ldots,w_k,x}(A(x)-\bar g)^2 + =2^{-k}\left(\E_xg(x)^2-\bar g^2\right) + \le 2^{-k}\exp(2b^{0.65}). +\end{equation} + +For fixed first-stage shifts, let +\[ + \mathcal G=\{x\in H:A(x)\ge(1-\delta)\bar g\}, + \qquad \beta=1-\frac{|\mathcal G|}{|H|}. +\] +Chebyshev's inequality, $\bar g\ge1/2$, and +Equation~\eqref{eq:sampler-second-moment} give +\[ + \E_{w_1,\ldots,w_k}\beta + \le 4b^{1/5}2^{-k}\exp(2b^{0.65}) + \le \exp(-2c_*s). +\] +The last inequality holds for all sufficiently large $b$, uniformly in +$s\ge b^{0.69}$. Indeed, its logarithm is at most +\[ + \log 8+\tfrac15\log b+2b^{0.65}-\tfrac12s\log 2, +\] +and the first three terms are at most $s\log 2/4$ once $b$ is large. +Markov's inequality therefore yields +\begin{equation} +\label{eq:sampler-small-bad-set} + \Prob\{\beta>\exp(-c_*s)\}\le\exp(-c_*s). +\end{equation} +Fix any first-stage shifts for which +$\beta\le\exp(-c_*s)$. + +For $0\le j\le j_*$, let +\[ + N_j(x)= + \sum_{\eta\in\{0,1\}^j} + \ind_{\mathcal G}\left(x-\sum_{i=1}^j\eta_iw_{k+i}\right). +\] +Here we count bit labels with multiplicity, even if their subset sums +coincide. Thus, for a fresh shift $w=w_{k+j+1}$, +\[ + N_{j+1}(x)=N_j(x)+N_j(x-w). +\] +Set $\theta=c_*s/2$ and introduce the potential +\[ + \Phi_j=\E_{x\in H}\exp(-\theta N_j(x)). +\] +Since $N_0=\ind_{\mathcal G}$, the bound on $\beta$ gives +\[ + \Phi_0=\beta+(1-\beta)e^{-\theta} + \le e^{-2\theta}+e^{-\theta}, + \qquad \log\Phi_0\le-\theta+1. +\] +Most importantly, conditional on every shift chosen so far, +\begin{equation} +\label{eq:sampler-potential-square} + \E_w\Phi_{j+1} + =\E_{x,w}e^{-\theta N_j(x)}e^{-\theta N_j(x-w)} + =\Phi_j^2. +\end{equation} +The last equality uses the bijection $(x,w)\mapsto(x,x-w)$ of +$H\times H$. It makes no independence assumption about the subset-sum +labels themselves. + +Conditional Markov's inequality applied to +Equation~\eqref{eq:sampler-potential-square} shows that +\[ + \Prob\left\{ + \log\Phi_{j+1}>2\log\Phi_j+\theta\delta + \,\middle|\,\text{previous shifts} + \right\} + \le e^{-\theta\delta}. +\] +There are $j_*\le b$ steps. Except on an event of conditional probability +at most $b e^{-\theta\delta}$, the recursion holds at every step and gives +\begin{equation} +\label{eq:sampler-potential-iterate} + \log\Phi_j + \le 2^j(-\theta+1)+(2^j-1)\theta\delta + \qquad(0\le j\le j_*). +\end{equation} +For each $x$, its single summand in the potential implies +\[ + \Phi_j\ge |H|^{-1}e^{-\theta N_j(x)}. +\] +Combining this with Equation~\eqref{eq:sampler-potential-iterate} yields +\begin{equation} +\label{eq:sampler-uniform-good-fraction} + \frac{N_{j_*}(x)}{2^{j_*}} + \ge 1-\delta-\frac1\theta + -\frac{\log|H|}{\theta 2^{j_*}} + \ge 1-2\delta + \qquad(x\in H). +\end{equation} +For the last inequality, use +$\theta\ge(c_*/2)b^{0.69}$, $j_*\ge b^{0.69}/2$, and +$\log|H|\le b^4$. Uniformly under these bounds, +$1/\theta\le\delta/2$ and +$\log|H|/(\theta2^{j_*})\le\delta/2$ once $b_0$ is sufficiently large. + +Finally, the full average factors as +\[ + 2^{-s}\sum_{\eps\in\{0,1\}^s} + g\left(x-\sum_{i=1}^s\eps_iw_i\right) + =2^{-j_*}\sum_{\eta\in\{0,1\}^{j_*}} + A\left(x-\sum_{i=1}^{j_*}\eta_iw_{k+i}\right). +\] +On labels counted by $N_{j_*}(x)$, the summand is at least +$(1-\delta)\bar g$; all other summands are nonnegative. Thus +Equation~\eqref{eq:sampler-uniform-good-fraction} bounds this expression +below by +\[ + (1-2\delta)(1-\delta)\bar g + \ge(1-3\delta)\bar g, +\] +simultaneously for every $x$. By +Equation~\eqref{eq:sampler-small-bad-set}, the total failure probability +is at most +\[ + e^{-c_*s}+b e^{-\theta\delta} + \le e^{-c_*b^{0.69}} + +b e^{-(c_*/2)b^{0.59}}, +\] +as asserted. All thresholds used above are absolute. +\end{proof} + +\begin{remark}[Uniformity and offsets] +The first stage controls only the proportion of bad base points. A direct +union bound over $H$ would be insufficient under the allowed bound +$\log|H|\le b^4$. The second-stage potential instead pays +$\log|H|$ in Equation~\eqref{eq:sampler-uniform-good-fraction}, divided by +an exponentially large number of labels. Once its conclusion holds, the +base point can be replaced by any element of $H$, including an offset +determined by other random columns. In particular, suppose a subset is +chosen independently of the column values and its size lies in the range +of Lemma~\ref{lem:sampler}. One may condition on that subset, apply the +lemma to its columns, and then insert arbitrary offsets formed from the +complementary columns. This observation will justify the biased-bit +mixtures below. +\end{remark} diff --git a/preprints/A-single-lattice-covering-bound-of-order-n-log-n-September-23-2026/paper.pdf b/preprints/A-single-lattice-covering-bound-of-order-n-log-n-September-23-2026/paper.pdf new file mode 100644 index 0000000000000000000000000000000000000000..8d6ac5fbd03d0592c44223ae49b22b94457b6f73 --- /dev/null +++ b/preprints/A-single-lattice-covering-bound-of-order-n-log-n-September-23-2026/paper.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:c7dc74641f89bce3f456410280429a47b3e214365a9467b494e14ee6b2348cbb +size 448381 diff --git a/preprints/A-singular-normal-affine-surface-with-free-tangent-sheaf-September-23-2026/README.md b/preprints/A-singular-normal-affine-surface-with-free-tangent-sheaf-September-23-2026/README.md new file mode 100644 index 0000000000000000000000000000000000000000..44fe9b33a99d51ed6b096d21d0b8309518c1c2a3 --- /dev/null +++ b/preprints/A-singular-normal-affine-surface-with-free-tangent-sheaf-September-23-2026/README.md @@ -0,0 +1,18 @@ +# [A singular normal affine surface with free tangent sheaf](paper.pdf) + +**Author:** OpenAI + +**Date:** September 23, 2026 + +## Citation + +```bibtex +@misc{OAI:A-singular-normal-affine-surface-with-free-tangent-sheaf-September-23-2026, + author = {{OpenAI}}, + title = {{A singular normal affine surface with free tangent sheaf}}, + howpublished = {OpenAI Math Release preprint + \href{https://github.com/openai/math/blob/main/preprints/A-singular-normal-affine-surface-with-free-tangent-sheaf-September-23-2026/paper.pdf}{OAI:A-singular-normal-affine-surface-with-free-tangent-sheaf-September-23-2026}}, + year = {2026} +} +``` + diff --git a/preprints/A-singular-normal-affine-surface-with-free-tangent-sheaf-September-23-2026/build/01-introduction.tex b/preprints/A-singular-normal-affine-surface-with-free-tangent-sheaf-September-23-2026/build/01-introduction.tex new file mode 100644 index 0000000000000000000000000000000000000000..7d24136cea7cecc8aa98cdc2b57bac7c0395b036 --- /dev/null +++ b/preprints/A-singular-normal-affine-surface-with-free-tangent-sheaf-September-23-2026/build/01-introduction.tex @@ -0,0 +1,238 @@ +\begin{abstract} +We construct a singular normal affine complex surface whose tangent +sheaf is free of rank two. This disproves the Lipman--Zariski conjecture +in characteristic zero. +\end{abstract} + +\section{Introduction} + +For a finitely generated integral $\CC$-algebra $A$, write +\[ +\Der_{\CC}(A)=\Hom_A(\Omega^1_{A/\CC},A). +\] +The module $\Der_{\CC}(A)$ consists of the algebraic vector fields on +$\Spec A$. On a smooth variety of dimension $d$, the tangent sheaf is +locally free of rank $d$. The Lipman--Zariski conjecture asks for the +converse: a complex algebraic variety with locally free tangent sheaf +should be nonsingular. In affine terms, it asserts that local freeness +of $\Der_{\CC}(A)$ forces $A$ to be regular. We prove the following. + +\begin{theorem}\label{thm:main} +There exist a finitely generated normal integral $\CC$-algebra $A$ of Krull dimension two and a maximal ideal $\m\subset A$ such that +\[ +\Der_{\CC}(A)\simeq A^{\oplus2}, +\qquad A_{\m}\text{ is not regular}. +\] +\end{theorem} + +Here freeness holds on the entire affine surface, which can be chosen +smooth away from the marked point. We first construct an isolated +Gorenstein analytic surface singularity. The passage to algebraic geometry +preserves its completed local ring, from which tangent freeness descends. + +\subsection{Context and significance} + +Lipman's study of free derivation modules established that local freeness +of the tangent sheaf forces normality~\cite[Theorem~3]{Lipman1965}. +The conjecture thus asks whether the same hypothesis excludes the remaining +normal singularities. In positive characteristic the corresponding assertion +fails even for hypersurfaces: over a perfect field of characteristic $p$, +the singular surface $xy-z^p=0$ has tangent sheaf freely generated by +$x\partial_x-y\partial_y$ and $\partial_z$ +\cite[\S1]{GrafPositive2022}. The construction here has a different source +of tangent freeness and takes place entirely over $\CC$. + +Several positive results explain both the strength of tangent freeness +and the role of surfaces in the problem. Hochster proved that a reduced +finitely generated characteristic-zero algebra with a nonnegative grading +and degree-zero part equal to the base field is a polynomial algebra if +its derivation module is free at the irrelevant maximal ideal +\cite[\S1, Theorem]{Hochster1977}. Van Straten and Steenbrink used +extension of differential forms to prove the isolated-singularity case +in dimension at least three~\cite[Theorem~1.3 and~\S1.6]{VanStratenSteenbrink1985}. +Their extension argument leaves the surface case as a distinct problem. + +Hypersurfaces were treated by Scheja and Storch~\cite[Satz~9.3]{SchejaStorch1972}; +K\"allstr\"om established the characteristic-zero algebraic locally complete-intersection +case~\cite[Corollary~1.3(1), corrected version]{Kallstrom2011}. +Extension theorems for differential forms also give strong results defined +by the singularities of the canonical divisor. Greb, Kebekus, Kov\'acs and +Peternell proved the conjecture for Kawamata log terminal spaces +\cite[Theorem~6.1]{GKKP2011}. The log-canonical case was proved by +Druel~\cite[Theorem~1.1]{Druel2014} and by Graf and Kov\'acs +\cite[Corollary~1.3]{GrafKovacs2014}. +These results relate tangent freeness to the behavior of differential +forms on a resolution. Another distinction concerns the Lie bracket: +J\"order proved smoothness for a normal complex space whose tangent sheaf +locally admits a basis of pairwise commuting vector fields +\cite[Theorem~1.1]{Jorder2014}. Thus no tangent frame near our singular +point can be chosen commuting. + +For normal complex surfaces, the cohomology and configuration of the +exceptional curves give further criteria. Graf proved smoothness when +the geometric genus $p_g$ is at most one, and also when $p_g=2$ and the +exceptional divisor is not a tree of rational curves +\cite[Theorem~1.1]{GrafLowGenus2021}. Bergner and Graf then obtained the +criterion +\[ + p_g-g-b\le2, +\] +where $g$ is the sum of the genera of the exceptional curves and $b$ is +the first Betti number of the resolution's dual graph +\cite[Theorem~1]{BergnerGraf2020}. +The analytic singularity constructed here has one exceptional curve of genus $21$ +and self-intersection $-10$. Its canonical discrepancy is $-5$, and +$p_g\ge33$, so $p_g-g-b\ge12$. Remark~\ref{analytic:invariants} +proves these calculations. They place the analytic example beyond the +log-canonical and geometric-genus criteria just described. + +Global hypotheses provide other positive cases. Biswas, Gurjar and Kolte +prove smoothness for a normal affine surface whose tangent bundle on +the smooth locus is trivial and whose smooth locus has logarithmic +Kodaira dimension at most one~\cite[Theorem~0.1(1)]{BiswasGurjarKolte2014}. +Consequently, the smooth locus $U$ of the affine surface in +Theorem~\ref{thm:main} has logarithmic Kodaira dimension two +(Corollary~\ref{algebra:log-kodaira}). Here logarithmic Kodaira dimension +means the Iitaka dimension of $K_{\overline U}+D$ on a smooth projective +compactification with reduced simple-normal-crossing boundary $D$. +This global consequence uses the algebraic endpoint of the construction, +not just the analytic germ. In contrast, Graf's compact-surface result +identifies compact complex surfaces with free rank-two tangent sheaf as +complex two-tori~\cite[Corollary~1.3]{GrafLowGenus2021}. + +The example also gives a negative answer in characteristic zero to the +derivation-blowup question of Barajas, Ch\'avez-Mart\'inez and +Romano-Vel\'azquez~\cite[Introduction, Question~1]{BarajasChavezRomano2026}. +The blowup at a module is the universal proper birational modification +making its pullback modulo torsion locally free. As those authors observe, +a singular variety with free tangent sheaf cannot be resolved by iterating +this operation at the module of derivations: every step is the identity. +For our normal surface, inserting normalization does not change this +conclusion (Corollary~\ref{algebra:derivation-blowups}). + +The neighborhood of the exceptional curve is essential to the argument. +Hochster's theorem excludes obtaining the example from a singular +positively graded cone with free tangent sheaf. Instead, we start with a +split line bundle and change the gluing maps of its zero-section +neighborhood while preserving the normal bundle. This freedom supplies +the required vector fields. Their dual volume form has a fifth-order +pole along the exceptional curve, so holomorphic extension across that +curve is not an input to the construction. + +\subsection{Main ideas and proof structure} + +The construction first produces an analytic surface singularity and then +passes to an affine algebraic surface. The analytic step requires two +vector fields on a punctured neighborhood that remain linearly +independent everywhere. We build those fields on a resolution, where the +exceptional set is a smooth negative curve. Their determinant vanishes +along that curve to a prescribed order, and nowhere else near it. + +Let $C$ be a smooth curve and $L$ a positive line bundle with +$L^{\otimes4}\simeq K_C$. Its dual bundle $L^{-1}$ provides the initial +neighborhood of $C$. In compatible base and fiber coordinates $(z,t)$, +that neighborhood carries the meromorphic two-form +\[ +\sigma=t^{-5}\,dt\wedge dz. +\] +We seek two vector fields of the form +\[ +v_i=t\bigl(A_i(t,z)t\partial_t+B_i(t,z)\partial_z\bigr), +\qquad i=1,2, +\] +with $\sigma(v_1,v_2)=1+O(t)$. The leading vertical coefficients form a +pair $r=(r_1,r_2)$ of sections of $L$ with six simple common zeros, +whose reduced divisor is denoted by $Z$. These common zeros create the +main compatibility problem: each requires a pointwise correction to make +the leading determinant coefficient nonzero along $C$, but the corrections must also satisfy +global residue constraints to glue the vector fields. The curve and trace +construction produces data satisfying both requirements. + +\paragraph{A curve and normalized residues.} +Section~\ref{curve:section} constructs $C$ on $\PP^1\times\PP^1$, with +fourth-order contact at $Z$ with an auxiliary elliptic curve $\Gamma$. +A two-torsion relation on $\Gamma$ controls the section dimensions on $C$ and supplies +six vectors in a two-dimensional quadratic space. The resulting curve +has genus $21$, and $\deg L=10$. + +Section~\ref{trace:section} transports these vectors to $C$ by a trace +pairing for a degree-four map $C\to\PP^1$. Residue duality compares +principal parts on the two curves and fixes the scalar and sign of the +pairing. Proposition~\ref{prop:residue-data} then gives a meromorphic +pair $P$ of sections of $L^2$ whose local expansions enforce the determinant +condition and whose residue sums cancel the gluing obstructions. + +\paragraph{Finite vector-field jets.} +In Section~\ref{jets:section}, we modify the neighborhood by finitely +many holomorphic flows and choose compatible coefficients of the fields. +At order two the obstruction separates into a symmetric residue sum and +one antisymmetric sum; the normalized residue data cancel both. +Thereafter, we extend the vector fields and their desired determinant +as separate but compatible finite Taylor expansions. Choosing the +determinant coefficients before solving for the field coefficients leaves +the freedom needed to remove the remaining obstructions. Exactness of a +differential and cohomology vanishing finish the finite induction. +Proposition~\ref{prop:jets} supplies finite jets +with determinant $t^5(1+O(t))$ in ordinary coordinates, to sufficient +order for the holomorphic lifting below. + +\paragraph{Holomorphic lifting and contraction.} +Section~\ref{analytic:section} realizes the finite transition maps as +a Hausdorff surface neighborhood $Y$ of $C$. Set +$\cI=\OO_Y(-C)$ and $\cF=\cT_Y(-\log C)$, the sheaf of vector fields +tangent to $C$. Grauert's criterion contracts the curve to a point +$x$ of a normal surface $X$~\cite{Grauert1962}. Cohomology of the +successive ideal quotients and the theorem on formal +functions~\cite{Grauert1960} give the cohomological vanishing needed to +lift these jets to holomorphic fields after shrinking, without requiring +an infinite sequence of neighborhood +changes to converge. Their determinant gives a frame on +$Y\setminus C\simeq X\setminus\{x\}$, which normality extends to a +free tangent sheaf on $X$. Adjunction proves that $x$ is singular. + +\paragraph{The affine algebraic endpoint.} +Section~\ref{algebra:section} combines approximation with the finite +determinacy of equations together with their relations. The latter +method appears in Hironaka's equivalence theorem +\cite[\S3, Theorem~3.3]{Hironaka1969} and in Artin's algebraization +of isolated singularities~\cite[Theorems~1.10 and~3.8--3.9]{Artin1969}. +We prove the required finite-determinacy statement with a relation matrix: +the approximated equations and matrix retain their exact zero product. +This produces an algebraic local ring with the same completion as +$\OO_{X,x}$. A finite Jacobian presentation identifies its completed +derivation module with the continuous derivations of that completion. +Faithful flatness descends freeness and the local ring properties. +Finally, shrinking to one affine neighborhood gives the globally free +module in Theorem~\ref{thm:main}, while preserving the singular point. + +\begin{table}[htbp] +\centering +\small +\begin{tabular}{@{}p{0.19\textwidth}p{0.73\textwidth}@{}} +\toprule +Notation & Meaning and normalization \\ +\midrule +$C$, $Z$ & Smooth genus-$21$ curve and its six marked points. \\ +$L$, $r$ & $\deg L=10$, $L^4\simeq K_C$; the two sections in $r$ have +common zero divisor $Z$. \\ +$\Gamma$, $P$ & Auxiliary elliptic curve; meromorphic pair of sections +of $L^2$ supplying the residue identities. \\ +$Y$, $\cI$, $\cF$ & Surface neighborhood of $C$; $\cI=\OO_Y(-C)$ and +$\cF=\cT_Y(-\log C)$. The jets lie in $\cF/\cI^9\cF$. \\ +$\pi:Y\to X$ & Contraction of $C$ to the singular point $x$; +$C^2=-10$. \\ +$A$, $\mathfrak p$ & Affine algebra and marked singular point; +\mbox{$\widehat{A_{\mathfrak p}}\simeq\widehat{\OO_{X,x}}$}. \\ +\bottomrule +\end{tabular} +\caption{The main objects in the construction, in their order of use.} +\label{tab:notation} +\end{table} + +\subsection{Conventions} + +Line bundles on curves are written additively when convenient: $aL+bH$ means $L^{\otimes a}\otimes H^{\otimes b}$, and $H^q(L)$ means $H^q(C,L)$. A divisor and its associated line bundle may share a symbol. For two rows $X=(X_1,X_2)$ and $Y=(Y_1,Y_2)$, put +\[ +X\wedge Y=X_1Y_2-X_2Y_1. +\] +The symmetric bilinear form associated with a quadratic form $Q$ is normalized by $\langle x,x\rangle=Q(x)$. A residue on an annular overlap means the contour integral divided by $2\pi i$. Logarithmic residue and adjunction use the convention that the normal differential comes first. All germs and varieties are over $\CC$. We use standard facts about algebraic curve cohomology and adjunction as in~\cite{Hartshorne1977}, and about coherent analytic sheaves and normal spaces as in~\cite{GrauertRemmert1984}. diff --git a/preprints/A-singular-normal-affine-surface-with-free-tangent-sheaf-September-23-2026/build/02-curve.tex b/preprints/A-singular-normal-affine-surface-with-free-tangent-sheaf-September-23-2026/build/02-curve.tex new file mode 100644 index 0000000000000000000000000000000000000000..9f4b580e499d0e5d87ab5a46787f4ab74de76b5d --- /dev/null +++ b/preprints/A-singular-normal-affine-surface-with-free-tangent-sheaf-September-23-2026/build/02-curve.tex @@ -0,0 +1,440 @@ +\section{A curve with prescribed residue data} +\label{curve:section} + +We use additive notation for line bundles. If $B$ is a line bundle and +$D$ is a divisor, $B+D$ means $B\otimes\OO(D)$; restrictions of bundles +will occasionally be left implicit. We first construct a marked curve +and a fourth root of its canonical bundle. An auxiliary elliptic curve +will control the directions and the normalization of the residue data. +The output is collected in Proposition~\ref{prop:curve-data}: besides +$L^4\simeq K_C$, we need precise section dimensions and evaluation maps +at six marked points. These conditions will make the low-order gluing +obstructions computable. + +\subsection{The elliptic model and its quadratic form} + +Let $\Gamma$ initially be a smooth elliptic curve. Choose a degree-two +line bundle $F$ and a line bundle $D_0$ of exact order four, and set +$S=F+D_0$. Both $F$ and $S$ are generated by their two-dimensional +spaces of sections. The product of their associated maps embeds +$\Gamma$ in +\[ + \Sigma=\PP^1\times\PP^1. +\] +Indeed, after choosing an origin on $\Gamma$, the two fiber involutions +have the form $p\mapsto a-p$ and $p\mapsto a+d_0-p$, where $d_0\ne0$ +represents $D_0$. Two distinct points cannot be interchanged by both +involutions, and a point cannot be fixed by both. The product map is +therefore injective and immersive, hence is a closed embedding. Its +two projection degrees show that its image has bidegree $(2,2)$. +We now also write $F=(1,0)$ and $S=(0,1)$ for the ruling bundles on +$\Sigma$. + +On $\Gamma$ put +\[ + \Delta=2D_0,\qquad \Phi=F+\Delta,\qquad M=3\Phi. +\] +Thus $\Delta$ is nontrivial of order two, and +\[ + \Gamma|_\Gamma=2F+2S=4F+\Delta, + \qquad M=3F+\Delta,\qquad 2M=6F. +\] +Consequently +\begin{equation}\label{curve:bundle-identities} +\begin{split} + 4M&=(2\Gamma+4F)|_\Gamma,\\ + 2M-(\Gamma+F)|_\Gamma&=\Phi,\\ + (\Gamma+2F)|_\Gamma-2M&=\Delta. +\end{split} +\end{equation} +Fix isomorphisms in these identities compatibly. In particular, +viewing $\Phi$ as $2M-(\Gamma+F)|_\Gamma$, the first identity induces +the identification $2\Phi=2F|_\Gamma$ that we use below. + +Since $K_\Sigma+\Gamma=0$, choose a nowhere vanishing logarithmic +two-form $\Omega$ with pole divisor $\Gamma$. Its residue +$\omega=\Res_\Gamma\Omega$ is a nowhere vanishing differential on +$\Gamma$. Our residue convention places the normal differential +first: locally $\Res_{y=0}((dy/y)\wedge dz)=dz$. +Choose a basis $u=(u_1,u_2)$ of $H^0(\Sigma,F)$, and write +\[ + \eta_\Gamma=u_1\,du_2-u_2\,du_1 + \in H^0(\Gamma,K_\Gamma+2F). +\] +The expression is independent of the local frame for $F$: in that +frame it is the indicated coefficient differential times the square +of the frame. + +There is a unique element of $\Sym^2H^0(\Gamma,\Phi)$ whose image under +multiplication is +\begin{equation}\label{curve:quadratic} + -\frac{\eta_\Gamma}{\omega}\in H^0(\Gamma,2\Phi) + =H^0(\Gamma,2F). +\end{equation} +It defines a nondegenerate quadratic form $Q$ on +$H^0(\Gamma,\Phi)^*$. To see both existence and nondegeneracy, let +$\iota_F$ and $\iota_\Phi$ be the two degree-two involutions. One has +$\iota_\Phi=t_\Delta\iota_F$, where $t_\Delta$ is translation by the +nonzero point of order two corresponding to $\Delta$. The four +ramification points of $F|_\Gamma$, namely the fixed points of +$\iota_F$, are paired without fixed points by $\iota_\Phi$. They form +two distinct reduced fibers of the $\Phi$-map. Their divisor is +therefore the pullback of the zero divisor of a binary quadratic with +two distinct roots. Scaling this quadratic gives exactly +\eqref{curve:quadratic}. Uniqueness follows because pullback from the +surjective $\Phi$-map is injective on sections of $\OO_{\PP^1}(2)$. + +For $q\in\Gamma$, evaluation in a local frame of $\Phi$ gives a +nonzero element of $H^0(\Gamma,\Phi)^*$, defined up to a nonzero scalar. +The corresponding projective direction will be denoted by +$[\operatorname{ev}_q]$. By \eqref{curve:quadratic}, this direction is +isotropic precisely when $q$ is a ramification point of $F|_\Gamma$. +For any quadratic form in this section, we use its polar form +\[ + x\mathbin{\cdot}y=\frac{Q(x+y)-Q(x)-Q(y)}2, + \qquad x^2=Q(x). +\] + +The six vectors below encode both a pointwise normalization and a total +residue. If $w=\sum_jw_j$ and $w_j^2+w_j\mathbin{\cdot}w=1$ at +each marked point, then +\[ + \sum_{j=1}^6(w_j^2-1)=-w^2. +\] +The curve we construct has $\deg L=10$. Thus the choice $w^2=90$ +will give the total $-9\deg L$ required by the order-two obstruction +in Section~\ref{jets:section}, while retaining all six pointwise +normalizations. + +\begin{lemma}\label{curve:six-vectors} +In a two-dimensional nondegenerate complex quadratic space, one can +choose three distinct nonisotropic directions, avoiding any prescribed +finite set of projective directions, and nonzero vectors +$w_1,\ldots,w_6$ that use each of these directions twice, with the two +vectors in each pair equal, such that +\begin{equation}\label{curve:vector-conditions} + w=\sum_{j=1}^6w_j,\qquad + w^2=90,\qquad w_j^2+w_j\mathbin{\cdot}w=1 + \quad(1\le j\le6). +\end{equation} +\end{lemma} + +\begin{proof} +Choose coordinates in which $Q(x,y)=xy$, so that +$(x,y)\mathbin{\cdot}(x',y')=(xy'+yx')/2$. Set +\[ + h=\sqrt{90}>0,\quad R=\sqrt{1+h^2/4}>0,\quad + b=\frac{2h}{R},\quad + d(z)=\left(Rz-\frac h2,\frac Rz-\frac h2\right). +\] +Direct expansion gives +\[ + d(z)^2+d(z)\mathbin{\cdot}(h,h)=R^2-h^2/4=1. +\] +For a sufficiently small general nonzero complex number $\lambda$, +let $z_1,z_2,z_3$ be the roots of +\[ + X^3-bX^2+b\lambda X-\lambda=0. +\] +Vieta's formulas give +\[ + \sum_{i=1}^3z_i=b,\qquad + \sum_{i=1}^3z_i^{-1}=b,\qquad + 2\sum_{i=1}^3d(z_i)=(h,h). +\] +Thus taking each $d(z_i)$ twice gives \eqref{curve:vector-conditions}. + +For completeness, these vectors have the required distinct directions. +As $\lambda\to0$, two roots approach zero and the third approaches +$b$. The roots are distinct for general $\lambda$: the cubic +discriminant has nonzero leading term $-4b^3\lambda$ at zero. +Near zero the projective ratio of the two entries of $d(z)$ is +\[ + \frac{z(Rz-h/2)}{R-hz/2}, +\] +whose derivative at zero is $-h/(2R)\ne0$. The two small roots +therefore give distinct nonisotropic directions. At $z=b$ the entries +are $3h/2$ and $(R^2-h^2)/(2h)$, both nonzero, so the third limiting +direction is different from their common limit. These properties +persist for sufficiently small general $\lambda$. Finally apply +simultaneously to all six vectors an isometry +$(x,y)\mapsto(\nu x,\nu^{-1}y)$. On nonisotropic projective +directions this multiplies the ratio by $\nu^2$. Only finitely many +values of $\nu\in\CC^*$ are excluded by the prescribed finite set, +and all equations in \eqref{curve:vector-conditions} are preserved. +\end{proof} + +Apply Lemma~\ref{curve:six-vectors} to the quadratic form $Q$, excluding +the branch values of the $\Phi$-map. Each of its three directions is +then $[\operatorname{ev}_q]$ at two distinct points of $\Gamma$. +Label these points $q_1,\ldots,q_6$ so that +$[w_j]=[\operatorname{ev}_{q_j}]$, and put +\begin{equation}\label{curve:marked-divisor} + Z=q_1+\cdots+q_6\in|3\Phi|=|M|. +\end{equation} +Choose a section $\beta$ of $M$ with divisor $Z$. +Nonisotropy implies that $F|_\Gamma$ is unramified at all six points. +Their $F$-values are not all equal, since a degree-two fiber cannot +contain six distinct points. + +\subsection{Fourth-order contact on the quadric} + +We have chosen the six points and their quadratic-space data on the +elliptic curve. We now construct the curve $C$ through those points. +Fourth-order contact will transfer the short evaluation schemes from +$C$ to $\Gamma$, where their section dimensions are easy to compute. + +Write $\gamma$ for the section of $\OO_\Sigma(\Gamma)$ defining +$\Gamma$. We choose a section +\[ + \kappa\in H^0(\Sigma,2\Gamma+4F),\qquad + \kappa|_\Gamma=\beta^4, +\] +using the fixed identification in \eqref{curve:bundle-identities}. +Such sections exist: restriction to $\Gamma$ has kernel +$\OO_\Sigma(\Gamma+4F)=\OO_\Sigma(6,2)$, whose first cohomology +vanishes. The affine space of choices is +\[ + \kappa_0+\gamma H^0(\Sigma,\Gamma+4F). +\] +A general choice defines a smooth curve $C$. Indeed, the associated +linear system is without base points off $\Gamma$, so Bertini's +Theorem gives smoothness there. On $\Gamma\setminus Z$ every choice +is nonzero. At each $q_j$, global generation of +$\OO_\Sigma(6,2)$ allows the normal derivative to vary nontrivially. +Avoiding six proper affine hyperplanes makes all those derivatives +nonzero. Their complements and the Bertini open meet in the same +irreducible affine parameter space; independence of the six evaluation +functionals is unnecessary. + +More explicitly, near $q_j$ choose a local equation $y=0$ for +$\Gamma$, a coordinate $z$ on $\Gamma$ vanishing at $q_j$, and frames +in which $\kappa$ has coefficient $k(y,z)$ and $\beta$ has coefficient +$b(z)$. The frames may be chosen compatibly with +\eqref{curve:bundle-identities}, so that +\[ + k(0,z)=b(z)^4,\qquad b'(0)\ne0,\qquad k_y(0,0)\ne0. +\] +The implicit function theorem expresses $C$ as $y=\varphi(z)$, where +\begin{equation}\label{curve:contact-expansion} + \varphi(z)=-\frac{b'(0)^4}{k_y(0,0)}z^4+O(z^5). +\end{equation} +Thus the contact order is exactly four and +$\operatorname{div}_C(\gamma|_C)=4Z$, with the same six marked points +now regarded as points of $C$. + +The bidegree of $C$ is $(8,4)$. The sequence for its defining divisor, +together with $H^1(\Sigma,\OO_\Sigma(-8,-4))=0$, gives +$H^0(C,\OO_C)=\CC$. Hence the smooth curve $C$ is connected and +irreducible. Put +\begin{equation}\label{curve:line-bundles} + H=F|_C,\qquad L=H+Z,\qquad\theta=2L, +\end{equation} +and let $\xi$ be the canonical section of $\OO_C(Z)$. +The divisor equality above specifies the isomorphism +$4\OO_C(Z)\simeq\OO_C(\Gamma)$ by $\xi^4\mapsto\gamma|_C$. +Together with adjunction, this gives +\begin{equation}\label{curve:canonical-root} + 4L=4H+4Z\simeq(\Gamma+4F)|_C\simeq K_C. +\end{equation} +We fix the last isomorphism by the following normalization. For a +local section $s$ of $\Gamma+4F$, it is +\begin{equation}\label{curve:adjunction-normalization} + s|_C\longmapsto + \Res_C\left(\frac{s\gamma\Omega}{\kappa}\right), +\end{equation} +with the normal differential first. This formula fixes the scalar in +\eqref{curve:canonical-root} for the later residue computations. + +\subsection{Section counts and evaluation maps} + +We give the jet calculation underlying all the special-divisor +dimensions. The section spaces below are algebraic section spaces; +equivalently, they are holomorphic section spaces on these projective +curves, by the comparison theorem of Serre~\cite[Theorem~1]{SerreGAGA1956}. + +\begin{lemma}\label{curve:adjoint-jets} +For $0\le j\le3$, set +\[ + \mathcal A_j=\Gamma+(4-j)F=(6-j,2),\qquad + N_j=\mathcal A_j|_\Gamma=(8-j)F+\Delta. +\] +Restriction identifies $H^0(\Sigma,\mathcal A_j)$ with +$H^0(C,K_C-jH)$ and maps it onto $H^0(\Gamma,N_j)$. +For $1\le\ell\le4$, evaluation on the length-$\ell$ subschemes at +the six marked points has the same rank on $C$ as on $\Gamma$. +The ranks needed here are +\begin{equation}\label{curve:residual-table} +\begin{array}{c|c|c|c|c|c} + j&\ell& h^0(\Gamma,N_j)&N_j-\ell Z + &h^0(\Gamma,N_j-\ell Z)&\text{evaluation rank}\\ \hline + 0&1&16&5F&10&6\\ + 1&2&14&\Phi&2&12\\ + 2&2&12&\Delta&0&12\\ + 3&2&10&-F+\Delta&0&10 +\end{array} +\end{equation} +\end{lemma} + +\begin{proof} +The restriction kernels on $\Sigma$ are respectively +\[ + \mathcal A_j-C=(-2-j,-2),\qquad + \mathcal A_j-\Gamma=(4-j,0). +\] +Product cohomology on $\PP^1\times\PP^1$ gives $H^0=H^1=0$ for +the first kernel and $H^1=0$ for the second. Adjunction therefore +gives the two assertions about restriction. + +By \eqref{curve:contact-expansion}, the ideals of the two length-$\ell$ +subschemes in the local surface are +\[ + (y-\varphi(z),z^\ell)=(y,z^\ell)\qquad(\ell\le4). +\] +Thus these are the same embedded subscheme, and restrictions of the +surface line bundle and its sections to them agree. The two +surjectivity statements just proved imply equality of evaluation ranks. + +On the elliptic curve, $\deg F=2$, $Z=3F+\Delta$, and $2Z=6F$. +These identities give every residual bundle in +\eqref{curve:residual-table}. A positive-degree line bundle on an +elliptic curve has as many sections as its degree; a negative-degree +bundle has no section. The degree-zero bundle $\Delta$ is nontrivial, +so both its zeroth and first cohomology vanish. Subtracting the +residual section dimensions from $h^0(\Gamma,N_j)$ gives the stated +ranks. In particular, the twelve-dimensional surface system for +$j=3$ has a two-dimensional kernel on restriction to $\Gamma$, and +the resulting ten-dimensional system injects into its length-twelve +evaluation space. +\end{proof} + +\begin{proposition}\label{prop:curve-data} +The construction above produces a smooth connected projective curve +$C$, a reduced divisor $Z=q_1+\cdots+q_6$, line bundles $H,L,\theta$, +and a basis $u=(u_1,u_2)$ of $H^0(C,H)$ with the following properties. +\begin{enumerate} +\item The numerical and canonical data are +\[ + \begin{gathered} + g(C)=21,\qquad \deg H=4,\qquad\deg L=10,\\ + L=H+Z,\qquad\theta=2L,\qquad K_C=4L, + \end{gathered} +\] +where the last isomorphism is normalized by +\eqref{curve:adjunction-normalization} and $\xi^4\mapsto\gamma|_C$. +The curve $C$ has exactly fourth-order contact with the auxiliary +elliptic curve $\Gamma$ at every $q_j$. +\item The pencil $u$ defines a finite degree-four morphism +$f:C\to\PP^1$ unramified at every $q_j$, and the values $f(q_j)$ are +not all equal. The zero divisor of $u_1$ can be chosen to be a reduced +fiber disjoint from $Z$. +\item The section dimensions are +\begin{equation}\label{curve:section-dimensions} +\begin{gathered} + h^0(C,jH)=j+1\quad(0\le j\le3),\\ + h^0(C,L)=h^0(C,H+2Z)=2,\qquad + h^0(C,\theta)=3,\qquad h^0(C,\OO_C(Z))=1,\\ + h^0(C,\theta+H)=6. +\end{gathered} +\end{equation} +More precisely, +\begin{equation}\label{curve:section-multiplication} +\begin{split} + H^0(C,L)&=\xi H^0(C,H),\\ + H^0(C,H+2Z)&=\xi^2H^0(C,H),\\ + H^0(C,\theta)&=\xi^2H^0(C,2H). +\end{split} +\end{equation} +In particular $r=\xi u$ is a basis of $H^0(C,L)$ with simple common +zero divisor exactly $Z$, and every section of $\theta$ vanishes +twice along $Z$. +\item For every $k\ge2$, the evaluation map +\begin{equation}\label{curve:evaluation-surjectivity} + H^0(C,(k+1)L)\longrightarrow H^0(Z,((k+1)L)|_Z) +\end{equation} +is surjective. +\item The auxiliary data on $\Gamma\subset\Sigma$, including the +compatible bundle identities, the logarithmic volume $\Omega$, and +the quadratic form $Q$, are as in \eqref{curve:bundle-identities} and +\eqref{curve:quadratic}; the divisor $Z$ belongs to $|M|$. +The six nonzero vectors $w_j\in H^0(\Gamma,\Phi)^*$ have the +projective evaluation directions at $q_j$, use three distinct +directions, and satisfy \eqref{curve:vector-conditions}, with +$w^2=90=9\deg L$. +\end{enumerate} +\end{proposition} + +\begin{proof} +The existence and contact assertions were established above. From +$C=(8,4)$ and $K_C=(6,2)|_C$, intersection on the quadric gives +\[ + \deg H=4,\qquad \deg K_C=6\cdot4+2\cdot8=40, + \qquad g(C)=21. +\] +Since $\deg Z=6$, one has $\deg L=10$. The morphism given by $H$ is +the first projection, restricted to the irreducible curve $C$, hence +is finite of degree four. Tangency at the marked points identifies +its differential there with that of $F|_\Gamma$, which is nonzero. +Its marked values are the corresponding $F|_\Gamma$-values, already +seen not to be all equal. Choose a regular fiber away from these +finitely many values. A determinant-one change of the basis $u$ +makes this fiber the zero divisor of $u_1$. Such a change preserves +$u_1\,du_2-u_2\,du_1$ on both curves, and hence preserves the +quadratic form and every normalization already chosen. + +For $0\le j\le3$, the sequence restricting $\OO_\Sigma(j,0)$ to +$C$ has kernel $\OO_\Sigma(j-8,-4)$, with $H^0=H^1=0$ by product +cohomology. This proves $h^0(C,jH)=j+1$. For any line bundle $B$ +on $C$ and effective divisor $D$, Riemann--Roch and Serre duality give +\begin{equation}\label{curve:dimension-formula} + h^0(C,B+D)-h^0(C,B) + =\deg D-\operatorname{rank}\left( + H^0(C,K_C-B)\longrightarrow H^0(D,(K_C-B)|_D)\right). +\end{equation} +Apply this with $B=jH$ and use the ranks in +Lemma~\ref{curve:adjoint-jets}. Its first three rows give +\[ + h^0(C,\OO_C(Z))=1, + \quad h^0(C,H+2Z)=2, + \quad h^0(C,2H+2Z)=3. +\] +The inclusions of $H^0(C,H)$ into the spaces for $H+Z$ and $H+2Z$ +then give $h^0(C,L)=2$. The fourth row gives +\[ + h^0(C,3H+2Z)=h^0(C,3H)+12-10=6. +\] +This proves \eqref{curve:section-dimensions}, and equality of the +dimensions in the natural multiplication inclusions proves +\eqref{curve:section-multiplication}. Because the pencil $u$ has no +base point and $Z$ is reduced, the assertion about the common zeros +of $r$ follows as well. + +Finally, the divisor sequence for $N=(k+1)L$, followed by Serre +duality, shows that \eqref{curve:evaluation-surjectivity} is +surjective if and only if the natural inclusion +\[ + H^0(C,(3-k)L)\longrightarrow H^0(C,(3-k)L+Z) +\] +is an isomorphism. For $k=2$ this is the inclusion from $L$ to +$L+Z=H+2Z$, between two-dimensional spaces. For $k=3$ it is the +inclusion from $\OO_C$ to $\OO_C(Z)$, between one-dimensional +spaces. For $k\ge4$ both spaces vanish, since the target bundle has +degree $10(3-k)+6\le-4$. The final assertion follows from +Lemma~\ref{curve:six-vectors} and \eqref{curve:marked-divisor}. +\end{proof} + +For later use, Riemann--Roch and Serre duality also give +\begin{equation}\label{curve:boundary-cohomology} + h^1(C,L)=h^0(C,3L)=12,\qquad h^0(C,4L)=21. +\end{equation} +Indeed, $h^0(C,L)-h^1(C,L)=10+1-21=-10$ and +$K_C-L=3L$. + +One additional consequence will be useful when removing a +two-dimensional obstruction. In local frames compatible with +$L=H+Z$, with $\xi$ represented by a parameter $z$ at a marked point, +one has $r=zu$ and +$r'(0)=u(0)$. The resulting six nonzero row vectors $r'(q_j)$ span +$\CC^2$: their projective directions are the pencil values $f(q_j)$, +which are not all equal. Changing the frames or the local parameters +only rescales each of these rows. diff --git a/preprints/A-singular-normal-affine-surface-with-free-tangent-sheaf-September-23-2026/build/03-trace.tex b/preprints/A-singular-normal-affine-surface-with-free-tangent-sheaf-September-23-2026/build/03-trace.tex new file mode 100644 index 0000000000000000000000000000000000000000..875de4b0d1d720cbf1189f999800aa59db5a41bf --- /dev/null +++ b/preprints/A-singular-normal-affine-surface-with-free-tangent-sheaf-September-23-2026/build/03-trace.tex @@ -0,0 +1,545 @@ +\section{The trace form and the residue data} +\label{trace:section} + +The goal of this section is Proposition~\ref{prop:residue-data}: a +meromorphic pair with prescribed poles and two normalized residue +identities. The trace form translates the quadratic-space data on the +elliptic curve into those identities on $C$. The comparison must preserve +its scalar and sign, since both enter the first gluing obstruction. + +Retain the data and the normalizations of Proposition~\ref{prop:curve-data}. +In particular, $\Omega$ is the fixed logarithmic volume on $\Sigma$, its +normal-first residue is the nowhere vanishing differential $\omega$ on +$\Gamma$, and $4L\simeq K_C$ is the specified adjunction isomorphism. +Put +\[ + \theta=2L,\qquad \Lambda=\theta+H=3H+2Z, + \qquad f:C\longrightarrow\PP^1. +\] +For the chosen basis $u=(u_1,u_2)$ of the ruling pencil, write +$\eta_C=u_1\,du_2-u_2\,du_1\in H^0(C,K_C+2H)$. +Although the individual derivatives depend on a frame of $H$, this +alternating expression does not. We use the analogous notation +$\eta_\Gamma$ on $\Gamma$. + +\subsection{A two-dimensional quotient of a perfect trace form} + +The pairing below is a concrete instance of finite-morphism duality, +as developed by Grothendieck and presented in Hartshorne's +\emph{Residues and Duality}~\cite[Chapter~III, \S6]{Hartshorne1966}. We check its local +perfection directly so that the trace normalization remains explicit. + +\begin{lemma}\label{trace:pairing} +The rank-four vector bundle $E=f_*\Lambda$ has a perfect symmetric pairing +\begin{equation}\label{trace:form} + E\otimes E\longrightarrow\OO_{\PP^1},\qquad + (T_1,T_2)\longmapsto + \Tr_{C/\PP^1}\!\left(\frac{T_1T_2}{\eta_C}\right). +\end{equation} +Its splitting type is $(3,0,0,-3)$. Multiplication by $\xi^2$ identifies +$\OO_{\PP^1}(3)$ with its positive line subbundle $\mathcal P$. +The form on global sections has radical +$\xi^2H^0(C,3H)$, and induces a nondegenerate symmetric form +$\langle\ ,\ \rangle$ on +\begin{equation}\label{trace:W} + W=H^0(C,\Lambda)/\xi^2H^0(C,3H),\qquad \dim W=2. +\end{equation} +For every $x\in\PP^1$, evaluation identifies this quadratic space with +$(\mathcal P^\perp/\mathcal P)_x$. + +Suppose $q\in Z$, and choose local frames $\mathfrak f$ of $H$ and +$\tau$ of $\Lambda$, and a parameter $z$ at $q$. Write, using the fixed +isomorphism $2\Lambda\simeq K_C+2H$, +\[ + \tau^2=s(z)\,dz\,\mathfrak f^2, + \qquad \eta_C=\mathcal E(z)\,dz\,\mathfrak f^2. +\] +Coefficient evaluation $\ell_q:T\mapsto(T/\tau)(q)$ descends to $W$. +Its square in the dual form is +\begin{equation}\label{trace:dual-evaluation} + \ell_q^2=\frac{\mathcal E(q)}{s(q)}. +\end{equation} +In particular, it is nonzero and nonisotropic. +\end{lemma} + +\begin{proof} +The morphism $f$ is finite and flat of degree four. The identity +$2\Lambda=K_C+2H$ identifies $\Lambda^2$ with the relative dualizing +bundle. We verify directly the normalization and perfection of its trace +pairing. Locally at a ramification point one can choose parameters with +$x=w^e$ on a single component over a target disk. A frame of $H$ pulled +back from that disk can be chosen so that +$\eta_C=dx\,\mathfrak f^2$. If +$\tau^2=a(w)\,dw\,\mathfrak f^2$, with $a$ a unit, the local pairing is +\[ + (b,c)\longmapsto + \Tr\!\left(\frac{a(w)bc}{e w^{e-1}}\right). +\] +On the basis $1,w,\ldots,w^{e-1}$, the pairing without the unit $a$ has +matrix +\[ + \Tr\!\left(\frac{w^{i+j}}{e w^{e-1}}\right) + =\begin{cases}1&i+j=e-1,\\0&i+j\ne e-1, + \end{cases} + \qquad 0\le i,j0, +\end{equation} +where $b,h$ are holomorphic on the overlap in the disk coordinate. +Use the time-one flow of $-D$, so that its action on vector fields is +$\exp(\operatorname{ad}D)$. Successive modifications are applied to +rows already expressed on the outside; all formulas below are relative +to the original split comparison. Every generator vanishes on the +zero section and changes its equation only by $t\mapsto t+O(t^2)$, +so the zero section and its conormal bundle are preserved. + +\begin{lemma}\label{jets:bracket} +For $D$ as in \eqref{jets:generator} and +$V=t^{\ell+1}(AE_t+B\partial_z)$ one has +\begin{align} + [D,V]=t^{k+\ell+1}\bigl(&((\ell+1-k)bA+hA'-Bb')E_t\notag\\ + &+((\ell+1)bB+hB'-khA-Bh')\partial_z\bigr). + \label{jets:general-bracket} +\end{align} +Moreover, +\begin{equation}\label{jets:divergence} + \mathcal L_D\sigma=t^k\bigl((k-4)b+h'\bigr)\sigma. +\end{equation} +In particular, for $k\ne4$ the choice $b=h'/(4-k)$ preserves $\sigma$ +exactly. The first effect of an order-$k$ modification on +\eqref{jets:fields} is +\begin{equation}\label{jets:leading-action} + \Delta A_k=(1-k)br+hr',\qquad \Delta B_k=-khr. +\end{equation} +\end{lemma} + +\begin{proof} +Use $E_t(t^a)=at^a$ and $[E_t,\partial_z]=0$ in the two components of +the bracket. For the divergence, the coefficient of $dt\wedge dz$ in +$\sigma$ is $t^{-5}$, so +\[ + \operatorname{div}_{\sigma}D + =t^5\left(\partial_t(t^{k-4}b) + +\partial_z(t^{k-5}h)\right) + =t^k((k-4)b+h'). +\] +Setting $\ell=0$, $A=r$, $B=0$ gives the last formula. +\end{proof} + +All calculations can initially be made with finite Taylor series. +They nevertheless prescribe actual transition germs: on a compact +subannulus every $D$ in \eqref{jets:generator} is holomorphic near its +zero section and vanishes there, so its time-one flow and inverse exist +after shrinking in the transverse direction. A finite composition has +the computed Taylor series. There are no intersections between +different disk charts, hence no additional triple-overlap equations. +The passage from these germs to a Hausdorff surface neighborhood will +be made in the next section. + +\subsection{The first two orders} + +At order one use $h=m=m_j$, $b=m'/3$ on the $j$th overlap, and choose +\begin{equation}\label{jets:first-order} +\begin{array}{c|cc} + &A_1&B_1\\ \hline + U_j&p&mr\\ + U_0&P&0. +\end{array} +\end{equation} +By \eqref{jets:leading-action}, the first transition adds +$(mr',-mr)$ to the inside row, giving exactly $(P,0)$. +The degree-one coefficient of $A\wedge B$ is zero on both charts. + +At order two write $U=B_2$. To give the determinant its required +leading coefficient, solve on each chart +\begin{equation}\label{jets:second-horizontal} + r\wedge U=1-A_1\wedge B_1. +\end{equation} +On a disk write $r=z\rho$, with $\rho$ nowhere zero as a row after +shrinking. By \eqref{jets:residue-input}, the right side of +\eqref{jets:second-horizontal} vanishes at $z=0$, so division by $z$ +reduces it to a surjective wedge map with $\rho$. +On $U_0$ the row $r$ has no common zero, and the corresponding +surjection of coherent sheaves is surjective on sections by +Steinness. Thus the required holomorphic horizontal rows exist. +After the first transition, their horizontal difference on an overlap +wedges to zero with $r$: the determinant target is compatible because +that transition preserves $\sigma$. +The difference is therefore a unique holomorphic multiple of $r$. +An order-two transition with $b=h'/2$ removes it, since its horizontal +effect is $-2hr$. Keep the chosen outside row $U$ fixed throughout this +matching. + +It remains to match the vertical lifts $A_2$. The obstruction belongs +to $H^1(C,L^3)^2$. Its four Serre-duality tests use $r_1,r_2$, a basis +of $H^0(C,K_C\otimes L^{-3})=H^0(C,L)$, against the two entries of +the vertical difference. The following calculation evaluates all four. + +\begin{lemma}\label{jets:second-obstruction} +The order-two vertical obstruction vanishes. +\end{lemma} + +\begin{proof} +Write $D^{(2)}=A_2$; this notation denotes the vertical row, not a +transition generator. For $i,j\in\{1,2\}$ set +\begin{equation}\label{jets:psi} + \Psi_{ij}=\left(r_iD_j^{(2)}+\frac14r_i'U_j + +\frac34r_j'U_i\right)dz. +\end{equation} +Once the horizontal rows have matched, the difference of these +expressions for the transported inside lift and the outside lift is +exactly $r_i$ times the $j$th vertical difference, multiplied by $dz$. +The derivative terms therefore change neither the obstruction nor its +four tests. + +First compute the total for the outside lift. Put +$\epsilon_{12}=1$, $\epsilon_{21}=-1$, and +$\epsilon_{11}=\epsilon_{22}=0$. +Under a split change with $a=(\log g)'$, formula +\eqref{jets:split-coefficients} gives +\[ + \widetilde r_i=g^{-1}r_i,\qquad + \partial_{\widetilde z}\widetilde r_i=g^{-5}(r_i'-ar_i),\qquad + \widetilde U_i=gU_i,\qquad + \widetilde D_i^{(2)}=g^{-3}(D_i^{(2)}+aU_i). +\] +Substituting in \eqref{jets:psi}, and expressing both differentials in +the old coordinate, yields +\begin{equation}\label{jets:psi-transformation} + \widetilde\Psi_{ij}-\Psi_{ij} + =\frac34(r_iU_j-r_jU_i)d\log g + =\frac34\epsilon_{ij}\,d\log g + \quad\hbox{on }U_0. +\end{equation} +The last equality uses $r\wedge U=1$ on the outside. +Use the section $r_1=\xi u_1$ of $L$. Its coefficient transforms by +$\widetilde r_1=g^{-1}r_1$, so +$\Psi_{ij}+\frac34\epsilon_{ij}d\log r_1$ is an intrinsic differential +on $U_0$. There it has precisely the four possible simple poles at +the reduced zero fiber of $u_1$, each of residue +$\frac34\epsilon_{ij}$. The residue theorem on the complement of +small disks therefore gives total contour residue $-3\epsilon_{ij}$ +around $Z$ for this intrinsic differential. At each of the six +points of $Z$, the section $r_1$ has a simple zero. Subtracting these +six logarithmic residues yields +\begin{equation}\label{jets:outside-residue} + \sum_{q\in Z}\Res_q\Psi_{ij}^{\mathrm{out}} + =-\frac34\epsilon_{ij}\deg L. +\end{equation} +Only contour integrals at $Z$ were used, so this argument does not +assume meromorphic extension of the outside lifts. + +Next compute the transported inside total. Before transport, +$\Psi_{ij}$ is holomorphic on the disk. The order-two transition +changes its rows by +\[ + \delta D^{(2)}=hr'-\frac12h'r,\qquad \delta U=-2hr. +\] +In \eqref{jets:psi} this is the exact differential +\begin{equation}\label{jets:psi-exact} + \delta\Psi_{ij}=-\frac12d(hr_ir_j), +\end{equation} +so it has zero contour residue. +It suffices to calculate the effect of the order-one transition, +including its second exponential term. + +The bracket of $t(m'E_t/3+m\partial_z)$ with the inside order-one +term $t^2(pE_t+mr\partial_z)$ contributes +\[ + \left(\frac13m'p+mp'-\frac13mm''r, + \frac23mm'r+m^2r'-mp\right) +\] +to $(A_2,U)$. Half of its second bracket with the leading term +$trE_t$ contributes +\[ + \left(\frac23mm'r'+\frac12m^2r''+\frac16mm''r, + -\frac13mm'r-m^2r'\right). +\] +Thus the full effect is +\begin{align} + \Delta A_2&=\frac13m'p+mp'+\frac23mm'r' + +\frac12m^2r''-\frac16mm''r, + \notag\\ + \Delta U&=-mp+\frac13mm'r. + \label{jets:second-expansion} +\end{align} +The $m^2r'$ horizontal terms cancel in this sum. + +Fix one disk and abbreviate $R=mr$, $m=c/z$, so $r=zR/c$. +Substitution in \eqref{jets:second-expansion} gives +\[ + \begin{aligned} + \Delta A_2&=-\frac{cR}{z^3}+\frac{cR'}{3z^2} + +\frac{cR''}{2z}-\frac{cp}{3z^2}+\frac{cp'}z, + \\ + \Delta U&=-\frac{cp}z-\frac{cR}{3z^2}. + \end{aligned} +\] +It follows, with every coefficient on the right evaluated at $0$, that +\begin{align} + F_{ij}:=\Res_0(r_i\Delta A_{2,j}\,dz) + &=-\frac13R_ip_j-\frac23R_iR_j'-R_i'R_j,\notag\\ + H_{ij}:=\Res_0(r_i'\Delta U_j\,dz) + &=-R_ip_j-\frac13R_iR_j'-\frac23R_i'R_j. + \label{jets:laurent-residues} +\end{align} +For example, the double pole $-R_iR_j/z^2$ in the first expression +contributes $-R_i'R_j-R_iR_j'$ to its residue; this accounts for both +derivative terms. The complete residue of the change in +\eqref{jets:psi} is consequently +\begin{equation}\label{jets:full-residue-matrix} + \begin{aligned} + T_{ij}&=F_{ij}+\frac14H_{ij}+\frac34H_{ji}\\ + &=-\frac7{12}R_ip_j-\frac9{12}R_jp_i + -\frac54R_iR_j'-\frac{17}{12}R_i'R_j. + \end{aligned} +\end{equation} + +Decompose this matrix into half its sum and half its difference with +its transpose. Directly from \eqref{jets:full-residue-matrix}, +\begin{align} + T_{ij}^{\mathrm{sym}} + &=-\frac23\bigl(R_ip_j+R_jp_i + +2R_iR_j'+2R_i'R_j\bigr),\notag\\ + T_{ij}^{\mathrm{alt}} + &=\frac1{12}\epsilon_{ij} + \bigl((R\wedge p)(0)+(R\wedge R')(0)\bigr). + \label{jets:symmetric-alternating} +\end{align} +To identify the symmetric term intrinsically, observe that +\[ + P=p+mr'=\frac{R(0)}z+p(0)+2R'(0)+O(z). +\] +Since $P_iP_j$ is a meromorphic section of $L^4=K_C$, its local +expression is $P_iP_j\,dz$, and +\[ + \Res_0(P_iP_j\,dz) + =R_ip_j+R_jp_i+2R_iR_j'+2R_i'R_j. +\] +Thus the sum of every symmetric entry in +\eqref{jets:symmetric-alternating} is zero, by the residue theorem +on $C$. For the alternating entry, +\eqref{jets:residue-input} gives $R(0)\wedge p(0)=-1$ and therefore +\[ + \sum_{q\in Z}T_{ij}^{\mathrm{alt}} + =\frac1{12}\epsilon_{ij}\sum_{\nu=1}^6(W_\nu-1) + =-\frac34\epsilon_{ij}\deg L. +\] +In particular, the sums of the four entries $(11),(12),(21),(22)$ +of the transported inside matrix are respectively +\[ + 0,\qquad -\frac34\deg L,\qquad + \frac34\deg L,\qquad 0. +\] +They coincide with the four outside totals in +\eqref{jets:outside-residue}. Hence every Serre-duality pairing of the +vertical difference vanishes. The corresponding class in +$H^1(C,L^3)^2$ is zero, so holomorphic vertical corrections on the +charts complete the order-two matching. +\end{proof} + +\subsection{Higher orders and determinant targets} + +From order three onward, each step extends both the fields and the +determinant they are required to have. Write the determinant equation as +\begin{equation}\label{jets:target-equation} + A\wedge B=t^2G,\qquad G=1+G_1t+G_2t^2+\cdots. +\end{equation} +The field equation in degree $k$ uses $G_{k-2}$. To compare the next +horizontal rows, this target coefficient must already be compatible +on the overlaps. We therefore choose each target coefficient before +imposing its field equation, and keep track of these two extensions +separately. + +The target is the local bivector $G\sigma^{-1}$. On a realization $Y$, +with $\cI=\OO_Y(-C)$ and $\cF=\cT_Y(-\log C)$, the relevant line bundle is +\[ + \mathcal M=\cI^4\det\cF=\cI^5\det\cT_Y, + \qquad \mathcal M|_C=L^4\otimes\cT_C\simeq\OO_C. +\] +Its local frame is $\sigma^{-1}$, and its degree-$k$ associated graded +is $L^k$. Only its leading trivialization on $C$ is fixed initially; +the successive targets extend that section through finite orders. +After transitions cease to preserve $\sigma$, $G$ is therefore not +an invariant scalar function. To be explicit, if +$q=\operatorname{div}_{\sigma}D$, then +\begin{equation}\label{jets:target-transport} + [D,G\sigma^{-1}]=(D(G)-qG)\sigma^{-1},\qquad + G\longmapsto\exp(D-q)G. +\end{equation} +Here $D-q$ denotes the differential operator $f\mapsto D(f)-qf$. +An order-$k$ transition changes a target with constant term one first +in degree $k$ of $G$, with contribution +$-((k-4)b+h')$. Once lower coefficients are compatible, a leading +degree-$k$ discrepancy is a cocycle in $L^k$. + +The factor $t^2$ in \eqref{jets:target-equation} and the equality +$B_0=0$ determine the order of the choices. For the field changes +at step $k$, the first determinant equations they can affect are +\[ +\begin{array}{c|c|c} + \text{changed field coefficient} + &\text{degree in }A\wedge B&\text{target coefficient}\\ \hline + \text{global addition to }A_{k-1}&k&G_{k-2}\\ + B_k&k&G_{k-2}\\ + A_k&k+1&G_{k-1}. +\end{array} +\] +The target coefficients in the last column remain fixed during these +changes. In particular, we can still adjust $A_{k-1}$ to solve the +degree-$k$ equation without changing any determinant equation imposed +at an earlier step. A new target coefficient $G_k$ will be chosen +only after the fields match through index $k$; its field equation +occurs in degree $k+2$. + +We will maintain the following invariant after step $n\geq2$: +the field rows through index $n$ match; the determinant equations +\eqref{jets:target-equation} are imposed through degree $n$ in +$A\wedge B$; and the coefficients $G_0=1,G_1,\ldots,G_n$ are chosen +and compatible through degree $n$ under the transition maps. +Thus target compatibility is known two determinant degrees beyond +the equations already imposed: the equations involving $G_{n-1}$ +and $G_n$ are still to be solved. +The preceding construction establishes this invariant for $n=2$, +with $G=1$, since both transitions preserve $\sigma$ exactly. + +Suppose now that the invariant holds through step $k-1$, where +$k\geq3$. The coefficient needed in the current determinant equation, +$G_{k-2}$, has already been chosen. On each chart the horizontal row +must solve +\begin{equation}\label{jets:horizontal-induction} + r\wedge B_k=G_{k-2} + -\sum_{a=1}^{k-1}A_a\wedge B_{k-a}. +\end{equation} +This is an equation in $L^{k-2}$. Indeed, under +\eqref{jets:split-coefficients}, the only additional terms in the +transformation of the sum on the right are multiples of +\[ + \sum_{a=1}^{k-1}B_a\wedge B_{k-a}=0; +\] +the terms cancel in pairs and any middle self-wedge vanishes. + +\paragraph{Making the right side divisible at $Z$.} +We may add any row $a\in H^0(C,L^k)^2$ to the previously chosen +vertical coefficient $A_{k-1}$. Differences of vertical lifts have +this tensor type by \eqref{jets:homogeneous-extension}. +Such a change preserves matching through index $k-1$: every +previous nonsplit transition has positive order, so its action on the +new term $t^kaE_t$ only changes indices at least $k$. +It also preserves all determinant equations already imposed, because +$B_0=0$; its first determinant contribution is $a\wedge B_1$ in +degree $k$. In particular, it need not preserve the as yet unimposed +equation whose target is $G_{k-2}$. + +At each $q_j$ the row $B_1(q_j)=R_j(0)$ is nonzero. +The interpolation in Proposition~\ref{prop:curve-data}(4) gives the +surjection +\[ + H^0(C,L^k)\longrightarrow H^0(Z,L^k|_Z)\qquad(k\geq3). +\] +It permits arbitrary values of both entries of $a$ at all six points. +The scalar maps +$a(q_j)\mapsto a(q_j)\wedge B_1(q_j)$ are surjective, so this +freedom makes the right side of \eqref{jets:horizontal-induction} +vanish at every $q_j$. +Since $r=z\rho$ has only a simple common zero, no derivative +condition is required. As at order two, division by $z$ on each +disk and Steinness on $U_0$ now give holomorphic solutions $B_k$. + +Although this global addition preserves all lower-order matching, +the previous transitions can act on it in the degree currently being solved. +The first transition contributes, at row index $k$, +\begin{equation}\label{jets:old-transition-effect} + \delta A_k=\frac{k-1}{3}m'a+ma',\qquad + \delta B_k=-ma. +\end{equation} +All other previous generators, and repeated brackets, affect later +indices. The displayed horizontal contribution satisfies +$r\wedge(-ma)=a\wedge(mr)$, exactly the corresponding determinant +change on the disk. We include these terms in the horizontal and +vertical comparisons below. + +\paragraph{Matching the horizontal rows.} +Choose any holomorphic vertical lifts of the newly selected $B_k$. +After transport by the previous transitions, the field rows already +agree at every lower index. The difference of their degree-$k$ +determinants is therefore $r\wedge\delta B_k$, since +$\delta A_k\wedge B_0=0$. Target compatibility through degree +$k-2$ in $G$ makes this difference zero. Thus the horizontal +discrepancy is a holomorphic multiple of $r$ on each overlap, and +\eqref{jets:leading-action} removes it by a choice of $h$ in an +order-$k$ transition. Use $b=h'$ when $k=3$, and $b=0$ when +$k\geq4$. +The new transition first affects $G$ in degree $k$, so cannot alter +the target comparison just used in degree $k-2$. + +\paragraph{Matching the vertical rows.} +After horizontal matching, the remaining class is in +$H^1(C,L^{k+1})^2$. +For $k\geq4$ it vanishes, since +\begin{equation}\label{jets:vertical-vanishing} + H^1(C,L^{k+1})\simeq H^0(C,L^{3-k})^*=0. +\end{equation} +Holomorphic vertical corrections on the charts therefore solve the +matching. They do not change the current determinant equation, +because $B_0=0$. + +At $k=3$ the obstruction is $H^1(C,K_C)^2$, tested by the constants +in the two entries. There is additional overlap freedom to kill it. +On the $j$th overlap replace $h$ by $h+\lambda_j/z$ and compensate +on its disk by +\begin{equation}\label{jets:third-compensation} + B_3\longmapsto B_3+\frac{3\lambda_jr}{z}. +\end{equation} +This is holomorphic and its wedge with $r$ is zero. It preserves +both the current determinant equation and horizontal matching: +the transition adds the opposite horizontal row $-3\lambda_jr/z$. +Keep the vertical lift $A_3$ unchanged in the chosen disk coordinate. +This prescribes a holomorphic section of $\mathcal E_3$ on that disk; +in another split coordinate its vertical component acquires the term +$a\,\delta B_3$ dictated by \eqref{jets:split-coefficients}. +Changing a degree-three input row affects its transport under old +positive-order transitions only at later indices. +The change in the transported vertical difference at index three is +therefore exactly +\begin{equation}\label{jets:third-residue} + \lambda_j\left(\frac{r'}z+\frac{2r}{z^2}\right),\qquad + \Res_0\left[\lambda_j\left(\frac{r'}z+\frac{2r}{z^2}\right)dz\right] + =3\lambda_jr'(0). +\end{equation} +The two entries here are coefficients of canonical differentials, +since $L^4=K_C$. The six rows $r'(q_j)$ span $\CC^2$: their +projective directions are those of the pencil values $u(q_j)$, which +are not all the same. Hence the map +$(\lambda_1,\ldots,\lambda_6)\mapsto\sum_j3\lambda_jr'(q_j)$ +has rank two and can cancel both obstruction entries. +The resulting vertical class splits, completing order three. + +The correction \eqref{jets:third-compensation} can change the +degree-four coefficient of $A\wedge B$. This causes no conflict: +$G_2=0$ has been selected, but its determinant equation is first +imposed at step four. The degree-three equation already imposed is +unchanged. Moreover the modified order-three transition still has +$b=h'$ and so preserves $\sigma$ exactly, leaving the transformation +law of the chosen targets intact. + +\paragraph{Extending the determinant target.} +It remains to choose $G_k$ compatibly after the order-$k$ transition. +For $k=3$ all transitions preserve $\sigma$, so we retain $G=1$. +At $k=4$ the earlier transitions still preserve it exactly; thus the +first target discrepancy is $-h'$ in degree four, by +\eqref{jets:target-transport} with $b=0$. +The horizontal coefficient of an order-four generator has type +$L^4\otimes\cT_C=\OO_C$. More explicitly, its split +transformation is $\widetilde h=g^{4-k}h$, which equals $h$ at +$k=4$. Consequently $h'dz=dh$ is an intrinsic exact differential +on the overlap. Each such differential has zero contour residue. +The sole obstruction to splitting this $L^4=K_C$ cocycle is its +pairing with the constant section of $\OO_C$, so it vanishes. +There are holomorphic chart coefficients $G_4$ accounting for the +discrepancy, with $G_1=G_2=G_3=0$. +Transport by a previous positive-order modification changes a +degree-four term only in degrees at least five. Equivalently, conjugating +the order-four generator by the earlier transitions leaves its degree-four +part unchanged. Thus the scalar transformation law of $h$, and the residue +calculation using $dh$, apply intrinsically at this order. + +For $k>4$, once the lower target coefficients are compatible, the +new leading discrepancy lies in $L^k$, and +\begin{equation}\label{jets:target-vanishing} + H^1(C,L^k)\simeq H^0(C,L^{4-k})^*=0. +\end{equation} +Thus it always splits and defines $G_k$. +In every case this target extension leaves all equations already +imposed unchanged, since $G_k$ is used only in degree $k+2$ of +$A\wedge B$. This completes the induction invariant. + +We now choose where to stop. A row of index $k$ contributes degree +$k+1$ in the logarithmic frame $(E_t,\partial_z)$, so matching through +index seven prescribes a section modulo $\cI^9\cF$. This is the +threshold needed in Section~\ref{analytic:section}: the successive +quotients of the unprescribed tail are extensions of $L^{n-4}$ by +$L^n$, and both line bundles have vanishing first cohomology for +$n\geq9$. Lemma~\ref{analytic:actual-lifting} turns these graded +vanishings into holomorphic lifting on a realized neighborhood. + +\begin{proposition}[Finite compatible jets]\label{prop:jets} +For the data of Propositions~\ref{prop:curve-data} and +\ref{prop:residue-data}, there are finitely many holomorphic +disk-to-outside transition germs, each a composition of flows +\eqref{jets:generator} of orders $1,\ldots,7$, together with +holomorphic local rows $A_k,B_k$ for $0\leq k\leq7$, such that: +the transitions fix $C$ and preserve its conormal bundle $L$; +the vector fields \eqref{jets:fields} match through row index seven; +and $A\wedge B=t^2(1+O(t))$ on every chart. + +On any surface neighborhood $Y$ realizing these transition germs, +write $\cI=\OO_Y(-C)$ and $\cF=\cT_Y(-\log C)$. +The rows give two sections of $\cF/\cI^9\cF$, lying in +$\cI\cF/\cI^9\cF$. Their determinant has leading term +$t^4E_t\wedge\partial_z$, equivalently +$t^5\partial_t\wedge\partial_z$, with coefficient one. +\end{proposition} + +\begin{proof} +Perform the first two steps and then the induction through $k=7$. +Only finitely many flows occur, and each is an actual holomorphic +germ on an annular neighborhood of the zero section, as explained +after Lemma~\ref{jets:bracket}. Terms with row indices $0,\ldots,7$ +have logarithmic-frame coefficients of degrees $1,\ldots,8$. +Matching those terms is precisely matching modulo $\cI^9\cF$. +Finally \eqref{jets:determinant} and the degree-two determinant +equation give +\[ + v_1\wedge v_2=t^4(1+O(t))E_t\wedge\partial_z + =t^5(1+O(t))\partial_t\wedge\partial_z. +\] +This statement uses only finite compatible jets; analytic realization +and the lifting of these quotient sections are addressed next. +\end{proof} diff --git a/preprints/A-singular-normal-affine-surface-with-free-tangent-sheaf-September-23-2026/build/05-analytic.tex b/preprints/A-singular-normal-affine-surface-with-free-tangent-sheaf-September-23-2026/build/05-analytic.tex new file mode 100644 index 0000000000000000000000000000000000000000..7a872221352a42d316474b6e7ae9128d040b7cb1 --- /dev/null +++ b/preprints/A-singular-normal-affine-surface-with-free-tangent-sheaf-September-23-2026/build/05-analytic.tex @@ -0,0 +1,425 @@ +\section{Analytic realization and contraction} +\label{analytic:section} + +We now realize the finite gluing data of Proposition~\ref{prop:jets} by +an actual complex surface. The two prescribed vector fields will then +be lifted from their finite jets by a cohomology vanishing argument. +The realization uses only the finite list of genuine holomorphic flow +maps already constructed. The formal-functions theorem used afterwards +computes a cohomology group on that realized neighborhood. +Throughout this section, powers of an ideal denote ordinary ideal +powers, and cohomology is analytic sheaf cohomology. We use coherence +of analytic structure and ideal sheaves as in +\cite[Chapter~II, Theorems~3.19 and~4.29]{Demailly2012}. + +\subsection{A Hausdorff neighborhood from the finite overlap maps} + +\begin{lemma}\label{analytic:realization} +The overlap maps of Proposition~\ref{prop:jets} define, after shrinking +their domains, a smooth Hausdorff complex surface neighborhood $Y$ of +$C$. Its conormal bundle is $L$, and hence +\[ + C^2=-10. +\] +Writing $\cI=\OO_Y(-C)$ and $\cF=\cT_Y(-\log C)$, the prescribed rows +define two global sections +\[ + \bar v_1,\bar v_2\in H^0(Y,\cF/\cI^9\cF). +\] +In divisor coordinates $(t,z)$, their local representatives have +logarithmic determinant $t^4(1+O(t))$ and ordinary determinant +$t^5(1+O(t))$. +\end{lemma} + +\begin{proof} +Let $N=L^{-1}$ be the split normal bundle. All changes to its +disk-to-outside identifications are finite compositions of flows of +holomorphic vector fields +\[ + D=t^k\bigl(b(z)t\partial_t+h(z)\partial_z\bigr),\qquad k\geq1, +\] +on punctured disk neighborhoods. On a compact annulus in any such +overlap, these vector fields vanish along the zero section. +The local existence theorem for holomorphic differential equations, +applied uniformly on that compact annulus, therefore gives the +required time-one maps and their inverses after shrinking the fiber +radius. There are only finitely many flows. Shrinking successively +gives their compositions on neighborhoods of the same compact +annular bands. Denote the resulting inside-to-outside maps by +$\Phi_j$, $1\leq j\leq6$. + +Here is a quotient construction that also verifies the separation +property. Choose mutually disjoint coordinate disks $D_j$ about the +six points, and in each choose radii +\[ + 00$. +The first assertion follows from the cohomology sequence. + +Because $C$ is a Cartier divisor and $\cF$ is locally free, +\[ + \cI^n\cF/\cI^{n+1}\cF\simeq\cF|_C\otimes L^n. +\] +The quotient +$\mathcal E/\cI^r\mathcal E=\cI^9\cF/\cI^{9+r}\cF$ +has a finite filtration with precisely these graded terms for +$9\leq n<9+r$. Their first cohomology vanishes by the first +assertion. Successive cohomology sequences give the second +assertion; the quotient is supported on the compact curve $C$. +\end{proof} + +\begin{lemma}\label{analytic:actual-lifting} +After shrinking $Y$ about $C$, the sections $\bar v_1,\bar v_2$ lift +to holomorphic sections $v_1,v_2$ of $\cF$. They form a frame for +$\cT_{Y\setminus C}$. +\end{lemma} + +\begin{proof} +The intersection matrix of $C$ is the negative definite matrix +$(-10)$. Grauert's contraction criterion therefore contracts $C$ +to a point $x$ of a normal complex surface: +\[ + \pi:Y\longrightarrow X,\qquad \pi^{-1}(x)=C, + \qquad Y\setminus C\simeq X\setminus\{x\}, +\] +where the fiber equality is set-theoretic, and +$\pi_*\OO_Y=\OO_X$. +We use the criterion in \cite[\S4.8(e), pp.~366--367]{Grauert1962}. +The contraction is proper and is a biholomorphism off the +exceptional curve. It is the holomorphic reduction in that theorem, +whose target is normal when the source is normal; these properties +are part of the reduction and contraction +statements in \cite[\S2, p.~337 and Definition~3, +pp.~339--340]{Grauert1962}. We choose a sufficiently small Stein +representative $X$ about $x$ and replace $Y$ by its inverse image. +In particular $X\setminus\{x\}$ is smooth. + +We prove $R^1\pi_*\mathcal E=0$ for +$\mathcal E=\cI^9\cF$, keeping track of the two ideal filtrations. +Let $\mathfrak m_x$ be the ideal sheaf of $x$ in $X$ and put +$\mathcal J=\mathfrak m_x\OO_Y$. Both $\mathcal J$ and $\cI$ +vanish exactly on $C$, and $\cI$ is radical. The analytic +Nullstellensatz~\cite[Chapter~II, Theorem~4.22]{Demailly2012} +and Noetherianity give an inclusion +$\cI^a\subseteq\mathcal J$ locally at each point of $C$ for some +positive integer $a$. A finite cover of the compact curve gives a +single exponent. After shrinking the representative, +\begin{equation}\label{analytic:cofinal} + \cI^a\subseteq\mathcal J\subseteq\cI, + \qquad + \cI^{ar}\subseteq\mathcal J^r\subseteq\cI^r + \quad(r\geq1). +\end{equation} +Such a shrinking may be made on $X$: for a proper map, every open +neighborhood of its compact fiber contains the inverse image of a +sufficiently small neighborhood of the image point. + +The proper analytic direct-image theorem makes +$R^1\pi_*\mathcal E$ coherent. The analytic theorem on formal +functions identifies its completed stalk with +\begin{equation}\label{analytic:formal-functions} + \widehat{(R^1\pi_*\mathcal E)_x} + \simeq + \varprojlim_{r\geq1}H^1(Y,\mathcal E/\mathcal J^r\mathcal E). +\end{equation} +We use \cite[\S6, Haupts\"atze~I and~IIa, +p.~59]{Grauert1960}: the hypotheses are a proper holomorphic map +and a coherent analytic sheaf. In particular no flatness of +$\pi$ is required. The quotients in +\eqref{analytic:formal-functions} are supported on the infinitesimal +fibers over $x$, so their cohomology is independent of a further +shrinking of the representative. + +For clarity, the right side vanishes by an explicit factorization, +without identifying the individual cohomology groups for the two +filtrations. The inclusions in \eqref{analytic:cofinal} give +\[ + \mathcal J^{ar}\mathcal E + \subseteq\cI^{ar}\mathcal E + \subseteq\mathcal J^r\mathcal E. +\] +Hence the transition map from index $ar$ to index $r$ in +\eqref{analytic:formal-functions} factors through +\[ + H^1(Y,\mathcal E/\cI^{ar}\mathcal E)=0 +\] +by Lemma~\ref{analytic:graded-vanishing}. The inverse system is +therefore pro-zero, and its inverse limit is zero. Since the stalk +$(R^1\pi_*\mathcal E)_x$ is a finite module over the Noetherian +local ring $\OO_{X,x}$, its zero completion implies that the stalk +itself is zero, by Nakayama's lemma. Away from $x$ the map $\pi$ +is an isomorphism, so the same higher direct image vanishes there. +We obtain $R^1\pi_*\mathcal E=0$. + +The sheaf $\pi_*\mathcal E$ is also coherent. Cartan's +Theorem~B on the Stein space $X$, together with the Leray spectral +sequence for $\pi$, now gives +\[ + H^1(Y,\mathcal E)=0. +\] +The cohomology sequence of +\[ + 0\longrightarrow\mathcal E\longrightarrow\cF + \longrightarrow\cF/\cI^9\cF\longrightarrow0 +\] +therefore lifts both $\bar v_i$ to actual holomorphic sections $v_i$ +of $\cF$. + +It remains to check that arbitrary such lifts retain the required +frame. In each divisor chart, let $v_i^{(8)}$ be the polynomial +representative of the prescribed jet. It belongs to $\cI\cF$, and +\[ + v_i-v_i^{(8)}\in\cI^9\cF. +\] +Thus the change in their wedge belongs to +$\cI^{10}\det\cF$. By Lemma~\ref{analytic:realization}, the +logarithmic determinant of the lifted pair is still +$t^4(1+O(t))$; its leading coefficient is $1$ even at the common +zeros of the leading row $r$. Its ordinary determinant is +\begin{equation}\label{analytic:ordinary-determinant} + v_1\wedge v_2=t^5 u(t,z)\,\partial_t\wedge\partial_z, + \qquad u(0,z)=1. +\end{equation} +The unit condition holds on a neighborhood of each point of $C$. +Compactness gives one neighborhood on which all these local factors +are nonvanishing. Properness allows a further shrinking of $X$ +whose inverse image lies in this neighborhood. Equation +\eqref{analytic:ordinary-determinant} then proves that $v_1,v_2$ +frame $\cT_{Y\setminus C}$. +\end{proof} + +\subsection{The singular point and its derivations} + +\begin{proposition}\label{prop:analytic-counterexample} +There is a normal complex surface germ $(X,x)$ with an isolated +singular point and free analytic derivation sheaf of rank two: +\[ + \cT_X=\mathcal{H}om_{\OO_X}(\Omega_X^1,\OO_X) + \simeq\OO_X^{\oplus2}. +\] +It has a resolution $\pi:Y\to X$ whose exceptional set is the +smooth curve $C$ of genus $21$ and self-intersection $-10$. +\end{proposition} + +\begin{proof} +Take the contraction constructed in +Lemma~\ref{analytic:actual-lifting}. Suppose first that $x$ were a +smooth point. A sufficiently small neighborhood of $x$ would admit +a nowhere-vanishing holomorphic two-form $\omega$. Its pullback +would be holomorphic on $Y$ and nonvanishing off $C$. Thus +\[ + \operatorname{div}(\pi^*\omega)=aC\qquad(a\geq0), + \quad\hbox{and hence}\quad K_Y\cdot C=aC^2\leq0. +\] +Adjunction, however, gives +\[ + K_Y\cdot C=2g(C)-2-C^2=40+10=50, +\] +a contradiction. The point $x$ is singular, and it is the only +singular point after the chosen shrinking. + +Write $X^\circ=X\setminus\{x\}$ and +$j:X^\circ\hookrightarrow X$. The two fields of +Lemma~\ref{analytic:actual-lifting} descend to a holomorphic frame +$w_1,w_2$ on $X^\circ$. Normality in dimension two gives the +Hartogs extension property~\cite[Chapter~II, Corollary~7.8]{Demailly2012} +\begin{equation}\label{analytic:hartogs} + \OO_X\simeq j_*\OO_{X^\circ}. +\end{equation} +This property extends the fields as derivations, as follows. Embed +a representative of $(X,x)$ in a smooth analytic germ with +coordinate functions $x_1,\ldots,x_e$. For each $i$ and $\alpha$, +the holomorphic function $w_i(x_\alpha)$ on $X^\circ$ extends +uniquely to a function $a_{i\alpha}$ on $X$. If $f$ is any local +defining equation of the embedding, then +\[ + \sum_{\alpha=1}^e + a_{i\alpha}\frac{\partial f}{\partial x_\alpha}=0 +\] +on $X^\circ$ and hence on $X$, by density and reducedness. The +tuple $(a_{i\alpha})_\alpha$ consequently defines a section of +$\mathcal{H}om_{\OO_X}(\Omega_X^1,\OO_X)$ extending $w_i$. We +continue to denote that extension by $w_i$. + +For any local analytic derivation $w$ on $X$, the frame on +$X^\circ$ gives unique holomorphic coefficients $h_1,h_2$ there +such that $w=h_1w_1+h_2w_2$. Both coefficients extend by +\eqref{analytic:hartogs}. The extended equality holds on $X$, +since applying its difference to the embedding coordinates gives +holomorphic functions vanishing on the dense open set $X^\circ$. +This proves surjectivity of the map +\[ + \OO_X^{\oplus2}\longrightarrow\cT_X, + \qquad(h_1,h_2)\longmapsto h_1w_1+h_2w_2. +\] +If a pair lies in its kernel, both coefficients vanish on +$X^\circ$ because $w_1,w_2$ are a frame there, and therefore vanish +on $X$. The map is also injective, proving the assertion. +\end{proof} + +\begin{remark}\label{analytic:invariants} +The construction also determines its canonical discrepancy and a +lower bound for its geometric genus. The two-form $\omega^\circ$ +dual to the frame $w_1,w_2$ generates the canonical sheaf on +$X^\circ$. Extending reflexively across the normal surface point +gives $\omega_X\simeq\OO_X$; equivalently, we may choose the +canonical divisor $K_X=0$. Since a normal surface is Cohen--Macaulay, +this reflexive canonical sheaf is its dualizing sheaf, so $X$ is +also Gorenstein; this surface criterion is recalled in +\cite[\S2.4, Definition~2.18 and the following paragraph]{FlennerZaidenberg2001}. +By +\eqref{analytic:ordinary-determinant}, its pullback is +\[ + \pi^*\omega^\circ=t^{-5}u(t,z)^{-1}\,dt\wedge dz. +\] +It is meromorphic on $Y$ with divisor exactly $-5C$. Thus +\[ + K_Y=\pi^*K_X-5C, +\] +so the discrepancy of $C$ is $-5$. + +For the geometric genus, put +$\OO_{nC}=\OO_Y/\cI^n$. Riemann--Roch and the established +equality $h^0(C,L)=2$ give +\[ + h^1(C,L)=h^0(C,L^3)=12. +\] +In the sequence +\[ + 0\longrightarrow L\longrightarrow\OO_{2C} + \longrightarrow\OO_C\longrightarrow0, +\] +the constant section of $\OO_C$ lifts to $\OO_{2C}$. Since +$H^2(C,L)=0$, its cohomology sequence gives +\[ + h^1(\OO_{2C})=h^1(C,L)+h^1(C,\OO_C)=12+21=33. +\] +For every $n\geq1$, the sequence with kernel $L^n$ similarly shows +that $H^1(\OO_{(n+1)C})\to H^1(\OO_{nC})$ is surjective. +The theorem on formal functions, with the cofinal filtrations in +\eqref{analytic:cofinal}, identifies their inverse limit with the +completion of $(R^1\pi_*\OO_Y)_x$. This stalk has finite length, +since the coherent higher direct image is supported at $x$, and +thus equals its completion. The inverse limit maps onto +$H^1(\OO_{2C})$, proving +\[ + p_g(X,x)=\dim_\CC (R^1\pi_*\OO_Y)_x\geq33. +\] +The exceptional curve has genus $g=21$ and its dual graph has first +Betti number $b=0$. Consequently $p_g-g-b\geq12$. +\end{remark} diff --git a/preprints/A-singular-normal-affine-surface-with-free-tangent-sheaf-September-23-2026/build/06-algebraization.tex b/preprints/A-singular-normal-affine-surface-with-free-tangent-sheaf-September-23-2026/build/06-algebraization.tex new file mode 100644 index 0000000000000000000000000000000000000000..0bfcb72392f1bc6747816bd5fb8e0263d05d4527 --- /dev/null +++ b/preprints/A-singular-normal-affine-surface-with-free-tangent-sheaf-September-23-2026/build/06-algebraization.tex @@ -0,0 +1,508 @@ +\section{Algebraization and the affine counterexample} +\label{algebra:section} + +We now pass from the analytic germ to an affine variety. Approximating +the defining equations does not by itself retain their relations. +We approximate the equations and a finite relation matrix together; +their exact zero product, together with sufficiently close approximation, +will recover the same complete local ring. Tangent freeness then descends +from the continuous derivations of that ring. This follows the +equation-and-relation determinacy method of Hironaka +\cite[\S3, Theorem~3.3]{Hironaka1969}, used in Artin's +isolated-singularity algebraization +\cite[Theorems~3.8--3.9]{Artin1969}. We prove the finite-determinacy +statement needed here, including fixed estimates that avoid a +complete-intersection hypothesis. + +\subsection{Finite determinacy of equations with relations} + +Put +\[ + P=\CC[[x_1,\ldots,x_e]],\qquad + \mathfrak n=(x_1,\ldots,x_e),\qquad + I=(f_1,\ldots,f_s),\qquad R=P/I, +\] +where $f=(f_1,\ldots,f_s)\in\mathfrak nP^s$ is a row. Write +$\mathfrak m_R=\mathfrak nR$, and let an overbar denote reduction modulo $I$. +Choose an $s\times t$ matrix $D$ whose columns generate all the relations +on $f$. Thus +\begin{equation} +\label{algebra:ideal-presentation} + P^t\xrightarrow{D}P^s\xrightarrow{f}I\longrightarrow0 +\end{equation} +is exact. Define +\[ + \psi:R^s\longrightarrow R^t,\qquad v\longmapsto v\bar D, + \qquad K=\ker\psi, + \qquad + J=\sum_{j=1}^e R\,\overline{\partial_j f}. +\] +Differentiating $fD=0$ and reducing modulo $I$ shows that $J\subset K$. +Right-exact reduction of \eqref{algebra:ideal-presentation} gives a +presentation of $I/I^2$; its dual therefore identifies +\begin{equation} +\label{algebra:conormal-dual} + K=\Hom_R(I/I^2,R),\qquad T^1=K/J. +\end{equation} +In particular, no exactness on the left after reduction has been assumed. + +\begin{lemma}[Finite determinacy with a relation matrix] +\label{algebra:finite-determinacy} +Suppose that $T^1$ in \eqref{algebra:conormal-dual} has finite length. +There exist integers $N,n_0$ depending only on $f,D$ such that the following +holds. If a row $g\in P^s$ and an $s\times t$ matrix $B$ satisfy +\[ + gB=0,\qquad + g-f\in\mathfrak n^{n_0}P^s,\qquad + B-D\in\mathfrak n^NP^{s\times t}, +\] +then there are a continuous $\CC$-algebra automorphism $\Phi$ of $P$ and +an invertible matrix $U\in\operatorname{GL}_s(P)$ such that +\[ + f=\Phi(g)U. +\] +One can take $\Phi(x_j)\equiv x_j\pmod{\mathfrak n^2}$ and +$U\equiv1\pmod{\mathfrak n}$. Consequently $P/(g)\simeq P/(f)$ as +complete local $\CC$-algebras. +\end{lemma} + +\begin{proof} +We improve the equations one order at a time. The exact relation +$gB=0$ makes the equation error, reduced modulo $I$, nearly belong to +$K$. Artin--Rees lets us correct it into $K$; the finite length of +$K/J$ then expresses the sufficiently high-order corrected class by +coordinate derivatives. A change of generators removes the remaining +$I$-valued error. We arrange these corrections so that the equation +precision increases at every step while the relation precision stays +fixed. The resulting coordinate and generator changes converge in $P$. + +We first choose the fixed bounds. For a map +$u:M\to M'$ of finite modules over a Noetherian local ring with maximal +ideal $\mathfrak a$, the Artin--Rees lemma~\cite[Tag~00IN]{StacksCompletion}, applied to +$\operatorname{im}u\subset M'$ gives a constant $a\ge0$ such that +\[ + \operatorname{im}u\cap\mathfrak a^rM' + \subset u(\mathfrak a^{r-a}M)\qquad(r\ge a). +\] +Choose nonnegative constants $\alpha,\beta,d$ so that +\begin{align} + \operatorname{im}\psi\cap\mathfrak m_R^rR^t + &\subset\psi(\mathfrak m_R^{r-\alpha}R^s), + \label{algebra:ar-image}\\ + K\cap\mathfrak m_R^rR^s + &\subset\mathfrak m_R^{r-\beta}K, + \label{algebra:ar-kernel}\\ + I^{\oplus s}\cap\mathfrak n^rP^s + &\subset\{fC:C\in\mathfrak n^{r-d}P^{s\times s}\} + \label{algebra:ar-ideal} +\end{align} +whenever the corresponding exponents are nonnegative. In +\eqref{algebra:ar-ideal}, Artin--Rees is applied to the map +$P^{s\times s}\to P^s$, $C\mapsto fC$, with image $I^{\oplus s}$; it is the +filtration induced from $P^s$ that is relevant. + +Choose $\ell\ge0$ with $\mathfrak m_R^\ell K\subset J$ and set +\begin{equation} +\label{algebra:constants} + c_0=\beta+\ell,\qquad c=c_0+d,\qquad + N\ge\alpha+1,\qquad + n_0\ge\max\{c+N,c+2,2c+1\}. +\end{equation} +We first describe one improvement step. Suppose $n\ge n_0$ and +\[ + h=g-f\in\mathfrak n^nP^s, + \qquad B-D\in\mathfrak n^NP^{s\times t}, + \qquad gB=0. +\] +The exact relation, reduced modulo the fixed ideal $I$, gives +\begin{equation} +\label{algebra:high-order-relation} + \bar h\bar D + =\bar g\bar D + =-\bar g(\bar B-\bar D) + =-\bar h(\bar B-\bar D) + \in\mathfrak m_R^{n+N}R^t. +\end{equation} +The factor $\bar h$ on the right is what provides order $n+N$. +By \eqref{algebra:ar-image}, there is +\[ + \bar\epsilon\in\mathfrak m_R^{n+N-\alpha}R^s + \subset\mathfrak m_R^{n+1}R^s, + \qquad \psi(\bar\epsilon)=\psi(\bar h). +\] +Hence +\[ + \kappa=\bar h-\bar\epsilon + \in K\cap\mathfrak m_R^nR^s + \subset\mathfrak m_R^{n-\beta}K + \subset\mathfrak m_R^{n-c_0}J. +\] +Here the last inclusion uses $\mathfrak m_R^\ell K\subset J$, +with the remaining factor of order $n-\beta-\ell=n-c_0$. +Lift coefficients expressing $\kappa$ in the displayed generators of +$J$ to elements $a_j\in\mathfrak n^{n-c_0}$, and lift +$\bar\epsilon$ to $\epsilon\in\mathfrak n^{n+1}P^s$. Then +\[ + w=h-\sum_{j=1}^e a_j\partial_jf-\epsilon + \in I^{\oplus s}\cap\mathfrak n^{n-c_0}P^s. +\] +The order of this ideal-valued remainder need only be $n-c_0$, since +the lifted derivative combination may have lower order than $h$: +a derivative of a defining equation can be a unit. +Using \eqref{algebra:ar-ideal}, write $w=fC$ with +$C\in\mathfrak n^{n-c}P^{s\times s}$. We have proved +\begin{equation} +\label{algebra:linear-correction} + h=\sum_{j=1}^e a_j\partial_jf+fC+\epsilon, + \qquad + a_j,C\in\mathfrak n^{n-c}, + \qquad \epsilon\in\mathfrak n^{n+1}P^s. +\end{equation} +Thus the loss $d$ for lifting the ideal-valued remainder has been included +in the fixed constant $c$. + +Put $q=n-c$, and define +\begin{equation} +\label{algebra:one-step} + \phi(x_j)=x_j-a_j,\qquad M=(1+C)^{-1},\qquad + g_{\mathrm{new}}=\phi(g)M,\qquad + B_{\mathrm{new}}=(1+C)\phi(B). +\end{equation} +Our bounds ensure +\[ + q\ge\max\{N,2\},\qquad 2q\ge n+1, + \qquad n+q-1\ge n+1. +\] +In particular, $\phi$ is a formal automorphism with identity linear part, +and $M$ is invertible. Formal Taylor expansion gives +\[ + \phi(f)-f+\sum_j a_j\partial_jf\in\mathfrak n^{2q}P^s, + \qquad + \phi(h)-h\in\mathfrak n^{n+q-1}P^s. +\] +Combining these estimates with \eqref{algebra:linear-correction} gives +\[ + \phi(g)\equiv f(1+C)\pmod{\mathfrak n^{n+1}P^s}, + \qquad + g_{\mathrm{new}}-f\in\mathfrak n^{n+1}P^s. +\] +The relations in \eqref{algebra:one-step} transform exactly: +\[ + g_{\mathrm{new}}B_{\mathrm{new}} + =\phi(g)(1+C)^{-1}(1+C)\phi(B)=0. +\] +Moreover, substitution by $\phi$ changes any power series by an element +of $\mathfrak n^q$, and $C$ has order at least $q$. Therefore +\[ + B_{\mathrm{new}}-D + =C\phi(B)+(\phi(B)-B)+(B-D) + \in\mathfrak n^NP^{s\times t}. +\] +The fixed relation precision $N$ is preserved, while the equation +precision increases from $n$ to $n+1$. + +Starting with $(g_{n_0},B_{n_0})=(g,B)$, repeat this step for +$n=n_0,n_0+1,\ldots$. Write $\phi_n,M_n$ for its corrections. +For completeness, the cumulative coordinate and generator changes obey +\[ + \Phi_{n_0}=\id,\quad U_{n_0}=1,\qquad + \Phi_{n+1}=\phi_n\circ\Phi_n,\qquad + U_{n+1}=\phi_n(U_n)M_n, +\] +so that $g_n=\Phi_n(g)U_n$ at every stage. Since +\[ + (\phi_n-\id)(P)\subset\mathfrak n^{n-c}, + \qquad M_n-1\in\mathfrak n^{n-c}P^{s\times s}, +\] +the consecutive images $\Phi_n(x_j)$ and the consecutive matrices +$U_n$ differ in orders tending to infinity. They converge in the +complete ring $P$ to $\Phi(x_j)$ and $U$. The resulting substitution +$\Phi$ has identity linear part and hence is an automorphism, while +$U\equiv1\pmod{\mathfrak n}$ is invertible. Taking limits in +$g_n=\Phi_n(g)U_n$ yields $f=\Phi(g)U$, as required. +\end{proof} + +The original matrix $D$ must generate all the relations, in order that +\eqref{algebra:conormal-dual} hold. No such assertion is needed for the +approximating matrix $B$: its exact zero product and its closeness to +$D$ are sufficient. Thus Lemma~\ref{algebra:finite-determinacy} concerns +solutions of the finite system $gB=0$, not arbitrary truncations of a +list of defining equations. + +\subsection{The algebraic neighborhood} + +\begin{proposition}[Algebraization of an isolated analytic germ] +\label{prop:algebraization} +Let $(X,x)$ be a normal complex analytic germ of dimension $d\ge1$, +smooth away from $x$, and suppose that its holomorphic tangent module +is free of rank $d$. There exist a finitely generated normal integral +$\CC$-algebra $A$ and a maximal ideal $\mathfrak p\subset A$ such that +\[ + \dim A=d,\qquad A/\mathfrak p=\CC,\qquad + \widehat{A_{\mathfrak p}}\simeq\widehat{\OO_{X,x}},\qquad + \Der_{\CC}(A)\simeq A^d. +\] +The affine variety $\Spec A$ can be chosen smooth away from +$\mathfrak p$. If $x$ is singular, $A_{\mathfrak p}$ is nonregular. +\end{proposition} + +\begin{proof} +\emph{The completed analytic presentation.} +Embed the germ in a smooth analytic germ and write +\[ + S=\OO_{X,x}=\CC\{x_1,\ldots,x_e\}/I_{\mathrm{an}}. +\] +Choose a finite row $f$ generating $I_{\mathrm{an}}$ and a finite matrix +$D$ generating all its relations over the convergent power-series ring. +Completion is exact on finite modules, so they give the presentation +\eqref{algebra:ideal-presentation} over $P=\CC[[x_1,\ldots,x_e]]$, +with $R=\widehat S=P/(f)$. Complex analytic local rings are excellent +\cite[Proposition~2.1(1)]{Cutkosky2017}; +thus the completion of the normal local ring $S$ is a normal local +domain. Also +\begin{equation} +\label{algebra:completion-invariants} + \dim R=\dim S=d, + \qquad R\text{ is regular if and only if }S\text{ is regular}. +\end{equation} +We use here the usual exactness and permanence properties of completion +\cite{StacksCompletion}. + +The convergent version of \eqref{algebra:conormal-dual} is the conormal +dual of the analytic embedding. On the smooth locus the tangent-to-normal +map is surjective: a smooth analytic subspace of a complex manifold is +locally a regular embedding. Its cokernel is consequently a coherent +module supported at the isolated point $x$, and has finite length. +Flat completion identifies this cokernel with the module $T^1=K/J$ +above. Lemma~\ref{algebra:finite-determinacy} therefore applies to $f,D$. + +We also record precisely which differentials are being completed. The +module of holomorphic differentials of $S$ has the finite Jacobian +presentation with generators $dx_j$ and relations $df_i$. Its dual is +the holomorphic tangent module. Flat completion of this kernel yields +\begin{equation} +\label{algebra:continuous-tangent} + \Der^{\mathrm{cont}}_{\CC}(R,R) + =\ker\left(R^e\longrightarrow R^s, + (v_j)\longmapsto\sum_{j=1}^e v_j\overline{\partial_jf}\right) + \simeq R^d. +\end{equation} +Here ``continuous'' refers to the maximal-ideal topology. A continuous +derivation of a power-series quotient is determined by its values on +the $x_j$, and the displayed relations are precisely the conditions for +it to descend from $P$. All completed differential modules used below +are these finite continuous differential modules. + +\emph{Approximation of equations and relations.} +Let +\[ + H=\bigl(\CC[x_1,\ldots,x_e]_{(x_1,\ldots,x_e)}\bigr)^h +\] +be the henselization, whose completion is $P$. This is precisely a +henselization of a finite-type algebra over the field $\CC$ at a prime, +as covered by Artin's Theorem~1.10. Apply that approximation theorem +over $H$ to the polynomial system in the $s+st$ entries of a row $G$ +and a matrix $B$, +\begin{equation} +\label{algebra:artin-system} + \sum_{i=1}^s G_i B_{i\nu}=0\qquad(1\le\nu\le t), +\end{equation} +with formal solution $(G,B)=(f,D)$. It gives a row $g$ and a matrix $B$ +with entries in $H$, satisfying $gB=0$ exactly and agreeing with $f,D$ +to any prescribed finite order +\cite[Theorem~1.10]{Artin1969}; see also~\cite{StacksApproximation}. Choose that +order at least $\max\{n_0,N\}$, with $n_0,N$ from +Lemma~\ref{algebra:finite-determinacy}. The lemma gives +\begin{equation} +\label{algebra:formal-equivalence} + P/(g)\simeq R. +\end{equation} + +The henselization is the filtered colimit of pointed \'etale +neighborhoods of the origin, with the same residue field +\cite{StacksHenselization}. Thus the finitely many entries of $g,B$ +are represented on a single such neighborhood. After passing to a +later neighborhood, the finitely many identities +\eqref{algebra:artin-system} hold there as well. If this stage is +expressed over the local polynomial ring, finite presentation and +clearing finitely many denominators spread it to a neighborhood of +the origin in affine space; shrinking preserves the marked point +and makes the morphism \'etale. After an affine shrinking the +identities hold in its coordinate ring. We obtain an affine +\'etale neighborhood $\Spec E\to\mathbb A^e_{\CC}$ with a marked point +$\mathfrak q$ over the origin, residue field $\CC$, and a row $g$ of +elements of $E$ vanishing at $\mathfrak q$. Its completed ambient +local ring is $P$. Put +\[ + A'=E/(g),\qquad \mathfrak p'=\mathfrak q/(g),\qquad + T=A'_{\mathfrak p'}. +\] +Then $A'$ is of finite type over $\CC$, and exactness of completion +together with \eqref{algebra:formal-equivalence} gives +\begin{equation} +\label{algebra:completed-local-ring} + \widehat T=P/(g)\simeq R. +\end{equation} +In particular, this construction preserves the complete local ring, +including its dimension and its normality. + +\emph{Descent of tangent freeness and local ring properties.} +Because the ambient neighborhood is \'etale over $\mathbb A^e_{\CC}$, +its differentials have basis $dx_1,\ldots,dx_e$ near the marked point. +Completing the finite presentation of $\Omega^1_{T/\CC}$ therefore +gives the continuous Jacobian presentation of $P/(g)$. For a finitely +presented module, flat base change commutes with its dual: this follows +by expressing the dual as the kernel of a map between finite free +modules. The continuous chain rule under +\eqref{algebra:completed-local-ring} and +\eqref{algebra:continuous-tangent} consequently give +\begin{equation} +\label{algebra:tangent-descent} + \Der_{\CC}(T)\otimes_T\widehat T + \simeq\Der^{\mathrm{cont}}_{\CC}(\widehat T,\widehat T) + \simeq\widehat T^{d}. +\end{equation} +This is an equality obtained from finite presentations; it does not +identify ordinary algebraic differentials of a power-series ring with +its continuous differentials. + +Let $M=\Der_{\CC}(T)$. By \eqref{algebra:tangent-descent}, its residue +vector space has dimension $d$. Lift a basis to $d$ elements of $M$. +Nakayama's lemma gives a surjection $T^d\to M$, which after completion +is a surjection between free modules of rank $d$ over the local ring +$\widehat T$, and hence an isomorphism. Faithful flatness of completion +annihilates its kernel, proving $M\simeq T^d$. + +The ring $T$ injects into its domain completion, so it is a domain. +Normality also descends. Indeed, if $a/b\in\operatorname{Frac}(T)$ +is integral over $T$, it is integral over the normal domain +$\widehat T$ and hence belongs to $\widehat T$. Thus +$a\in b\widehat T\cap T=bT$, where the last equality follows from +faithful flatness, and $a/b\in T$. Finally, completion preserves both +Krull dimension and the dimension of the maximal ideal modulo its +square. Equations \eqref{algebra:completion-invariants} and +\eqref{algebra:completed-local-ring} show that $T$ has dimension $d$ +and is nonregular whenever $S$ is nonregular. + +The isolated-singularity property can also be retained. Let +$\mathfrak j_{\mathrm{an}}$ be the $d$th Fitting ideal of the +holomorphic differential module of $S$. The analytic Jacobian +criterion identifies its zero set with the nonsmooth locus, so some +power of the maximal ideal of $S$ is contained in +$\mathfrak j_{\mathrm{an}}$; in the smooth case this ideal is the unit +ideal. Fitting ideals commute with base change. The completed +Jacobian presentations and \eqref{algebra:completed-local-ring} thus +imply, by contraction from the faithfully flat completion, that +\begin{equation} +\label{algebra:isolated-descent} + \mathfrak m_T^b\subset + \operatorname{Fitt}_d(\Omega^1_{T/\CC}) + \quad\text{for some }b\ge0. +\end{equation} +On the reduced pure-$d$-dimensional neighborhood constructed below, +the algebraic Jacobian criterion interprets this containment as +nonsmoothness supported at the marked point. + +\emph{An integral normal affine neighborhood with a free tangent module.} +The finite module $\Der_{\CC}(A')$ localizes at $\mathfrak p'$ to +$\Der_{\CC}(T)$, because $\Omega^1_{A'/\CC}$ is finitely presented. +Represent the basis just obtained after inverting an element outside +$\mathfrak p'$. It defines a map from a free module of rank $d$ to +the derivation module whose kernel and cokernel vanish at +$\mathfrak p'$. Their supports are closed, so a further principal +shrinking makes this map an isomorphism. + +Since $T$ is a domain, exactly one irreducible component of $\Spec A'$ +passes through $\mathfrak p'$, and the support of the nilradical avoids +$\mathfrak p'$. Remove the other components and that support. The +remaining neighborhood is reduced and irreducible. Its normal locus +is open, since schemes of finite type over $\CC$ are excellent, and +contains $\mathfrak p'$ by the normality of $T$; restrict to this +normal locus as well. This irreducible neighborhood has dimension +$d$, since its marked point is closed and its local ring has dimension +$d$. Its nonsmooth locus is closed and, by +\eqref{algebra:isolated-descent}, has no positive-dimensional component +through $\mathfrak p'$. Remove any components of that locus that do +not contain $\mathfrak p'$. + +All the restrictions just made are open neighborhoods of +$\mathfrak p'$. Choose a principal affine neighborhood contained in +their intersection, and write it as $\Spec A$, with marked maximal +ideal $\mathfrak p$. The algebra $A$ is finitely generated, normal +and integral, its dimension is $d$, and +\[ + A_{\mathfrak p}=T,\qquad + \Der_{\CC}(A)\simeq A^d. +\] +The marked residue field and completed local ring are unchanged, and +the variety is smooth away from that point. This proves all the +assertions. +\end{proof} + +\begin{proof}[Proof of Theorem~\ref{thm:main}] +Proposition~\ref{prop:analytic-counterexample} supplies a singular +normal complex analytic surface germ, smooth away from its marked +point, whose holomorphic tangent module is free of rank two. +Apply Proposition~\ref{prop:algebraization} and fix any of the +approximations and affine neighborhoods furnished by its proof. Its +coordinate ring $A$ is a finitely generated normal integral +$\CC$-algebra of dimension two, with +\[ + \Der_{\CC}(A)\simeq A^2, + \qquad A_{\mathfrak p}\text{ nonregular}. +\] +Thus the derivation module is free, and in particular projective, +whereas $A$ is not regular. This is the required counterexample over +$\CC$. +\end{proof} + +\subsection{The smooth locus and derivation blowups} + +Let $X=\Spec A$ be the affine surface just constructed, and let +$U=X_{\mathrm{reg}}$ be its smooth locus. Recall that the logarithmic +Kodaira dimension of a smooth quasiprojective surface is +\[ + \overline\kappa(U)=\kappa(K_{\overline U}+D), +\] +where $\overline U$ is a smooth projective compactification and its +reduced boundary $D=\overline U\setminus U$ is a simple normal-crossing +divisor. This value is independent of the compactification. + +\begin{corollary}\label{algebra:log-kodaira} +The smooth locus $U$ of the affine surface $X=\Spec A$ constructed in +Theorem~\ref{thm:main} has $\overline\kappa(U)=2$. +\end{corollary} + +\begin{proof} +The free tangent sheaf on $X$ restricts to a trivial tangent bundle on +$U$. If $\overline\kappa(U)\le1$, the theorem of Biswas, Gurjar and +Kolte~\cite[Theorem~0.1(1)]{BiswasGurjarKolte2014} would make the normal +affine surface $X$ smooth, contrary to Theorem~\ref{thm:main}. +The only remaining value of logarithmic Kodaira dimension for a surface +is two. +\end{proof} + +For a finite $A$-module $M$ of generic rank $r$, viewed as a coherent +sheaf on $X$, its blowup is universal among proper birational morphisms +from integral varieties to $X$ for which the pullback of $M$ modulo +torsion is locally free of rank $r$. Every such morphism factors uniquely +through the blowup. We use Villamayor's description, as recalled in +\cite[\S1.6, Theorem~1.17]{BarajasChavezRomano2026}. + +\begin{corollary}\label{algebra:derivation-blowups} +Starting with the singular surface $X=\Spec A$ of +Theorem~\ref{thm:main}, every iteration of blowing up at the derivation +module is isomorphic to $X$. The same holds if each blowup is followed +by normalization. Neither procedure resolves its singular point. +\end{corollary} + +\begin{proof} +Since $\Der_{\CC}(A)\simeq A^2$, the identity of $X$ already has the +universal property just stated. Its blowup at the derivation module +is therefore $X$ itself, and the operation remains unchanged at every +iteration. Normalization also leaves $X$ unchanged because $A$ is +normal. The relation with the Lipman--Zariski conjecture was noted in +\cite[Introduction, after Question~1]{BarajasChavezRomano2026}; the +surface above gives a negative answer to that question in characteristic +zero. +\end{proof} diff --git a/preprints/A-singular-normal-affine-surface-with-free-tangent-sheaf-September-23-2026/build/bibliography.tex b/preprints/A-singular-normal-affine-surface-with-free-tangent-sheaf-September-23-2026/build/bibliography.tex new file mode 100644 index 0000000000000000000000000000000000000000..c1b029d6473559ac5e5d99f69ca0758274db6c11 --- /dev/null +++ b/preprints/A-singular-normal-affine-surface-with-free-tangent-sheaf-September-23-2026/build/bibliography.tex @@ -0,0 +1,221 @@ +\begin{thebibliography}{99} + +\bibitem{Artin1969} +M.~Artin, +\emph{Algebraic approximation of structures over complete local rings}, +Publ. Math. 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Algebra \textbf{337} (2011), no.~1, 169--180. +\href{https://doi.org/10.1016/j.jalgebra.2011.05.003}{doi:10.1016/j.jalgebra.2011.05.003}. +Corrected version: \href{https://arxiv.org/abs/1003.4241v6}{arXiv:1003.4241v6}, 2018. + +\bibitem{Lipman1965} +J.~Lipman, +\emph{Free derivation modules on algebraic varieties}, +Amer. J. Math. \textbf{87} (1965), no.~4, 874--898. +\href{https://doi.org/10.2307/2373252}{doi:10.2307/2373252}. + +\bibitem{SchejaStorch1972} +G.~Scheja and U.~Storch, +\emph{Differentielle Eigenschaften der Lokalisierungen analytischer Algebren}, +Math. Ann. \textbf{197} (1972), 137--170. +\href{https://doi.org/10.1007/BF01419591}{doi:10.1007/BF01419591}. + +\bibitem{SerreDuality1955} +J.-P.~Serre, +\emph{Un th\'eor\`eme de dualit\'e}, +Comment. Math. Helv. \textbf{29} (1955), 9--26. +\href{https://doi.org/10.1007/BF02564268}{doi:10.1007/BF02564268}. + +\bibitem{SerreGAGA1956} +J.-P.~Serre, +\emph{G\'eom\'etrie alg\'ebrique et g\'eom\'etrie analytique}, +Ann. Inst. Fourier \textbf{6} (1956), 1--42. +\href{https://doi.org/10.5802/aif.59}{doi:10.5802/aif.59}. + +\bibitem{StacksCompletion} +The Stacks Project Authors, +\emph{The Stacks Project}, +\href{https://stacks.math.columbia.edu/tag/00IN}{Tag~00IN} +(Artin--Rees), +\href{https://stacks.math.columbia.edu/tag/00MA}{Tag~00MA} +(exactness on finite modules), +\href{https://stacks.math.columbia.edu/tag/00MC}{Tag~00MC} +(faithful flatness of completion), +\href{https://stacks.math.columbia.edu/tag/07NV}{Tag~07NV} +(dimension and completion), +\href{https://stacks.math.columbia.edu/tag/07NY}{Tag~07NY} +(regularity and completion), and +\href{https://stacks.math.columbia.edu/tag/0C23}{Tag~0C23} +(normality of completion). +Accessed 23~September 2026. + +\bibitem{StacksApproximation} +The Stacks Project Authors, +\emph{The Stacks Project}, +\href{https://stacks.math.columbia.edu/tag/07QY}{Tag~07QY} +(approximation over henselian local G-rings) and +\href{https://stacks.math.columbia.edu/tag/07QZ}{Tag~07QZ} +(approximation in pointed \'etale neighborhoods). +Accessed 23~September 2026. + +\bibitem{StacksHenselization} +The Stacks Project Authors, +\emph{The Stacks Project}, +\href{https://stacks.math.columbia.edu/tag/04GN}{Tag~04GN} +(construction of henselization), +\href{https://stacks.math.columbia.edu/tag/06LJ}{Tag~06LJ} +(completion of a henselization), and +\href{https://stacks.math.columbia.edu/tag/0A1W}{Tag~0A1W} +(algebraic elements of a completion). +Accessed 23~September 2026. + +\bibitem{VanStratenSteenbrink1985} +D.~van Straten and J.~Steenbrink, +\emph{Extendability of holomorphic differential forms near isolated +hypersurface singularities}, +Abh. Math. Sem. Univ. Hamburg \textbf{55} (1985), 97--110. +\href{https://doi.org/10.1007/BF02941491}{doi:10.1007/BF02941491}. + +\end{thebibliography} diff --git a/preprints/A-singular-normal-affine-surface-with-free-tangent-sheaf-September-23-2026/build/paper.tex b/preprints/A-singular-normal-affine-surface-with-free-tangent-sheaf-September-23-2026/build/paper.tex new file mode 100644 index 0000000000000000000000000000000000000000..17133863409064e37fb0fd8dcaf1271b29c5e2eb --- /dev/null +++ b/preprints/A-singular-normal-affine-surface-with-free-tangent-sheaf-September-23-2026/build/paper.tex @@ -0,0 +1,80 @@ +\documentclass[11pt,a4paper]{article} +\usepackage[margin=28mm]{geometry} +\usepackage[T1]{fontenc} +\usepackage{lmodern} +\usepackage{amsmath,amssymb,amsthm,mathtools} +\usepackage{enumitem,booktabs,array} +\usepackage{microtype,needspace} +\usepackage{xcolor} +\usepackage[colorlinks=true,linkcolor=blue!40!black,citecolor=blue!40!black,urlcolor=blue!40!black]{hyperref} +\hypersetup{pdftitle={A singular normal affine surface with free tangent sheaf},pdfauthor={OpenAI}} +\pdfinfoomitdate=1 +\pdftrailerid{} +\pdfsuppressptexinfo=-1 +\pdfgentounicode=1 +\pdfglyphtounicode{mapsto}{007C} +\pdfglyphtounicode{arrowhookleft}{21AA} +\pdfglyphtounicode{parenleftbig}{0028} +\pdfglyphtounicode{parenrightbig}{0029} +\pdfglyphtounicode{parenleftBig}{0028} +\pdfglyphtounicode{parenrightBig}{0029} +\pdfglyphtounicode{parenleftbigg}{0028} +\pdfglyphtounicode{parenrightbigg}{0029} +\pdfglyphtounicode{parenleftBigg}{0028} +\pdfglyphtounicode{parenrightBigg}{0029} +\pdfglyphtounicode{bracketleftbigg}{005B} +\pdfglyphtounicode{bracketrightbigg}{005D} +\pdfglyphtounicode{braceleftBigg}{007B} +\pdfglyphtounicode{vextendsingle}{23D0} +\pdfglyphtounicode{summationtext}{2211} +\pdfglyphtounicode{summationdisplay}{2211} +\pdfglyphtounicode{integraltext}{222B} +\pdfglyphtounicode{uniondisplay}{22C3} +\pdfglyphtounicode{radicalBig}{221A} +\pdfglyphtounicode{hatwide}{02C6} +\pdfglyphtounicode{hatwider}{02C6} +\pdfglyphtounicode{tildewide}{02DC} +\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{\CC}{\mathbb C} +\newcommand{\PP}{\mathbb P} +\newcommand{\OO}{\mathcal O} +\newcommand{\cI}{\mathcal I} +\newcommand{\cF}{\mathcal F} +\newcommand{\cT}{\mathcal T} +\newcommand{\m}{\mathfrak m} +\DeclareMathOperator{\Der}{Der} +\DeclareMathOperator{\Hom}{Hom} +\DeclareMathOperator{\Spec}{Spec} +\DeclareMathOperator{\Tr}{Tr} +\DeclareMathOperator{\Res}{Res} +\DeclareMathOperator{\Sym}{Sym} +\DeclareMathOperator{\ord}{ord} +\DeclareMathOperator{\Span}{span} +\DeclareMathOperator{\id}{id} +\DeclareMathOperator{\diver}{div} +\setlist{topsep=4pt,itemsep=2pt,parsep=0pt} +\setlength{\emergencystretch}{1.5em} +\title{A singular normal affine surface\\ with free tangent sheaf} +\author{OpenAI} +\date{September 23, 2026} +\begin{document} +\maketitle +\input{01-introduction.tex} +\input{02-curve.tex} +\input{03-trace.tex} +\input{04-jets.tex} +\input{05-analytic.tex} +\input{06-algebraization.tex} +\Needspace{8\baselineskip} +\phantomsection +\addcontentsline{toc}{section}{References} +\input{bibliography.tex} +\end{document} diff --git a/preprints/A-singular-normal-affine-surface-with-free-tangent-sheaf-September-23-2026/paper.pdf b/preprints/A-singular-normal-affine-surface-with-free-tangent-sheaf-September-23-2026/paper.pdf new file mode 100644 index 0000000000000000000000000000000000000000..63348e95ff71111a4cae01f60b038191c8599e2c --- /dev/null +++ b/preprints/A-singular-normal-affine-surface-with-free-tangent-sheaf-September-23-2026/paper.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:48b1a74d3131e97b2cb8ed8f5c1d29458e8f2e92b6446d5d9d0161c4ab3612db +size 505802 diff --git a/preprints/A-smooth-three-torus-diffeomorphism-with-simple-Lebesgue-spectrum-September-23-2026/README.md b/preprints/A-smooth-three-torus-diffeomorphism-with-simple-Lebesgue-spectrum-September-23-2026/README.md new file mode 100644 index 0000000000000000000000000000000000000000..d5362b2d91bfeaf99fb090bfe27968e6753c5e48 --- /dev/null +++ b/preprints/A-smooth-three-torus-diffeomorphism-with-simple-Lebesgue-spectrum-September-23-2026/README.md @@ -0,0 +1,20 @@ +# [A smooth three-torus diffeomorphism with simple Lebesgue spectrum](paper.pdf) + +**Author:** OpenAI + +**Date:** September 23, 2026 + +## Citation + +To cite this manuscript: + +```bibtex +@misc{OAI:A-smooth-three-torus-diffeomorphism-with-simple-Lebesgue-spectrum-September-23-2026, + author = {{OpenAI}}, + title = {{A smooth three-torus diffeomorphism with simple Lebesgue spectrum}}, + howpublished = {OpenAI Math Release preprint + \href{https://github.com/openai/math/blob/main/preprints/A-smooth-three-torus-diffeomorphism-with-simple-Lebesgue-spectrum-September-23-2026/paper.pdf}{OAI:A-smooth-three-torus-diffeomorphism-with-simple-Lebesgue-spectrum-September-23-2026}}, + year = {2026} +} +``` + diff --git a/preprints/A-smooth-three-torus-diffeomorphism-with-simple-Lebesgue-spectrum-September-23-2026/build/main.tex b/preprints/A-smooth-three-torus-diffeomorphism-with-simple-Lebesgue-spectrum-September-23-2026/build/main.tex new file mode 100644 index 0000000000000000000000000000000000000000..8161aabcb47a1869eef2826275292e285074175a --- /dev/null +++ b/preprints/A-smooth-three-torus-diffeomorphism-with-simple-Lebesgue-spectrum-September-23-2026/build/main.tex @@ -0,0 +1,59 @@ +\documentclass[11pt]{article} +\usepackage[T1]{fontenc} +\usepackage{lmodern} +\usepackage[margin=1.05in]{geometry} +\usepackage{amsmath,amssymb,amsthm,mathtools} +\usepackage{microtype} +\usepackage{enumitem} +\usepackage{xcolor} +\PassOptionsToPackage{hyphens}{url} +\usepackage{hyperref} +\hypersetup{colorlinks=true,linkcolor=blue!45!black,citecolor=blue!45!black,urlcolor=blue!45!black,pdftitle={A smooth three-torus diffeomorphism with simple Lebesgue spectrum},pdfauthor={OpenAI}} +\newcommand{\T}{\mathbb T} +\newcommand{\R}{\mathbb R} +\newcommand{\Z}{\mathbb Z} +\newcommand{\N}{\mathbb N} +\newcommand{\C}{\mathbb C} +\newcommand{\eps}{\varepsilon} +\newcommand{\1}{\mathbf 1} +\newcommand{\dd}{\,\mathrm d} +\DeclareMathOperator{\ee}{e} +\DeclareMathOperator{\Rea}{Re} +\DeclareMathOperator{\Ima}{Im} +\DeclareMathOperator{\supp}{supp} +\DeclareMathOperator{\dist}{dist} +\DeclareMathOperator{\Leb}{Leb} +\newcommand{\calH}{\mathcal H} +\newcommand{\calC}{\mathcal C} +\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} +\setlist{itemsep=2pt,topsep=5pt} +\setlength{\emergencystretch}{1em} +\title{A smooth three-torus diffeomorphism with simple Lebesgue spectrum} +\author{OpenAI} +\date{September 23, 2026} +\begin{document} +\maketitle +\begin{abstract} +We solve the probability-preserving form of Banach's simple Lebesgue-spectrum problem in smooth dynamics. We construct a $C^\infty$ diffeomorphism of the three-torus that preserves standard volume and whose Koopman operator has simple Lebesgue spectrum on the entire mean-zero $L^2$ space. +\end{abstract} +\tableofcontents +\input{sections/01-introduction} +\input{sections/02-passages} +\input{sections/03-signals} +\input{sections/04-blocks} +\input{sections/05-completion} +\input{sections/06-consequences} +\begingroup +\small +\bibliographystyle{plain} +\bibliography{references} +\endgroup +\end{document} diff --git a/preprints/A-smooth-three-torus-diffeomorphism-with-simple-Lebesgue-spectrum-September-23-2026/build/references.bib b/preprints/A-smooth-three-torus-diffeomorphism-with-simple-Lebesgue-spectrum-September-23-2026/build/references.bib new file mode 100644 index 0000000000000000000000000000000000000000..e75e6328fe332a1e6b2a01b5a1714a5dc6388356 --- /dev/null +++ b/preprints/A-smooth-three-torus-diffeomorphism-with-simple-Lebesgue-spectrum-September-23-2026/build/references.bib @@ -0,0 +1,168 @@ +@book{Ulam1960, + author = {Ulam, S. M.}, + title = {{\href{https://www.dbraulibrary.org.in/RareBooks/A\%20collection\%20of\%20mathematical\%20problems.pdf}{A Collection of Mathematical Problems}}}, + series = {Interscience Tracts in Pure and Applied Mathematics}, + volume = {8}, + publisher = {Interscience Publishers}, + address = {New York}, + year = {1960}, + note = {\S 6, p.~76} +} + +@article{Rokhlin1949, + author = {Rokhlin, V. A.}, + title = {{\href{https://www.mathnet.ru/eng/rm8607}{Selected topics from the metric theory of dynamical systems}}}, + journal = {Uspekhi Matematicheskikh Nauk}, + volume = {4}, + number = {2(30)}, + pages = {57--128}, + year = {1949}, + note = {In Russian; see \S 4, no.~7, p.~107. English translation in American Mathematical Society Translations, Series 2, 49 (1966), 171--240}, + url = {https://www.mathnet.ru/eng/rm8607} +} + +@article{Rokhlin1960, + author = {Rokhlin, V. A.}, + title = {{\href{https://www.mathnet.ru/eng/rm6735}{New progress in the theory of transformations with invariant measure}}}, + journal = {Russian Mathematical Surveys}, + volume = {15}, + number = {4}, + pages = {1--22}, + year = {1960}, + url = {https://www.mathnet.ru/eng/rm6735} +} + +@misc{MultipleMixing2026, + author = {{OpenAI}}, + title = {{Rokhlin's multiple-mixing problem for one transformation}}, + howpublished = {OpenAI Math Release preprint + \href{https://github.com/openai/math/blob/main/preprints/Rokhlins-multiple-mixing-problem-for-one-transformation-September-23-2026/paper.pdf}{OAI:Rokhlins-multiple-mixing-problem-for-one-transformation-September-23-2026}}, + year = {2026} +} + +@article{Mane1981, + author = {Ma{\~n}{\'e}, Ricardo}, + title = {{\href{https://doi.org/10.1017/S0143385700001188}{A proof of {Pesin}'s formula}}}, + journal = {Ergodic Theory and Dynamical Systems}, + volume = {1}, + pages = {95--102}, + year = {1981}, + doi = {10.1017/S0143385700001188}, + url = {https://doi.org/10.1017/S0143385700001188}, + note = {Errata: Ergodic Theory and Dynamical Systems 3(1) (1983), 159--160, \href{https://doi.org/10.1017/S0143385700001863}{doi:10.1017/S0143385700001863}} +} + +@article{Prikhodko2020, + author = {Prikhod'ko, A. A.}, + title = {{\href{https://www.mathnet.ru/eng/sm8147}{On ergodic flows with simple {Lebesgue} spectrum}}}, + journal = {Sbornik: Mathematics}, + volume = {211}, + number = {4}, + pages = {594--615}, + year = {2020}, + doi = {10.1070/SM8147}, + url = {https://www.mathnet.ru/eng/sm8147} +} + +@article{HelsonParry1978, + author = {Helson, Henry and Parry, William}, + title = {{\href{https://doi.org/10.1007/BF02385994}{Cocycles and spectra}}}, + journal = {Arkiv f{\"o}r Matematik}, + volume = {16}, + pages = {195--206}, + year = {1978}, + doi = {10.1007/BF02385994}, + url = {https://doi.org/10.1007/BF02385994} +} + +@article{MathewNadkarni1984, + author = {Mathew, J. and Nadkarni, M. G.}, + title = {{\href{https://doi.org/10.1112/blms/16.4.402}{A measure preserving transformation whose spectrum has {Lebesgue} component of multiplicity two}}}, + journal = {Bulletin of the London Mathematical Society}, + volume = {16}, + number = {4}, + pages = {402--406}, + year = {1984}, + doi = {10.1112/blms/16.4.402}, + url = {https://doi.org/10.1112/blms/16.4.402} +} + +@article{Guenais1999, + author = {Guenais, M{\'e}lanie}, + title = {{\href{https://doi.org/10.1017/S0143385799126579}{Morse cocycles and simple {Lebesgue} spectrum}}}, + journal = {Ergodic Theory and Dynamical Systems}, + volume = {19}, + number = {2}, + pages = {437--446}, + year = {1999}, + doi = {10.1017/S0143385799126579}, + url = {https://doi.org/10.1017/S0143385799126579} +} + +@article{Fayad2001, + author = {Fayad, Bassam R.}, + title = {{\href{https://doi.org/10.1112/S0024610701002447}{Partially mixing and locally rank one smooth transformations and flows on the torus {$\mathbb T^d$}, {$d\ge3$}}}}, + journal = {Journal of the London Mathematical Society}, + volume = {64}, + number = {3}, + pages = {637--654}, + year = {2001}, + doi = {10.1112/S0024610701002447}, + url = {https://doi.org/10.1112/S0024610701002447} +} + +@article{FFK2021, + author = {Fayad, Bassam and Forni, Giovanni and Kanigowski, Adam}, + title = {{\href{https://arxiv.org/abs/1609.03757}{{Lebesgue} spectrum of countable multiplicity for conservative flows on the torus}}}, + journal = {Journal of the American Mathematical Society}, + volume = {34}, + number = {3}, + pages = {747--813}, + year = {2021}, + doi = {10.1090/jams/970}, + url = {https://arxiv.org/abs/1609.03757} +} + +@article{AFK2023, + author = {Abdedou, Fatna and Fayad, Bassam and Kessi, Arezki}, + title = {{\href{https://www.aimsciences.org/article/doi/10.3934/dcds.2023063}{Analytic reparametrizations of translation toral flows with countable {Lebesgue} spectrum}}}, + journal = {Discrete and Continuous Dynamical Systems}, + volume = {43}, + number = {10}, + pages = {3706--3727}, + year = {2023}, + doi = {10.3934/dcds.2023063}, + url = {https://www.aimsciences.org/article/doi/10.3934/dcds.2023063} +} + +@misc{ElAbdalaoui2023, + author = {el Abdalaoui, el Houcein}, + title = {Spectral ergodic {Banach} problem and flat polynomials}, + year = {2023}, + howpublished = {arXiv:1508.06439v8}, + note = {Preprint; first posted 2015, version 8 dated 19 June 2023}, + url = {https://arxiv.org/abs/1508.06439v8} +} + +@article{AnosovKatok1970, + author = {Anosov, D. V. and Katok, A. B.}, + title = {New examples in smooth ergodic theory. {Ergodic} diffeomorphisms}, + journal = {Trudy Moskovskogo Matematicheskogo Obshchestva}, + volume = {23}, + pages = {3--36}, + year = {1970}, + note = {English translation: Transactions of the Moscow Mathematical Society 23 (1970), 1--35}, + url = {https://www.mathnet.ru/eng/mmo237} +} + +@book{Hormander2003, + author = {H{\"o}rmander, Lars}, + title = {{\href{https://link.springer.com/book/10.1007/978-3-642-61497-2}{The Analysis of Linear Partial Differential Operators {I}: Distribution Theory and Fourier Analysis}}}, + edition = {Second}, + series = {Classics in Mathematics}, + publisher = {Springer}, + address = {Berlin}, + year = {2003}, + doi = {10.1007/978-3-642-61497-2}, + url = {https://link.springer.com/book/10.1007/978-3-642-61497-2} +} diff --git a/preprints/A-smooth-three-torus-diffeomorphism-with-simple-Lebesgue-spectrum-September-23-2026/build/sections/01-introduction.tex b/preprints/A-smooth-three-torus-diffeomorphism-with-simple-Lebesgue-spectrum-September-23-2026/build/sections/01-introduction.tex new file mode 100644 index 0000000000000000000000000000000000000000..f879819514cf24dd4042d9d786e32b9dbafc3999 --- /dev/null +++ b/preprints/A-smooth-three-torus-diffeomorphism-with-simple-Lebesgue-spectrum-September-23-2026/build/sections/01-introduction.tex @@ -0,0 +1,144 @@ +\section{Introduction} +\label{sec:introduction} +Let $\T=\R/\Z$, let $\mu$ be normalized Lebesgue measure on $\T^3$, +and put $\ee(t)=\exp(2\pi i t)$. For an invertible measure-preserving +transformation $T$, we use the convention $U_Tg=g\circ T$. +The closed subspace of complex square-integrable functions of integral zero +is denoted by $L^2_0(\T^3,\mu)$. + +On a probability space the constants form an invariant subspace, so the +probability-preserving form of Banach's simple Lebesgue-spectrum problem +asks for an invertible transformation whose Koopman operator on the +\emph{entire} orthogonal complement of the constants is unitarily +equivalent to multiplication by $w$ on $L^2(S^1,m)$, where $m$ is +normalized Haar measure. Equivalently, that restriction has a bilateral +orthonormal orbit which is complete. We realize this spectral model with +the standard smooth volume on a compact manifold. + +\begin{theorem}\label{thm:main} +There exist a $C^\infty$ volume-preserving diffeomorphism +$T\colon\T^3\to\T^3$ and a real function $f\in L^2_0(\T^3,\mu)$ such that +\[ + \{f\circ T^n:n\in\Z\} +\] +is an orthonormal basis of $L^2_0(\T^3,\mu)$. +Consequently $T$ is ergodic, and $U_T$ on this space is unitarily equivalent +to multiplication by the coordinate $w$ on $L^2(S^1,m)$. +\end{theorem} + +The invariant measure in Theorem~\ref{thm:main} is the standard smooth +volume itself. The generator $f$ is only required to belong to $L^2$. +In particular, the assertion concerns the whole mean-zero space, including +both its spectral type and its multiplicity. +Section~\ref{sec:consequences} derives mixing, zero entropy, and vanishing +Lyapunov exponents. Mixing of every order uses, in addition, the separate +multiple-mixing theorem of \cite{MultipleMixing2026}. + +The formulation above is distinct from the historical real-line question +recorded by Ulam and attributed to Banach \cite[\S 6, p.~76]{Ulam1960}. +Rokhlin asked for ergodic automorphisms with simple, or at least +finite-multiplicity, Lebesgue spectrum +\cite[\S 4, no.~7, p.~107]{Rokhlin1949}. Helson and Parry constructed +cocycles over aperiodic probability-preserving transformations whose +associated unitary operators have Lebesgue spectrum +\cite[p.~196, Theorem~(a)]{HelsonParry1978}. Mathew and Nadkarni constructed a +probability-preserving transformation with a Lebesgue \emph{component} +of multiplicity two \cite{MathewNadkarni1984}. Guenais connected the +Morse-cocycle question with flat polynomials and constructed a countable +abelian group action with simple spectrum of mixed discrete and Haar +type \cite{Guenais1999}. These results distinguish a Lebesgue component, +or a related group action, from one bilateral Lebesgue orbit spanning +the whole centered space. + +Simple spectrum in smooth dynamics was already achieved by Fayad, who +constructed $C^\infty$ volume-preserving torus diffeomorphisms with +simple spectrum \cite{Fayad2001}; that result did not assert Lebesgue +spectral type. +In another setting, an infinite-measure conservative transformation +with simple Lebesgue spectrum is presented in +\cite[Theorem~2.2]{ElAbdalaoui2023}; the probability-space conclusion +does not follow from it. Theorem~\ref{thm:main} gives that conclusion +for the full mean-zero space of a smooth probability-preserving map. + +Related existence results in continuous time have different spectral +consequences. Prikhod'ko constructs a finite-measure flow with simple +Lebesgue spectrum \cite[Theorem~3]{Prikhodko2020}. In the smooth category, +Fayad, Forni and Kanigowski obtain a real analytic conservative flow on +$\T^2$ with one singularity and countably infinite Lebesgue multiplicity +\cite[Theorem~1]{FFK2021}; Abdedou, Fayad and Kessi obtain real analytic +reparametrizations of minimal translation flows on $\T^5$ with the same +multiplicity \cite[Theorem~1.1]{AFK2023}. Taking a nonzero time map of a +simple Lebesgue flow yields countably infinite multiplicity: its spectral +model is multiplication by $\ee(t\xi)$ on $L^2(\R,\dd\xi)$, and the +substitution $\xi=(x+k)/|t|$, with $x\in[0,1)$ and $k\in\Z$, gives one +Lebesgue copy for each $k$. Thus these continuous-time results do not +supply Theorem~\ref{thm:main}. + +Successive smooth changes of coordinates have a substantial history in +smooth dynamics, including the approximation-by-conjugacy constructions +of Anosov and Katok \cite{AnosovKatok1970}. We use explicit +volume-preserving passages between integrable twists rather than an +external conjugacy theorem. Their quantitative spectral-density +transport, and the finite Fourier signals built on it, are proved below. + +\subsection*{The construction} +The approximating maps are integrable twists. In a volume-preserving +coordinate system on the torus, they have the form +\[ + S(\theta,u,a)=(\theta+\phi(u),u,a), +\] +where $\phi$ is a smooth real circle function that is strictly increasing +on an arc whose complement can be made arbitrarily short. A finite angular +Fourier expansion supported on that arc has an explicit absolutely +continuous spectral measure. We refer to its individual harmonics as +\emph{packets}, and keep a finite label, or \emph{color}, on disjoint groups +of packets. The colors carry the coefficients of an observable that we +want to approximate by translates of a single vector. + +There are three steps. A slow passage in the coordinate $u$ replaces the +twist by an arbitrarily close twist whose angular coordinate is the old +$u$. Stationary phase locates the new angular indices and determines the +spectral density at a much finer scale. The spare coordinate $a$ supplies +a small adjustable variation in the speed (the \emph{jitter}). A large integer added to the +lift of $\phi$ separates the angular indices of different packets without +changing the old map. This separation is what allows later perturbations +to distinguish colors even when their temporal spectra overlap. + +Next, a positive Fourier construction and a finite prime sieve produce a +small sinusoidal bias in each color density. Its coefficient depends on +the normalized angular index. A neutral passage between successive biased +passages makes residue classes carry equal shares of each color. The +bias strength can be chosen before the angular scale grows; the passage +parameter is then free to enforce any required smooth or spectral +accuracy. + +Finally, many small biases are grouped into blocks. If color $r$ has +spectral density $p_r$ and target coefficient $b_r$, the center at a +temporal frequency $z$ is the density-weighted mean +\[ + p(z)=\sum_r p_r(z),\qquad + \beta(z)=\frac{\sum_r b_rp_r(z)}{p(z)}\quad(p(z)>0). +\] +Centering each block at a smooth approximation to this current mean +makes the loss in $\int\sqrt p$ quadratic +in the block's strength budget, while the same block provides an estimator +of the color label. Repetition makes prediction arbitrarily accurate with +arbitrarily little loss in $\int\sqrt p$. This separation of the two +estimates is the central finite construction. When the vector's norm is +close to one, keeping $\int\sqrt p$ close to one makes whitening a small +change in physical $L^2$, while restoring the spectral density exactly +to one. A diagonal limit preserves the orthonormal orbit and makes its +span contain every member of a dense family of observables. + +\subsection*{Order of the proof} +Section~\ref{sec:passages} proves the analytic passage rules, including the +comparison of jitter laws along a rapidly oscillating phase. +Section~\ref{sec:signals} constructs the positive laws that transmit the +small biases. Section~\ref{sec:blocks} proves the finite prediction result, +and Section~\ref{sec:completion} prepares arbitrary observables and takes +the smooth limit. Section~\ref{sec:consequences} derives the dynamical +consequences. Every passage acts on finite data. Its integer parameter +is chosen last, after the current coordinate chart, cutoffs, jitter, +and all desired error bounds have been fixed. No uniform bound on the +number of packets or on the derivatives of successive coordinate charts +is assumed. diff --git a/preprints/A-smooth-three-torus-diffeomorphism-with-simple-Lebesgue-spectrum-September-23-2026/build/sections/02-passages.tex b/preprints/A-smooth-three-torus-diffeomorphism-with-simple-Lebesgue-spectrum-September-23-2026/build/sections/02-passages.tex new file mode 100644 index 0000000000000000000000000000000000000000..0f8e0b4db727162800da0f1efad01cdcebbfe5bf --- /dev/null +++ b/preprints/A-smooth-three-torus-diffeomorphism-with-simple-Lebesgue-spectrum-September-23-2026/build/sections/02-passages.tex @@ -0,0 +1,505 @@ +\section{Smooth passages and spectral densities}\label{sec:passages} + +A volume-preserving coordinate system, also called a chart below, means +a smooth volume-preserving identification of the physical torus with $\T^3$. +All coordinate systems in the construction preserve normalized volume. +An intermediate transformation has the form +\begin{equation}\label{eq:twist} + S(\theta,u,a)=(\theta+\phi(u),u,a), + \qquad (\theta,u,a)\in\T^3, +\end{equation} +where $\phi\in C^\infty(\T;\R)$ has $\phi'>0$ on an open arc +$I$. We call $I$ a good arc. Derivatives on an arc are understood in an +increasing real lift of that arc. + +A packet is a function $A(u,a)\ee(m\theta)$, where +$m\in\Z\setminus\{0\}$ and $A$ is smooth with compact support in +$I\times\T$. A finite packet list is disjoint if, for any two entries with +the same $m$, their coefficient supports are disjoint. Entries with +different $m$ need no support restriction. Packets may be grouped into +colors; we write $F_r$ for the sum of color $r$. Disjointness implies +\begin{equation}\label{eq:packet-orthogonality} + \langle U_S^n F_r,U_S^k F_l\rangle=0 + \quad(r\ne l,\ n,k\in\Z). +\end{equation} +Indeed $U_S^n(A\ee(m\theta))=\ee(nm\phi(u))A\ee(m\theta)$, +and either angular orthogonality or disjoint coefficient supports applies. +The spectral measure of one packet is +\begin{equation}\label{eq:packet-spectral-measure} + (m\phi\bmod1)_*\bigl(|A(u,a)|^2\dd u\dd a\bigr). +\end{equation} +It has a smooth density because its coefficient support is compactly +inside the good arc. We use the inner product linear in its first +variable, so a vector of density $p$ has +$\langle U_S^nF,F\rangle=\int_\T\ee(nz)p(z)\dd z$. +A measurable spectral multiplier $v(z)$ acts on its +coefficient by multiplication by $v(m\phi(u)\bmod1)$, whenever the result +belongs to $L^2$. These statements follow directly from the displayed +formula for $U_S^n$, first for trigonometric polynomials and then by +$L^2$ approximation. The pushforward formula, absolute continuity, +and multiplier rule also apply to nonsmooth coefficients in $L^2$ +supported in the good arc; their densities need not be smooth. + +For later comparisons, if two vectors have spectral densities $p_F,p_G$ +for the same transformation, then +\begin{equation}\label{eq:density-norm-bound} + \|p_F-p_G\|_1 + \le (\|F\|_2+\|G\|_2)\|F-G\|_2. +\end{equation} +To see this, test their spectral measures against an arbitrary function +of absolute value at most one and use the contraction property of its +spectral multiplier. The total variation norm here and below is the +$L^1$ norm of densities, without a factor $1/2$. + +\begin{lemma}[Passage]\label{lem:passage} +Fix \eqref{eq:twist}, its physical volume-preserving coordinate chart, +and a finite disjoint packet list. Choose a positive integer $c$ so large +that the compact ranges $m(c+\phi(\T))$, for the distinct old indices $m$, +are pairwise disjoint and avoid zero. Set $V=c+\phi$. +Fix $J\in C^\infty(\T;\R)$ with $J'>0$ on an open arc $I'$ and fix +$0\le\psi\le1$ in $C_c^\infty(I')$. +For integers $N\to\infty$, choose $\Delta_N$ so that +\begin{equation}\label{eq:passage-rounding} + |\Delta_N|\le N^{-1},\qquad + k_N:=N\left(\int_\T V(u)\dd u+\Delta_N\right)\in\Z, + \qquad H_N'=N(V+\Delta_N). +\end{equation} +Put $\sigma_N(a)=N^{-1}+N^{-2}J(a)$. Then +\begin{equation}\label{eq:passage-map} + T_N(\theta,u,a)= + \bigl(\theta+H_N(u+\sigma_N(a))-H_N(u),\, + u+\sigma_N(a),\,a\bigr) +\end{equation} +is a volume-preserving smooth torus diffeomorphism, and both $T_N$ and +$T_N^{-1}$ converge smoothly to $S$ and $S^{-1}$, also in the fixed +physical coordinates. In the new coordinates +\begin{equation}\label{eq:passage-chart} + (\theta',u',a')=C_N(\theta,u,a) + =(u,a,\theta-H_N(u)), +\end{equation} +it is the twist of speed $\sigma_N$ and good arc $I'$. + +Here are its packet and density properties. For an old packet indexed +by $\alpha$, with index $m_\alpha$ and amplitude $A_\alpha$, define +\begin{equation}\label{eq:stationary-image} + E_\alpha=\{(m_\alpha V(u),a):(u,a)\in\supp A_\alpha\}. +\end{equation} +One may choose pairwise disjoint, arbitrarily small fixed neighborhoods +of these compact sets, and smooth cutoffs $\chi_\alpha(s,a)$ supported +in those neighborhoods and equal to one near $E_\alpha$. Retain the +finitely many new packets +\begin{align} + G_{\alpha,j}(\theta',u',a') + &=\ee(j\theta')\ee(m_\alpha a') + \psi(u')\chi_\alpha(j/N,u')C_{\alpha,j}(u'), + \label{eq:retained-packets}\\ + C_{\alpha,j}(a) + &=\int_\T A_\alpha(u,a)\ee(m_\alpha H_N(u)-ju)\dd u. + \label{eq:exact-passage-coefficient} +\end{align} +All their angular indices are nonzero for sufficiently large $N$. +For each $\alpha$, +\begin{equation}\label{eq:transport-norm} + \left\|\sum_jG_{\alpha,j}\circ C_N + -\psi(a)A_\alpha(u,a)\ee(m_\alpha\theta)\right\|_2=o(1). +\end{equation} +They are disjoint as packets, and their moduli are independent of the +new spare coordinate $a'$. Every constant attached to an old packet is +preserved by this linear transport and these cutoffs. + +For a single old packet write, with zero extension outside its branch, +\begin{equation}\label{eq:W-definition} + W(s,a)=\frac{|A(u,a)|^2}{|m\phi'(u)|},\qquad s=mV(u). +\end{equation} +Let $K(s,y)$ be the density of the pushforward of +$W(s,a)\psi(a)^2\dd a$ under $a\mapsto sJ(a)$. It is continuous and +compactly supported in $(s,y)$, with its $s$ support away from zero. +For any fixed positive integer $P$ and residue $h\pmod P$, the unfolded +temporal density of the retained packets with $j\equiv h\pmod P$ differs +in $L^1(\R)$ by $o(1)$ from +\begin{equation}\label{eq:fine-density-unfolded} + Q_{N,h}(t)=\sum_{k\in\Z}K(t,Nt-h-kP). +\end{equation} +Consequently their density on $\T$ differs in $L^1(\T)$ by $o(1)$ from +$\sum_{l\in\Z}Q_{N,h}(z+l)$. The assertions hold after summing over +any of the finitely many colors. All $o(1)$ terms refer to the old data, +$c,J,\psi,P$, and support neighborhoods fixed before $N$ tends to +infinity. In particular, $N$ may be required to exceed any prescribed +bound. +\end{lemma} + +\begin{proof} +\emph{Geometry and smooth convergence.} +Choose $k_N$ by rounding $N\int V$ to an integer and integrate +\eqref{eq:passage-rounding}. Then $H_N(u+1)-H_N(u)=k_N$, so the map +$u\mapsto H_N(u)\bmod1$ is globally defined. The map $C_N$ is a smooth +torus diffeomorphism of Jacobian one, with inverse +$(\theta',u',a')\mapsto(a'+H_N(\theta'),\theta',u')$. +Direct substitution gives +\[ + C_NT_NC_N^{-1}(\theta',u',a') + =(\theta'+\sigma_N(u'),u',a'). +\] +This proves the global assertions, except convergence. The integral +identity +\[ + H_N(u+\sigma_N(a))-H_N(u) + =N\int_0^{\sigma_N(a)}(V(u+t)+\Delta_N)\dd t + =V(u)+O_{C^k}(N^{-1}) +\] +holds for each fixed $k$, with constants depending on the fixed $c,J$ +and $\phi$. Since $c$ is an integer, addition of $V$ and of $\phi$ agree +on $\T$. The inverse of \eqref{eq:passage-map} is +\[ + (\theta,u,a)\longmapsto + (\theta-H_N(u)+H_N(u-\sigma_N(a)),u-\sigma_N(a),a), +\] +and the same estimate proves inverse convergence. Composing with a +fixed physical chart and its inverse preserves convergence in every +fixed smooth norm. No bound uniform over future coordinate charts is +being asserted. + +\emph{Fourier coefficients.} +Work on a compact +subarc of $I$ containing all old coefficient supports in its interior. +For $s=j/N$ the phase in \eqref{eq:exact-passage-coefficient}, divided +by $N$, has critical equation +\begin{equation}\label{eq:critical-equation} + m(V(u)+\Delta_N)=s +\end{equation} +and second derivative $m\phi'(u)$. It has at most one critical point +on this arc. With $u_*=u_*(s-m\Delta_N)$ on that branch, set +\[ + L_{m,j}(a)= + \frac{A(u_*,a)}{\sqrt{N|m\phi'(u_*)|}} + \ee\left(mH_N(u_*)-ju_*+\frac{\operatorname{sgn}(m)}8\right), +\] +and set this to zero wherever the amplitude at the critical point is +zero or there is no point on the branch. Uniformly for $j/N$ in any +fixed compact interval and $a\in\T$, +\begin{equation}\label{eq:stationary-uniform} + C_{m,j}(a)-L_{m,j}(a)=o(N^{-1/2}). +\end{equation} + +For general stationary-phase background see \cite{Hormander2003}; +we give the compact-support one-dimensional argument needed here. +Enlarge the support arc twice inside $I$. Critical points +near the original support then range over a fixed compact interval on +which the absolute second derivative is bounded below. A smooth change +of variable, depending smoothly on the critical point, reduces the +divided phase to its critical value plus +$\operatorname{sgn}(m)y^2/2$; its Jacobian at zero is +$|m\phi'(u_*)|^{-1/2}$. After putting $v=\sqrt N y$, the integral on +$|v|\le D$ converges uniformly to the constant amplitude times the +quadratic exponential integral there. The remaining local tails are +$O(N^{-1/2}/D)$, uniformly in the parameters, by integrating against +$\ee(\operatorname{sgn}(m)v^2/2)$: the rescaled amplitudes have uniformly +bounded suprema and total variation. The part away from the critical +point is $O(N^{-1})$ by integration by parts. Finally +\[ + \lim_{D\to\infty}\int_{-D}^D\ee(\pm v^2/2)\dd v + =\exp(\pm i\pi/4); +\] +this follows by inserting a Gaussian factor, evaluating the Gaussian +integral, and removing the factor using the same tail bound. +Taking first $N\to\infty$, then $D\to\infty$, proves +\eqref{eq:stationary-uniform}. When the potential critical point is +outside the enlarged support interval, the integral is uniformly +nonstationary. This also covers zero amplitudes and the ends of the +branch. + +There are $O(N)$ indices in a fixed bounded $j/N$ range. Outside a +sufficiently large such range, repeated integration by parts gives, +for any fixed positive integer $q$, +$|C_{m,j}(a)|\le C_q|j|^{-q}$ uniformly in $a$ and $N$. +Indeed the phase derivative has size comparable to $|j|$, and its +higher derivatives are $O(N)$ with $N=O(|j|)$ there. Thus +\eqref{eq:stationary-uniform} implies +\[ + \sum_j\int_\T |C_{m,j}-L_{m,j}|^2\dd a=o(1). +\] +The $L^2$ norm of the leading array is bounded by a Riemann sum. +Cauchy--Schwarz therefore gives the useful mass estimate +\begin{equation}\label{eq:summed-stationary-mass} + \sum_j\int_\T + \left||C_{m,j}(a)|^2 + -\frac1N W(j/N-m\Delta_N,a)\right|\dd a=o(1). +\end{equation} + +\emph{Cutoffs and transport.} +The map $(u,a)\mapsto(mV(u),a)$ is injective on $I\times\T$. +It takes disjoint compact supports with the same $m$ to disjoint +compact sets. Our choice of $c$ separates the images for different +$m$. The stated cutoffs $\chi_\alpha$ therefore exist. For large $N$ +they equal one on the support of the corresponding leading array, +since its $s$ coordinate is shifted by only $m_\alpha\Delta_N$. +Because each cutoff is one on the shifted leading-array support, the +deleted exact coefficient mass is bounded by the summed squared +$C_{m,j}-L_{m,j}$ error, which tends to zero by the preceding estimate +and Parseval's identity. The same argument is valid after +multiplication by $\psi$, proving \eqref{eq:transport-norm} and +\eqref{eq:summed-stationary-mass} for the retained arrays with the +factor $\psi^2$ on the right. The cutoffs make separation exact, not +merely asymptotic. Different old $m$ never contribute to the same +retained $j$, and at a shared $j$ the remaining supports in $u'=a$ +are disjoint. The factor $\ee(ma')$ in +\eqref{eq:retained-packets} proves the asserted independence of the +new spare-coordinate modulus, also after combining packets at one +index. Linearity proves the statement about attached constants. + +\emph{Fine temporal density.} +We first remove rounding before any fine-scale sampling. Smoothness and +compact support of the fixed $W$ give +\begin{equation}\label{eq:rounding-mass-removal} + \frac1N\sum_j\int_\T + |W(j/N-m\Delta_N,a)-W(j/N,a)|\dd a=o(1). +\end{equation} +For example, a uniform bound on $\partial_s W$ and $O(N)$ nonzero +summands give an $O(|m|/N)$ bound. The same holds with $\psi^2$ and +for any residue subset. Pushforward contracts total variation, so +\eqref{eq:summed-stationary-mass} and +\eqref{eq:rounding-mass-removal} allow us to replace the retained +squared coefficient measures by +$N^{-1}W(j/N,a)\psi(a)^2\dd a$, at total error $o(1)$, before +pushing to temporal frequency. In particular, the actual lattice +$j/N$ is never translated by $m\Delta_N$. + +On the retained arc, the change of variable $y=sJ(a)$ gives +\begin{equation}\label{eq:K-jacobian} + K(s,y)=\frac{W(s,a)\psi(a)^2}{|s|J'(a)}, + \qquad a=J|_{I'}^{-1}(y/s), +\end{equation} +with zero extension. The compact support of $\psi$ inside $I'$ makes +this extension continuous (in fact smooth away from $s=0$). +The real temporal frequency of a new angular index $j$ is +\begin{equation}\label{eq:output-frequency} + t=j\sigma_N(a)=j/N+(j/N)J(a)/N. +\end{equation} +The density of the pushforward of +$N^{-1}W(j/N,a)\psi(a)^2\dd a$ under this map is exactly +$K(j/N,Nt-j)$. The factor $N$ in the density change of variable +cancels the coefficient mass $N^{-1}$. + +Choose $Y$ so that $K(s,y)=0$ for $|y|>Y$. For fixed $t$, at most +$2Y+2$ integers contribute to either +$\sum_j K(j/N,Nt-j)$ or $\sum_jK(t,Nt-j)$; on every contributing +term $|j/N-t|\le Y/N$. Uniform continuity of $K$, and a fixed +bounded interval containing the $t$ supports for large $N$, prove +that replacing $j/N$ by $t$ costs $o(1)$ in $L^1(\R)$. +The same proof works after restriction to $j\equiv h\pmod P$. +This is \eqref{eq:fine-density-unfolded}. Folding modulo one +contracts the $L^1$ norm and finishes the proof. +\end{proof} + +It is useful to emphasize the normalization in +\eqref{eq:fine-density-unfolded}. The periodization +$\sum_k K(t,y-kP)$ is a density relative to length on the circle of +length $P$, so its mean value is +\begin{equation}\label{eq:residue-mean} + \frac1P\int_\T W(t,a)\psi(a)^2\dd a. +\end{equation} +For $P=1$, folding $t=z+l$ leaves the fast phase $Nz$ unchanged, +since $N$ is an integer. For general $P$, the folded expression has +phase $N(z+l)-h$ modulo $P$; no divisibility condition on $N$ is +imposed. + +\begin{lemma}[Fast-variable averaging]\label{lem:fast-average} +Let $P>0$, and let $G(t,y)$ be continuous, compactly supported in $t$, +and $P$-periodic in $y$. Then, uniformly in real offsets $\beta_N$, +\begin{equation}\label{eq:fast-average} + \int_\R G(t,Nt+\beta_N)\dd t + \longrightarrow \frac1P\int_\R\int_0^P G(t,y)\dd y\dd t. +\end{equation} +In particular, suppose $w\ge0$ is continuous and compactly supported +in a compact set $K\subset\R\setminus\{0\}$. For continuous +integrable densities $\rho,\widetilde\rho$ define +\[ + \mathcal P_{s,P}\rho(y) + =\sum_{k\in\Z}|s|^{-1}\rho((y+kP)/s). +\] +Assume these periodizations are jointly continuous for $s\in K$. +Then +\begin{align}\label{eq:law-diagonal-comparison} + &\lim_{N\to\infty} + \int_\R w(s) + |\mathcal P_{s,P}\rho(Ns+\beta_N) + -\mathcal P_{s,P}\widetilde\rho(Ns+\beta_N)|\dd s\\ + &\hspace{12mm}= + \frac1P\int_\R w(s)\int_0^P + |\mathcal P_{s,P}\rho(y)-\mathcal P_{s,P}\widetilde\rho(y)| + \dd y\dd s + \le\frac{\|w\|_1}{P}\|\rho-\widetilde\rho\|_1.\notag +\end{align} +Joint continuity is automatic, for example, if the densities are +continuous with bounds $C/(1+y^2)$, or are continuous and compactly +supported. Neither density need have mass one. +\end{lemma} + +\begin{proof} +Partition the $t$ axis into intervals of length $P/N$ on each of +which $Nt+\beta_N$ traverses one period. On each interval replace +the slow argument $t$ of $G$ by an endpoint. Uniform continuity on +the fixed compact support bounds the integrated error by a constant +times its modulus of continuity at $P/N$, with an additional +$O(N^{-1})$ boundary error. The resulting averages are Riemann sums +for the right side of \eqref{eq:fast-average}. This proves the first +assertion, uniformly in the offset. Apply it to the continuous +absolute difference in \eqref{eq:law-diagonal-comparison}. +The integral over one period of that difference is bounded by +\[ + \sum_k\int_0^P |s|^{-1} + |\rho((y+kP)/s)-\widetilde\rho((y+kP)/s)|\dd y + =\|\rho-\widetilde\rho\|_1. +\] +Finally, on a compact set of nonzero $s$, quadratic tail bounds give +a uniform summable bound $C'/(1+k^2)$ on the periodizing summands, +for $0\le y\le P$. Uniform convergence proves joint continuity. +Compactly supported densities are handled in the same way with +only finitely many nonzero summands on that compact set. +\end{proof} + +In the special case $|A(u,a)|$ is independent of $a$, +\eqref{eq:W-definition} has the form $W(s,a)=w(s)$. +If $\widetilde\rho\dd y=J_*(\psi^2\dd a)$, then +$K(s,y)=w(s)|s|^{-1}\widetilde\rho(y/s)$. Thus replacing the +realized jitter law by a regular target law $\rho$ in the +fine-density prescription costs, after taking $N$ sufficiently +large, at most +$\|w\|_1\|\rho-\widetilde\rho\|_1+o(1)$ for the full density. +One sums this estimate over the fixed packet list and folds modulo +one. This is an averaged estimate along the fast variable, not a +pointwise estimate of periodized densities from total variation. +It is applied only after \eqref{eq:rounding-mass-removal}, so the +family being averaged is fixed. + +\begin{lemma}[Realization of jitter laws]\label{lem:law-approx} +For any strictly positive smooth probability density $\rho$ on +$\R$ and any $\eps>0$, there are a smooth real circle function $J$, +an open arc $I'$ with $|\T\setminus I'|<\eps$ and $J'>0$ on $I'$, +and $0\le\psi\le1$ in $C_c^\infty(I')$, such that +\[ + J_*(\psi^2\dd a)=\widetilde\rho(y)\dd y, + \qquad \widetilde\rho\in C_c^\infty(\R), + \qquad \|\rho-\widetilde\rho\|_1<\eps. +\] +Here the cutoff law is a subprobability measure; no renormalization +is made. Also $\int_\T(1-\psi^2)\dd a<\eps$. +\end{lemma} + +\begin{proof} +Let $F(y)=\int_{-\infty}^y\rho(v)\dd v$. Strict positivity makes +$F$ a smooth diffeomorphism from $\R$ to $(0,1)$. Choose +$0<\delta<\min(1/8,\eps/4)$ and a smooth function $\xi$ compactly +supported in $(\delta/2,1-\delta/2)$ and equal to one on +$[\delta,1-\delta]$. In the circle coordinate $00$. Choose $\psi\in C_c^\infty(I')$ with +$0\le\psi\le1$ and $\psi=1$ on $[2\delta,1-2\delta]$. +Change of variables gives +\[ + \widetilde\rho(y)=\rho(y)\psi(F(y))^2\le\rho(y),\qquad + \|\rho-\widetilde\rho\|_1 + =1-\int_\T\psi^2\dd a\le4\delta<\eps. +\] +Its support is compact because that of $\psi$ is compactly inside +$(0,1)$. All the claims follow. +\end{proof} + +\begin{lemma}[Weighted neutral refresh]\label{lem:neutral} +Given a twist, its fixed physical chart, a finite disjoint smooth +packet list with color densities $p_r$, a positive integer $P$, +and a positive tolerance, one can perform a passage with the +following simultaneous properties. Its physical map and inverse +lie in any prescribed smooth neighborhood of the old ones. Its +new good-arc complement is arbitrarily small. Every color vector +is transported within the prescribed $L^2$ tolerance, retaining +constant tags and disjointness. The new packet moduli are +independent of the new spare coordinate. The output densities +$p'_r$ and the densities $p'_{r,h}$ contributed by output indices +$j\equiv h\pmod P$ satisfy +\begin{equation}\label{eq:neutral-densities} + \|p'_r-p_r\|_1<\eps, + \qquad \|p'_{r,h}-p_r/P\|_1<\eps + \quad(0\le h0$ may be prescribed arbitrarily. +The integer $c$ may first be required to exceed any fixed bound, +and the retained supports may use any sufficiently small fixed +enlargements of \eqref{eq:stationary-image}. If colors were +disjoint in their old angular indices alone, their output angular +index sets are still disjoint. The final passage integer $N$ may +exceed any prescribed bound. +\end{lemma} + +\begin{proof} +Fix the old finite data and choose $c$ as in Lemma~\ref{lem:passage}, +also satisfying its prescribed lower bound. Identify all but a +small interval of the $a$ circle with an interval in $\R$. +Choose an open arc $I'$ whose complement is as small as requested, +and $0\le\psi\le1$ compactly supported there and equal to one +except on a set of sufficiently small length. The boundedness of +the finitely many old amplitudes allows us to make both their +$L^2$ cutoff errors and the removed squared masses arbitrarily +small. Fix this cutoff before proceeding. + +For a positive parameter $L$, take a smooth periodic function +$J_L$ equal to $La+\beta$ on $I'$, extending it smoothly across +the complementary interval. Such an extension exists by taking +a smooth cutoff equal to one on the closure of $I'$ in a slightly +larger real coordinate interval. In particular $J_L'=L$ on $I'$. +Put $f_s(a)=W(s,a)\psi(a)^2$, extended by zero in this real +coordinate. The periodization of its pushforward by $sJ_L$ is +\begin{equation}\label{eq:neutral-riemann-sum} + \sum_{k\in\Z}\frac1{|s|L} + f_s\left(\frac{(y+kP)/s-\beta}{L}\right). +\end{equation} +This is $1/P$ times a Riemann sum for $\int f_s(a)\dd a$, of mesh +$P/(|s|L)$, with an arbitrary lattice offset and possibly reversed +orientation. The error is bounded by +\begin{equation}\label{eq:neutral-riemann-error} + \frac{\|\partial_a f_s\|_{L^1(\R)}}{|s|L}. +\end{equation} +Indeed compare each lattice value times the mesh with the integral +on its cell, using the fundamental theorem of calculus and summing +the absolute derivative. Since $s$ ranges over a fixed compact set +away from zero, this bound tends to zero uniformly in $s$ and $y$. +It remains valid for arbitrary fixed dependence of the original +amplitude on $a$. + +Choose $L$ large enough for these uniform errors to be as small as +needed for the finite packet list. Now $J_L$ and all its norms are +fixed. Choose the support cutoffs in Lemma~\ref{lem:passage}, and +then take $N$ sufficiently large. Its fine-density rule, together +with \eqref{eq:neutral-riemann-sum}, approximates each unfolded +residue density by $P^{-1}\int W(s,a)\psi(a)^2\dd a$ in $L^1$. +Folding modulo one gives $p_r/P$, up to the small cutoff loss; +the integer $mc$ in $mV$ has no effect on this folding. The +estimate is uniform in the periodic phase, so the folding offsets +$Nl$ cause no difficulty. Summing the residue estimates gives the +total density estimate, upon initially making each error smaller +by the fixed factor $P$ if necessary. Norm transport, exact +support properties, and smooth proximity follow from the same +passage lemma. All constraints on $N$ are lower bounds imposed +after the physical chart, the amplitudes, $c$, the cutoff, $L$, +and the support neighborhoods have been fixed. +\end{proof} + +\begin{remark}\label{rem:finite-family-passages} +A common passage can handle finitely many packet lists, requiring +disjointness only within each list. Choose $c$ using the union of +their old indices, choose the neutral cutoff and slope to meet all +their finitely many mass and density bounds, and choose $N$ above +all the resulting lower bounds. Support cutoffs are chosen +separately for each list, disjoint within that list. The norm and +density conclusions then hold for every list; no orthogonality +between different lists is asserted. +\end{remark} + +All these operations preserve real data. Include together the +packets $(m,A)$ and $(-m,\overline A)$, attach conjugate tags, and +choose $\chi_{-m}(s,a)=\chi_m(-s,a)$. The exact coefficients obey +$C_{-m,-j}=\overline{C_{m,j}}$, and all other cutoffs are real. +Thus conjugate packets and conjugate colors can be retained at +every passage. diff --git a/preprints/A-smooth-three-torus-diffeomorphism-with-simple-Lebesgue-spectrum-September-23-2026/build/sections/03-signals.tex b/preprints/A-smooth-three-torus-diffeomorphism-with-simple-Lebesgue-spectrum-September-23-2026/build/sections/03-signals.tex new file mode 100644 index 0000000000000000000000000000000000000000..abab379dcaf0a2f383a19c8c376b0a6106b9f9b4 --- /dev/null +++ b/preprints/A-smooth-three-torus-diffeomorphism-with-simple-Lebesgue-spectrum-September-23-2026/build/sections/03-signals.tex @@ -0,0 +1,459 @@ +\section{Small Fourier signals} +\label{sec:signals} + +The purpose of this section is to modulate a packet's spectral density +according to its angular index. The size of the modulation will depend on +a fixed band of \emph{normalized} indices, but not on the scale of those +indices. The integer added to the old speed is allowed to depend on the +actual, finite set of indices. + +Fix numbers $00$ with the following property. Any fixed +$0<\gamma\le\gamma_0$ can be used, independently of $B$. + +Let $S$ be an intermediate map, and let +\[ + F_r=\sum_{m\in\mathcal M_r} A_{r,m}(u,a)\ee(m\theta) +\] +be finite smooth packet data in its good arc. Suppose that the finite +sets $\mathcal M_r\subset\Z\setminus\{0\}$ are mutually disjoint, +$|A_{r,m}(u,a)|$ is independent of $a$, and +$b\le |m|/B\le D_0$. Write $p_{r,m}$ for the spectral density of the +indicated packet. Given any $\eps,\delta_x,\delta_z>0$ and prescribed +positive bounds for the physical $L^2$ transport errors, a passage produces +finite output packet sums $F_{r,m}^+$, approximating the corresponding +input packets in physical $L^2$, whose spectral densities satisfy +\begin{equation} + \sum_{r,m}\left\|p_{r,m}^+ + -p_{r,m}(z)\left[1+2\gamma\1_{\gcd(m,P)=1} + \Rea\{d(m/B)\ee(-Nz)\}\right]\right\|_{L^1(\T)} + <\eps. + \label{eq:signal-conclusion} +\end{equation} +The new scale is $B^+=NcB$, and the new speed is $\sigma_N$ from +Lemma~\ref{lem:passage}. For every point $(u',a')$ in the coefficient +support of a retained output packet of index $j$ descending from +$(r,m)$, there is $(u,a)\in\supp A_{r,m}$ such that +\begin{equation} + \left|\frac{j}{B^+}-\frac mB\right|<\delta_x, + \qquad + \dist_\T\bigl(j\sigma_N(u'),m\phi(u)\bigr)<\delta_z. + \label{eq:signal-support-closeness} +\end{equation} +Its modulus is independent of the new spare coordinate. Distinct old +angular indices have disjoint output angular-index supports. +Conjugate symmetry is preserved when present. + +The passage and its inverse can lie in any prescribed smooth +neighborhood of $S$ and $S^{-1}$ in physical coordinates, and the new +good arc can have arbitrarily small complement. The choices can be made +in the order $c$, a bounded smooth jitter and cutoffs, and finally the +integer $N$. After the preceding choices, every sufficiently large +integer $N$ works. In particular an arbitrary lower bound on $N$ can +be imposed. +\end{lemma} + +\begin{proof} +\emph{A positive law.} +We first construct a positive law without using $B$. We use the +positive-exponent Fourier convention +\[ + \widehat q(v)=\int_\R q(R)\ee(vR)\,\dd R. +\] +Choose $0<\alpha0$ such that +\begin{equation} + \frac{c_0}{1+R^2}\le q_0(R)\le\frac{C_0}{1+R^2} + \qquad(R\in\R). + \label{eq:signal-quadratic-tails} +\end{equation} + +Choose $D>b$ such that $\supp d\subset[-D,D]$, and then an integer +$L>\max(2,D/b)$. Let $P$ be the product of all primes at most $L$. +The strict inequality in this choice will allow a small dilation of +the Fourier arguments. Define +\[ + g(R)=\int_\R d(v)\ee(-vR)\,\dd v, + \qquad + C_P(h)=\sum_{k\in(\Z/P\Z)^\times}\ee(-kh/P) + \quad(0\le h0$ so small that +\[ + \gamma_0 C_*G\le c_0/2, + \qquad 2\gamma_0\|d\|_\infty\le1/2; +\] +a zero factor imposes no restriction. Then, for every permitted +$\gamma$, +\begin{equation} + \frac{q_0(R)}{2P}\le Q_h(R)\le\frac{3q_0(R)}{2P}, + \qquad \int_\R Q_h(R)\,\dd R=\frac1P. + \label{eq:signal-positive-law} +\end{equation} +Thus $Q_h(R)\,\dd R$ defines a probability law on +$\R\times\{0,\ldots,P-1\}$. Finite Fourier inversion gives, for every +$v\in\R$ and $k\in\Z$, +\begin{equation} + \sum_{h=0}^{P-1}\int_\R Q_h(R)\ee(vR+kh/P)\,\dd R + =\widehat q_0(v)\1_{k\equiv0\ (P)} + +\gamma d(v)\1_{\gcd(k,P)=1}. + \label{eq:signal-joint-fourier} +\end{equation} +Indeed the sum over $h$ pairs $C_P(h)$ with the character at $k$, +leaving precisely the indicator that $k$ is a unit. All choices up to +this point are independent of $B$, the map, and its finite packet list. + +\emph{Moments and the finite sieve.} +For fixed $B$ and an integer $c>0$, consider the random variable +\begin{equation} + J_*=(R/B+h/P)/c + \label{eq:signal-ideal-jitter} +\end{equation} +under this law. Its density is +\[ + \rho_{B,c}(y)=cB\sum_{h=0}^{P-1}Q_h\bigl(B(cy-h/P)\bigr). +\] +It is a strictly positive smooth probability density. The bounds +\eqref{eq:signal-positive-law} imply quadratic upper and lower tails, +with constants now allowed to depend on $B,c$. + +We calculate the periodized density of $sJ_*$. Write $x=m/B$ and +$s=mct$. For its positive integer Fourier moments, the exact formula is +\begin{equation} + M_\ell(t)=\frac1P\sum_{h=0}^{P-1}\ee(\ell mt h/P) + \bigl[\widehat q_0(\ell xt) + +\gamma C_P(h)d(\ell xt)\bigr],\qquad \ell\ge1. + \label{eq:signal-exact-moment} +\end{equation} +At $t=1$, \eqref{eq:signal-joint-fourier} applies. Its baseline term +vanishes since $|\ell x|\ge b>\alpha$. If $\ell\ge2$ and +$d(\ell x)\ne0$, then $\ell\le D/b1$. It follows that +\begin{equation} + M_1(1)=\gamma d(x)\1_{\gcd(m,P)=1}, + \qquad M_\ell(1)=0\quad(\ell\ge2). + \label{eq:signal-ideal-moments} +\end{equation} + +For finite $c$ the discrete phases in +\eqref{eq:signal-exact-moment} need not be residue characters. We +estimate their errors before using the sieve. Choose +$\tau\in(1/2,1)$ so that $D/(\tau b)\alpha$. +The perturbation also vanishes identically when +$\ell>D/(\tau b)$. Hence nonintegral discrete phases cannot introduce +an unbounded collection of harmonics. For the fixed remaining finite +set, put $C_1=P^{-1}\sum_h|C_P(h)|$. The elementary bound +$|\ee(v)-\ee(w)|\le2\pi|v-w|$ gives +\begin{equation} + |M_\ell(t)-M_\ell(1)| + \le\gamma C_1\ell + \bigl[D_0\|d'\|_\infty+2\pi|m|\|d\|_\infty\bigr]|t-1|. + \label{eq:signal-finite-c} +\end{equation} +We may thus make the sum of all moment errors arbitrarily small by +choosing $c$ after the actual finite set of old indices. In particular +$c$ is allowed to grow with $\max|m|$; $\gamma$ does not need to decrease. +The same choice can make the ranges $m(c+\phi(\T))$ for distinct old +indices disjoint and bounded away from zero. + +To pass from moments to densities, let +\[ +\mathcal P_s\rho(y) + =\sum_{k\in\Z}\frac1{|s|}\rho((y+k)/s),\qquad y\in\T. +\] +Once $c$ has been chosen, all the relevant $s$ ranges are fixed compact +sets away from zero. No bound on these sets uniform over future +passages is needed. +For $s$ in a fixed compact subset of $\R\setminus\{0\}$, the quadratic +upper tail of $\rho_{B,c}$ makes this a jointly continuous function of +$(s,y)$, with uniformly summable tails. Its Fourier coefficients in +the positive-exponent convention are the moments just computed. +The compact bandwidth shows that its Fourier series is a finite sum +on the relevant branches. Fourier uniqueness and +\eqref{eq:signal-ideal-moments}--\eqref{eq:signal-finite-c} therefore +give, uniformly on those branches and in $y$, +\begin{equation} + \mathcal P_s\rho_{B,c}(y) + =1+2\gamma\1_{\gcd(m,P)=1}\Rea\{d(m/B)\ee(-y)\} + +o_{c\to\infty}(1). + \label{eq:signal-periodic-density} +\end{equation} +The error here is for the fixed old list and scale $B$. +The negative exponent follows directly from our convention: +$\int\mathcal P_s\rho(y)\ee(\ell y)\,\dd y=M_\ell$. + +\emph{Realization and support.} +We now realize this law by a passage. Lemma~\ref{lem:law-approx} +provides a bounded smooth circle function $J$, strictly increasing on +a good arc, and a smooth cutoff $\psi$ compactly supported in that arc, +such that the subprobability law +\[ + \widetilde\rho=J_*(\psi(a)^2\,\dd a) +\] +is arbitrarily close to $\rho_{B,c}$ in total variation. Its missing +mass is included in this error. The good arc has as small a complement +as requested. Since $|A_{r,m}|$ is independent of $a$, its stationary +weight is a function of $s$ alone: +\[ + w_{r,m}(s)=\frac{|A_{r,m}(u,a)|^2}{|m\phi'(u)|}, + \qquad m(c+\phi(u))=s, +\] +extended by zero. For such a nonnegative compactly supported weight, +Lemma~\ref{lem:fast-average} and contraction of total variation under +pushforward and periodization give +\begin{align} + &\lim_{N\to\infty}\int_\R w_{r,m}(s) + |\mathcal P_s\widetilde\rho(Ns) + -\mathcal P_s\rho_{B,c}(Ns)|\,\dd s \notag\\ + &\qquad=\int_\R w_{r,m}(s)\int_\T + |\mathcal P_s\widetilde\rho(y) + -\mathcal P_s\rho_{B,c}(y)|\,\dd y\,\dd s \notag\\ + &\qquad\le\|w_{r,m}\|_1\|\widetilde\rho-\rho_{B,c}\|_{L^1(\R)}. + \label{eq:signal-law-comparison} +\end{align} +Here densities of subprobabilities are used in the $L^1$ norm. +The realized law has a continuous compactly supported density; the +ideal law has the uniform tail bound established above. These facts +supply the continuity and domination required for fast averaging. +The independence of the weight from $a$ is essential in this step. + +Lemma~\ref{lem:passage} identifies the output density, up to an +arbitrarily small $L^1$ error, as the folding to $\T$ of +$w_{r,m}(s)\mathcal P_s\widetilde\rho(Ns)$. +In using that lemma the rounding correction is removed at the +coefficient-mass level: its replacement of $w_{r,m}(s)$ by +$w_{r,m}(s-m\Delta_N)$ costs $o(1)$ in the summed masses. It does +not translate the fast argument $Ns$. Alternatively, +\eqref{eq:signal-finite-c} also holds uniformly with $t=t_N$. +Since $cm$ and $N$ are integers, +\[ + p_{r,m}(z)=\sum_{k\in\Z}w_{r,m}(z+k), + \qquad \ee(-N(z+k))=\ee(-Nz). +\] +Now \eqref{eq:signal-periodic-density}, +\eqref{eq:signal-law-comparison}, and the passage density formula +prove \eqref{eq:signal-conclusion}. Choose $c$ first to make its +error sufficiently small, then the law approximation, and then $N$. +The finite packet list allows all errors to be summed. + +For completeness we give the simultaneous support estimates, which +will also apply to neutral passages. Put +$M=\max_{r,m}|m|$, with $M=0$ for an empty packet list. +By the support cutoff in +Lemma~\ref{lem:passage}, at each retained point of an output packet at +index $j$ there is a point $u$ in its old support for which +\begin{equation} + j/N=m(c+\phi(u)+\Delta_N)+\eta, + \qquad |\eta|\le\eta_0, + \label{eq:signal-retained-support} +\end{equation} +where the image enlargement $\eta_0$ can be as small as desired. +If $x'=j/(NcB)$ and $z'=j/N+(j/N)J(a)/N$ is the new temporal +frequency, then +\begin{align} + |x'-m/B| + &\le\frac{D_0(\|\phi\|_\infty+1)}c+\frac{\eta_0}{cB}, + \label{eq:signal-index-drift}\\ + \dist_\T(z',m\phi(u)) + &\le\frac MN+\eta_0 + +\frac{[M(c+\|\phi\|_\infty+1)+\eta_0]\|J\|_\infty}{N}. + \label{eq:signal-temporal-drift} +\end{align} +Choose $c$ for the first bound and index separation, then $\eta_0$ +and the bounded $J$, and finally $N$ for the second bound. A large +bound on $J$ only increases the eventual requirement on $N$. +These choices also give physical smooth closeness of the passage and +its inverse and the required $L^2$ approximations, by +Lemma~\ref{lem:passage}. + +Every new coefficient descending from old index $m$ has the form +$\ee(ma_{\rm new})$ times a scalar function of the new speed +coordinate. Because different old indices have disjoint retained +new index ranges, their spare-coordinate phases are never added at +one new index. Thus the resulting packet moduli are independent of +the new spare coordinate. All cutoffs can be chosen in conjugate +pairs, so conjugation is retained as well. +\end{proof} + +We next remove the coprimality restriction by placing a neutral passage +before transmission. The output residue classes of the neutral +passage, rather than the input residue classes, are the ones tested in +the transmission. + +\begin{lemma}[Refresh and transmission] +\label{lem:refresh-transmit} +Fix $b,D_0,d,P,\gamma$ as in Lemma~\ref{lem:signal}, and put +\begin{equation} + \lambda=\gamma\frac{|(\Z/P\Z)^\times|}{P}>0. + \label{eq:signal-lambda} +\end{equation} +Suppose finite colored packet data have mutually disjoint angular-index +sets and moduli independent of the spare coordinate. Suppose their +normalized indices lie in a fixed compact subset of +$\{x:b<|x|0$ is arbitrary and $N$ is the second passage integer. +The physical map and inverse, the physical packet vectors, and the +normalized-index and temporal-frequency displacements can all be +controlled with arbitrary prescribed positive error bounds. Output +packet moduli remain independent of the spare coordinate, and colors +remain disjoint in angular indices. The final $N$ can exceed any +prescribed bound. Conjugate-symmetric data can be treated +conjugate-symmetrically. + +If different colors start in separated compact normalized-index +sets, any fixed sufficiently small enlargements of those sets can +contain their outputs. The same conclusion holds through any fixed +finite number of such pairs, with positive margins around the band +$b<|x|0$ and any neighborhood of the physical map +and its inverse in the smooth topology, there are a finite sequence of +passages, a final intermediate map $S'$, final packets $F'_r$, and a +smooth function $\Psi:\T\to\C$, such that +\begin{align} + \sum_r\|F'_r-F_r\|_2&<\eps, + \label{eq:prediction-transport}\\ + \int_\T\sqrt{p'(z)}\,\dd z + &\ge \int_\T\sqrt{p(z)}\,\dd z-\tau, + \qquad p'=\sum_r p'_r, + \label{eq:prediction-hellinger}\\ + \left\|\sum_r b_rF'_r- + \Psi(U_{S'})\sum_r F'_r\right\|_2&<\eps. + \label{eq:prediction-error} +\end{align} +Here the differences in \eqref{eq:prediction-transport} are measured +in the original physical $L^2$ space. All the physical maps and their +inverses can be kept in the prescribed neighborhood. The final good +arc $I'$ satisfies $|\T\setminus I'|<\kappa$. +The output colors again have disjoint nonzero angular-index sets, +moduli independent of the spare coordinate, and conjugation symmetry. +The multiplier can be chosen with +$\Psi(-z)=\overline{\Psi(z)}$. +\end{proposition} + +We first record the precise averaging assertion needed in the proof. +In the following lemma, a Laurent polynomial in $w_1,\ldots,w_n$ means +a finite linear combination of the monomials +$w^\nu=\prod_i w_i^{\nu_i}$, with $\nu\in\Z^n$. + +\begin{lemma}[Finite phase tests] +\label{lem:phase-tests} +Fix finitely many expressions +\[ + P_j(z,w)=\sum_{\nu\in\mathcal E_j}c_{j,\nu}(z)w^\nu, + \qquad c_{j,\nu}\in L^1(\T), +\] +where each $\mathcal E_j$ is finite. For every $\eta>0$, successively +large positive integers $L_1,\ldots,L_n$ can be chosen so that, +simultaneously for all $j$, +\begin{equation} + \left|\int_\T P_j\bigl(z,\ee(L_1z),\ldots,\ee(L_nz)\bigr)\,\dd z + -\int_\T\int_{\T^n}P_j\bigl(z,\ee(y_1),\ldots,\ee(y_n)\bigr) + \,\dd y\,\dd z\right|<\eta. + \label{eq:phase-tests} +\end{equation} +At each selection, $L_i$ may additionally be required to exceed any +finite lower bound determined by the preceding choices and the +then-current construction data. +\end{lemma} + +\begin{proof} +Let $D$ bound the absolute values of all the exponents in all the +$\mathcal E_j$. The integral of a nonconstant monomial on the product +torus is zero. On the one-dimensional diagonal its contribution is a +Fourier coefficient of $c_{j,\nu}$ at +$-\sum_i\nu_iL_i$. +The Fourier coefficients of an $L^1$ function tend to zero: this +follows by approximation in $L^1$ by trigonometric polynomials, whose +sufficiently high Fourier coefficients vanish. Since the coefficient +list is finite, choose $R$ so that the sum of the bounds for all these +nonconstant contributions is less than $\eta$ whenever their +frequencies have absolute value greater than $R$. +Choose successively +\[ + L_j>R+D\sum_{iR. +\] +This proves the assertion. No independence assertion about the +distribution of the entire diagonal is made or needed. +\end{proof} + +\begin{proof}[Proof of Proposition~\ref{prop:prediction}] +Zero colors may be omitted. If all colors vanish, a neutral passage +with zero output packets and $\Psi=0$ proves the assertion, including +the requested new good arc. Hence assume that a nonzero color is +present. +Dividing all tags by $\max(1,\max_r|b_r|)$ and reducing the requested +prediction tolerance reduces the proof to $|b_r|\le1$. The transport +and square-root tolerances remain independently adjustable. We work +under this normalization until the end. + +\paragraph{The data fixed at the beginning of a block.} +For the moment suppose that the current packet mass is at most $2$. +Write its actual densities as $p_r$, put $p=\sum_rp_r$, and on +$\{p>0\}$ set +\[ + t_r=\frac{p_r}{p},\qquad + \beta=\sum_r t_rb_r. +\] +Set $\beta=0$ on $\{p=0\}$. Then $|\beta|\le1$ and +$\beta(-z)=\overline{\beta(z)}$. Fix $0<\delta\le1/100$. +There is a smooth circle function $q$ such that +\begin{equation} + |q|\le2,\qquad q(-z)=\overline{q(z)},\qquad + \int_\T\sqrt p\,|\beta-q|^2\,\dd z<\delta. + \label{eq:block-centering} +\end{equation} +Indeed, convolve $\beta$ with an even nonnegative smooth approximate +identity. The convolutions have absolute value at most $1$, have +the required symmetry, and converge almost everywhere to $\beta$. +Since $\sqrt p\in L^1$, dominated convergence gives +\eqref{eq:block-centering}. In particular this choice does not require +any uniform bound on derivatives of $q$. + +Let $A$ be any fixed bounded function on the temporal circle. In the +iteration, $A$ will be the sum of the estimators from $k$ already +completed blocks. Both $q$ and $A$ are fixed before any frequencies +for the present block are chosen. The densities $p_r$ in every model +below are these actual \emph{pre-block} densities. + +\paragraph{Anchoring, with the number of passages fixed first.} +Choose an integer $c_0$ large enough that the full compact ranges +$m(c_0+\phi(\T))$ for distinct old indices are separated and avoid +zero, as required by Lemma~\ref{lem:passage}. In particular this +separates the compact packet images +\[ + m(c_0+\phi(u)),\qquad + u\in\operatorname{proj}_u\supp A_{r,m}, +\] +for all distinct old indices $m$. Let $K_r$ be the union of the +images belonging to color $r$. +The $K_r$ are disjoint compact sets and +$K_{r^*}=-K_r$. Choose slightly larger compact neighborhoods of them, +still disjoint and off zero, and then disjoint open neighborhoods with +compact closures. We will retain enough room between these two +families of neighborhoods for all the support movements in the block. +There is a smooth compactly supported function on $\R\setminus\{0\}$ +such that +\begin{equation} + d(-x)=\overline{d(x)},\qquad + d(x)=b_r-q(x\bmod1) + \quad\hbox{on a neighborhood of the enlarged }K_r. + \label{eq:block-band-function} +\end{equation} +For example, use disjoint smooth cutoffs equal to one near these sets, +chosen compatibly with reflection, and multiply the $r$th cutoff by +$b_r-q(x\bmod1)$. Fix $00$, it suffices to choose every +$\eta_i0\}$; all their +integrands are multiplied by $p_r$ or $\sqrt p$ and are extended by +zero on its complement. We have +\begin{equation} + \mathbb E H_r=1, + \qquad + \mathbb E(H_rH_s) + =(1+2\lambda^2\Rea(a_r\overline{a_s}))^n, + \qquad \mathbb E R=1. + \label{eq:block-second-moment-identity} +\end{equation} +For real $y$ and integer $n\ge1$, the binomial expansion gives +\[ + |(1+y)^n-1-ny| + \le\sum_{j=2}^n\frac{(n|y|)^j}{j!} + \le\frac{(n|y|)^2}{2}\exp(n|y|). +\] +Apply this with $y=2\lambda^2\Rea(a_r\overline{a_s})$; +then $n|y|\le18\delta$. Since $\sum_rt_r=1$, the sum of the +linear terms in \eqref{eq:block-second-moment-identity} is +$2n\lambda^2|\sum_rt_ra_r|^2$. Therefore +\begin{equation} + \mathbb E(R-1)^2 + \le2\delta|\beta-q|^2+162e^{18\delta}\delta^2. + \label{eq:block-second-moment-bound} +\end{equation} +This bound is uniform in the number of colors and in all the later +geometric parameters. + +For every $x\ge0$, +\begin{equation} + \sqrt x\ge1+\frac{x-1}{2}-(x-1)^2. + \label{eq:block-square-root-bound} +\end{equation} +In fact, on setting $t=\sqrt x$, the difference between the two sides +equals $(t-1)^2(t^2+2t+1/2)$. Multiplying +\eqref{eq:block-square-root-bound} by $\sqrt p$, taking the +independent-phase expectation, and using +\eqref{eq:block-centering} and \eqref{eq:block-second-moment-bound} +shows that the averaged polynomial lower bound is at least +\[ + \int_\T\sqrt p\,\dd z + -2\delta^2-162e^{18\delta}\delta^2\int_\T\sqrt p\,\dd z + \ge \int_\T\sqrt p\,\dd z-300\delta^2. +\] +Here $\int\sqrt p\le\sqrt2$ and $\delta\le1/100$ were used. + +The expression +\[ + \sqrt p\,[1+(R-1)/2-(R-1)^2] +\] +is a Laurent polynomial in the phases with fixed integrable +coefficients. Lemma~\ref{lem:phase-tests} therefore transfers this +lower bound, up to any prescribed error, to the diagonal phases +$Z_i=\ee(L_i z)$. On that diagonal +$pR=\sum_r h_r^{(n)}$. +There is no need to apply the lemma to $\sqrt R$ itself. +Finally, for nonnegative densities $g,h$ on the unit circle, +\begin{equation} + \left|\int_\T\sqrt g-\int_\T\sqrt h\right| + \le\left(\int_\T|g-h|\right)^{1/2}, + \label{eq:block-density-hellinger-error} +\end{equation} +by $(\sqrt g-\sqrt h)^2\le|g-h|$ and Cauchy--Schwarz. +Thus the actual output density $p^{\rm out}=\sum_rg_r^{(n)}$ +satisfies, for any chosen $\nu>0$ and sufficiently accurate +construction, +\begin{equation} + \int_\T\sqrt{p^{\rm out}}\,\dd z + \ge\int_\T\sqrt p\,\dd z-300\delta^2-\nu. + \label{eq:block-hellinger-step} +\end{equation} + +\paragraph{Prediction and a previous estimator.} +With the same formal phases, put +\[ + D=q+\frac1{n\lambda}\sum_{i=1}^n Z_i. +\] +For a fixed color $r$ and fixed $z$, $H_r$ is a positive probability +density on the product phase torus, and under this density the phases +remain independent. Direct integration of one factor gives +\[ + \mathbb E_{H_r}Z_i=\lambda a_r, + \qquad + \mathbb E_{H_r}|Z_i-\lambda a_r|^2 + =1-\lambda^2|a_r|^2. +\] +Consequently +\begin{equation} + \mathbb E_{H_r}D=b_r, + \qquad + \mathbb E_{H_r}|D-b_r|^2 + =\frac{1-\lambda^2|a_r|^2}{n\lambda^2} + \le\frac2\delta. + \label{eq:block-estimator-moments} +\end{equation} +For the arbitrary fixed bounded previous estimator $A$, this means +that the cross term against $A-kb_r$ vanishes \emph{at each $z$}: +\[ + \mathbb E_{H_r}|A+D-(k+1)b_r|^2 + =|A-kb_r|^2+\mathbb E_{H_r}|D-b_r|^2. +\] +After multiplying by $p_r$, summing, and integrating, the increase +in this mean square is at most $4/\delta$. + +The expression +\[ + \sum_r p_rH_r|A+q+(n\lambda)^{-1}\textstyle\sum_iZ_i-(k+1)b_r|^2 +\] +is another finite Laurent polynomial with the fixed integrable +coefficients in \eqref{eq:block-fixed-data}. Apply +Lemma~\ref{lem:phase-tests} to it simultaneously with the preceding +square-root lower-bound test. Write +\begin{equation} + D(z)=q(z)+\frac1{n\lambda}\sum_{i=1}^n\ee(L_i z) + \label{eq:block-diagonal-estimator} +\end{equation} +for the resulting actual multiplier. Its norm is bounded by +$2+1/\lambda$, independently of the choices of $L_i$. In passing +from model to actual color densities, the error in the prediction +integral is at most $W E_n$, where +\begin{equation} + W=(\|A\|_\infty+2+1/\lambda+k+1)^2. + \label{eq:block-estimator-error-weight} +\end{equation} +This finite number is known before choosing the anchor accuracy or +any transmission frequencies. For example, choose +\[ + e_0<\min\{(\nu/4)^2,\nu/(4W)\}, +\] +and make the two phase-test errors less than $\nu/2$. +Together with \eqref{eq:block-error-propagation} this proves +\begin{equation} + \sum_r\int_\T p_r^{\rm out}|A+D-(k+1)b_r|^2\,\dd z + \le\sum_r\int_\T p_r|A-kb_r|^2\,\dd z + +\frac4\delta+\nu, + \label{eq:block-prediction-step} +\end{equation} +simultaneously with \eqref{eq:block-hellinger-step}. +The same $e_0$ and support accuracies can be imposed before the +anchor. Thus a small strength entails a more demanding finite +accuracy requirement, without a circular parameter choice. + +\paragraph{Iteration of blocks.} +All color vectors can be transported with an arbitrarily small sum +of physical norm errors at every passage. Reserve a total such +budget smaller than +\[ + \min\{\eps,\sqrt2-\sqrt{3/2}\}. +\] +Since the sum of the color errors bounds the error of their sum, +all intermediate packet sums then have squared norm less than $2$. +This supplies the mass bound used above at every step, with room to +spare. Smooth map and inverse errors can likewise be apportioned +among the finitely many blocks and, after $n$ is chosen in a block, +among its $2n+1$ passages. Choosing a basic neighborhood specified +by finitely many smooth norms inside the requested neighborhood +makes this allocation explicit. Every passage may also make the +new excluded arc arbitrarily small; apply this in particular at the +last passage. + +Set $\nu\le\delta^2$ in each block. Start with $A_0=0$, and let +$A_{k+1}=A_k+D_{k+1}$ after block $k+1$. At the beginning of each +new block choose a \emph{new} $q$ from that block's actual input +densities. That $q$ is then held fixed throughout its block. +The band function, sieve modulus, strength, and length may all +change from block to block. The estimates depend only on +$|b_r|\le1$, $|q|\le2$, the mass bound $2$, and +\eqref{eq:block-budget}; they require no uniform lower bound for +the strengths and no uniform derivative bound for the successive +centering functions. + +After $K$ blocks, telescoping +\eqref{eq:block-hellinger-step} and +\eqref{eq:block-prediction-step} gives +\begin{align} + \int_\T\sqrt{p^{(K)}}\,\dd z + &\ge\int_\T\sqrt{p^{(0)}}\,\dd z-400K\delta^2, + \label{eq:block-total-hellinger}\\ + \sum_r\int_\T p_r^{(K)} + |A_K/K-b_r|^2\,\dd z + &\le\frac5{K\delta}. + \label{eq:block-total-prediction} +\end{align} +For the second estimate the initial error is zero, and +$4/\delta+\delta^2\le5/\delta$. +Given the desired prediction tolerance $\eps$, first choose +$\delta>0$ sufficiently small and then put +\[ + K=\left\lceil\frac{10}{\eps^2\delta}\right\rceil. +\] +We can require simultaneously $\delta\le1/100$ and +\[ + 400\left(\frac{10\delta}{\eps^2}+\delta^2\right)<\tau. +\] +Then $K\delta$ is large enough for +\eqref{eq:block-total-prediction}, and $K\delta^2$ is small enough +for \eqref{eq:block-total-hellinger}. These choices precede all +blocks and do not depend on their future strengths or lengths. + +Set $\Psi=A_K/K$. Disjoint angular-index supports make all +different color cyclic subspaces orthogonal, so the spectral +density formula gives exactly +\[ + \left\|\sum_r b_rF_r^{(K)}- + \Psi(U_{S^{(K)}})\sum_rF_r^{(K)}\right\|_2^2 + =\sum_r\int_\T p_r^{(K)}|b_r-\Psi|^2\,\dd z. +\] +This proves the required norm estimate. Every $D$ in +\eqref{eq:block-diagonal-estimator} is smooth and has conjugation +symmetry, hence so does $\Psi$. Restoring the original scale of +the tags proves the proposition in its stated form. +\end{proof} diff --git a/preprints/A-smooth-three-torus-diffeomorphism-with-simple-Lebesgue-spectrum-September-23-2026/build/sections/05-completion.tex b/preprints/A-smooth-three-torus-diffeomorphism-with-simple-Lebesgue-spectrum-September-23-2026/build/sections/05-completion.tex new file mode 100644 index 0000000000000000000000000000000000000000..0346647a77183e2afc54bba17e8167a6f1c3f2ec --- /dev/null +++ b/preprints/A-smooth-three-torus-diffeomorphism-with-simple-Lebesgue-spectrum-September-23-2026/build/sections/05-completion.tex @@ -0,0 +1,440 @@ +\section{Preparation, whitening, and the smooth limit} +\label{sec:completion} + +The finite prediction result encodes a target observable in a packet +sum while retaining nearly flat total spectral density. We prepare +arbitrary observables for that result, whiten the resulting vector to +exactly flat density, and diagonalize the stages to obtain a smooth map +whose one orbit is complete. + +All vector comparisons in this section take place in the fixed physical +space $L^2(\T^3)$, even when intermediate coordinates change. A neighborhood +of a diffeomorphism will mean a neighborhood of the pair consisting of the +map and its inverse in the $C^\infty$ topology. For a unitary $U$, write +\[ + \calC(U,f)=\overline{\operatorname{span}\{U^nf:n\in\Z\}}. +\] +We use the packet spectral calculus from Section~\ref{sec:passages}: +for a measurable function $\alpha$ of the additive spectral variable $z$, the +coefficient of $\alpha(U_S)F$ at angular index $m$ is +$\alpha(m\phi(u)\bmod1)A_m(u,a)$. The notation is used only when the resulting +vector has finite $L^2$ norm. + +\begin{lemma}[A flat vector]\label{lem:flat-vector} +For every intermediate twist and every good arc $I$, there is a real +centered vector $v$ of spectral density $1$, using two nonzero angular +indices and coefficients supported in a compact subinterval of $I$. +\end{lemma} +\begin{proof} +Choose a positive integer $m$ and $u_00$. In every prescribed neighborhood of $S$ there is an +admissible pair $(S',f')$, with good arc $I'$, and a finite Laurent +polynomial $P$ such that +\begin{equation}\label{eq:stage-conclusion} + \|f'-f\|_2<\tau,\qquad + |\T\setminus I'|<\rho,\qquad + \|h-P(U_{S'})f'\|_2 + <|\T\setminus I|^{1/2}+\eta. +\end{equation} +\end{proposition} + +\begin{proof} +We first prepare smooth packets for $f$ and $h$, then put constant tags +on disjoint packets, and finally use Proposition~\ref{prop:prediction}. +Every accuracy appearing below may be made arbitrarily small after the +finite data on which it depends have been fixed. + +At a fixed map we use the density estimate +\eqref{eq:density-norm-bound}, which also applies to the nonsmooth +$L^2$ coefficients of an admissible vector. Across a passage, density +preservation instead comes from Lemma~\ref{lem:neutral}. + +\paragraph{Converting the angular averages.} +In the current coordinates write +\[ + h=h_\perp+g(u,a),\qquad + g_0(a)=\int_\T g(u,a)\dd u,\qquad r(u,a)=g(u,a)-g_0(a). +\] +Here $h_\perp$ has zero $\theta$-average, $r$ has zero $u$-average +for each $a$, and $\int g_0=0$. Fourier truncation and smoothing on the +good arc give a real smooth packet sum $F$ arbitrarily close to $f$. +By \eqref{eq:density-norm-bound}, its density is arbitrarily close to $1$ +in $L^1$. + +Let $\beta=|\T\setminus I|$. Fiberwise Parseval gives +\[ + \int_\T|h_\perp(\theta,u,a)|^2\dd\theta + \le \int_\T|h(\theta,u,a)|^2\dd\theta\le1. +\] +Thus cutting $h_\perp$ off smoothly inside $I$, and then retaining +finitely many of its nonzero angular modes, loses at most +$\sqrt\beta+\eps$ in norm, for arbitrary $\eps>0$. For example, the +cutoff may be chosen with +$\int_\T(1-\chi)^2\dd u<\beta+\eps^2$. +Denote the resulting packet sum by $H_\perp$. +The simultaneous-list version of Lemma~\ref{lem:neutral}, stated in +Remark~\ref{rem:finite-family-passages}, transports the $F$ and +$H_\perp$ lists through a common passage even when their supports +overlap. We require density preservation only for the $F$ list; no +orthogonality between the two lists is asserted. + +The next passage turns variation in the old $u$ coordinate into +nonzero angular modes; the passage after it does the same for the old +$a$ coordinate. There are two remaining terms, and neither will be +discarded. Before +the first passage integer $N_1$ is chosen, fix a finite nonzero +$u$-Fourier truncation $r_K$ of $r$. Choose the new good arc in the +$a$ variable and its smooth cutoff $\chi_1(a)$ so that replacing $r_K$ +by $\chi_1r_K$ costs arbitrarily little. These are fixed functions of +the old slow variables. The cutoff can simultaneously lose arbitrarily +little mass from the finitely many existing packets. Choose the broad +neutral jitter and then $N_1$, as in Lemma~\ref{lem:neutral}. In the new +coordinates +\[ + (u,a,b),\qquad b=\theta-H_1(u), +\] +the function $\chi_1(a)r_K(u,a)$ is already a finite sum of nonzero +angular modes supported in the new good arc. The former packets are +transported with arbitrarily small norm error, and the density of the +$F$ packets changes by arbitrarily little in $L^1$. Choose $N_1$ also so that every retained transported old index lies +outside $[-K,K]$: its quotient by $N_1$ stays in a fixed compact set +away from zero. This keeps the added modes of $r_K$ disjoint in angular +index from the transported test packets. The function $g_0(a)$ has been left entirely untouched. + +Now fix a finite nonzero $a$-Fourier truncation $(g_0)_L$ of $g_0$. +It has no constant term because $\int g_0=0$. Apply a second neutral +passage to all existing packets. Its new coordinates are +\[ + (a,b,d),\qquad d=u-H_2(a). +\] +Choose its good arc and cutoff $\chi_2(b)$ before its final integer +$N_2$. The term $\chi_2(b)(g_0)_L(a)$ is now a smooth packet sum in +nonzero angular modes. Its cutoff loses arbitrarily little norm and +cannot introduce an angular constant. Choose $N_2$ also so that the transported indices lie outside +$[-L,L]$, keeping the newly added modes disjoint from them. The existing packets are +transported with arbitrarily small norm and density errors. + +Relabel the coordinates of the resulting twist as $(\theta,u,a)$. +For every $\eps>0$ the above choices therefore give real smooth packet +sums $F,H$ such that +\begin{equation}\label{eq:prepared-pair} + \|F-f\|_2<\eps,\qquad + \|H-h\|_2<\sqrt\beta+\eps,\qquad + \|p_F-1\|_1<\eps. +\end{equation} +The functions that needed new cutoffs were fixed before the relevant +passage integer was chosen. In particular, this argument has not +assumed uniform control of coefficients arising only after that choice. + +\paragraph{Constant ratios on rectangles.} +Enlarge the finite angular index list so that it contains the modes of +both $F$ and $H$, and write their coefficients as $F_m,H_m$. For almost +every sufficiently small real $t$, +\begin{equation}\label{eq:ratio-null-set} + \int_{\{F_m+tH_m=0\}}|H_m|^2\dd u\,\dd a=0 + \quad\hbox{for every }m. +\end{equation} +To see this, at a point where $H_m\ne0$ at most one real $t$ solves +$F_m+tH_m=0$; Fubini applied to $|H_m|^2\dd u\,\dd a\,\dd t$ proves +the assertion. There are only finitely many indices. Replace $F$ by +$F+tH$ with such a sufficiently small real $t$. Its norm and density +changes are arbitrarily small by \eqref{eq:density-norm-bound}, and +reality is preserved. + +Define $R_m=H_m/F_m$ where $F_m\ne0$, and zero otherwise. In view of +\eqref{eq:ratio-null-set}, $H_m=R_mF_m$ almost everywhere. With +$\dd\nu_m=|F_m|^2\dd u\,\dd a$, one has +$\int|R_m|^2\dd\nu_m=\|H_m\|_2^2$. Truncating $R_m$ to a bounded +function therefore changes $R_mF_m$ arbitrarily little in $L^2$. +Finite rectangular step functions are dense in $L^2(\nu_m)$: continuous +functions are dense for this finite measure, and uniform continuity +approximates a continuous function by a fine rectangular grid. Grid +boundaries have $\nu_m$-measure zero, since $\nu_m$ is absolutely +continuous. + +Approximate each bounded ratio by such a step function, and replace the +indicators of the finitely many grid rectangles by smooth interior +cutoffs. Trim enough to obtain positive gaps between distinct retained +rectangles, but lose arbitrarily little in both the $F$ norm and the +tagged norm. This is possible after the finitely many step values are +known. Where a ratio has been truncated, a zero step value is allowed; +we do not need to remove the corresponding $F$ mass. Performing the +construction at positive indices and reflecting it at negative indices +gives packets $F_r$ and constants $b_r$ with conjugate symmetry, such +that +\[ + F^{\mathrm{tag}}=\sum_r F_r\quad\hbox{approximates }f, + \qquad + H^{\mathrm{tag}}=\sum_r b_rF_r\quad\hbox{approximates }h +\] +with the errors in \eqref{eq:prepared-pair}, up to arbitrarily small +additions. Different rectangles in $(m,u,a)$ are disjoint with gaps; +each is its own color. The density of $F^{\mathrm{tag}}$ remains +arbitrarily close to $1$ in $L^1$. The number +\[ + M=\max\bigl(1,\max_r|b_r|\bigr) +\] +is finite. All subsequent approximation errors for tagged vectors are +chosen after $M$ is known. + +\paragraph{Separation by angular index alone.} +Two further neutral passages supply exactly the hypotheses of +Proposition~\ref{prop:prediction}. Here are the support details. +Write the single old angular index of rectangle $r$ as $m_r$, and its +compact coefficient support as $K_r$. Choose $c$ so that different +$m_r$ have separated ranges, and form +\[ + E_r=\{(m_r(c+\phi(u)),a):(u,a)\in K_r\}. +\] +The compact sets $E_r$ are pairwise disjoint: for a fixed $m_r$ the map +$(u,a)\mapsto(m_r(c+\phi(u)),a)$ is injective, and different indices +were separated by $c$. Choose smooth cutoffs $\xi_r(s,a)$ supported +in disjoint neighborhoods of these sets and equal to one near them. +On the first passage retain the exact coefficients +\begin{equation}\label{eq:first-separation} + C_{j,r}(a)=\chi(a)\xi_r(j/N,a) + \int_\T A_r(u,a)\ee(m_rH(u)-ju)\dd u. +\end{equation} +Here $\chi$ is the new good-arc cutoff; its loss was made small before +choosing the broad neutral jitter and $N$. Lemma~\ref{lem:passage} +shows that the support restrictions delete arbitrarily little norm as +$N$ increases. There are finitely many retained $j$, all nonzero. In +coordinates $(u,a,b)$ the resulting packets are +\[ + C_{j,r}(a)\ee(ju)\ee(m_rb). +\] +Their moduli are independent of the new spare coordinate $b$. +Different old indices cannot share $j$; at a shared $j$, distinct colors +have disjoint compact $a$ supports with positive gaps. + +Fix this finite output before making the second passage. Choose its +$c_2$ to separate the ranges belonging to distinct $j$. For each fixed +$j$, strict monotonicity of the current speed $\sigma$ on its good arc +makes the compact sets +\[ + Q_{j,r}=\{j(c_2+\sigma(a)):a\in\supp C_{j,r}\} +\] +disjoint as $r$ varies. Thus all nonempty $Q_{j,r}$ have disjoint open +neighborhoods. Restrict the second-passage coefficient at new index +$k$ by a smooth cutoff of $k/N_2$ supported in its designated +neighborhood and equal to one near $Q_{j,r}$. Taking $N_2$ last makes +the deleted norm arbitrarily small, while the retained finite +$k$-index sets are \emph{exactly} disjoint for distinct pairs $(j,r)$. +An individual output term has the form +\[ + B_{k,j,r}(b)\ee(ka)\ee(jd),\qquad d=u-H_2(a), +\] +so its modulus is independent of the final spare coordinate $d$. +No interference can spoil this property, because different pairs have +disjoint final index sets. + +Every integral and cutoff used here commutes with the constant tag +$b_r$. Consequently, if the sum of the individual color transport +errors is less than $\varepsilon/(1+M)$, both the untagged and the +tagged physical errors are less than $\varepsilon$. Neutral density +preservation and \eqref{eq:density-norm-bound} keep the total density +arbitrarily close to $1$. At this point the packet colors are disjoint +in angular indices, their moduli are independent of the spare +coordinate, and their mass can be required to be at most $3/2$. + +\paragraph{Prediction and whitening.} +Apply Proposition~\ref{prop:prediction} to the tags $b_r/M$, with +prediction tolerance divided by $M$. Its physical transport errors +are likewise chosen with the factor $M$ included. Write $G$ for the +final real packet sum, $p$ for its density, and $D(U)G$ for the +resulting approximation to the tagged vector, with the factor $M$ +restored. Here $U$ is the Koopman operator of the final intermediate +map. We can ensure +\begin{equation}\label{eq:prediction-ready} + \|G-f\|_2\longrightarrow0, + \qquad + \|h-D(U)G\|_2<\sqrt\beta+\eta/2, + \qquad + \int_\T\sqrt p\,\dd z\longrightarrow1 +\end{equation} +as the preparation and prediction tolerances tend to zero. For the +last assertion, if the input density is $p_0$, then +\[ + \left|\int\sqrt{p_0}-1\right| + \le\|\sqrt{p_0}-1\|_2 + \le\|p_0-1\|_1^{1/2}, +\] +and Proposition~\ref{prop:prediction} gives an arbitrarily small loss +in $\int\sqrt p$. Its upper limit is at most one by +$\int\sqrt p\le\|G\|_2$ and $\|G-f\|_2\to0$. + +Reality implies $p(-z)=p(z)$ almost everywhere. Set $E=\{p=0\}$, +choose the flat real vector $v$ of Lemma~\ref{lem:flat-vector} for this +last map, and define +\begin{equation}\label{eq:whitening} + f'=\left(\frac{\1_{E^c}}{\sqrt p}\right)(U)G+\1_E(U)v. +\end{equation} +The first term is well defined in $L^2$, with squared norm $|E^c|$. +The terms have disjoint spectral supports and densities $\1_{E^c}$ +and $\1_E$, respectively. Indeed, $\1_E(U)G=0$, even if $v$ uses some +of the same angular modes. The total density of $f'$ is exactly one. +Both multipliers preserve reality, nonzero angular indices, and compact +support in the good arc. Hence the resulting pair is admissible. +Directly, including the contribution from $E$, +\begin{equation}\label{eq:whitening-distance} + \|f'-G\|_2^2 + =\int_\T(\sqrt p-1)^2\dd z + =\|G\|_2^2+1-2\int_\T\sqrt p\,\dd z. +\end{equation} +This tends to zero by \eqref{eq:prediction-ready}, and therefore +$\|f'-f\|_2$ can be made less than the independently specified $\tau$. + +Whitening also preserves the prediction. In the measurable spectral +calculus, +\[ + G=(\sqrt p)(U)f',\qquad + D(U)G=(D\sqrt p)(U)f'. +\] +Since $f'$ has density one and $D(U)G\in L^2$, the function +$D\sqrt p$ belongs to $L^2(\T,\dd z)$. Trigonometric polynomials are +dense in this space, so $D(U)G$ belongs to $\calC(U,f')$ and is +approximated by a finite Laurent polynomial in $U$ applied to $f'$. +Choose that approximation with error less than $\eta/2$. This proves +the last inequality in \eqref{eq:stage-conclusion}; it asserts an +approximation to $h$, not exact membership of $h$ at this finite stage. + +Finally, each neutral passage and Proposition~\ref{prop:prediction} +allows an arbitrarily small physical perturbation of both map and +inverse. There are finitely many such operations here, so they can +all be kept in the prescribed neighborhood. The last new good arc +may have complement less than $\rho$. All choices of small physical +errors are made after the finite tags, cutoffs, and coordinate changes +that affect their bounds have been fixed. Thus neither large tags nor +large derivatives of an earlier coordinate change obstruct the +arbitrary closeness claimed in the proposition. +\end{proof} + +\begin{proof}[Proof of Theorem~\ref{thm:main}] +Choose a smooth periodic speed $\phi_0$ which agrees with an increasing +affine function on a nonempty closed interval inside $(0,1)$, and let +$I_0$ be an open subinterval where $\phi_0'>0$. Start with the twist +$S_0$ and its flat vector $f_0$ from Lemma~\ref{lem:flat-vector}. +Choose a countable family of real smooth centered functions, bounded +by one, whose real linear span is dense in real $L^2_0(\T^3)$; the +physical torus sines and cosines will do. List them as $(h_n)_{n\ge1}$ +so that each occurs infinitely often. + +We record the continuity needed to retain approximations. If +volume-preserving maps $T_i$ and their inverses converge uniformly to +$T$ and $T^{-1}$, then for every fixed $k\in\Z$, +\begin{equation}\label{eq:strong-koopman} + U_{T_i}^kg\longrightarrow U_T^kg\quad\hbox{in }L^2 + \quad(g\in L^2). +\end{equation} +Uniform convergence of the corresponding fixed iterates follows by +composition on the compact torus. For continuous $g$ this proves +uniform convergence of the compositions; for general $g$, approximation +by continuous functions and unitarity prove \eqref{eq:strong-koopman}. +Consequently, for every fixed Laurent polynomial $P$ the map +\[ + (T,g)\longmapsto P(U_T)g +\] +is continuous into physical $L^2$ when maps and inverses have the +$C^\infty$ topology and vectors have the $L^2$ topology. Explicitly, +if $P(w)=\sum_k c_kw^k$, its difference is bounded by +\[ + \sum_k|c_k|\|g_i-g\|_2+ + \sum_k|c_k|\|U_{T_i}^kg-U_T^kg\|_2. +\] + +Inductively apply Proposition~\ref{prop:stage} with $h=h_n$ and +$\eta=2^{-n}$, obtaining $(S_n,f_n)$, a good arc $I_n$, and a finite +Laurent polynomial $P_n$, such that +\begin{equation}\label{eq:stage-polynomial} + |\T\setminus I_n|<4^{-n},\qquad + \|h_n-P_n(U_{S_n})f_n\|_2 + <|\T\setminus I_{n-1}|^{1/2}+2^{-n}. +\end{equation} +Use the arbitrary closeness in that proposition also to impose +\[ + \|f_n-f_{n-1}\|_2<2^{-n}, + \qquad + d_k(S_n,S_{n-1})+d_k(S_n^{-1},S_{n-1}^{-1})<2^{-n} + \quad(0\le k\le n), +\] +where $d_k$ are ambient $C^k$ sup-norm distances induced by a fixed +smooth embedding of the torus. At the same time impose +the finitely many open conditions +\begin{equation}\label{eq:retain-polynomial} + \|P_j(U_{S_n})f_n-P_j(U_{S_{n-1}})f_{n-1}\|_2<2^{-n} + \qquad(1\le jj}2^{-n} + =|\T\setminus I_{j-1}|^{1/2}+2^{1-j}. +\] +Along the infinitely many occurrences of any fixed test the right side +tends to zero. Hence every test belongs to $\calC(U_T,f)$. Their +complex linear span is dense in complex $L^2_0(\T^3)$, whereas every +orbit vector is centered. Thus +\[ + \calC(U_T,f)=L^2_0(\T^3). +\] +Together with \eqref{eq:limit-orthonormal}, this says that the bilateral +orbit of $f$ is an orthonormal basis of the entire centered space. +With $w=\ee(z)$, the assignment $U_T^nf\mapsto w^n$ extends to a +unitary map onto $L^2(S^1,m)$ intertwining $U_T$ with multiplication +by $w$. + +Multiplication by $w$ has no nonzero invariant vector, since +$(w-1)g(w)=0$ forces $g=0$ almost everywhere. Consequently every +$T$-invariant $L^2$ function is constant, which proves ergodicity. +The manifold $\T^3$ is compact, connected, smooth, and without +boundary, and its normalized flat volume has strictly positive smooth +density. These are all the properties required in +Theorem~\ref{thm:main}. +\end{proof} diff --git a/preprints/A-smooth-three-torus-diffeomorphism-with-simple-Lebesgue-spectrum-September-23-2026/build/sections/06-consequences.tex b/preprints/A-smooth-three-torus-diffeomorphism-with-simple-Lebesgue-spectrum-September-23-2026/build/sections/06-consequences.tex new file mode 100644 index 0000000000000000000000000000000000000000..265d6d96b95751d0c8e1d1bc73ae4b6225c8aff1 --- /dev/null +++ b/preprints/A-smooth-three-torus-diffeomorphism-with-simple-Lebesgue-spectrum-September-23-2026/build/sections/06-consequences.tex @@ -0,0 +1,65 @@ +\section{Dynamical consequences} +\label{sec:consequences} + +The spectral model gives ordinary mixing directly. The passage to higher +orders uses the separate multiple-mixing theorem; the entropy and Lyapunov +conclusions do not depend on that additional input. + +\begin{corollary}\label{cor:dynamical-profile} +For the diffeomorphism $T$ and normalized standard volume $\mu$ of +Theorem~\ref{thm:main}, every integer $k\ge2$ and every fixed choice of +measurable sets $A_1,\ldots,A_k\subseteq\T^3$ satisfy +\[ + \mu\left(\bigcap_{i=1}^k T^{-t_i}A_i\right) + \longrightarrow\prod_{i=1}^k\mu(A_i) +\] +whenever $t_1<\cdots1$, the final lower bound is automatic.} +\label{fig:observation-flow} +\end{figure} diff --git a/preprints/A-spectral-gap-throughout-the-high-temperature-Sherrington-Kirkpatrick-phase-September-24-2026/build/main.tex b/preprints/A-spectral-gap-throughout-the-high-temperature-Sherrington-Kirkpatrick-phase-September-24-2026/build/main.tex new file mode 100644 index 0000000000000000000000000000000000000000..8785cc4247627df0aea3429a49750240e442d4a9 --- /dev/null +++ b/preprints/A-spectral-gap-throughout-the-high-temperature-Sherrington-Kirkpatrick-phase-September-24-2026/build/main.tex @@ -0,0 +1,26 @@ +\documentclass[11pt]{article} +\input{style} +\title{A spectral gap throughout the high-temperature\texorpdfstring{\\}{ }Sherrington--Kirkpatrick phase} +\author{OpenAI} +\date{September 24, 2026} +\begin{document} +\maketitle +\begin{abstract} +For every fixed inverse temperature $0<\beta<1$, we prove that the unscaled +spectral gap of single-site heat-bath dynamics for the zero-field Gaussian +Sherrington--Kirkpatrick model is bounded away from zero with probability +tending to one over the disorder. Equivalently, the Gibbs law satisfies a +dimension-free Poincar\'e inequality for all functions. +\end{abstract} +{\small\tableofcontents} +\clearpage +\input{sections/introduction} +\input{sections/observation} +\input{sections/matrices} +\input{sections/entropy} +\input{sections/stability} +\input{sections/residuals} +\input{sections/consequences} +\input{sections/terminal} +\input{references} +\end{document} diff --git a/preprints/A-spectral-gap-throughout-the-high-temperature-Sherrington-Kirkpatrick-phase-September-24-2026/build/references.tex b/preprints/A-spectral-gap-throughout-the-high-temperature-Sherrington-Kirkpatrick-phase-September-24-2026/build/references.tex new file mode 100644 index 0000000000000000000000000000000000000000..dbeaba0df557bd4981e3d80b6db090a2bd6d77d5 --- /dev/null +++ b/preprints/A-spectral-gap-throughout-the-high-temperature-Sherrington-Kirkpatrick-phase-September-24-2026/build/references.tex @@ -0,0 +1,161 @@ +\phantomsection +\addcontentsline{toc}{section}{References} +\begin{thebibliography}{99} + +\bibitem{ABXY2024} +A.~Adhikari, C.~Brennecke, C.~Xu, and H.-T.~Yau, +\emph{Spectral gap estimates for mixed $p$-spin models at high temperature}, +Probability Theory and Related Fields \textbf{189} (2024), nos.~3--4, +879--907. +\href{https://doi.org/10.1007/s00440-024-01261-9}{doi:10.1007/s00440-024-01261-9}. + +\bibitem{ALR1987} +M.~Aizenman, J.~L.~Lebowitz, and D.~Ruelle, +\emph{Some rigorous results on the Sherrington--Kirkpatrick spin glass model}, +Communications in Mathematical Physics \textbf{112} (1987), 3--20. +\href{https://doi.org/10.1007/BF01217677}{doi:10.1007/BF01217677}. + +\bibitem{AJKPV2022} +N.~Anari, V.~Jain, F.~Koehler, H.~T.~Pham, and T.-D.~Vuong, +\emph{Entropic independence: optimal mixing of down-up random walks}, +Proceedings of the 54th Annual ACM SIGACT Symposium on Theory of Computing +(STOC 2022), 1418--1430. 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S.~Oveis Gharan, +\emph{Rank-1-perturbed trickledown theorems: Mixing time of Glauber dynamics +for the Sherrington--Kirkpatrick model up to $\beta\leq\frac12+\varepsilon$}, +\href{https://arxiv.org/abs/2609.13138v1}{arXiv:2609.13138v1}, +September 11, 2026. + +\bibitem{Bolthausen2014} +E.~Bolthausen, +\emph{An iterative construction of solutions of the TAP equations for the +Sherrington--Kirkpatrick model}, Communications in Mathematical Physics +\textbf{325} (2014), no.~1, 333--366. +\href{https://doi.org/10.1007/s00220-013-1862-3}{doi:10.1007/s00220-013-1862-3}. + +\bibitem{BSXY2024} +C.~Brennecke, A.~Schertzer, C.~Xu, and H.-T.~Yau, +\emph{The two point function of the SK model without external field at +high temperature}, Probability and Mathematical Physics +\textbf{5} (2024), 131--175. +\href{https://doi.org/10.2140/pmp.2024.5.131}{doi:10.2140/pmp.2024.5.131}. + +\bibitem{Celentano2024} +M.~Celentano, +\emph{Sudakov--Fernique post-AMP, and a new proof of the local convexity of 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Wang, \emph{Optimal Mixing of Glauber Dynamics for the +Sherrington--Kirkpatrick Model at $\beta<1/2$}, +\href{https://arxiv.org/abs/2608.22159v2}{arXiv:2608.22159v2}, +August 28, 2026. + +\bibitem{Wu2006} +L.~Wu, \emph{Poincar\'e and transportation inequalities for Gibbs measures +under the Dobrushin uniqueness condition}, Annals of Probability +\textbf{34} (2006), no.~5, 1960--1989. +\href{https://doi.org/10.1214/009117906000000368}{doi:10.1214/009117906000000368}. + +\end{thebibliography} diff --git a/preprints/A-spectral-gap-throughout-the-high-temperature-Sherrington-Kirkpatrick-phase-September-24-2026/build/sections/consequences.tex b/preprints/A-spectral-gap-throughout-the-high-temperature-Sherrington-Kirkpatrick-phase-September-24-2026/build/sections/consequences.tex new file mode 100644 index 0000000000000000000000000000000000000000..04626a1ac579b1a13bec38a333faea6ee2ae2767 --- /dev/null +++ b/preprints/A-spectral-gap-throughout-the-high-temperature-Sherrington-Kirkpatrick-phase-September-24-2026/build/sections/consequences.tex @@ -0,0 +1,243 @@ +\section{Posterior estimates at a stable field}\label{sec:consequences} + +The residual rules now yield the two posterior estimates used in +Section~\ref{sec:observation}. We first select a finite collection of +small-residual regions, then compare the primary iterates with the exact +root. Throughout, $J$ satisfies the required fixed-length word diagnostics, +and $h_0$ satisfies the whole stable-field implication through residual +threshold $\rho_0\sqrt n$. Denote the unique zero of $F_{h_0}$ by $r$ and +set $t_*=\tanh r$, $q_*=\av{t_*^2}$. All expectations and forms here are +at $\mu_{h_0}$, and $R_l=m^{l-1}-m^l$ denotes the residual from +Section~\ref{sec:residuals}. + +\subsection{Selecting a small-residual region} +The primary recursion need not converge. The finite-depth fact we need is +that two consecutive residuals are small on most of the Gibbs mass, with +a separate bound after square reweighting. Smooth cutoffs make that selection +compatible with discrete differentiation. +\begin{lemma}\label{lem:residual-selection} +For each sufficiently small fixed $\rho>0$, there is a fixed depth $d$ and +functions $C_k$, $2\le k\le d$, with the following properties: +\begin{enumerate} +\item $0\le C_k\le1$, and $C_k$ is supported where +$\max(\norm{R_k},\norm{R_{k-1}})\le\rho\sqrt n$. It equals one where both +norms are at most $\rho\sqrt n/2$, and $\norm{dC_k}\le C_\rho/\sqrt n$ +pointwise. +\item Set $\widehat C_k=C_k\prod_{i=2}^{k-1}(1-C_i)$ and +$D=1-\sum_{k=2}^d\widehat C_k$. Then $0\le D\le1$ and +\[ + \Prob_{\mu_{h_0}}(D>0)\le\frac{C_\rho}{n}. +\] +For $N=\E f^2>0$ and $\nu=f^2\mu_{h_0}/N$, +\[ + \Prob_\nu(D>0)\le\frac{C_\rho}{\sqrt n} + \left(1+\frac{\D_{h_0}(f)}{N}\right). +\] +\end{enumerate} +All constants are uniform over the field and matrix under the stated +diagnostics. The depth and the required diagnostic length may depend on +$\rho$. +\end{lemma} + +\begin{proof} +We use the scalar residual pairing $S_l=R_l^{\mathsf T}m^l/\sqrt n$ to +control the total squared increment. First, the primary derivative bounds +give $\norm{dS_l}\le C_l$ pointwise. In the product rule, the linear +terms are controlled by the operator norms of $\nabla R_l$ and +$\nabla m^l$, multiplied by $\norm{m^l}/\sqrt n$ and +$\norm{R_l}/\sqrt n$. The cross term for coordinate $i$ has size at most +$2\norm{d_i R_l}\norm{d_i m^l}/\sqrt n$; summing its square uses the +bounded column norms and gives the same scale. + +Apply Lemma~\ref{lem:residual} with target $m^l/\sqrt n$ and test $G=S_l$. +It gives +\[ + \E S_l^2\le C_l\bigl(\E S_l^2+C_l\bigr)^{1/2}, + \qquad\text{hence}\qquad \E S_l^2\le C_l'. +\] +For the square-density version use $H_0=S_l/\sqrt n$, which is bounded with +bounded difference norm. Dividing the resulting estimate by $N\sqrt n$ +gives +\begin{equation}\label{eq:weighted-S} + \E_\nu\left(\frac{S_l}{\sqrt n}\right)^2 + \le\frac{C_l}{\sqrt n}\left(1+\frac{\D_{h_0}(f)}{N}\right). +\end{equation} +We now use the squared-increment identity +\eqref{eq:residual-telescoping}. Choose a fixed $d$ with +$\lfloor d/2\rfloor\rho^2/4>2$, followed by $\eps>0$ with $2d\eps<1$. +Whenever $\max_{l\le d}|S_l|/\sqrt n\le\eps$, telescoping gives +$\sum_{l=1}^d\norm{R_l}^2/n<2$. If every consecutive pair contained a +residual larger than $\rho\sqrt n/2$, the disjoint pairs +$(1,2),(3,4),\ldots$ would force the sum to exceed this bound. At least +one pair therefore meets the required threshold. + +To implement the selection smoothly, apply a scalar cutoff equal to one +on $[0,1/4]$ and zero on $[1,\infty)$ to each of +$\norm{R_k}^2/(\rho^2n)$ and $\norm{R_{k-1}}^2/(\rho^2n)$, and multiply +the results to define $C_k$. The primary operator and column bounds, +with the product rule, give +$\norm{d(\norm{R_l}^2/n)}\le C_l/\sqrt n$, hence the claimed derivative +bound. Finally $D=\prod_{k=2}^d(1-C_k)$: positive deficit implies +$\max_{l\le d}|S_l|/\sqrt n>\eps$. The preceding two moment bounds and +a finite union bound give the ordinary and square-reweighted estimates. +\end{proof} + +\subsection{A directional covariance bound} +On each selected region, the implicit test vector converts the residual +identity into a comparison with $t_*$. Its auxiliary scalar error is +estimated first; only then do we test in an arbitrary Euclidean direction. +\begin{proposition}\label{prop:directional} +There is a constant $C$ such that, at every field satisfying the above +stable-field implication, +\begin{equation}\label{eq:directional} + \norm{\Cov_{\mu_{h_0}}(G,x)} + \le C\bigl(\Var_{\mu_{h_0}}(G)+\D_{h_0}(G)\bigr)^{1/2} +\end{equation} +for every $G$. The constant is uniform in $n,h_0$, and $G$ on the specified +matrix events; only a fixed finite diagnostic length is required. +\end{proposition} + +\begin{proof} +Fix an index $k\ge2$ and work on the small-residual region +\eqref{eq:small-region}, with the buffered implicit construction +\eqref{eq:implicit}. Abbreviate $m=m^k$, $p=m-t_*$, and put +\[ + a=j(1-b_{k-1}-q_*),\qquad + L_* =\sqrt n\bigl(\chi-(1-q_*)\bigr). +\] +Here $\chi=\av{A_i}$ and $A_i=\Phi(h_i^k;r_i,a)$ are defined globally by +\eqref{eq:phi-data}, even though the implicit vector is only needed locally. +Thus $|L_*|\le C\sqrt n$ and $\norm{dL_*}\le C_k$ pointwise, by the primary +derivative bounds and $\norm{db_{k-1}}=O(n^{-1/2})$. + +For either of the two allowed sources $w_0$, the following pointwise identity +holds on the buffered region: +\begin{align}\label{eq:implicit-comparison} + &\bigl(h^k-h_0-(J-jb_{k-1}I)m\bigr)^{\mathsf T}w + -j(b_k-b_{k-1})nc_1 \notag\\ + &\quad=\sum_i\bigl[(h_i^k-r_i-am_i)-a\partial_k\bigr]w_{0,i} + +j(1-q_*-\chi)p^{\mathsf T}w-jc_1R_k^{\mathsf T}p. +\end{align} +Here the zeroth-order term in the first sum acts by multiplication on +$w_{0,i}$. For the verification, subtract the root equation to write the +vector on the left as $h^k-r-am-Jp+j(1-q_*)p$, then substitute +$Jw=y^{\rm imp}+j\chi w+jc_1m^{k-1}$. Apply \eqref{eq:phi-stein} to +the contribution $A(y^{\rm imp}-jc_1t_*)$, holding its auxiliary arguments +and parameters fixed, and use $nc_1=\sum_i\partial_k w_i$. The remaining +scalar terms cancel by +\[ + p^{\mathsf T}(m^{k-1}+t_*)+n(b_k-b_{k-1}) + =n(1-b_{k-1}-q_*)+R_k^{\mathsf T}p, +\] +which follows by expanding $p=m-t_*$ and $R_k=m^{k-1}-m$. + +Lemma~\ref{lem:terminal-residual} controls the tested left-hand side of +\eqref{eq:implicit-comparison}. The last term on the right also has tested +expectation at most $C\norm{G}_*$: apply Lemma~\ref{lem:residual} to target +$p/\sqrt n$ and multiplier $\sqrt n c_1$. The latter is bounded with bounded +difference norm on the buffer. Meanwhile the size estimates for the implicit +solution and Lemma~\ref{lem:root} give +\begin{equation}\label{eq:middle-error} + \abs{j(1-q_*-\chi)p^{\mathsf T}w}\le C_0\rho|L_*|. +\end{equation} +The source size and stability margins determine $C_0$ independently of +$k$ and the selection depth. This lets us choose $\rho$ first so that +$C_0\rho<1/2$ and all stability and flip-buffer conditions hold. The +selection depth in Lemma~\ref{lem:residual-selection} is chosen afterward. +This order will permit absorption of the scalar error without a +self-dependent choice of depth. + +First control the averaged coefficient by taking +$w_{0,i}=\partial_{r_0}\Phi(h_i^k;r_i,a)/\sqrt n$. +The second identity in \eqref{eq:implicit-source-actions} makes the first +sum in \eqref{eq:implicit-comparison} equal to $L_*$. +Test with $G=C_k^2L_*$. The cutoff and difference bounds give +\[ + \norm{C_k^2L_*}_*\le C+\norm{C_kL_*}_{L^2(\mu_{h_0})}. +\] +Indeed its difference vector is bounded pointwise: $|L_*|=O(\sqrt n)$ +multiplies $\norm{d(C_k^2)}=O(n^{-1/2})$, and $\norm{dL_*}=O(1)$. +Absorbing \eqref{eq:middle-error} now yields +\begin{equation}\label{eq:susceptibility-average} + \norm{C_kL_*}_{L^2(\mu_{h_0})}\le C_k'. +\end{equation} + +Now test a direction by taking $w_0=Ae$, where $e$ is any deterministic +vector with $\norm e\le1$. By the first identity in +\eqref{eq:implicit-source-actions}, the first sum in +\eqref{eq:implicit-comparison} is $p^{\mathsf T}e$. +Test with $\widehat C_kG$. The residual estimates control the left side +and the last term by $C\norm G_*$; the middle term is bounded by +\eqref{eq:susceptibility-average} and Cauchy--Schwarz, since +$0\le\widehat C_k\le C_k$. Thus +\[ + \abs{\E[\widehat C_kG(m^k-t_*)^{\mathsf T}e]}\le C\norm G_*. +\] +Summing Lemma~\ref{lem:residual} with the constant target $e$ over +$l=1,\ldots,k$ gives the same bound for +$\E[\widehat C_kG(x-m^k)^{\mathsf T}e]$. +Sum over $k$. The deficit has contribution at most +\[ + 2\sqrt n\,\norm G_{L^2(\mu_{h_0})} + \Prob_{\mu_{h_0}}(D>0)^{1/2}\le C\norm G_*. +\] +Thus $|\E[G(x-t_*)^{\mathsf T}e]|\le C\norm G_*$ uniformly over +$\norm e\le1$. Centering $G$ and taking the supremum over this seed +class gives \eqref{eq:directional}. The uncentered estimate also records +the ordinary-mean consequence: take $G=1$ to obtain +$\norm{\E_{\mu_{h_0}}x-t_*}\le C$. This norm is the unnormalized +Euclidean norm. It compares the mean itself with $t_*$; no +approximation of the derivative of the mean map is used here. +The local construction required $n$ above a fixed threshold. In the +remaining dimensions, the elementary bounds +$\norm{\Cov(G,x)}^2\le n\Var(G)$ and +$\norm{\E_{\mu_{h_0}}x-t_*}\le2\sqrt n$ extend both conclusions after +enlarging $C$. +\end{proof} + +\subsection{Means under square reweighting} +For the change of observation law we need a different estimate, valid for +arbitrary square densities. It uses only ordinary recipes and trades a +small fixed leading error for a constant depending on that tolerance. +\begin{proposition}\label{prop:weighted-mean} +For every sufficiently small fixed $\delta>0$, there is a constant $C_\delta$ +such that, for $N=\E_{\mu_{h_0}}f^2>0$ and $\nu=f^2\mu_{h_0}/N$, +\begin{equation}\label{eq:weighted-mean} + \frac{\norm{\E_\nu x-t_*}}{\sqrt n} + \le C\delta+\frac{C_\delta}{\sqrt n} + \left(1+\frac{\D_{h_0}(f)}{N}\right). +\end{equation} +The leading constant $C$ is independent of $\delta$ and of the selection +depth. This estimate requires only ordinary word diagnostics, of a fixed +length that may depend on $\delta$. +\end{proposition} + +\begin{proof} +Select the cutoffs with a fixed threshold $\rho\le\delta$ small enough +for Lemma~\ref{lem:root}. On each cutoff support, the residual-to-root +comparison gives +\[ + \norm{m^k-t_*}\le C\delta\sqrt n. +\] +Here the constant comes from the recurrence and the inverse bound in +Lemma~\ref{lem:root}, and is independent of $k$. +For $\norm e\le1$, sum the square-density version of +Lemma~\ref{lem:residual} with target $e$ through $k$, using multiplier +$\widehat C_k$. After division by $N$, this bounds +\[ + \abs{\E_\nu[\widehat C_k(x-m^k)^{\mathsf T}e]} + \le C_\delta\left(1+\frac{\D_{h_0}(f)}{N}\right). +\] +The cutoff weights sum to at most one, so all root-comparison terms +together cost at most $C\delta\sqrt n$. The leftover mass contributes +at most $2\sqrt n\Prob_\nu(D>0)$, which is controlled by the second +part of Lemma~\ref{lem:residual-selection}. Add the terms, divide by +$\sqrt n$, and take the supremum over $e$ to obtain +\eqref{eq:weighted-mean}. The leading $C$ comes from root stability; +only $C_\delta$ and the finite diagnostic length depend on the selected +tolerance. Any dimensions below a fixed threshold required by this +construction are covered by the elementary bound +$\norm{\E_\nu x-t_*}/\sqrt n\le2$ after increasing $C_\delta$; this +leaves the leading constant unchanged. The argument has used no implicit +test vector. +\end{proof} diff --git a/preprints/A-spectral-gap-throughout-the-high-temperature-Sherrington-Kirkpatrick-phase-September-24-2026/build/sections/entropy.tex b/preprints/A-spectral-gap-throughout-the-high-temperature-Sherrington-Kirkpatrick-phase-September-24-2026/build/sections/entropy.tex new file mode 100644 index 0000000000000000000000000000000000000000..559d6f5a678b2c819355bd8ba23f71877a7aad78 --- /dev/null +++ b/preprints/A-spectral-gap-throughout-the-high-temperature-Sherrington-Kirkpatrick-phase-September-24-2026/build/sections/entropy.tex @@ -0,0 +1,337 @@ +\section{A scalar entropy inequality} +\label{sec:entropy} + +The stability argument needs a scalar criterion that forces an +empirical field law toward a Gaussian law. We first bound relative +entropy below by a squared discrepancy involving the field and its +hyperbolic tangent. We then identify the Gaussian law compatible +with the scalar consistency equations. Together with the equality +case of the first estimate, this characterizes the consistent +equality laws. + +Throughout this section, $N(d,s)$ denotes the Gaussian law with mean +$d$ and \emph{variance} $s$. All scalar expectations are taken with +respect to the indicated probability law. + +\begin{lemma}[Scalar entropy inequality] +\label{lem:scalar-entropy} +Let $00$. Let $P$ be a probability law on $\R$ +with finite second moment, and set +\[ + m(y)=\tanh y,\qquad q=\E_P m^2,\qquad b=1-q,\qquad + a=\E_P[(y-d)m],\qquad S=s+jq. +\] +If $s\ge jq$, then +\begin{equation} + D(P\Vert N(d,s))\ge \frac{j(a-sb)^2}{2sS}. + \label{eq:scalar-entropy} +\end{equation} +Equality holds if and only if $P=N(d,s)$. +\end{lemma} + +\begin{proof} +The second-moment assumption makes the right-hand side finite, so +it suffices to consider the case $D:=D(P\Vert N(d,s))<\infty$. In this case $P$ is +absolutely continuous, and hence $q>0$: otherwise $P$ would be the +point mass at zero. Also $q<1$, because $\tanh^2 y<1$ for every finite +$y$. We use the following dimensionless parameters: +\begin{equation} + k=\frac{jq}{s}\in(0,1],\qquad C=\frac{ja}{s},\qquad + B=jb\in(0,1),\qquad a_0=1-B\in(0,1). + \label{eq:entropy-scalars} +\end{equation} +In these parameters, multiplying the desired inequality by $j$ +gives the target +\begin{equation} + jD\ge \frac{(C-B)^2}{2(1+k)}. + \label{eq:entropy-target} +\end{equation} + +We will combine two entropy bounds. A monotone change of variables +handles one range of the discrepancy $C-B$; comparison with a Gaussian +of a different variance handles the remaining range. We retain +strictness in both arguments to identify all equality cases. + +\paragraph{A change-of-variables bound.} +For the change of variables, fix $u<1$. The map +\[ + T_u(y)=y-u\tanh y +\] +is an increasing $C^1$ diffeomorphism of $\R$ onto itself, with derivative +$T_u'(y)=1-u(1-m(y)^2)>0$. Change of variables in relative entropy yields +\begin{align*} + D + &=D((T_u)_\#P\Vert N(d,s)) + +\frac{2ua-u^2q}{2s} + +\E_P\log\bigl(1-u(1-m^2)\bigr)\\ + &\ge \frac{2ua-u^2q}{2s} + +\E_P\log\bigl(1-u(1-m^2)\bigr). +\end{align*} +The entropy identity is well defined: for fixed $u<1$, $\log T_u'$ +is bounded, and the second-moment assumption makes the difference +of the Gaussian log densities integrable. To estimate the logarithmic +term, apply concavity on the segment joining $1$ and $1-u$. For +$0\le v\le1$, this gives +\[ + \log(1-uv)\ge v\log(1-u). +\] +Consequently, with $\ell(u)=-\log(1-u)-u$, +\begin{equation} + jD\ge u(C-B)-\frac{k u^2}{2}-B\ell(u),\qquad u<1. + \label{eq:entropy-transport} +\end{equation} +Completing the square isolates the amount by which this bound can +exceed the target in \eqref{eq:entropy-target}: +\begin{equation} + jD-\frac{(C-B)^2}{2(1+k)} + \ge h(u)-\frac{\bigl(C-B-(1+k)u\bigr)^2}{2(1+k)}, + \qquad h(u)=\frac{u^2}{2}-B\ell(u). + \label{eq:entropy-slack} +\end{equation} + +We first locate a range in which the extra term $h(u)$ is strictly +positive: +\begin{equation} + h(u)>0\quad\text{for }u\le a_0,\ u\ne0, + \qquad h(a_0)\ge\frac{a_0^3}{6}. + \label{eq:entropy-slack-positive} +\end{equation} +For the negative range, integrate $v/(1+v)\le v$ over +$0\le v\le -u$ to obtain $\ell(u)\le u^2/2$. Since $B<1$, this +proves strict positivity for $u<0$. For the positive range, use +$\ell(u)=\sum_{r\ge2}u^r/r$: it shows that $\ell(u)/u^2$ is increasing +on $00$ for $00$, so +\eqref{eq:entropy-target} is strict. If $u_*>a_0$, put +\[ + C_0=1+ka_0, + \qquad C_1=C_0+\sqrt{\frac{1+k}{3}}\,a_0^{3/2}. +\] +In this case $C>C_0$. The endpoint choice $u=a_0$ in +\eqref{eq:entropy-slack} still gives +\[ + jD-\frac{(C-B)^2}{2(1+k)} + \ge \frac{a_0^3}{6}-\frac{(C-C_0)^2}{2(1+k)}. +\] +This proves strictness throughout $C_00$. Comparison with the +Gaussian $N(d,sv)$ gives the exact identity +\[ + D=D(P\Vert N(d,sv))+\frac{v-1-\log v}{2}, +\] +and hence $2D\ge v-1-\log v$. Cauchy--Schwarz gives +\[ + v\ge\frac{a^2}{sq}=\frac{C^2}{jk}>1, +\] +where the last inequality follows from $C\ge C_1>C_0>1$ and $jk<1$. +The function $v\mapsto v-1-\log v$ is increasing for $v\ge1$, so +\[ + 2jD\ge \frac{C^2}{k}-j-j\log\frac{C^2}{jk} + \ge \frac{C^2}{k}-1-\log\frac{C^2}{k}. +\] +The second inequality removes $j$ from the lower bound. Indeed, +with $C,k$ held fixed, the derivative of +$C^2/k-\lambda-\lambda\log(C^2/(\lambda k))$ is +$-\log(C^2/(\lambda k))\le0$ for $j\le\lambda\le1$. +Thus the remaining comparison with the target reduces to +\[ + f(C):=\frac{C^2}{k}-1-\log\frac{C^2}{k} + -\frac{(C-B)^2}{1+k}>0 + \quad(C\ge C_1). +\] +We estimate $f$ at $C_0$ and control its increase from $C_0$ to +$C_1$. Direct simplification at $C_0$ gives +\begin{align*} + f(C_0) + &=\frac1k-1+\log k + +2a_0-a_0^2-2\log(1+ka_0)\\ + &\ge -\frac{2a_0^3}{3}. +\end{align*} +Indeed $1/k-1+\log k\ge0$ for $00). +\end{align*} +For completeness, the first inequality follows from +\[ + \frac1k-\frac1{1+ka_0}-\frac{a_0}{1+a_0} + =\frac{(1-k)(1+a_0+ka_0^2)}{k(1+ka_0)(1+a_0)}\ge0. +\] +The derivative bounds compensate for the possible negative value +at $C_0$. Write $\Delta=C_1-C_0\ge a_0^{3/2}/\sqrt3$ and integrate +them to obtain +\begin{align*} + f(C_1) + &\ge -\frac{2a_0^3}{3} + +\frac{2a_0}{1+a_0}\Delta+\frac{\Delta^2}{2}\\ + &\ge a_0^3\left( + -\frac23+\frac{2}{\sqrt3(1+a_0)\sqrt{a_0}}+\frac16 + \right)>0. +\end{align*} +The strict inequality uses $02/3$. Since $f'$ is positive at $C_0$ and increasing +thereafter, $f(C)>0$ for every $C\ge C_1$. + +The two ranges together prove strictness in +\eqref{eq:entropy-target} whenever $C\ne B$. To finish the equality +case, suppose $C=B$. The target then has right-hand side zero, so +equality is equivalent to $D=0$, or $P=N(d,s)$. Conversely, Gaussian +integration by parts gives $a=s\E_P\sech^2 y=sb$ for this law, +which verifies equality. +\end{proof} + +We next impose the scalar consistency equations used in the +application. Here the Gaussian parameters themselves depend on +$P$: for given $t\ge0$ and $\sigma\ge0$, set +\begin{equation} + M=\E_P\tanh y,\qquad q=\E_P\tanh^2 y,\qquad + d=t+jM,\qquad s=t+\sigma^2+jq. + \label{eq:scalar-consistency} +\end{equation} +These equations give $s\ge jq$, the comparison required by +Lemma~\ref{lem:scalar-entropy}. When $s>0$, that lemma restricts +equality to Gaussian laws. The next result identifies which +Gaussian laws satisfy the consistency equations when $\sigma=0$, +and also treats the degenerate case at $t=0$. + +\begin{lemma}[The consistent Gaussian law] +\label{lem:gaussian-equality} +Fix $00$, equality in \eqref{eq:scalar-entropy} holds precisely at +$P_*(t)$. + +The function $t\mapsto s_*(t)$ is Lipschitz with constant $(1-j)^{-1}$, +and +\begin{equation} + t\le s_*(t)\le\frac{t}{1-j},\qquad + q_*(t):=\E_{P_*(t)}\tanh^2 y + =\frac{s_*(t)-t}{j}\le\frac{t}{1-j}. + \label{eq:scalar-fixed-point-bounds} +\end{equation} +In particular $s_*(0)=q_*(0)=0$, and $s_*(t),q_*(t)>0$ for $t>0$. +\end{lemma} + +\begin{proof} +We first show that a consistent Gaussian must have equal mean and +variance. We then solve the resulting one-variable fixed-point +equation and prove the bounds in the statement. + +Two Gaussian symmetry identities provide the first step. For $s>0$, +the density $\rho_s$ of $N(s,s)$ satisfies +\[ + \frac{\rho_s(y)-\rho_s(-y)}{\rho_s(y)+\rho_s(-y)}=\tanh y. +\] +Pairing the integrals over $y$ and $-y$ therefore gives +\begin{equation} + \E_{N(s,s)}\tanh y=\E_{N(s,s)}\tanh^2 y. + \label{eq:gaussian-nishimori} +\end{equation} +Second, if $d\ge0$ and $s>0$, pairing the positive and negative +half-lines shows that +\begin{equation} + \E_{N(d,s)}[\tanh y\,\sech^2 y]\ge0. + \label{eq:gaussian-odd-sign} +\end{equation} +Here the integrand is odd and nonnegative for $y\ge0$, while the density +at $y$ is at least its value at $-y$ when $d\ge0$. + +Now suppose a Gaussian law $N(d,s)$ with $s>0$ satisfies +\eqref{eq:scalar-consistency} with $\sigma=0$. We begin by locating +its mean. For fixed $s$, the map +\[ + d\longmapsto t+j\E_{N(d,s)}\tanh y +\] +is a contraction with Lipschitz constant at most $j$, because its +derivative is $j\E_{N(d,s)}\sech^2 y\in[0,j]$. Iteration from zero +stays nonnegative, so its unique fixed point $d$ is nonnegative. + +To compare this nonnegative mean with the variance, keep $s$ fixed +and define +\[ + H_s(d)=\E_{N(d,s)}[\tanh y-\tanh^2 y]. +\] +For $d\ge0$, Gaussian differentiation and +\eqref{eq:gaussian-odd-sign} give +\[ + H_s'(d)=\E_{N(d,s)}[(1-m^2)(1-2m)]\in[-1,1]. +\] +Indeed the integrand is bounded below by $-1$, and its expectation is +at most $\E(1-m^2)\le1$ after subtracting +$2\E[m(1-m^2)]\ge0$. Thus $H_s$ is 1-Lipschitz on $[0,\infty)$. +By \eqref{eq:gaussian-nishimori}, $H_s(s)=0$. On the other hand, +consistency says $d-s=j(M-q)=jH_s(d)$. Consequently +\[ + |H_s(d)|\le |d-s|=j|H_s(d)|, +\] +forcing $H_s(d)=0$ and $d=s$. The consistent Gaussian law must therefore +satisfy \eqref{eq:scalar-fixed-point}. + +It remains to solve \eqref{eq:scalar-fixed-point}. Set +\[ + F(s)=\E_{N(s,s)}\tanh y\quad(s>0),\qquad F(0)=0. +\] +It is continuous at zero by bounded convergence. For $s>0$, +differentiating both the Gaussian mean and variance gives +\begin{align*} + F'(s) + &=\E_{N(s,s)}\left[(\tanh y)' + +\tfrac12(\tanh y)''\right]\\ + &=\E_{N(s,s)}[(1-m^2)(1-m)]\in[0,1]. +\end{align*} +The lower bound is pointwise; the upper bound follows from +\eqref{eq:gaussian-odd-sign}. Hence $F$ is nonnegative and +1-Lipschitz on $[0,\infty)$, with $F(s)\le s$. +Consequently, $s\mapsto t+jF(s)$ is a contraction of +$[0,\infty)$ into itself and has exactly one fixed point $s_*(t)$. +The identity \eqref{eq:gaussian-nishimori} then verifies both +consistency equations for $N(s_*(t),s_*(t))$. +If a consistent law has $s=0$, then $0=t+jq$ forces $t=q=0$, +and hence the law is the point mass at zero and $d=0$; this agrees +with the solution just constructed. + +Existence and uniqueness are now established, including the +degenerate case. To obtain the dependence on time, compare the +fixed-point equations at $t,t'$: +\[ + |s_*(t)-s_*(t')| + \le |t-t'|+j|s_*(t)-s_*(t')|, +\] +which proves the asserted Lipschitz bound. The size estimates follow +from $0\le F(s)\le s$, which gives $t\le s_*(t)\le t/(1-j)$, and +\eqref{eq:gaussian-nishimori} gives +$q_*(t)=F(s_*(t))=(s_*(t)-t)/j\le t/(1-j)$. +For $t>0$, the variance $s_*(t)$ is positive, so its Gaussian law is +nondegenerate and $q_*(t)>0$. The equality assertion follows from +Lemma~\ref{lem:scalar-entropy}. +\end{proof} diff --git a/preprints/A-spectral-gap-throughout-the-high-temperature-Sherrington-Kirkpatrick-phase-September-24-2026/build/sections/introduction.tex b/preprints/A-spectral-gap-throughout-the-high-temperature-Sherrington-Kirkpatrick-phase-September-24-2026/build/sections/introduction.tex new file mode 100644 index 0000000000000000000000000000000000000000..c25b7d41d77eb21295b28b538a53b57f59a04c98 --- /dev/null +++ b/preprints/A-spectral-gap-throughout-the-high-temperature-Sherrington-Kirkpatrick-phase-September-24-2026/build/sections/introduction.tex @@ -0,0 +1,254 @@ +\section{The theorem and the structure of the proof}\label{sec:introduction} + +How quickly does local resampling remove fluctuations in a densely coupled +spin system? For the Sherrington--Kirkpatrick model~\cite{SK1975}, the signs +of the interactions matter: each spin interacts weakly with every other spin, +but the sum of the absolute interaction strengths grows with the dimension. +We study heat-bath dynamics, which updates a spin from its conditional Gibbs +law, in the tradition of Glauber's local stochastic dynamics~\cite{Glauber1963}. +Our result is a dimension-independent Poincar\'e inequality throughout the +high-temperature phase, at each fixed inverse temperature below one. It controls every real observable on a single +event of disorder probability tending to one. + +Fix $0<\beta<1$ and put $j=\beta^2$. Let $J$ be a real symmetric +$n\times n$ matrix with zero diagonal and independent off-diagonal entries +$J_{ij}\sim N(0,j/n)$, $i0$ is fixed. + +\item \textbf{Two estimates at a good field.} +For every $G$, Proposition~\ref{prop:directional} proves +\begin{equation}\label{eq:input-directional} + \norm{\Cov_{\mu_h}(G,x)}^2 + \le C\bigl(\Var_{\mu_h}(G)+\D_h(G)\bigr). +\end{equation} +For $N=\E_{\mu_h}f^2>0$ and $\nu=f^2\mu_h/N$, +Proposition~\ref{prop:weighted-mean} proves, for each sufficiently small fixed $\delta>0$, +\begin{equation}\label{eq:input-weighted} + \frac{\norm{\E_\nu x-t_*(h)}}{\sqrt n} + \le C\delta+\frac{C_\delta}{\sqrt n} + \left(1+\frac{\D_h(f)}{N}\right). +\end{equation} +Here $t_*(h)=\tanh r(h)$, where $r(h)$ is the unique solution of +\[ + r-h+\{j(1-n^{-1}\|\tanh r\|^2)I-J\}\tanh r=0. +\] +Existence and uniqueness at the fields under consideration are part of +Lemma~\ref{lem:root}. The constant $C$ is independent of $\delta$; $C_\delta$ may depend +on it. Taking $f=1$ gives the same comparison for $\E_{\mu_h}x$. + +\item \textbf{A terminal gap.} +For a sufficiently large fixed $T$, Proposition~\ref{prop:terminal-gap} +gives $\Var_{\mu_{Y(T)}}f\le2\D_{Y(T)}(f)$ outside an event of ordinary +observation probability at most $e^{-a n}$. +\end{enumerate} + +The transfer uses the two estimates for different purposes. Directional +covariance controls posterior variance loss, while square-reweighted means +bound the entropy of the changed observation law. Their combination controls +the contribution of the bad paths before the terminal gap is applied. +Section~\ref{sec:observation} proves this implication first, so the later +technical sections have a precise target. + +\subsection{Organization} +After the observation transfer in Section~\ref{sec:observation}, +Section~\ref{sec:matrices} constructs the matrix diagnostics used by the +adaptive field calculations. Sections~\ref{sec:entropy} +and~\ref{sec:stability} identify the scalar equality law and establish +stability along observations. Section~\ref{sec:residuals} develops the +finite recursions and their weak residual rules; Section~\ref{sec:consequences} +uses them to prove the two posterior estimates. Section~\ref{sec:terminal} +closes the argument with the fixed-time terminal gap. diff --git a/preprints/A-spectral-gap-throughout-the-high-temperature-Sherrington-Kirkpatrick-phase-September-24-2026/build/sections/matrices.tex b/preprints/A-spectral-gap-throughout-the-high-temperature-Sherrington-Kirkpatrick-phase-September-24-2026/build/sections/matrices.tex new file mode 100644 index 0000000000000000000000000000000000000000..cd1bd35de46568faec7e31223caee3a2f03fb29b --- /dev/null +++ b/preprints/A-spectral-gap-throughout-the-high-temperature-Sherrington-Kirkpatrick-phase-September-24-2026/build/sections/matrices.tex @@ -0,0 +1,870 @@ +\section{Uniform random-matrix estimates}\label{sec:matrices} + +The field analysis uses random-matrix estimates in two forms. Finite +field recursions produce products whose diagonal coefficients may be +chosen after the disorder is observed. Lemma~\ref{lem:words} controls +such products of bounded length, subject to a separate positivity +margin whenever an inverse occurs. We also obtain fixed-vector +bilinear estimates and a simultaneous determinant bound for the +stability argument in Section~\ref{sec:stability}. Their expectation +calculations use the same inverse identity. We begin with fixed +deterministic coefficients and then use concentration and finite meshes +to obtain the required simultaneous estimates. + +Throughout this section, $j\in(0,1)$ is fixed. A centered GOE matrix +$W$ has independent entries on and above the diagonal, with +\[ + \E W_{ab}^{2}=j/n\quad(a0$ so that, with $a_+=1+\eta$, +\begin{equation}\label{eq:matrix-subcritical} + j a_+^2<1. +\end{equation} +For a nonnegative diagonal matrix $A\le a_+I$, put +\begin{equation}\label{eq:K} + \begin{gathered} + \chi_A=\frac1n\Tr A,\qquad B_A=j\chi_A,\qquad + K_A(M)=K_A(1;M),\\ + K_A(z;M)=(I-zAM+z^2B_AA)^{-1}. + \end{gathered} +\end{equation} +The associated symmetric matrix is +\[ + S_A(z;M)=I+z^2B_AA-zA^{1/2}MA^{1/2}. +\] +The symmetric matrix lets us control the inverse without imposing a +positive lower bound on the individual diagonal entries of $A$. +Whenever $S_A(z;M)$ is invertible, the following identities hold: +\begin{align} + K_A(z;M) + &=I+A^{1/2}S_A(z;M)^{-1}A^{1/2}(zM-z^2B_AI), + \label{eq:K-symmetric-representation}\\ + K_A(z;M)A&=A^{1/2}S_A(z;M)^{-1}A^{1/2}. + \label{eq:KA-symmetric-representation} +\end{align} +To verify the first identity, apply +$(I-UV)^{-1}=I+U(I-VU)^{-1}V$. For the second, multiply the two +sides using $S_A=I-A^{1/2}(zM-z^2B_AI)A^{1/2}$. These formulas +will also let us truncate the inverse while retaining bounded +matrix factors. + +\subsection{Words and their predictions} + +A \emph{word} is an ordered product of diagonal matrices, copies of a +symmetric matrix $M$, and at most one factor $K_A(M)$. Its length is +the number of factors. Consecutive diagonal factors may always be +combined, and an absent diagonal factor is interpreted as $I$. +To each word $Q$ we associate a diagonal matrix $\mathsf P(Q)$ by +the contraction rule below. The estimates will compare the diagonal +of $Q$ with this explicitly defined matrix. + +Begin with a word without $K_A$. Consider every complete +noncrossing pairing of its occurrences of $M$, in their linear +order. For each pairing, remove an innermost pair by the rule +\begin{equation}\label{eq:noncrossing-contraction} + MDM\ \longmapsto\ j\frac{\Tr D}{n}I, +\end{equation} +where $D$ is diagonal after previous removals. Combine adjacent +diagonals and continue. Sum the resulting diagonal matrices over +all complete noncrossing pairings. If there is no complete pairing, +the prediction is zero; if there is no occurrence of $M$, it is the +word itself. The result for a fixed pairing does not depend on the +order of removal: two available innermost pairs are disjoint, and +their removals commute. +For words without an inverse factor, this is the operator-valued +semicircular moment rule with covariance map +$D\mapsto jn^{-1}\Tr(D)I$; see Speicher~\cite{Speicher1998}. + +For example, four occurrences of $M$ have two complete noncrossing +pairings, giving +\[ + \begin{split} + \mathsf P(MD_1MD_2MD_3M) + ={}&j^2\frac{\Tr D_1}{n}\frac{\Tr D_3}{n}D_2\\ + &+j^2\frac{\Tr D_2}{n}\frac{\Tr(D_1D_3)}n I. + \end{split} +\] +The first term pairs the first two and last two occurrences; the second +pairs the outside occurrences around the middle pair. + +For a word containing $K_A$, replace that factor by the formal series +\begin{equation}\label{eq:K-formal-series} + K_A(z;M)=\sum_{r\ge0}(zAM-z^2B_AA)^r, +\end{equation} +and apply the preceding rule coefficient by coefficient. Ordinary +occurrences of $M$ outside this factor are left unscaled. We prove +below that the resulting series $\mathsf P(Q;z)$ is a polynomial; +we define $\mathsf P(Q)=\mathsf P(Q;1)$. + +The corresponding trace prediction is +$\mathsf p(Q)=n^{-1}\Tr\mathsf P(Q)$. To see its cyclic invariance, +draw an ordinary trace word on a circle, with a chord for each +pair in a noncrossing pairing. Its contribution is $j$ for each chord, +multiplied by, for each face, the average of the product of the +diagonal matrices on that face's boundary arcs. Removing an +innermost chord gives exactly \eqref{eq:noncrossing-contraction}, and +the average over the last face is the final normalized trace. This +description is unchanged by rotating the circle. For words with +$K_A(z;M)$, apply this argument coefficient by coefficient in +\eqref{eq:K-formal-series}. + +\begin{lemma}[Simultaneous word estimates]\label{lem:words} +Fix $L\in\mathbb N$, $M_0<\infty$, and $c>0$. There are constants +$C,c_1>0$ and $n_0$, depending only on $j,a_+,L,M_0,c$, such that the +following event has probability at least $1-Ce^{-c_1n}$ for +$n\ge n_0$. Simultaneously for all words $Q$ of length at most $L$, +all their real diagonal factors of norm at most $M_0$, and all +$0\le A\le a_+I$, one has +\begin{equation}\label{eq:word-estimates} + \norm{Q}\le C,\qquad + \left(\sum_{a\ne b}|Q_{ab}|^4\right)^{1/4}\le C,\qquad + \norm{\diag(Q-\mathsf P(Q))}_2\le C. +\end{equation} +For a word containing $K_A(J)$, the assertion is restricted to +parameters satisfying $S_A(1;J)\succeq cI$. The same conclusion +holds with $W$ in place of $J$, with the corresponding stability +restriction. Thus all diagonal parameters in this assertion may +be chosen after the random matrix has been observed. + +If a word has $r$ ordinary occurrences of $M$ outside its one +$K_A(z;M)$ factor, then $\mathsf P(Q;z)$ has degree at most $r$. +\end{lemma} + +We prove the simultaneous estimate in three steps. For a fixed +deterministic diagonal, Gaussian comparison gives stability along +$0\le z\le1$. We then calculate the expected diagonal of a +truncated word. Finally, concentration makes the word estimates +simultaneous in the diagonal parameters. The first step supplies +stability for each deterministic choice; it does not remove the +separate stability restriction for adaptive $A$ in +Lemma~\ref{lem:words}. + +\begin{lemma}[Stability for a deterministic diagonal] +\label{lem:matrix-path-stability} +There is $c_0>0$, depending only on $j,a_+$, with the following +property. For every fixed deterministic $0\le A\le a_+I$, and every +fixed $R>2\sqrt j$, there are constants $C,c_2>0$ and $n_0$, +uniform in $A$, such that +\begin{equation}\label{eq:deterministic-path-stability} + \Prob\left\{\norm W\le R\ \hbox{and}\quad + S_A(z;W)\succeq c_0I\text{ for all }0\le z\le1\right\} + \ge1-Ce^{-c_2n}\qquad(n\ge n_0). +\end{equation} +In addition, for every fixed $\varepsilon>0$, +$\Prob\{\norm W>2\sqrt j+\varepsilon\}\le +C_\varepsilon e^{-c_\varepsilon n}$ for sufficiently large $n$. +\end{lemma} + +\begin{proof} +We first bound the expected quadratic process that defines the +smallest eigenvalue of $S_A$. Concentration will give a margin at +each fixed $z$, and a finite mesh will extend it along the path. + +We use two standard Gaussian facts. A real-valued +$L$-Lipschitz function of a standard Gaussian vector satisfies +$\Prob\{|F-\E F|>u\}\le2\exp(-u^2/(2L^2))$. +Also, if two centered Gaussian processes have ordered squared +increments, the process with larger squared increments has the +larger expected supremum (Sudakov--Fernique). The latter comparison remains valid +after the same deterministic function is added to both processes; +see the deterministic-shift comparison in Vitale~\cite{Vitale2000}. +Both facts apply by finite nets and continuity to the compact +index sets below. The linear map from the independent standard +Gaussian coordinates to $W$, in Frobenius norm, has norm +$\sqrt{2j/n}$. + +For the comparison, write $w=A^{1/2}p$ for $\norm p=1$ and consider +\[ + X_p=w^TWw,\qquad + Y_p=2\sqrt{j/n}\,\norm w\,g^Tw, +\] +where $g$ is standard Gaussian in $\R^n$. If $w'=A^{1/2}p'$, direct +calculation gives +\begin{align*} + &\E(Y_p-Y_{p'})^2-\E(X_p-X_{p'})^2\\ + &\qquad=\frac{2j}{n} + \left[(\norm w^2-\norm{w'}^2)^2 + +2(\norm w\norm{w'}-w^Tw')^2\right]\ge0. +\end{align*} +Consequently, for each $z\in[0,1]$, +\[ + \E\sup_{\norm p=1}\{zX_p-z^2j\chi_A\norm w^2\} + \le \E\sup_{\norm p=1}\{zY_p-z^2j\chi_A\norm w^2\}. +\] +Set $T_g=\sqrt{j/n}\norm{A^{1/2}g}$. Since +$g^Tw=p^TA^{1/2}g\le\norm{A^{1/2}g}$, the last supremum is at most +\[ + \sup_{0\le v\le\sqrt{a_+}} + (2zT_gv-z^2j\chi_Av^2). +\] +The remaining supremum depends on the Gaussian vector only through +$T_g$. The variable $T_g^2$ has mean $j\chi_A$ and variance +$2j^2\Tr(A^2)/n^2\le C/n$. The inequality +$|\sqrt x-\sqrt y|\le\sqrt{|x-y|}$ therefore shows, uniformly in +$A$, that $\E|T_g-\sqrt{j\chi_A}|=o(1)$. On replacing $T_g$ by +$\sqrt{j\chi_A}$ in the preceding display, the expression becomes +$2d-d^2$, with +$0\le d=z\sqrt{j\chi_A}v\le\sqrt j\,a_+<1$. +Thus, with $\delta=(1-\sqrt j\,a_+)^2>0$, +\begin{equation}\label{eq:quadratic-process-margin} + \E\sup_{\norm p=1} + \{zp^TA^{1/2}WA^{1/2}p-z^2j\chi_Ap^TAp\} + \le1-\delta+o(1), +\end{equation} +uniformly in $A,z$. +The supremum in \eqref{eq:quadratic-process-margin} is +$a_+$-Lipschitz in $W$ for Frobenius norm. Gaussian concentration +gives an exponentially small probability that it exceeds +$1-\delta/2$, uniformly for each fixed $A,z$. + +To pass from a fixed $z$ to the entire interval, we also need a +bound on $\norm W$. Taking $A=I$ in the comparison without +deterministic offsets gives +\[ + \E\lambda_{\max}(W)\le2\sqrt{j/n}\E\norm g\le2\sqrt j. +\] +The same holds for $-W$. Concentration of these two eigenvalues +proves the stated norm tail. On $\norm W\le R$, the function of +$z$ inside the supremum in \eqref{eq:quadratic-process-margin} is +uniformly Lipschitz, with constant at most $a_+R+2ja_+^2$. +A sufficiently fine constant mesh of $[0,1]$, followed by a union +bound, therefore gives \eqref{eq:deterministic-path-stability}, for +example with $c_0=\delta/4$. +\end{proof} + +\subsection{Truncation and the loop recursion} + +The inverse in a word need not exist for every realization of the +matrix. We therefore replace each word by a globally defined +bounded function of $W$, agreeing with the original word whenever +the required norm and stability conditions hold. This permits +concentration and integration by parts before those conditions +are imposed. + +Fix a sufficiently large constant $R$. Let $\Pi_R$ be metric +projection, in the Hilbert space of real symmetric matrices with +Frobenius norm, onto $\{M:\norm M\le R\}$. This is a nonexpansive +map and is equivariant under orthogonal conjugation. Put +$\widehat W=\Pi_R(W)$. Choose a smooth compactly supported real +function $f$ which equals $x^{-1}$ on the interval +\[ + \left[\tfrac12\min(c,c_0),\,2+c+c_0+ja_+^2+a_+R\right]. +\] +This interval contains, with slack, every stable spectrum needed +both for deterministic-parameter estimates and for the restriction +$S_A(1;M)\succeq cI$ in Lemma~\ref{lem:words}. Replace $K_A(z;W)$ +by +\begin{equation}\label{eq:truncated-K} + \widehat K_A(z) + =I+A^{1/2}f(S_A(z;\widehat W))A^{1/2} + (z\widehat W-z^2B_AI). +\end{equation} +Replace every ordinary $W$ in a word by $\widehat W$ as well; denote +the resulting word by $\widehat Q_\theta(z)$, where $\theta$ lists +all its diagonal parameters. These truncated factors equal the +original factors on the stability events where they will be used. + +For each fixed length and fixed bounds on the diagonal parameters, +there is a constant $C$ such that +\begin{align} + \norm{\widehat Q_\theta(z)}&\le C, + &\norm{\widehat Q_\theta(z;W)-\widehat Q_\theta(z;W')}_{\Frob} + &\le C\norm{W-W'}_{\Frob}, + \label{eq:truncated-word-lipschitz}\\ + \Lip_{\Frob}\bigl(\widehat Q_\theta(z;\cdot) + -\widehat Q_{\theta'}(z;\cdot)\bigr) + &\le C\sqrt\Delta + &&\text{if }\norm{\theta-\theta'}_\infty\le\Delta\le1. + \label{eq:truncated-word-increment-lipschitz} +\end{align} +The constants are uniform for $z\in[0,1]$ and for all matrix sizes. +Here the Lipschitz constants concern maps into matrices with +Frobenius norm. + +We verify both estimates with constants independent of dimension. +On the operator-norm ball, all factors are bounded, and their +differentials in a perturbation $E$ are bounded in Frobenius norm +by $C\norm E_{\Frob}$. The only functional-calculus factor is +handled by writing $f$ as a Fourier integral and using +\[ + \dd(e^{itS})[E] + =it\int_0^1 e^{it(1-u)S}E e^{ituS}\,\dd u. +\] +The Fourier transform of $f$ decays faster than any power, so its +first two weighted absolute moments are finite. This formula and +the corresponding difference formula give +\[ + \norm{\dd f(S)[E]}_{\Frob}\le C\norm E_{\Frob},\qquad + \norm{(\dd f(S)-\dd f(S'))[E]}_{\Frob} + \le C\norm{S-S'}\norm E_{\Frob}. +\] +Changing the diagonal parameters by at most $\Delta$ changes their +square roots in norm by at most $\sqrt\Delta$. Consequently the +factors, and their matrix differentials, change by respectively +$C\sqrt\Delta$ and $C\sqrt\Delta\norm E_{\Frob}$. The product rule +proves the same differential assertions for a word. Integrating +along the line segment between two matrices in the convex +operator-norm ball proves \eqref{eq:truncated-word-lipschitz} and +\eqref{eq:truncated-word-increment-lipschitz} there. Composition with +the nonexpansive projection $\Pi_R$ proves them everywhere. + +These Lipschitz estimates first give the moment bounds needed for +the expectation calculation. For a fixed parameter tuple, +\begin{equation}\label{eq:word-entry-and-trace-variance} + \Var(\widehat Q_{ab})\le C/n,\qquad + \Var\left(\frac1n\Tr\widehat Q\right)\le C/n^2. +\end{equation} +The normalized trace gains a factor $n^{-1/2}$ in its Frobenius +Lipschitz constant. The same estimates for a difference of two +parameter tuples gain a factor $\Delta$ on the right-hand sides. +Fourth centered entry moments are bounded by $C/n^2$, or +$C\Delta^2/n^2$ for such a difference, by integrating the Gaussian +concentration tail. + +\begin{lemma}[Fixed-parameter diagonal expectations] +\label{lem:word-bias} +For the truncated words just defined, uniformly in deterministic +diagonal parameters in the indicated bounded ranges, in +$z\in[0,1]$, and in the diagonal index $i$, +\begin{equation}\label{eq:word-bias} + \E\widehat Q_\theta(z)_{ii} + =\mathsf P(Q_\theta;z)_{ii}+O(n^{-1}). +\end{equation} +For words without $K_A$, the prediction has no $z$ dependence. +The formal prediction for a word with $r$ ordinary $W$ letters is a +polynomial of degree at most $r$. +\end{lemma} + +\begin{proof} +The calculation has two parts. We first justify integration by +parts for the truncated words and compute the inverse factor +alone. We then obtain a recursion in the number of ordinary +$W$ factors; the same recursion determines the formal +prediction exactly and the expectation up to $O(n^{-1})$. + +All constants in this proof depend only on the fixed length and +parameter bounds. Lemma~\ref{lem:matrix-path-stability} shows that, +for every fixed deterministic parameter tuple, the truncations +agree with the actual matrices for all $z\in[0,1]$, except on an +event of probability $Ce^{-c_2n}$. Gaussian integration by parts +can be applied to the globally Lipschitz truncated factors, using +weak derivatives. More explicitly, for a scalar Lipschitz +function $H$, +\begin{equation}\label{eq:GOE-integration-by-parts} + \E[W_{ab}H(W)] + =\frac jn\E\bigl[\dd H(W)[E_{ab}+E_{ba}]\bigr]. +\end{equation} +When $a=b$, the elementary direction on the right is doubled, +as required by the diagonal variance $2j/n$. + +For \eqref{eq:GOE-integration-by-parts}, remove the projection +from the distinguished ordinary $W$ factor. The resulting +expectation error is exponentially small, since the other factors +are bounded and Gaussian norm tails control the removed factor. +On the event of agreement, the differential of the inverse is +\begin{equation}\label{eq:inverse-differential} + \dd K_A(z;W)[E]=zK_A(z;W)AEK_A(z;W). +\end{equation} +We may use the expressions on the right, with truncated factors, +on the whole probability space. The error is exponentially small +even after the index sums: all factors and their true truncated +differentials are bounded, there are at most a fixed polynomial +number of indices, and the disagreement event has exponentially +small probability. More precisely, the elementary direction +$E_{ab}+E_{ba}$ has Frobenius norm at most $2$. Both the true weak +differential and its formal replacement are therefore bounded by +$C$ in each entry; their difference vanishes on the agreement event. +Each such differential error has expected absolute value at most +$Ce^{-c_2n}$. Expanding the diagonal entry of a length-$L$ word and +applying the product rule uses at most +$(L-1)n^{L-1}$ scalar contributions before any index identifications; +the factor $j/n$ in integration by parts only improves this bound. +Thus their total error is at most a fixed polynomial in $n$ times +$e^{-c_2n}$, hence at most $C'e^{-c'n}$. Subsequent words have +bounded lengths depending only on the original length. We +therefore suppress hats during the loop calculation, with this +convention understood. + +We now compute the base case, in which the only nondiagonal +factor is $K=K_A(z;W)$. Write +$a_i=A_{ii}$, $k_i(z)=\E K_{ii}$, and +$r(z)=n^{-1}\sum_i a_i k_i(z)$. Differentiating $K_{bi}$ in +\eqref{eq:GOE-integration-by-parts} gives +\begin{align*} + \E(WK)_{ii} + &=\frac{zj}{n}\E\left[(\Tr KA)K_{ii}\right] + +\frac{zj}{n}\sum_b\E[(KA)_{bi}K_{bi}]+O(e^{-c'n})\\ + &=zjr(z)k_i(z)+O(n^{-1}). +\end{align*} +For the second sum, the absolute value of the sum over $b$ is at +most $\norm{KAe_i}\norm{Ke_i}\le C$. The factorization in the +first term follows from \eqref{eq:word-entry-and-trace-variance}; +its covariance is at most $C n^{-3/2}$, which is more than enough. +The inverse identity $K=I+zAWK-z^2B_AAK$ now yields +\begin{equation}\label{eq:K-loop-equations} + \E(WK)_{ii}=zjr k_i+O(n^{-1}),\qquad + k_i=1+z^2ja_i(r-\chi_A)k_i+O(n^{-1}). +\end{equation} + +The loop equations must select the solution near $k_i=1$. +We establish this by continuation from $z=0$, where the inverse +is the identity. Put +$h(z)=r(z)-\chi_A$. The truncated $k_i$ are uniformly bounded and +continuous in $z$. If $|h(z)|\le\delta_1$, the second equation in +\eqref{eq:K-loop-equations} gives +$\max_i|k_i(z)-1|\le C\delta_1+C/n$. Choose $\delta_1>0$ small, +using \eqref{eq:matrix-subcritical}, so that throughout this region +and for large $n$, +\[ + \left|z^2j\frac1n\sum_i a_i^2 k_i(z)\right| + \le1-\delta_2 +\] +for some fixed $\delta_2>0$. Averaging the second equation in +\eqref{eq:K-loop-equations} with weights $a_i$ gives +\[ + h(z)=z^2j\left(\frac1n\sum_i a_i^2k_i(z)\right)h(z)+O(n^{-1}), +\] +and hence $|h(z)|\le C/n$ while $|h(z)|\le\delta_1$. Since +$h(0)=0$, continuity rules out a first exit from that region for +large $n$. It follows uniformly in $z$ that +\begin{equation}\label{eq:K-and-WK-bias} + \E K_{ii}=1+O(n^{-1}),\qquad + \E(WK)_{ii}=zB_A+O(n^{-1}). +\end{equation} + +To complete the base case, we verify the formal identity +$\mathsf P(K_A(z;W))=I$ independently of the expectation calculation. +The cancellation takes place at each coefficient of the series. +In the expansion +\eqref{eq:K-formal-series}, a block $-z^2B_AA$ may be regarded as +the negative contraction of two adjacent blocks $zAW,zAW$, since +their contraction is $z^2j\chi_AA=z^2B_AA$. +Fix a completely noncrossing-paired chain $(AW)^{2m}$ with $m>0$. +Let $\mathcal A$ be the nonempty set of its pairs which join two +originally adjacent occurrences of $W$. It is nonempty because +every nonempty noncrossing pairing of a linear order has an +adjacent pair. All pairs in $\mathcal A$ are disjoint. The terms +of \eqref{eq:K-formal-series} which give this paired chain are in +bijection with subsets $T\subseteq\mathcal A$: contract the pairs +in $T$ into the negative deterministic blocks, and pair the +remaining occurrences normally. The magnitude of the final +diagonal contribution is unchanged, and its sign is $(-1)^{|T|}$. +Thus the total multiplier is +$\sum_{T\subseteq\mathcal A}(-1)^{|T|}=0$. This argument is finite +at each power of $z$ and proves the asserted formal identity. +Multiplication by surrounding diagonal matrices proves both +claims of the lemma for words with no ordinary $W$ letters. + +For the induction step, select an ordinary occurrence of $W$ in +a general word and apply integration by parts at that occurrence. +We describe the resulting finite recursion directly, without +requiring convergence of the formal series. In each product-rule +term, mark the partner of the selected occurrence: for a derivative +falling on an ordinary factor, this is the $W$ being differentiated. +If the derivative falls on $K_A$, replace that factor by +$zK_AA\,\boxed W\,K_A$ and take the boxed occurrence as the +partner. In either case, list the two marked positions in their +linear order, so the marked product reads +\[ + U\,\boxed W\,V\,\boxed W\,Q. +\] +The two orientations in \eqref{eq:GOE-integration-by-parts} +contribute respectively +\begin{equation}\label{eq:word-loop-contractions} + \frac jn\E\bigl[(UQ)_{ii}\Tr V\bigr],\qquad + \frac jn\E(UV^TQ)_{ii}. +\end{equation} +There is an additional factor $z$ when the derivative fell on +$K_A$. The second expression in +\eqref{eq:word-loop-contractions} is $O(n^{-1})$ by operator-norm +bounds, including when it contains two copies of $K_A$. +The first expression is +\begin{equation}\label{eq:word-loop-leading} + j\left(\frac1n\E\Tr V\right)\E(UQ)_{ii}+O(n^{-1}), +\end{equation} +again by \eqref{eq:word-entry-and-trace-variance}. The index sums +explain the two terms. The opposite orientation +identifies the two interior indices and gives $\Tr V$; the equal +orientation reverses the interior path and gives $V^T$. + +Consequently $\E\widehat Q_{ii}$ is the finite sum of the leading +products \eqref{eq:word-loop-leading} over all ordinary partners, +plus $z$ times the corresponding leading product for the one +$K_A$ factor if present, with an $O(n^{-1})$ remainder. Each word +$V$ and $UQ$ in this recursion has fewer ordinary $W$ letters than +the original word. Each contains at most one $K_A$: if the +derivative fell on $K_A$, the splitting places one of its two +copies in each word. Extra diagonal factors $A$ cause no problem, +since at most one is added per decrease of the induction index. + +The decrease in the number of ordinary letters holds whether the +selected occurrence lies before or after the inverse. The two +possible orders can be checked explicitly. If the original word is +$P K_A R W T$, the contraction with $K_A$ contributes +\[ + zj\,\E\left[(P K_A A T)_{ii}\frac{\Tr(K_A R)}n\right]. +\] +If it is $P W R K_A T$, that contribution is +\[ + zj\,\E\left[(P K_A T)_{ii}\frac{\Tr(R K_A A)}n\right]. +\] +Here $P,R,T$ contain no inverse factor. The displayed outer and +inner words each have one inverse factor, even though the original +distinguished occurrence may lie anywhere among the ordinary letters. + +It remains to compare this expectation recursion with the formal +prediction. Sort the noncrossing pairings by the partner of the +selected ordinary occurrence. The inside and outside may then be paired +independently; their predictions are respectively +$\mathsf p(V)$ and $\mathsf P(UQ)$. If the partner occurs inside +the series for $K_A(z;W)$, marking that occurrence in the formal +series is exactly the formal differentiation identity +$\dd K_A[E]=zK_AAEK_A$. Thus the entire sum over positions in that +series is the single $z$-weighted product just described. + +We can now induct on the number $r$ of ordinary $W$ letters. +An ordinary partner leaves a total of $r-2$ such letters in the +two smaller words; a partner in $K_A$ leaves $r-1$. The base +prediction is constant. The recursion therefore proves that +the formal prediction is a polynomial of degree at most $r$. +It also proves \eqref{eq:word-bias}: use the induction hypothesis +for the two expectations in each leading product, and average +the diagonal induction hypothesis to handle its normalized +trace. There are only finitely many products and all factors +are uniformly bounded. The same induction, with no inverse +factor, proves the statement for ordinary words. +\end{proof} + +\subsection{Making the word estimates simultaneous} + +\begin{proof}[Proof of Lemma~\ref{lem:words}] +We first make the centered estimates uniform over the diagonal +parameters, then add the expectation estimate from +Lemma~\ref{lem:word-bias}. Fix a letter pattern and let $\theta$ +range over its parameter box, which has at most $C_Ln$ real +coordinates. Work initially with the truncated words at $z=1$. Conjugation by any +diagonal sign matrix preserves the law of $W$ and conjugates +$\widehat Q_\theta$ by that same sign matrix. The diagonal +parameters commute with the conjugation, and both the projection +and the functional calculus respect it. Thus +$\E(\widehat Q_\theta)_{ab}=0$ whenever $a\ne b$. + +Define the two matrix seminorms +\[ + N_2(M)=\norm{\diag M}_2,\qquad + N_4(M)=\left(\sum_{a\ne b}|M_{ab}|^4\right)^{1/4}. +\] +Both are $1$-Lipschitz with respect to Frobenius norm. +The entry moment bounds following +\eqref{eq:word-entry-and-trace-variance} give +\begin{equation}\label{eq:centered-word-norm-expectations} + \E N_k(\widehat Q_\theta-\E\widehat Q_\theta)\le C + \quad(k=2,4). +\end{equation} +Indeed, sum the $n$ second moments for $k=2$, and the at most +$n^2$ fourth moments for $k=4$, then apply Jensen's inequality. +For two parameter tuples at distance at most $\Delta$, the same +argument, using \eqref{eq:truncated-word-increment-lipschitz}, gives +\[ + \E N_k\bigl((\widehat Q_\theta-\E\widehat Q_\theta) + -(\widehat Q_{\theta'}-\E\widehat Q_{\theta'})\bigr) + \le C\sqrt\Delta. +\] +The random norms in this last display concentrate around their +means at Gaussian scale $C\sqrt{\Delta/n}$, again by +\eqref{eq:truncated-word-increment-lipschitz}. The centered +expectations are deterministic and do not change Lipschitz +constants. + +To control every parameter tuple, approximate it successively +on finer meshes. For each $\ell\ge0$, choose a sup-norm mesh +$\mathcal N_\ell$ of spacing $2^{-\ell}$ in the parameter box. It may be chosen with +\[ + |\mathcal N_\ell|\le\exp(C_L(\ell+1)n). +\] +For a tuple $\theta$, choose nearest mesh points +$\theta_\ell\in\mathcal N_\ell$. Then +$\norm{\theta_\ell-\theta_{\ell-1}}_\infty\le C2^{-\ell}$. +For all pairs of mesh points at this distance, concentration and +a union bound show that their centered increments have both +$N_2$ and $N_4$ norm at most +\[ + C'\sqrt{\ell+1}\,2^{-\ell/2}, +\] +except with probability $Ce^{-c(\ell+1)n}$, provided $C'$ is +chosen sufficiently large. Specifically, the squared ratio +of this threshold to the Gaussian scale is a constant multiple +of $(\ell+1)n$, and its constant can be chosen to dominate the +logarithm of the number of mesh pairs. The initial mesh is +bounded in the same way using +\eqref{eq:centered-word-norm-expectations}. Summing the failure +probabilities over $\ell$, and summing the convergent increment +bounds along each chain, proves, with probability +$1-Ce^{-c_1n}$, +\begin{equation}\label{eq:simultaneous-centered-words} + \sup_\theta N_k(\widehat Q_\theta-\E\widehat Q_\theta)\le C + \quad(k=2,4). +\end{equation} +At each fixed $n$, continuity of the truncated word and its +expectation in $\theta$ justifies taking the limit of the mesh +chain. There are only finitely many letter patterns of length +at most $L$, so their events may be intersected. + +The centered words are now controlled on one event for every +parameter tuple. Lemma~\ref{lem:word-bias} supplies the remaining +bias estimate, +$N_2(\E\widehat Q_\theta-\mathsf P(Q_\theta))\le C/\sqrt n$, +uniformly in $\theta$. Sign symmetry makes the off-diagonal of +$\E\widehat Q_\theta$ identically zero. Combining these facts +with \eqref{eq:simultaneous-centered-words} proves the desired +diagonal and off-diagonal assertions for the truncated words. +Their operator norms are already bounded by construction. +On $\norm W\le R$ and at parameters with $S_A(1;W)\succeq cI$, +the truncations equal the exact words, proving the GOE version. + +The GOE estimate transfers to the zero-diagonal model through +$J=W-D_{\diag W}$. This last step uses a Frobenius bound on the +removed diagonal. Since +\[ + \norm{D_{\diag W}}_{\Frob}^{2} + \ \stackrel{\mathrm{law}}=\ \frac{2j}{n}\chi_n^2, +\] +it is bounded by a fixed constant except with exponentially +small probability. The norm tail for $W$, and a sufficiently +large choice of $R$, ensure also that both $W$ and $J$ lie in the +operator-norm ball except with exponentially small probability. +For every parameter tuple, \eqref{eq:truncated-word-lipschitz} +then bounds the Frobenius norm of the difference of the truncated +word evaluated at $W$ and at $J$ by a fixed constant. Both +$N_2$ and $N_4$ of this difference are bounded by its Frobenius +norm. The prediction is unchanged by this substitution. At +parameters with $S_A(1;J)\succeq cI$, the truncated factors +evaluated at $J$ equal the exact factors. This proves all of +\eqref{eq:word-estimates}. The polynomial assertion was proved +in Lemma~\ref{lem:word-bias}. +\end{proof} + +\subsection{Bilinear forms and regularized determinants} + +We next extract two forms of the preceding estimates. The first +controls bilinear forms for a prescribed finite family of +deterministic vectors. The second controls a regularized +determinant simultaneously over its diagonal parameters and +bounded finite-rank perturbations. Their quantifiers differ, +and we keep the two statements separate. + +\begin{lemma}[Bilinear deterministic equivalents]\label{lem:bilinear} +Fix an integer $m\ge1$, a vector bound $M_1<\infty$, a tolerance +$\varepsilon>0$, and $R>2\sqrt j$. For every deterministic +$0\le A\le a_+I$ and deterministic vectors +$v_1,\ldots,v_m\in\R^n$ with $\norm{v_r}\le M_1$, there is an +event of probability at least $1-Ce^{-c_3n}$ on which +\eqref{eq:deterministic-path-stability} holds and, with +\[ + G_A=A^{1/2}S_A(1;W)^{-1}A^{1/2}=K_A(W)A, +\] +one has, for every $r,s$, +\begin{align} + |v_r^T(G_A-A)v_s|&\le\varepsilon, + \label{eq:bilinear-G}\\ + |v_r^T(G_AW-B_AA)v_s|&\le\varepsilon, + \label{eq:bilinear-GW}\\ + |v_r^T(WG_AW-B_AI-B_A^2A)v_s|&\le\varepsilon. + \label{eq:bilinear-WGW} +\end{align} +Here $C,c_3,n_0$ may depend on $j,a_+,m,M_1,\varepsilon,R$ but +are uniform in $A$, the vectors, and $n\ge n_0$. The assertion +is for each deterministic choice of these parameters; it does +not claim simultaneity over all vector choices. +\end{lemma} + +\begin{proof} +For each of the specified bilinear forms, we calculate its +expectation and then apply concentration. Use the truncations +above with a cutoff agreeing with the inverse on the path-stability +event. Sign conjugation makes the off-diagonal expectations of +each relevant matrix vanish. +Equation~\eqref{eq:K-and-WK-bias} at $z=1$ gives +\[ + \E\widehat K_{ii}=1+O(n^{-1}),\qquad + \E(\widehat W\widehat K)_{ii}=B_A+O(n^{-1}). +\] +The identities, valid on the event of agreement, +\[ + G_AW=K_A(W)(I+B_AA)-I, + \qquad + WG_AW=WK_A(W)(I+B_AA)-W +\] +therefore give the three diagonal expectation predictions +$A$, $B_AA$, and $B_AI+B_A^2A$, with entrywise errors +$O(n^{-1})$. Their use with truncated factors changes +expectations only exponentially little, by the same argument +as in Lemma~\ref{lem:word-bias}. In particular, their errors +in any of the specified bilinear forms are $O(n^{-1})$: +for a diagonal error matrix $E$, one has +$|v_r^TEv_s|\le\norm E\norm{v_r}\norm{v_s}$. + +The expectation calculation has identified the three required +predictions. To control fluctuations, each truncated bilinear form +is $C M_1^2$-Lipschitz in $W$ for Frobenius norm, by +\eqref{eq:truncated-word-lipschitz}. Gaussian +concentration gives failure probability $Ce^{-c_3n}$ at any +fixed positive tolerance. Intersect these events for the +$3m^2$ forms and with Lemma~\ref{lem:matrix-path-stability}. +For large $n$ the expectation errors are less than half the +specified tolerance, proving the result. +\end{proof} + +\begin{lemma}[A simultaneous regularized determinant bound] +\label{lem:logdet} +Fix $\varepsilon>0$, an integer $r_0\ge0$, and $M_2<\infty$. +There is $\gamma_0>0$, depending only on $j,\varepsilon$, such +that for every fixed $0<\gamma\le\gamma_0$ and all sufficiently +large $n$, an event of probability at least $1-Ce^{-c_4n}$ has +the following property. Simultaneously for all diagonal +$0\le V\le I$ and all real matrices $E$ with +$\rank E\le r_0$ and $\norm E\le M_2$, put +\[ + b=\frac1n\Tr V,\qquad B=jb,\qquad + L=I+(BI-W)V+E. +\] +Then +\begin{equation}\label{eq:logdet-upper-bound} + \frac1n\log\det(L^TL+\gamma I)\le jb^2+\varepsilon. +\end{equation} +The constants $C,c_4$ and the lower threshold on $n$ may depend +on $j,\varepsilon,\gamma,r_0,M_2$. In particular, $V$ and $E$ +may depend on $W$. +\end{lemma} + +\begin{proof} +We first compute the mean log determinant with $E=0$ and a fixed +deterministic $V$. Regularization will then provide a global +Lipschitz bound, allowing a finite mesh in $V$ and finally the +finite-rank perturbation. Set +\[ + L_z=I+z^2BV-zWV, + \qquad K(z)=(I-zVW+z^2BV)^{-1}. +\] +Since $V\le I$, we use the proof of +Lemma~\ref{lem:matrix-path-stability} with the diagonal upper bound +equal to $1$; its stability and inverse-norm constants here depend +only on $j$. +On the path-stability event from +Lemma~\ref{lem:matrix-path-stability} with $A=V$, the determinant +of $L_z$ equals the positive determinant of $S_V(z;W)$ by +$\det(I-UV)=\det(I-VU)$, and both +$L_z^{-1}$ and $K(z)$ have uniformly bounded operator norm. +Indeed $L_z=K(z)^{-T}$, and +\eqref{eq:K-symmetric-representation} applies. Differentiating +the normalized log determinant on this event gives +\begin{align} + \frac{\dd}{\dd z}\frac1n\log\det L_z + &=\frac1n\Tr\bigl[L_z^{-1}(2zBV-WV)\bigr]\notag\\ + &=2zB\frac1n\Tr(K(z)V) + -\frac1n\Tr(WK(z)V). + \label{eq:logdet-derivative} +\end{align} +Here $VL_z^{-1}=K(z)V$, which follows by multiplying +$(I-zVW+z^2BV)V=V L_z$, and cyclicity of trace was used. + +The derivative is expressed in the two matrices whose diagonal +expectations were computed in \eqref{eq:K-and-WK-bias}. +Let $\mathcal E_V$ be the path-stability event intersected with +$\norm W\le R$, for a fixed $R>2\sqrt j$. This one event applies +to the entire interval of $z$. Equations~\eqref{eq:K-and-WK-bias}, and +replacement by the truncated factors off $\mathcal E_V$, show +uniformly in $z,V$ that the expectation of the right-hand side +of \eqref{eq:logdet-derivative}, multiplied by +$\one_{\mathcal E_V}$, is +\[ + 2zBb-zBb+O(n^{-1})=zjb^2+O(n^{-1}). +\] +All differentiated quantities are bounded on $\mathcal E_V$, +so integration and expectation may be interchanged. Since +$L_0=I$, it follows that +\begin{equation}\label{eq:unregularized-logdet-mean} + \E\left[\one_{\mathcal E_V}\frac2n\log\det L_1\right] + =jb^2+O(n^{-1}). +\end{equation} +On $\mathcal E_V$, the singular values of $L_1$ are bounded +below by a positive constant. Therefore +\[ + 0\le\frac1n\log\det(L_1^TL_1+\gamma I) + -\frac2n\log\det L_1 + \le C\gamma. +\] +Off $\mathcal E_V$, the absolute value of the regularized +normalized log determinant is bounded by +\[ + |\log\gamma|+C+2\log(1+\norm W), +\] +for $0<\gamma\le1$. The norm tails of $W$ and +\eqref{eq:deterministic-path-stability} make its expected +contribution exponentially small, for each fixed $\gamma$. +Consequently, uniformly in deterministic $V$, +\begin{equation}\label{eq:regularized-logdet-mean} + \E\left[\frac1n\log\det(L_1^TL_1+\gamma I)\right] + \le jb^2+C\gamma+O(n^{-1})+O_\gamma(e^{-c n}). +\end{equation} + +The mean bound is uniform in deterministic $V$. To obtain a +single event for all $V$, we use the concentration supplied by +regularization. Define on all real matrices +\[ + F_\gamma(M)=\frac1n\log\det(M^TM+\gamma I). +\] +Its Frobenius gradient is +$2n^{-1}M(M^TM+\gamma I)^{-1}$. Each singular value of this +gradient is at most $1/(n\sqrt\gamma)$, and hence +\begin{equation}\label{eq:regularized-logdet-lipschitz} + \Lip_{\Frob}(F_\gamma)\le\frac1{\sqrt{n\gamma}}. +\end{equation} +For fixed $V$, the map $W\mapsto L_1$ has Frobenius Lipschitz +constant at most one. Gaussian concentration therefore gives +\begin{equation}\label{eq:regularized-logdet-concentration} + \Prob\{|F_\gamma(L_1)-\E F_\gamma(L_1)|>u\} + \le2\exp\left(-\frac{\gamma n^2u^2}{4j}\right). +\end{equation} + +Choose $\gamma_0\le1$ small enough that $C\gamma_0$ in +\eqref{eq:regularized-logdet-mean} is at most +$\varepsilon/8$. For fixed $\gamma\le\gamma_0$ and sufficiently +large $n$, the expectation bound and +\eqref{eq:regularized-logdet-concentration} bound all points +of any fixed sup-norm mesh of the cube $[0,1]^n$ by +$jb^2+\varepsilon/2$, with failure $\exp(-c_\gamma n^2)$. +Indeed a mesh of spacing $\Delta>0$ has at most +$(C/\Delta)^n$ points, whose logarithm is only of order $n$. +On $\norm W\le R$, if $\norm{V-V'}\le\Delta$, then +\[ + |b-b'|\le\Delta,\qquad + \norm{(I+(BI-W)V)-(I+(B'I-W)V')}_{\Frob} + \le\sqrt n(2j+R)\Delta. +\] +By \eqref{eq:regularized-logdet-lipschitz}, the corresponding +$F_\gamma$ values differ by at most +$(2j+R)\Delta/\sqrt\gamma$, whereas $|jb^2-j(b')^2|\le2j\Delta$. +Choose $\Delta$, depending only on +$j,R,\gamma,\varepsilon$, so that these errors sum to at most +$\varepsilon/4$. Including the exponentially likely norm +event proves, simultaneously for all $V$, +\[ + F_\gamma(I+(BI-W)V)\le jb^2+3\varepsilon/4. +\] + +We have obtained the simultaneous bound for $E=0$ with room for +the perturbation. The same Lipschitz estimate +\eqref{eq:regularized-logdet-lipschitz} gives, for every allowed $E$, +\[ + |F_\gamma(M+E)-F_\gamma(M)| + \le\frac{\norm E_{\Frob}}{\sqrt{n\gamma}} + \le M_2\sqrt{\frac{r_0}{n\gamma}}. +\] +This is at most $\varepsilon/4$ for large $n$, uniformly even +when $E$ is chosen from the observed matrix. It proves +\eqref{eq:logdet-upper-bound} and the lemma. +\end{proof} diff --git a/preprints/A-spectral-gap-throughout-the-high-temperature-Sherrington-Kirkpatrick-phase-September-24-2026/build/sections/observation.tex b/preprints/A-spectral-gap-throughout-the-high-temperature-Sherrington-Kirkpatrick-phase-September-24-2026/build/sections/observation.tex new file mode 100644 index 0000000000000000000000000000000000000000..c029d28b842066e692fa099a5499f5ef31824028 --- /dev/null +++ b/preprints/A-spectral-gap-throughout-the-high-temperature-Sherrington-Kirkpatrick-phase-September-24-2026/build/sections/observation.tex @@ -0,0 +1,212 @@ +\section{From posterior estimates to a spectral gap}\label{sec:observation} + +We now prove the transfer announced in the introduction. Its inputs are +a covariance bound, a square-reweighted mean bound, and a terminal gap. +The stopping rule and good-disorder event must be independent of the test +function, so we keep the exceptional-path estimate explicit. The argument +uses the Gaussian-channel formulation of localization +\cite{ElAlaouiMontanari2021} and the variance/Dirichlet-form framework +of~\cite{CE2025}; the finite-spin identities are derived here. + +\subsection{Posterior martingales and Dirichlet forms} +Let $\mathcal F_t=\sigma(Y(s):s\le t)$ and write $\E_t$ for expectation under +$\mu_{Y(t)}$. Bayes' formula gives this posterior: the likelihood of an +observation path given $X=x$ is proportional to +$\exp(x^{\mathsf T}Y(t)-tn/2)$. The process +\[ + I(t)=Y(t)-\int_0^t\E_s x\,\dd s +\] +is the innovation Brownian motion in $\mathcal F_t$: conditional +expectation makes it a continuous martingale, and its quadratic +covariation is $tI_n$. The Brownian martingale characterization therefore +applies. Applying It\^o's formula to the finitely many posterior weights +gives, for every spin function $g$, +\begin{equation}\label{eq:filtering} + \dd(\E_t g)=\Cov_t(g,x)^{\mathsf T}\dd I(t). +\end{equation} +To verify the cancellation, differentiate the posterior expectation with +respect to $Y$; its gradient is $\Cov_t(g,x)$. Each likelihood weight +$\exp(x^{\mathsf T}Y(t)-tn/2)$ satisfies a stochastic exponential +equation. The quotient rule for their weighted sums then gives +\eqref{eq:filtering}: subtracting $\E_t x\,\dd t$ from $\dd Y$ removes +the finite-variation term. + +At every deterministic time, +\begin{equation}\label{eq:dirichlet-monotonicity} + \E\D_{Y(t)}(f)\le\D_0(f). +\end{equation} +This is conditional-variance contraction: the $i$th summand on the left +is $\E\Var(f(X)\mid X_{-i},\mathcal F_t)$, and the one on the right is +$\E\Var(f(X)\mid X_{-i})$. The conditional variance decomposition gives +the inequality after averaging over observations. In this section, an +expectation without a field subscript refers to that experiment, with +$J$ held fixed. + +\subsection{Stopping before stability fails} +Stop before the first possible loss of the field estimates. More precisely, +fix the terminal horizon $T$, take the observation stopping time $\tau_0$ +from Corollary~\ref{cor:stopping}, and set $\tau=T\wedge\tau_0$. This +rule does not use the test function. Before $\tau$, stability holds through +residual threshold $\rho_0\sqrt n$, so both posterior estimates apply. +On the specified matrix event, $\Prob(\tau_0\le T)\le e^{-an}$. + +For a fixed $J$, choose the canonical terminal good set +\[ + \mathcal T_n(J)=\{h:\Var_{\mu_h}(g)\le2\D_h(g)\text{ for every }g\}. +\] +Membership means positivity of a finite-dimensional symmetric quadratic +form with coefficients continuous in $h$, so this set is Borel. +Proposition~\ref{prop:terminal-gap} controls the probability that $Y(T)$ +misses it. Declare the stopped path bad if $\tau_0\le T$, or if +$\tau_0>T$ and $Y(T)\notin\mathcal T_n(J)$. The event is +$\mathcal F_\tau$-measurable: before $T$ only stopping can make it bad, +and at $T$ the remaining condition is determined by $Y(T)$. The two +failure probabilities give, on the typical matrix events, +\begin{equation}\label{eq:ordinary-bad} + \Prob(\mathrm{bad})\le 2e^{-a n} +\end{equation} +for a fixed $a>0$, after decreasing the rate if necessary. + +Suppose $\E_{\mu_0}f=0$ and $\E_{\mu_0}f^2=1$. Set +\[ + M_t=\E_t f,\quad N_t=\E_t f^2,\quad V_t=N_t-M_t^2, + \quad d_0=\D_0(f). +\] +By \eqref{eq:filtering} and optional stopping, $u(t)=\E V_{t\wedge\tau}$ is +absolutely continuous and satisfies, almost everywhere, +\[ + u'(t)=-\E\bigl[\one_{\{t<\tau\}}\norm{\Cov_t(f,x)}^2\bigr]. +\] +The directional estimate bounds the loss term before stopping. Together +with \eqref{eq:dirichlet-monotonicity}, applied at the deterministic time +$t$, it gives $u'(t)\ge-C_{\mathrm d}(u(t)+d_0)$, with fixed +$C_{\mathrm d}$. Solving this scalar inequality from $u(0)=1$ yields +\begin{equation}\label{eq:variance-lower} + \E V_\tau\ge v-(1-v)d_0,\qquad v=e^{-C_{\mathrm d}T}>0. +\end{equation} +All uses of optional stopping here are at bounded times and for bounded +random variables at each fixed $n$. + +\subsection{Changing the law of the observations} +The variance lower bound alone does not control how much variance lies on +bad paths. For that purpose change the initial spin law to $f^2\mu_0$, +which is a probability law by normalization, and call the resulting +observation law $\Prob^f$. Its density on $\mathcal F_t$ is $N_t$, and +its spin posterior is $\nu_t=f^2\mu_{Y(t)}/N_t$. Since $\mu_{Y(t)}$ has +full support and $f$ is nonzero, $N_t>0$. Bounded stopping gives the +density $N_\tau$ on $\mathcal F_\tau$. +Let +\[ + a_t=\E_{\nu_t}x-\E_t x. +\] +Equation~\eqref{eq:filtering} with $g=f^2$ gives +$\dd N_t/N_t=a_t^{\mathsf T}\dd I(t)$. Under $\Prob^f$, $I$ has drift $a_t$. +The drift-energy entropy identity, in the form discussed +by Lehec~\cite{Lehec2013}, has a direct finite-mixture proof here. +It\^o's formula for $\log N_t$ yields +\begin{equation}\label{eq:stopped-entropy} + D\bigl(\Prob^f|_{\mathcal F_\tau}\,\Vert\, + \Prob|_{\mathcal F_\tau}\bigr) + =\E^f\log N_\tau + =\frac12\E^f\int_0^\tau\norm{a_t}^2\,\dd t. +\end{equation} +The stopped identity needs no limiting integrability argument. At fixed +$n$, the drift satisfies $\norm{a_t}\le2\sqrt n$ and $\tau\le T$. +Novikov's condition therefore holds, and the stochastic integral in the +log-density formula has zero mean. + +Before stopping, applying \eqref{eq:input-weighted} to $f$ and to $1$ gives +\[ + b_t:=\frac{\norm{a_t}}{\sqrt n} + \le2C_{\mathrm w}\delta+\frac{C_\delta}{\sqrt n} + \left(2+\frac{\D_{Y(t)}(f)}{N_t}\right). +\] +Since $b_t\le2$, we may use $b_t^2\le2b_t$. At deterministic $t$, the density +identity and \eqref{eq:dirichlet-monotonicity} imply +\begin{equation}\label{eq:weighted-dirichlet} + \E^f\frac{\D_{Y(t)}(f)}{N_t} + =\E\D_{Y(t)}(f)\le d_0. +\end{equation} +Write $\mathcal H_\tau$ for the entropy in \eqref{eq:stopped-entropy}. +The preceding estimates give the quantitative bound +\begin{equation}\label{eq:entropy-bound} + \frac{\mathcal H_\tau}{n} + \le 2C_{\mathrm w}T\delta + +\frac{C_\delta T}{\sqrt n}(2+d_0). +\end{equation} +The separation of constants is essential: $C_{\mathrm w}$ is independent +of $\delta$, whereas $C_\delta$ may grow when $\delta$ decreases. +We next convert this entropy bound into a bound on the bad-path probability. +If an event has probabilities $p$ and $q$ under two laws, the log-sum +inequality for the event and its complement bounds relative entropy below +by $p\log(1/q)-\log2$. With $p=\Prob^f(\mathrm{bad})$, +\eqref{eq:ordinary-bad} therefore gives, for large $n$, +\[ + p\le\frac{\mathcal H_\tau+\log2}{an-\log2}. +\] +If the ordinary bad probability is zero, absolute continuity gives $p=0$. +For $d_0\le1$, \eqref{eq:entropy-bound} therefore implies +\begin{equation}\label{eq:weighted-bad-bound} + p\le\frac{4C_{\mathrm w}T}{a}\delta + +\frac{6C_\delta T}{a\sqrt n} + +\frac{2\log2}{an}. +\end{equation} +Choose $\delta>0$ once so that the first term is at most $v/8$. +With this choice fixed, enlarge $n$ until the last two terms sum to at +most $v/8$. The resulting bound holds simultaneously for all normalized +$f$ with $d_0\le1$: +\begin{equation}\label{eq:weighted-bad} + \Prob^f(\mathrm{bad})\le v/4. +\end{equation} + +Figure~\ref{fig:observation-flow} records how the two posterior estimates +will control the stopped variance and its exceptional contribution. +\input{figures/observation-flow} + +\subsection{Completion of the theorem} +We can now close the argument, keeping the order of choices explicit. +First choose $T$ from the terminal estimate. Next choose the stable-field +constants, the ordinary failure rate $a$, and the finite diagnostic length +for the directional estimate; these choices determine $C_{\mathrm d}$ +and $v$. Root stability and the operator bound on $J$ determine the +leading weighted-mean constant $C_{\mathrm w}$ independently of selection +depth. We may therefore select $\delta$ in +\eqref{eq:weighted-bad-bound}, then its diagnostic length and +$C_\delta$, and finally $n_0$. Every depth and tolerance is fixed before +$n$ grows. + +With the choices made, the proof needs a finite intersection of matrix +events. Include the directional word event and the word events for the +selected weighted-mean construction, each with its fixed length and +coefficient bounds. Lemma~\ref{lem:words} accommodates each fixed +$M_0$, and Lemma~\ref{lem:implicit-construction} supplies an actual +inverse margin independent of depth. Add the events of +Propositions~\ref{prop:stable-fields} and~\ref{prop:terminal-gap}. +Their intersection still has probability tending to one, and all constants +and the threshold $n_0$ are uniform on it and over the test functions. + +For such a matrix and a normalized $f$ with $d_0\le1$, use $V_\tau\le N_\tau$ +and the stopped density identity to obtain +\[ + \E[\one_{\mathrm{bad}}V_\tau] + \le\E[\one_{\mathrm{bad}}N_\tau] + =\Prob^f(\mathrm{bad})\le v/4. +\] +On a good path $\tau=T$ and the terminal gap applies, so +\[ + \E[\one_{\mathrm{good}}V_\tau] + \le2\E\D_{Y(T)}(f)\le2d_0. +\] +Combining these estimates with \eqref{eq:variance-lower} gives +\[ + v-(1-v)d_0\le2d_0+v/4, + \qquad\text{hence}\qquad + d_0\ge\frac{3v}{4(3-v)}\ge\frac v4. +\] +The case $d_0>1$ already satisfies this bound. Center and rescale any +nonconstant function to obtain Theorem~\ref{thm:main} once the three +inputs have been proved, with the finite constant +$C_\beta=4e^{C_{\mathrm d}T}$. We do not optimize this constant. +For the clock conversion, the self-adjoint operator +$H=\sum_i(I-P_i^0)$ has form $\D_0$, and one discrete attempt is +$I-H/n$. Its gap is therefore the asserted unscaled gap divided by $n$. diff --git a/preprints/A-spectral-gap-throughout-the-high-temperature-Sherrington-Kirkpatrick-phase-September-24-2026/build/sections/residuals.tex b/preprints/A-spectral-gap-throughout-the-high-temperature-Sherrington-Kirkpatrick-phase-September-24-2026/build/sections/residuals.tex new file mode 100644 index 0000000000000000000000000000000000000000..c90f0abdb2e6af1bb20b66cbd8f1671eb78a4d92 --- /dev/null +++ b/preprints/A-spectral-gap-throughout-the-high-temperature-Sherrington-Kirkpatrick-phase-September-24-2026/build/sections/residuals.tex @@ -0,0 +1,1069 @@ +\section{Weak residual identities and discrete differentiation} +\label{sec:residuals} + +We derive the residual identities needed for the two posterior estimates. +Moments of a fixed finite sequence of primary iterates will locate two +consecutive small increments. Ordinary test-vector recipes then control +means under square reweighting; one additional implicit recipe supplies +the linear-response test for the directional covariance estimate. The +proof proceeds from discrete derivative bounds to a weak residual rule, +and ends with the terminal identity for that implicit test. + +This section is deterministic. We fix $J$ satisfying the word estimates of +Lemma~\ref{lem:words}, fix an external field $h_0\in\R^n$, and take all +expectations with respect to $\mu_{h_0}$. We fix the depth, the finite +construction, and +its coefficient bounds independently of $n$. These choices determine a +finite required word length, which can increase when the choices change. +Constants +in derivative estimates can depend on these choices and on the corresponding +word bounds; they are uniform in $n$, $h_0$, the spin configuration, and the +permitted seeds. The size and inverse bounds for the initial implicit +construction below have the stronger, depth-independent uniformity explicitly +stated there. + +We use the discrete derivatives, product rule +\eqref{eq:discrete-product}, and norm $\norm G_*$ introduced above, +with $dg=(d_i g)_{i=1}^n$ for scalar $g$. All vector functions such as +$\tanh h^1$ are understood coordinatewise. + +\subsection{Primary fields} + +The following recursion has the Onsager-corrected form of iterative TAP +constructions, as in Bolthausen~\cite{Bolthausen2014}. Here it starts from +a Gibbs spin sample and is used only through a fixed finite depth. +Define +\begin{equation} + \begin{gathered} + m^0=x,\quad b_0=0,\qquad + m^l=\tanh h^l,\quad b_l=\av{1-(m^l)^2},\quad R_l=m^{l-1}-m^l,\\ + h^{l+1}=h_0+Jm^l-jb_lm^{l-1}\quad(l\geq1). + \end{gathered} + \label{eq:amp} +\end{equation} +The finite-depth objective is visible in the scalar pairings +$S_l=R_l^{\mathsf T}m^l/\sqrt n$. Expanding the squared increment gives +\begin{equation}\label{eq:residual-telescoping} + \frac{\norm{R_l}^2}{n}=b_l-b_{l-1}-\frac{2S_l}{\sqrt n}. +\end{equation} +Since $0\le b_l\le1$, control of $S_l/\sqrt n$ through a sufficiently +large fixed depth forces two consecutive residuals to be small. We will +prove the moment bounds +\[ + \E S_l^2\le C_l,\qquad + \E_\nu\left(\frac{S_l}{\sqrt n}\right)^2 + \le\frac{C_l}{\sqrt n}\left(1+\frac{\D_{h_0}(f)}{N}\right), + \quad N=\E f^2>0,\quad \nu=\frac{f^2\mu_{h_0}}{N}. +\] +These bounds will supply the ordinary and square-reweighted selection +estimates in Section~\ref{sec:consequences}; convergence of the primary +recursion is not required. + +The first residual explains what is needed from a test vector $U$. +Conditional covariance in one spin gives the exact identity +\[ + \E G R_1^{\mathsf T}U=\E\sum_i v_i^1d_i(GU_i). +\] +The product rule bounds its right side by $C\norm G_*$ whenever +$\norm U$ and $\sum_i|d_iU_i|$ are bounded pointwise. Thus the essential +property of our tests is a bound on the sum of their diagonal derivatives, +in addition to a bound on their Euclidean norm. The Onsager corrections +will preserve this property through the finite constructions below. +We begin with the primary fields, whose derivatives enter those tests. +We use +$\norm{Q}_{\ell^4}=(\sum_{i,j}|Q_{ij}|^4)^{1/4}$ for the unnormalized +entrywise norm. + +\begin{lemma}[Primary differentiation]\label{lem:primary-derivatives} +Let +\begin{equation} + H_1=J,\qquad M_0=I,\qquad + M_l=D_{1-(m^l)^2}H_l,\qquad + H_{l+1}=JM_l-jb_lM_{l-1}. + \label{eq:primary-formal} +\end{equation} +For each fixed $l\geq1$, +\begin{align} + &\norm{\nabla h^l-H_l}_{\Frob} + +\norm{\nabla m^l-M_l}_{\Frob}\leq C,\label{eq:primary-error}\\ + &\norm{\nabla h^l}+\norm{\nabla m^l} + +\norm{\nabla h^l}_{\ell^4} + +\norm{\diag\nabla h^l}_2\leq C, + \qquad \norm{db_l}\leq Cn^{-1/2}. + \label{eq:primary-bounds} +\end{align} +More generally, let $g_i$ be a smooth function of a fixed list of coordinates +$h_i^l$ and averages $b_l$, with uniformly bounded derivatives through the +orders being used. Deterministic site labels are permitted, provided the +same bounds hold uniformly in $i$. Its derivative matrix has bounded +operator norm, bounded diagonal $\ell^2$ norm, and bounded entrywise +$\ell^4$ norm. +\end{lemma} + +\begin{proof} +We first bound the formal matrices in \eqref{eq:primary-formal}, then +compare them with the actual discrete derivatives. Expanding $H_{l+1}$ +produces the chain +\[ + J D_{1-(m^l)^2}J\cdots D_{1-(m^1)^2}J +\] +and the terms obtained by contracting disjoint pairs of originally adjacent +$J$'s. Contracting $J D_{1-(m^s)^2}J$ contributes +$-j b_s I$. This description follows inductively from the two terms defining +$H_{l+1}$: either keep its first link, or contract that link with the next +one. Multiplicities are retained when different contractions happen to give +the same matrix. + +The contraction terms cancel the diagonal predictions. To see this, fix +a complete noncrossing pairing of the original chain. Reinserting each +contracted adjacent pair identifies its contribution with the prediction of +this full pairing, multiplied by the signs of the contractions. The +originally adjacent pairs belonging to a pairing are mutually disjoint, and +every nonempty complete noncrossing pairing contains at least one such pair. +Summing over their subsets gives $(1-1)^a=0$, where $a\geq1$ is their +number. An odd chain has no complete pairing. Lemma~\ref{lem:words} +therefore bounds the diagonal $\ell^2$ norm of $H_l$, and bounds its operator +and off-diagonal $\ell^4$ norms term by term. Its full $\ell^4$ norm is +bounded as well, since $\norm{\diag H_l}_4\leq\norm{\diag H_l}_2$. + +We now compare these formal bounds with the actual derivatives. For a +scalar function $\psi$ with bounded second derivative, Taylor's theorem gives +\[ + \abs{d_j\psi(h_i^l)-\psi'(h_i^l)d_jh_i^l} + \leq C|d_jh_i^l|^2. +\] +The Frobenius norm of the remainder is at most +$C\norm{\nabla h^l}_{\ell^4}^2$. This proves the magnetization part of +\eqref{eq:primary-error} once the field part is known. The same argument +for $\psi(z)=\sech^2z$ gives an operator bound for its site derivative +matrix $Q_l$. Averaging its rows yields +\[ + db_l=n^{-1}Q_l^{\mathsf T}\one,\qquad + \norm{db_l}\leq n^{-1/2}\norm{Q_l}. +\] +In differentiating $b_lm^{l-1}$, the term from $db_l$ is a rank-one +matrix of Frobenius norm at most +$\norm{m^{l-1}}\norm{db_l}\leq C$. The discrete product error is +$\nabla m^{l-1}$ times a diagonal whose diagonal has norm at most +$2\norm{db_l}$. Its Frobenius norm is bounded, since +$\norm{\nabla m^{l-1}}\leq C$. Starting at +$\nabla h^1=J$, these estimates close the induction. The resulting bounded +Frobenius perturbation preserves the operator, diagonal $\ell^2$, and +entrywise $\ell^4$ bounds of the formal matrices. + +For the final assertion, apply the same comparison to a smooth function +of several arguments. Expand along the segment between the arguments at +$x$ and $x^{(j)}$. The preceding bounds control the primary linear terms. +A linear term involving $db_s$ is a +rank-one matrix with a bounded-entry left vector, hence has bounded +Frobenius norm. The squared-average remainder is controlled by +\[ + n\sum_j |d_jb_s|^4\leq n\norm{db_s}^4\leq C. +\] +Mixed remainders are bounded by sums of pure squares, and primary quadratic +remainders are controlled by the entrywise $\ell^4$ bounds. This proves +the assertion. +\end{proof} + +We have in particular bounded the Euclidean change of every primary +vector under a single spin flip, at each fixed depth. This is the estimate +that will keep the local constructions inside their buffers. + +\subsection{Finite recipes for small vectors} + +The residuals will be tested against vectors built by finitely many +operations. We call these vectors ``small'' when their Euclidean norm is +bounded; the word does not refer to coordinatewise magnitude. The following +definition specifies the operations and the meaning of their site partials. + +\begin{definition}[Ordinary admissible recipe]\label{def:small-recipe} +Choose finitely many deterministic seeds $s\in\R^n$ of bounded Euclidean +norm. Seeds may depend on $J$ and $h_0$, but are fixed as functions of $x$. +Choose a finite tuple of scalar functions $\theta$ satisfying +\[ + \sup_x\bigl(|\theta_\alpha(x)|+\norm{d\theta_\alpha(x)}\bigr)\leq C +\] +for each component. Such scalar functions will be called parameters; they +may include the averages $b_l$. + +Given auxiliary fields $y^1,\ldots,y^{a-1}$ already constructed, a source +has the form +\begin{equation} + w_i^a=\sum_{s}\phi_{a,s,i}\bigl((h_i^l)_{l\in I_a},\theta\bigr)s_i + +\sum_{b0$, \eqref{eq:implicit} has a unique solution +on $\Omega_{2\rho}$. There are constants independent of the primary +depth $k$ such that, throughout this region, +\begin{equation} + \norm w+\norm{y^{\rm imp}}+\sqrt n|c_1|\leq C, + \qquad + I+j\chi A-A^{1/2}JA^{1/2}\succeq cI. + \label{eq:implicit-size} +\end{equation} +On $\Omega_{3\rho/2}$, for all sufficiently large $n$ depending on the +fixed depth, the solution also satisfies +\begin{equation} + \norm{\nabla w}_{\Frob}+\norm{\nabla y^{\rm imp}}_{\Frob} + +\sqrt n\norm{dc_1}\leq C_k. + \label{eq:implicit-frobenius} +\end{equation} +\end{lemma} + +\begin{proof} +We first control the coefficients and solve the implicit linear system. +Only then do we differentiate it, using the larger region to cover spin +flips from the smaller one. The uniform kernel and quotient bounds proved +above will control the coefficients even when some $A_i$ are small. + +Put $\ell_i=(\partial_z\Phi/\Phi)(h_i^k;r_i,a)$. +At $z=r_0$ this ratio equals $-\tanh r_0$, for every $a$. Comparison +with $z=r_0,a=0$, followed by \eqref{eq:implicit-near-root}, gives +\begin{equation} + \norm{A-D_{1-(m^k)^2}}_{\Frob}\leq C\rho\sqrt n, + \qquad \norm{\ell+t_*}\leq C\rho\sqrt n, + \qquad |\chi-b_k|\leq C\rho. + \label{eq:implicit-coefficients} +\end{equation} +In particular, the comparisons put $A$ among the diagonals allowed by +the stable-field implication, once $\rho$ is decreased. + +Let $u_0=w_0/A$ coordinatewise and $p_0=A\partial_k u_0$. The quotient +bounds just proved give $\norm{u_0}+\norm{p_0}\leq C$; their local +derivatives have coordinatewise bounds proportional to $|e_i|$ in the +first case and to $n^{-1/2}$ in the second. Direct differentiation with +the specified partial convention gives +\[ + \partial_k w=D_\ell w+p_0,\qquad + c_1=\av{\ell w+p_0}. +\] +We can now eliminate $y^{\rm imp}$ and $c_1$ and solve for the source +alone. The resulting linear equation is +\begin{equation} + \left[I-A(J-j\chi I) + +\frac jn A(m^{k-1}+t_*)\ell^{\mathsf T}\right]w + =A\left[u_0-j(m^{k-1}+t_*)\av{p_0}\right]. + \label{eq:implicit-linear-system} +\end{equation} +Write the matrix on the left as $I+AL_2$. Its symmetrically normalized +version $I+A^{1/2}L_2A^{1/2}$ differs in operator norm by $O(\rho)$ from +\[ + I+jb_k A-A^{1/2}JA^{1/2} + -\frac{2j}{n}A^{1/2}m^k(m^k)^{\mathsf T}A^{1/2}. +\] +For the rank term, use \eqref{eq:implicit-near-root} and +\eqref{eq:implicit-coefficients} in the elementary inequality +$\norm{uv^{\mathsf T}-u'v'^{\mathsf T}} +\leq\norm{u-u'}\norm v+\norm{u'}\norm{v-v'}$. +The displayed symmetric matrix has the fixed positive margin supplied +by \eqref{eq:stable-field}. Decrease $\rho$ until the perturbation is less +than half that margin. The normalized matrix then has bounded inverse, +even though it need not itself be symmetric. The identity +\[ + (I+AL_2)^{-1} + =I-A^{1/2}(I+A^{1/2}L_2A^{1/2})^{-1}A^{1/2}L_2 +\] +gives the same bound for the unnormalized inverse, since $\norm{L_2}$ is +bounded. The right side of \eqref{eq:implicit-linear-system} is bounded +in norm, because $|\av{p_0}|\leq\norm{p_0}/\sqrt n$. +This establishes the size bounds. To obtain the last assertion of +\eqref{eq:implicit-size}, remove the negative rank term from the stable-field +matrix and change $b_k$ to $\chi$. The constants used to solve the system +and obtain these bounds depend only on the fixed stability and operator +margins. + +It remains to bound the discrete derivatives. Compare the equations at +$x$ and $x^{(i)}$. On the stated inner region both configurations belong +to $\Omega_{2\rho}$ for large $n$, by +Lemma~\ref{lem:primary-derivatives}. In every coefficient-difference term, +retain the solution vector at the common point $x$. In particular, +\begin{align*} + d_i c_1 + &=\av{(d_i\ell)w(x)+d_i p_0} + +n^{-1}\ell(x^{(i)})^{\mathsf T}d_iw,\\ + d_i\{c_1(m^{k-1}+t_*)\} + &=c_1(x)d_i m^{k-1} + +(m^{k-1}(x^{(i)})+t_*)d_ic_1. +\end{align*} +Use the same product arrangement for $A$ and $\chi$ in +\[ + w=A\{u_0+(J-j\chi I)w-jc_1(m^{k-1}+t_*)\}. +\] +For clarity, put $v=m^{k-1}+t_*$ and +$\mathcal B=u_0+(J-j\chi I)w-jc_1v$, and use primes for evaluation at +$x^{(i)}$ in the next display. With +$\zeta_i=n^{-1}\{(d_i\ell)^{\mathsf T}w+\one^{\mathsf T}d_ip_0\}$, +the exact equation is +\begin{align} + &\left[I-A'(J-j\chi'I)+\frac jn A'v'(\ell')^{\mathsf T}\right]d_iw + \notag\\ + &\qquad=(d_iA)\mathcal B+A'd_iu_0 + -jA'w\,d_i\chi-jc_1A'd_im^{k-1}-jA'v'\zeta_i. + \label{eq:implicit-flip-system} +\end{align} +Thus the left matrix is precisely the matrix in +\eqref{eq:implicit-linear-system} evaluated at $x^{(i)}$. The bracket +$\mathcal B$ on the right is consequently one common-point +vector for all columns. This allows the following estimates to control +the Frobenius norm of the entire right side. + +We bound its terms in turn. A site coefficient difference +times a common vector $s$ of bounded norm has bounded Frobenius norm, +since +\[ + \sum_{i,j}s_i^2|d_j\psi_i|^2 + \leq \norm s^2\max_i\sum_j|d_j\psi_i|^2\leq C. +\] +This applies to $A,\ell,u_0,p_0$, using their coefficient bounds and +Lemma~\ref{lem:primary-derivatives}; the seed factors are retained for +$u_0,p_0$. The bracket multiplying $d_iA$ is bounded in norm by the +already established size estimates, so no bound on $1/A_i$ is needed. +Also $\norm{d\chi}\leq C/\sqrt n$, because $A_i$ is a primary site +function with an average parameter of that derivative scale. If +$\zeta_i=\av{(d_i\ell)w+d_i p_0}$, then +\[ + \norm\zeta\leq n^{-1/2} + \bigl(\norm{D_w\nabla\ell}_{\Frob} + +\norm{\nabla p_0}_{\Frob}\bigr) + \leq C/\sqrt n. +\] +Its multipliers $m^{k-1}(x^{(i)})+t_*$ have norm at most $2\sqrt n$, +which still gives a bounded Frobenius norm after summing the squared +column norms. Finally +$|c_1|\norm{\nabla m^{k-1}}_{\Frob}\leq C$, by its operator bound and +$|c_1|\leq C/\sqrt n$. Applying the uniformly bounded flipped inverses +column by column proves $\norm{\nabla w}_{\Frob}\leq C_k$. +Returning to the two displayed difference identities gives +$\norm{dc_1}\leq C_k/\sqrt n$ and the required bound for +$\nabla y^{\rm imp}$. +The operator in \eqref{eq:implicit-linear-system}, the regions +$\Omega_\rho$, and the required buffer inclusions do not depend on the +directional seed $e$. All bounds for its right side and its derivatives +use only $\norm e\leq1$; indeed $u_0=e$ and $p_0=0$ for that source. +The inverse margins, the threshold on $n$, and all these estimates are +therefore uniform over the full Euclidean unit ball of seeds. +\end{proof} + +\paragraph{Local recipes and their buffers.} +A local recipe starts with this implicit field and adds only ordinary +auxiliaries. Such +recipes are evaluated on the inner +region $\Omega_\rho$, with all estimates also available on its fixed +finite flip-neighborhoods. To define them as functions on the whole cube, +keep the actual solution through $\Omega_{2\rho}$ and assign finite values +to $y^{\rm imp},c_1$ outside that region, defining sources by their stated +expressions. The implicit equation is required only inside the buffer. +Every test +in a local statement is supported on $\Omega_\rho$. For a fixed recipe, +once $n$ is sufficiently large, every difference used in the proof stays +strictly inside the buffer. The arbitrary extension therefore does not +enter the estimates. + +The buffer sizes are chosen before the construction depth; the threshold +on dimension is chosen afterward. To verify this order, choose a finite +recipe and let $q$ be the largest number of successive spin flips used by +any difference in its proof. The induction below adds only finitely many +operations at each smaller residual index, so $q$ is finite. The primary +bounds give, for every involved level $l$, +\[ + \|R_l(x^{(i)})-R_l(x)\| + \le 2\|\nabla R_l(x)e_i\|\le C_l. +\] +With $C=\max_l C_l$, a path of at most $q$ flips increases either +residual defining $\Omega_\rho$ by at most $qC$. Thus +$n>(2qC/\rho)^2$ puts every required neighborhood of +$\Omega_{3\rho/2}$ strictly inside $\Omega_{2\rho}$. Here $q,C$, and the +threshold on $n$ may depend on the already fixed recipe and depth; +$\rho$ still depends only on the earlier stability and size margins. + +\subsection{Trace control for small derivatives} + +The primary derivative bounds do not yet give the diagonal trace control +needed for integration by parts. For small vectors, the source structure +and its correction terms supply this additional cancellation. We prove +it together with the size and derivative bounds used to construct later +auxiliary fields. + +\begin{lemma}[Small-vector differentiation]\label{lem:small-derivatives} +Every source, auxiliary field, and terminal vector $U$ in a fixed ordinary +admissible recipe satisfies +\begin{equation} + \norm U+\norm{\nabla U}_{\Frob} + +\norm{\diag\nabla U}_1\leq C. + \label{eq:small-derivative-bound} +\end{equation} +The same holds locally for recipes with the single initial implicit field +above. In these recipes the initial partial $\partial_k w$ is also a +small source. For every primary coefficient and auxiliary coefficient +in \eqref{eq:small-field}, respectively, +\begin{equation} + \begin{split} + \alpha_l=\av{\partial_lw^a}:&\quad + |\alpha_l|+\norm{d\alpha_l}\leq C/\sqrt n,\\ + \gamma_b=\av{\partial_{y^b}w^a}:&\quad + |\gamma_b|+\norm{d\gamma_b}\leq C. + \end{split} + \label{eq:small-average-bounds} +\end{equation} +All local statements include the required fixed flip buffers. Constants +can depend on the entire finite recipe and its primary depth. +\end{lemma} + +\begin{proof} +The proof has two parts. We bound the errors introduced by discrete +differentiation, then cancel the trace predictions of the retained formal +terms. Write $\norm E_{\rm nuc}$ for the sum of the singular values of +$E$. The error estimates repeatedly use +\begin{equation} + |\Tr(PE)|\leq\norm P\norm E_{\rm nuc},\qquad + \norm{BC}_{\rm nuc}\leq\norm B_{\Frob}\norm C_{\Frob},\qquad + \norm E_{\rm nuc}\leq\sum_j\norm{E_{\cdot j}}. + \label{eq:nuclear-tools} +\end{equation} +Every bounded-length diagnostic word $P$ has bounded operator norm and +bounded off-diagonal entrywise $\ell^4$ norm. For the errors arising before +we solve the differentiated implicit +equations, $P$ may contain one inverse $K$ from \eqref{eq:K}. Its required +margin is supplied by the last assertion of \eqref{eq:implicit-size}. + +\paragraph{Source errors.} +Consider one source term $\phi_i s_i$. Retain the derivative +\begin{equation} + D_\phi\nabla s+\sum_lD_{s\partial_l\phi}\nabla h^l, + \label{eq:retained-source} +\end{equation} +and put parameter derivatives into the error rather than into the formal +expression. Set +\[ + Q_{ij}=\sum_l|d_jh_i^l|,\qquad + \Theta_j=\sum_\alpha|d_j\theta_\alpha|. +\] +The finite sums have the bounds +\[ + \max_i\sum_jQ_{ij}^2+\norm Q_{\ell^4}^2 + +\norm{\diag Q}_2^2\leq C, + \qquad \norm\Theta\leq C. +\] +Taylor expansion of $d_j\phi_i$ has linear parameter terms and a remainder +bounded entrywise by +\begin{equation} + C\{Q_{ij}^2+Q_{ij}\Theta_j+\Theta_j^2\}. + \label{eq:source-taylor-error} +\end{equation} +After multiplication by $s_i$, the linear parameter terms have bounded +nuclear norm: each is a rank-one matrix with a bounded-norm left vector +and a bounded-norm right vector. For the $Q^2$ part, the matrix before +multiplication by $D_s$ has bounded Frobenius norm. Its product with +$D_s$ has bounded nuclear norm by \eqref{eq:nuclear-tools} and +$\norm{D_s}_{\Frob}=\norm s\leq C$. +For the mixed part the sum of column norms is at most +\[ + C\sum_j\Theta_j\Bigl(\sum_i s_i^2Q_{ij}^2\Bigr)^{1/2} + \leq C\norm\Theta + \Bigl(\sum_i s_i^2\sum_jQ_{ij}^2\Bigr)^{1/2}\leq C. +\] +For the last part it is at most $C\norm s\sum_j\Theta_j^2\leq C$. +These separate estimates also apply to the combined remainder in +\eqref{eq:source-taylor-error}: entry by entry, assign each summand in +the bound its proportional share of the actual remainder. + +The product rule also has a remainder with entries bounded by +\[ + C|d_js_i|\{Q_{ij}+\Theta_j\}. +\] +Assume $\norm{\nabla s}_{\Frob}\leq C$, which is zero for a seed, +holds for the initial implicit field by +Lemma~\ref{lem:implicit-construction}, and is inductive for later fields. +The $\Theta$ part has bounded sum of column norms by Cauchy--Schwarz. +For the $Q$ part, test against $P$ and split the trace into diagonal and +off-diagonal terms. H\"older's inequality gives +\begin{align*} + \sum_{i\ne j}|P_{ji}|\,|d_js_i|Q_{ij} + &\leq \norm{P_{\rm off}}_{\ell^4} + \norm{\nabla s}_{\Frob}\norm Q_{\ell^4}\leq C,\\ + \sum_i|P_{ii}|\,|d_is_i|Q_{ii} + &\leq\norm P\norm{\diag\nabla s}_2\norm{\diag Q}_2\leq C. +\end{align*} +The first line uses exponents $4,2,4$. In particular, the argument does +not assume the diagonal $\ell^1$ bound for $\nabla s$ that we are still +proving. Bounded entries of $Q$ also give a bounded Frobenius norm for +this error. We have therefore controlled each source error both in +Frobenius norm and in trace against the indicated words. + +\paragraph{Average corrections and size bounds.} +Next we control the averaged coefficients in the correction terms. +Differentiating a coefficient function in a primary argument gives another +source with the same inputs and bounds. Apply the size and Frobenius +parts of the preceding estimates to that source, then average: +\[ + |\av u|\leq\norm u/\sqrt n,\qquad + \norm{d\av u}\leq\norm{\nabla u}_{\Frob}/\sqrt n, +\] +which proves the first line of \eqref{eq:small-average-bounds}. +The implicit coefficient $c_1$ already has these bounds from its direct +construction. For an auxiliary partial the scale is different: it is a +bounded coefficient vector, rather than a small vector. Averaging its +difference estimate +$|d_j\phi_i|\leq C(Q_{ij}+\Theta_j)$ and using the row bounds above +gives $|\av\phi|+\norm{d\av\phi}\leq C$. This proves the second +line, including the self-coefficient $\chi$ when it occurs. + +For a primary correction $\alpha_lm^{l-1}$ retain +$\alpha_l\nabla m^{l-1}$. The dropped rank-one matrix has nuclear norm +at most $\norm{m^{l-1}}\norm{d\alpha_l}\leq C$. The dropped product +error is $\nabla m^{l-1}$ times a diagonal of signed entries +$2d_i\alpha_l$. Its nuclear norm is bounded by +\[ + 2\norm{\nabla m^{l-1}}_{\Frob}\norm{d\alpha_l} + \leq C\sqrt n\,n^{-1/2}\leq C. +\] +This also holds for $m^0=x$. For an auxiliary correction $\gamma_bw^b$, +the analogous rank-one and product errors have bounded nuclear norm using +$\norm{w^b}+\norm{\nabla w^b}_{\Frob}\leq C$ and +$|\gamma_b|+\norm{d\gamma_b}\leq C$. +When primary derivatives in retained terms are replaced by +$H_l,M_l$, their Frobenius errors from +Lemma~\ref{lem:primary-derivatives} are multiplied either by a +small-vector diagonal or by an $O(n^{-1/2})$ scalar. In the former case +\eqref{eq:nuclear-tools} applies; in the latter use +$\norm E_{\rm nuc}\leq\sqrt n\norm E_{\Frob}$. Both give bounded +nuclear norm. + +We can now close the induction for the size and Frobenius parts of +\eqref{eq:small-derivative-bound}. In \eqref{eq:small-field}, the terms are a +bounded-norm leading term, primary corrections bounded by +$C\sqrt n\,n^{-1/2}$, and bounded-norm auxiliary corrections. +The retained source derivative \eqref{eq:retained-source} is +Frobenius-bounded: use the preceding auxiliary derivative bounds and +$\norm{D_{s\partial_l\phi}\nabla h^l}_{\Frob} +\leq\norm{s\partial_l\phi}\norm{\nabla h^l}$. +The retained primary correction is bounded in Frobenius norm by +$|\alpha_l|\norm{\nabla m^{l-1}}_{\Frob}\leq C$. +Thus the size and Frobenius induction closes before we use or prove +the diagonal $\ell^1$ assertion. + +\paragraph{Solving the differentiated self-dependence.} +The initial implicit source needs one additional step before the trace +cancellation. Retain its site partials, freeze the parameters and +averages, and replace the primary derivatives by their formal matrices. +This gives +\begin{equation} + \begin{split} + \nabla w&=A\nabla y^{\rm imp}+D_{\partial_k w}H_k+E_1,\\ + \nabla y^{\rm imp}&=(J-j\chi I)\nabla w-jc_1M_{k-1}+E_2. + \end{split} + \label{eq:implicit-formal-differentiation} +\end{equation} +The errors have bounded Frobenius norm and bounded trace against words +containing at most one $K$. The parameter $c_1$ causes no exception: +write the corresponding source term as +$-j(\sqrt n c_1)A(t_*/\sqrt n)$, so that it is exactly of the source +form above. Its difference bound follows from +\eqref{eq:implicit-frobenius}. +Solving \eqref{eq:implicit-formal-differentiation} yields +\begin{equation} + \nabla w + =K\{D_{\partial_k w}H_k-jc_1AM_{k-1}+E_1+AE_2\}, + \qquad K=(I-AJ+j\chi A)^{-1}. + \label{eq:implicit-derivative-solved} +\end{equation} +Testing a propagated error against an ordinary word now puts at most +one $K$ into the test word. The preceding error estimates therefore give +both bounded trace and bounded Frobenius norm. Substitution gives the same +conclusion for +$\nabla y^{\rm imp}$. Subsequent fields are ordinary: propagating these +errors multiplies +them only by ordinary words. Induction therefore bounds the trace of +every final error against an ordinary word. Since the recipe is fixed +and finite, the required increases in word length are finite as well. + +\paragraph{Cancellation of the formal traces.} +All differentiated errors are now controlled. To finish, we must bound +the diagonal $\ell^1$ norm of the retained formal derivative. A single +ordinary field already illustrates the cancellation. From +$w_i=\phi(h_i^1)s_i$ we obtain +$y=Jw-j\av{\phi'(h^1)s}x$. If $P=D_{\phi'(h^1)s}$, its retained +derivative is $JPJ-j\av{\phi'(h^1)s}I$, whose two diagonal predictions +cancel. We now show that the correction at each node gives this same +cancellation along every dependency path, including paths through the +implicit field. + +At the point $x$, freeze the local partial vectors and all their averages. Index +each primary or auxiliary field by a node $a$, and attach its source to +the same node: for example, $h^l$ has source $m^{l-1}$, and $y^a$ has +source $w^a$. Let $p_{ad}$ be the partial of the source at node $a$ +with respect to the field at node $d$. Introduce a formal variable $z$, +and denote the formal source and field derivatives by +$\mathfrak S_a$ and $\mathfrak H_a$, respectively. They obey +\begin{equation} + \mathfrak S_a=\sum_dD_{p_{ad}}\mathfrak H_d, + \qquad + \mathfrak H_a=zJ\mathfrak S_a + -z^2j\sum_d\av{p_{ad}}\mathfrak S_d. + \label{eq:formal-graph} +\end{equation} +At the spin leaf, the source derivative for $h^1$ is $I$ and its field +derivative is $zJ$. A terminal source uses just the first equation. +At $z=1$ these are precisely the retained formal derivatives. In the +implicit case the only cyclic node is $y^{\rm imp}$ and its source $w$; +closing it gives +\begin{equation} + \mathfrak S_{\rm imp} + =K(z)\{D_{\partial_k w}\mathfrak H_{h^k} + -z^2jc_1A\mathfrak S_{h^k}\}, + \quad K(z)=(I-zAJ+z^2j\chi A)^{-1}. + \label{eq:formal-self-loop} +\end{equation} +Every other dependency is acyclic and linear. Solving the single cyclic +node therefore leaves at most one inverse $K(z)$ in each product of the +closed formulas. + +The averaged term in \eqref{eq:formal-self-loop} is essential even for +$k=2$. Write $D=D_{1-(m^1)^2}$ and $P=D_{\partial_2w}$, so that +$b_1=n^{-1}\Tr D$ and $c_1=n^{-1}\Tr P$. Then +\[ + \mathfrak H_{h^2}=z^2(JDJ-jb_1I),\qquad + \mathfrak S_{h^2}=zDJ, +\] +and the implicit derivative is +$K(z)\{z^2P(JDJ-jb_1I)-z^3jc_1ADJ\}$. +At degree four, the two noncrossing pairings of $AJAJPJDJ$ have +predictions +\[ + j^2\chi b_1 AP, + \qquad j^2c_1\frac{\Tr(AD)}n A. +\] +The first is canceled by the terms from $-j\chi A$ in the +quadratic coefficient of $K(z)$ and $-jb_1I$ in +$\mathfrak H_{h^2}$. The second is canceled by the degree-four +term $-jc_1AJADJ$ from the averaged correction. This calculation exhibits +the first cancellation involving both the +self-dependent node and its primary input. We now account for every +degree by following dependency paths. + +We work coefficient by coefficient. An uncontracted +dependency path, followed backwards from a field to the spin leaf, has +successive links $JD_{p_{ad}}$ and ends with the $J$ link of $h^1$. +A terminal source contributes an initial partial diagonal and then such +a path. The choices of input nodes index these paths separately, preserving +all multiplicities. Expanding \eqref{eq:formal-graph} is +inclusion--exclusion over disjoint originally adjacent link pairs: the +first term keeps the first link, while the second contracts it with the +next link, with factor $-j\av{p_{ad}}$, and continues from the input +node's source. That continuation skips exactly the link just contracted. +Every contraction has degree two in $z$, equal to the degree of the +replaced links. Consequently a coefficient of degree $L$ involves only +uncontracted paths of length $L$, of which there are finitely many even +with the self-dependence. + +To compute the trace prediction, test the formal derivative by a bounded +diagonal $D$. Fix a nonempty path and a full noncrossing pairing of its +links. Reinserting contracted adjacent pairs restores that same +full-pairing prediction. +Each contraction contributes its minus sign. Conversely, all subsets of +the originally adjacent pairs of that full pairing occur, with precisely +these signs. Such a full pairing has at least one originally adjacent +pair, so its total coefficient is zero. Paths of odd length have no full +pairing. There is no empty path contributing a derivative of a small +seed, because all seeds are fixed with respect to $x$; parameter +derivatives were already put into the error. Thus the trace prediction +vanishes as a formal power series. The +polynomial-prediction assertion for single-inverse words in +Lemma~\ref{lem:words} then makes it zero at $z=1$ in the closed formulas. + +To pass from this zero prediction to a bounded trace, we use the +small-vector structure. Every product in a closed formal derivative +of a small vector contains +either a small-vector diagonal or a scalar of size $O(n^{-1/2})$: +a path giving a nonzero derivative must enter the primary nodes before +reaching the spin leaf. Entry from a small source uses +either its primary partial, a small vector, or the average of that partial +in a correction term. The latter has the indicated scalar size. +For the implicit starting node these are exactly +$\partial_k w$ and $c_1$ in \eqref{eq:formal-self-loop}. + +For a term containing a small diagonal $D_s$, rotate the tested trace +to put that factor first. The word estimate for the remaining factors +and Cauchy--Schwarz give +\[ + \left|\Tr(D_sQ)-\sum_i s_i\,[\mathsf P(Q)]_{ii}\right| + \leq\norm s\norm{\diag(Q-\mathsf P(Q))}_2\leq C. +\] +The normalized-trace prediction is cyclically invariant by +Lemma~\ref{lem:words}, so the rotation does not change the prediction +being cancelled. For a term with a scalar of size $O(n^{-1/2})$, its +unscaled trace error is at most $C\sqrt n$ by the diagonal $\ell^2$ +word estimate, and multiplication gives the same $O(1)$ bound. +The closed formula has finitely many terms. Combining their errors +with the zero prediction and the already controlled differentiated errors +gives +\[ + |\Tr(D\nabla U)|\leq C\qquad(\norm D\leq1,\ D\text{ diagonal}). +\] +Taking $D_{ii}$ to be the sign of $d_iU_i$ proves the remaining assertion +$\norm{\diag\nabla U}_1\leq C$. All arguments were pointwise inside +the prescribed buffers, which proves the local version as well. +\end{proof} + +\subsection{The weak residual rule} + +We now apply the diagonal $\ell^1$ estimate to the exact conditional +integration-by-parts identity at the first primary field. The scalar +integral identity then reduces each higher residual to earlier ones. +We carry out this induction for ordinary tests and square-density tests +together. + +\begin{lemma}[Weak residual rule]\label{lem:residual} +Let $U$ be a terminal vector in a fixed recipe admissible from level +$p\geq1$. For an ordinary recipe and every scalar $G$, +\begin{equation} + \left|\E G R_p^{\mathsf T}U\right|\leq C\norm G_*. + \label{eq:weak-residual} +\end{equation} +For a local recipe with implicit starting index $k$, the same conclusion +holds when $p\leq k$ and $G$ is supported on $\Omega_\rho$. + +There is also an ordinary square-density version. If +$\sup_x(|H_0(x)|+\norm{dH_0(x)})\leq C_0$, then for every $f$, +\begin{equation} + \left|\E f^2H_0 R_p^{\mathsf T}U\right| + \leq C\{\E f^2+\D_{h_0}(f)\}. + \label{eq:square-residual} +\end{equation} +Constants are uniform in $f,G,H_0$ subject to the stated bound, and may +depend on $p$, the recipe, its coefficient bounds, and the finite set of +word diagnostics needed in the proof. +\end{lemma} + +\begin{proof} +We induct strongly on $p$ over all finite admissible recipes, proving +the two assertions together. Each step may enlarge the original recipe, +but it adds only finitely many auxiliary operations and invokes only +smaller residual indices. Consequently every particular instance of the +induction uses finitely many operations and a finite word-length bound. + +At the base level $p=1$, condition on all spins except the one being +differentiated. Its conditional mean is $m_i^1$ and its conditional +variance is $v_i^1$. A function of this single spin is affine, with linear +coefficient $d_i$ of the function. Taking its conditional covariance +with the spin gives the exact identity +\begin{equation} + \E\sum_i(x_i-m_i^1)GU_i + =\E\sum_i v_i^1d_i(GU_i). + \label{eq:conditional-ibp} +\end{equation} +By \eqref{eq:discrete-product}, its right side is +\[ + \E\sum_i v_i^1Gd_iU_i + +\E\sum_i v_i^1(d_iG)\{U_i-2x_id_iU_i\}. +\] +The diagonal derivative bound makes the first term at most $C\E|G|$. +For the second term, the coefficient vector has pointwise norm at most +$\norm U+2\norm{\diag\nabla U}_2\leq C$. +Weighted Cauchy--Schwarz and $0p$ use the terminal source +$m^{l-1}/\sqrt n$, admissible from level $p$, and multiplier +$\sqrt n\alpha_l$. The multiplier bounds follow from +Lemma~\ref{lem:small-derivatives}. The remaining term is handled by the +identity +\begin{equation} + R_p^{\mathsf T}m^{p-1} + =n(b_p-b_{p-1})-R_p^{\mathsf T}m^p. + \label{eq:residual-inner-product} +\end{equation} +This follows from +$(m^{p-1}-m^p)^{\mathsf T}(m^{p-1}+m^p) +=\norm{m^{p-1}}^2-\norm{m^p}^2$. +The first term on the right of \eqref{eq:residual-inner-product} +cancels $a_pn\alpha_p$ in +\eqref{eq:residual-induction-expansion}; the remaining term is handled +with the admissible source $m^p/\sqrt n$ and multiplier +$\sqrt n\alpha_p$. This completes the induction for both +\eqref{eq:weak-residual} and +\eqref{eq:square-residual}. In the local version it stops at level $k$; +each added source respects the support and buffer requirements of the +initial implicit construction. +\end{proof} + +The last step puts the implicit source into the form needed for the +posterior comparison. The following identity is a direct application of +the weak residual rule, with the averaged correction retained. + +\begin{lemma}[Terminal residual identity]\label{lem:terminal-residual} +For the implicit construction at index $k$ and every $G$ supported on +$\Omega_\rho$, +\begin{equation} + \left|\E G\left[ + \{h^k-h_0-(J-jb_{k-1}I)m^k\}^{\mathsf T}w + -j(b_k-b_{k-1})n c_1\right]\right| + \leq C\norm G_*. + \label{eq:terminal-residual} +\end{equation} +\end{lemma} + +\begin{proof} +The vector in braces is $JR_k-jb_{k-1}(R_{k-1}+R_k)$. +To express its pairing with the implicit source in terms of earlier +residuals, use symmetry of $J$, the equation +$Jw=y^{\rm imp}+j\chi w+jc_1m^{k-1}$, and +\eqref{eq:residual-inner-product}. The bracketed expression becomes +\[ + R_k^{\mathsf T}y^{\rm imp} + +j(\chi-b_{k-1})R_k^{\mathsf T}w + -jb_{k-1}R_{k-1}^{\mathsf T}w + -jc_1R_k^{\mathsf T}m^k. +\] +Apply Lemma~\ref{lem:residual} to the first three terms, absorbing their +bounded parameters into the test. For the last term, use source +$m^k/\sqrt n$ and multiplier $\sqrt n c_1$. The bounds needed for this +multiplier are precisely \eqref{eq:implicit-size} and +\eqref{eq:implicit-frobenius}. +\end{proof} diff --git a/preprints/A-spectral-gap-throughout-the-high-temperature-Sherrington-Kirkpatrick-phase-September-24-2026/build/sections/stability.tex b/preprints/A-spectral-gap-throughout-the-high-temperature-Sherrington-Kirkpatrick-phase-September-24-2026/build/sections/stability.tex new file mode 100644 index 0000000000000000000000000000000000000000..bf1a400dba38a3a0d8cacaabcde3f1ef9238e255 --- /dev/null +++ b/preprints/A-spectral-gap-throughout-the-high-temperature-Sherrington-Kirkpatrick-phase-September-24-2026/build/sections/stability.tex @@ -0,0 +1,1207 @@ +\section{Stable fields along Gaussian observations} +\label{sec:stability} + +The observation argument requires control of every approximate solution +of the field equation at the fields visited by the observation process. +We first prove this stability statement, including perturbations of the +diagonal variance matrix, and then derive a deterministic existence, +uniqueness, and inverse estimate for the field equation. + +For a symmetric matrix $J$, a field $h_0\in\R^n$, and $y\in\R^n$, define +\begin{equation} + \begin{split} + m(y)&=\tanh y,\qquad q(y)=\av{m(y)^2},\qquad + b(y)=1-q(y),\\ + V_y&=D_{1-m(y)^2},\qquad B(y)=jb(y),\\ + F_{J,h_0}(y)&=y-h_0+(B(y)I-J)m(y). + \end{split} + \label{eq:field-map} +\end{equation} +For fixed $J$, abbreviate $F_h=F_{J,h}$; we also suppress $h$ when +there is no ambiguity. Hyperbolic functions act coordinatewise. To measure +stability while allowing the diagonal factor to vary, define, for each +nonnegative diagonal matrix $A$, +\begin{equation} + \mathsf H_J(y,A) + =I+A^{1/2}\left(B(y)I-J-\frac{2j}{n}m(y)m(y)^{\mathsf T}\right)A^{1/2}. + \label{eq:field-hessian} +\end{equation} +The matrix $\mathsf H_J(y,V_y)$ is the symmetric version of the +possibly nonsymmetric derivative $F'_{J,h_0}(y)$. The external field +$h_0$ enters the residual $F_{J,h_0}(y)$ but not +$\mathsf H_J(y,A)$. This separation will let us use the same matrix +witness when the observation field moves. + +\begin{proposition}[Stability along observations] +\label{prop:stable-fields} +Fix $j\in(0,1)$ and $00$, depending only on $j,T$, and events +$\mathcal J_n$ for the zero-diagonal Gaussian matrix $J$, such that +$\Prob(J\in\mathcal J_n)\longrightarrow1$ and the following assertion +holds for every $J\in\mathcal J_n$. Let $X\sim\mu_0$, let $B_0$ be an +independent standard $n$-dimensional Brownian motion, and put +$Y(t)=tX+B_0(t)$. With conditional probability at least $1-e^{-an}$, +simultaneously for $0\le t\le T$, $y\in\R^n$, and all diagonal $A$, +\begin{equation} + \left. + \begin{gathered} + \norm{F_{J,Y(t)}(y)}\le2\rho_0\sqrt n,\\ + 0\le A\le(1+\eta)I,\qquad + \norm{A-V_y}_{\Frob}\le\epsilon_A\sqrt n + \end{gathered} + \right\} + \quad\Longrightarrow\quad + \mathsf H_J(y,A)\succ cI. + \label{eq:stable-field} +\end{equation} +We may require $j(1+\eta)^2<1$ and +$\norm J\le2\sqrt j+\theta$ on $\mathcal J_n$, for any sufficiently +small fixed $\theta>0$ chosen in the proof. +\end{proposition} + +\begin{corollary}[Stopping formulation]\label{cor:stopping} +For every $J\in\mathcal J_n$ there is an observation stopping time $\tau_0$ +such that +\[ + \Prob(\tau_0\le T\mid J)\le e^{-an}, +\] +and, for every $t\in[0,T]$ with $t<\tau_0$, the implication \eqref{eq:stable-field} holds +at $Y(t)$ with residual threshold $\rho_0\sqrt n$ in place of +$2\rho_0\sqrt n$. The stopping rule depends on $J$ and the observations, +and is independent of any test function. +\end{corollary} + +\begin{proof} +We express failure of stability through a witness set that is fixed +before observing the path. Let $\mathcal W_J$ consist of pairs $(y,A)$ satisfying +$0\le A\le(1+\eta)I$ and $\norm{A-V_y}_{\Frob}\le\epsilon_A\sqrt n$ +for which $\mathsf H_J(y,A)\not\succ cI$. Define the least +normalized residual among these witnesses by +\[ + d_J(h)=\inf_{(y,A)\in\mathcal W_J} + \frac{\norm{F_{J,h}(y)}}{\sqrt n}, + \qquad \inf\varnothing=+\infty. +\] +If $\mathcal W_J$ is nonempty, $d_J$ is finite and $n^{-1/2}$-Lipschitz, +because $F_{J,h}(y)=F_{J,0}(y)-h$. Let +\[ + \tau_0=\inf\{t\in[0,T]:d_J(Y(t))\le\rho_0\}, +\] +with value $+\infty$ when the set is empty. Continuity makes the first +hitting time an observation stopping time. +Before the hit, every violating witness has residual greater than +$\rho_0\sqrt n$. At a hit, the definition of the infimum supplies a +violating witness with residual strictly below $2\rho_0\sqrt n$; +attainment of the infimum is unnecessary. +Proposition~\ref{prop:stable-fields} therefore bounds the hitting probability. +\end{proof} + +The proof of Proposition~\ref{prop:stable-fields} has three parts. A +Gaussian integral bounds the weight of unstable configurations; its +scalar rate is controlled by the entropy inequality, and its conditional +matrix factor is controlled near the equality laws. A local-volume +estimate then shows that any unstable approximate zero would contribute +too much to that integral. Finally we transfer the resulting estimate to +the original observation law, uniformly over the time interval. + +\subsection{A Gaussian integral and its scalar rate} + +We carry out the integral calculation first under a planted Gaussian +law, whose relation to the original law will be proved when we assemble +the path estimate. Under this law, +\begin{equation} + \wt J=W+\frac jn\one\one^{\mathsf T},\qquad + h_0=t\one+\sqrt t\,g, + \label{eq:planted-field} +\end{equation} +where $W$ is centered GOE of scale $j$ and $g$ is independent standard +Gaussian. We retain the diagonal of $\wt J$ during this calculation. +For a Borel set $\mathcal E\subseteq\R^n\times\mathrm{Sym}_n$, define +\begin{equation} + I_{t,\sigma}(\mathcal E) + =(2\pi\sigma^2)^{-n/2}\int + \one_{\{(y,\wt J)\in\mathcal E\}} + \exp\left\{-\frac{\norm{F_{\wt J,h_0}(y)}^2}{2\sigma^2} + +\frac{njb(y)^2}{2}\right\}\,\dd y, + \qquad \sigma>0. + \label{eq:field-integral} +\end{equation} +The weight $e^{njb(y)^2/2}$ compensates for the exponential scale of +the local determinant: Lemma~\ref{lem:local-field-volume} will use the +regularized determinant bound to turn an approximate zero into a lower +bound for this integral. The sets to which we apply the integral depend +on $y,\wt J$ and are independent of $h_0$. Thus the observation can be +averaged while retaining the restriction to $\mathcal E$. + +For fixed $y,t,\sigma$, introduce the scalar quantities +\begin{equation} + \begin{gathered} + m=\tanh y,\quad q=\av{m^2},\quad b=1-q,\quad B=jb, + \quad M=\av m,\\ + d=t+jM,\qquad s=t+\sigma^2+jq,\qquad S=s+jq,\\ + a_y=\av{(y-d\one)m},\qquad z=y-d\one+Bm,\qquad + \psi=\frac{j(a_y-sb)^2}{2sS}. + \label{eq:scalar-parameters} + \end{gathered} +\end{equation} +Write $\varphi_{d,s}$ for the density of $N(d,s)$, with $s$ denoting +the variance, and define +\begin{equation} + G_{t,\sigma,n}(y) + =\prod_{i=1}^n\varphi_{d,s}(y_i)\,e^{n\psi}. + \label{eq:scalar-density} +\end{equation} +The Gaussian parameters in this expression depend on $y$ through the +preceding averages; the product is therefore not a fixed product density. + +\begin{lemma}[Conditional Gaussian reduction] +\label{lem:gaussian-reduction} +With the preceding definitions, and with $g'$ an independent standard +Gaussian vector, +\begin{equation} + \E I_{t,\sigma}(\mathcal E) + =\int \sqrt{\frac sS}\,G_{t,\sigma,n}(y) + \Prob\left((y,\wt J)\in\mathcal E\,\middle|\, + Wm+\sqrt{t+\sigma^2}\,g'=z\right)\,\dd y. + \label{eq:conditional-integral} +\end{equation} +The conditional distribution in this identity is the usual Gaussian +conditional distribution and is defined for every $y$, since +$\sigma>0$. +\end{lemma} + +\begin{proof} +We first evaluate the integral without its indicator. For fixed $m$, +the GOE covariance is +\[ + \Cov(Wm)=jqI+\frac jnmm^{\mathsf T}. +\] +Convolution with the heat kernel of variance $\sigma^2$ and with +$\sqrt t\,g$ therefore produces the Gaussian vector +$Wm+\sqrt{t+\sigma^2}\,g'$, of covariance +$sI+jmm^{\mathsf T}/n$. Its determinant is $s^nS/s$, and its inverse +is +\[ + \frac1sI-\frac{j}{nsS}mm^{\mathsf T}. +\] +Moreover, +$F_{\wt J,h_0}(y)=z-Wm-\sqrt t\,g$. The Gaussian density at $z$, +multiplied by $e^{njb^2/2}$, differs from +$\prod_i\varphi_{d,s}(y_i)$ by the prefactor $\sqrt{s/S}$ and the +exponential whose logarithm divided by $n$ is +\[ + -\frac{Ba_y}{s}-\frac{B^2q}{2s} + +\frac{j(a_y+Bq)^2}{2sS}+\frac{jb^2}{2} + =\frac{j(a_y-sb)^2}{2sS}. +\] +This calculation gives the identity for the unrestricted integral. +To retain the indicator, condition the joint Gaussian pair formed by +$W$ and the convolved observation on the value $z$ of the latter. +The resulting conditional probability is exactly the factor in +\eqref{eq:conditional-integral}. All integrands are nonnegative, so +Tonelli's theorem justifies the interchange of integrations. +\end{proof} + +The scalar entropy inequality controls the remaining integral in +\eqref{eq:conditional-integral}. To apply it, replace empirical averages +in \eqref{eq:scalar-parameters} by integration against a probability +measure $P$ with finite second moment, keeping the notation +$M,q,b,d,s,S,a_y,\psi$. Where needed, write $a(P)$ instead of $a_y$. +Lemma~\ref{lem:scalar-entropy} then gives +\begin{equation} + \psi(P,t,\sigma)-D(P\Vert N(d,s))\le0 + \qquad (s>0). + \label{eq:rate-nonpositive} +\end{equation} +At $\sigma=0$, Lemma~\ref{lem:gaussian-equality} identifies all equality +cases: they form the curve +\[ + P_t=N(s_*(t),s_*(t)),\qquad + s_*(t)=t+j\E_{P_t}\tanh y, +\] +with the point-mass convention $P_0=\delta_0$. In particular, +$s_*(t)\le t/(1-j)$ and +$q(P_t)=(s_*(t)-t)/j\le t/(1-j)$. + +\begin{lemma}[Uniform scalar rate bounds] +\label{lem:scalar-rates} +Fix $j\in(0,1)$ and $T<\infty$. +\begin{enumerate}[label=\textup{(\roman*)}] +\item For every $s_{\min}>0$ and $L<\infty$, one can choose $R<\infty$ +such that, for all sufficiently large $n$, uniformly over +$0\le t\le T$ and $0<\sigma\le1$, +\[ + \int_{\{s\ge s_{\min},\,\av{y^2}>R^2\}} + G_{t,\sigma,n}(y)\,\dd y\le e^{-Ln}. +\] +For every $\varepsilon>0$, the unrestricted integral over +$\{s\ge s_{\min}\}$ is at most $e^{\varepsilon n}$ for all +sufficiently large $n$. +\item Fix $t_0>0$. Outside any prescribed open neighborhood of the +equality curve $\{(P_t,t):t_0\le t\le T\}$, there are +$a,\sigma_*>0$ such that the integral of $G_{t,\sigma,n}$ is at most +$e^{-an}$, uniformly over $t_0\le t\le T$ and +$0<\sigma\le\sigma_*$, for all sufficiently large $n$. +The neighborhood can impose weak closeness of the empirical law and +closeness of any finite list of continuous, at most linearly growing +averages, including all parameters in +\eqref{eq:scalar-parameters}. +\item For every $q_0>0$, there are $t_0,\sigma_*,a>0$ such that +\[ + \int_{\{q\ge q_0\}}G_{t,\sigma,n}(y)\,\dd y\le e^{-an} +\] +uniformly over $0\le t\le t_0$ and $0<\sigma\le\sigma_*$, for all +sufficiently large $n$. +\item Fix $t_0>0$ and $\nu>0$. For all sufficiently large $R_1$ there +is $a>0$ such that +\[ + \int_{\{\av{y^2\one_{\{|y|>R_1\}}}>\nu\}} + G_{t,\sigma,n}(y)\,\dd y\le e^{-an}, +\] +uniformly over $t_0\le t\le T$ and $0<\sigma\le1$, for all +sufficiently large $n$. +\end{enumerate} +All constants are independent of $n,t,\sigma$ in the indicated ranges. +In \textup{(ii)}, neighborhoods are interpreted on each fixed +second-moment ball; the complement of a sufficiently large such ball +is covered by \textup{(i)}. +\end{lemma} + +\begin{proof} +We first obtain a Gaussian tail envelope, then prove a compact-set +bound by a finite cover. Strict negativity away from the equality laws +will give the second and third assertions; an exponential tilt will give +the fourth. + +Throughout, $|d|\le T+j$ and $s\le T+1+j$. Also +$a(P)^2\le q\int(y-d)^2\,\dd P$ and $jq/S\le1/2$. The elementary +inequality $(r-u)^2\le\frac32r^2+3u^2$ consequently gives +\[ + n\psi\le\frac{3jq}{4sS}\sum_i(y_i-d)^2+\frac{3nj}{2} + \le\frac{3}{8s}\sum_i(y_i-d)^2+\frac{3nj}{2}. +\] +On $s\ge s_{\min}$, it follows that +\begin{equation} + G_{t,\sigma,n}(y)\le + \exp\left(Cn-c\sum_i y_i^2\right), + \label{eq:scalar-tail-envelope} +\end{equation} +where $c>0$ and $C<\infty$ depend only on $j,T,s_{\min}$. +On $\sum_i y_i^2>nR^2$, use half of the remaining Gaussian square +for integration and the other half for the required exponential decay. +This proves the first assertion of \textup{(i)}. + +The estimates on bounded second moments follow from one compactness +argument, which we now give in full. Put +\[ + \mathcal P_R=\left\{P:\int y^2\,\dd P\le R^2\right\}. +\] +Tightness and preservation of the second-moment bound under weak limits +make this set weakly compact. Integration of a continuous function of at +most linear growth is continuous on it: the second-moment bound makes +the contribution of $|y|>K$ uniformly $O(K^{-1})$, while bounded +truncations are weakly continuous. In particular, $d,s,a(P),\psi$ are +continuous on +$\mathcal P_R\times[0,T]\times[0,1]$ where $s\ge s_{\min}$. + +For every compact subset $\mathcal C$ of that region, +\begin{equation} + \limsup_{n\to\infty}\frac1n\log + \sup_{t,\sigma}\int_{\{(P_y,t,\sigma)\in\mathcal C\}} + G_{t,\sigma,n}(y)\,\dd y + \le\sup_{\mathcal C}\bigl\{\psi-D(P\Vert N(d,s))\bigr\}, + \label{eq:compact-rate-bound} +\end{equation} +where $P_y=n^{-1}\sum_i\delta_{y_i}$; empty integrals are zero. +To establish this bound, fix a strict finite upper bound $r$ for its +right-hand side. At each $\xi=(P,t,\sigma)\in\mathcal C$, choose a +bounded continuous test function $\varphi_\xi$ from the variational +formula for relative entropy so that +\[ + \psi(\xi)-\int\varphi_\xi\,\dd P + +\log\int e^{\varphi_\xi}\,\dd N(d,s)0$, so the +preceding compactness argument and the tail envelope apply without +change, including at $t=0$. + +It remains to control the contribution of large coordinates in +\textup{(iv)}. Fix $t_0,\nu>0$, and take +$0<\lambda<1/[8(T+1+j)]$. Let +$h_{R_1}(y)=y^2\one_{\{|y|>R_1\}}$. On a fixed second-moment ball, +repeat the proof of \eqref{eq:compact-rate-bound}, this time inserting +$e^{\lambda\sum_i h_{R_1}(y_i)}$. Begin with any small positive rate +bound $\varepsilon$ in that proof and its finite collection of tests +$\varphi_\xi$. Uniformly for $|d|\le T+j$ and +$t_0\le s\le T+1+j$, +\[ + \int\bigl(e^{\lambda h_{R_1}(y)}-1\bigr)\,\dd N(d,s)(y) + \longrightarrow0\qquad(R_1\to\infty), +\] +by Gaussian tails; the choice of $\lambda$ leaves a uniformly negative +quadratic exponent. The same holds with the integrand multiplied by +any of the finitely many bounded $e^{\varphi_\xi}$. The additional +logarithmic moment can therefore be made arbitrarily +small in every member of the finite cover. For sufficiently large $R_1$, +the tilted integral on the ball is at most $e^{2\varepsilon n}$. On +$\av{h_{R_1}(y)}>\nu$, remove the tilt at cost +$e^{-\lambda\nu n}$. Choosing $2\varepsilon<\lambda\nu/2$ gives a +strictly negative rate there. The integral off the ball is estimated +without the tilt, using \textup{(i)} with a larger negative rate. +This proves \textup{(iv)}. +\end{proof} + +\subsection{The conditional Hessian} + +The scalar bounds concentrate the integral near the equality curve. +We now show that, there, the conditional Hessian is positive +with an exponential probability bound. The conclusion is simultaneous +over all admissible nearby diagonal matrices, which will be needed to +exclude every unstable approximate zero. + +\begin{lemma}[Conditional stability near the scalar equality curve] +\label{lem:conditional-field-stability} +Fix $00$ with +$j(1+\eta)^2<1$. There are $c_*,\epsilon_*,a_*,\sigma_*>0$, +a neighborhood $\mathcal U$ of +$\{(P_t,t):t_0\le t\le T\}$, a tail tolerance $\nu>0$, and a finite +$R_1$ such that the following holds. Suppose +\[ + \av{y^2}\le R^2,\qquad (P_y,t)\in\mathcal U,\qquad + \av{y^2\one_{\{|y|>R_1\}}}\le\nu, + \qquad t_0\le t\le T,\quad0<\sigma\le\sigma_*. +\] +Under the Gaussian conditioning in +\eqref{eq:conditional-integral}, with probability at least +$1-e^{-a_*n}$, simultaneously for all diagonal $A$ satisfying +\[ + 0\le A\le(1+\eta)I,\qquad + \norm{A-V_y}_{\Frob}\le\epsilon_*\sqrt n, +\] +one has $\mathsf H_{\wt J}(y,A)\succeq c_*I$. +All constants are uniform over the displayed ranges and sufficiently +large $n$. The tail cutoff $R_1$ may be required to be arbitrarily +large after choosing $\nu$; $\epsilon_*$ is then decreased if +necessary. +\end{lemma} + +\begin{proof} +We first compute the Gaussian regression and the finite-rank correction +at an equality law. We then show that the empirical-law and diagonal +hypotheses preserve its positive margin, uniformly over all admissible +diagonals. + +\paragraph{Gaussian regression and whitening.} +In a sufficiently small neighborhood of the equality curve, $s$ and +$q$ have positive lower bounds depending on $j,t_0,T$. +Indeed $s_*(t)\ge t_0$, and +$q(P_t)=\E_{P_t}\tanh^2 y$ is positive and continuous there. Set +\[ + u=\frac{m}{\sqrt{nq}},\qquad + Z_1=\frac{y-d\one}{\sqrt{ns}},\qquad P_u=I-uu^{\mathsf T}. +\] +The norms of $\one/\sqrt n,u,Z_1$ are bounded uniformly by the assumed +second-moment bound. Write +\[ + W=P_uWP_u+uw^{\mathsf T}+wu^{\mathsf T}+\alpha uu^{\mathsf T}. +\] +The three blocks in this decomposition are independent before +conditioning, with $\Cov(w)=(j/n)P_u$ and $\Var(\alpha)=2j/n$. +The projection of the +observation in \eqref{eq:conditional-integral} perpendicular to $u$ +observes $\sqrt{nq}\,w$ with independent noise of covariance +$(t+\sigma^2)P_u$; projecting along $u$ observes +$\sqrt{nq}\,\alpha$ with noise variance $t+\sigma^2$. The elementary +Gaussian conditional-mean and conditional-variance formulas therefore +give the following representation, using a fresh centered GOE $W_0$: +\begin{equation} + \begin{split} + P_uWP_u&=P_uW_0P_u,\\ + w&=\bar w+\kappa P_uW_0u,\qquad + \bar w=j\sqrt{q/s}\,Z_1-\frac{ja_y}{s}u,\qquad + \kappa=\sqrt{\frac{t+\sigma^2}{s}},\\ + \alpha&=\frac{2j(a_y+Bq)}S + +\sqrt{\frac{t+\sigma^2}S}\,u^{\mathsf T}W_0u. + \end{split} + \label{eq:conditional-W} +\end{equation} +For example, the perpendicular conditional mean is +$j\sqrt{q/s}\,P_uZ_1$, which equals the displayed $\bar w$ because +$u^{\mathsf T}Z_1=a_y/\sqrt{qs}$. The parallel conditional mean follows +from $u^{\mathsf T}z=\sqrt{n/q}(a_y+Bq)$. The parallel residual +variance is $(2j/n)(t+\sigma^2)/S$. These conditional means and +residual variances verify all three blocks +of \eqref{eq:conditional-W}. + +For the moment, fix a deterministic admissible $A$. We compare the +conditional Hessian with the positive matrix built from the fresh GOE +by setting +\[ + \chi=\frac1n\Tr A,\qquad B_A=j\chi,\qquad + S_A=I+B_AA-A^{1/2}W_0A^{1/2},\qquad + \mathcal Y(p)=S_A^{-1/2}A^{1/2}p. +\] +Lemma~\ref{lem:matrix-path-stability} gives $S_A\succeq c_0I$ except +with exponentially small probability, where $c_0>0$ is uniform in +$A$. Let $G_A=A^{1/2}S_A^{-1}A^{1/2}$. Lemma~\ref{lem:bilinear}, +applied to the three deterministic vectors +$\one/\sqrt n,u,Z_1$, says that their Gram entries after $\mathcal Y$ +are as close as desired to their $A$-weighted Gram entries, and that +the extra vector $\mathcal Y(W_0u)$ has Gram entries as close as desired +to those of +\begin{equation} + B_A\mathcal Y(u)+\sqrt{B_A}\,N, + \label{eq:extra-whitened-vector} +\end{equation} +where $N$ is a unit vector orthogonal to the three deterministic +images in this comparison. Specifically, its mixed entries are +$B_Ap^{\mathsf T}Au$ and its squared norm is +$B_A+B_A^2u^{\mathsf T}Au$. Here $N$ describes only Gram entries; +this notation does not assert a coupling with an additional random or +deterministic vector. The comparison concerns finite Gram matrices; the +associated symmetric finite-rank eigenvalues depend continuously on their +entries, as we verify when returning to the empirical parameters. Each fixed +Gram tolerance has failure probability at most $Ce^{-a n}$ uniformly +over the bounded deterministic vectors and over $A$. Also +$u^{\mathsf T}W_0u$ is smaller than any fixed tolerance with +exponentially high probability, since its variance is $2j/n$. + +\paragraph{The correction at an equality law.} +We now compute the finite-rank perturbation in the ideal Gram comparison. +Set $\sigma=0$, take $s=s_*(t)$, and replace each weighted average +$n^{-1}\sum_i A_{ii}f(y_i)$ by +$\E_{N(s,s)}[\sech^2 y\,f(y)]$. Evaluate all other empirical +averages under the same law. Thus $a_y=sb$, $B_A=B=jb$, and +$\kappa^2=1-jq/s$. Write +\[ + X_1=\mathcal Y(\one/\sqrt n),\qquad + U=\mathcal Y(u),\qquad Z_2=\mathcal Y(Z_1), +\] +using the Gram interpretation just described. Subtracting +$W_0$ from \eqref{eq:conditional-W} gives exactly +\begin{equation} + \begin{split} + W-W_0={}&u\bar w^{\mathsf T}+\bar w u^{\mathsf T} + -(1-\kappa)\{u(W_0u)^{\mathsf T}+(W_0u)u^{\mathsf T}\}\\ + &+\{\alpha+(1-2\kappa)u^{\mathsf T}W_0u\}uu^{\mathsf T}. + \end{split} + \label{eq:conditional-rank-difference} +\end{equation} +At equality, $\bar w=j\sqrt{q/s}\,Z_1-Bu$ and $\alpha=2B$ after +discarding the vanishing Gaussian scalar. Whiten by $S_A$, and use +\eqref{eq:extra-whitened-vector} with $B_A=B$. The perturbation to +subtract from the identity consists of the planted term +$jX_1X_1^{\mathsf T}$, the term $2jqUU^{\mathsf T}$, and the whitened +version of \eqref{eq:conditional-rank-difference}. The terms +$-2BUU^{\mathsf T}$ and $+2BUU^{\mathsf T}$ cancel. Eliminating the +orthogonal direction $N$ by Schur complement adds +$(1-\kappa)^2BUU^{\mathsf T}$ to the perturbation. Its total +$UU^{\mathsf T}$ coefficient becomes +\[ + 2jq-2(1-\kappa)B+(1-\kappa)^2B + =jq(2-B/s). +\] +Consequently the Schur complement is the identity minus +\begin{equation} + j\{X_1X_1^{\mathsf T}+(2-B/s)M_1M_1^{\mathsf T} + +M_1G_1^{\mathsf T}+G_1M_1^{\mathsf T}\}, + \qquad M_1=\sqrt q\,U,\quad G_1=Z_2/\sqrt s. + \label{eq:limiting-rank-perturbation} +\end{equation} + +To prove positivity of this Schur complement, it suffices to examine +its three-vector Gram matrix. Put +$e=\E_{N(s,s)}[m^2(1-m^2)]$. The Gram matrix of +$(X_1,M_1,G_1)$ equals +\begin{equation} + H=\begin{pmatrix} + b&e&-2e\\ + e&e&b-3e\\ + -2e&b-3e&b/s-2b+6e + \end{pmatrix}. + \label{eq:limiting-Gram} +\end{equation} +We derive the entries of this matrix explicitly. The density of +$N(s,s)$ satisfies +$p(y)/p(-y)=e^{2y}$. Pairing $y$ and $-y$ shows that every integrable +odd function $f$ satisfies $\E f(y)=\E[f(y)\tanh y]$. This gives +$\E[m(1-m^2)]=e$. Gaussian integration by parts gives +\[ + \frac1s\E[(y-s)(1-m^2)]=-2e,\qquad + \frac1s\E[(y-s)m(1-m^2)]=b-3e. +\] +Applying it twice, and using +$(1-m^2)''=4m^2(1-m^2)-2(1-m^2)^2$, gives the final diagonal entry +$b/s-2b+6e$. The remaining entries are immediate, proving +\eqref{eq:limiting-Gram}. + +The coefficient matrix in \eqref{eq:limiting-rank-perturbation} is +\[ + C=j\begin{pmatrix}1&0&0\\0&2-B/s&1\\0&1&0\end{pmatrix}. +\] +Set $d_0=1-jb$ and $g_1=je$. The first and third columns of $I-HC$ +are $(d_0,-g_1,2g_1)^{\mathsf T}$ and +$(-g_1,-g_1,d_0+3g_1)^{\mathsf T}$. Adding $B/s$ times the third +column to the second makes the second column +$(0,d_0+g_1,0)^{\mathsf T}$. Hence +\begin{equation} + \det(I-HC)=(d_0+2g_1)(d_0+g_1)^2>0. + \label{eq:positive-rank-determinant} +\end{equation} +The same determinant is obtained from the identity minus +\eqref{eq:limiting-rank-perturbation} on any ambient space containing +its three vectors. To determine its sign as a quadratic form, first +subtract only +$jX_1X_1^{\mathsf T}$; the result is positive definite because +$j\norm{X_1}^2=jb<1$. The remaining two-vector perturbation has at +most one positive eigenvalue, since its coefficient block +$j\left(\begin{smallmatrix}2-B/s&1\\1&0\end{smallmatrix}\right)$ +has one positive and one negative eigenvalue. The resulting matrix can +consequently have at most one nonpositive +direction. The positive determinant in +\eqref{eq:positive-rank-determinant} excludes both a zero eigenvalue +and a single negative eigenvalue. Hence the matrix is positive definite. +The Schur complement then also gives positivity before $N$ was eliminated. + +\paragraph{Perturbing the scalar law and the diagonal.} +The equality calculation supplies the positive margin we need. We first +explain why the hypotheses put the actual weighted Gram entries near +these equality values. + +Each deterministic vector has coordinates $n^{-1/2}$ times +one of $1,m/\sqrt q,(y-d)/\sqrt s$. Products of two such coordinate +functions are bounded by $C(1+y^2)$. For any such product $f$, the +tail and Frobenius bounds imply +\begin{equation} + \left|\frac1n\sum_i(A_{ii}-(V_y)_{ii})f(y_i)\right| + \le C_{R_1}\frac{\norm{A-V_y}_{\Frob}}{\sqrt n}+C\nu. + \label{eq:A-moment-comparison} +\end{equation} +On $|y_i|\le R_1$, use Cauchy--Schwarz and boundedness of $f$; outside, +use $|A_{ii}-(V_y)_{ii}|\le2+\eta$ and the tail hypothesis, with +$R_1\ge1$. The corresponding $V_y$-weighted functions are bounded +and continuous, even when multiplied by $y^2$, because +$\sech^2 y$ decays exponentially. Their averages are therefore close +to the equality-law averages in a sufficiently small weak +neighborhood. The parameters $d,s,q,a_y$ are continuous on the +second-moment ball, so their denominators and coefficients are close +as well. Finally $|B_A-B|\le j\norm{A-V_y}_{\Frob}/\sqrt n$. + +We retain the positivity uniformly for $t\in[t_0,T]$ and under +small Gram errors. If $Q$ has finitely many columns, the nonzero +eigenvalues of +$QCQ^{\mathsf T}$ agree with those of +$(Q^{\mathsf T}Q)^{1/2}C(Q^{\mathsf T}Q)^{1/2}$, including +multiplicity. The largest eigenvalue, with zero adjoined, is therefore +a continuous function of the Gram matrix and the coefficients. +All equality parameters vary continuously on the compact interval +$[t_0,T]$, and $s,q$ stay positive there. The exact calculation thus +gives a uniform positive whitened margin. Choose a fixed tolerance +$\xi>0$ on coefficients and Gram entries small enough to preserve, +say, half of it. Lemmas~\ref{lem:matrix-path-stability} +and~\ref{lem:bilinear}, the Gaussian scalar estimate, and +\eqref{eq:A-moment-comparison} show that this tolerance is achieved +for each fixed admissible $A$ with failure probability at most +$Ce^{-a n}$, after choosing the scalar neighborhood, tail tolerance, +$\sigma_*$, and Frobenius distance sufficiently small. Returning +from whitening multiplies the margin by at least $c_0$. +The omitted mass $(B-B_A)A$ has norm at most +$j(1+\eta)\norm{A-V_y}_{\Frob}/\sqrt n$ and can be absorbed by +decreasing that distance again. + +At this stage the margin is proved for each fixed $A$. To make it +simultaneous, we use that its failure exponent is uniform over the +bounded deterministic vector norms. Those bounds were fixed before +choosing the Frobenius distance. Also \eqref{eq:conditional-W} implies +$\norm W\le K_1$ with failure $Ce^{-a_1n}$ for some uniform $K_1$: +the norms of the deterministic means are bounded, and the remaining +terms are controlled by $\norm{W_0}$. +On this event, $A\mapsto\mathsf H_{\wt J}(y,A)$ is uniformly +$1/2$-H\"older in diagonal sup norm. Indeed +$\norm{A^{1/2}-\widehat A^{1/2}}\le +\norm{A-\widehat A}^{1/2}$, while the middle factor in +\eqref{eq:field-hessian} has bounded norm. + +Choose a fixed mesh size $\zeta>0$ small enough that the H\"older +error is below one quarter of the margin. Given $A$ with normalized +Frobenius distance at most $\epsilon$ from $V_y$, let +$\mathcal S_A=\{i:|A_{ii}-(V_y)_{ii}|>\zeta\}$. It has at most +$\alpha n$ elements, where $\alpha=\epsilon^2/\zeta^2$. +Outside $\mathcal S_A$ replace $A_{ii}$ by $(V_y)_{ii}$; on $\mathcal S_A$, round it +within $\zeta$ to a grid of $[0,1+\eta]$, including the endpoints. +The resulting $\widehat A$ obeys +\[ + \norm{A-\widehat A}\le\zeta,\qquad + \norm{\widehat A-V_y}_{\Frob}\le2\epsilon\sqrt n. +\] +For fixed $y$, these approximants are deterministic and number at +most +\[ + \sum_{k\le\alpha n}\binom nk N_\zeta^k, + \qquad N_\zeta\le C(1+1/\zeta). +\] +Its logarithm divided by $n$ tends to zero as $\alpha\downarrow0$; +for instance it is at most +$-\alpha\log\alpha-(1-\alpha)\log(1-\alpha) + +\alpha\log N_\zeta+o(1)$ when $\alpha<1/2$. +Choose $\epsilon_*$ so small that $2\epsilon_*$ meets all moment +tolerances and this counting rate is less than half the fixed-$A$ +failure exponent. A union bound, followed by the H\"older comparison, +proves the desired uniform margin. + +For later use, record the order that preserves the concentration +exponents. First fix $R,t_0,T,\eta$, +the limiting margin, and the Gram tolerance $\xi$. This fixes the +vector norm bounds, conditional concentration exponents, $K_1$, and +the mesh $\zeta$. Next take the equality neighborhood and tail +tolerance $\nu$ small. Choose any sufficiently large finite $R_1$, +then take $\epsilon_*$ small enough for +\eqref{eq:A-moment-comparison} and the union bound. Finally decrease +$\sigma_*$ so all scalar coefficients lie within the required +tolerance. No concentration exponent is lost when $\epsilon_*$ is +decreased. This also proves the last assertion of the lemma. +\end{proof} + +\subsection{From a small integral to exclusion of approximate zeros} + +The conditional estimate controls all nearby diagonals for each fixed +$y$ satisfying the scalar hypotheses. Combined with the scalar rate +bounds, it will make the expected integral over unstable configurations +exponentially small. To exclude approximate zeros at any $y$, the next +deterministic lemma gives the complementary lower bound: an approximate +zero satisfying a regularized determinant bound contributes a definite +exponential amount to the integral over a surrounding ball. The +regularization makes a sign assumption on the Jacobian unnecessary. + +\begin{lemma}[Local volume at an approximate zero] +\label{lem:local-field-volume} +Fix $j,K<\infty$, $r>0$, and $\varepsilon>0$. Suppose +$\norm{\wt J}\le K$, and a set $E\subseteq\R^n$ contains the ball +$\{y':\norm{y'-y}\le r\sqrt n\}$. Put +\[ + L=F'_{\wt J,h_0}(y) + =I+(BI-\wt J)V_y-\frac{2j}{n}mm^{\mathsf T}V_y,\qquad + Q=L^{\mathsf T}L+\gamma I. +\] +If +\begin{equation} + \frac1n\log\det Q\le jb(y)^2+\varepsilon_1,\qquad + \norm{F_{\wt J,h_0}(y)}\le\rho\sqrt n, + \label{eq:local-volume-hypotheses} +\end{equation} +then +\begin{equation} + (2\pi\sigma^2)^{-n/2}\int_E + e^{-\norm{F_{\wt J,h_0}(y')}^2/(2\sigma^2)+njb(y')^2/2}\,\dd y' + \ge e^{-\varepsilon n} + \label{eq:local-volume-conclusion} +\end{equation} +for all sufficiently large $n$, provided the constants are chosen in +this order: first $\varepsilon_1>0$ sufficiently small in terms of +$\varepsilon$; then any fixed $\gamma>0$; then $\sigma>0$ +sufficiently small in terms of $j,K,r,\varepsilon,\gamma$; and finally +$\rho>0$ sufficiently small in terms of those constants and $\sigma$. +The bound is uniform in $y,h_0,n$ under the stated hypotheses. +\end{lemma} + +\begin{proof} +We integrate over a Gaussian ellipsoid centered at the approximate +zero. Its covariance is chosen to control the linearized residual. +Let $g''$ be standard Gaussian and put +$\Delta=\sigma Q^{-1/2}g''$, and restrict to +$\mathcal B=\{\norm{g''}\le2\sqrt n\}$. For +$2\sigma/\sqrt\gamma\le r$, all resulting points $y+\Delta$ belong +to $E$. Changing variables in the integral gives the lower bound +\begin{equation} + (\det Q)^{-1/2}\Prob(\mathcal B) + \E\left[\exp\left\{ + \frac{\norm{g''}^2}{2} + -\frac{\norm{F(y+\Delta)}^2}{2\sigma^2} + +\frac{njb(y+\Delta)^2}{2}\right\}\,\middle|\,\mathcal B\right]. + \label{eq:volume-change-variables} +\end{equation} +Here and for the remainder of this proof $F=F_{\wt J,h_0}$. +The conditioning has probability tending to one, and all estimates +below remain valid under it with a uniform factor, since +$\Prob(\mathcal B)\ge1/2$ for large $n$. + +We must control the nonlinear residual on this ellipsoid uniformly +in its center. Coordinatewise Taylor expansion gives +\[ + m(y+\Delta)=m+V_y\Delta+r_m,\qquad + |(r_m)_i|\le C\Delta_i^2. +\] +Writing $b(y+\Delta)=b+\ell_b+r_b$, its linear part is +$\ell_b=-2\av{m(1-m^2)\Delta}$, and +$|r_b|\le C\norm\Delta^2/n$. Thus, when the product $jb,m$ in +$F$ is expanded, its remainder is bounded in norm by +\[ + C\left(\norm{r_m}+\frac{\norm\Delta^2}{\sqrt n}\right). +\] +For the product of first differences use +$|b(y+\Delta)-b(y)|\le C\norm\Delta/\sqrt n$ and +$\norm{m(y+\Delta)-m(y)}\le\norm\Delta$. The remaining term +$-\wt Jm$ has remainder bounded by $K\norm{r_m}$. +Since the covariance of $\Delta$ is bounded above by +$\sigma^2I/\gamma$, Gaussian fourth moments give +\[ + \E\sum_i\Delta_i^4\le Cn\sigma^4/\gamma^2,\qquad + \E\norm\Delta^4\le Cn^2\sigma^4/\gamma^2. +\] +Combining these Taylor estimates with the Gaussian moments, and writing +$R_F=F(y+\Delta)-F(y)-L\Delta$, yields +\begin{equation} + \E[\norm{R_F}^2\mid\mathcal B] + \le Cn\sigma^4/\gamma^2,\qquad + \E[|b(y+\Delta)^2-b(y)^2|\mid\mathcal B] + \le C\sigma/\sqrt\gamma. + \label{eq:volume-remainders} +\end{equation} +All constants depend only on $j,K$. + +The choice of ellipsoid now controls the linear term: the singular +values of $LQ^{-1/2}$ are at most one. Hence +$\norm{L\Delta}\le\sigma\norm{g''}$ pointwise, including on +$\mathcal B$. Expanding +$F(y+\Delta)=F(y)+L\Delta+R_F$ and applying Cauchy--Schwarz to its +cross terms yields +\begin{align*} + &\frac1n\E\left[ + \frac{\norm{g''}^2}{2} + -\frac{\norm{F(y+\Delta)}^2}{2\sigma^2} + \,\middle|\,\mathcal B\right]\\ + &\hspace{1cm}\ge + -C\left\{ + \frac\rho\sigma+\frac\sigma\gamma + +\left(\frac\rho\sigma\right)^2 + +\left(\frac\sigma\gamma\right)^2 + +\frac\rho\gamma\right\}. +\end{align*} +The last mixed term may also be absorbed into the two squares. +Apply Jensen's inequality to the conditional expectation in +\eqref{eq:volume-change-variables}. Its logarithm divided by $n$ is +at least +\[ + -\frac{1}{2n}\log\det Q+\frac{jb(y)^2}{2} + -C\left\{\frac\rho\sigma+\frac\sigma\gamma + +\left(\frac\rho\sigma\right)^2 + +\left(\frac\sigma\gamma\right)^2 + +\frac\sigma{\sqrt\gamma}\right\}+o(1). +\] +The determinant hypothesis cancels $jb(y)^2/2$, with a loss of at +most $\varepsilon_1/2$. Choose the constants in the order in the +statement; the other errors then have total size smaller than +$\varepsilon$. This proves the lower bound. +\end{proof} + +\subsection{Proof of the observation-path statement} + +\begin{proof}[Proof of Proposition~\ref{prop:stable-fields}] +We assemble the scalar, matrix, and local-volume estimates under the +planted law \eqref{eq:planted-field}. After excluding unstable +approximate zeros at a fixed time, we remove the auxiliary diagonal, +control the whole path by a fixed mesh, and undo the planting. + +Let $J$ be $\wt J$ with its diagonal zeroed. Choose +$\eta,\theta,q_0>0$ sufficiently small that +$j(1+\eta)^2<1$ and, whenever $q\le2q_0$, +\begin{equation} + I+(j-3jq-2\sqrt j-2\theta)A\succeq c_{\mathrm{sm}}I + \qquad(0\le A\le(1+\eta)I) + \label{eq:small-time-matrix-margin} +\end{equation} +for some $c_{\mathrm{sm}}>0$. These choices are possible because at +$\eta=\theta=q=0$ the lower scalar bound is +$1+j-2\sqrt j=(1-\sqrt j)^2>0$. On +\begin{equation} + \norm J\le2\sqrt j+\theta,\qquad + \max_i|\wt J_{ii}|\le\theta, + \label{eq:planted-norm-restriction} +\end{equation} +we have $\norm{\wt J}\le2\sqrt j+2\theta$. Since +$mm^{\mathsf T}/n\preceq qI$, \eqref{eq:small-time-matrix-margin} +implies $\mathsf H_{\wt J}(y,A)\succeq c_{\mathrm{sm}}I$ for all +$q\le2q_0$. Choose $t_0\in(0,T]$ small enough for +Lemma~\ref{lem:scalar-rates}\textup{(iii)}. + +For the remaining times $t\in[t_0,T]$, the conditional Hessian lemma +provides the margin near the equality curve. Choose a second-moment +ball whose complement has a fixed negative rate, using +Lemma~\ref{lem:scalar-rates}\textup{(i)} with $s_{\min}=t_0$. +Apply Lemma~\ref{lem:conditional-field-stability} on this ball. Take +its equality neighborhood and tail tolerance, choose its tail cutoff +$R_1$ sufficiently large for +Lemma~\ref{lem:scalar-rates}\textup{(iv)}, and then choose its +diagonal-distance radius. Decrease $\sigma_*>0$ so that +Lemma~\ref{lem:scalar-rates}\textup{(ii)} excludes the complement of +that equality neighborhood, and so that the small-time scalar bound +also holds. The scalar regions excluded by these choices all have fixed +strictly +negative rates, uniformly for $0<\sigma\le\sigma_*$. On the region +left over, the conditional matrix margin fails with probability at +most $e^{-a_*n}$. + +Let $\epsilon_*>0$ denote the full diagonal-distance radius just chosen. +Choose $c_E>0$ smaller than both $c_*/2$ and $c_{\mathrm{sm}}/2$. +For $t\ge t_0$, define +\[ + E_t(W)=\left\{y:\exists A,\quad + 0\le A\le(1+\eta)I,\quad + \norm{A-V_y}_{\Frob}\le\epsilon_*\sqrt n,\quad + \lambda_{\min}(\mathsf H_{\wt J}(y,A))\le c_E\right\}. +\] +For $0\le t0$ such that +\begin{equation} + \E I_{t,\sigma}(E_t)\le e^{-a_0n} + \quad(0\le t\le T, 0<\sigma\le\sigma_*) + \label{eq:bad-integral-rate} +\end{equation} +for all sufficiently large $n$, uniformly in $t,\sigma$. +For small times, this follows from the scalar bound and +Lemma~\ref{lem:gaussian-reduction}. For larger times, split the +conditional integral into the excluded scalar regions and their complement. +The excluded regions have negative rates. On the complement multiply +the conditional probability $e^{-a_*n}$ by the unrestricted scalar +integral, whose rate is at most any prescribed positive number, by +Lemma~\ref{lem:scalar-rates}\textup{(i)}. Taking that number below +$a_*/2$ proves \eqref{eq:bad-integral-rate}. Finitely many pieces can +be absorbed by decreasing $a_0$. + +We now use local volume to exclude every unstable approximate zero. +An individual witness will force a ball of the excluded set to contribute +too much to the integral. Suppose +\begin{equation} + \norm{F_{\wt J,h_0}(y)}\le\rho\sqrt n,\qquad + \norm{A-V_y}_{\Frob}\le\tfrac12\epsilon_*\sqrt n,\qquad + \lambda_{\min}(\mathsf H_{\wt J}(y,A))\le\tfrac12c_E + \label{eq:bad-approximate-point} +\end{equation} +for some $0\le A\le(1+\eta)I$, and assume +\eqref{eq:planted-norm-restriction}. There is a constant $r>0$, +independent of this witness and of $n,t$, such that +$\norm{y'-y}\le r\sqrt n$ implies $y'\in E_t(W)$. +For $t\ge t_0$, the same $A$ witnesses instability throughout a +sufficiently small ball. Indeed, +\[ + \norm{V_{y'}-V_y}_{\Frob}\le2\norm{y'-y},\qquad + |b(y')-b(y)|\le2\norm{y'-y}/\sqrt n, +\] +and +\[ + \norm{m(y')m(y')^{\mathsf T}/n-m(y)m(y)^{\mathsf T}/n} + \le2\norm{y'-y}/\sqrt n +\] +preserve the full diagonal-distance constraint and the bound +$\lambda_{\min}\le c_E$ if $r$ is small enough. For $t2q_0$; the same Lipschitz bound for $q$ then gives +$q(y')\ge q_0$ for a smaller $r$. + +Set $L=F'_{\wt J,h_0}(y)$. It is a bounded-rank, bounded-norm +perturbation of +\[ + I+(BI-W)V_y: + \quad L=I+(BI-W)V_y + -\frac jn\one\one^{\mathsf T}V_y + -\frac{2j}{n}mm^{\mathsf T}V_y. +\] +Here $n^{-1}\Tr V_y=b(y)$, so the determinant rate in +Lemma~\ref{lem:logdet} matches the weight in \eqref{eq:field-integral}. +That lemma says that, for any chosen $\varepsilon_1>0$, one may choose +a fixed $\gamma>0$ sufficiently +small so that, except on an event of probability $Ce^{-a_1n}$, +simultaneously at every $y$, +\[ + \frac1n\log\det(L^{\mathsf T}L+\gamma I) + \le jb(y)^2+\varepsilon_1. +\] +The lemma's rank-perturbation allowance applies here with two rank-one +terms of norms at most $j$ and $2j$. +Choose $\varepsilon0$ +sufficiently small for it. A witness +\eqref{eq:bad-approximate-point} on the determinant event forces +$I_{t,\sigma}(E_t)\ge e^{-\varepsilon n}$. Markov's inequality and +\eqref{eq:bad-integral-rate} show that the probability of such a +witness, under \eqref{eq:planted-norm-restriction}, is at most +$Ce^{-a_2n}$, uniformly in $t\in[0,T]$. Here the uniformity of the +negative rates permits the later choice of +$\sigma$ required by the local-volume estimate. + +The integral argument used a full GOE matrix. We now pass to its +zero-diagonal version, retaining slack in both residual and Hessian bounds. +Choose a fixed +$\delta_D>0$ so small that +\[ + \delta_D\le\theta,\qquad + \delta_D\le\rho/2,\qquad + (1+\eta)\delta_D\le c_E/4. +\] +The diagonal entries of $\wt J$ are independent +$N(j/n,2j/n)$, so +$\Prob(\max_i|\wt J_{ii}|>\delta_D)\le Ce^{-a_Dn}$ for large $n$. +On its complement, +\[ + \frac{\norm{F_{\wt J,h_0}(y)-F_{J,h_0}(y)}}{\sqrt n} + \le\delta_D,\qquad + \norm{\mathsf H_{\wt J}(y,A)-\mathsf H_J(y,A)} + \le(1+\eta)\delta_D. +\] +Thus a zero-diagonal witness with residual at most +$\rho\sqrt n/2$, radius $\epsilon_*\sqrt n/2$, and Hessian minimum +at most $c_E/4$ would give a witness +\eqref{eq:bad-approximate-point}. We have proved an exponentially +small fixed-time failure probability for the zero-diagonal matrix, +restricted only to $\norm J\le2\sqrt j+\theta$. + +To pass from a fixed time to the whole interval, we control observation +increments on a deterministic mesh. Take a mesh of size $\Delta>0$ +that includes both endpoints. For a scalar +Brownian motion, the reflection bound implies that +$\sup_{0\le u\le\Delta}|B(u)|^2/\Delta$ has an exponential moment +at a fixed positive argument. The coordinate Brownian motions are +independent. Exponential Markov applied to the sum of their squared +suprema consequently gives, for each desired $\kappa_0>0$, a mesh +$\Delta$ small enough that +\[ + \Prob\left(\max_k\sup_{t_k\le t\le t_{k+1}} + \frac{\norm{B_0(t)-B_0(t_k)}}{\sqrt n}>\kappa_0\right) + \le Ce^{-a_3n}. +\] +The deterministic drift contributes at most $\Delta\sqrt n$ to each +increment. Choose the mesh so the full normalized increment is at +most $\rho/4$. Set $\rho_0=\rho/8$, +$\epsilon_A=\epsilon_*/2$, and $c=c_E/4$. A pathwise violation of +\eqref{eq:stable-field} then gives a fixed-time violation at a mesh +point: the Hessian and its witness do not use the field, and the +residual increases by at most $\rho\sqrt n/4$. A union bound over +the fixed finite mesh proves an annealed path failure bound +$Ce^{-a_4n}$ under the planted law, restricted to +$\norm J\le2\sqrt j+\theta$. + +We have obtained a path estimate under planting. To recover the +original disorder law, introduce the partition-function density +\[ + Z(J)=\sum_x e^{x^{\mathsf T}Jx/2},\qquad + L(J)=Z(J)/\E Z(J). +\] +Size-bias the original joint law of $J,X$, with $X\sim\mu_0$, by +$L(J)$. Its density in $J,X$ is proportional to +$e^{x^{\mathsf T}Jx/2}$ times the original Gaussian density. +Gaussian tilting makes $X$ uniform and, conditionally on $X$, makes +the off-diagonal entries of $J$ independent with means +$jX_iX_j/n$ and variance $j/n$. Add independent diagonal entries with law $N(j/n,2j/n)$---a centered +GOE diagonal plus $jI/n$---and conjugate by $D_X$; the resulting matrix and observation have exactly +the law \eqref{eq:planted-field}. All witness conditions are +equivariant under this conjugation, because $m(D_Xy)=D_Xm(y)$, +$V_{D_Xy}=V_y$, and diagonal $A$ commutes with $D_X$. Conditional +observation paths given $J$ are unchanged by size bias. + +Undoing this change of law requires a lower bound for $L(J)$ on +typical original matrices. For independent uniform signs $\varepsilon_i$, +Gaussian averaging gives +\[ + \frac{\E Z^2}{(\E Z)^2} + =e^{-j/2}\E\exp\left\{\frac j{2n} + \left(\sum_i\varepsilon_i\right)^2\right\} + \le(1-j)^{-1/2}. +\] +To obtain the inequality, introduce an independent standard Gaussian +$g$ and use +$\E_\varepsilon e^{\sqrt{j/n}g\sum_i\varepsilon_i} + =\cosh(\sqrt{j/n}g)^n\le e^{jg^2/2}$. +Paley--Zygmund then yields a constant $p_j>0$ with +$\Prob(Z\ge\E Z/2)\ge p_j$. As a function of the independent +standard off-diagonal Gaussians, $\log Z$ has gradient squared at +most $(j/n)\binom n2\le jn/2$. Gaussian concentration about a median +therefore has scale $O(\sqrt n)$. The preceding positive-probability +lower bound implies that this median is at least +$\log\E Z-O_j(\sqrt n)$: otherwise its upper concentration tail +would be smaller than $p_j$. It follows that, for every fixed +$\delta>0$, +\begin{equation} + \Prob\{L(J)0$. Equation~\eqref{eq:partition-lower-tail} and the usual GOE norm +bound, including diagonal removal, show that the restrictions on +$L(J)$ and $\norm J$ hold with probability tending to one. Intersect +these events to obtain $\mathcal J_n$ and finish the proof. + +We finish by checking measurability of the witness events used in the +argument. At fixed $y$, the matrix-witness sets are projections over +the compact diagonal range. For existence +of $y,A,t$ along a continuous path, additionally restrict +$\norm y$, $\norm J$, and $\sup_{t\le T}\norm{Y(t)}$ by integers. +The witness inequalities are closed and the witness parameter space +is compact, so these restricted projections are closed. Taking a +countable union removes the bounds. This also justifies the +conditional probabilities and the restricted integrals above. +\end{proof} + +\subsection{A unique root and quantitative inversion} + +The preceding proposition supplies stability along random observations +at fixed subcritical $j$. The following consequence is deterministic: +its hypothesis requires positivity throughout the region of small +residual. From that whole-region assumption we obtain a unique zero +and a bound on the distance to it. This separates the probabilistic +construction of stable fields from the inversion used later. + +\begin{lemma}[Root and inverse bound at a stable field] +\label{lem:root} +Fix $j,K,c,r_0>0$. Let $J$ be symmetric with $\norm J\le K$, and +let $h_0\in\R^n$. Suppose +\begin{equation} + \mathsf H_J(y,V_y)\succeq cI + \quad\text{whenever}\quad + \norm{F_{J,h_0}(y)}\le r_0\sqrt n. + \label{eq:root-stability-assumption} +\end{equation} +Then $F_{J,h_0}$ has exactly one zero $r\in\R^n$, and +\begin{equation} + \norm{y-r}\le C\norm{F_{J,h_0}(y)} + \quad\text{if}\quad\norm{F_{J,h_0}(y)}1+\max_i\left(|(h_0)_i|+\sum_j|J_{ij}|+j\right). +\] +Every critical point lies in the interior of $[-a,a]^n$, and on each +face of this box the negative gradient $-\nabla\Phi$ points strictly +inward. The minimum of $\Phi$ on the box is therefore attained at an +interior critical point. This proves existence. + +To prove uniqueness, consider the negative gradient flow from any +point of the box. The inward-pointing condition keeps the trajectory +in the box for all positive times. Its potential decreases and is bounded +below, so +$\int_0^\infty\norm{\nabla\Phi(m(t))}^2\,\dd t<\infty$. +The squared gradient is uniformly continuous in time, because the +gradient and Hessian are bounded on the box. It follows that the +gradient tends to zero: a nonzero limsup would, by uniform +continuity, force disjoint intervals of fixed positive integral. +Every limit point of a trajectory is therefore critical. +The critical set is finite, since it is a compact set of isolated +zeros. The limit set of a trajectory is connected: it is the +intersection of the nested compact connected closures of its tails. +Each trajectory therefore converges to one critical point. + +It remains to show that all trajectories converge to the same point. +The positive Hessian makes every critical point a strict local +attractor. For clarity, in a sufficiently small ball about it one has +$(m-m_*)^{\mathsf T}\nabla\Phi(m)\ge c_*\norm{m-m_*}^2$, +which gives exponential contraction of the distance along the flow +and invariance of the ball. Continuous dependence of the flow on +its starting point, up to the time it enters that ball, shows that +each attraction basin is relatively open in $[-a,a]^n$. These +disjoint nonempty basins cover the connected box. Hence there is +only one basin and one critical point, proving uniqueness of $r$. + +We now derive the distance estimate. Suppose +$\norm{F(y)}0$ and $C<\infty$ such that, for every +$n$, every symmetric zero-diagonal $J$ satisfying +$\max_{i,j}|U_{ij}|\le u_0$, every $h_0\in\R^n$, and every real function $f$ +on the discrete cube, +\begin{equation} + \E_{\mu_{h_0}}[(Hf)^2] + \ge (1-Cr_4(U))\D_{h_0}(f) + -\E_{\mu_{h_0}}\bigl[a^{\mathsf T}Ua + +C|a|^{\mathsf T}Q|a|\bigr]. + \label{eq:terminal-curvature} +\end{equation} +Here $|a|$ denotes coordinatewise absolute value. +\end{lemma} + +\begin{proof} +The curvature bound is obtained by summing an exact two-spin calculation. +Fix $i\ne j$, condition on the other spins, and call the remaining spins +$x,y$. Their conditional density is proportional to +$\exp(\alpha x+\gamma y+J_{ij}xy)$. Introduce the scalar parameters +\[ + a=\tanh\alpha,\quad b=\tanh\gamma,\quad + A=1-a^2,\quad B=1-b^2,\quad u=\tanh J_{ij}. +\] +The letters $a,b,A,B$ have these scalar meanings only during the pair +calculation. Use $\E_0$ for the product law of means $a,b$, and +$\E_{ij}$ for the interacting conditional law. The identity +$e^{J_{ij}xy}=\cosh(J_{ij})(1+uxy)$ gives their exact density ratio: +\begin{equation} + \frac{\dd\mu_{ij}}{\dd\mu_{0,ij}}(x,y) + =\frac{1+uxy}{1+uab}. + \label{eq:pair-density} +\end{equation} +Expand $f$ in the four-dimensional space of functions of $x,y$. Its +two gradients then have the unique representation +\[ + g_i=p+r(y-b),\qquad g_j=s+r(x-a) +\] +with real $p,s,r$. The shared coefficient $r$ is the mixed-spin term +of this expansion. The conditional means and variances satisfy +\begin{align} + x-m_i^1&=\frac{(x-a)(1-uxy)}{1+uay},& + v_i^1&=\frac{A(1-u^2)}{(1+uay)^2},\notag\\ + y-m_j^1&=\frac{(y-b)(1-uxy)}{1+ubx},& + v_j^1&=\frac{B(1-u^2)}{(1+ubx)^2}. + \label{eq:pair-conditional-identities} +\end{align} + +For a function $F$ of one spin, write $F(z)=\overline F+z\,dF$, and set +\[ + F_1(y)=\frac{g_i(y)}{1+uay},\qquad + F_2(x)=\frac{g_j(x)}{1+ubx}. +\] +The identity $(1-uxy)^2(1+uxy)=(1-u^2)(1-uxy)$ and independence under +$\E_0$ give the following exact expression for the cross term: +\begin{equation} + \frac{\E_{ij}[(x-m_i^1)g_i\,(y-m_j^1)g_j]}{AB} + =\frac{1-u^2}{1+uab} + \bigl(dF_1dF_2-u\overline F_1\overline F_2\bigr). + \label{eq:pair-cross-exact} +\end{equation} +Both terms follow from the one-spin identities +$\E_0[(x-a)F_2(x)]=A\,dF_2$ and +$\E_0[(x-a)xF_2(x)]=A\overline F_2$, together with their $y$ +counterparts. Thus the cross term has been reduced to four scalar +coefficients. + +The four coefficients appearing here are +\begin{align*} + dF_1&=\frac{r(1+uab)-uap}{1-u^2a^2},& + \overline F_1&=\frac{p-(b+ua)r}{1-u^2a^2},\\ + dF_2&=\frac{r(1+uab)-ubs}{1-u^2b^2},& + \overline F_2&=\frac{s-(a+ub)r}{1-u^2b^2}. +\end{align*} +Consequently the right-hand side of \eqref{eq:pair-cross-exact} is +\begin{equation} + K\bigl[(-u+u^2ab)ps+u^2bA\,rp+u^2aB\,rs+Rr^2\bigr], + \label{eq:pair-cross-coefficients} +\end{equation} +where +\begin{align*} + K&=\frac{1-u^2}{(1+uab)(1-u^2a^2)(1-u^2b^2)},\\ + R&=(1+uab)^2-u(b+ua)(a+ub)\\ + &=1+uab+u^2(a^2b^2-a^2-b^2)-u^3ab. +\end{align*} +The mixed coefficients $rp$ and $rs$ begin at order $u^2$: their +first-order terms have canceled. Keeping this cancellation is what will +leave only a quartic row cost after Young's inequality. + +For $|u|\le1/4$ and $|a|,|b|\le1$, all denominators above are bounded +away from zero. There is thus a universal constant $C_1$ such that +\[ + |K-1|+|R-1|\le C_1|u|. +\] +Consequently the $ps$ coefficient in \eqref{eq:pair-cross-coefficients} +is $-u+O(u^2)$, both mixed coefficients are $O(u^2)$, and the $r^2$ +coefficient is $1+O(|u|)$. These constants are uniform in the conditional +means $a,b$: all the displayed rational functions and their required +derivatives are bounded on the compact parameter range. + +Return temporarily to the notation $a_i=v_i^1g_i$ and $a_j=v_j^1g_j$ for +the two gradient weights. Equations \eqref{eq:pair-density} and +\eqref{eq:pair-conditional-identities} imply +\begin{equation} + \frac{\E_{ij}[a_i a_j]}{AB} + =\E_0\bigl[w_u(x,y)g_i(y)g_j(x)\bigr],\qquad + w_u(x,y)=\frac{(1-u^2)^2(1+uxy)} + {(1+uab)(1+uay)^2(1+ubx)^2}. + \label{eq:pair-gradient-product} +\end{equation} +Uniformly in $a,b,x,y$, one has +\[ + c_1\le w_u(x,y)\le C_2,\qquad + |w_u(x,y)-1|\le C_2|u|\qquad (|u|\le1/4) +\] +for universal positive constants $c_1,C_2$. Expanding +$g_i(y)g_j(x)$ and using that the centered factors have zero product-law +mean gives +\begin{equation} + \frac{\E_{ij}[a_i a_j]}{AB} + =ps+e_0ps+e_1rp+e_2rs+e_3r^2, + \qquad |e_\ell|\le C_3|u|. + \label{eq:pair-gradient-coefficients} +\end{equation} +To obtain the four error bounds, expand the product into its $ps$, +$rp$, $rs$, and $r^2$ terms. Each centered spin is bounded by two, and +the uniform estimate for $w_u-1$ controls every resulting coefficient. + +Adding $u$ times \eqref{eq:pair-gradient-coefficients} to +\eqref{eq:pair-cross-coefficients}, and decreasing the universal $u_0>0$ +if necessary, yields +\begin{align*} + &\frac{\E_{ij}[(x-m_i^1)g_i\,(y-m_j^1)g_j] + +u\E_{ij}[a_i a_j]}{AB}\\ + &\hspace{15mm}\ge + \frac12r^2-C_4u^2|ps| + -C_4u^2|r|(|p|+|s|)\\ + &\hspace{15mm}\ge -C_5u^2|ps|-C_5u^4(p^2+s^2). +\end{align*} +The last inequality is Young's inequality, absorbing the mixed terms +into $r^2/2$. + +We must now express the two scalar costs in the heat-bath weights that +can be summed over pairs. The lower bound for $w_u$ in +\eqref{eq:pair-gradient-product}, followed by product independence, +gives +\begin{align*} + \E_{ij}|a_i a_j| + &\ge c_1AB\,\E_0|g_i(y)g_j(x)|\\ + &=c_1AB\,\E_0|g_i(y)|\,\E_0|g_j(x)| + \ge c_1AB|ps|. +\end{align*} +Similarly, the density in \eqref{eq:pair-density} and the ratios +$v_i^1/A$, $v_j^1/B$ are bounded above and below by universal positive +constants for $|u|\le u_0$. Hence +\[ + \E_{ij}[v_i^1g_i^2]\ge c_2A\E_0g_i^2 + \ge c_2Ap^2\ge c_2ABp^2, + \qquad + \E_{ij}[v_j^1g_j^2]\ge c_2ABs^2. +\] +The comparison constants do not deteriorate when a conditional variance +$A$ or $B$ is small. Substituting these bounds into the preceding scalar +inequality proves the pair estimate +\begin{align} + \E_{ij}[(x-m_i^1)g_i\,(y-m_j^1)g_j] + \ge{}&-u\E_{ij}[a_i a_j]-C_6u^2\E_{ij}|a_i a_j|\notag\\ + &-C_6u^4\E_{ij}[v_i^1g_i^2+v_j^1g_j^2]. + \label{eq:terminal-pair-bound} +\end{align} + +To assemble the curvature form, first condition on all but spin $i$ in +the diagonal term of $\E(Hf)^2$. Their sum is +$\sum_i\E[v_i^1g_i^2]=\D_{h_0}(f)$. Next average +\eqref{eq:terminal-pair-bound} over the outside spins and sum over +ordered pairs $i\ne j$. The signed and absolute pair costs become the +two quadratic forms in \eqref{eq:terminal-curvature}. The quartic cost +is bounded by +\[ + 2C_6\,r_4(U)\sum_i\E[v_i^1g_i^2]. +\] +Increasing $C$ proves the lemma. +\end{proof} + +The curvature inequality is useful when the few coordinates with appreciable +conditional variance occupy a small principal submatrix. Bandeira, El Alaoui, +and R\"odder~\cite[Theorem~1.1]{BER2026} combine uniform bounds on small +principal submatrices with a sufficiently large constant field to prove +polynomial mixing. Here the matrix bounds enter the signed curvature form +directly. We next obtain matrix control uniform over every such coordinate +set; the set will later depend on the spin configuration. + +\begin{lemma}[Small principal submatrices] +\label{lem:terminal-matrices} +Fix $j\in(0,1)$. There is a finite constant $M$, depending only on $j$, +with the following property. For every $\varepsilon>0$ there is +$\alpha\in(0,1/2)$ such that, with probability tending to one over $J$, +\begin{gather} + \max_{i,j}|J_{ij}|\le n^{-1/4},\qquad + \|J\|,\|U\|,\|Q\|\le M,\qquad + r_4(U)\le Mn^{-1/2}, + \label{eq:terminal-global-matrix}\displaybreak[1]\\ + \|U_{SS}\|\le\varepsilon,\qquad + \|Q_{SS}\|\le\varepsilon + \quad\text{for every }S\subseteq\{1,\ldots,n\} + \text{ with }|S|\le\alpha n. + \label{eq:terminal-sparse-matrix} +\end{gather} +The same $M$ works for all $\varepsilon$. +\end{lemma} + +\begin{proof} +For any deterministic unit vector $z$, the centered Gaussian variable +$z^{\mathsf T}Jz$ has variance +\[ + \frac{4j}{n}\sum_{it\}\le + 2\exp\{-nt^2/(4j)\}. + \label{eq:terminal-quadratic-tail} +\end{equation} +To pass from fixed quadratic forms to norms, choose a $1/4$-net of the +unit sphere in $\R^k$ with at most $9^k$ points. Every symmetric +$k\times k$ matrix $B$ then satisfies +\[ + \|B\|\le2\max_{z\text{ in the net}}|z^{\mathsf T}Bz|; +\] +to see this, approximate a unit vector maximizing the absolute quadratic +form and bound the change by $2\|B\|/4$. Taking $k=n$ in +\eqref{eq:terminal-quadratic-tail} proves $\|J\|\le M_0$ with probability +at least $1-e^{-c n}$ for a sufficiently large constant $M_0=M_0(j)$. + +For a fixed row $i$ and a fixed set $S$ of at most $k$ indices, Gaussian +independence within that row gives +\begin{equation} + \Prob\left\{\sum_{\ell\in S}J_{i\ell}^2>t\right\} + \le \exp\left\{-\frac{nt}{4j}+\frac{k}{2}\log2\right\}. + \label{eq:terminal-row-tail} +\end{equation} +Use exponential Markov with parameter $n/(4j)$ to obtain this estimate: +each nonzero Gaussian square contributes the factor $\sqrt2$ to the +moment generating function. Choosing the full index set and taking a +union bound over rows bounds every row square sum by a fixed +$M_1=M_1(j)$, with failure at most $e^{-cn}$. Separately, the Gaussian +entry tail and a union bound give +\[ + \Prob\{\max_{i,\ell}|J_{i\ell}|>n^{-1/4}\} + \le 2n^2\exp\{-\sqrt n/(2j)\}=o(1). +\] +Since $|\tanh z|\le|z|$, these bounds imply $\|Q\|\le M_1$ and +$r_4(U)\le M_1n^{-1/2}$, using the maximal absolute row-sum bound for +the operator norm of a symmetric matrix. Moreover, +\[ + |\tanh z-z|\le |z|^3/3, + \qquad + \|U-J\|\le \frac13\max_i\sum_\ell|J_{i\ell}|^3 + \le\frac{M_1}{3}n^{-1/4}. +\] +The scalar inequality follows by integrating +$|1-\sech^2 z|=\tanh^2 z\le z^2$. This proves +\eqref{eq:terminal-global-matrix}, with a fixed $M$. + +The global bounds have fixed $M$; we now select the small-set fraction +$\alpha$. Put $k=\lfloor\alpha n\rfloor$. Bounding sets of size exactly +$k$ suffices, because a smaller set can be enlarged and restriction cannot +increase a principal-submatrix norm. The $\binom nk$ possible sets have +logarithmic count at most $n\mathfrak h(k/n)$, where +\[ + \mathfrak h(t)=-t\log t-(1-t)\log(1-t), +\] +and $\mathfrak h(t)\to0$ as $t\downarrow0$. Applying the sphere-net +argument on each support and using \eqref{eq:terminal-quadratic-tail} +gives +\[ + \Prob\left\{\max_{|S|=k}\|J_{SS}\|>\varepsilon/2\right\} + \le 2\binom nk9^k + \exp\{-n\varepsilon^2/(64j)\}. +\] +Choose $\alpha>0$ sufficiently small that this tends to zero +exponentially in $n$. Then $\|U-J\|=o(1)$ proves the assertion for $U$. + +Finally, \eqref{eq:terminal-row-tail}, with $t=\varepsilon$, and a union +bound over all rows and all $k$-element sets give +\[ + \Prob\left\{\max_i\max_{|S|=k}\sum_{\ell\in S}J_{i\ell}^2 + >\varepsilon\right\} + \le n\binom nk + \exp\left\{-\frac{n\varepsilon}{4j}+\frac{k}{2}\log2\right\}. +\] +Decreasing $\alpha$ further makes this probability exponentially small. +On its complement, the maximal row sum of every $Q_{SS}$ is at most +$\varepsilon$, which proves the remaining assertion. +\end{proof} + +We apply these matrix bounds to a noisy observation. The following statement +allows any initial spin law, with the interaction still drawn at a fixed +$j\in(0,1)$; the exceptional probability concerns the observation at +that fixed matrix. + +\begin{proposition}[Terminal observation gap] +\label{prop:terminal-gap} +Fix $j\in(0,1)$. There are finite $T>0$, $c>0$, and events +$\mathcal E_n^{\mathrm{term}}$ depending only on $J$, with +$\Prob_J(\mathcal E_n^{\mathrm{term}})\to1$, such that the following holds +for every sufficiently large $n$ and every $J\in\mathcal E_n^{\mathrm{term}}$. +Let $X$ have any probability law on $\{-1,1\}^n$, possibly depending on +$J$, and let $Z$ be an independent standard Gaussian vector. Set +$Y=TX+\sqrt T\,Z$. Then, conditionally on $J$, except on an event of +probability at most $e^{-cn}$, the inequality +\begin{equation} + \Var_{\mu_Y}(f)\le 2\D_Y(f) + \label{eq:terminal-poincare} +\end{equation} +holds simultaneously for every real function $f$ on the cube. +All constants and matrix events are independent of the law of $X$. +\end{proposition} + +\begin{proof} +Choose the tolerances in the order needed to absorb the curvature costs. +Let $C$ be the constant in Lemma~\ref{lem:curvature}, and choose +$\varepsilon>0$ with $(1+C)\varepsilon\le1/8$. +Lemma~\ref{lem:terminal-matrices} supplies $\alpha$ and the fixed global +bound $M$. Finally choose the fixed observation time $T$ large enough that +\begin{equation} + \frac{16M^2}{T^2}\le\frac\alpha2, + \qquad + (1+C)M\bigl(2\sech(T/4)+\sech^2(T/4)\bigr)\le\frac18, + \qquad + e^{-T/8}\le\frac{\alpha}{4(e-1)}. + \label{eq:terminal-T-choice} +\end{equation} +Let $\mathcal E_n^{\mathrm{term}}$ be the events supplied by that lemma. +For all sufficiently large $n$, their bounds also ensure +$\max|U_{ij}|\le u_0$ and $Cr_4(U)\le1/8$. + +First count coordinates whose observation field remains small. Given +$X$, the indicators $\one_{\{|Y_i|\alpha n/2\mid X,J\} + \le e^{-\alpha n/2}(1+p_T(e-1))^n + \le e^{-\alpha n/4}. +\] +The same bound holds after averaging over $X$. + +Suppose now that $B_Y\le\alpha n/2$. For every spin configuration $x$, +$\|Jx\|^2\le M^2n$, and consequently +\[ + \#\{i:|(Jx)_i|>T/4\}\le\frac{16M^2n}{T^2}\le\frac\alpha2 n. +\] +It follows, simultaneously for all $x$, that +\begin{equation} + S(x)=\{i:|Y_i+(Jx)_i|0$, every monomial of weight at most $-r$ is +divisible by $p^r$, so +\[ + \mathcal F_{-r}A\subseteq p^rA. +\] +By Proposition~\ref{prop:polynomial}, $A$ is a polynomial ring, and +Equation~\eqref{con:candidate} shows that $p$ is a nonzero nonunit. +Therefore $\bigcap_{r>0}p^rA=0$: in a unique +factorization domain, a nonzero element cannot be divisible by +arbitrarily high powers of an irreducible factor of $p$. +This proves separation. In particular, every nonzero $a\in A$ has a +well-defined degree +\[ + \deg a=\min\{d\in\mathbb Z:a\in\mathcal F_dA\} +\] +and a nonzero initial form in $\mathcal F_dA/\mathcal F_{d-1}A$. + +The element $x$ is homogeneous of weight zero in $P$, and +$H=H_*-u$ has highest weight part $H_*$, of weight one. +For every nonzero $q\in P$, the highest weight part of $qH$ is +$(\operatorname{in}q)H_*$: the polynomial ring is a graded domain, +so the product of the highest parts does not vanish. +Consequently the ideal of initial relations is exactly $(H_*)$. +For completeness, the natural graded map $P\to\gr A$ sends a homogeneous +polynomial of weight $d$ to its class in +$\mathcal F_dA/\mathcal F_{d-1}A$, which may at this stage be zero. +This map is surjective. +A homogeneous polynomial of weight $d$ lies in its kernel exactly +when its image in $A$ has a representative of weight at most $d-1$; +subtracting that representative gives a relation with the original +polynomial as its highest part. Thus the kernel is the ideal of +initial relations, proving \eqref{gr:presentation}. + +To prove that this quotient is a domain, use the polynomial +coordinates $x,y,z,u$ of $P[p^{-1}]$ from the preceding section. +The identity $xy-z(z+1)=p(H+u)$ becomes +\[ + pH_*=xy-z(z+1). +\] +The polynomial on the right is irreducible over +$k[p,p^{-1},x,z,u]$: as a polynomial in $y$, it is primitive, +because $x$ and $z(z+1)$ are coprime, and it has degree one over +the fraction field. Thus $H_*$ is irreducible after inverting $p$. +It is not divisible by $p$ before localization, since its reduction +modulo $p$ is +\[ + (s^2-u^2)^2F-\bigl(1+2s(s^2-u^2)\bigr)J\ne0. +\] +Any factorization of $H_*$ in $P$ has a factor that becomes a unit after +inverting $p$, hence is a scalar times a power of $p$. +Since $p\nmid H_*$, this factor is already a unit in $P$. +Thus $H_*$ is irreducible in $P$, and hence prime by unique factorization. +Finally, $H_*$ is not associated to any +of the five variables, so their classes in $G$ are nonzero. +These classes are the initial forms of the corresponding elements +of $A$, proving the asserted degrees. +\end{proof} + +\subsection{A quadric and its distinguished ideal} + +From now on write $\tau$ for the class of $p$ in $G$, to distinguish +the graded ring from $A$. Consider the quadric ring +\begin{equation} +\label{gr:quadric} + B=k[x,y,z,u]/\bigl(xy-z(z+1)\bigr). +\end{equation} +The formulas +\[ + x=s^2-u^2+\tau F,\qquad + y=s+x(x+u^2),\qquad z=sx+\tau J +\] +define elements of degree zero in $G$. They satisfy the relation in +\eqref{gr:quadric}; after inverting $\tau$, the inverse coordinate +formulas from the preceding section give an isomorphism +\begin{equation} +\label{gr:localization} + G[\tau^{-1}]=B[\tau,\tau^{-1}], + \qquad \deg B=0,\quad \deg\tau=-1. +\end{equation} +In particular, the induced map $B\to G$ is injective: its composition +with localization is the inclusion of $B$ in this Laurent polynomial +ring. We henceforth regard $B$ as a subring of $G$. + +Define elements and an ideal of $B$ by +\begin{equation} +\label{gr:ideal} + \begin{aligned} + s_B&=y-x(x+u^2),\\ + f_0&=x-s_B^2+u^2,\qquad + g_0=z-s_Bx,\qquad I=(f_0,g_0). + \end{aligned} +\end{equation} +In \eqref{gr:localization} we have $s=s_B$, $F=f_0/\tau$, and +$J=g_0/\tau$. These identities give the more precise description +\begin{equation} +\label{gr:modification} + G=B[\tau,I/\tau] + =B[\tau,f_0/\tau,g_0/\tau] + \subset B[\tau,\tau^{-1}]. +\end{equation} +This is the algebraic construction of an affine modification, with +divisor $\tau=0$ and center $(\tau,I)$ in $B[\tau]$; see +\cite[Definition~1.1 and Proposition~1.1(b)]{KalimanZaidenberg1999}. +We will only need its explicit +homogeneous pieces. + +\begin{lemma} +\label{gr:positive-piece} +For each integer $\ell>0$, the degree-$\ell$ piece of $G$ is +\[ + G_\ell=\tau^{-\ell}I^\ell. +\] +\end{lemma} + +\begin{proof} +A monomial in the generators in \eqref{gr:modification}, with +coefficient in $B$, has the form +\[ + b\,\tau^r(f_0/\tau)^i(g_0/\tau)^j, + \qquad r,i,j\ge0. +\] +It has degree $\ell$ exactly when $i+j-r=\ell$. +In that case it equals $\tau^{-\ell}b f_0^i g_0^j$, whose numerator +belongs to $I^{\ell+r}\subseteq I^\ell$. +Since the presentation is homogeneous, such monomials span $G_\ell$. +Conversely, every element of $I^\ell$ is a $B$-linear combination of +the products $f_0^i g_0^j$ with $i+j=\ell$, and dividing these +products by $\tau^\ell$ gives elements of $G_\ell$. +\end{proof} + +\subsection{What a coordinate would force} + +The ideal $I$ records the positive part of the grading. We now show +why a hypothetical coordinate system containing $p$ supplies a nonzero +locally nilpotent derivation of $B$ fixing a nonzero element of $I$. + +\begin{proposition} +\label{prop:coordinate-obstruction} +If $p$ is a coordinate of $A$, then there exist a nonzero locally +nilpotent derivation $E$ of the ring $B$ in \eqref{gr:quadric} and +an element $0\ne h\in I$ such that $E(h)=0$. +\end{proposition} + +\begin{proof} +Suppose $A=k[p,q_1,q_2,q_3]$ for a polynomial coordinate system. +At least one $q_i$ has strictly positive filtration degree. +Indeed, if all three had degree at most zero, then so would every +polynomial in $p,q_1,q_2,q_3$, contrary to $\deg F=1$. +Relabel so that $\deg q_1>0$, and take +\[ + D=\frac{\partial}{\partial q_2}. +\] +This is a nonzero locally nilpotent derivation of $A$ satisfying +$D(p)=D(q_1)=0$. + +We give the details of its passage to $G$, because the filtration +has both positive and negative indices. +Write $\omega(t)$ for the assigned weight of a generator +$t\in\{p,s,u,F,J\}$, and put +\begin{equation} +\label{gr:shift} + m=\max_{D(t)\ne0}\bigl(\deg D(t)-\omega(t)\bigr). +\end{equation} +This maximum is defined: a nonzero derivation cannot kill all +algebra generators. The product rule shows that a monomial of +weight at most $d$ has derivative in $\mathcal F_{d+m}A$. +Every element of $\mathcal F_dA$ has a representative that is a sum +of such monomials, by the definition of the quotient filtration. +Thus +\[ + D(\mathcal F_dA)\subseteq\mathcal F_{d+m}A + \qquad(d\in\mathbb Z). +\] +It follows that $D$ induces a homogeneous derivation $D_0$ of $G$ +of degree $m$. +For $a\in\mathcal F_dA$, write $[a]_d$ for its class in +$\mathcal F_dA/\mathcal F_{d-1}A$. Explicitly, +\begin{equation} +\label{gr:iterate} + D_0([a]_d)=[D(a)]_{d+m},\qquad + D_0^r([a]_d)=[D^r(a)]_{d+rm}\quad(r\ge0). +\end{equation} +The first formula is well-defined because $D$ also sends +$\mathcal F_{d-1}A$ into $\mathcal F_{d+m-1}A$; the second follows +by induction. It proves that $D_0$ is locally nilpotent, first on +homogeneous elements and then on finite sums of them. +Moreover, equality is attained in \eqref{gr:shift} for some +generator $t$, whose initial form has degree $\omega(t)$ by +Proposition~\ref{prop:graded}. Its image under $D_0$ is the nonzero +initial form of $D(t)$, so $D_0\ne0$. +Finally, \eqref{gr:iterate} shows that $D_0$ kills both $\tau$ and +the nonzero initial form $\eta$ of $q_1$. The latter is homogeneous +of degree $\ell=\deg q_1>0$. + +Because $D_0(\tau)=0$, extending $D_0$ to +$G[\tau^{-1}]=B[\tau,\tau^{-1}]$ preserves local nilpotence. +Homogeneity implies that there is a derivation $E:B\to B$ with +\begin{equation} +\label{gr:descent} + D_0(b)=\tau^{-m}E(b)\qquad(b\in B). +\end{equation} +Indeed, every degree-$m$ element of the Laurent ring is uniquely +$\tau^{-m}$ times an element of $B$. +The derivation $E$ is nonzero: otherwise $D_0$ would kill both $B$ +and $\tau$, hence the whole Laurent ring and its subring $G$. +It is locally nilpotent because +\[ + E^r(b)=\tau^{rm}D_0^r(b) + \qquad(b\in B, r\ge0). +\] +Here, as in \eqref{gr:iterate}, the formula follows by induction, +using that $D_0$ fixes $\tau$. + +By Lemma~\ref{gr:positive-piece}, write +$\eta=\tau^{-\ell}h$ with $0\ne h\in I^\ell$. +Equations \eqref{gr:descent} and $D_0(\eta)=0$ give +\[ + 0=D_0(\eta)=\tau^{-m-\ell}E(h). +\] +Thus $E(h)=0$, and $I^\ell\subseteq I$ because $\ell>0$. +\end{proof} + +Proposition~\ref{prop:coordinate-obstruction} has removed the +original coordinate system from the problem. It remains to show +that no nonzero locally nilpotent derivation of $B$ can have a +nonzero invariant in $I$. We will make this ideal principal after +pulling it back to the complement of the zero section in a line bundle, +where a second grading gives the contradiction. diff --git a/preprints/A-stable-coordinate-that-is-not-a-coordinate-in-four-variables-October-5-2026/build/source/sections/04-bundle.tex b/preprints/A-stable-coordinate-that-is-not-a-coordinate-in-four-variables-October-5-2026/build/source/sections/04-bundle.tex new file mode 100644 index 0000000000000000000000000000000000000000..099c3373bed9f142dcf150f4585f5a837d012192 --- /dev/null +++ b/preprints/A-stable-coordinate-that-is-not-a-coordinate-in-four-variables-October-5-2026/build/source/sections/04-bundle.tex @@ -0,0 +1,233 @@ +\section{Lifting the obstruction to a line bundle}\label{sec:bundle} + +Proposition~\ref{prop:coordinate-obstruction} reduces noncoordinateness to +excluding a nonzero locally nilpotent derivation $E$ of $B$ with a nonzero +invariant in $I=(f_0,g_0)$. We pull $I$ back to the complement of the zero +section in a line bundle over $\Spec B$; on this complement the ideal +becomes principal. Every additive action on the base lifts linearly to +the line bundle and preserves this complement. Consequently, an invariant +in $I$ will force the lifted action to fix one specific polynomial. This use of the quadric and its line bundle +adapts the construction of \cite[Sections~4--6]{Cancellation2026}; all the +lifting and principalization arguments needed here are proved below. + +Put $Y=\Spec B$ and define +\begin{equation}\label{bun:C} + C=k[a,d,b,c,u]/(ac-bd-1). +\end{equation} +The assignment +\begin{equation}\label{bun:base-map} + x=ab,\qquad y=dc,\qquad z=db,\qquad u=u +\end{equation} +defines a homomorphism $B\longrightarrow C$, since +$xy=abdc=(db)(ac)=z(z+1)$. The ring $C$ is a domain: the polynomial +$ac-bd-1$, viewed as a primitive linear polynomial in $c$ over +$k[a,d,b,u]$, is irreducible. We give $C$ the \emph{fiber weights} +\begin{equation}\label{bun:fiber-weights} + \operatorname{wt}(a,d,b,c,u)=(1,1,-1,-1,0). +\end{equation} +The relation is homogeneous of weight zero, and the image of $B$ has weight +zero. These weights describe the action of $\G_m$ that scales the fibers +of the following bundle. Geometrically this is the familiar quotient of +$\mathrm{SL}_2$ by a diagonal torus, times the affine $u$-line; see +\cite[p.~8431]{BlancVanSanten2019}. We use explicit charts throughout. + +\begin{lemma}\label{bun:charts} +The map $B\longrightarrow C$ is injective. The morphism +$\Spec C\longrightarrow Y$ is the complement of the zero section in a line +bundle $\mathcal L\longrightarrow Y$. It is trivial over the open cover +\[ + U_0=D(z+1),\qquad U_1=D(z) +\] +of $Y$, with respective invertible fiber coordinates $a$ and $d$. +\end{lemma} + +\begin{proof} +Over $U_0$, the identity $ac=z+1$ makes $a$ invertible. The formulas +\begin{equation}\label{bun:chart-a} + c=\frac{z+1}{a},\qquad b=\frac{x}{a},\qquad + d=\frac{ay}{z+1} +\end{equation} +give an isomorphism +$C[(z+1)^{-1}]=B[(z+1)^{-1}][a,a^{-1}]$. +In particular the localized map from $B$ is injective, and so is the +original map because $B$ is a domain. Over $U_1$, the corresponding formulas +are +\begin{equation}\label{bun:chart-d} + b=\frac{z}{d},\qquad c=\frac{y}{d},\qquad a=\frac{xd}{z}, +\end{equation} +and give $C[z^{-1}]=B[z^{-1}][d,d^{-1}]$. +The opens cover $Y$ because $z$ and $z+1$ generate the unit ideal. On their +overlap the fiber coordinates satisfy +\begin{equation}\label{bun:transition} + d=\frac{y}{z+1}\,a, +\end{equation} +where $y/(z+1)$ is a unit with inverse $x/z$. Gluing the trivial line bundles +by this transition gives $\mathcal L$; removing their zero sections gives +precisely the two charts of $\Spec C$ above. Both fiber coordinates have +weight one, so the scaling agrees with \eqref{bun:fiber-weights}. +\end{proof} + +The two generators of $I$ acquire a common factor in $C$. Define +\begin{equation}\label{bun:v} + v=a^2(x+u^2)-d^2=a^3b+a^2u^2-d^2\in C. +\end{equation} + +\begin{lemma}\label{bun:principalization} +The extended ideal $IC$ is the principal ideal $(v)$. +\end{lemma} + +\begin{proof} +In $C[a^{-1}]$, put $\xi=d/a$ and $j=x+u^2-\xi^2$. The determinant relation +and \eqref{bun:base-map} give +\[ + z=\xi x,\qquad y=\xi+\xi^2x,\qquad + s_B=y-x(x+u^2)=\xi-xj. +\] +Consequently +\begin{equation}\label{bun:ideal-chart} + f_0=(1+2\xi x-x^2j)j,\qquad g_0=x^2j,\qquad v=a^2j. +\end{equation} +The two coefficients of $j$ generate the unit ideal, as the identity +\[ + (1-2\xi x)(1+2\xi x-x^2j) + +x^2(4\xi^2+j-2\xi xj)=1 +\] +shows. Thus $IC[a^{-1}]=(j)=(v)$. + +It remains to check primes of $C$ containing $a$. Modulo such a prime, +$bd=-1$, while $x=0$ and $z=-1$. Hence $g_0=z-s_Bx$ has nonzero residue +$-1$, and $v$ has nonzero residue $-d^2$. Both are units in the corresponding +local ring. The ideals $IC$ and $(v)$ therefore agree at every prime, which +proves the assertion. +\end{proof} + +Figure~\ref{bun:figure} records the geometric meaning of this ideal +equality: the inverse image of $V(I)$ is cut out by the single function $v$. + +\begin{figure}[ht] +\centering +\begin{tikzcd}[column sep=large,row sep=large] + V(v) \arrow[r,hook] \arrow[d] & \Spec C \arrow[d] \\ + V(I) \arrow[r,hook] & Y=\Spec B +\end{tikzcd} +\caption{A Cartesian square of closed subschemes. Pulling $I$ to the +complement of the zero section of $\mathcal L$ gives the principal ideal +$(v)$. In particular, every $h\in I$ pulls back to a multiple of $v$.} +\label{bun:figure} +\end{figure} + +We next lift additive actions from $Y$ to $\mathcal L$. The elementary +reason that a lift exists is that adjoining an affine-line parameter does +not change line bundles on $Y$. + +\begin{lemma}\label{bun:homotopy} +Every line bundle on $\A^1\times Y$ is isomorphic to the pullback of its +restriction to $\{0\}\times Y$. +\end{lemma} + +\begin{proof} +The two charts of Lemma~\ref{bun:charts} have coordinate rings +\begin{align*} + B[(z+1)^{-1}]&=k[x,t,u,(xt+1)^{-1}], + &t&=\frac{y}{z+1},\quad z=xt,\\ + B[z^{-1}]&=k[x,t,u,(xt-1)^{-1}], + &t&=\frac{y}{z},\quad z=xt-1. +\end{align*} +Here $t$ denotes a separate coordinate on each chart. Each ring, and its +polynomial extension by the affine-line coordinate $T$, is a noetherian +unique factorization domain. A line bundle on the spectrum of such a +ring is trivial: its divisor class is represented by a Cartier divisor, +and unique factorization makes every divisor principal. + +Let $\mathcal M$ be a line bundle on $\A^1\times Y$. Choose trivializations +on $\A^1\times U_0$ and $\A^1\times U_1$. Its transition function belongs +to $B[z^{-1},(z+1)^{-1}][T]^*$. A polynomial over a domain is a unit only +when it is a constant unit, so this transition function is independent of +$T$. Its specialization at $T=0$ is therefore the same transition +function. The chosen trivializations identify $\mathcal M$ with the +pullback of its restriction at zero. +\end{proof} + +The lifting argument has a useful general form. Its normalization step +is the standard one for linearizing a line bundle under an additive action; +compare \cite[Lemma~2.9]{Brion2015}. + +\begin{lemma}\label{bun:action-lift} +Let $X$ be an affine integral $k$-variety such that every line bundle on +$\A^1\times X$ is the pullback of its restriction at zero. Every +$\G_a$-action on $X$ lifts to any line bundle on $X$ by maps linear on +fibers, commuting with fiber scaling. +\end{lemma} + +\begin{proof} +Write $R=k[X]$, let $\mathcal M$ be the line bundle, and let +$\alpha:\A^1\times X\longrightarrow X$ be the action. If +$\pi:\A^1\times X\longrightarrow X$ is projection, then +$\alpha^*\mathcal M$ and $\pi^*\mathcal M$ restrict to $\mathcal M$ at +parameter zero. The hypothesis therefore supplies an isomorphism +\[ + \eta:\pi^*\mathcal M\longrightarrow\alpha^*\mathcal M. +\] +At zero, $\eta$ is an automorphism of $\mathcal M$, hence multiplication +by a unit of $R$. Multiplying $\eta$ by the inverse pullback of that unit +normalizes it to the identity at zero. + +The isomorphism $\eta$ gives a fiberwise linear lift of $\alpha$. To check +the group law, compare, over $\A^2\times X$, the map on fibers obtained +by applying parameters $S$ and then $T$ with the map obtained by applying +$T+S$. Both are global isomorphisms of the same source and target line bundles, +with fiber maps from $\mathcal M_x$ to +$\mathcal M_{\alpha(T+S,x)}$. Their ratio is an automorphism of the source +line bundle. Its endomorphism sheaf is canonically $\mathcal O$, so this +automorphism is multiplication by a global unit in $R[T,S]$. +Since $R$ is a domain, this unit is independent of $T,S$. At $(T,S)=(0,0)$ +it equals one by the normalization of $\eta$. Thus the group law holds. +The resulting action is linear on fibers, so it commutes with fiber +scaling and preserves the complement of the zero section. +\end{proof} + +We will also use the following elementary property of locally nilpotent +derivations; see \cite[Proposition~2.1(3)]{KalimanMakarLimanov2007}. + +\begin{lemma}\label{bun:factorial-closure} +Let $R$ be a domain over a field of characteristic zero and let $\delta$ be a locally +nilpotent derivation of $R$. If $r,s\in R$ are nonzero and $\delta(rs)=0$, +then $\delta(r)=\delta(s)=0$. +\end{lemma} + +\begin{proof} +The exponential $\exp(T\delta)$ takes values in $R[T]$. If $rs$ is fixed, +then +\[ + \exp(T\delta)(r)\,\exp(T\delta)(s)=rs. +\] +Degrees in $T$ add under multiplication of nonzero polynomials over a +domain. Both factors on the left therefore have degree zero, and their +coefficients of $T$ give the assertion. +\end{proof} + +\begin{proposition}\label{prop:lift} +Let $E$ be a nonzero locally nilpotent derivation of $B$ such that +$I\cap\ker E\ne0$. There is a nonzero locally nilpotent derivation +$\widetilde E$ of $C$ that extends $E$, preserves fiber weights, and satisfies +$\widetilde E(v)=0$. +\end{proposition} + +\begin{proof} +The exponential of $E$ defines an additive-group action on $Y$. +By Lemmas~\ref{bun:homotopy} and~\ref{bun:action-lift}, it lifts to +$\mathcal L$ by fiberwise linear maps. Restrict to the complement of the +zero section, namely $\Spec C$. +Differentiating this action at the identity gives a derivation +$\widetilde E$ of $C$. The action law expresses its pullback on each +function as $\sum_{r\ge0}T^r\widetilde E^{\,r}/r!$; because the action is +algebraic, the sum is finite for each function. Hence $\widetilde E$ is +locally nilpotent. It extends $E$ and is nonzero, and commutation with +fiber scaling says exactly that it preserves the weights +\eqref{bun:fiber-weights}. + +Finally choose $0\ne h\in I\cap\ker E$. The inclusion $B\subset C$ and +Lemma~\ref{bun:principalization} give a factorization $h=vq$ in $C$, with +both factors nonzero. Since $\widetilde E(h)=0$, +Lemma~\ref{bun:factorial-closure} yields $\widetilde E(v)=0$. +\end{proof} diff --git a/preprints/A-stable-coordinate-that-is-not-a-coordinate-in-four-variables-October-5-2026/build/source/sections/05-rigidity.tex b/preprints/A-stable-coordinate-that-is-not-a-coordinate-in-four-variables-October-5-2026/build/source/sections/05-rigidity.tex new file mode 100644 index 0000000000000000000000000000000000000000..70744d8b3ece5a65b4415ba31af0125b896b0eda --- /dev/null +++ b/preprints/A-stable-coordinate-that-is-not-a-coordinate-in-four-variables-October-5-2026/build/source/sections/05-rigidity.tex @@ -0,0 +1,125 @@ +\section{Rigidity and the coordinate obstruction}\label{sec:rigidity} + +Proposition~\ref{prop:lift} converts the hypothetical invariant in $I$ into +a locally nilpotent derivation of $C$ that fixes $v$ and preserves fiber +weights. We now rule out such a derivation. An auxiliary grading isolates +the difference of squares $a^2u^2-d^2$ inside $v$. Its two factors force +three generators to be invariant, leaving a coefficient whose required +fiber weight is impossible. + +\begin{proposition}\label{prop:rigidity} +There is no nonzero locally nilpotent derivation of +$C=k[a,d,b,c,u]/(ac-bd-1)$ that preserves the fiber weights +\eqref{bun:fiber-weights} and annihilates +$v=a^3b+a^2u^2-d^2$. +\end{proposition} + +\begin{proof} +Suppose that $\widetilde E$ is such a derivation. Give $C$ the auxiliary +grading in the second row of the following table: +\begin{equation}\label{rig:gradings} +\begin{array}{c|rrrrr} + &a&d&b&c&u\\ \hline + \text{fiber weight}&1&1&-1&-1&0\\ + \text{auxiliary degree}&0&1&-1&0&1 +\end{array} +\end{equation} +The two gradings are compatible because $ac-bd-1$ is homogeneous of degree +zero for each. The auxiliary degrees of the two parts of $v$ are +\[ + \deg_{\mathrm{aux}}(a^3b)=-1,\qquad + \deg_{\mathrm{aux}}(a^2u^2-d^2)=2. +\] + +Decompose $\widetilde E$ into its homogeneous components for auxiliary +degree shift, and let $E'$ be the nonzero component with largest shift, +say $m$. There are only finitely many components: a derivation is +determined by its values on the five homogeneous generators, and each +value is a finite sum of homogeneous terms. Each component is a +derivation by the product rule. Moreover, $E'$ is locally nilpotent. +Indeed, for homogeneous $r\in C$ of auxiliary degree $e$, the component +of auxiliary degree $e+nm$ in $\widetilde E^{\,n}(r)$ is $(E')^n(r)$. +For sufficiently large $n$ the former expression vanishes, so the latter +vanishes as well. Homogeneous elements span $C$, proving local +nilpotence on every element. Since the gradings in +\eqref{rig:gradings} are compatible, $E'$ still preserves fiber weights. + +Taking the component of degree $m+2$ in $\widetilde E(v)=0$ gives +\[ + E'(a^2u^2-d^2)=0. +\] +The factorization +\[ + a^2u^2-d^2=(au-d)(au+d) +\] +and Lemma~\ref{bun:factorial-closure} imply that $E'$ annihilates both +$au-d$ and $au+d$. Thus $E'(d)=E'(au)=0$, and another application of that +lemma gives +\begin{equation}\label{rig:fixed} + E'(a)=E'(d)=E'(u)=0. +\end{equation} +All the factors used here are nonzero; for example, after inverting $a$ +one has +\begin{equation}\label{rig:a-chart} + C[a^{-1}]=k[a,a^{-1},d,u][b],\qquad c=\frac{1+bd}{a}, +\end{equation} +where their nonvanishing is immediate. + +It remains to control the possible values of $E'$ on $b,c$. +Differentiating $ac-bd=1$ and using \eqref{rig:fixed} yields +$aE'(c)=dE'(b)$. Define +\[ + q=cE'(b)-bE'(c)\in C. +\] +Multiplication by $a$ and $d$, respectively, then gives +\begin{equation}\label{rig:q} + E'(b)=aq,\qquad E'(c)=dq. +\end{equation} +The element $q$ is nonzero, since otherwise $E'$ would kill every generator. +We claim that $q$ belongs to the polynomial subring $k[a,d,u]$. + +The elements $a,d,u$ are algebraically independent by +\eqref{rig:a-chart}. Put $K=k(a,d,u)$ and localize $C$ at the nonzero +elements of $k[a,d,u]$. Because these elements lie in $\ker E'$, the +localized derivation remains locally nilpotent. The resulting algebra is +$K[b]$, with $c=(1+bd)/a$. A locally nilpotent $K$-derivation of $K[b]$ +sends $b$ into $K$: if its value $P(b)$ had degree at least one, iterating +$P(b)\,d/db$ on $b$ would never give zero. More explicitly, differentiation +of a nonconstant polynomial multiplies its nonzero leading coefficient +by its positive degree, and multiplication by $P$ then leaves the result +nonconstant. Equation~\eqref{rig:q} therefore gives $q\in K$. + +To compute $C\cap K$ inside the fraction field of $C$, use both coordinate +charts: +\[ + C[a^{-1}]=k[a,a^{-1},d,u][b],\qquad + C[d^{-1}]=k[a,d,d^{-1},u][c]. +\] +An element of the first polynomial ring lying in $K$ has no $b$ terms, +and an element of the second lying in $K$ has no $c$ terms. Hence +\[ + C\cap K\subset + k[a,a^{-1},d,u]\cap k[a,d,d^{-1},u]=k[a,d,u]. +\] +The last equality follows from unique factorization in $k[a,d,u]$: +a reduced denominator that is both a power of $a$ and a power of $d$ +is a unit. The reverse inclusion is immediate, proving the claim. + +Finally, preservation of fiber weights makes $E'(b)$ have weight $-1$. +By \eqref{rig:q}, the nonzero polynomial $q$ has weight $-2$. This is +impossible in $k[a,d,u]$, whose generators have weights $1,1,0$. +The contradiction proves the proposition. +\end{proof} + +\begin{proof}[Proof of Theorem~\ref{thm:main}] +By Proposition~\ref{prop:polynomial}, the algebra $A$ is a polynomial ring +in four variables and its element $p$ becomes the polynomial $f$ stated +in the theorem. Proposition~\ref{prop:stable} makes $p$ a coordinate of +$A[w]$. If $p$ were a coordinate of $A$, +Proposition~\ref{prop:coordinate-obstruction} would produce a nonzero +locally nilpotent derivation $E$ of $B$ with $I\cap\ker E\ne0$. +Proposition~\ref{prop:lift} would then supply a derivation forbidden by +Proposition~\ref{prop:rigidity}. Thus $p$ is not a coordinate of $A$. +Transporting these two properties through the polynomial presentation +proves the assertion for $f$. +\end{proof} diff --git a/preprints/A-stable-coordinate-that-is-not-a-coordinate-in-four-variables-October-5-2026/stable-coordinate-four-variables.pdf b/preprints/A-stable-coordinate-that-is-not-a-coordinate-in-four-variables-October-5-2026/stable-coordinate-four-variables.pdf new file mode 100644 index 0000000000000000000000000000000000000000..7a91dab1307844bd16f3b37b6130d9b296ccc4b8 --- /dev/null +++ b/preprints/A-stable-coordinate-that-is-not-a-coordinate-in-four-variables-October-5-2026/stable-coordinate-four-variables.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:52d49d150cf01ef08c0eb878d1fde75b17125437a276a9616b4df7f1adeedcaa +size 338818 diff --git a/preprints/A-strict-four-row-permanent-inequality-and-permutation-moments-September-26-2026/README.md b/preprints/A-strict-four-row-permanent-inequality-and-permutation-moments-September-26-2026/README.md new file mode 100644 index 0000000000000000000000000000000000000000..e99f4caecc5840d5b46f458bdd4632b0bd3a6e8a --- /dev/null +++ b/preprints/A-strict-four-row-permanent-inequality-and-permutation-moments-September-26-2026/README.md @@ -0,0 +1,16 @@ +# [A strict four-row permanent inequality and permutation moments](main.pdf) + +OpenAI · September 26, 2026 + +## Citation + +```bibtex +@misc{OAI:A-strict-four-row-permanent-inequality-and-permutation-moments-September-26-2026, + author = {{OpenAI}}, + title = {{A strict four-row permanent inequality and permutation moments}}, + howpublished = {OpenAI Math Release preprint + \href{https://github.com/openai/math/blob/main/preprints/A-strict-four-row-permanent-inequality-and-permutation-moments-September-26-2026/main.pdf}{OAI:A-strict-four-row-permanent-inequality-and-permutation-moments-September-26-2026}}, + year = {2026} +} +``` + diff --git a/preprints/A-strict-four-row-permanent-inequality-and-permutation-moments-September-26-2026/build/main.tex b/preprints/A-strict-four-row-permanent-inequality-and-permutation-moments-September-26-2026/build/main.tex new file mode 100644 index 0000000000000000000000000000000000000000..6f2b65dcd168ce28617d522b341830297623033d --- /dev/null +++ b/preprints/A-strict-four-row-permanent-inequality-and-permutation-moments-September-26-2026/build/main.tex @@ -0,0 +1,48 @@ +\documentclass[11pt]{article} +\pdfinfoomitdate=1 +\pdftrailerid{} +\pdfsuppressptexinfo=15 +\usepackage[T1]{fontenc} +\usepackage{lmodern,microtype,amsmath,amssymb,amsthm,mathtools,booktabs} +\usepackage[margin=1.05in]{geometry} +\usepackage{tikz} +\usetikzlibrary{arrows.meta,positioning} +\usepackage[colorlinks=true,linkcolor=blue,citecolor=blue,urlcolor=blue]{hyperref} +\hypersetup{pdftitle={A strict four-row permanent inequality and permutation moments},pdfauthor={OpenAI}} +\newtheorem{theorem}{Theorem}[section] +\newtheorem{lemma}[theorem]{Lemma} +\newtheorem{proposition}[theorem]{Proposition} +\newtheorem{corollary}[theorem]{Corollary} +\theoremstyle{remark}\newtheorem{remark}[theorem]{Remark} +\DeclareMathOperator{\Tr}{Tr} +\DeclareMathOperator{\tr}{tr} +\DeclareMathOperator{\TV}{TV} +\DeclareMathOperator{\op}{op} +\DeclareMathOperator{\Res}{Res} +\DeclareMathOperator{\Ind}{Ind} +\newcommand{\E}{\mathbb E} +\newcommand{\Pp}{\mathbb P} +\newcommand{\one}{\mathbf1} +\newcommand{\sgn}{\mathrm{sgn}} +\newcommand{\HS}{\mathrm{HS}} +\newcommand{\norm}[1]{\left\lVert#1\right\rVert} +\title{A strict four-row permanent inequality and permutation moments} +\author{OpenAI} +\date{September 26, 2026} +\begin{document} +\maketitle +\begin{abstract} +We prove a permanent inequality with exponent strictly below two for laws on $S_4$ sufficiently close to uniform and with exactly uniform coordinate marginals. Applied to a four-row recursion, it shows that an absolute number of coordinate sweeps brings the full permutation of the Thorp shuffle on $N=2^d$ cards to total-variation distance tending to zero from uniform as $N\to\infty$. Thus the mixing time has the optimal order $O(\log N)$ in physical shuffles. +\end{abstract} +\input{sections/01-introduction} +\input{sections/02-permanent} +\input{sections/03-recursion} +\input{sections/04-multiplicities} +\input{sections/05-sparse} +\input{sections/06-tree} +\input{sections/07-amplification} +\begingroup\small +\bibliographystyle{plain} +\bibliography{references} +\endgroup +\end{document} diff --git a/preprints/A-strict-four-row-permanent-inequality-and-permutation-moments-September-26-2026/build/references.bib b/preprints/A-strict-four-row-permanent-inequality-and-permutation-moments-September-26-2026/build/references.bib new file mode 100644 index 0000000000000000000000000000000000000000..0e7ad80ea852ca23f7016a93480082add5d0186b --- /dev/null +++ b/preprints/A-strict-four-row-permanent-inequality-and-permutation-moments-September-26-2026/build/references.bib @@ -0,0 +1,166 @@ +@article{BereleRegev1987, + author = {Berele, A. and Regev, A.}, + title = {Hook 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year={1990}, doi={10.1007/BF01045887}} + +@article{cll2006, + author = {Carlen, Eric A. and Lieb, Elliott H. and Loss, Michael}, + title = {An inequality of {Hadamard} type for permanents}, + journal = {Methods and Applications of Analysis}, + volume = {13}, + number = {1}, + pages = {1--18}, + year = {2006}, + eprint = {math/0508096v1}, + archivePrefix = {arXiv}, + url = {https://arxiv.org/abs/math/0508096v1}, + note = {arXiv version 1, 4 August 2005} +} + +@article{liebeck-shalev2004, + author = {Liebeck, Martin W. and Shalev, Aner}, + title = {Fuchsian groups, coverings of {Riemann} surfaces, subgroup growth, random quotients and random walks}, + journal = {Journal of Algebra}, + volume = {276}, + number = {2}, + pages = {552--601}, + year = {2004}, + url = {https://www.ma.imperial.ac.uk/~mwl/fuchs.pdf} +} + +@article{mason-niese2016, + author = {Mason, Sarah K. and Niese, Elizabeth}, + title = {Quasisymmetric $(k,l)$-hook {Schur} functions}, + journal = {Annals of Combinatorics}, + volume = {22}, + pages = {167--199}, + year = {2018}, + doi = {10.1007/s00026-018-0376-2}, + eprint = {1606.06942v1}, + archivePrefix = {arXiv}, + url = {https://arxiv.org/abs/1606.06942v1}, + note = {Author version: arXiv:1606.06942v1, 22 June 2016} +} + +@misc{orellana-zabrocki2000, + author = {Orellana, Rosa C. and Zabrocki, Mike}, + title = {Some remarks on the characters of the general {Lie} superalgebra}, + year = {2000}, + eprint = {math/0008152v1}, + archivePrefix = {arXiv}, + url = {https://arxiv.org/abs/math/0008152v1}, + note = {Version 1, 18 August 2000} +} + + +@article{Thorp1973, + author={Thorp, Edward O.}, + title={Nonrandom shuffling with applications to the game of {Faro}}, + journal={Journal of the American Statistical Association}, + volume={68}, number={344}, pages={842--847}, year={1973}, + doi={10.1080/01621459.1973.10481434}} + +@article{Morris2008, + author={Morris, Ben}, title={The mixing time of the {Thorp} shuffle}, + journal={SIAM Journal on Computing}, volume={38}, number={2}, + pages={484--504}, year={2008}, doi={10.1137/050636231}, + url={https://arxiv.org/abs/math/0507307v1}, + note={Author version: arXiv:math/0507307v1, 15 July 2005}} + +@article{MontenegroTetali2006, + author={Montenegro, Ravi and Tetali, Prasad}, + title={Mathematical aspects of mixing times in {Markov} chains}, + journal={Foundations and Trends in Theoretical Computer Science}, + volume={1}, number={3}, pages={237--354}, year={2006}, + doi={10.1561/0400000003}, + url={https://www.stat.berkeley.edu/~aldous/260-FMIE/Papers/montenegro_tetali.pdf}} + +@article{Morris2009, + author={Morris, Ben}, + title={Improved mixing time bounds for the {Thorp} shuffle and {$L$}-reversal chain}, + journal={Annals of Probability}, volume={37}, number={2}, + pages={453--477}, year={2009}, doi={10.1214/08-AOP409}, + url={https://arxiv.org/abs/0802.0339v1}, + note={Author version: arXiv:0802.0339v1, 4 February 2008}} + +@article{Morris2013, + author={Morris, Ben}, title={Improved mixing time bounds for the {Thorp} shuffle}, + journal={Combinatorics, Probability and Computing}, volume={22}, + number={1}, pages={118--132}, year={2013}, doi={10.1017/S0963548312000478}, + url={https://arxiv.org/abs/0912.2759v1}, + note={Author version: arXiv:0912.2759v1, 14 December 2009}} + +@article{audenaert2007, + author={Audenaert, Koenraad M. R.}, + title={On the {Araki--Lieb--Thirring} inequality}, + journal={International Journal of Information and Systems Sciences}, + volume={4}, number={1}, pages={78--83}, year={2008}, + eprint={math/0701129v2}, + archivePrefix={arXiv}, + url={https://arxiv.org/abs/math/0701129v2}, + note={Author version: arXiv:math/0701129v2, 29 March 2007; Theorem 1}} + +@article{BristielCaputo2024, + author = {Bristiel, Alexandre and Caputo, Pietro}, + title = {Entropy inequalities for random walks and permutations}, + journal = {Annales de l'Institut Henri Poincar{\'e}, Probabilit{\'e}s et Statistiques}, + volume = {60}, + number = {1}, + pages = {54--81}, + year = {2024}, + doi = {10.1214/22-AIHP1267}, + eprint = {2109.06009v3}, + archivePrefix = {arXiv}, + url = {https://arxiv.org/abs/2109.06009v3}, + note = {Author version: arXiv:2109.06009v3, 10 May 2022; Corollary 1.14 and Section 4.3} +} diff --git a/preprints/A-strict-four-row-permanent-inequality-and-permutation-moments-September-26-2026/build/sections/01-introduction.tex b/preprints/A-strict-four-row-permanent-inequality-and-permutation-moments-September-26-2026/build/sections/01-introduction.tex new file mode 100644 index 0000000000000000000000000000000000000000..e8cf333a18639d4c78f7ce2bd55c7271ac4938a6 --- /dev/null +++ b/preprints/A-strict-four-row-permanent-inequality-and-permutation-moments-September-26-2026/build/sections/01-introduction.tex @@ -0,0 +1,68 @@ +\section{Introduction} +A product of independent random switches can make each card uniform while leaving substantial dependence between cards. The Thorp shuffle illustrates this distinction particularly well. On $N=2^d$ positions, with $d\ge1$, one coordinate sweep visits the binary-coordinate directions in the fixed order $1,\ldots,d$. In each direction it independently swaps the two cards on every edge with probability $1/2$, leaving them in place otherwise. Every individual card is uniform after this sweep. The question is how many sweeps randomize the entire permutation. + +We study that question through a four-row decomposition. Removing two coordinate directions leaves four smaller independent sweeps, one in each row of a $4$-by-$N/4$ array. The last two directions act within its columns. Each smaller sweep contracts according to the representations of its own row group, but restricting a representation of $S_N$ to the four row groups creates multiplicities. Those multiplicities accumulate as the decomposition is repeated. The main point is an inequality that makes their total cost strictly smaller than the available representation dimension. + +For a function $f:\{1,2,3,4\}\to[0,\infty)$, write +\[ + \|f\|_p=\left(\frac14\sum_{j=1}^4 f(j)^p\right)^{1/p}. +\] +If $u$ denotes uniform measure on $S_4$, then +$\E_u\prod_i f_i(\pi(i))$ is the permanent of the matrix $(f_i(j))$ divided by $4!$. Carlen, Lieb and Loss proved that it is at most $\prod_i\|f_i\|_2$ and characterized equality \cite[Theorem~1.1]{cll2006}. For nonzero nonnegative functions on four points, equality requires that every function be constant. We will use both the bound and this equality statement. + +\begin{samepage} +\begin{theorem}[A robust four-row inequality] +There are absolute constants $p_0\in(4/3,2)$ and $\varepsilon>0$ with the following property.\label{thm:permanent} Let $\nu$ be a probability measure on $S_4$ such that +\[ + \|\nu-u\|_{\TV}<\varepsilon, + \qquad + \nu\{\pi:\pi(i)=j\}=\frac14\quad(1\le i,j\le4). +\] +Then every four nonnegative functions on $\{1,2,3,4\}$ satisfy +\begin{equation}\label{eq:permanent} + \E_\nu\prod_{i=1}^4f_i(\pi(i))\le\prod_{i=1}^4\|f_i\|_{p_0}. +\end{equation} +\end{theorem} +\end{samepage} + +Here total variation is one half the sum of the absolute differences of the probability masses. The marginal assumption in the theorem is exact. Indeed, setting three functions equal to one in \eqref{eq:permanent}, and varying the fourth around the constant one, forces its coordinate marginal to be uniform. Thus this hypothesis describes a structural feature of the inequality, as well as the cancellation used in its proof. + +Permanents with row or column $\ell^p$ constraints have a broader history. Carlen--Lieb--Loss discuss the below-two problem and Caputo's conjecture, and Samorodnitsky obtained further bounds in that range \cite{cll2006,Samorodnitsky2008}. Bristiel and Caputo subsequently proved the sharp permanent bound for every $p\ge1$ \cite[Corollary~1.14]{BristielCaputo2024}. In the present normalization, their result gives +\[ + \E_u\prod_i f_i(\pi(i))\le\prod_i\|f_i\|_p + \quad\text{for every }p\ge p_c:=\frac{4\log4}{\log24}. +\] +The identity matrix shows that this uniform-law threshold is necessary. + +The assertion needed here is stability under small perturbations of the permutation law that preserve every coordinate marginal. We prove this nonoptimal robust form directly from the exponent-two theorem: a strict quadratic gap treats functions near constants, while the equality classification and compactness treat all remaining functions. The local calculation explains the endpoint $4/3$ in Theorem~\ref{thm:permanent}; the final exponent is chosen closer to two and is not optimized. Thus a below-two bound for the uniform law is not itself the new ingredient in the recursion. + +To state the application, let $\rho_\lambda$ be the irreducible unitary representation of $S_N$ indexed by a partition $\lambda\vdash N$, and write $D_\lambda$ for its dimension and $\lambda'$ for the conjugate partition. If $\mu_N$ is the law of one sweep, put +\[ + T_N(\lambda)=\sum_{g\in S_N}\mu_N(g)\rho_\lambda(g), + \qquad B_N(\lambda)=T_N(\lambda)T_N(\lambda)^*. +\] +Traces are ordinary, unnormalized matrix traces. Unmarked logarithms are natural. + +\begin{theorem}[A fixed moment with a degree saving]\label{thm:moment} +There exist an even integer $r\ge2$ and $\eta>0$, independent of $N$, such that for every dyadic $N$ and every $\lambda\vdash N$, +\begin{equation}\label{eq:moment} + D_\lambda\Tr B_N(\lambda)^r\le D_\lambda^{1-\eta}. +\end{equation} +Consequently there is an absolute integer $q$ for which the total-variation distance of $q$ independent forward sweeps from uniform tends to zero as $N\to\infty$, uniformly over the starting deck. +\end{theorem} + +A physical Thorp shuffle consists of independent fair switches on the pairs differing in one binary coordinate, followed by cyclic rotation of the coordinates. Tracking those deterministic rotations makes $d$ physical shuffles exactly one sweep. Theorem~\ref{thm:moment} therefore gives an $O(d)$ upper bound. The opposite order follows from the elementary support estimate +\begin{equation}\label{eq:support-intro} + \|\mu_t-U_{S_N}\|_{\TV}\ge1-\frac{2^{tN/2}}{N!}, +\end{equation} +where $\mu_t$ is the law after $t$ physical shuffles: there are only $2^{tN/2}$ coin strings. In particular, distance at most $1/4$ requires +$t\ge\lceil(2/N)\log_2(3N!/4)\rceil=2d-O(1)$. + +The shuffle originates in Thorp's study of nonrandom shuffling and Faro \cite{Thorp1973}. For $N=2^d$, Morris obtained an $O(d^{44})$ upper bound \cite{Morris2008}, and Montenegro and Tetali improved it to $O(d^{29})$ \cite[Theorems~6.14--6.15]{MontenegroTetali2006}. Morris subsequently proved $O((\log N)^4)$ for every even deck size \cite{Morris2009} and $O(d^3)$ for dyadic decks \cite{Morris2013}. The present proof reaches logarithmic order through a four-row moment recursion. + +The proof has three stages. Section~\ref{sec:permanent} proves the robust inequality and tensors it over columns. In Section~\ref{sec:recursion}, a positive trace comparison reduces the sweep moment to four smaller moments, multiplied by powers of squared restriction multiplicities. The power is +$\theta=1-1/p_0<1/2$: this strict improvement over one half is the feature the dimension comparison needs. Section~\ref{sec:multiplicities} bounds the multiplicities by removing the first few rows and columns of a Young diagram. Horizontal and vertical Pieri rules control the removed strips. For every fixed width $h$, the total number of boxes outside the first $h$ rows and columns of the four child diagrams is at most the corresponding number in the parent. We deduce this inequality from positive hook-Schur specialization, originating with Berele--Regev \cite{BereleRegev1987}; the formulas and tableau conventions used here are recorded in \cite{orellana-zabrocki2000,mason-niese2016}. + +Finally, Section~\ref{sec:sparse} stops the recursion for diagrams close to a row or column. Section~\ref{sec:tree} sums the costs at every other scale, including the immediate children at which recursion has stopped, and compares the result with the hook-length formula. The strict inequality $\theta<1/2$ is exactly the margin that absorbs these costs. Section~\ref{sec:amplification} converts the resulting moment to mixing. This last step follows the finite-group Fourier approach of Diaconis--Shahshahani \cite{DiaconisShahshahani1981}, with matrix norms retained because a sweep is generally nonnormal. + +The argument uses ordinary complex representation theory of symmetric groups: branching, induction and restriction, the Littlewood--Richardson and Pieri rules, and the hook-length formula \cite{James1978,Sagan2001,Stanley1999}. We give the additional multiplicity and dimension estimates in full. The positive trace comparison is the Araki--Lieb--Thirring inequality \cite{araki1990}. No shuffle moment or mixing theorem from another source is an input here. diff --git a/preprints/A-strict-four-row-permanent-inequality-and-permutation-moments-September-26-2026/build/sections/02-permanent.tex b/preprints/A-strict-four-row-permanent-inequality-and-permutation-moments-September-26-2026/build/sections/02-permanent.tex new file mode 100644 index 0000000000000000000000000000000000000000..d79fab05f20fe616876bd40c7f00253a08512e46 --- /dev/null +++ b/preprints/A-strict-four-row-permanent-inequality-and-permutation-moments-September-26-2026/build/sections/02-permanent.tex @@ -0,0 +1,84 @@ +\section{The strict permanent inequality}\label{sec:permanent} +We first prove Theorem~\ref{thm:permanent}. The two parts of the proof address different possible failures of a uniform improvement: functions close to the equality cases, and functions bounded away from them. + +\begin{proof}[Proof of Theorem~\ref{thm:permanent}] +A zero function makes the assertion immediate. By homogeneity, normalize each nonzero function to have $\|f_i\|_2=1$. The resulting space of quadruples is compact. The exponent-two Carlen--Lieb--Loss inequality is strict everywhere on this space except at the single quadruple $f_i\equiv1$. Its normalization follows directly from +\[ + \operatorname{perm}(f_i(j))\le\frac{4!}{4^2} + \prod_i\left(\sum_j f_i(j)^2\right)^{1/2}. +\] +The equality classification for nonzero nonnegative columns gives exactly the stated exceptional quadruple \cite[Theorem~1.1 and equation~(1.4)]{cll2006}. + +All scalar expectations and inner products on four points use uniform measure: +$\E_jh(j)=\tfrac14\sum_jh(j)$ and $\langle g,h\rangle=\tfrac14\sum_jg(j)h(j)$. +Fix $p_*\in(4/3,2)$. Near that quadruple, the means are positive, so renormalize by the means and write $f_i=1+g_i$, with $\E_jg_i(j)=0$. Let +\[ + V=\sum_i\|g_i\|_2^2,\qquad b=\max_i\|g_i\|_\infty. +\] +For $i\ne j$, uniform sampling without replacement gives +\[ + \E_u g_i(\pi(i))g_j(\pi(j)) + =-\frac13\langle g_i,g_j\rangle. +\] +Consequently the total quadratic term is +\begin{equation}\label{eq:quadratic} + \sum_{i1/6$, first choose $b_*>0$ and $\varepsilon_*>0$ small enough that the error terms in \eqref{eq:lhs-local} and \eqref{eq:rhs-local} fit inside this strict gap. The desired inequality then holds for every $p\in[p_*,2]$ whenever $b0$. Small positive $t$ contradicts the inequality. + +\begin{lemma}[Tensorization]\label{lem:tensorization} +Let $\pi_1,\ldots,\pi_b$ be independent random permutations, each with a law satisfying Theorem~\ref{thm:permanent} for the same $p_0$. If $F_i$ is a nonnegative function on $\{1,2,3,4\}^b$, then +\[ + \E\prod_{i=1}^4 F_i(\pi_1(i),\ldots,\pi_b(i)) + \le\prod_{i=1}^4 + \left(4^{-b}\sum_{x\in\{1,2,3,4\}^b}F_i(x)^{p_0}\right)^{1/p_0}. +\] +\end{lemma} +\begin{proof} +Induct on $b$. Fix the first $b-1$ coordinates and apply Theorem~\ref{thm:permanent} to the last coordinate. The four resulting functions on $b-1$ coordinates are +\[ + G_i(x)=\left(\frac14\sum_{j=1}^4 F_i(x,j)^{p_0}\right)^{1/p_0}. +\] +The induction hypothesis applies to their product, and its $p_0$th-power sums are precisely the full product-measure sums in the assertion. +\end{proof} + +For later use, let $U$ be a positive semidefinite stochastic matrix of order $s$. Its entries are nonnegative and its eigenvalues lie in $[0,1]$. H\"older's inequality, applied entry by entry with powers $2-p_0$ and $p_0-1$, gives +\begin{equation}\label{eq:interpolation} + \sum_{x,y}U(x,y)^{p_0} + \le\left(\sum_{x,y}U(x,y)\right)^{2-p_0} + \left(\sum_{x,y}U(x,y)^2\right)^{p_0-1} + \le s^{2-p_0}(\Tr U)^{p_0-1}. +\end{equation} +The final inequality uses $\Tr U^2\le\Tr U$. Taking the $p_0$th root with the uniform counting normalization gives +\[ + \left(s^{-2}\sum_{x,y}U(x,y)^{p_0}\right)^{1/p_0} + \le s^{-1}(\Tr U)^{1-1/p_0}. +\] +Section~\ref{sec:recursion} applies this estimate to four color-string transition matrices. Each contributes a factor $s^{-1}$ and a trace power $\theta=1-1/p_0$. The strict inequality $\theta<1/2$ will remain fixed throughout the proof. diff --git a/preprints/A-strict-four-row-permanent-inequality-and-permutation-moments-September-26-2026/build/sections/03-recursion.tex b/preprints/A-strict-four-row-permanent-inequality-and-permutation-moments-September-26-2026/build/sections/03-recursion.tex new file mode 100644 index 0000000000000000000000000000000000000000..26be70a686f6d7ee0e1e499c24f8301a7b329788 --- /dev/null +++ b/preprints/A-strict-four-row-permanent-inequality-and-permutation-moments-September-26-2026/build/sections/03-recursion.tex @@ -0,0 +1,176 @@ +\section{From columns to a four-row moment recursion}\label{sec:recursion} +Our objective is to express a moment on $N$ cards in terms of moments on $N/4$ cards. We first fix the operator conventions, then isolate the cost of the interaction between the four rows. + +Positions are binary vectors. A permutation sends input positions to output positions, and $gh$ applies $h$ first. In each coordinate direction, the average of all optional edge swaps is an orthogonal projection $\Pi_i$. With chronological order $1,\ldots,d$, +\[ + T_N=\Pi_d\cdots\Pi_1,\qquad B_N=T_NT_N^*. +\] +To check physical time, let $R(x_1,\ldots,x_d)=(x_2,\ldots,x_d,x_1)$ and let $S_t$ be the switches in the first coordinate. A physical step is $RS_t$. If $Y_t$ is the accumulated permutation, then $X_t=R^{-t}Y_t$ satisfies +\[ + X_{t+1}=(R^{-t}S_{t+1}R^t)X_t. +\] +The conjugated directions visit all coordinates in a fixed cyclic order, and $R^d=I$. Relabeling that order as $1,\ldots,d$ gives exactly $T_N$ after $d$ steps. + +These are group-algebra elements as well as operators in any representation. The Fourier transform convention is $\widehat\mu(\lambda)=\sum_g\mu(g)\rho_\lambda(g)$; convolution multiplies these matrices in the same order. For a matrix $A$, the unnormalized Schatten norm is $\|A\|_p^p=\Tr|A|^p$. + +\begin{lemma}[Annihilation and the finite-size gap]\label{lem:gap} +Every sweep kills the sign representation and every shape with more than $N/2$ rows. For fixed dyadic $N\ge2$, its operator norm on each nontrivial irreducible representation is strictly less than one. +\end{lemma} +\begin{proof} +Each layer averages a subgroup $H\cong(S_2)^{N/2}$. The sign character averages to zero. Young's rule says that an irreducible with $H$-fixed vectors must dominate the partition $(2^{N/2})$, so it has at most $N/2$ rows. For the norm assertion, equality in a product of orthogonal projections on a unit vector would force that vector to be fixed by every layer. It would then be fixed by every coordinate-edge transposition. Edge transpositions of a connected graph generate its full symmetric group, whereas a nontrivial irreducible has no invariant vector. +\end{proof} + +Take $N=4m$, using the last two coordinates as the row index. The first $d-2$ directions are independent sweeps within the four rows; the final two directions form a two-axis sweep independently in each of the $m$ columns. Let $M$ be the latter operator. Then +\begin{equation}\label{eq:split} + B_N=MAM^*,\qquad A=B_m^{\otimes4}. +\end{equation} +The row group is $H=(S_m)^4$. The column group is $(S_4)^m$; its factors act on disjoint columns. Figure~\ref{fig:fourrows} records the two operations. + +\begin{figure}[ht] +\centering +\begin{tikzpicture}[x=0.60cm,y=0.55cm,>=Stealth] +\foreach \i in {0,...,3}{\foreach \j in {0,...,7}{\fill (\j,-\i) circle (1.7pt);}} +\foreach \i in {0,...,3}{\draw[->,blue,thick] (-.6,-\i+.16)--(7.6,-\i+.16);} +\foreach \j in {0,...,7}{\draw[->,red,thick] (\j+.15,.6)--(\j+.15,-3.6);} +\node[blue,anchor=west] at (8,-1) {four sweeps on $m$ positions}; +\node[red,anchor=west] at (8,-2) {$m$ sweeps on four positions}; +\end{tikzpicture} +\caption{The four rows are indexed by the last two processed bits. The horizontal sweeps occur first; the vertical two-axis sweeps occur last. The number of columns shown is schematic.}\label{fig:fourrows} +\end{figure} + +\paragraph{A positive trace for each row type.} +Fix an even integer $r\ge2$, and put +\[ + F_r(\lambda)=D_\lambda\Tr B_{|\lambda|}(\lambda)^r. +\] +We suppress the subscript $r$ while deriving the recursion. Set $X=A^{r/2}$ and $Y=(M^*M)^{r/2}$. The Araki--Lieb--Thirring inequality for positive operators, with exponent $r/2\ge1$ and outer power two, gives +\begin{equation}\label{eq:alt} + F(\lambda)\le D_\lambda\Tr(XYXY)_\lambda + =D_\lambda\|Y^{1/2}X_\lambda Y^{1/2}\|_{\HS}^2. +\end{equation} +Indeed $MAM^*$ and $A^{1/2}M^*MA^{1/2}$ have the same nonzero eigenvalues, after which the stated positive-matrix inequality applies \cite{araki1990}; see also the formulation in \cite[Theorem~1]{audenaert2007}. + +The row operator $X=A^{r/2}$ is a positive element of the group algebra of +$H$. Let $\mathcal I(\lambda)$ be the set of tuples +$\boldsymbol\alpha=(\alpha_1,\ldots,\alpha_4)$, $\alpha_i\vdash m$, +occurring in $\Res_H V_\lambda$. For each tuple let +$X_{\boldsymbol\alpha}$ be $X$ multiplied by its central isotype +projection in the group algebra of $H$. This projection commutes with $X$, +so the summand is positive in every unitary representation. On $V_\lambda$ +it acts on all copies of that row type, not on a chosen copy. +The Hilbert--Schmidt triangle inequality in \eqref{eq:alt} gives +\begin{equation}\label{eq:triangle} + F(\lambda)^{1/2}\le + \sum_{\boldsymbol\alpha\in\mathcal I(\lambda)} + \bigl(D_\lambda\Tr(X_{\boldsymbol\alpha}Y + X_{\boldsymbol\alpha}Y)_\lambda\bigr)^{1/2}. +\end{equation} +We now estimate one term of this sum. In every irreducible representation, +$\Tr(X_{\boldsymbol\alpha}Y X_{\boldsymbol\alpha}Y)$ is the squared +Hilbert--Schmidt norm of $Y^{1/2}X_{\boldsymbol\alpha}Y^{1/2}$ and is +nonnegative. The regular representation of $S_N$ contains $D_\lambda$ +copies of $V_\lambda$. Its trace therefore bounds the quantity inside +the corresponding square root in \eqref{eq:triangle}. Working in this +larger representation will allow us to express the bound as a probability. + +\paragraph{From coefficients to an overlap probability.} +Fix the tuple $\boldsymbol\alpha$. Write +$X_{\boldsymbol\alpha}=\sum_{p\in H}\xi(p)p$, where +$\xi(p)=\prod_i\xi_i(p_i)$. Each row factor has Fourier block +$B_m(\alpha_i)^{r/2}$ on $\alpha_i$ and zero blocks elsewhere. +Plancherel on $S_m$ gives +\[ + m!\sum_{p_i}|\xi_i(p_i)|^2 + =D_{\alpha_i}\Tr B_m(\alpha_i)^r=F(\alpha_i). +\] +If one of these quantities is zero, the tuple contributes nothing. +Otherwise the squared coefficients define probability laws +$u_i(p_i)=|\xi_i(p_i)|^2/\sum|\xi_i|^2$ on the four row groups. +Write $u=\bigotimes_i u_i$ for their independent product. + +Because $r/2$ is an integer, $Y=(M^*M)^{r/2}$ is convolution by a +probability law $\omega$ on the column group. Expanding the regular trace gives +\[ + N!\sum_{p,q\in H}\sum_{b,c}\xi(p)\xi(q)\omega(b)\omega(c) + \one_{\{p b q c=e\}}. +\] +Bound each coefficient product by +$|\xi(p)\xi(q)|\le(|\xi(p)|^2+|\xi(q)|^2)/2$. +The two square terms have equal sums: cyclically interchange $(p,b)$ +and $(q,c)$ in the identity $pbqc=e$. For fixed $p,b,c$, there is +at most one possible $q$, and it lies in $H$ exactly when +$b^{-1}p^{-1}c^{-1}\in H$. +Self-adjointness gives $\xi(p^{-1})=\overline{\xi(p)}$, so $u$ is +invariant under inversion; the real probability coefficients of the +self-adjoint $Y$ have the same symmetry. We may thus invert each of +the three independent variables. Using the preceding Plancherel +normalizations yields +\begin{equation}\label{eq:coefficient-trace} + \Tr_{\rm reg}(X_{\boldsymbol\alpha}Y + X_{\boldsymbol\alpha}Y) + \le\prod_i F(\alpha_i)\, + \frac{N!}{(m!)^4}\Pp\{bpc\in H\}, +\end{equation} +where $p\sim u$, and $b,c$ are independent column permutations of law +$\omega$. The remaining problem is to bound this overlap probability +uniformly in the squared-coefficient laws $u_i$. + +\paragraph{Two independent colorings.} +Let $a(z)$ be the row index of position $z$, and define +\[ + \eta(z)=a(c^{-1}z),\qquad \zeta(z)=a(bz). +\] +The condition $bpc\in H$ is equivalent to +$\zeta(pz)=\eta(z)$ for every position $z$: substitute $z=cw$ in +$a(bpcw)=a(w)$. In each column, either coloring uses the four colors +once. Different columns are independent, and the two colorings are +independent because $b$ and $c$ are. + +For row $i$, define a transition matrix on all $4^m$ color strings by +\[ + U_i(x,y)=\Pp_{p_i\sim u_i}\{y(p_i z)=x(z)\text{ for every position }z + \text{ in row }i\}. +\] +Thus $U_i$ averages the permutation action of the row group on strings. +Conditional on the two colorings, the four row permutations are +independent. With $\eta_i,\zeta_i$ denoting their restrictions to row $i$, +\[ + \Pp\{bpc\in H\}=\E\prod_i U_i(\eta_i,\zeta_i). +\] +The matrices $U_i$ are stochastic and symmetric, since each $u_i$ is +invariant under inversion. The particular origin of $u_i$ gives the +additional positivity needed in \eqref{eq:interpolation}. Fourier +inversion writes +$\xi_i(g)=(D_{\alpha_i}/m!)\Tr(Q\rho_{\alpha_i}(g^{-1}))$ with +$Q=B_m(\alpha_i)^{r/2}\ge0$. Hence $\xi_i$, and then +$|\xi_i|^2$, are positive-definite functions. Their Fourier transforms +are positive semidefinite, so the average of $u_i$ in the color-string +representation is positive semidefinite as well. + +The four-site law of $M$ sends every site exactly uniformly to the four sites. Its adjoint does too, and the same remains true of every positive power of $M^*M$. Moreover the support of $M^*M$ contains the identity and the edge transpositions of the square, which generate $S_4$. Its convolution powers therefore converge to uniform. Choose the even integer $r$ sufficiently large that the law of each column of $Y$ satisfies Theorem~\ref{thm:permanent}. Every larger even $r$ does too, once a tail threshold for convergence has been chosen. + +\paragraph{Closing the recursion.} +The variables defining $(\eta_i,\zeta_i)$ consist of $2m$ independent +four-color column assignments. Apply Lemma~\ref{lem:tensorization} to +these assignments with the four nonnegative functions $U_i$, and then +\eqref{eq:interpolation} with $s=4^m$. This gives +\[ + \Pp\{bpc\in H\} + \le\prod_i\left(s^{-2}\sum_{x,y}U_i(x,y)^{p_0}\right)^{1/p_0} + \le s^{-4}\prod_i(\Tr U_i)^\theta, + \quad\theta=1-1/p_0<\frac12. +\] +Since $N!/(m!)^4\le4^N=s^4$, the exponential factors cancel exactly. The only remaining inflation is a power of the four color-string traces. Write $c^\alpha_{\boldsymbol\beta}$ for the multiplicity of $\bigotimes_{i=1}^4V_{\beta_i}$ in the restriction of $V_\alpha$ to $\prod_{i=1}^4S_{|\beta_i|}$. Section~\ref{sec:multiplicities} proves +\begin{equation}\label{eq:trace-C} + \Tr U_i\le C(\alpha_i),\qquad + C(\alpha)=\sum_{\boldsymbol\beta}(c^{\alpha}_{\boldsymbol\beta})^2. +\end{equation} +The sum includes every weak four-part composition of $m$ and every +irreducible tuple for its Young subgroup. Combining this trace bound +with \eqref{eq:coefficient-trace} and \eqref{eq:triangle} gives +\begin{equation}\label{eq:recursion} + F(\lambda)^{1/2}\le + \sum_{\boldsymbol\alpha\in\mathcal I(\lambda)} + \prod_{i=1}^4\left(F(\alpha_i)C(\alpha_i)^\theta\right)^{1/2}. +\end{equation} +All restriction multiplicities have been retained; they are accounted for by the regular trace and by $C(\alpha)$, rather than discarded during a choice of one copy. diff --git a/preprints/A-strict-four-row-permanent-inequality-and-permutation-moments-September-26-2026/build/sections/04-multiplicities.tex b/preprints/A-strict-four-row-permanent-inequality-and-permutation-moments-September-26-2026/build/sections/04-multiplicities.tex new file mode 100644 index 0000000000000000000000000000000000000000..b1f0ee2a9250a098cc819f7f02479d240286e9c6 --- /dev/null +++ b/preprints/A-strict-four-row-permanent-inequality-and-permutation-moments-September-26-2026/build/sections/04-multiplicities.tex @@ -0,0 +1,151 @@ +\section{Squared multiplicities and hook deficits}\label{sec:multiplicities} +We write $\chi_\alpha$ for the character of $V_\alpha$ and +$\langle f,g\rangle_{S_m}=(m!)^{-1}\sum_{x\in S_m}f(x)\overline{g(x)}$ for the normalized character inner product. The cost $C(\alpha)$ in \eqref{eq:recursion} measures how many copies can occur when colors divide a row into four parts. We first justify this interpretation, then bound the cost by boxes outside a thin hook. + +\begin{lemma}[Color-string trace]\label{lem:color-trace} +Let $\alpha\vdash m$, let $Q\ge0$ be a nonzero operator on $V_\alpha$, and define the probability +\[ + u(g)=\frac{D_\alpha}{m!\Tr Q^2} + |\Tr(Q\rho_\alpha(g))|^2\quad(g\in S_m). +\] +Let $U$ be its average action on strings of length $m$ over four colors. Then +\[ + \Tr U\le\langle\chi_\alpha^2,4^{\#\mathrm{cycles}}\rangle_{S_m} + =\sum_{\boldsymbol\beta}(c^{\alpha}_{\boldsymbol\beta})^2=C(\alpha). +\] +\end{lemma} +\begin{proof} +Schur orthogonality normalizes $u$ to have mass one. Write +$V_\alpha\otimes\overline{V_\alpha}=\bigoplus_\nu V_\nu\otimes\mathbb C^{a_\nu}$, and let $P_\nu$ be its isotypic projection. Character orthogonality gives +\[ + \sum_g u(g)\chi_\nu(g) + =\frac{D_\alpha\Tr((Q\otimes\bar Q)P_\nu)} + {D_\nu\Tr Q^2}. +\] +The two commuting positive operators $Q\otimes I$ and $I\otimes\bar Q$ satisfy +\[ + Q\otimes\bar Q\le\tfrac12(Q^2\otimes I+I\otimes\bar Q^2). +\] +The partial trace of $P_\nu$ over either factor commutes with the irreducible action on the other. Its trace is $a_\nu D_\nu$, so it equals +$(a_\nu D_\nu/D_\alpha)I$. The preceding character average is therefore at most $a_\nu$. + +A permutation fixes exactly $4^{\#\mathrm{cycles}(g)}$ color strings. This is a permutation character, so its expansion in irreducible characters has nonnegative integer coefficients. Apply the preceding inequality to that expansion. It yields the first asserted bound, since $a_\nu$ is the multiplicity of $\nu$ in $\chi_\alpha^2$. + +Finally the color-string representation is the direct sum, over all color-class sizes $m_1+\cdots+m_4=m$, of +$\Ind_{S_{m_1}\times\cdots\times S_{m_4}}^{S_m}\one$. +Frobenius reciprocity identifies its inner product with $\chi_\alpha^2$ as the sum of the dimensions of the commuting algebras of the corresponding restrictions of $V_\alpha$. Those dimensions are the sums of squared multiplicities displayed in the statement. +\end{proof} + +For a partition $\alpha\vdash m$, put +\[ + k(\alpha)=m-\max(\alpha_1,\alpha'_1),\qquad + t_h(\alpha)=|\{(i,j)\in\alpha:i>h,\ j>h\}|\quad(h\ge1). +\] +The latter is the number of boxes outside the union of the first $h$ rows and first $h$ columns. + +\begin{lemma}[The cost of restriction]\label{lem:restriction-cost} +There is an absolute $C$ such that, for every $\alpha\vdash m$ and integer $h\ge1$, +\begin{equation}\label{eq:restriction-cost} + \log C(\alpha)\le t_h(\alpha)\log4 + +C(1+h)(1+\sqrt{k(\alpha)})\log(2m). +\end{equation} +\end{lemma} +\begin{proof} +The central estimate for a residual shape $\delta\vdash t$ is +\begin{equation}\label{eq:squared-lr} + (c^{\delta}_{\gamma_1,\ldots,\gamma_4})^2 + \le\binom{t}{|\gamma_1|,\ldots,|\gamma_4|}\le4^t. +\end{equation} +To prove it, let the multiplicity on the left before squaring be $c$. Restriction and induction dimensions give, respectively, +\[ + c\prod_iD_{\gamma_i}\le D_\delta, + \qquad + cD_\delta\le + \binom{t}{|\gamma_1|,\ldots,|\gamma_4|}\prod_iD_{\gamma_i}. +\] +Multiplying and cancelling the positive dimensions proves \eqref{eq:squared-lr}. It is essential here to combine one restriction estimate and one induction estimate. + +Both $C(\alpha)$ and $t_h(\alpha)$ are invariant under conjugation, so orient $\alpha$ with its first row of length $m-k$, where $k=k(\alpha)$. Remove its first $h$ rows and then the first $h$ columns of the remainder. The residual straight diagram $\delta$ has size $t=t_h(\alpha)$. Deleting a first row leaves an interlacing partition, so reversing that deletion adds a horizontal strip. Deleting a first column gives the vertical analogue. + +More explicitly, let $\ell\le2h$ be the number of nonempty removed +lines, let their lengths be $a_1,\ldots,a_\ell$, and let $\xi_j$ be +the trivial character of $S_{a_j}$ for a removed row and the sign +character for a removed column. Then +\[ + W=\Ind_{S_t\times S_{a_1}\times\cdots\times S_{a_\ell}}^{S_m} + (V_\delta\otimes\xi_1\otimes\cdots\otimes\xi_\ell) +\] +contains $V_\alpha$ by the Pieri rules. Thus each multiplicity in a restriction of +$V_\alpha$ is bounded by the corresponding multiplicity in $W$. +We will use this comparison only for tuples $\boldsymbol\beta$ +that occur in the original restriction of $V_\alpha$. For those +tuples, the first-row inequality in the Littlewood--Richardson rule says +$\alpha_1\le\sum_i\beta_{i,1}$, and hence +\begin{equation}\label{eq:tail-budget} + \sum_i(|\beta_i|-\beta_{i,1})\le k. +\end{equation} +Every intermediate diagram in a Pieri chain ending at such a $\beta_i$ is contained in it, and has at most $k$ boxes below its first row. The number of partitions of any size at most $m$ satisfying that condition is at most +\[ + Q_m(k)=(m+1)^{1+2\lceil\sqrt{k}\rceil}. +\] +One way to see this is to choose the first-row length, and encode the tail by its row lengths and column lengths along a Durfee square of side at most $\sqrt k$. Padding the two lists to length $\lceil\sqrt k\rceil$ gives the displayed upper bound. For $k=0$, only the first-row length is needed. + +Fix such a final tuple $\boldsymbol\beta$, and put +$H_{\boldsymbol\beta}=\prod_{i=1}^4S_{|\beta_i|}$. +Mackey's formula for $\Res_{H_{\boldsymbol\beta}}W$ indexes the decomposition by +allocations of the residual factor and the $\ell$ strip factors +among these classes. There are at most $(m+1)^{4(\ell+1)}$ +allocation matrices. For each allocation, restrict $V_\delta$ to +the four allocated subsets, then add the allocated horizontal or +vertical strips within each color class by Pieri. A chain ending +at $\beta_i$ consists of diagrams contained in $\beta_i$, so every +one obeys the small-tail bound just counted. Choosing the residual +tuple and all subsequent diagrams costs at most +$Q_m(k)^{4(\ell+1)}$. Each Pieri transition is multiplicity free, +and \eqref{eq:squared-lr} bounds the residual multiplicity by +$4^{t/2}$. Therefore +\[ + c^\alpha_{\boldsymbol\beta} + \le \operatorname{mult}_{\boldsymbol\beta}(\Res_{H_{\boldsymbol\beta}}W) + \le (m+1)^{4(\ell+1)}Q_m(k)^{4(\ell+1)}4^{t/2}. +\] +There are at most $Q_m(k)^4$ possible final occurring tuples. +Squaring this bound and summing over them gives $4^t$ times +$\exp\{C(1+h)(1+\sqrt k)\log(2m)\}$, proving +\eqref{eq:restriction-cost}. No tail estimate for the other +constituents of $W$ is required. +\end{proof} + +The next fact lets us add hook costs along an entire level of the recursion. + +\begin{lemma}[Hook deficits under restriction]\label{lem:hook-deficit} +If $c^{\lambda}_{\alpha_1,\ldots,\alpha_4}>0$, then for every integer $h\ge1$, +\begin{equation}\label{eq:hook-deficit} + \sum_i t_h(\alpha_i)\le t_h(\lambda), + \qquad \sum_i k(\alpha_i)\le k(\lambda). +\end{equation} +\end{lemma} +\begin{proof} +For the assertion about $k$, orient the parent with its longest side as first row. The first-row inequality gives a bound by its row deficit on the sum of the child row deficits; each child's $k$ is no larger than its row deficit. If the parent's longest side is a column, transpose every partition, which preserves Littlewood--Richardson multiplicities. + +Write $s_\nu$ for the ordinary Schur function, and $s_\nu(X\mid Y)$ for its hook-Schur specialization \cite[Definition~1(a)]{orellana-zabrocki2000}. For the assertion about $t_h$, take $h$ ordinary variables $X$, $h$ transpose variables $Y$, and an unlimited ordinary alphabet $Z$. Variables in $X,Y$ have degree zero and those in $Z$ have degree one. The positive hook-Schur tableau rule uses ordinary entries weakly along rows and strictly down columns, and transpose entries strictly along rows and weakly down columns \cite[Section~2]{mason-niese2016}. For $s_\nu(X\mid Y)$, order all $X$ letters before all $Y$ letters, so the ordinary entries occupy a subdiagram. The rule gives +$s_\nu(X\mid Y)\ne0$ exactly when $\nu_{h+1}\le h$. +For completeness, a tableau over $h$ ordinary and $h$ transpose letters cannot fill an $(h+1)$-by-$(h+1)$ square: the ordinary subdiagram has at most $h$ rows, and the transpose-filled remainder has at most $h$ columns in each remaining row. Conversely, fill the first $h$ rows with their ordinary row letters and the remaining boxes with their transpose column letters. This realizes every diagram in that hook. + +Writing $p_j$ for the $j$th power sum, the supersymmetric substitution is +\[ + p_j\longmapsto p_j(X)+(-1)^{j-1}p_j(Y). +\] +Applied to the ordinary Schur coproduct, it gives +\[ + s_\lambda(X+Z\mid Y) + =\sum_{\nu\subseteq\lambda}s_\nu(X\mid Y)s_{\lambda/\nu}(Z). +\] +Its coefficients are nonnegative. The largest subdiagram of $\lambda$ in the $(h,h)$ hook is the union of its first $h$ rows and first $h$ columns. The smallest degree in $Z$ is therefore exactly $t_h(\lambda)$; the corresponding skew Schur function is nonzero on an unlimited alphabet. + +Apply this substitution to +$\prod_i s_{\alpha_i}=\sum_\lambda c^{\lambda}_{\boldsymbol\alpha}s_\lambda$. +The product on the left has minimum $Z$-degree $\sum_i t_h(\alpha_i)$. Every summand on the right has nonnegative coefficients, so an occurring parent's minimum degree cannot be smaller. This is \eqref{eq:hook-deficit}. +\end{proof} + +Equations~\eqref{eq:recursion} and \eqref{eq:restriction-cost} now give a recursive estimate in which the leading charge is $\theta\log4$ times a hook deficit. The other charges involve only $(1+h)(1+\sqrt k)\log(2m)$. We next prove where this recursion can stop, before summing those charges. diff --git a/preprints/A-strict-four-row-permanent-inequality-and-permutation-moments-September-26-2026/build/sections/05-sparse.tex b/preprints/A-strict-four-row-permanent-inequality-and-permutation-moments-September-26-2026/build/sections/05-sparse.tex new file mode 100644 index 0000000000000000000000000000000000000000..a5c7f340c17d2ebdf04cdfefb94de86ad6bb5516 --- /dev/null +++ b/preprints/A-strict-four-row-permanent-inequality-and-permutation-moments-September-26-2026/build/sections/05-sparse.tex @@ -0,0 +1,243 @@ +\section{Sparse stopping by encounter forests}\label{sec:sparse} +The four-row recursion will stop when a diagram is close to one row or column. We prove the needed bound directly from the paths of a small collection of cards. Branching expresses the trace for $s$ tracked cards as a binomial sum over first-row deficits. Inverting that sum cancels contributions supported on fewer than all the tracked cards. To preserve this cancellation, we couple all subsets of the tracked cards in one sweep. + +Throughout this section, $n=2^d$, $T=T_n$, $B=TT^*$, and $B_\lambda=B_n(\lambda)$. Empty diagrams have dimension one. + +\begin{lemma}[Sparse stopping] +\label{lem:sparse} +There is an absolute $N_0$ such that, if $n=2^d\ge N_0$ and +$1\le k\le n^{1/100}$, then +\[ + \sum_{\mu\vdash k}D_\mu\, + \Tr B_{(n-k,\mu)}\le n^{-3k/10}. +\] +Consequently, for $n\ge N_0$, if +$k(\lambda)=n-\max\{\lambda_1,\lambda'_1\}\le n^{1/100}$, +then for every integer $r\ge4$, +\[ + D_\lambda\Tr B_\lambda^r\le1. +\] +There is a fixed even $r_0\ge4$ for which the latter assertion holds +for every power of two $n\ge2$ and every such $\lambda$. +\end{lemma} + +\begin{proof} +\emph{Isolating the first-row deficit.} +Let $\mathcal I_s$ be the set of ordered injections from $[s]$ into +the $n$ positions, with $\mathcal I_0$ a singleton. The sweep induces +a transition matrix $T_s$ on this set. Write +\[ + a_s=\Tr_{\mathbb C^{\mathcal I_s}}(TT^*) + =\sum_{x,y\in\mathcal I_s}T_s(x,y)^2. +\] +This is the ordinary, unnormalized trace on the indicated permutation module. + +The permutation module on $\mathcal I_s$ is +$\operatorname{Ind}_{S_{n-s}}^{S_n}\mathbf1$. By repeated branching, +the multiplicity of $V_\lambda$ in it is the number of standard +tableaux of skew shape $\lambda/(n-s)$. Assume $s\le k$ and +$n\ge2k$. Every occurring partition has the form +$\lambda=(n-j,\mu)$ with $0\le j\le s$ and $\mu\vdash j$. +The skew shape consists of the tail $\mu$ and a row of $s-j$ cells +whose columns exceed $n-s\ge k\ge\mu_1$. These components share +neither a row nor a column. Choosing the $j$ entries of the tail and +then a standard tableau on it gives multiplicity +$\binom{s}{j}D_\mu$. Hence +\[ + a_s=\sum_{j=0}^s\binom{s}{j}b_j, + \qquad + b_j=\sum_{\mu\vdash j}D_\mu\, + \Tr B_{(n-j,\mu)}. +\] +Binomial inversion gives +\begin{equation} +\label{eq:sparse-inverse} + b_k=\sum_{s=0}^k(-1)^{k-s}\binom{k}{s}a_s\ge0, +\end{equation} +where positivity follows from the positive semidefiniteness of each +$B_\lambda$. It remains to bound this inverse without losing its +cancellation. + +For fixed initial and final positions of one card, there is exactly +one possible path through the successive coordinate layers: at layer +$i$, its first $i$ processed coordinates have their final values and +its remaining coordinates still have their initial values. Thus fixed +endpoints for $s$ cards prescribe all their paths. If two tracked cards +use the same switch, the two prescribed coin requirements either +disagree, making the endpoints impossible, or agree, saving one coin. +If $e$ is the number of switches shared by two tracked cards along a +realized sweep, the transition probability to its realized endpoint is +therefore $2^{-ds+e}=n^{-s}2^e$. Averaging first over the realized +endpoint and then over a uniform initial injection gives the exact +identity +\begin{equation} +\label{eq:sparse-trace} + a_s=\frac{(n)_s}{n^s}\,\mathbb E\,2^e, + \qquad (n)_s=n(n-1)\cdots(n-s+1). +\end{equation} + +Choose $k$ distinct uniform initial positions $X_1,\ldots,X_k$, +independently of the coins of one complete sweep. For this realized +sweep, form a graph $G$ on all $n$ initial positions, joining two +positions if their cards share a switch. Each vertex has degree $d$. +The graph is simple: after two cards meet at coordinate $i$, their +positions differ at that coordinate forever, so they cannot meet at a +later coordinate. Let $G_X$ be the graph induced on the selected tags, +and let $e(S)$ count its edges with both endpoints in $S\subseteq[k]$. +The marginal initial positions of every such subset are a uniform +injection. Thus, with +\[ + w(S)=\frac{(n)_{|S|}}{n^{|S|}} + =\prod_{i\in S}\left(1- + \frac{\#\{j\in S:jn/2$, every matching projection vanishes +on $V_\lambda$. Indeed, its fixed-space dimension is the +multiplicity of $V_\lambda$ in +$\operatorname{Ind}_{S_2^{n/2}}^{S_n}\mathbf1$. Repeated horizontal +Pieri shows that every constituent is obtained by adding $n/2$ +horizontal strips of size two. Each strip adds at most one cell to +the first column, so every constituent has first-column height at +most $n/2$. Thus $B_\lambda=0$ in the case under consideration. +For $k=0$, the trivial representation has the claimed value one +and the sign representation is annihilated by a matching projection. + +Finally, Lemma~\ref{lem:gap} gives operator norm strictly less than +one on every nontrivial irreducible at each fixed $n$. +There are only finitely many powers of two below $N_0$ and finitely +many irreducibles at each of them. Increasing $r$ to one fixed even +$r_0\ge4$ therefore gives the stated stopping estimate also at these +remaining sizes. Larger powers preserve the estimate because $B$ +is a positive semidefinite contraction. +\end{proof} diff --git a/preprints/A-strict-four-row-permanent-inequality-and-permutation-moments-September-26-2026/build/sections/06-tree.tex b/preprints/A-strict-four-row-permanent-inequality-and-permutation-moments-September-26-2026/build/sections/06-tree.tex new file mode 100644 index 0000000000000000000000000000000000000000..05282444b1afa665f0878ee2d170d3cfec33fc65 --- /dev/null +++ b/preprints/A-strict-four-row-permanent-inequality-and-permutation-moments-September-26-2026/build/sections/06-tree.tex @@ -0,0 +1,141 @@ +\section{Summing the costs and comparing dimensions}\label{sec:tree} +We now assemble the recursion. The purpose of the sparse estimate is to prevent costs from accumulating indefinitely on diagrams with a long row or column. The remaining costs can be summed before we compare them with the dimension of the root. + +Write $\log^+x=\max\{0,\log x\}$. Fix a root $\lambda\vdash N$, with $K=k(\lambda)>0$. A node with diagram $\alpha\vdash m$ is expanded only if $m$ exceeds a fixed cutoff $M$ and $k(\alpha)>m^{.01}$. Otherwise it is stopped. Choose the even moment $r$ large enough for Lemma~\ref{lem:sparse} and, by Lemma~\ref{lem:gap}, large enough that $F(\alpha)\le1$ for every stopped diagram of size at most $M$. These choices are possible simultaneously. The numerical cutoff will be fixed below before $r$ is finally chosen. + +Squaring \eqref{eq:recursion} and bounding a sum by its number of terms times its largest term gives +\begin{equation}\label{eq:log-recursion} + \log F(\alpha)\le2\log|\mathcal I(\alpha)| + +\max_{\boldsymbol\beta\in\mathcal I(\alpha)} + \sum_{i=1}^4\{\log F(\beta_i)+\theta\log C(\beta_i)\}. +\end{equation} +We use $\log0=-\infty$. If $F(\alpha)=0$, no bound is needed. Iterating this inequality amounts to taking the maximum over finite four-ary trees of occurring restrictions. Each expanded node contributes its choice cost $2\log|\mathcal I(\alpha)|$, and each of its four immediate children contributes $\theta\log C(\beta_i)$. The latter charge remains present even if the child is stopped. + +The tail count in the proof of Lemma~\ref{lem:restriction-cost} gives +\[ + \log|\mathcal I(\alpha)| + \le C(1+\sqrt{k(\alpha)})\log(2m). +\] +To apply it, orient the parent along its longest side, use the first-row inequality, and count each child's tail with at most the parent's deficit. Child sizes are prescribed. Thus this choice cost fits within the lower-order term of \eqref{eq:restriction-cost}. We may charge it at the parent, including the root, without altering the estimates below. + +For $H>2$, put +\[ + R_H(\lambda)=\sum_{(i,j)\in\lambda}\log^+\frac{\min(i,j)}H. +\] +This quantity will count, box by box, the levels at which a root box +can lie outside the chosen hook. The budget below adds an entropy +term $\tfrac12K\log(N/K)$. Lemma~\ref{lem:hook-comparison} will +compare $R_H(\lambda)+K\log(N/K)$ with $\log D_\lambda$, with +explicit errors. We therefore seek a coefficient below one in front +of the budget, leaving room to absorb its lower-order costs. + +\begin{lemma}[Uniform tree budget]\label{lem:tree-budget} +Fix $\delta\in(0,1/2)$ and put $a=1/2-\delta$. For every $H>2$ and $\epsilon>0$, a sufficiently large fixed cutoff $M$ makes every such recursion tree satisfy +\begin{equation}\label{eq:tree-budget} + \log F(\lambda)\le\epsilon K+ + \frac\theta a\left\{ + R_H(\lambda)+\frac K2\log\frac NK\right\}. +\end{equation} +The cutoff and all constants in this statement are independent of the final even moment $r$. +\end{lemma} +\begin{proof} +At a level with node size $m$, let $q_m$ be the number of charged nodes and let their deficits be $k_1,\ldots,k_{q_m}$. By Lemma~\ref{lem:hook-deficit}, their sum is at most $K$. Expanded nodes satisfy $k_i>m^{.01}$, so their number is at most $K/m^{.01}$. A charged child of size $m$ has an expanded parent of size $4m$, and there are four children per parent. Consequently, after changing one absolute constant, +\begin{equation}\label{eq:charged-count} + q_m\le C K m^{-.01},\qquad q_m\le N/m, + \qquad\sum_i k_i\le K. +\end{equation} +The root also satisfies these bounds when expanded. A stopped root already has $F\le1$ and satisfies the conclusion directly. + +At that level choose +\[ + h_m=\left\lceil m^a\sqrt{K/N}\right\rceil. +\] +The lower-order terms in the choice and restriction costs are bounded by +\[ + C\sum_i(1+h_m)(1+\sqrt{k_i})\log(2m). +\] +Cauchy--Schwarz gives $\sum_i\sqrt{k_i}\le\sqrt{q_mK}$. The terms arising from the ceiling and the constant one are therefore at most +$CKm^{-.005}\log(2m)$. For the terms involving $m^a$, use both bounds in \eqref{eq:charged-count}: +\[ + m^a\sqrt{K/N}\sqrt{q_mK}\le Km^{-\delta}, +\] +\[ + m^a\sqrt{K/N}\,q_m + \le CKm^{-\delta-.005}. +\] +For the second inequality, $q_m\le\sqrt{(N/m)(CK/m^{.01})}$. These costs are summable geometric tails because level sizes differ by a factor four. All charged nodes have size greater than $M/4$, including the stopped children of the last expanded level. Choosing $M$ large makes their total at most $\epsilon K$. + +It remains to sum the leading hook terms. At every level, iteration of Lemma~\ref{lem:hook-deficit} bounds the sum of $t_{h_m}$ over charged nodes by $t_{h_m}(\lambda)$. Follow one root box $(i,j)$ and write $s=\min(i,j)$. It can contribute only if +\[ + s>h_m\ge m^a\sqrt{K/N}. +\] +The eligible sizes $m$ form part of a geometric progression of ratio four, all larger than $M/4$. By increasing $M$ so that $(M/4)^a\ge4^a H$, their number is at most +\[ + \frac{\log^+(s/H)+\tfrac12\log(N/K)}{a\log4}. +\] +Indeed the number is bounded by the positive part of one plus the logarithm of the ratio of the upper endpoint +$(s\sqrt{N/K})^{1/a}$ to the lower endpoint $M/4$, divided by $\log4$; the choice of $M$ absorbs the extra one. Replacing $\log(s/H)$ by its positive part can only enlarge the bound. + +Boxes in the longest first row or column never contribute, since $h_m\ge1$. There are exactly $K$ remaining boxes. Multiply their level counts by $\theta\log4$ and add the lower-order costs. This proves \eqref{eq:tree-budget}. +\end{proof} + +We have reduced the operator recursion to a purely diagrammatic comparison. The next lemma compares $R_H(\lambda)$ with the hook-length formula, retaining the first-row entropy needed when $K$ is small relative to $N$. + +\begin{lemma}[Hook comparison]\label{lem:hook-comparison} +Orient $\lambda\vdash N$ so its first row is a longest side, put $K=N-\lambda_1>0$, and let $\zeta=(\lambda_2,\lambda_3,\ldots)\vdash K$. For $H>2$, +\begin{align} + \log D_\lambda + &\ge K\log(N/K)+\log D_\zeta-E_{N,K},\label{eq:row-hook}\\ + R_H(\lambda)&\le\log D_\zeta+4K/H,\label{eq:rank-hook} +\end{align} +where $E_{N,K}=O(K/N)$ for $K0$ such that +$\log D_\lambda\ge cK$ for all sufficiently large $N$ and every nontrivial, nonsign $\lambda$. +\end{lemma} +\begin{proof} +The hook-length formula separates the first row exactly: +\[ + D_\lambda=\binom NKD_\zeta + \prod_{j=1}^{\lambda_1} + \left(1+\frac{\lambda'_j-1}{\lambda_1-j+1}\right)^{-1}. +\] +If $K0$, +\[ + \log D_\zeta\ge R_H-t\log\frac{4eK}{tH} + \ge R_H-\frac{4K}H. +\] +The last inequality follows by maximizing $t\log(4eK/(tH))$ over $t>0$. For $t=0$, the assertion is immediate. + +Finally we prove $\log D_\lambda\ge cK$ without losing uniformity in the shape. If $K0$ so small that +\[ + b:=\frac\theta{1/2-\delta}<1. +\] +The universal degree constant $c$ in Lemma~\ref{lem:hook-comparison} is independent of this choice. Choose $H$ sufficiently large and $\epsilon$ sufficiently small that +$\epsilon+4b/H<(1-b)c/4$. Then choose the cutoff $M$ required by Lemma~\ref{lem:tree-budget}, increasing it further so that the errors $E_{N,K}$ in Lemma~\ref{lem:hook-comparison} are less than $(1-b)cK/(4b)$ whenever $N>M$. This is possible uniformly: the first error divided by $K$ is $O(1/N)$, and the second is $O(\log N/\sqrt N)$ in its stated range. + +Only now choose the even moment $r$. It must make the four-site column law admissible for Theorem~\ref{thm:permanent}, enforce the sparse stopping rule, and make $F_r\le1$ for all nontrivial shapes of all dyadic sizes at most $M$. These are finitely many or fixed-threshold requirements, by Lemmas~\ref{lem:gap} and \ref{lem:sparse}; all persist on increasing $r$. + +For an expanded root, \eqref{eq:tree-budget}, \eqref{eq:row-hook} and \eqref{eq:rank-hook} give +\[ + \log F_r(\lambda) + \le b\log D_\lambda+\epsilon K+4bK/H+bE_{N,K} + \le\frac{1+b}{2}\log D_\lambda. +\] +Here we discarded the additional negative term +$-bK\log(N/K)/2$. Thus $\eta=(1-b)/2>0$ works at every expanded root. At a stopped root $F_r\le1\le D_\lambda^{1-\eta}$. The row representation has $F_r=1$, while the column representation is killed by Lemma~\ref{lem:gap}. This proves the moment assertion for every dyadic $N$. +\end{proof} diff --git a/preprints/A-strict-four-row-permanent-inequality-and-permutation-moments-September-26-2026/build/sections/07-amplification.tex b/preprints/A-strict-four-row-permanent-inequality-and-permutation-moments-September-26-2026/build/sections/07-amplification.tex new file mode 100644 index 0000000000000000000000000000000000000000..459934d87641323a6073ce2c5c28596245f9a917 --- /dev/null +++ b/preprints/A-strict-four-row-permanent-inequality-and-permutation-moments-September-26-2026/build/sections/07-amplification.tex @@ -0,0 +1,43 @@ +\section{Independent forward sweeps}\label{sec:amplification} +The estimate just proved concerns the positive operator $B_N=T_NT_N^*$. Actual repeated shuffles use powers of $T_N$. We make that distinction explicit in the final argument. + +First, the degree estimates above imply, for any fixed $A>0$, +\begin{equation}\label{eq:degree-sum} + \sum_{\lambda\ne(N),(1^N)}D_\lambda^{-A}\longrightarrow0. +\end{equation} +Here are the elementary summability details. There are at most $2p(k)$ partitions with $k(\lambda)=k$, because deleting the longer first row or column leaves a partition of $k$. The partition generating function gives $p(k)\le e^{3\sqrt k}$: evaluate it at $x=e^{-1/\sqrt k}$, and bound its logarithm by +\[ + \sum_{j\ge1}\frac1{j(e^{j/\sqrt k}-1)} + \le\sqrt k\sum_{j\ge1}j^{-2}\le2\sqrt k. +\] +Lemma~\ref{lem:hook-comparison} bounds $D_\lambda^{-A}$ by $e^{-Ac k}$ for all sufficiently large $N$. The resulting bound $2e^{3\sqrt k-Ack}$ is summable in $k$. For each fixed $k\ge1$, the first-row hook identity gives $D_\lambda\to\infty$ as $N\to\infty$, uniformly over those finitely many tails. Dominated convergence proves \eqref{eq:degree-sum}. This elementary argument suffices here; more precise fixed-exponent degree sums are known \cite{liebeck-shalev2004}. + +From \eqref{eq:moment}, +\[ + \|B_N(\lambda)\|_{\op}^r\le\Tr B_N(\lambda)^r + \le D_\lambda^{-\eta}. +\] +Thus, for any integer $q\ge r$, Schatten H\"older and the operator norm bound give +\[ + \|T_N(\lambda)^q\|_{\HS}^2 + \le\|T_N(\lambda)\|_{\op}^{2(q-r)} + \|T_N(\lambda)\|_{2r}^{2r} + =\|B_N(\lambda)\|_{\op}^{q-r}\Tr B_N(\lambda)^r. +\] +Multiplying by $D_\lambda$ and using the moment theorem yields +\begin{equation}\label{eq:forward-power} + D_\lambda\|T_N(\lambda)^q\|_{\HS}^2 + \le D_\lambda^{1-\eta q/r}. +\end{equation} +Choose once and for all an integer $q>r/\eta$. Equation~\eqref{eq:degree-sum} shows that the sum of \eqref{eq:forward-power} over all nonsign, nontrivial representations tends to zero. The sign block is zero. + +For clarity, finite-group Plancherel and Cauchy--Schwarz state that any probability $\sigma$ on $S_N$ satisfies +\[ + 4\|\sigma-U_{S_N}\|_{\TV}^2 + \le N!\sum_g|\sigma(g)-1/N!|^2 + =\sum_{\lambda\ne(N)}D_\lambda + \|\widehat\sigma(\lambda)\|_{\HS}^2. +\] +Apply this to $\sigma=\mu_N^{*q}$, whose Fourier matrix is $T_N(\lambda)^q$. This proves convergence to uniform after $q$ forward sweeps. Translating the law by any deterministic starting deck preserves total variation, so the result is uniform over starting states. A sweep is $d$ physical shuffles, giving the asserted $O(d)$ bound. The support estimate \eqref{eq:support-intro} supplies the matching lower order. + +No step identifies $(T_NT_N^*)^q$ with $T_N^q(T_N^*)^q$. The positive moment supplies singular-value information; H\"older is the separate passage to forward evolution. diff --git a/preprints/A-strict-four-row-permanent-inequality-and-permutation-moments-September-26-2026/main.pdf b/preprints/A-strict-four-row-permanent-inequality-and-permutation-moments-September-26-2026/main.pdf new file mode 100644 index 0000000000000000000000000000000000000000..16c09b6b638a96cbdb10dae277a6679903089f34 --- /dev/null +++ b/preprints/A-strict-four-row-permanent-inequality-and-permutation-moments-September-26-2026/main.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:1fc78f640d9422a5ea8b626e5cbcbb0edf7b24845f0a0057790f10a866dc74e3 +size 410537 diff --git a/preprints/A-strict-inverse-first-power-bound-for-univalent-functions-September-24-2026/README.md b/preprints/A-strict-inverse-first-power-bound-for-univalent-functions-September-24-2026/README.md new file mode 100644 index 0000000000000000000000000000000000000000..929fd82d556125c5c0959f4b7a388acb4d37beeb --- /dev/null +++ b/preprints/A-strict-inverse-first-power-bound-for-univalent-functions-September-24-2026/README.md @@ -0,0 +1,17 @@ +# [A strict inverse-first-power bound for univalent functions](paper.pdf) + +**Author:** OpenAI +**Date:** September 24, 2026 + +## Citation + +```bibtex +@misc{OAI:A-strict-inverse-first-power-bound-for-univalent-functions-September-24-2026, + author = {{OpenAI}}, + title = {{A strict inverse-first-power bound for univalent functions}}, + howpublished = {OpenAI Math Release preprint + \href{https://github.com/openai/math/blob/main/preprints/A-strict-inverse-first-power-bound-for-univalent-functions-September-24-2026/paper.pdf}{OAI:A-strict-inverse-first-power-bound-for-univalent-functions-September-24-2026}}, + year = {2026} +} +``` + diff --git a/preprints/A-strict-inverse-first-power-bound-for-univalent-functions-September-24-2026/build/figures/rank-pairing.tex b/preprints/A-strict-inverse-first-power-bound-for-univalent-functions-September-24-2026/build/figures/rank-pairing.tex new file mode 100644 index 0000000000000000000000000000000000000000..7e38a5c449daad4f21710d6128245b6bf2e6c01b --- /dev/null +++ b/preprints/A-strict-inverse-first-power-bound-for-univalent-functions-September-24-2026/build/figures/rank-pairing.tex @@ -0,0 +1,22 @@ +\begin{figure}[tbp] +\centering +\begin{tikzpicture}[x=1cm,y=.65cm, + point/.style={circle,fill,inner sep=1.5pt}, + >={Stealth[length=5pt]}] + \node at (0,5.6) {Saddles}; + \node at (3.5,5.6) {Finite maxima}; + \foreach \j/\sy/\my in {1/4.1/4.8,2/2.3/3.7,3/.7/2.9}{ + \node[point,label=left:{$s_{\j}$}] (s\j) at (0,\sy) {}; + \node[point,label=right:{$m_{\j}$}] (m\j) at (3.5,\my) {}; + \draw[->,semithick] (s\j) -- (m\j); + } + \node[point,label=right:{$m_4$}] at (3.5,1.6) {}; + \draw[->] (5.6,0) -- (5.6,5.1) node[above] {$V_\xi^F$}; +\end{tikzpicture} +\caption{Schematic rank ordering on one good fiber \(G_F^{-1}(\xi)\). +Both lists are ordered by decreasing critical value, and equal ranks +are paired. Distinct critical values put each paired maximum strictly +above its saddle. Heights represent values rather than spatial +positions; the displayed counts and gaps are illustrative.} +\label{fig:ranks} +\end{figure} diff --git a/preprints/A-strict-inverse-first-power-bound-for-univalent-functions-September-24-2026/build/main.tex b/preprints/A-strict-inverse-first-power-bound-for-univalent-functions-September-24-2026/build/main.tex new file mode 100644 index 0000000000000000000000000000000000000000..3a7100c0fe2186a2a4deefd37af6e1764b788328 --- /dev/null +++ b/preprints/A-strict-inverse-first-power-bound-for-univalent-functions-September-24-2026/build/main.tex @@ -0,0 +1,57 @@ +\documentclass[11pt,a4paper]{article} +\usepackage[margin=30mm]{geometry} +\usepackage[T1]{fontenc} +\usepackage{lmodern} +\usepackage{amsmath,amssymb,amsthm,mathtools} +\usepackage{microtype} +\usepackage{needspace} +\usepackage{tikz} +\usetikzlibrary{arrows.meta,positioning} +\usepackage[hyphens]{url} +\usepackage[hidelinks]{hyperref} +\hypersetup{pdftitle={A strict inverse-first-power bound for univalent functions},pdfauthor={OpenAI}} +\input{glyphtounicode} +\input{glyphtounicode-cmex} +\pdfgentounicode=1 +\pdfglyphtounicode{negationslash}{0338} +\newtheorem{theorem}{Theorem}[section] +\newtheorem{lemma}[theorem]{Lemma} +\newtheorem{proposition}[theorem]{Proposition} +\newtheorem{corollary}[theorem]{Corollary} +\theoremstyle{remark} +\newtheorem{remark}[theorem]{Remark} +\newcommand{\DD}{\mathbb D} +\newcommand{\HH}{\mathbb H} +\newcommand{\C}{\mathbb C} +\newcommand{\R}{\mathbb R} +\newcommand{\E}{\mathbb E} +\newcommand{\KK}{\mathcal K} +\newcommand{\SSS}{\mathcal S} +\newcommand{\ind}{\mathbf 1} +\newcommand{\im}{\operatorname{Im}} +\newcommand{\re}{\operatorname{Re}} +\newcommand{\supp}{\operatorname{supp}} +\newcommand{\rank}{\operatorname{rank}} +\title{A strict inverse-first-power bound for univalent functions} +\author{OpenAI} +\date{September 24, 2026} +\begin{document} +\maketitle +\begin{abstract} +We prove a uniform upper bound for inverse-first-power integral means +of normalized univalent disk maps with exponent strictly below \(1/4\). +Consequently, the bounded universal integral-means spectrum satisfies +\(B_b(-1)<1/4\), disproving Kraetzer's conjectured spectrum at \(p=-1\). +\end{abstract} + +\input{sections/00-introduction} +\input{sections/01-compact-model} +\input{sections/02-weighted-law} +\input{sections/03-escape} +\input{sections/04-strict-pairing} +\input{sections/05-strict-transport} +\input{sections/06-equality-barrier} + +\bibliographystyle{alphaurl} +\bibliography{references/references} +\end{document} diff --git a/preprints/A-strict-inverse-first-power-bound-for-univalent-functions-September-24-2026/build/references/references.bib b/preprints/A-strict-inverse-first-power-bound-for-univalent-functions-September-24-2026/build/references/references.bib new file mode 100644 index 0000000000000000000000000000000000000000..bbbb6b3cd5d9dbbf9991fb393b605dfb327e1433 --- /dev/null +++ b/preprints/A-strict-inverse-first-power-bound-for-univalent-functions-September-24-2026/build/references/references.bib @@ -0,0 +1,170 @@ +@article{Kraetzer, + author = {Kraetzer, Philipp}, + title = {Experimental bounds for the universal integral means spectrum of conformal maps}, + journal = {Complex Variables, Theory and Application}, + volume = {31}, + number = {4}, + pages = {305--309}, + year = {1996}, + doi = {10.1080/17476939608814969}, + url = {https://doi.org/10.1080/17476939608814969} +} + +@phdthesis{KraetzerThesis, + author = {Kraetzer, Philipp}, + title = {Algorithmische Methoden der konformen Abbildungen auf fraktale Gebiete}, + school = {Technische Universit{\"a}t Berlin}, + year = {2000}, + doi = {10.14279/depositonce-231}, + url = {https://doi.org/10.14279/depositonce-231} +} + +@article{CJ, + author = {Carleson, Lennart and Jones, Peter W.}, + title = {On coefficient problems for univalent functions and conformal dimension}, + journal = {Duke Mathematical Journal}, + volume = {66}, + number = {2}, + pages = {169--206}, + year = {1992}, + doi = {10.1215/S0012-7094-92-06605-1}, + url = {https://doi.org/10.1215/S0012-7094-92-06605-1} +} + +@article{Makarov, + author = {Makarov, Nikolai G.}, + title = {Fine structure of harmonic measure}, + journal = {St. Petersburg Mathematical Journal}, + volume = {10}, + number = {2}, + pages = {217--268}, + year = {1999}, + url = {https://www.mathnet.ru/eng/aa980}, + note = {Translation of Algebra i Analiz 10(2) (1998), 1--62} +} + +@article{HedenmalmShimorin, + author = {Hedenmalm, H{\aa}kan and Shimorin, Serguei}, + title = {Weighted {Bergman} spaces and the integral means spectrum of conformal mappings}, + journal = {Duke Mathematical Journal}, + volume = {127}, + number = {2}, + pages = {341--393}, + year = {2005}, + doi = {10.1215/S0012-7094-04-12725-3}, + eprint = {math/0406345}, + archivePrefix = {arXiv}, + url = {https://doi.org/10.1215/S0012-7094-04-12725-3} +} + +@article{BeliaevSmirnov, + author = {Beliaev, Dmitri and Smirnov, Stanislav}, + title = {Random conformal snowflakes}, + journal = {Annals of Mathematics}, + volume = {172}, + number = {1}, + pages = {597--615}, + year = {2010}, + doi = {10.4007/annals.2010.172.597}, + eprint = {math/0701463}, + archivePrefix = {arXiv}, + url = {https://annals.math.princeton.edu/2010/172-1/p12} +} + +@article{Beliaev, + author = {Beliaev, Dmitri}, + title = {Integral means spectrum of random conformal snowflakes}, + journal = {Nonlinearity}, + volume = {21}, + number = {7}, + pages = {1435--1442}, + year = {2008}, + doi = {10.1088/0951-7715/21/7/003}, + eprint = {0710.4175}, + archivePrefix = {arXiv}, + url = {https://doi.org/10.1088/0951-7715/21/7/003} +} + +@article{Sola, + author = {Sola, Alan}, + title = {An Estimate of the Universal Means Spectrum of Conformal Mappings}, + journal = {Computational Methods and Function Theory}, + volume = {6}, + number = {2}, + pages = {423--436}, + year = {2006}, + doi = {10.1007/BF03321620}, + url = {https://link.springer.com/article/10.1007/BF03321620} +} + +@book{Folland, + author = {Folland, Gerald B.}, + title = {Real Analysis: Modern Techniques and Their Applications}, + edition = {Second}, + publisher = {John Wiley \& Sons}, + address = {New York}, + year = {1999}, + isbn = {0-471-31716-0} +} + +@book{Duren, + author = {Duren, Peter L.}, + title = {Univalent Functions}, + series = {Grundlehren der mathematischen Wissenschaften}, + volume = {259}, + publisher = {Springer}, + address = {New York}, + year = {1983} +} + +@article{Sard, + author = {Sard, Arthur}, + title = {The measure of the critical values of differentiable maps}, + journal = {Bulletin of the American Mathematical Society}, + volume = {48}, + pages = {883--890}, + year = {1942}, + doi = {10.1090/S0002-9904-1942-07811-6}, + url = {https://doi.org/10.1090/S0002-9904-1942-07811-6} +} + +@article{Hopf, + author = {Hopf, Heinz}, + title = {{\"U}ber die {Drehung} der {Tangenten} und {Sehnen} ebener {Kurven}}, + journal = {Compositio Mathematica}, + volume = {2}, + pages = {50--62}, + year = {1935}, + url = {https://numdam.org/item/CM_1935__2__50_0/} +} + +@book{Moschovakis, + author = {Moschovakis, Yiannis N.}, + title = {Descriptive Set Theory}, + edition = {Second}, + series = {Mathematical Surveys and Monographs}, + volume = {155}, + publisher = {American Mathematical Society}, + address = {Providence, RI}, + year = {2009}, + doi = {10.1090/surv/155}, + url = {https://www.math.ucla.edu/~ynm/lectures/dst2009/dst2009.pdf} +} + +@mastersthesis{SolaThesis, + author = {Sola, Alan}, + title = {{Bergman} space methods and integral means spectra of univalent functions}, + type = {Licentiate thesis}, + school = {Royal Institute of Technology}, + address = {Stockholm}, + year = {2007}, + url = {https://www.diva-portal.org/smash/get/diva2:11935/FULLTEXT01.pdf} +} + +@misc{BrennanMethod, + author = {{OpenAI}}, + title = {{Brennan's conjecture and sharp inverse-square integral means}}, + howpublished = {OpenAI Math Release preprint + \href{https://github.com/openai/math/blob/main/preprints/Brennans-conjecture-and-sharp-inverse-square-integral-means-September-24-2026/paper.pdf}{OAI:Brennans-conjecture-and-sharp-inverse-square-integral-means-September-24-2026}}, + year = {2026} +} diff --git a/preprints/A-strict-inverse-first-power-bound-for-univalent-functions-September-24-2026/build/sections/00-introduction.tex b/preprints/A-strict-inverse-first-power-bound-for-univalent-functions-September-24-2026/build/sections/00-introduction.tex new file mode 100644 index 0000000000000000000000000000000000000000..15cbd452de44374c2455d48200883208b9c5c208 --- /dev/null +++ b/preprints/A-strict-inverse-first-power-bound-for-univalent-functions-September-24-2026/build/sections/00-introduction.tex @@ -0,0 +1,146 @@ +\section{Introduction}\label{sec:introduction} + +For a univalent function, the derivative can become large or small near +the boundary even when the image domain is bounded. Integral means +measure the rate of this concentration. Their universal spectrum records +the largest possible growth exponent over a class of conformal maps; +it brings questions about derivatives, coefficients, and the geometry +of image boundaries into a common framework. + +Let \(\DD=\{z\in\C:|z|<1\}\), and let \(\SSS\) consist of all +injective holomorphic maps \(f:\DD\to\C\) with \(f(0)=0\) and +\(f'(0)=1\). Write \(\SSS_b\) for the subclass of maps with bounded +image. The bound on the image may depend on \(f\), and no boundary +regularity is imposed. For \(p\in\R\) and \(0\le r<1\), define +\begin{align*} + M_p[f'](r)&=\frac1{2\pi}\int_{-\pi}^{\pi}|f'(re^{i\theta})|^p\,d\theta,\\ + \beta_f(p)&=\limsup_{r\uparrow1} + \frac{\log M_p[f'](r)}{\log(1/(1-r))}, + &B_b(p)&=\sup_{f\in\SSS_b}\beta_f(p). +\end{align*} +The derivative of a univalent map never vanishes, so all real powers +are defined. The supremum in \(B_b\) is taken after each map's limiting +exponent. Multiplying a map by a nonzero constant leaves that exponent +unchanged, so normalization of the derivative does not change the +bounded-class spectrum. + +\subsection{The conjecture and the uniform bound} + +Kraetzer's experiments with conformal maps onto bounded domains led +to the prediction +\begin{equation}\label{eq:kraetzer} + B_b(p)= + \begin{cases} + p^2/4,&|p|\le2,\\ + |p|-1,&|p|\ge2. + \end{cases} +\end{equation} +The quadratic part is stated in his 1996 paper and explicitly +restated in his thesis \cite{Kraetzer,KraetzerThesis}; the full +quadratic--linear formulation also appears in Beliaev's discussion +of numerical spectra \cite{Beliaev}. We prove a strict upper bound +at the negative interior exponent \(p=-1\). + +\begin{theorem}[Uniform inverse-first-power bound]\label{thm:main} +There exist constants \(0<\varepsilon<1/4\) and \(C<\infty\) such that +\begin{equation}\label{eq:uniform} + M_{-1}[f'](r)\le C(1-r)^{-1/4+\varepsilon} + \qquad\text{for every }f\in\SSS\text{ and every }1/2\le r<1. +\end{equation} +Consequently \(B_b(-1)<1/4\), and Kraetzer's conjectured identity +\eqref{eq:kraetzer} is false. +\end{theorem} + +Both constants are independent of the map, including its image size. +The proof gives existence of a positive gap but does not evaluate it +numerically. The conclusion concerns the inverse first power; it does +not settle the proposed value at \(p=1\). + +\subsection{Context and the role of strictness} + +The distinction between bounded and unrestricted disk maps matters. +Carleson and Jones connected coefficient problems with conformal +dimension \cite{CJ}. Makarov's comparison theorem identifies the +unrestricted spectrum with the maximum of the bounded spectrum and +the Koebe contribution \(3p-1\) \cite[Theorem 5.4]{Makarov}. +In particular, the bounded spectrum in \eqref{eq:kraetzer} should +not be replaced by the unrestricted disk spectrum at positive exponents. +Hedenmalm and Shimorin developed a weighted Bergman-space approach +to bounds for these spectra, keeping this distinction explicit +\cite{HedenmalmShimorin}. Their upper bound at \(p=-1\) for the +unrestricted normalized disk class is \(0.403\) +\cite[Table 1]{HedenmalmShimorin}. Sola subsequently obtained the +bound \(0.388\) by related Bergman-space methods +\cite{Sola}; see also the author's account in +\cite[p.~6]{SolaThesis}. These estimates remain above the +conjectured value \(1/4\); Theorem~\ref{thm:main} establishes a +uniform exponent strictly below it. + +Random conformal snowflakes provide another approach. Beliaev and +Smirnov constructed random conformal maps whose averaged spectrum +approaches the universal spectrum at any prescribed exponent, and +related lower bounds to positive integral operators +\cite[Sections 3--4]{BeliaevSmirnov}. Beliaev further investigated +these operators numerically, distinguishing the numerical estimates +from rigorous error control \cite{Beliaev}. The common feature with +the present argument is the use of a law on conformal maps and a +positive operator. Here the operator samples inverse derivatives +through affine changes of base point, and its growth controls an +upper bound uniformly over the entire class \(\SSS\). + +The affine-law and saddle-to-maximum construction is also used for +inverse-square means in the companion article +\emph{Brennan's conjecture and sharp inverse-square integral means} +\cite{BrennanMethod}. The two arguments share the compact +normalization of half-plane maps, dyadic sampling, and a comparison +of critical points over a common target. We prove every +inverse-first-power estimate and every transport assertion needed +here; the companion's inverse-square endpoint theorem is not an +input. + +The obstacle specific to the present theorem is strictness at the +candidate exponent \(1/4\). A non-strict comparison of saddle and +maximum weights only rules out larger exponents. We retain the exact +operator growth parameter when constructing the probability law. +For almost every target, distinct critical points have distinct +values, so the rank pairing sends each saddle to a strictly higher +maximum. This strict increase forces a vanishing Jacobian in the +remaining equality case. A harmonic lower barrier then excludes +that case. The strict increase can be arbitrarily small from one +pair to another; no common pointwise gain is required. + +\subsection{Proof outline} + +Section~\ref{sec:operator} transfers disk maps to the compact family +of univalent maps on the upper half-plane \(\HH=\{x+iy:y>0\}\), +normalized by \(F(i)=0\), +\(F'(i)=1\). A positive operator \(L\) samples \(1/|F'|\) at two +affine subscales. The exponential growth rate of \(\|L^n\|\), denoted +by \(2^\beta\), bounds all the disk means in Theorem~\ref{thm:main}. +If \(\beta\ge1/4\), Section~\ref{sec:law} constructs a probability +law whose inverse-derivative weighted rebasings scale exactly as +\(y^{-\beta}\). Differentiating this identity fixes two moments of +the logarithmic derivative. + +For a target \(\xi\in\C\), the potential +\(y^{(\beta+2)/3}/|F(x+iy)-\xi|\) turns those moment identities into +a geometric comparison. Section~\ref{sec:escape} proves compactness +of its positive superlevels for almost every map and target. +Section~\ref{sec:pairing} identifies its critical points through a +smooth map \(G_F\), proves the target-generic absence of degeneracies +and ties, and constructs the measurable rank pairing. The nullity of singular +values is the planar instance of Sard's theorem \cite{Sard}; injectivity +supplies the separate exclusion of tied critical values. The planar +index count follows the tangent-turning argument of Hopf \cite{Hopf}. +The moment identities make the expected normalized signed Jacobian +of \(G_F\) positive when \(\beta>1/4\), and zero when \(\beta=1/4\). + +Section~\ref{sec:transport} changes variables through \(G_F\) and +compares the paired weights on expanding source rectangles. It also +transfers exceptional-target nullity to the fixed normalized point, +where area-genericity alone would be insufficient. +The comparison forces the expected signed Jacobian to be nonpositive; +strictness of the pairing forces the Jacobian itself to vanish almost +surely in the remaining equality case. +Section~\ref{sec:equality} rules out the vanishing-Jacobian equality +case and converts \(\beta<1/4\) into the uniform estimate. diff --git a/preprints/A-strict-inverse-first-power-bound-for-univalent-functions-September-24-2026/build/sections/01-compact-model.tex b/preprints/A-strict-inverse-first-power-bound-for-univalent-functions-September-24-2026/build/sections/01-compact-model.tex new file mode 100644 index 0000000000000000000000000000000000000000..3d1739ae96bf2fe619974b114e24e551d55c0ab8 --- /dev/null +++ b/preprints/A-strict-inverse-first-power-bound-for-univalent-functions-September-24-2026/build/sections/01-compact-model.tex @@ -0,0 +1,152 @@ +\section{A compact model and its growth operator}\label{sec:operator} + +This section replaces the boundary-growth problem by growth of a +single positive operator on a compact family of normalized maps. + +\subsection{Univalence estimates} + +We use the classical quarter and derivative-distortion theorems in +exactly the following form \cite[Chapter 2]{Duren}. + +\begin{lemma}[Classical univalence estimates]\label{lem:classical} +For every \(f\in\SSS\) and \(|z|=r<1\), +\begin{equation}\label{eq:koebe} + \frac{1-r}{(1+r)^3}\le |f'(z)|\le + \frac{1+r}{(1-r)^3},\qquad + B(0,1/4)\subset f(\DD). +\end{equation} +If \(F\) is univalent on the Euclidean disk \(B(z_0,s)\), then +\begin{equation}\label{eq:local-quarter} + B\bigl(F(z_0),s|F'(z_0)|/4\bigr) + \subset F(B(z_0,s)). +\end{equation} +\end{lemma} +\begin{proof} +The first assertions are the quoted classical estimates for normalized +univalent disk maps. Apply the quarter theorem to +\((F(z_0+s\zeta)-F(z_0))/(sF'(z_0))\) to obtain the last assertion. +\end{proof} + +Let \(\HH=\{x+iy:y>0\}\), and define +\[ + \KK=\{F:\HH\to\C\text{ univalent}:F(i)=0,\ F'(i)=1\}. +\] +For \(F\in\KK\), set \(Q_F=1/F'\). +With \(m(z)=(z-i)/(z+i)\), the maps of \(\KK\) are exactly +\(F=2i f\circ m\), \(f\in\SSS\). + +\begin{lemma}[Compact normalized half-plane family]\label{lem:compact} +The locally uniform topology makes \(\KK\) compact metrizable. +Evaluation of \(F,Q_F\) and all their derivatives is jointly +continuous in \(F,z\), and uniformly bounded on each compact +source set. Moreover, uniformly in \(F\in\KK\), +\begin{equation}\label{eq:strip} + |Q_F(x+iy)|\le C/y\quad(|x|\le2,\ 00\quad(|m(z)|\ge1/2). +\end{equation} +\end{lemma} +\begin{proof} +Integrating the derivative bound in the disk gives local uniform +boundedness. Montel's theorem gives convergent subsequences. +Limits retain the normalization by convergence of derivatives +and remain injective by Hurwitz's theorem: for a fixed source point, +subtract its value and apply Hurwitz on the punctured domain. +The nonconstant normalization excludes the identically zero +alternative. A countable compact exhaustion metrizes this topology. +Interior holomorphic convergence, together with nonvanishing +derivatives, gives the asserted continuity and bounds. + +The identities +\[ + 1-|m(z)|^2=\frac{4y}{|z+i|^2}, + \qquad |Q_F(z)|=\frac{|z+i|^2}{4|f'(m(z))|} +\] +and \eqref{eq:koebe} imply the strip estimate. +The image under \(f\) of the disk of radius \(1/2\) contains +\(B(0,1/8)\) by \eqref{eq:local-quarter}. Injectivity gives +the second assertion, with \(c=1/4\) for \(F=2i f\circ m\). +\end{proof} + +\subsection{Affine rebasings and dyadic samples} + +Identify a point \(z=x+iy\in\HH\) with the affine map +\[ + z\cdot w=x+yw,\qquad z^\#=(-x+i)/y. +\] +This is a group with identity \(i\) and inverse \(z^\#\). +For \(F\in\KK\), define +\[ + (T_zF)(w)=\frac{F(z\cdot w)-F(z)}{yF'(z)}. +\] +Then \(T_zF\in\KK\), and direct substitution gives +\begin{equation}\label{eq:action} + T_w(T_zF)=T_{z\cdot w}F,\qquad + Q_{T_zF}(w)=\frac{Q_F(z\cdot w)}{Q_F(z)}. +\end{equation} +On \(C(\KK)\), with its supremum norm, put +\[ + (A_z\Phi)(F)=|Q_F(z)|\,\Phi(T_zF),\qquad + L=\tfrac12(A_{-1/2+i/2}+A_{1/2+i/2}). +\] +These are bounded positive operators. For each continuous \(\Phi\), +\(A_z\Phi\) is jointly continuous in \(F,z\); rebasings send compact +source sets to compact sets locally in the parameter \(z\). +Equation~\eqref{eq:action} gives \(A_zA_w=A_{z\cdot w}\). +In particular, if +\(v=2^{-n}\) and \(u_j=-1+(2j+1)v\), then +\begin{equation}\label{eq:dyadic} + L^n=v\sum_{j=0}^{2^n-1}A_{u_j+iv},\qquad + \|L^n\|=\max_{F\in\KK}(L^n1)(F). +\end{equation} +The abscissae are the sums \(\sum_{\ell=1}^n\pm2^{-\ell}\); +positivity proves the norm identity. +Testing the affine map \(F(z)=z-i\), and using \eqref{eq:strip}, +we obtain \(1\le\|L^n\|\le C2^n\). +The same operator will control every map. Submultiplicativity defines +\begin{equation}\label{eq:rho} + \rho=\lim_{n\to\infty}\|L^n\|^{1/n}, + \qquad \beta=\log_2\rho,\qquad 0\le\beta\le1. +\end{equation} +Indeed, writing \(n=qN+r\), \(0\le r\rho\), the series +\[ + R_\tau=\sum_{n\ge0}\tau^{-n}L^n,\qquad + g_\tau=R_\tau1,\qquad M_\tau=\max g_\tau +\] +converges in operator norm. Since +\[ + Lg_\tau=\tau(g_\tau-1) + \le\tau(1-1/M_\tau)g_\tau,\qquad 1\le g_\tau\le M_\tau, +\] +positivity gives +\(\|L^n\|\le M_\tau[\tau(1-1/M_\tau)]^n\). +Thus \(\rho\le\tau(1-1/M_\tau)\), so +\(M_\tau\ge\tau/(\tau-\rho)\to\infty\). +At a maximizer \(F_\tau\) of \(g_\tau\), the functional +\(\mu_\tau=\delta_{F_\tau}R_\tau/M_\tau\) is a probability and +\[ + \mu_\tau L=\tau\mu_\tau-\frac{\tau}{M_\tau}\delta_{F_\tau}. +\] +A weak limit as \(\tau\downarrow\rho\) gives the required \(\mu\). + +\emph{Horizontal and scale averages.} The composition law gives +\(A_{u+iv}A_{2+i}=A_{u+2v+iv}\): right translation by \(2\) +shifts the grid in \eqref{eq:dyadic} by one cell of width \(2v\). +Using \(\mu=\rho^{-n}\mu L^n\), all but the two endpoint terms cancel: +\[ + \mu A_{2+i}-\mu + =\rho^{-n}v\bigl(\mu A_{1+v+iv}-\mu A_{-1+v+iv}\bigr). +\] +The norm of this difference is at most \(2C\rho^{-n}\) by +\eqref{eq:strip}, and tends to zero because \(\beta>0\). +Therefore \(\mu A_{2+i}=\mu\). +It follows that \(t\mapsto\mu A_{t+ih}\) has period two for every +fixed \(h>0\). + +Define +\(\bar\mu=\tfrac12\int_0^2\mu A_{x+i}\,dx\) on continuous tests. +Periodicity gives \(\bar\mu A_{s+i}=\bar\mu\) for all \(s\in\R\). +Moreover, +\[ + \rho\bar\mu + =\tfrac12\int_0^2\mu L A_{x+i}\,dx + =\tfrac12\int_{-1/2}^{3/2}\mu A_{t+i/2}\,dt + =\bar\mu A_{i/2}. +\] +The middle equality uses \(t=-1/2+x/2\) and \(t=1/2+x/2\) +for the two summands of \(L\); the last uses period two. +Consequently \(h^\beta\bar\mu A_{ih}\) is multiplicatively +two-periodic in \(h>0\). +Each of these functionals is horizontally invariant on the right, +because \(A_{ih}A_{s+i}=A_{hs+i}A_{ih}\). +The functional +\[ + \nu=\int_1^2 h^\beta\bar\mu A_{ih}\,\frac{dh}{h} +\] +has finite, strictly positive mass by compact-family continuity. +Periodicity and the substitution \(u=hy\) give +\(\nu A_{iy}=y^{-\beta}\nu\). Horizontal invariance persists, +so normalization of \(\nu\) gives \eqref{eq:law} for continuous tests. +For each fixed \(x+iy\), the two sides define finite Borel measures +(one is a weighted pushforward). Equality on continuous tests makes +them equal as measures, proving the assertion for all nonnegative +Borel tests. +\end{proof} + +The law now supplies the differential moments needed for the signed +Jacobian comparison. For \(z=x+iy\), define +\begin{equation}\label{eq:qk} + q_F(z)=iy\,\frac{Q_F'(z)}{Q_F(z)} + =-iy\,\frac{F''(z)}{F'(z)}=a_F(z)+ib_F(z), + \qquad k=\frac{\beta+2}{3}. +\end{equation} +Thus \(3(k-1)=\beta-1\) and \(3/4\le k\le1\). +Equation~\eqref{eq:action} implies +\(q_{T_zF}(w)=q_F(z\cdot w)\). + +\begin{lemma}[Logarithmic-derivative moments]\label{lem:moments} +The law satisfies +\begin{equation}\label{eq:moments} + \E a_F(i)=-\beta,\qquad + \E|q_F(i)|^2=\beta(\beta+1). +\end{equation} +\end{lemma} +\begin{proof} +The constant test in \eqref{eq:law} gives +\(\E|Q_F(x+iy)|=y^{-\beta}\). +Since \(\log|Q_F|\) is harmonic, +\[ + \partial_y|Q_F|=\frac{|Q_F|a_F}{y},\qquad + \Delta|Q_F|=|Q_F|\left|\frac{Q_F'}{Q_F}\right|^2. +\] +Differentiate at \(i\), where \(Q_F(i)=1\). +Lemma~\ref{lem:compact} and local nonvanishing give the uniform +smoothness bounds needed to differentiate under expectation. +\end{proof} diff --git a/preprints/A-strict-inverse-first-power-bound-for-univalent-functions-September-24-2026/build/sections/03-escape.tex b/preprints/A-strict-inverse-first-power-bound-for-univalent-functions-September-24-2026/build/sections/03-escape.tex new file mode 100644 index 0000000000000000000000000000000000000000..5eb9b8b48fa321eb2f3f2a265b5e741fda28ea3f --- /dev/null +++ b/preprints/A-strict-inverse-first-power-bound-for-univalent-functions-September-24-2026/build/sections/03-escape.tex @@ -0,0 +1,149 @@ +\section{Compact superlevels for almost every target}\label{sec:escape} + +The affine law must next control escape in the source half-plane. +Our objective is compactness of positive superlevels, which will make +the subsequent critical-point counts finite above each positive value. +For \(F\in\KK\) and \(\xi\in\C\), put +\[ + V_\xi^F(x+iy)=\frac{y^k}{|F(x+iy)-\xi|}, +\] +with value \(+\infty\) at the possible pole. +The choice \(3k=\beta+2\) lets a submean estimate for \(|1/F|^3\) +convert the weighted area tail below into decay of \(V_0^F\) at +source infinity. +We write \(dA\) for planar Lebesgue area. + +\begin{proposition}[Compact positive superlevels]\label{prop:compact-levels} +For \(P\)-almost every \(F\), and then for area-almost every +\(\xi\in\C\), every closed positive superlevel +\(\{V_\xi^F\ge\delta\}\), \(\delta>0\), is compact in \(\HH\). +\end{proposition} + +We prove this by controlling infinity and then the real boundary. +The order of these two controls is useful: the first supplies +a horizontal preimage-length bound for the second. That bound is +uniform in the height, with the map and compact target set fixed. + +\begin{lemma}[The weighted area tail]\label{lem:area-tail} +Writing \(S_0=\{z\in\HH:|m(z)|\ge1/2\}\), almost every \(F\) satisfies +\begin{equation}\label{eq:area-tail} + I(F):=\int_{S_0}y^\beta|F(z)|^{-3}\,dA(z)<\infty. +\end{equation} +\end{lemma} +\begin{proof} +For fixed \(z\ne i\), apply \eqref{eq:law} to +\(\Phi(F)=|F(z^\#)|^{-3}\). +The identity +\((T_zF)(z^\#)=-F(z)Q_F(z)/y\) gives +\[ + \E\bigl[|F'(z)|^2|F(z)|^{-3}\bigr] + =y^{-\beta-3}\E|F(z^\#)|^{-3}. +\] +The integral of the left side over \(S_0\) is uniformly finite: +by injectivity and \eqref{eq:strip}, it is bounded by +\(\int_{|\omega|\ge c}|\omega|^{-3}\,dA(\omega)\). +The involution \(z\mapsto z^\#\) preserves \(|m(z)|\), inverts +the height, and has absolute Jacobian \(y^{-3}\). +Changing variables on the right leaves exactly the weight \(y^\beta\). +Tonelli proves finite expectation of \(I\), hence the assertion. +\end{proof} + +\begin{lemma}[Escape at infinity and horizontal length]\label{lem:infinity} +If \(I(F)<\infty\), then +\begin{equation}\label{eq:infinity} + \frac{y^k}{|F(x+iy)|}\longrightarrow0 + \qquad\text{as } |x|+y\longrightarrow\infty. +\end{equation} +For every compact target set \(K\), its preimage restricted to +\(y\ge\epsilon>0\) is compact, its whole preimage has bounded height, +and +\begin{equation}\label{eq:horizontal} + \sup_{s>0}\int_\R\ind_K(F(x+is))\,dx<\infty. +\end{equation} +\end{lemma} +\begin{proof} +If \(w=x'+iy'\in B(z,y/2)\), then \(y/2\le y'\le3y/2\) and +\(|x|+y\le |x'|+3y'\). +Thus these disks escape every bounded \(|x'|+y'\) region as +their centers do. The complement of \(S_0\) is bounded, so +the disks are eventually contained in \(S_0\) and avoid the zero of \(F\). +Submean for \(|1/F|^3\), together with \(3k=\beta+2\), gives +\[ + \left(\frac{y^k}{|F(z)|}\right)^3 + \le C_\beta\int_{B(z,y/2)} + (\im w)^\beta|F(w)|^{-3}\,dA(w). +\] +The right side tends to zero as a tail of \eqref{eq:area-tail}. +For \(|F(z)|\) bounded and \(y\ge\epsilon\), this excludes source +infinity, proving strip compactness. It also excludes unbounded +heights in the whole preimage of \(K\). + +Choose a nonnegative smooth compactly supported \(\psi\ge\ind_K\) +and put \(H(y)=\int_\R\psi(F(x+iy))\,dx\). +For fixed \(s>0\), the support in \(y\ge s/2\) is compact by the +strip assertion. Hence differentiation and integration by parts give +\begin{align*} + H(s)&=\int_s^\infty(y-s)H''(y)\,dy\\ + &=\int_{y>s}(y-s)|F'(z)|^2(\Delta\psi)(F(z))\,dA(z). +\end{align*} +Here the integral of \(\partial_{xx}(\psi\circ F)\) is zero and +\(\Delta(\psi\circ F)=|F'|^2(\Delta\psi)\circ F\). +If \(Y_\psi\) bounds the preimage heights of \(\supp\psi\), the +absolute value is at most +\[ + Y_\psi\int_\HH |F'(z)|^2|(\Delta\psi)(F(z))|\,dA(z) + \le Y_\psi\int_\C|\Delta\psi|\,dA. +\] +The last inequality uses injectivity, and the bound is independent +of \(s\). This proves \eqref{eq:horizontal}. +\end{proof} + +\begin{proof}[Proof of Proposition~\ref{prop:compact-levels}] +Fix \(F\) satisfying Lemma~\ref{lem:area-tail}. The preceding +lemma controls source infinity. It remains to control approach to the +real boundary, uniformly in the horizontal coordinate for almost every +target. +For omitted targets, meaning \(\xi\notin F(\HH)\), restrict to +\(|\xi|\le R\) and fix \(\delta>0\). +At scale \(h=2^{-\ell}\), \(\ell\ge0\), partition the line into +cells \([jh,(j+1)h)\). Mark a cell if some point \(z=x+iy\) +with \(x\) in that cell and \(h\le y\le2h\) satisfies +\(|F(z)-\xi|\le y^k/\delta\) for at least one such omitted target. +At a witness, \eqref{eq:local-quarter} on \(B(z,y)\) gives +\(y|F'(z)|\le4|F(z)-\xi|\). +For every \(x'\) in the same cell, the rebased point +\((x'-x)/y+i h/y\) belongs to the compact rectangle +\([-1,1]+i[1/2,1]\). Lemma~\ref{lem:compact} applied to \(T_zF\) +therefore gives +\begin{equation}\label{eq:cell-cover} + |F(x'+ih)-\xi|\le C_\delta h^k. +\end{equation} +This bound holds for every witness and its target. + +Each marked cell's entire segment at height \(h\) is mapped into +\(\{|\omega|\le R+C_\delta\}\). +Equation~\eqref{eq:horizontal} thus bounds the number of marked +cells by \(C/h\). +Disks of radius \(C_\delta h^k\) centered at their midpoint images +cover all targets witnessed at this scale, and have summed area +\(O(h^{2k-1})\). +The sum over dyadic scales is finite because \(k>1/2\). +The limsup of these measurable covering unions has area zero. +Taking integer \(R>0\) and \(\delta=1/n\), \(n\ge1\), shows that +for almost every omitted target, +\[ + \sup_x V_\xi^F(x+iy)\longrightarrow0\qquad(y\downarrow0). +\] +Indeed a failure at arbitrarily small heights gives witnesses +in infinitely many dyadic bands for one such threshold. + +If \(\xi\in F(\HH)\), a small closed disk about \(\xi\) contained +in \(F(\HH)\) has compact inverse image under the homeomorphism +onto the image. Below its minimum preimage height, +\(|F(z)-\xi|\) is uniformly positive, giving the same limit. +At heights bounded below, \eqref{eq:infinity} gives +\(|F(z)|\to\infty\) and \(V_\xi^F(z)\to0\) at source infinity. +Finally \(\{V_\xi^F\ge\delta\}\) is relatively closed, since it is +\(\{|F(z)-\xi|\le y^k/\delta\}\), also at a pole. +The two escape exclusions prove its compactness. +\end{proof} diff --git a/preprints/A-strict-inverse-first-power-bound-for-univalent-functions-September-24-2026/build/sections/04-strict-pairing.tex b/preprints/A-strict-inverse-first-power-bound-for-univalent-functions-September-24-2026/build/sections/04-strict-pairing.tex new file mode 100644 index 0000000000000000000000000000000000000000..f740149bd681dae40a163150efef1001e4f2384a --- /dev/null +++ b/preprints/A-strict-inverse-first-power-bound-for-univalent-functions-September-24-2026/build/sections/04-strict-pairing.tex @@ -0,0 +1,270 @@ +\section{Critical points and strict pairing}\label{sec:pairing} + +For \(P\)-almost every map and area-almost every target, our next goal is to pair +each saddle of the potential with a higher non-pole local maximum. +Compact superlevels and nondegeneracy will supply finite critical-point +counts above positive levels. + +\subsection{The critical map} + +Define the smooth real map and its normalized signed Jacobian by +\begin{equation}\label{eq:G} + G_F(z)=F(z)-\frac{iyF'(z)}k,\qquad + J_F(z)=\frac{\det DG_F(z)}{|F'(z)|^2}. +\end{equation} +Its real derivative columns are +\((F'/k)(k+q_F)\) and \((iF'/k)(k-1+q_F)\). +Consequently +\begin{equation}\label{eq:Jacobian} + k^2J_F=k(k-1)+(2k-1)a_F+|q_F|^2,\qquad + k^2\E J_F(i)=\frac{(4\beta-1)(\beta+2)}9. +\end{equation} +The second identity uses \eqref{eq:moments} and \(k=(\beta+2)/3\). + +Off the possible pole, write \(\ell=\log V_\xi^F\) and +\(r_0=F'/(F-\xi)\). Then +\[ + \Delta\ell=-k/y^2,\qquad + (\ell_x,\ell_y)=(-\re r_0,\ k/y+\im r_0). +\] +Critical points are therefore exactly the preimages \(G_F(z)=\xi\). +A pole is not such a preimage. At a critical point, +\begin{equation}\label{eq:critical-value} + V_\xi^F(z)=k y^{k-1}|Q_F(z)|,\qquad + r_0'=\frac{k(k+q_F)}{y^2}. +\end{equation} +Since \(3(k-1)=\beta-1\), cubing the critical value produces +\(k^3y^{\beta-1}|Q_F(z)|^3\). This is the weight, up to the constant +\(k^3\), that the target change of variables in +Section~\ref{sec:transport} will assign to each critical point. +Direct differentiation yields +\begin{equation}\label{eq:hessian} + \operatorname{Hess}\ell + =\frac{k}{y^2} + \begin{pmatrix}-(k+a_F)&b_F\\b_F&k-1+a_F\end{pmatrix}, + \qquad + \det\operatorname{Hess}\ell=-\frac{k^4}{y^4}J_F. +\end{equation} +Its trace is negative. Thus \(J_F>0\) gives a nondegenerate saddle, +and \(J_F<0\) a nondegenerate local maximum. We call the latter +a \emph{finite maximum}, excluding a pole. + +\subsection{Good targets} + +Let \(\mathcal G\subset\KK\times\C\) be the set of pairs +\((F,\xi)\) for which all positive closed superlevels +of \(V_\xi^F\) are compact, \(\xi\) is a regular value of \(G_F\), +and distinct preimages under \(G_F\) have distinct \(V_\xi^F\)-values. +We call these pairs \emph{good}. + +\begin{lemma}[Generic targets and Borel goodness]\label{lem:good} +Goodness is Borel in \((F,\xi)\). For \(P\)-almost every \(F\), +area-almost every \(\xi\) is good. +\end{lemma} +\begin{proof} +The superlevel assertion is Proposition~\ref{prop:compact-levels}. +For any fixed \(F\), singular values of \(G_F\) have area zero. +This is the plane case of Sard's theorem \cite[Theorem 4.1]{Sard}. +Here is the elementary estimate needed for this smooth plane map. +On a compact source box, cover singular points by grid squares +of side \(h\). Linearize at one singular point of each square. +Bounded second derivatives put its image in an \(O(h^2)\)-neighborhood +of a segment of length \(O(h)\). Its area is \(O(h^3)\). +There are \(O(h^{-2})\) squares, giving total area tending to zero. +Exhaust the source by compact boxes. + +Cover the nonsingular locus by countably many inverse-function +charts. On an overlap where two branches \(z_1(\xi),z_2(\xi)\) +are distinct, the target differential of their critical log values is +\[ + d_\xi\bigl[\log V_\xi^F(z_j(\xi))\bigr](v) + =\re\frac{v}{F(z_j(\xi))-\xi}. +\] +The source derivative vanishes by criticality. +The two linear functionals differ by injectivity of \(F\). +Hence the difference of log values has nonzero differential, and +its zero set is locally a smooth curve. Countably many chart pairs +give an area-null tie set. + +For measurability, use +\(C_m=\{|x|\le m,\ 1/m\le y\le m\}\). +For every integer \(n\ge1\), require the open set +\(\{V_\xi^F>1/n\}=\{|F(z)-\xi|0\), +\begin{equation}\label{eq:count} + \#\{z:\ z\text{ is a finite maximum},\ V_\xi^F(z)\ge h\} + \ \ge\ + \#\{z:\ z\text{ is a saddle},\ V_\xi^F(z)\ge h\}. +\end{equation} +Both counts are finite. +\end{lemma} +\begin{proof} +The fiber \(G_F^{-1}(\xi)\) is closed and discrete, and the closed +positive superlevel is compact. This proves finiteness. There can +be no accumulation at a pole either, because a pole is not on the fiber. + +First count in \(U=\{V_\xi^F>\lambda\}\) for a positive regular +level \(\lambda\). We will show that each component is bounded by +one simple closed curve and has total gradient index one, giving +\begin{equation}\label{eq:index-count} + \#\text{components} + =\#\text{finite maxima}-\#\text{saddles}+\#\text{poles} + \quad\text{in }U. +\end{equation} +The boundary of \(U\) is the compact smooth one-manifold +\(\{V_\xi^F=\lambda\}\), away from any pole. +It is a finite disjoint union of smooth simple closed curves. +For detail, the components are open in this compact manifold, so +there are finitely many. Each component is one orbit of the smooth +unit tangent field obtained by rotating \(\nabla\ell/|\nabla\ell|\). +The orbits are complete and open. A nonperiodic single orbit would +be a bijective local homeomorphism from \(\R\) to the compact +component, which is impossible. Its least period gives a simple +closed curve. + +The bounded Jordan interior of each curve has closure in \(\HH\): +the region below the curve's minimum height lies in the unbounded +complement. Everywhere inside, \(V_\xi^F>\lambda\). Otherwise +\(\ell\) would have an interior minimum, contrary to +\(\Delta\ell=-k/y^2<0\). If a pole is present, remove a sufficiently +small neighborhood where \(\ell\) is large before taking this minimum. +Thus curves cannot be nested, and their interiors are exactly the +components of \(U\). Indeed each component meets the level boundary, +has its superlevel side locally inside one of the curves, and cannot +leave that interior without crossing the level. + +On each positively oriented curve, \(\nabla\ell\) points inward +and has winding one. We recall the tangent-turning argument +of Hopf \cite[Section 2]{Hopf}. +Parametrize a simple regular curve by \(\gamma:[0,1]\to\C\), +with matching endpoint derivatives, starting at a leftmost point. +The positive tangent initially points down. The unit secant direction +from \(\gamma(s)\) to \(\gamma(t)\), \(0\le s\le t\le1\), +extends continuously to the diagonal by the tangent, and to +\((0,1)\) by the negative initial tangent. +Lift its angle continuously on this triangle. +On \((0,t)\) it lies in the closed right semicircle and changes +angle by \(\pi\). On \((s,1)\) it is the negative of the direction +at \((0,s)\), giving another \(\pi\). +The diagonal therefore records tangent winding one, unchanged for +the inward normal. + +Remove disjoint small disks about the finitely many critical points +and possible pole in each component. Stokes applied to the closed +angle form +\[ + \frac{X_1\,dX_2-X_2\,dX_1}{|X|^2},\qquad X=\nabla\ell, +\] +shows that the sum of interior indices is one. +At a nondegenerate critical point the index is the sign of the +Hessian determinant, hence \(+1\) for a maximum and \(-1\) for +a saddle. At the pole \(z_0\), +\(\ell=-\log|z-z_0|+\text{a smooth function}\), giving index \(+1\). +Summing over components proves \eqref{eq:index-count}. +There is at most one pole; if present it belongs to a component. +Thus the finite-maximum count is at least the saddle count. +This also covers the empty superlevel. + +Finally choose a regular \(\lambda0,\ (F,G_F(z))\in\mathcal G\} +\] +there is a Borel assignment \(R_F(z)\) to a finite maximum over +the same target, injective on each target fiber, such that +\begin{equation}\label{eq:strict-value} + V_{G_F(z)}^F(R_F(z))>V_{G_F(z)}^F(z). +\end{equation} +It is equivariant: +\[ + R_F(t\cdot w)=t\cdot R_{T_tF}(w) +\] +on the corresponding domains. +\end{proposition} +\begin{proof} +On each good fiber order saddles by decreasing critical value, +and separately order finite maxima the same way. +Each point has finitely many predecessors. A nonempty list has a +first point: choose any one point and inspect the finite nonempty +set at or above its value. Repeating gives integer ranks for all points. +Lemma~\ref{lem:count} supplies at least as many maxima as saddles +at or above each saddle's value. Pair the saddle of rank \(j\) +with the maximum of rank \(j\). +No ties are allowed, so the paired value is strictly larger, +as illustrated in Figure~\ref{fig:ranks}. +Different ranks give different partners. Equation~\eqref{eq:V-action} +preserves ranks, proving equivariance. + +We prove the measurability assertion directly using the nonsingularity +of each root. +The nonsingular solution graph \(G_F(z)=\xi,\ J_F(z)\ne0\) +has countably many continuous local root branches \(b_j(F,\xi)\), +defined on open parameter sets. +To obtain a branch near \((F_0,\xi_0,z_0)\), set +\(M=DG_{F_0}(z_0)\). On a sufficiently small closed source ball, +\[ + w\longmapsto w-M^{-1}(G_F(w)-\xi) +\] +has Lipschitz constant at most \(1/2\) for all nearby parameters. +Shrinking the parameter neighborhood makes the displacement of +the center less than half the radius. The map is a contraction +of the ball into itself, with a unique interior nonsingular root. +Compactness and uniqueness imply continuous dependence on the +parameters. These branches form relatively open graph patches; +second countability of \(\KK\times\C\times\HH\) gives a countable cover. + +At each parameter activate branch \(j\) only when it is defined +and its value differs from every earlier defined branch value. +This Borel test enumerates each root once. +The signs of \(J\), critical values \eqref{eq:critical-value}, +and comparisons of such values are Borel. On good fibers, +counting active predecessors of the same sign gives Borel integer +ranks. Selecting the unique active negative-sign branch of the +same rank as a given active positive-sign branch is therefore Borel. +Substitute \(\xi=G_F(z)\) to obtain the claimed assignment. +\end{proof} + +\input{figures/rank-pairing} diff --git a/preprints/A-strict-inverse-first-power-bound-for-univalent-functions-September-24-2026/build/sections/05-strict-transport.tex b/preprints/A-strict-inverse-first-power-bound-for-univalent-functions-September-24-2026/build/sections/05-strict-transport.tex new file mode 100644 index 0000000000000000000000000000000000000000..9ede76921b8280c1695cf80ea6de80a6f1a0ac4d --- /dev/null +++ b/preprints/A-strict-inverse-first-power-bound-for-univalent-functions-September-24-2026/build/sections/05-strict-transport.tex @@ -0,0 +1,118 @@ +\section{Strict weighted transport}\label{sec:transport} + +The pairing will imply \(\E J_F(i)\le0\), with equality only if +\(J_F(i)=0\) almost surely. Comparing this conclusion with +Equation~\eqref{eq:Jacobian} will reduce the contradiction to a +rigidity argument. + +Put \(J_{F,\pm}=\max(\pm J_F,0)\) and +\(\omega_F(z)=y^{\beta-1}|Q_F(z)|\). +For any fixed \(F\) and nonnegative Borel \(h\), local change of variables +gives +\begin{equation}\label{eq:area-formula} + \int_\HH h(z)\omega_F(z)J_{F,\pm}(z)\,dA(z) + = + \int_\C\! + \sum_{\substack{G_F(z)=\xi\\ \pm J_F(z)>0}} + h(z)y^{\beta-1}|Q_F(z)|^3\,dA(\xi). +\end{equation} +Indeed cover the indicated sign locus by countably many local +diffeomorphism charts, disjointify into Borel pieces, and use absolute +Jacobian \(|J_F||F'|^2\). No global properness of \(G_F\) is required. +At a preimage, \eqref{eq:critical-value} and \(3(k-1)=\beta-1\) give +\begin{equation}\label{eq:target-weight} + y^{\beta-1}|Q_F(z)|^3=(V_\xi^F(z)/k)^3. +\end{equation} + +\begin{lemma}[Goodness at the normalized point]\label{lem:base-good} +The positive-Jacobian mass at the base point is supported on good fibers: +\begin{equation}\label{eq:base-good} + \E\bigl[J_{F,+}(i) + \ind_{\{(F,G_F(i))\text{ not good}\}}\bigr]=0. +\end{equation} +\end{lemma} +\begin{proof} +For almost every \(F\), bad targets have area zero by +Lemma~\ref{lem:good}. Formula~\eqref{eq:area-formula} therefore gives +\[ + \int_{\{|x|\le1,\ 1\le y\le2\}} + \omega_F(z)J_{F,+}(z) + \ind_{\{(F,G_F(z))\text{ not good}\}}\,dA(z)=0. +\] +The product of \(J_{F,+}\) and the indicator equals its base-point +version for \(T_zF\), by \eqref{eq:G-action}--\eqref{eq:V-action}. +Take expectations and apply \eqref{eq:law} and Tonelli. +The spatial density is \(dx\,dy/y\); its integral on this box is +\(2\log2>0\). This proves \eqref{eq:base-good}. +\end{proof} + +The weighted assertion is essential: \(G_F(i)=-i/k\) for every \(F\), +so area-genericity alone says nothing about this fixed target. +We need only the weighted conclusion, not almost-sure goodness +at the base point without the Jacobian factor. + +\Needspace{10\baselineskip} +\begin{proposition}[Strict weighted transport]\label{prop:strict-transport} +Assume \(\beta\ge1/4\), let \(P\) be the law of +Proposition~\ref{prop:law}, and set \(k=(\beta+2)/3\). +Then \(\E J_F(i)\le0\). +Equality is possible only if \(J_F(i)=0\) almost surely. +\end{proposition} +\begin{proof} +For \(N\ge1\), restrict the pairing domain \(\mathcal D\) to +\(\mathcal D_N\) by requiring, for \(z'=R_F(z)=x'+iy'\), +\[ + 1/N\le y'/y\le N,\qquad |x'-x|\le Ny. +\] +These Borel sets increase to \(\mathcal D\), and their indicators +at \((F,z)\) equal their indicators at \((T_zF,i)\). +Define the equivariant Borel ratio +\[ + c_F(z)= + \left(\frac{V_{G_F(z)}^F(R_F(z))}{V_{G_F(z)}^F(z)}\right)^3>1 + \quad\text{on }\mathcal D, +\] +and set it to zero elsewhere. +For \(S>0\), \(Y>1\), use the boxes +\[ + B=\{|x|\le S,\ 1\le y\le Y\},\qquad + B'=\{|x|\le S+NY,\ 1/N\le y\le NY\}. +\] +Every included saddle in \(B\) has its partner in \(B'\). +On the target side of \eqref{eq:area-formula}, multiplication +by \(c_F\) makes its weight exactly the partner's weight +by \eqref{eq:target-weight}. Fiberwise injectivity of the pairing +therefore proves, for every \(F\), +\[ + \int_B\omega_FJ_{F,+}c_F\ind_{\mathcal D_N}(F,z)\,dA(z) + \le\int_{B'}\omega_FJ_{F,-}\,dA(z). +\] +Take expectations and apply \eqref{eq:law} to the equivariant +nonnegative tests. The resulting inequality is +\begin{align*} + 2S\log Y\, + \E[J_{F,+}(i)c_F(i)\ind_{\mathcal D_N}(F,i)] + \le + 2(S+NY)(\log Y+2\log N)\,\E J_{F,-}(i). +\end{align*} +Here \(J_F(i)\) is bounded by compact-family continuity, so +the right side is finite. At fixed \(N\), first let \(S\to\infty\), +then \(Y\to\infty\). Finally let \(N\to\infty\) by monotone +convergence. This gives +\[ + A:=\E[J_{F,+}(i)c_F(i)\ind_{\mathcal D}(F,i)] + \le d:=\E J_{F,-}(i). +\] +Lemma~\ref{lem:base-good} gives +\(b:=\E J_{F,+}(i)=\E[J_{F,+}(i)\ind_{\mathcal D}(F,i)]\). +Thus \(b\le A\le d\), proving \(\E J_F(i)\le0\). +If equality holds, then \(b=d\) and +\[ + 0\le\E[J_{F,+}(i)(c_F(i)-1)\ind_{\mathcal D}(F,i)] + =A-b\le0. +\] +The integrand is strictly positive wherever \(J_{F,+}(i)>0\) +on \(\mathcal D\). Equation~\eqref{eq:base-good} removes positive +mass outside this domain, so \(b=0\), and hence \(d=0\). +No uniform positive gap in \(c_F-1\) is needed. +\end{proof} diff --git a/preprints/A-strict-inverse-first-power-bound-for-univalent-functions-September-24-2026/build/sections/06-equality-barrier.tex b/preprints/A-strict-inverse-first-power-bound-for-univalent-functions-September-24-2026/build/sections/06-equality-barrier.tex new file mode 100644 index 0000000000000000000000000000000000000000..c45c51bc8a2511102bcbadcb2d81c012ef394260 --- /dev/null +++ b/preprints/A-strict-inverse-first-power-bound-for-univalent-functions-September-24-2026/build/sections/06-equality-barrier.tex @@ -0,0 +1,54 @@ +\section{The harmonic equality barrier and the uniform gap}\label{sec:equality} + +The moment identity and strict transport can coexist at +\(\beta\ge1/4\) only in the equality case \(\beta=1/4\). +We first rule out its deterministic consequence, an identically +vanishing Jacobian, and then transfer that obstruction back to the law. + +\begin{lemma}[The harmonic equality barrier]\label{lem:no-zero-J} +For \(k=3/4\), no \(F\in\KK\) has \(J_F\equiv0\). +\end{lemma} +\begin{proof} +By \eqref{eq:Jacobian}, such a map would satisfy +\[ + |q_F(z)+1/4|^2=1/4. +\] +Therefore \(|q_F(z)|\ge1/4\) and +\(|F''(z)/F'(z)|\ge1/(4y)\). +The pre-Schwarzian has no zeros, so +\(u=\log|F''/F'|\) is harmonic and \(u\ge-\log4-\log y\). +For \(0<\epsilon<1/2\), the harmonic function +\[ + b_\epsilon(x,y)=-\log4+ + \log(1/\epsilon)\cos(\pi x/2) + \frac{\sinh(\pi(1-y)/2)}{\sinh(\pi/2)} +\] +lies below \(u\) on the boundary of +\([-1,1]\times[\epsilon,1]\). +On its sides and top the added term is zero; on its bottom +the cosine-sinh factor is between zero and one. +The minimum principle gives \(u\ge b_\epsilon\) inside. +At \(i/2\), the coefficient of \(\log(1/\epsilon)\) is fixed +and positive, contradicting finiteness as \(\epsilon\downarrow0\). +\end{proof} + +\begin{proof}[Proof of Theorem~\ref{thm:main}] +Under \eqref{eq:contrary}, Equation~\eqref{eq:Jacobian} and +Proposition~\ref{prop:strict-transport} force +\(\beta=1/4\), \(k=3/4\), and \(J_F(i)=0\) almost surely. +Apply \eqref{eq:law} to the test \(|J_F(i)|\). +By \eqref{eq:G-action} and the strictly positive weight +\(|Q_F(z)|\), \(J_F(z)=0\) almost surely at every fixed \(z\). +A countable dense set and continuity give \(J_F\equiv0\) +for almost every \(F\), contradicting Lemma~\ref{lem:no-zero-J}. +Thus the single exponent in \eqref{eq:rho} satisfies \(\beta<1/4\). + +Choose \(b\) with \(\beta2\), our examples have unbounded block sensitivity and satisfy +\(\bs(f)\ge s(f)^\alpha\). +\end{abstract} +\input{sections/00-introduction} +\input{sections/01-construction} +\input{sections/02-recurrences} +\input{sections/03-separation} +\bibliographystyle{alpha} +\bibliography{references} +\end{document} diff --git a/preprints/A-superquadratic-separation-between-sensitivity-and-block-sensitivity-September-25-2026/build/references.bib b/preprints/A-superquadratic-separation-between-sensitivity-and-block-sensitivity-September-25-2026/build/references.bib new file mode 100644 index 0000000000000000000000000000000000000000..459d3ad0802ee08f72c4b29cc24094811fb66b86 --- /dev/null +++ b/preprints/A-superquadratic-separation-between-sensitivity-and-block-sensitivity-September-25-2026/build/references.bib @@ -0,0 +1,143 @@ +@inproceedings{Nisan1989, + author = {Nisan, Noam}, + title = {{CREW PRAMs} and Decision Trees}, + booktitle = {Proceedings of the Twenty-First Annual ACM Symposium on Theory of Computing}, + series = {STOC '89}, + pages = {327--335}, + publisher = {ACM}, + year = {1989}, + doi = {10.1145/73007.73038}, + note = {\href{https://doi.org/10.1145/73007.73038}{doi:10.1145/73007.73038}}, + url = {https://www.cs.huji.ac.il/~noam/crew.pdf} +} + +@article{Virza2011, + author = {Virza, Madars}, + title = {Sensitivity versus block sensitivity of {Boolean} functions}, + journal = {Information Processing Letters}, + volume = {111}, + number = {9}, + pages = {433--435}, + year = {2011}, + doi = {10.1016/j.ipl.2011.02.001}, + note = {\href{https://doi.org/10.1016/j.ipl.2011.02.001}{doi:10.1016/j.ipl.2011.02.001}}, + url = {https://doi.org/10.1016/j.ipl.2011.02.001} +} + +@article{NisanSzegedy1994, + author = {Nisan, Noam and Szegedy, Mario}, + title = {On the degree of {Boolean} functions as real polynomials}, + journal = {Computational Complexity}, + volume = {4}, + number = {4}, + pages = {301--313}, + year = {1994}, + doi = {10.1007/BF01263419}, + note = {\href{https://doi.org/10.1007/BF01263419}{doi:10.1007/BF01263419}}, + url = {https://doi.org/10.1007/BF01263419} +} + +@article{Huang2019, + author = {Huang, Hao}, + title = {Induced subgraphs of hypercubes and a proof of the {Sensitivity Conjecture}}, + journal = {Annals of Mathematics}, + volume = {190}, + number = {3}, + pages = {949--955}, + year = {2019}, + doi = {10.4007/annals.2019.190.3.6}, + note = {\href{https://doi.org/10.4007/annals.2019.190.3.6}{doi:10.4007/annals.2019.190.3.6}}, + url = {https://annals.math.princeton.edu/2019/190-3/p06} +} + +@article{Rubinstein1995, + author = {Rubinstein, David}, + title = {Sensitivity vs.\ block sensitivity of {Boolean} functions}, + journal = {Combinatorica}, + volume = {15}, + number = {2}, + pages = {297--299}, + year = {1995}, + doi = {10.1007/BF01200762}, + note = {\href{https://doi.org/10.1007/BF01200762}{doi:10.1007/BF01200762}}, + url = {https://doi.org/10.1007/BF01200762} +} + +@techreport{AmbainisSun2011, + author = {Ambainis, Andris and Sun, Xiaoming}, + title = {New separation between {$s(f)$} and {$bs(f)$}}, + institution = {Electronic Colloquium on Computational Complexity}, + number = {TR11-116}, + year = {2011}, + eprint = {1108.3494}, + archivePrefix = {arXiv}, + primaryClass = {cs.CC}, + note = {\href{https://arxiv.org/abs/1108.3494v1}{arXiv:1108.3494v1}}, + doi = {10.48550/arXiv.1108.3494}, + url = {https://arxiv.org/abs/1108.3494v1} +} + +@misc{Meiburg2026, + author = {Meiburg, Alexander}, + title = {{Block Sensitivity can exceed Spectral Sensitivity Squared}}, + year = {2026}, + eprint = {2608.00851}, + archivePrefix = {arXiv}, + primaryClass = {cs.CC}, + note = {\href{https://arxiv.org/html/2608.00851v1}{arXiv:2608.00851v1}, 1 August 2026}, + doi = {10.48550/arXiv.2608.00851}, + url = {https://arxiv.org/html/2608.00851v1} +} + +@article{Wellens2022, + author = {Wellens, Jake}, + title = {Relationships between the number of inputs and other complexity measures of {Boolean} functions}, + journal = {Discrete Analysis}, + year = {2022}, + note = {Paper No. 20, 21 pp.; arXiv:2005.00566v2}, + eprint = {2005.00566}, + archivePrefix = {arXiv}, + primaryClass = {math.CO}, + doi = {10.19086/da.57741}, + url = {https://arxiv.org/abs/2005.00566v2} +} + +@inproceedings{Tal2013, + author = {Tal, Avishay}, + title = {Properties and Applications of {Boolean} Function Composition}, + booktitle = {Proceedings of the 2013 ACM Conference on Innovations in Theoretical Computer Science}, + series = {ITCS '13}, + pages = {441--454}, + publisher = {ACM}, + year = {2013}, + doi = {10.1145/2422436.2422485}, + note = {\href{https://doi.org/10.1145/2422436.2422485}{doi:10.1145/2422436.2422485}}, + url = {https://web.vu.lt/mif/s.jukna/boolean/Tal2013.pdf} +} + + +@inproceedings{AaronsonEtAl2021, + author = {Aaronson, Scott and Ben-David, Shalev and Kothari, Robin and Rao, Shravas and Tal, Avishay}, + title = {Degree vs. Approximate Degree and Quantum Implications of {Huang}'s Sensitivity Theorem}, + booktitle = {Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing}, + series = {STOC '21}, + pages = {1330--1342}, + publisher = {ACM}, + year = {2021}, + doi = {10.1145/3406325.3451047}, + note = {\href{https://doi.org/10.1145/3406325.3451047}{doi:10.1145/3406325.3451047}}, + eprint = {2010.12629}, + archivePrefix = {arXiv}, + url = {https://arxiv.org/abs/2010.12629v1} +} + +@techreport{AmbainisPrusis2014, + author = {Ambainis, Andris and Pr{\=u}sis, Kri{\v{s}}j{\=a}nis}, + title = {A Tight Lower Bound on Certificate Complexity in Terms of Block Sensitivity and Sensitivity}, + institution = {Electronic Colloquium on Computational Complexity}, + number = {TR14-027}, + year = {2014}, + month = jul, + note = {Revision 1, July 31, 2014; \href{https://eccc.weizmann.ac.il/report/2014/027/revision/1/download}{revised report}}, + url = {https://eccc.weizmann.ac.il/report/2014/027/revision/1/download} +} diff --git a/preprints/A-superquadratic-separation-between-sensitivity-and-block-sensitivity-September-25-2026/build/sections/00-introduction.tex b/preprints/A-superquadratic-separation-between-sensitivity-and-block-sensitivity-September-25-2026/build/sections/00-introduction.tex new file mode 100644 index 0000000000000000000000000000000000000000..c0130588575857a7a68e3ff2dcc016dac5026bf4 --- /dev/null +++ b/preprints/A-superquadratic-separation-between-sensitivity-and-block-sensitivity-September-25-2026/build/sections/00-introduction.tex @@ -0,0 +1,139 @@ +\section{Introduction} +\label{sec:introduction} + +Sensitivity measures how many individual input bits can change a Boolean +function's value; block sensitivity allows disjoint sets of bits to be +flipped instead. We construct total Boolean functions for which the second +quantity grows faster than any constant multiple of the square of the first. + +For a positive integer \(n\), write \([n]=\{1,\ldots,n\}\). +If \(x\in\{0,1\}^n\) and \(B\subseteq[n]\), let \(x^B\) be obtained +by flipping precisely the coordinates in \(B\); abbreviate +\(x^{\{a\}}\) to \(x^a\). +For a total Boolean function \(f:\{0,1\}^n\to\{0,1\}\), define +\[ + s(f,x)=\bigl|\{a\in[n]:f(x^a)\ne f(x)\}\bigr|, + \qquad s(f)=\max_x s(f,x). +\] +Its block sensitivity \(\bs(f,x)\) is the largest number of pairwise +disjoint nonempty sets \(B_1,\ldots,B_m\subseteq[n]\) such that +\(f(x^{B_j})\ne f(x)\) for every \(j\), and +\(\bs(f)=\max_x\bs(f,x)\). + +The Sensitivity Conjecture asked whether block sensitivity is bounded by a +polynomial in sensitivity, uniformly over the number of inputs and the +function. The question goes back to Nisan's work on parallel computation, +which introduced block sensitivity and already recorded Rubinstein's +quadratic separation \cite[Section~2]{Nisan1989}. +Nisan and Szegedy studied the problem through the degree of the unique +multilinear real polynomial representing a Boolean function, and explicitly +suggested the quadratic inequality \(\bs(f)\le s(f)^2\) +\cite[Section~4]{NisanSzegedy1994}. +Rubinstein's construction \cite{Rubinstein1995} showed that a polynomial +upper bound would need degree at least two. Virza improved the separation +to \(\bs(f)=\tfrac12s(f)^2+\tfrac12s(f)\) \cite{Virza2011}, and +Ambainis and Sun subsequently obtained examples satisfying +\[ + \bs(f)=\frac23s(f)^2-\frac13s(f) +\] +\cite[Theorem~1]{AmbainisSun2011}. +These improvements left open whether any universal quadratic upper bound +could hold. + +Huang resolved the polynomial conjecture in 2019. His degree bound, +together with Tal's refinement of the degree--block-sensitivity comparison +\cite{Tal2013}, gives \(\bs(f)\le s(f)^4\) for every total Boolean function +\cite[Theorem~1.5]{Huang2019}. Wellens later improved its leading constant, +proving +\(\bs(f)\le\sqrt{2/3}\,s(f)^4+1\) +\cite[Corollary~4.2]{Wellens2022}. +These results settle the polynomial question. The theorem below rules out +its quadratic strengthening. + +\begin{theorem}\label{thm:main} +For every real \(C>0\), there are a positive integer \(n\) and a +nonconstant total Boolean function \(f:\{0,1\}^n\to\{0,1\}\) such that +\[ + \bs(f)>C\,s(f)^2. +\] +More precisely, for every integer \(d\ge1\) the construction below gives +a nonconstant total Boolean function \(f\) with +\[ + \frac{\bs(f)}{s(f)^2}\ge \frac{2^d}{4(d+2)^2}. +\] +\end{theorem} + +\paragraph{Method and relation to spectral sensitivity.} +Meiburg's spectral-sensitivity construction uses tournament-indexed rows +of positive conditions and one selected blocking coordinate in each +outgoing row \cite[Sections~3.1 and~5.2--5.5]{Meiburg2026}.\footnote{In the +attribution accompanying this construction, Meiburg credits GPT-5.6-Sol +with the gated-tournament ideas and describes his own contributions as +exposition, size and parameter optimization, and checking the faithfulness +of the Lean formalization.} +Random labels exclude small local configurations in which many accepting +subcubes lie close to the same accepting input. We adapt this architecture +to nested predicates, using a different forbidden-label condition and a +direct union bound. Simultaneous changes of two predicates then control +ordinary sensitivity under recursion. + +The distinction between the two sensitivity measures matters here. +The sensitivity graph joins adjacent cube inputs exactly when the function +changes value. Its maximum degree is \(s(f)\); spectral sensitivity +\(\lambda(f)\) is the operator norm of its adjacency matrix +\cite{AaronsonEtAl2021}. +Since \(\lambda(f)\le s(f)\), a lower bound on +\(\bs(f)/\lambda(f)^2\) alone does not supply the ordinary-sensitivity +separation in Theorem~\ref{thm:main}. +The needed estimates are proved directly below, without a spectral input. + +\paragraph{Construction and proof strategy.} +We recursively construct nested families of Boolean predicates: their +accepting sets decrease as the predicate index increases. At a recursive +step, the input is partitioned into disjoint child inputs arranged in +rows, and every pair of rows is given one directed edge. Each edge is +labelled by a position in its destination row. A clause associated with +row \(i\) has target conditions requiring the strongest child predicate +to accept every child in that row. For each edge from \(i\) to another row, +it also requires a weaker child predicate to reject the child selected by +the edge label. The parent predicate accepts when one of these clauses +holds. Varying the weaker predicate used in these gate conditions produces +a new nested family. + +Fix one parent predicate at a rejecting input, and consider clauses that +can become satisfied after one bit changes. A bit belongs to only one child +and hence affects only one condition of a fixed clause. The candidate +clauses therefore fail exactly one target condition or one gate condition, +and no other condition. The edge labels bound the number of candidates of +the first kind. The tournament allows at most three candidates of the +second kind, and all but at most one have their failed gate in a row whose +strongest child predicates all accept. Repairing such a gate must reverse +both its weaker predicate and the strongest predicate on the same child. +We consequently track the number of bits that reverse a fixed pair of +distinct predicates simultaneously, in addition to ordinary sensitivity. +For a gate repair that makes a clause hold for two rejecting parent +predicates, the two corresponding child gate predicates must both change, +even in the possible exceptional clause. + +The initial family consists of indicators of unions of Hamming balls +with successive integer radii around separated centres. One bit cannot +cross two of these radii, so all its joint sensitivities vanish. +Section~\ref{sec:construction} gives this family, the required edge +labelling, and common disjoint sensitive blocks at zero. +Section~\ref{sec:recurrences} then proves the four coupled sensitivity +bounds for arbitrary inputs. With a size parameter \(M\) chosen +sufficiently large for each fixed depth, their normalization in +Section~\ref{sec:separation} bounds sensitivity growth by essentially +a factor \(M\) per level, while the number of disjoint sensitive blocks +grows by \(2M^2+1\). An OR of disjoint copies balances the sensitivities +at zero and one, following the principle used by Ambainis and Sun +\cite[Section~4, Lemma~1]{AmbainisSun2011}. + +Finally, Lemma~\ref{lem:composition} amplifies a fixed seed with +\(f(0)=0\) and \(\bs(f,0)>s(f)^2\) into the fixed power separation of +Corollary~\ref{cor:power-separation}. Ambainis and Pr\={u}sis explicitly +noted that a seed with block sensitivity greater than sensitivity squared +would yield a superquadratic power separation by iteration +\cite[Section~1]{AmbainisPrusis2014}. Composition as an amplification method +is treated systematically by Tal~\cite{Tal2013}; we prove the sensitivity +upper bound and zero-input block lower bound used here. diff --git a/preprints/A-superquadratic-separation-between-sensitivity-and-block-sensitivity-September-25-2026/build/sections/01-construction.tex b/preprints/A-superquadratic-separation-between-sensitivity-and-block-sensitivity-September-25-2026/build/sections/01-construction.tex new file mode 100644 index 0000000000000000000000000000000000000000..de207d277f48fef7956018d61b26038580c0d19a --- /dev/null +++ b/preprints/A-superquadratic-separation-between-sensitivity-and-block-sensitivity-September-25-2026/build/sections/01-construction.tex @@ -0,0 +1,193 @@ +\section{The labelled tournament and nested predicates} +\label{sec:construction} + +We construct the recursive predicates and the blocks that will witness their +block sensitivity at zero. The sensitivity estimates require control at +every input, so we also define the ordinary and joint profiles passed from +one level to the next. + +Fix integers \(d\ge1\) and \(M\ge3\), and put +\[ + L=d+2,\qquad k=2M^2+1,\qquad r=\lceil\sqrt M\rceil, + \qquad t=16. +\] +We will choose \(M\) as a function of \(d\) in +Section~\ref{sec:separation}. +A tournament is an orientation of every pair of distinct vertices: +exactly one of \(i\to j\) and \(j\to i\) holds. +Use a tournament on \([k]\) in which every vertex has outdegree \(M^2\). +For example, place the vertices cyclically and direct an edge from +each vertex to the next \(M^2\) vertices. + +The tournament rows and selected blocking coordinates adapt the architecture +of Meiburg~\cite[Sections~3.1 and~5.2--5.5]{Meiburg2026}. +The specific label property needed for the ordinary-sensitivity argument is +the following elementary consequence of independent random labels. + +\begin{lemma}\label{lem:labelling} +The edges can be labelled by numbers \(a(i,j)\in[r]\) so that there is +no set \(I\subseteq[k]\) of size \(t\) admitting choices +\(m_j\in[r]\), \(j\in I\), with +\[ + a(i,j)=m_j \qquad\text{for every edge }i\to j\text{ within }I. +\] +\end{lemma} +\begin{proof} +Choose all edge labels independently and uniformly from \([r]\). +For fixed \(I\) and fixed choices \((m_j)_{j\in I}\), the probability +of all the stated equalities is \(r^{-\binom t2}\). +There are at most \(k^t r^t\) choices to consider, so the probability +of any violation is at most +\[ + k^{16}r^{-104}\le (3M^2)^{16}(\sqrt M)^{-104} + =3^{16}M^{-20}<1. +\] +Here \(k\le3M^2\), \(r\ge\sqrt M\), and \(M\ge3\). +Thus a labelling with the required property exists. +\end{proof} + +Fix such a labelling for the remainder of the construction. +For \(0\le\ell\le d\), set +\[ + H_\ell=d+1-\ell,\qquad N_\ell=LM(kr)^\ell. +\] +We construct predicates \(P_{\ell,q}:\{0,1\}^{N_\ell}\to\{0,1\}\), +\(1\le q\le H_\ell\), that are nested in the sense that +\[ + P_{\ell,1}(x)\ge P_{\ell,2}(x)\ge\cdots + \ge P_{\ell,H_\ell}(x)\quad\text{for every }x. +\] +Throughout, nesting concerns the predicate index, not the input bits. + +\subsection{Initial predicates} +Partition \([LM]\) into \(M\) blocks \(E_1,\ldots,E_M\), each of +size \(L\). Let \(z_j\) be the string that is one precisely on \(E_j\). +Write \(\operatorname{dist}(x,z)\) for Hamming distance and set +\[ + D(x)=\min_{j\in[M]}\operatorname{dist}(x,z_j),\qquad + P_{0,q}(x)=\mathbf1\{D(x)\le H_0-q\}. +\] +These predicates are nested. Their radii are the integers from \(d\) +down to zero, while distinct centres have distance \(2L=2d+4\). + +\subsection{Recursive clauses} +Suppose \(1\le\ell\le d\). Partition the input into \(kr\) disjoint +children \(x_{j,c}\in\{0,1\}^{N_{\ell-1}}\), arranged in \(k\) rows +\(j\in[k]\), each with \(r\) positions \(c\in[r]\). +For this level put \(H=H_\ell\) and \(Q=H+1=H_{\ell-1}\). +For \(q\in[H]\), let clause \(i\) at index \(q\) assert +\begin{equation}\label{eq:predicate-definition} + \begin{aligned} + P_{\ell-1,Q}(x_{i,c})&=1 &&\text{for all }c\in[r],\\ + P_{\ell-1,Q-q}(x_{j,a(i,j)})&=0 &&\text{for all }i\to j. + \end{aligned} +\end{equation} +Call the first line its \emph{target conditions} and the second line +its \emph{gate conditions}. Figure~\ref{fig:clause} depicts the target row +and one outgoing gate. Define \(P_{\ell,q}(x)=1\) if at least +one clause is satisfied, and zero otherwise. All indices are valid: +\(1\le Q-q\le H=Latex, every node/.style={font=\small}, + condition/.style={draw,rounded corners=2pt,inner sep=8pt,align=center}] + \node[condition] (target) at (0,0) + {Row \(i\): all \(r\) children\\[2pt] + \(P_{\ell-1,Q}(x_{i,c})=1\)}; + \node[condition] (gate) at (6,0) + {Row \(j\): child \(a(i,j)\)\\[2pt] + \(P_{\ell-1,Q-q}(x_{j,a(i,j)})=0\)}; + \draw[->] (target) -- node[above]{\(i\to j\)} (gate); + \node[below=5pt of target,align=center] {One target condition\\per child in row \(i\)}; + \node[below=5pt of gate,align=center] {One gate condition\\per outgoing neighbour \(j\)}; +\end{tikzpicture} +\caption{One recursive clause. The edge label selects a single child in +each outgoing neighbour's row. Raising \(q\) strengthens the gates; +all targets use the strongest child predicate \(Q\).} +\label{fig:clause} +\end{figure} + +\subsection{Sensitivity profiles and the initial bounds} +For \(b\in\{0,1\}\), define +\[ + S_{\ell,b}=\max_{\substack{1\le q\le H_\ell,\ x\in\{0,1\}^{N_\ell}\\ + P_{\ell,q}(x)=b}} + \bigl|\{a\in[N_\ell]:P_{\ell,q}(x^a)=1-b\}\bigr|. +\] +We also keep track of simultaneous changes of two distinct predicates: +\[ + J_{\ell,b}=\max_{\substack{1\le qd\), and the blocks \(E_j\) have the stated +property because \(0^{E_j}=z_j\). +Suppose the claim holds at level \(\ell-1\). At the all-zero input, +every target predicate is zero, so every clause fails at level \(\ell\). + +Enumerate the preceding-level block family as +\(B_1,\ldots,B_{Mk^{\ell-1}}\). +For each row \(i\) and each family index \(h\), take the union of the +copies of \(B_h\) in all \(r\) children of row \(i\). +Flipping this union makes the strongest predicate in each of those +children equal to one, so all targets for clause \(i\) hold. +All gate children of clause \(i\) lie in other rows and remain zero, +so its gates hold for every \(q\). +Thus every \(P_{\ell,q}\) is one after the flip. +Within a row these unions are disjoint because the preceding-level +blocks are disjoint in every child; unions from different rows are +also disjoint. There are \(k\cdot Mk^{\ell-1}=Mk^\ell\) of them, +each of size \(r\cdot Lr^{\ell-1}=Lr^\ell\), as required. +\end{proof} diff --git a/preprints/A-superquadratic-separation-between-sensitivity-and-block-sensitivity-September-25-2026/build/sections/02-recurrences.tex b/preprints/A-superquadratic-separation-between-sensitivity-and-block-sensitivity-September-25-2026/build/sections/02-recurrences.tex new file mode 100644 index 0000000000000000000000000000000000000000..7bdec637579b49049319b36528c109d0d1e0bdfd --- /dev/null +++ b/preprints/A-superquadratic-separation-between-sensitivity-and-block-sensitivity-September-25-2026/build/sections/02-recurrences.tex @@ -0,0 +1,149 @@ +\section{Sensitivity recurrences} +\label{sec:recurrences} + +The joint profiles bound simultaneous changes of a fixed pair of predicates +on the same child. The following estimates hold for every input, including +inputs far from the blocks used in Lemma~\ref{lem:blocks}. + +\begin{proposition}\label{prop:recurrences} +For $1\leq \ell\leq d$, write +$S'_b=S_{\ell-1,b}$ and $J'_b=J_{\ell-1,b}$. Then +\begin{equation}\label{eq:recurrences} +\begin{aligned} + S_{\ell,1}&\leq M^2S'_0+rS'_1, + &\qquad J_{\ell,1}&\leq M^2J'_0+rS'_1,\\ + S_{\ell,0}&\leq tS'_0+S'_1+3J'_1, + &\qquad J_{\ell,0}&\leq tS'_0+3J'_1. +\end{aligned} +\end{equation} +\end{proposition} + +\begin{proof} +Fix $\ell$, put $H=H_\ell$ and $Q=H+1$, and recall +Equation~\eqref{eq:predicate-definition}. For predicate index $q$, every +target tests $P_{\ell-1,Q}=1$ and every gate tests +$P_{\ell-1,Q-q}=0$ on its specified child. The targets of clause $i$ +lie in row $i$, and its gates lie in distinct other rows. Thus each +condition of a fixed clause uses a different child. + +All flips below are flips of individual coordinates of the original +Boolean input. Such a flip affects exactly one child. It may change +several nested predicates of that child; we make no restriction on the +number of thresholds it crosses. Nevertheless, at a fixed parent index +it can change at most one condition of any fixed clause. A child may +occur in several different clauses, so when counting possible flips +over clauses we use a union bound. + +\emph{Transitions from one.} +Fix an input where $P_{\ell,q}=1$, and choose a clause satisfied there. +Every flip taking this predicate to zero must break that fixed clause. +It either changes a gate predicate from zero to one or changes a target +predicate from one to zero. There are $M^2$ gate children and $r$ +target children, giving at most $M^2S'_0+rS'_1$ such flips. + +Next fix $q0$. Combining +\eqref{eq:final-sensitivity} and~\eqref{eq:final-block-sensitivity} yields +\begin{equation} +\label{eq:ratio-separation} + \frac{\bs(f)}{s(f)^2} + \geq\frac{\bs(f,0)}{s(f)^2} + \geq\frac{(k/M^2)^d}{4L^2} + \geq\frac{2^d}{4(d+2)^2}. +\end{equation} +The last expression tends to infinity as $d$ tends to infinity. +For any proposed constant $C$, first choose $d$ so that this expression +exceeds $C$, then choose $M\geq9^{d+1}$, then choose labels supplied by +Lemma~\ref{lem:labelling} and form $f$ as above. +The resulting finite-dimensional total nonconstant function satisfies +$\bs(f)>C s(f)^2$. Hence no universal quadratic constant exists. +\end{proof} + +\subsection{A fixed exponent by composition} + +The construction also supplies a fixed seed whose block sensitivity at +zero exceeds the square of its sensitivity. Repeated composition of that +seed gives a power separation. This is the composition-and-powering +principle studied by Tal~\cite{Tal2013}. We include the elementary composition facts +to make this additional consequence self-contained. + +\begin{lemma} +\label{lem:composition} +Let $f:\{0,1\}^a\to\{0,1\}$ and +$g:\{0,1\}^b\to\{0,1\}$ be total Boolean functions. +Their disjoint composition is the function on $ab$ bits defined by +\[ + (f\circ g)(x_1,\ldots,x_a) + =f\bigl(g(x_1),\ldots,g(x_a)\bigr), + \qquad x_i\in\{0,1\}^b. +\] +Then +\[ + s(f\circ g)\leq s(f)s(g). +\] +If $g(0)=0$, then $(f\circ g)(0)=f(0)$ and +\[ + \bs(f\circ g,0)\geq\bs(f,0)\bs(g,0). +\] +More explicitly, any disjoint families of $p$ and $q$ blocks sensitive at +zero for $f$ and $g$, respectively, yield a disjoint family of $pq$ blocks +sensitive at zero for $f\circ g$. +\end{lemma} + +\begin{proof} +For an input $x=(x_1,\ldots,x_a)$, put +$y=(g(x_1),\ldots,g(x_a))$. +A bit in child $x_i$ is sensitive for $f\circ g$ precisely when it is +sensitive for $g$ at $x_i$ and coordinate $i$ is sensitive for $f$ at $y$. +Thus +\[ + s(f\circ g,x) + =\sum_{i:\,f(y^{\{i\}})\ne f(y)}s(g,x_i) + \leq s(f,y)s(g) + \leq s(f)s(g). +\] + +Now suppose $g(0)=0$. Let $C_1,\ldots,C_p\subseteq[a]$ and +$D_1,\ldots,D_q\subseteq[b]$ be disjoint nonempty sensitive blocks for +$f$ and $g$ at zero. Identify the coordinates of the composition with +$[a]\times[b]$. For each $1\leq i\leq p$ and $1\leq j\leq q$, set +\[ + E_{i,j}=C_i\times D_j. +\] +These blocks are nonempty and pairwise disjoint: different $i$ use +disjoint sets of children, while for fixed $i$ different $j$ use +disjoint coordinate sets in each child. +Flipping $E_{i,j}$ at zero makes precisely the children indexed by $C_i$ +output one, because $g(0)=0$ and $D_j$ is sensitive there. +The outer input therefore becomes $0^{C_i}$, which changes the value of +$f$. Hence every $E_{i,j}$ is sensitive for $f\circ g$ at zero. +Choosing maximum families proves the stated block-sensitivity inequality; +if either maximum is zero, the inequality is immediate. +\end{proof} + +\begin{corollary} +\label{cor:power-separation} +There exist a real number $\alpha>2$ and nonconstant total Boolean +functions $F_m$, for integers $m\geq1$, such that +\[ + \bs(F_m,0)\geq s(F_m)^\alpha + \qquad\text{for every }m, + \qquad + \bs(F_m,0)\longrightarrow\infty. +\] +\end{corollary} + +\begin{proof} +Fix $d=9$, choose an integer $M\geq9^{10}$, and choose any admissible +labelling. Let $f$ be the function constructed in the proof of +Theorem~\ref{thm:main} for these fixed parameters. Put +\[ + A=22M^{10},\qquad B=M^2k^9. +\] +Here $L=11$, so~\eqref{eq:final-sensitivity} and +\eqref{eq:final-block-sensitivity} give +$s(f)\leq A$ and $\bs(f,0)\geq B$, while $f(0)=0$. +Moreover, +\[ + \frac{B}{A^2} + =\frac{(k/M^2)^9}{484} + \geq\frac{2^9}{484} + =\frac{128}{121}>1. +\] +In particular, $A>1$, $B>A^2$, and the fixed number +\[ + \alpha=\frac{\log B}{\log A} +\] +is greater than two. + +Set $F_1=f$ and, for $m\geq1$, set $F_{m+1}=f\circ F_m$ using disjoint +copies of $F_m$. Inductively, $F_m$ is a total function on $n^m$ bits, +where $n$ is the fixed seed dimension in~\eqref{eq:final-dimension}, +and $F_m(0)=0$. +Lemma~\ref{lem:composition} gives +\[ + s(F_m)\leq A^m, + \qquad + \bs(F_m,0)\geq B^m. +\] +The latter bound proves nonconstancy and tends to infinity. Since +$A^\alpha=B$, we also obtain +\[ + s(F_m)^\alpha\leq A^{m\alpha}=B^m\leq\bs(F_m,0). +\] +The same estimates give the explicit ratio bound +\[ + \frac{\bs(F_m)}{s(F_m)^2} + \geq\left(\frac{B}{A^2}\right)^m + \geq\left(\frac{128}{121}\right)^m.\qedhere +\] +\end{proof} diff --git a/preprints/A-superquadratic-separation-between-sensitivity-and-block-sensitivity-September-25-2026/paper.pdf b/preprints/A-superquadratic-separation-between-sensitivity-and-block-sensitivity-September-25-2026/paper.pdf new file mode 100644 index 0000000000000000000000000000000000000000..a85409e3049776597e8ca037c2412e76f337f79a --- /dev/null +++ b/preprints/A-superquadratic-separation-between-sensitivity-and-block-sensitivity-September-25-2026/paper.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:a589970798bc590e5a27cb35cfa3092a355a4048cfa383bc369177289a00a1f8 +size 370807 diff --git a/preprints/A-surface-counterexample-to-Shafarevich-holomorphic-convexity-September-23-2026/README.md b/preprints/A-surface-counterexample-to-Shafarevich-holomorphic-convexity-September-23-2026/README.md new file mode 100644 index 0000000000000000000000000000000000000000..2d5f88b0bbde815f3c158b8cbc7872cd2e7ed405 --- /dev/null +++ b/preprints/A-surface-counterexample-to-Shafarevich-holomorphic-convexity-September-23-2026/README.md @@ -0,0 +1,18 @@ +# [A surface counterexample to Shafarevich holomorphic convexity](paper.pdf) + +**Author:** OpenAI + +**Date:** September 23, 2026 + +## Citation + +```bibtex +@misc{OAI:A-surface-counterexample-to-Shafarevich-holomorphic-convexity-September-23-2026, + author = {{OpenAI}}, + title = {{A surface counterexample to Shafarevich holomorphic convexity}}, + howpublished = {OpenAI Math Release preprint + \href{https://github.com/openai/math/blob/main/preprints/A-surface-counterexample-to-Shafarevich-holomorphic-convexity-September-23-2026/paper.pdf}{OAI:A-surface-counterexample-to-Shafarevich-holomorphic-convexity-September-23-2026}}, + year = {2026} +} +``` + diff --git a/preprints/A-surface-counterexample-to-Shafarevich-holomorphic-convexity-September-23-2026/build/main.tex b/preprints/A-surface-counterexample-to-Shafarevich-holomorphic-convexity-September-23-2026/build/main.tex new file mode 100644 index 0000000000000000000000000000000000000000..c5389e23193d4b7f882073955ef94602f5825b0f --- /dev/null +++ b/preprints/A-surface-counterexample-to-Shafarevich-holomorphic-convexity-September-23-2026/build/main.tex @@ -0,0 +1,61 @@ +\documentclass[11pt]{article} +\pdfinfoomitdate=1 +\pdftrailerid{} +\pdfsuppressptexinfo=15 +\usepackage[T1]{fontenc} +\usepackage{lmodern} +\usepackage[a4paper,margin=29mm]{geometry} +\usepackage{amsmath,amssymb,amsthm,mathtools} +\usepackage{needspace,etoolbox} +\usepackage{microtype} +\usepackage{tikz} +\usetikzlibrary{arrows.meta,positioning,calc,decorations.pathreplacing} +\usepackage[numbers,sort&compress]{natbib} +\usepackage{xurl} +\usepackage[colorlinks=true,linkcolor=blue!45!black,citecolor=blue!45!black,urlcolor=blue!45!black]{hyperref} +\usepackage[nameinlink,noabbrev,capitalise]{cleveref} +\newtheorem{theorem}{Theorem}[section] +\newtheorem{proposition}[theorem]{Proposition} +\newtheorem{lemma}[theorem]{Lemma} +\newtheorem{corollary}[theorem]{Corollary} +\theoremstyle{definition} +\newtheorem{definition}[theorem]{Definition} +\theoremstyle{remark} +\newtheorem{remark}[theorem]{Remark} +\AtBeginEnvironment{theorem}{\Needspace{4\baselineskip}} +\AtBeginEnvironment{proposition}{\Needspace{4\baselineskip}} +\AtBeginEnvironment{lemma}{\Needspace{4\baselineskip}} +\newcommand{\C}{\mathbb C} +\newcommand{\F}{\mathbb F} +\newcommand{\Z}{\mathbb Z} +\newcommand{\PP}{\mathbb P} +\DeclareMathOperator{\im}{im} +\DeclareMathOperator{\Aut}{Aut} +\DeclareMathOperator{\End}{End} +\DeclareMathOperator{\id}{id} +\DeclareMathOperator{\cl}{cl} +\setlength{\emergencystretch}{2em} +\hyphenation{OpenAI} +\setcounter{tocdepth}{2} +\hypersetup{pdftitle={A surface counterexample to Shafarevich holomorphic convexity},pdfauthor={OpenAI}} +\title{A surface counterexample to Shafarevich holomorphic convexity} +\author{OpenAI} +\date{September 23, 2026} +\begin{document} +\maketitle +\begin{abstract} +We construct a smooth connected projective complex surface whose universal +cover is not holomorphically convex. This disproves the Shafarevich +conjecture on holomorphic convexity in complex dimension two. +\end{abstract} +\tableofcontents +\clearpage +\input{sections/01-introduction} +\input{sections/02-weighted} +\input{sections/03-family} +\input{sections/04-involution} +\input{sections/05-quotient} +\input{sections/06-surface} +\bibliographystyle{plainnat} +\bibliography{references} +\end{document} diff --git a/preprints/A-surface-counterexample-to-Shafarevich-holomorphic-convexity-September-23-2026/build/references.bib b/preprints/A-surface-counterexample-to-Shafarevich-holomorphic-convexity-September-23-2026/build/references.bib new file mode 100644 index 0000000000000000000000000000000000000000..52f77e753d8c18134c720af8f24896078b6fd59c --- /dev/null +++ b/preprints/A-surface-counterexample-to-Shafarevich-holomorphic-convexity-September-23-2026/build/references.bib @@ -0,0 +1,310 @@ +@article{BogomolovKatzarkov1998, + author = {Bogomolov, Fedor and Katzarkov, Ludmil}, + title = {Complex projective surfaces and infinite groups}, + journal = {Geometric and Functional Analysis}, + volume = {8}, + number = {2}, + year = {1998}, + pages = {243--272}, + doi = {10.1007/s000390050055}, + url = {https://arxiv.org/abs/alg-geom/9703002v2} +} + +@misc{OpenAIAbelianity2026, + author = {{OpenAI}}, + title = {{The abelianity conjecture for special compact K\"ahler manifolds}}, + howpublished = {OpenAI Math Release preprint + \href{https://github.com/openai/math/blob/main/preprints/The-abelianity-conjecture-for-special-compact-Kahler-manifolds-September-23-2026/paper.pdf}{OAI:The-abelianity-conjecture-for-special-compact-Kahler-manifolds-September-23-2026}}, + year = {2026} +} + +@misc{OpenAISemialgebraicCovers2026, + author = {{OpenAI}}, + title = {{Semialgebraic universal covers of normal projective varieties}}, + howpublished = {OpenAI Math Release preprint + \href{https://github.com/openai/math/blob/main/preprints/Semialgebraic-universal-covers-of-normal-projective-varieties-September-24-2026/paper.pdf}{OAI:Semialgebraic-universal-covers-of-normal-projective-varieties-September-24-2026}}, + year = {2026} +} + +@article{EyssidieuxKatzarkovPantevRamachandran2012, + author = {Eyssidieux, Philippe and Katzarkov, Ludmil and Pantev, Tony and Ramachandran, Mohan}, + title = {Linear {Shafarevich} conjecture}, + journal = {Annals of Mathematics}, + volume = {176}, + number = {3}, + year = {2012}, + pages = {1545--1581}, + doi = {10.4007/annals.2012.176.3.4}, + url = {https://annals.math.princeton.edu/2012/176-3/p04} +} + + +@article{GolodShafarevich1964, + author = {Golod, E. S. and Shafarevich, I. R.}, + title = {On the class field tower}, + journal = {Izvestiya Akademii Nauk SSSR. 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B.}, + title = {On the theorem concerning the infinite-dimensionality of an associative algebra}, + journal = {Izvestiya Akademii Nauk SSSR. Seriya Matematicheskaya}, + volume = {29}, + number = {1}, + year = {1965}, + pages = {209--214}, + note = {In Russian}, + url = {https://www.mathnet.ru/eng/im2905} +} + +@article{Golod1964, + author = {Golod, E. S.}, + title = {On nil-algebras and finitely approximable {$p$}-groups}, + journal = {Izvestiya Akademii Nauk SSSR. Seriya Matematicheskaya}, + volume = {28}, + number = {2}, + year = {1964}, + pages = {273--276}, + note = {In Russian}, + url = {https://www.mathnet.ru/eng/im2956} +} + + +@book{Shafarevich1974, + author = {Shafarevich, Igor R.}, + title = {Basic Algebraic Geometry}, + publisher = {Springer-Verlag}, + address = {Berlin, Heidelberg}, + year = {1974}, + doi = {10.1007/978-3-642-96200-4}, + url = {https://link.springer.com/book/10.1007/978-3-642-96200-4} +} + +@article{LasellRamachandran1996, + author = {Lasell, Brendon and Ramachandran, Mohan}, + title = {Observations on harmonic maps and singular varieties}, + journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure}, + series = {4}, + volume = {29}, + number = {2}, + year = {1996}, + pages = {135--148}, + doi = {10.24033/asens.1737}, + url = {https://www.numdam.org/item/ASENS_1996_4_29_2_135_0/} +} + +@article{KatzarkovRamachandran1998, + author = {Katzarkov, Ludmil and Ramachandran, Mohan}, + title = {On the universal coverings of algebraic surfaces}, + journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure}, + series = {4}, + volume = {31}, + number = {4}, + year = {1998}, + pages = {525--535}, + doi = {10.1016/S0012-9593(98)80105-5}, + url = {https://www.numdam.org/item/ASENS_1998_4_31_4_525_0/} +} + +@article{GurjarShastri1985, + author = {Gurjar, R. V. and Shastri, A. R.}, + title = {Covering spaces of an elliptic surface}, + journal = {Compositio Mathematica}, + volume = {54}, + number = {1}, + year = {1985}, + pages = {95--104}, + url = {https://www.numdam.org/item/CM_1985__54_1_95_0/} +} + +@article{Katzarkov1997, + author = {Katzarkov, Ludmil}, + title = {Nilpotent groups and universal coverings of smooth projective varieties}, + journal = {Journal of Differential Geometry}, + volume = {45}, + number = {2}, + year = {1997}, + pages = {336--348}, + doi = {10.4310/jdg/1214459801}, + eprint = {alg-geom/9510009}, + archivePrefix = {arXiv}, + url = {https://doi.org/10.4310/jdg/1214459801} +} + +@article{Eyssidieux2004, + author = {Eyssidieux, Philippe}, + title = {Sur la convexit\'e holomorphe des rev\^etements lin\'eaires r\'eductifs d'une vari\'et\'e projective alg\'ebrique complexe}, + journal = {Inventiones mathematicae}, + volume = {156}, + number = {3}, + year = {2004}, + pages = {503--564}, + doi = {10.1007/s00222-003-0345-0}, + url = {https://link.springer.com/article/10.1007/s00222-003-0345-0} +} + +@article{CampanaClaudonEyssidieux2015, + author = {Campana, Fr\'ed\'eric and Claudon, Beno\^{i}t and Eyssidieux, Philippe}, + title = {Repr\'esentations lin\'eaires des groupes k\"ahl\'eriens : factorisations et conjecture de {Shafarevich} lin\'eaire}, + journal = {Compositio Mathematica}, + volume = {151}, + number = {2}, + year = {2015}, + pages = {351--376}, + doi = {10.1112/S0010437X14007751}, + url = {https://doi.org/10.1112/S0010437X14007751} +} + +@misc{EyssidieuxFunar, + author = {Eyssidieux, Philippe and Funar, Louis}, + title = {Orbifold {K\"ahler} groups related to mapping class groups}, + year = {2025}, + howpublished = {Author-hosted preprint}, + note = {Revision dated September 30, 2025; initial version arXiv:2112.06726 (2021)}, + url = {https://www-fourier.univ-grenoble-alpes.fr/~funar/mcgokahgpv25.pdf} +} + +@misc{DengYamanoi2024, + author = {Deng, Ya and Yamanoi, Katsutoshi}, + title = {Reductive {Shafarevich} conjecture}, + year = {2024}, + eprint = {2306.03070v2}, + archivePrefix = {arXiv}, + primaryClass = {math.AG}, + doi = {10.48550/arXiv.2306.03070}, + note = {Version 2, May 29, 2024; with an appendix joint with Ludmil Katzarkov}, + url = {https://arxiv.org/abs/2306.03070v2} +} + +@misc{BakkerBrunebarbeTsimerman2026, + author = {Bakker, Benjamin and Brunebarbe, Yohan and Tsimerman, Jacob}, + title = {The linear {Shafarevich} conjecture for quasiprojective varieties and algebraicity of {Shafarevich} morphisms}, + year = {2026}, + howpublished = {Author-hosted preprint}, + note = {Revision dated September 9, 2026, 134 pp.; initial version arXiv:2408.16441 (2024)}, + url = {https://benjamin-bakker.github.io/Shafarevich.pdf} +} + +@article{ColtoiuJoita2011, + author = {Col{\c{t}}oiu, Mihnea and Joi{\c{t}}a, Cezar}, + title = {The disk property of coverings of 1-convex surfaces}, + journal = {Proceedings of the American Mathematical Society}, + volume = {140}, + number = {2}, + year = {2012}, + pages = {575--580}, + eprint = {1104.1077v1}, + archivePrefix = {arXiv}, + primaryClass = {math.CV}, + url = {https://arxiv.org/abs/1104.1077v1} +} + +@article{Campana1994, + author = {Campana, Fr{\'e}d{\'e}ric}, + title = {Remarques sur le rev{\^e}tement universel des vari{\'e}t{\'e}s k{\"a}hl{\'e}riennes compactes}, + journal = {Bulletin de la Soci{\'e}t{\'e} Math{\'e}matique de France}, + volume = {122}, + number = {2}, + year = {1994}, + pages = {255--284}, + doi = {10.24033/bsmf.2232}, + url = {https://www.numdam.org/item/10.24033/bsmf.2232.pdf} +} + +@article{Campana2004, + author = {Campana, Fr{\'e}d{\'e}ric}, + title = {Orbifolds, special varieties and classification theory}, + journal = {Annales de l'Institut Fourier}, + volume = {54}, + number = {3}, + year = {2004}, + pages = {499--630}, + doi = {10.5802/aif.2027}, + url = {https://www.numdam.org/item/10.5802/aif.2027.pdf} +} + +@article{Kollar1993, + author = {Koll{\'a}r, J{\'a}nos}, + title = {{Shafarevich} maps and plurigenera of algebraic varieties}, + journal = {Inventiones mathematicae}, + volume = {113}, + number = {1}, + year = {1993}, + pages = {177--215}, + doi = {10.1007/BF01244307}, + url = {https://doi.org/10.1007/BF01244307} +} diff --git a/preprints/A-surface-counterexample-to-Shafarevich-holomorphic-convexity-September-23-2026/build/sections/01-introduction.tex b/preprints/A-surface-counterexample-to-Shafarevich-holomorphic-convexity-September-23-2026/build/sections/01-introduction.tex new file mode 100644 index 0000000000000000000000000000000000000000..3ff00fb3d22c49a885ccf538c3183415063545bf --- /dev/null +++ b/preprints/A-surface-counterexample-to-Shafarevich-holomorphic-convexity-September-23-2026/build/sections/01-introduction.tex @@ -0,0 +1,170 @@ +\section{Introduction} +\label{sec:introduction} + +Shafarevich asked whether the universal cover of a complete algebraic +variety is holomorphically convex \citep{Shafarevich1974}. +We follow the formulation reproduced by +\citet[pp.~135--137]{LasellRamachandran1996}, who locate the question +on p.~407 of Shafarevich's book. +We use its usual smooth-projective formulation, called the +\emph{Shafarevich conjecture on holomorphic convexity}: the universal +cover of every smooth connected projective complex variety is +holomorphically convex. For a complex manifold $V$ and a nonempty compact set $K\subset V$, its \emph{holomorphic hull} is +\[ + \widehat K_V=\{v\in V: |g(v)|\leq\sup_K|g| + \text{ for every }g\in\mathcal O(V)\}. +\] +Holomorphic convexity means that every such hull is compact. We disprove the conjecture in complex dimension two. + +\begin{theorem}\label{thm:main} +There is a smooth connected projective complex surface $X$ containing a connected nodal curve $Z_0$ with smooth rational irreducible components such that +\[ + \im\bigl(\pi_1(Z_0)\longrightarrow\pi_1(X)\bigr) + \quad\text{is infinite}. +\] +In the simply connected universal cover of $X$, a connected component above $Z_0$ is a closed, connected, noncompact, locally finite union of compact rational curves. The holomorphic hull of any one of its points is noncompact. In particular, this universal cover is not holomorphically convex. +\end{theorem} + +\subsection{Earlier approaches and the obstruction} + +The conjecture connects projective geometry with the analytic geometry +of covering spaces. Gurjar and Shastri proved it for smooth projective +elliptic surfaces \citep[Theorem~2(ii)]{GurjarShastri1985}, and +Katzarkov proved it for smooth projective varieties with virtually +nilpotent fundamental group \citep{Katzarkov1997}. +Eyssidieux proved holomorphic convexity for the cover defined by the +intersection of the kernels of all reductive complex representations +of a fixed rank \citep{Eyssidieux2004}. Eyssidieux, Katzarkov, Pantev +and Ramachandran then established the universal-cover conclusion for +smooth projective varieties whose fundamental groups admit faithful +finite-dimensional complex representations +\citep{EyssidieuxKatzarkovPantevRamachandran2012}. +Campana, Claudon and Eyssidieux extended this linear Shafarevich +theorem to compact K\"ahler manifolds +\citep{CampanaClaudonEyssidieux2015}. + +Recent results extend this theory beyond smooth projective varieties. +Deng and Yamanoi, with an appendix joint with Katzarkov, proved the +fixed-rank reductive-cover conclusion for normal projective varieties +\citep[Theorem~C(i)]{DengYamanoi2024}. Bakker, Brunebarbe and Tsimerman +proved that, for a connected normal algebraic space whose fundamental +group has a finite-dimensional complex representation with finite +kernel, the universal cover is a dense analytic-Zariski open subset +of a holomorphically convex complex space +\citep[Theorem~1.1]{BakkerBrunebarbeTsimerman2026}. This last conclusion +concerns a partial compactification of the universal cover. + +The geometric reduction approach seeks to contract subvarieties for which +the fundamental group of the normalization has finite image in the ambient +fundamental group. Koll\'ar's rational Shafarevich maps +and Campana's meromorphic $\Gamma$-reduction address this problem through +sufficiently general points \citep[\S1]{Kollar1993} +\citep[Theorem~3.5 and Definition~3.8]{Campana1994}. +Such generic reductions do not by themselves provide a proper holomorphic +map from the entire universal cover to a Stein space. + +The obstruction used in the present construction is classical. +Lasell and Ramachandran \citep[pp.~136--137]{LasellRamachandran1996} +and Katzarkov and Ramachandran +\citep[pp.~527--528]{KatzarkovRamachandran1998} explain why a +connected curve can obstruct holomorphic convexity when its normalized +components have finite fundamental-group image but the curve itself has +infinite image. A connected lift then has compact irreducible +components but is itself noncompact. Such a curve is called an +\emph{infinite Nori string} +\citep[arXiv version, Definition~3]{ColtoiuJoita2011}. +Every holomorphic function is constant on each compact component, +and connectedness makes these constants agree. A closed infinite Nori +string therefore cannot lie in a holomorphically convex space. +For our rational components the individual fundamental groups are +trivial; the decisive issue is the image of the loops in the incidence +graph of their whole nodal union. + +Bogomolov and Katzarkov proposed constructions from degenerating +curves and quotients of their fundamental groups in which this +obstruction would occur if the resulting geometric quotient remained +infinite after all required relations +\citep[author manuscript, \S4.1, Lemma~4.2 and Conjecture~4.1]{BogomolovKatzarkov1998}. +That infinitude assertion remained conditional. Eyssidieux and Funar +subsequently proved holomorphic convexity for the associated +uniform-ramification construction from semistable families of curves +of genus at least two, outside explicit finite exceptional ranges +\citep[Theorems~1.2 and~6.18]{EyssidieuxFunar}. + +The construction here follows the same broad passage from a family +of curves to a group quotient and then to a projective surface. +Its input is a marked genus-zero family, with both horizontal marked +sections and vertical boundary components. A \emph{meridian} is the +boundary of a small disk transverse to a divisor. We kill the meridians of +one distinguished fiber and impose high powers of the remaining +boundary meridians only after compactification. The argument must +establish both infinitude of this particular quotient and its +survival in the final surface. The weighted presentation proves the +first assertion; finite detection of entire local boundary-group +images, followed by transfer to the compact rational fiber, proves +the second. These are the steps developed below. + +\subsection{Consequences and comparisons} + +By the linear Shafarevich theorem, the fundamental group of the surface +in \Cref{thm:main} has no faithful finite-dimensional complex +representation. Two further comparisons concern restrictions on the +surface and on its universal cover. + +A smooth compact K\"ahler manifold $M$ is \emph{special} in Campana's +sense if no holomorphic line bundle $L$ admits a nonzero map +$L\to\Omega_M^p$ with $\kappa(M,L)=p$ for any $1\le p\le\dim M$; +here $\kappa$ denotes Iitaka dimension +\citep[Theorem~2.27]{Campana2004}. The abelianity theorem for +special compact K\"ahler manifolds gives a virtually abelian +fundamental group, and hence a faithful finite-dimensional complex +representation. The compact K\"ahler linear theorem therefore makes +their ordinary universal covers holomorphically convex +\citep[Corollary~1.4]{OpenAIAbelianity2026}. +Consequently the surface in \Cref{thm:main} is not special. + +A subset of a complex projective variety is \emph{semialgebraic} +if it is given in affine charts by finite Boolean combinations of +real polynomial equalities and inequalities. The ordinary universal +cover of our surface is not biholomorphic to a semialgebraic open +subset of a projective variety, with openness in the complex +topology: such a presentation would imply holomorphic convexity by +\citep[Corollary~8.1]{OpenAISemialgebraicCovers2026}, contrary to +\Cref{thm:main}. + +\subsection{The construction in outline} + +We begin with a family of spheres with $48$ moving punctures. The family comes from multiplication by $4$ on the elliptic curve with an automorphism of order three. Near one parameter value, the punctures collide in $24$ pairs. A double base change and one blowup at each collision produce an unmarked main sphere with $24$ attached spheres, each carrying two marked points. Finite covers of these components, branched only at those points, are again rational. + +The main task is to retain an infinite fundamental-group image after filling in this fiber and all other missing fibers. Three features make this possible. + +First, we study the moving points on the elliptic curve before taking its sign quotient. Three explicit double covers show that monodromy changes each generator only in higher degree. Deep elements of the free parameter group therefore impose relations of arbitrarily high degree. A surjective map of the actual based fiber groups transfers these equalities to the sphere family. + +Second, the multiplication cover has deck group $(\Z/4)^2$. Successive differences under its two translations give a free basis with useful degrees. A calculation with the linear terms of the sign involution selects $24$ relations that impose all sign identifications on this $48$-generator group. These relations leave a strict margin in a weighted Golod--Shafarevich inequality. That margin accommodates the deep monodromy relations and large powers of the remaining boundary loops. The weighted infinitude method builds on \citet{GolodShafarevich1964}, \citet{Golod1964}, \citet{Vinberg1965}, \citet{Ershov2011}, and \citet{ErshovJaikinZapirain2013}. The group construction takes place in inverse limits of finite 2-groups, described in \Cref{sec:weighted}. + +Third, we choose the finite parameter cover and compactify the marked family. Only then do we impose high powers of the remaining boundary meridians so that their images are finite. We then choose a finite quotient that is injective on each entire local boundary-group image. In the associated cover, this lets the infinite quotient survive normalization and resolution, including every exceptional divisor. The resulting smooth projective surface retains a rational special fiber. A neighborhood deformation retraction and properness transfer the infinite image of a nearby fiber to a connected component of that rational fiber. + +The two reusable mechanisms are the economical involution presentation in \Cref{prop:involution} and the finite-local-group argument in \Cref{lem:local-detection}. The former retains enough generators relative to relations for the weighted infinitude criterion; the latter extends a homomorphism from the fundamental group of a divisor complement across a smooth projective compactification of a finite cover. + +We give the weighted algebra in \Cref{sec:weighted}, construct and compare the actual fiber groups in \Cref{sec:family}, and analyze the involution in \Cref{sec:involution}. \Cref{sec:quotient} constructs the infinite quotient and the compactified marked family. \Cref{sec:surface} completes the surface and its rational curve. + +\subsection{Why the rational curve is enough} + +The following elementary form of the compact-curve obstruction makes the last step explicit. It is the rational-curve case of the mechanism described in \citet[pp.~136--137]{LasellRamachandran1996}, \citet[pp.~527--528]{KatzarkovRamachandran1998}, and \citet[author manuscript, \S4.1]{BogomolovKatzarkov1998}. + +\begin{lemma}\label{lem:rational-hull} +Let $X$ be a connected complex manifold and let $Z\subset X$ be a compact connected nodal curve whose irreducible components are smooth rational curves. If the image of $\pi_1(Z)\to\pi_1(X)$ is infinite, then every connected component $W$ of the inverse image of $Z$ in the simply connected universal cover $\widetilde X$ is closed and noncompact. It is a locally finite union of compact rational curves, and every holomorphic function on $\widetilde X$ is constant on $W$. +Consequently $\widehat{\{w\}}_{\widetilde X}$ is noncompact for every $w\in W$. +\end{lemma} + +\begin{proof} +The restricted map $W\to Z$ is the connected covering corresponding to +\[ + \ker\bigl(\pi_1(Z)\longrightarrow\pi_1(X)\bigr). +\] +Thus it has infinitely many sheets. A fiber is a closed discrete infinite subset, so $W$ cannot be compact. It is closed in $\widetilde X$, since it is a connected component of the closed inverse image of $Z$. + +Each rational component of $Z$ is simply connected. Its inverse image therefore consists of compact rational curves mapping isomorphically onto it. Choose an evenly covered neighborhood in $X$ of each point of $Z$, small enough that it meets $Z$ in one smooth branch or in the two branches of a node. Each lifted neighborhood meets at most one or two lifted components. These neighborhoods show that the collection is locally finite in the ambient manifold $\widetilde X$; points outside the closed inverse image have a neighborhood meeting none of it. The connected space $W$ is locally path connected; a path in it has compact image and meets only finitely many of these curves. The maximum principle makes a holomorphic function constant on each compact curve, and the constants agree at their intersections. They therefore agree throughout $W$. + +For $w\in W$, this proves $W\subset\widehat{\{w\}}_{\widetilde X}$. A compact hull cannot contain the closed noncompact set $W$. +\end{proof} diff --git a/preprints/A-surface-counterexample-to-Shafarevich-holomorphic-convexity-September-23-2026/build/sections/02-weighted.tex b/preprints/A-surface-counterexample-to-Shafarevich-holomorphic-convexity-September-23-2026/build/sections/02-weighted.tex new file mode 100644 index 0000000000000000000000000000000000000000..623455012b14110e51b5210edfaa7fbefa1a52c7 --- /dev/null +++ b/preprints/A-surface-counterexample-to-Shafarevich-holomorphic-convexity-September-23-2026/build/sections/02-weighted.tex @@ -0,0 +1,96 @@ +\section{Weighted free pro-2 groups} +\label{sec:weighted} + +We will construct an infinite quotient by comparing the weighted number of generators with the weighted number of relations. This section records the precise version of the Golod--Shafarevich argument that we use, including convergence for infinitely many relators. + +\subsection{Magnus degrees and automorphisms} + +A pro-2 group is an inverse limit of finite groups of order a power of two. The free pro-2 group on a finite set is the pro-2 completion of the discrete free group on that set. Let $F_d$ be free pro-2 on $g_1,\ldots,g_d$, and assign positive integer weights $w_1,\ldots,w_d$. Its Magnus expansion sends +\[ + g_i\longmapsto1+U_i + \quad\text{in}\quad + A=\F_2\langle\!\langle U_1,\ldots,U_d\rangle\!\rangle, + \qquad \deg U_i=w_i. +\] +Here $A$ is the algebra of formal noncommutative power series, with the filtration $A_{\ge n}$ by weighted degree at least $n$. For $g\ne1$, let $\nu(g)$ be the least degree in $g-1$, and put $\nu(1)=\infty$. + +For clarity, this expansion is faithful and gives the pro-2 topology. Each truncated algebra is finite, and its group $1+A_{>0}/A_{\ge n}$ is a finite 2-group. Conversely the augmentation ideal of $\F_2[P]$ is nilpotent for every finite 2-group $P$. To see this, induct on $|P|$, choosing a central involution $z$ when $P\ne1$. The kernel of $\F_2[P]\to\F_2[P/\langle z\rangle]$ is generated by $z-1$ and has square zero. Nilpotence follows by induction. Thus every finite 2-quotient of the free group factors through some Magnus truncation. This proves faithfulness on the pro-2 completion and cofinality of the truncations. Positive finite weights give the same topology as ordinary degree. + +Multiplication of series gives +\begin{equation}\label{eq:degree-rules} + \nu(gh)\ge\min\{\nu(g),\nu(h)\},\qquad + \nu([g,h])\ge\nu(g)+\nu(h),\qquad + \nu(g^{2^e})\ge2^e\nu(g). +\end{equation} +We use either commutator convention, since only its degree matters. The last inequality follows from $g^{2^e}-1=(g-1)^{2^e}$ in characteristic two. + +\begin{lemma}[Substitution]\label{lem:substitution} +Suppose an automorphism $\alpha$ of $F_d$ satisfies +\[ + \nu\bigl(\alpha(g_i)g_i^{-1}\bigr)\ge w_i+a + \quad(1\le i\le d) +\] +for an integer $a\ge1$. Then its continuous action on $A$ satisfies +\[ + (\alpha-\id)(A_{\ge n})\subset A_{\ge n+a}. +\] +More generally, substitution of degree-raising operators for Magnus variables is well defined whenever the operator assigned to a variable of weight $w$ raises degree by at least $w$. +\end{lemma} + +\begin{proof} +The hypothesis says that $\alpha(U_i)-U_i$ has degree at least $w_i+a$. Expand the difference between a monomial and its substituted image. Every summand replaces at least one factor by an error of at least $a$ additional degrees. The estimate follows for polynomials and then for series by completeness. For operator substitution, a word of weighted degree $n$ raises degree by at least $n$; hence only finitely many words contribute in each finite truncation. Multiplication and composition respect these substitutions. +\end{proof} + +We also use the following basis criterion. The \emph{Frattini quotient} of a finitely generated pro-2 group $G$ is +\[ + G/\overline{G^2[G,G]}, +\] +an $\F_2$-vector space. Elements whose images span this space topologically generate $G$: otherwise their closed generated subgroup has proper image in a finite 2-quotient, and is contained in a maximal subgroup of index two there, contradicting the span. Consequently, $d$ elements of $F_d$ whose Frattini images form a basis are again a free pro-2 basis. Indeed, they define a surjective endomorphism of $F_d$, and every finitely generated profinite group is Hopfian. For the latter assertion, a surjective endomorphism permutes, by inverse image, the finite set of open normal subgroups of any fixed index. Its kernel therefore lies in every open normal subgroup and is trivial. + +\subsection{The infinitude inequality} + +The infinitude method originates in the work of \citet{GolodShafarevich1964}; \citet{Vinberg1965} developed its filtered form. For the weighted completed-algebra inequality and its pro-$p$ consequence, see \citet[Theorem~2.1 and Corollary~2.2]{Ershov2011}. The more general language of positive weighted deficiency is developed by \citet[arXiv version, Corollary~4.4]{ErshovJaikinZapirain2013}. We include the filtered proof needed here to specify exactly the completion and the infinite-relator convention. + +\begin{lemma}[Weighted infinitude criterion]\label{lem:weighted-gs} +Let $F_d$ have the weighted basis above, and let $(r_\lambda)$ be a finite or countable family of relators with positive integer bounds $v_\lambda$ satisfying +\[ + \nu(r_\lambda)\ge v_\lambda\ge1. +\] +Assume only finitely many $v_\lambda$ lie below any fixed bound. If, for some $00}/R_{\ge n},\qquad + (b_i)_i\longmapsto\sum_i U_i b_i. +\] +Its kernel has dimension at most $\sum_\lambda C(n-v_\lambda)$. Here is the filtered justification. Write $f_\lambda=\sum_i U_i f_{i\lambda}$. A vector in the kernel can be lifted to series whose first-letter sum lies in $I+A_{\ge n}$. Subtracting the first-letter decomposition of the degree-at-least-$n$ term does not change the truncated vector, so the sum may be taken in $I$. For a product $a f_\lambda h$, the positive-degree part of $a$ contributes first-letter coefficients that vanish in $R$; its constant part contributes +\[ + (\overline{f_{i\lambda}})_i\,\bar h. +\] +This vector has shifted degree at least $v_\lambda$. Only $\bar h$ modulo $R_{\ge n-v_\lambda}$ matters. These vectors therefore span a space of dimension at most $\sum_\lambda C(n-v_\lambda)$. Closure adds nothing to a span in a finite-dimensional truncation. Since $\dim R_{>0}/R_{\ge n}=C(n)-1$, we obtain +\begin{equation}\label{eq:cumulative-gs} + C(n)-\sum_i C(n-w_i)+\sum_\lambda C(n-v_\lambda)\ge1. +\end{equation} + +Suppose the stated pro-2 quotient were finite, of order $m$. Its image spans every truncated algebra $R/R_{\ge n}$: the variables are the images of $g_i-1$, and their products are linear combinations of group elements. The group map to each truncated unit group kills the relators, hence factors through that quotient. Thus $C(n)\le m$ for all $n$. +Multiply \eqref{eq:cumulative-gs} by $t^{n-1}$ and sum over $n\ge1$. Boundedness of $C(n)$ and convergence of the relator series justify the sums. This gives +\[ + \left(1-\sum_i t^{w_i}+\sum_\lambda t^{v_\lambda}\right) + \sum_{n\ge1}C(n)t^{n-1}\ge\frac1{1-t}, +\] +contradicting \eqref{eq:gs-criterion}. +\end{proof} + +The use of lower bounds $v_\lambda$ makes the criterion convenient: increasing an actual relator degree can only improve the inequality. In the construction below, the inexpensive relations impose a geometric involution. The remaining relations will be pushed arbitrarily far into the filtration. diff --git a/preprints/A-surface-counterexample-to-Shafarevich-holomorphic-convexity-September-23-2026/build/sections/03-family.tex b/preprints/A-surface-counterexample-to-Shafarevich-holomorphic-convexity-September-23-2026/build/sections/03-family.tex new file mode 100644 index 0000000000000000000000000000000000000000..cc7109103404717704b8c7cf465a8cde88512b0a --- /dev/null +++ b/preprints/A-surface-counterexample-to-Shafarevich-holomorphic-convexity-September-23-2026/build/sections/03-family.tex @@ -0,0 +1,469 @@ +\section{A family of paired punctures} +\label{sec:family} + +We construct a family of spheres with 48 marked points. Near one missing +parameter value these points form 24 disjoint pairs. The purpose of this +section is to relate the actual based monodromy of the family to an action +on a free group whose weighted Magnus degree it raises. We also identify +the quotient obtained by killing the loops around the pairs. + +\subsection{The family and its torus cover} + +Put +\[ + E=\C/(\Z+\Z\omega),\qquad \omega^3=1,\quad \omega\ne1, + \qquad \iota(z)=-z. +\] +Let $q:E\to E/\langle\iota\rangle=\PP^1$ be the quotient map. +Multiplication by $\omega$ and by $4$ induce maps $\bar\omega$ and $f$ +on $\PP^1$, respectively. Thus +\[ + q\circ[4]=f\circ q,\qquad q\circ[\omega]=\bar\omega\circ q. +\] +Our parameter curve is +\begin{equation}\label{eq:parameter-base} + B=\PP^1\setminus + \bigl(q(E[2])\cup q(\ker(1-\omega))\bigr). +\end{equation} +For $s\in B$, let $\mathcal A_s$ be the following subset of the fiber +sphere: +\begin{equation}\label{eq:sphere-markings} + \mathcal A_s=\{z\in\PP^1:f(z)=\bar\omega^j s + \text{ for some }j\in\{0,1,2\}\}. +\end{equation} + +\begin{lemma}\label{lem:family} +The complement defining $B$ consists of five points. The sets +$\mathcal A_s$ form 48 disjoint points varying as an \emph{\'{e}tale +multisection}: locally on $B$ they are the graphs of 48 distinct +holomorphic functions. The point $q(0)$ belongs to no $\mathcal A_s$. +If $e_0\in E[2]\setminus\{0\}$ and $s_*=q(e_0)$, then as $s\to s_*$ +the markings collide in 24 pairs, each at a simple ramification point +of $f$. +\end{lemma} + +\begin{proof} +The group $\ker(1-\omega)$ has order three. Its two nonzero points are +exchanged by $\iota$, and its intersection with $E[2]$ is $\{0\}$. +This gives the five excluded values. A fixed point of $\bar\omega$ +lifts to a point satisfying $\omega z=\pm z$; since $1+\omega$ is a +unit in $\Z[\omega]$, all such values are excluded by +\eqref{eq:parameter-base}. The three targets in +\eqref{eq:sphere-markings} are therefore distinct. They avoid +$q(E[2])$, which contains the critical values of the degree-16 map $f$. +Also $f(q(0))=q(0)$, so $q(0)$ gives a section of the complement. + +The three points $e_j=\omega^j e_0$ are the distinct nonzero elements +of $E[2]$. A point $p\in E$ with $[4]p=e_j$ is not two-torsion. +Consequently $q$ is unramified at $p$, $[4]$ is \'etale there, and $q$ +is simply ramified at $e_j$. Thus $f$ has simple ramification at +$q(p)$. The sixteen points above each $e_j$ are paired freely by +$\iota$, giving eight collisions for each $j$ and 24 in total. +\end{proof} + +Choose henceforth $e_0\in E[2]\setminus\{0\}$ and put $s_*=q(e_0)$. +Fix $s_0\in B$ sufficiently close to $s_*$, and choose +$x_0\in q^{-1}(s_0)$ close to $e_0$. The six points +$\pm\omega^j x_0$ lie, two at a time, in disjoint $\iota$-invariant +disks $D_j$ about $e_j$, none containing zero. Write +\[ + T=E\setminus\{\pm\omega^j x_0:0\le j\le2\},\qquad + F=\pi_1(T,0),\qquad \Gamma=\pi_1(B,s_0). +\] +The group $\Gamma$ is free of rank four. In the core +$E\setminus\bigcup_j\operatorname{int}D_j$, choose handle generators +$x,y$ representing the lattice basis $1,\omega$. Choose based +meridians $a_j,a_j^-$ about the two punctures in +$D_j$, with core access paths, so that its boundary loop is +$c_j=a_ja_j^-$. The choices and ordering can be made so that the +surface relation is $[x,y]c_0c_1c_2=1$. Hence $F$ is free on +\begin{equation}\label{eq:torus-free-basis} + x,y,a_0,a_1,a_2,c_0,c_1. +\end{equation} +Give the first five generators weight one and the last two weight +two. Let $\nu_F$ be the resulting Magnus valuation on the pro-2 +completion $\widehat F$. The surface relation gives $\nu_F(c_2)\ge2$. + +The multiplication map $[4]$ restricts to an unramified cover of $T$. +Its deck group and based covering subgroup are +\[ + H=(\Z/4)^2,\qquad + K=\ker\bigl(F\longrightarrow H\bigr), +\] +where the map records the two handle coordinates modulo four. +Thus $K$ is free of rank $1+16(7-1)=97$. Its source is +\[ + T^{(4)}=[4]^{-1}(T),\qquad K=\pi_1(T^{(4)},0), +\] +the torus with the 96 preimages of the six punctures removed. +Quotienting $T^{(4)}$ by $\iota$ gives +\[ + S=\PP^1\setminus\mathcal A_{s_0},\qquad + \Pi=\pi_1(S,q(0)). +\] +We identify $K$ with the based fundamental group of the source torus +through $[4]$ and denote by $p:K\to\Pi$ the map induced by $q$. +The group $\Pi$ is free of rank 47. + +The two fiber groups serve different purposes. The torus group $F$ has a +small basis on which monodromy can be controlled; the sphere group $\Pi$ +is the fundamental group of the geometric fiber used in the surface. +The covering subgroup $K$ will connect these two descriptions. Keeping +this connection based, rather than only up to an inner automorphism, +is essential when we impose monodromy relations. + +\subsection{Actual based monodromy} + +Following $x_0$ over a loop $m\in\Gamma$ ends at +$(-1)^{\epsilon(m)}x_0$; this defines a character +$\epsilon:\Gamma\to\Z/2$. Follow also the six points +$\pm\omega^j x_0$. Their motion extends to an isotopy $H_u$ of the +torus which fixes zero and commutes with $\iota$. Indeed, subdivide +the path into motions in disjoint small disks, prescribe a vector +field in one disk of each sign pair, extend it equivariantly to the +other, and use bump functions supported away from zero. Every $H_u$ +fixes all four points of $E[2]$ pointwise: commutation with $\iota$ +preserves this discrete fixed-point set, and an isotopy starting at the +identity cannot permute it. Define the +corrected endpoint and its based action by +\begin{equation}\label{eq:corrected-monodromy} + \iota^{\epsilon(m)}H_1, + \qquad \varphi_m\in\Aut(F). +\end{equation} +The correction fixes each of the six punctures individually; this +is what \emph{pure} means here. + +\begin{proposition}\label{prop:based-monodromy} +The actions $\varphi_m$ form a homomorphism +$\Gamma\to\Aut(F)$, are pure, and commute with $\iota_*$. +They preserve $K$. If $\beta_m\in\Aut(\Pi)$ is the actual based +monodromy of the punctured sphere family, then +\begin{equation}\label{eq:actual-monodromy-square} + p\circ\varphi_m|_K=\beta_m\circ p. +\end{equation} +Moreover, $p:K\to\Pi$ is surjective. +\end{proposition} + +\begin{proof} +Two extensions of the same point motion differ by an isotopy fixing +the initial punctures and zero. Their endpoint actions therefore +agree. The disk construction also extends homotopies of parameter +paths. With loop composition chosen to agree with composition of +the endpoint actions, uncorrected monodromy is a homomorphism. +All its maps commute with $\iota$, and $\epsilon$ is a character, +so the corrected actions also compose. + +The uncorrected endpoint acts trivially on the homology of the +closed torus; the correction acts there by a sign. It therefore +preserves the handle-coordinate kernel $K$ and lifts uniquely +through $[4]$ fixing zero. The lift commutes with $\iota$ by +uniqueness: both compositions lift the same map and fix zero. +Requiring this fixed basepoint also excludes any nontrivial deck +translation. Write $\widetilde H_u$ for the lift of $H_u$ through +$[4]$ starting at the identity; it fixes zero and gives the actual +motion of the $96$ punctures upstairs. The zero-fixed lift of the +corrected endpoint is $\iota^{\epsilon(m)}\widetilde H_1$, because +$[4]$ commutes with sign. After quotienting by sign, this endpoint +is the endpoint induced by $\widetilde H_u$, hence gives the actual +based sphere monodromy. This proves \eqref{eq:actual-monodromy-square}. + +For surjectivity, represent a loop in $S$ based at $q(0)$ by a path +which avoids the four branch values of $q$ except at its endpoints. +Its open interval lifts through the ordinary two-sheeted cover +away from those values. Both lifted endpoints tend to zero, +the unique point above $q(0)$. Thus the lift is a based loop in +the source punctured torus, as required. +\end{proof} + +\subsection{Three invariant double covers} + +We next prove that $\varphi_m-\id$ raises the weighted degree on +$\widehat F$. The only delicate first step is its action in degree +one. Three double covers of the moving torus complement make this +action explicit. + +\begin{lemma}\label{lem:degree-one} +Every $\varphi_m$, and also $\iota_*$, acts as the identity on +the weighted degree-one quotient +$V=\widehat F/\{g:\nu_F(g)\ge2\}$. +\end{lemma} + +\begin{proof} +The vector space $V$ over $\F_2$ has the five basis classes +$x,y,a_0,a_1,a_2$. Two characters of $F$ are obtained from the +handle coordinates modulo two. Uncorrected monodromy preserves +them, and $\iota$ preserves them modulo two. + +Choose an affine coordinate on the quotient sphere with +$q(0)=\infty$, using the same letter $q$ for its pullback to $E$. +It is an even meromorphic function with a double pole at zero. +Normalize its leading coefficient to one in an additive local +coordinate $v$ at zero, so that +$v(\iota z)=-v(z)$ and $q(z)=v(z)^{-2}+O(1)$. +For $j=0,1,2$, consider over the moving torus complement the cover +\begin{equation}\label{eq:character-cover} + Y_j^2=q(z)-\bar\omega^j s. +\end{equation} +Here $s$ and $\bar\omega^j s$ are expressed in the chosen affine +coordinate. The right side has simple zeros precisely at the +two removed $j$-punctures. The cover is unramified elsewhere, +including at zero: with $\eta=vY_j$, its local equation is +\[ + \eta^2=v^2(q(z)-\bar\omega^j s)=1+O(v^2). +\] +Thus the points $\eta=1$ and $\eta=-1$ over zero give two global +lifts of the fixed basepoint section throughout $B$. + +Let $\chi_j:F\to\F_2$ be the character of this double cover on the +fiber over $s_0$. A cover of the entire family with a lifted +basepoint section gives an extension of $\chi_j$ to the fundamental +group of its total space. Conjugating a fiber loop by the section +loop does not change its value in $\F_2$. Hence uncorrected based +monodromy preserves $\chi_j$. Sign has the lift +\[ + (z,Y_j)\longmapsto(-z,-Y_j). +\] +This lift fixes each selected sheet over zero, because both $v$ +and $Y_j$ change sign and $\eta=vY_j$ does not. It follows that +$\iota_*$ also preserves $\chi_j$, and therefore so does the +corrected action $\varphi_m$. + +The character $\chi_j$ takes value one on $a_j$ and $a_j^-$ and +zero on the other puncture meridians. In particular it vanishes +on all $c_i$ and factors through $V$. The three characters +$\chi_j$, together with the two handle characters, form a basis +of $V^*$: evaluation on the five displayed generators has an +invertible block triangular matrix. Their common kernel is therefore +$\{g:\nu_F(g)\ge2\}$. Since all five characters are preserved, +this kernel is preserved and the two induced actions on $V$ are trivial. +\end{proof} + +\begin{proposition}\label{prop:monodromy-raising} +For every $m\in\Gamma$, the induced operator +$\varphi_m-\id$ on the weighted completed Magnus algebra of +$\widehat F$ raises degree by at least one. +\end{proposition} + +\begin{proof} +By \Cref{lem:degree-one}, the changes of the five weight-one +generators in \eqref{eq:torus-free-basis} have valuation at least +two. It remains to check the two weight-two boundaries. +Fix one pair, writing $a=a_j$ and $a^-=a_j^-$. +Pureness gives $\varphi_m(a)=waw^{-1}$ for some $w\in F$. +Choose $d\in F$ with $\iota_*(a)=d a^-d^{-1}$. +Commutation with $\iota_*$ yields +\[ + \varphi_m(a^-)=w'a^-(w')^{-1},\qquad + w'=\varphi_m(d)^{-1}\iota_*(w)d. +\] +By \Cref{lem:degree-one}, $w'w^{-1}$ has valuation at least two. +Replacing $w'$ by $w$ in the conjugation of $a^-$ therefore changes +it by an element of valuation at least three. On the other hand, +conjugating $c_j=aa^-$ by $w$ changes it by valuation at least +$1+\nu_F(c_j)\ge3$. It follows that +\[ + \nu_F\bigl(\varphi_m(c_j)c_j^{-1}\bigr)\ge3. +\] +The degree-raising substitution rule of \Cref{lem:substitution} +now applies to the entire weighted free basis. +\end{proof} + +\subsection{Pinching the pairs and passing to the sphere} + +We have obtained a degree-raising action on the seven-generator +torus group. We now collapse the lifted core to obtain a free group +on the individual puncture pairs. This construction will identify +both the quotient map from $K$ and its compatibility with the +\emph{actual} sphere group $\Pi$. + +Write +\[ + T_{\mathrm{core}}=E\setminus\bigcup_j\operatorname{int}D_j, + \qquad T_{\mathrm{core}}^{(4)}=[4]^{-1}(T_{\mathrm{core}}). +\] +The lifted core $T_{\mathrm{core}}^{(4)}$ is connected, because the +handle loops in $T_{\mathrm{core}}$ map onto $H$. Its complementary disks are indexed by +$(h,j)\in H\times\{0,1,2\}$. For each $j$, choose the index-zero +disk to be the one reached by lifting the chosen access path of +$a_j$ from zero; use deck translations to label the others. +All four fixed points of sign lie in $T_{\mathrm{core}}^{(4)}$, since multiplication +by four sends them to zero, outside every $D_j$. + +Collapsing $T_{\mathrm{core}}^{(4)}$ kills its fundamental group, including every +disk boundary. By van Kampen, the resulting quotient of $K$ is +free on one positive meridian for each of the $48$ disks: in each +two-puncture disk, killing the boundary makes the negative meridian +the inverse of the positive one. Let $L$ be the pro-2 completion +of this free group, with generators $a_{h,j}$. Paths in the lifted +core disappear under collapse, so these generators do not depend +on the remaining choices of core access paths. Deck translations +induce the coordinate shifts $T_1,T_2$ on $L$. + +We also denote the corresponding elements of the factor $H$ in +$L\rtimes H$ by $T_1,T_2$. Define +\begin{equation}\label{eq:torus-substitution} + \begin{aligned} + \Phi:F&\longrightarrow L\rtimes H,\\ + x&\longmapsto T_1,& y&\longmapsto T_2,& + a_j&\longmapsto a_{0,j},& c_0,c_1&\longmapsto1. + \end{aligned} +\end{equation} +The shifts commute and have order four, so $c_2$ also maps to one. +On the core group, $\Phi$ is exactly the $H$-valued covering +character. It therefore kills $\pi_1(T_{\mathrm{core}}^{(4)},0)\subset K$. +A positive meridian in disk $(h,j)$ can be based using the lift +of $u a_j u^{-1}$, where $u$ is a core loop with handle coordinate +$h$; its image under $\Phi$ is exactly $a_{h,j}$. Thus +$\Phi|_K:K\to L$ is the core-collapse map just constructed, +followed by pro-2 completion. Its image contains the discrete free +group on the $a_{h,j}$ and is dense in $L$. + +Sign preserves the lifted core and pairs its complementary disks +freely: their centers map to nonzero two-torsion, whereas $[4]$ +kills the source two-torsion. Let $b_j\in H$ be the index of the +negative of the chosen index-zero center. Sign takes a positive +puncture in disk $(h,j)$ to the negative puncture in disk +$(b_j-h,j)$. Since both core access paths and disk boundaries +have been killed, its induced action is exactly +\begin{equation}\label{eq:pinched-involution} + \sigma(a_{h,j})=a_{b_j-h,j}^{-1}. +\end{equation} +If the index-zero center is represented by $v/4$, with +$v\bmod(\Z+\Z\omega)=e_j$, then $b_j$ is represented, up to +the deck-coordinate convention, by $-2v$ modulo +$4(\Z+\Z\omega)$. Since $e_j$ is nonzero two-torsion, +\begin{equation}\label{eq:odd-displacement} + b_j\bmod2H\ne0. +\end{equation} + +\begin{lemma}\label{lem:pinched-sphere} +Let $\Pi_{\mathrm{pair}}$ be the quotient of $\Pi$ obtained by +killing the boundary loops of its 24 two-puncture disks. +It is free of rank 24, and its pro-2 completion is naturally +\begin{equation}\label{eq:pinched-sphere-group} + \widehat{\Pi_{\mathrm{pair}}} + \simeq L/\overline{\langle\!\langle + \sigma(g)g^{-1}:g\in L + \rangle\!\rangle}. +\end{equation} +More precisely, let $\lambda:L\twoheadrightarrow R$ be a continuous +quotient of pro-2 groups satisfying $\lambda\sigma=\lambda$. +There is a homomorphism $\theta:\Pi\to R$ with dense image and +\begin{equation}\label{eq:pinching-square} + \theta p=\lambda\Phi|_K. +\end{equation} +For $m\in\Gamma$, the condition +\begin{equation}\label{eq:torus-equalizer-condition} + \lambda\Phi(\varphi_m(k))=\lambda\Phi(k) + \qquad(k\in K) +\end{equation} +implies $\theta\beta_m=\theta$ on the entire unpinched group +$\Pi$. +\end{lemma} + +\begin{proof} +Each sign-paired pair of torus disks projects to one two-puncture +disk on the sphere, and either disk maps homeomorphically to it. +The image $q(T_{\mathrm{core}}^{(4)})$ is the sphere exterior, a sphere with $24$ +holes. Killing its boundary loops kills its fundamental group, +so the quotient defining $\Pi_{\mathrm{pair}}$ is the quotient +induced by collapsing this exterior. Van Kampen identifies it +with the free group on one meridian per projected disk. + +The map $q$ commutes with the two core collapses, with the +basepoints in their respective cores. Upstairs there is one +cyclic generator per disk, and sign identifies them in pairs by +$a_{h,j}=a_{b_j-h,j}^{-1}$. This accounts for the entire quotient: +every fixed point of sign is already in the collapsed core, and +either disk in each remaining pair maps homeomorphically to the +corresponding sphere disk. Passing to pro-2 completions proves +\eqref{eq:pinched-sphere-group}. The based collapse diagram then +gives the dense map $\theta$ and the exact identity +\eqref{eq:pinching-square}. + +Finally, \Cref{prop:based-monodromy} and +\eqref{eq:torus-equalizer-condition} give +\[ + \theta\beta_m p + =\theta p\varphi_m|_K + =\lambda\Phi\varphi_m|_K + =\lambda\Phi|_K + =\theta p. +\] +The map $p$ is surjective, so $\theta\beta_m=\theta$. +This argument does not require the monodromy to preserve the +kernel of the pair-pinching map. +\end{proof} + +The maps just constructed are summarized in \Cref{fig:group-maps}. +\begin{figure}[htbp] +\centering +\begin{tikzpicture}[baseline=(current bounding box.center),>=Stealth] + \node (K) at (0,1) {$K$}; + \node (L) at (2.8,1) {$L$}; + \node (R) at (5.6,1) {$R$}; + \node (Pi) at (0,-0.4) {$\Pi$}; + \draw[->] (K) -- node[above] {$\Phi|_K$} (L); + \draw[->] (L) -- node[above] {$\lambda$} (R); + \draw[->] (K) -- node[left] {$p$} (Pi); + \draw[->] (Pi) -- node[below right] {$\theta$} (R); +\end{tikzpicture} +\caption{The actual sphere group $\Pi$ maps to every pro-$2$ quotient +$R$ of $L$ on which sign acts trivially. The triangle records +$\theta p=\lambda\Phi|_K$. The map $p$ is surjective and $\Phi|_K$ +has dense image. Monodromy is compared on $K$ and $\Pi$ before it is +passed to $R$.} +\label{fig:group-maps} +\end{figure} + + +We conclude with two properties of the finite geometric monodromy +which will control a parameter cover. Let $\gamma_*\in\Gamma$ be +the positively oriented local circuit around $s_*$ based at the +nearby point $s_0$. + +\begin{lemma}\label{lem:finite-geometric-monodromy} +The joint action of $\Gamma$ on the 48 markings and on the sign +of $x_0$ factors through a finite 2-group. The element +$\gamma_*$ has nontrivial sign, whereas $\gamma_*^2$ fixes every +marking and acts trivially on $\Pi_{\mathrm{pair}}$. +\end{lemma} + +\begin{proof} +Choose a positive representative in the source torus for each +marking. Within the $j$th block these representatives are indexed +by $H$. Along a parameter loop, their endpoints are obtained by +a translation in $H$ and the common sign $(-1)^{\epsilon(m)}$. +The three blocks are preserved individually. Thus the joint action +is contained in $H^3\rtimes\Z/2$, with sign acting by inversion; +this is a finite 2-group. + +The quotient $q$ is simply ramified at $e_0$, so $\gamma_*$ +exchanges the two lifts of the parameter. By the local simple +ramification in \Cref{lem:family}, each marking pair has equation +$w^2=t$ near its collision. Its motion under $\gamma_*$ is a +half twist and under $\gamma_*^2$ a full twist. These motions +are supported in the disjoint pair disks and fix the exterior. +A full twist acts on a disk group by conjugation by its boundary; +after that boundary is killed, it fixes the remaining cyclic +generator. Hence the induced action on +$\Pi_{\mathrm{pair}}$ is trivial. +\end{proof} + +The groups and their geometric roles are collected below. All ranks are +free-group ranks, with pro-$2$ completion understood only in the row +for $L$. +\begin{center} +\begin{tabular}{@{}llr@{}} +Group & Geometric description & Rank \\ +\hline +$\Gamma$ & Five-punctured parameter sphere & $4$ \\ +$F$ & Six-punctured torus & $7$ \\ +$K$ & Its degree-$16$ multiplication cover & $97$ \\ +$\Pi$ & Sphere with $48$ punctures & $47$ \\ +$L$ & Lifted torus after killing the core and pair boundaries & $48$ \\ +$\Pi_{\mathrm{pair}}$ & Sphere after killing the $24$ pair boundaries & $24$ +\end{tabular} +\end{center} +The next section keeps the presentation through $L$ in order to retain +the degree estimates coming from $F$. diff --git a/preprints/A-surface-counterexample-to-Shafarevich-holomorphic-convexity-September-23-2026/build/sections/04-involution.tex b/preprints/A-surface-counterexample-to-Shafarevich-holomorphic-convexity-September-23-2026/build/sections/04-involution.tex new file mode 100644 index 0000000000000000000000000000000000000000..f396c101e8a22d507e3de50c08a9536d48b4ff10 --- /dev/null +++ b/preprints/A-surface-counterexample-to-Shafarevich-holomorphic-convexity-September-23-2026/build/sections/04-involution.tex @@ -0,0 +1,318 @@ +\section{A weighted presentation for the pinched fiber} +\label{sec:involution} + +We now equip the free pro-\(2\) group \(L\) with a filtration adapted to +the covering shifts. In this filtration the involution relations from +the preceding section can be imposed with a sufficiently small contribution to the weighted +relation sum. We also show that the degree bounds on \(F\) survive the map +\(\Phi:F\to L\rtimes H\). Although the pinched sphere group is +already free of rank $24$, an arbitrary basis of that quotient would +not retain these degree bounds. Keeping $48$ generators and imposing +$24$ carefully chosen relations preserves the filtration needed to +control all later monodromy relations. + +\subsection{A basis adapted to the two translations} + +Recall that \(L\) is free on \(a_{h,j}\), where +\(h\in H=(\Z/4)^2\) and \(j\in\{0,1,2\}\). The commuting shifts +\(T_1,T_2\) translate the two coordinates of \(h\). The involution is +\[ + \sigma(a_{h,j})=a_{b_j-h,j}^{-1}, + \qquad b_j\bmod 2\ne0. +\] +Thus \(\sigma T_r\sigma=T_r^{-1}\) for \(r=1,2\). +For an automorphism \(T\), write +\(\delta_T(g)=T(g)g^{-1}\), and define +\begin{equation}\label{eq:shift-basis} + u_{i\ell,j}=\delta_{T_1}^{\,i}\delta_{T_2}^{\,\ell}(a_{0,j}), + \qquad 0\le i,\ell<4. +\end{equation} + +\begin{lemma}\label{lem:shift-basis} +The elements in \eqref{eq:shift-basis} form a free pro-\(2\) basis of +\(L\). Give \(u_{i\ell,j}\) weight \(i+\ell+1\), and denote the +resulting Magnus valuation by \(\nu_L\). Each of +\(T_1-\id\), \(T_2-\id\), and \(\sigma-\id\) raises the degree of the +completed Magnus algebra by at least one. +\end{lemma} + +\begin{proof} +In the Frattini quotient, the span of the generators in one block is +the permutation module +\[ + R=\F_2[H] + =\F_2[\xi,\eta]/(\xi^4,\eta^4), + \qquad T_1=1+\xi,\quad T_2=1+\eta. +\] +The images of \(u_{i\ell,j}\) are its monomial basis +\(\xi^i\eta^\ell\). The Frattini basis criterion therefore gives the +first assertion. We henceforth use the stated weights. + +The proof has three steps: control the last difference of a cyclic +shift, propagate the two translation bounds, and then treat sign. +We first establish the endpoint estimate needed for a cyclic shift. +Suppose \(T^4=\id\), put \(w_i=\delta_T^i(w_0)\), and assume +\begin{equation}\label{eq:cyclic-hypothesis} + \nu_L(w_i)\ge d+i\qquad(0\le i\le3), + \qquad d\ge1. +\end{equation} +Then +\begin{equation}\label{eq:cyclic-endpoint} + \nu_L(w_4)\ge d+4. +\end{equation} +This assertion does not assume that \(T\) preserves the filtration. +To prove it, take formal free generators \(z_0,z_1,z_2,z_3\) of weights +\(d,d+1,d+2,d+3\), and define the triangular automorphism +\[ + T'(z_i)=z_{i+1}z_i\quad(0\le i<3), + \qquad T'(z_3)=z_3. +\] +By \Cref{lem:substitution}, \(T'-\id\) raises algebra degree by one. +In characteristic two, +\[ + (T')^4-\id=(T'-\id)^4, +\] +so \(T'^4(z_0)z_0^{-1}\) has valuation at least \(d+4\). +Evaluation at \(z_i=w_i\) preserves this bound by +\eqref{eq:cyclic-hypothesis}. The positive word \(T'^3(z_0)\) contains +\(z_3\) exactly once, as its first letter. In passing from the +evaluated formal fourth iterate to the actual fourth iterate, the only +changed substitution is +\[ + T(w_3)=w_4w_3 + \quad\hbox{in place of}\quad + T'(z_3)=z_3. +\] +Consequently \(T^4(w_0)=w_4\,T'^4(z_0)|_{z_i=w_i}\). +Since \(T^4(w_0)=w_0\), the bound for the evaluated formal iterate +proves \eqref{eq:cyclic-endpoint}. + +For each fixed \(\ell,j\), apply this estimate to +\(w_i=u_{i\ell,j}\), \(T=T_1\), and \(d=\ell+1\). +For \(i<3\), the required difference is exactly +\(\delta_{T_1}(u_{i\ell,j})=u_{i+1,\ell,j}\); +\eqref{eq:cyclic-endpoint} supplies the case \(i=3\). +Thus \(T_1-\id\) raises degree by one on every basis element, and +therefore on the algebra. + +The same argument gives the \(T_2\) bounds on the initial sequence +\(u_{0\ell,j}\). They propagate along the \(T_1\)-sequences as follows. +If \(T_2(z)=vz\), commutativity of the shifts gives +\begin{equation}\label{eq:commuting-difference} + T_2(\delta_{T_1}z) + =T_1(v)(\delta_{T_1}z)v^{-1}. +\end{equation} +Suppose \(z\) has assigned weight \(d\), the element \(v\) has +valuation at least \(d+1\), and \(\delta_{T_1}z\) has valuation at +least \(d+1\). The already proved \(T_1\) bound gives +\(\nu_L(T_1(v)v^{-1})\ge d+2\). Conjugating +\(\delta_{T_1}z\) by \(v\) changes it only in degree at least +\(2d+2\ge d+2\). Equation \eqref{eq:commuting-difference} therefore +gives the next \(T_2\) bound. Induction proves that \(T_2-\id\) +raises degree by one on the full basis. + +It remains to treat \(\sigma\). Powers of either shift preserve the +filtration and differ from the identity by an operator raising degree +by one. Since +\(\sigma(a_{0,j})=T_1^{(b_j)_1}T_2^{(b_j)_2}(a_{0,j})^{-1}\), +and inversion has the same linear term in characteristic two, we have +\[ + \nu_L\bigl(\sigma(a_{0,j})a_{0,j}^{-1}\bigr)\ge2. +\] +We propagate this estimate along the two difference sequences. +For either shift \(T\), its inverse preserves the filtration, and +\[ + T^{-1}-T=T^{-1}(\id-T^2), + \qquad T^2-\id=(T-\id)^2. +\] +Hence, if \(\nu_L(z)\ge d\), the elements +\(\delta_{T^{-1}}z\) and \(\delta_Tz\) agree modulo degree \(d+2\). +If \(\sigma(z)=vz\), the identity +\(\sigma T=T^{-1}\sigma\) gives +\begin{equation}\label{eq:inverting-difference} + \sigma(\delta_Tz) + =T^{-1}(v)(\delta_{T^{-1}}z)v^{-1}. +\end{equation} +When \(\nu_L(v)\ge d+1\), the inverse-shift bound and the +commutator estimate show that the right side differs from +\(\delta_{T^{-1}}z\) only in degree at least \(d+2\). +The preceding comparison then replaces +\(\delta_{T^{-1}}z\) by \(\delta_Tz\). +Starting with \(a_{0,j}\), first apply this argument along +\(\delta_{T_2}\) and then along \(\delta_{T_1}\). +It proves that \(\sigma\) changes each basis element of weight \(w\) +only in degree at least \(w+1\). A final application of +\Cref{lem:substitution} proves the assertion on the completed algebra. +\end{proof} + +\subsection{Twenty-four relations impose the involution} + +There are 24 pairs of generators exchanged, up to inversion, by sign. +Choosing one relation per pair in the original basis would impose the +involution, but would lose the higher degrees gained from the shifts. +We must choose 24 relations whose degrees are measured in the new +basis. The Frattini quotient will identify those relations, and its +generation criterion will show that they impose the full group action. + +\begin{proposition}\label{prop:involution} +Let \(L\) be the free pro-\(2\) group on \(a_{h,j}\), with +\(h\in(\Z/4)^2\) and \(0\le j\le2\), and let +\(\sigma(a_{h,j})=a_{b_j-h,j}^{-1}\), where \(b_j\bmod2\ne0\). +Equip \(L\) with the basis and weights of \Cref{lem:shift-basis}. +There are 24 elements \(g\in L\), with assigned positive weights +\(w_g\), such that +\[ + \nu_L(g)\ge w_g,\qquad + \nu_L(r_g)\ge w_g+1, + \qquad r_g=\sigma(g)g^{-1}, +\] +and the closed normal subgroup generated by the \(r_g\) imposes +\(\sigma=\id\) on all of \(L\). The basis polynomial and the +polynomial of the chosen weights are +\begin{equation}\label{eq:weight-polynomials} + \begin{split} + P(t)&=\sum_{i,\ell,j}t^{i+\ell+1} + =3t(1+t+t^2+t^3)^2,\\ + G(t)&=\sum_g t^{w_g} + =3t(1+t)(1+t^2)^2. + \end{split} +\end{equation} +In particular, \(P=(1+t)G\), and these involution relations have +degree cost at most \(tG(t)\). +\end{proposition} + +\begin{proof} +The full Magnus degree bounds are already proved. We now use the +Frattini quotient to choose enough relations to impose the whole +involution; generation is detected in this linear quotient. +Consider one block of the Frattini quotient, identified with +\(R=\F_2[\xi,\eta]/(\xi^4,\eta^4)\) as above. +Give a homogeneous polynomial of degree \(n\) weight \(n+1\). +Write \(b=(b_1,b_2)\) for the shift of this block. +The action of \(\sigma\) on \(R\) is +\begin{equation}\label{eq:frattini-involution} + f(\xi,\eta)\longmapsto + (1+\xi)^{b_1}(1+\eta)^{b_2} + f\bigl((1+\xi)^{-1}-1,(1+\eta)^{-1}-1\bigr). +\end{equation} +Here inversion of a group generator disappears in the mod-\(2\) +Frattini quotient. Since +\((1+\xi)^{-1}-1=\xi+\xi^2+\cdots\), expansion of +\eqref{eq:frattini-involution} shows that the part of +\(\sigma-\id\) raising weight by exactly one is +\begin{equation}\label{eq:frattini-differential} + d=(\bar b_1\xi+\bar b_2\eta) + +\xi^2\frac{\partial}{\partial\xi} + +\eta^2\frac{\partial}{\partial\eta}, + \qquad (\bar b_1,\bar b_2)=b\bmod2. +\end{equation} +The first term means multiplication. + +Set \(u=\bar b_1\xi+\bar b_2\eta\), which is nonzero, and choose a +linear coordinate \(v\) independent of \(u\) over \(\F_2\). +The fourth-power relations become \(u^4=v^4=0\). +Moreover, the vector field +\(\xi^2\partial_\xi+\eta^2\partial_\eta\) sends any linear form +over \(\F_2\) to its square. In these coordinates, +\begin{equation}\label{eq:even-coordinate-differential} + R=\F_2[u,v]/(u^4,v^4), + \qquad d=u+u^2\frac{\partial}{\partial u} + +v^2\frac{\partial}{\partial v}. +\end{equation} +The \(u\)-part sends \(1\) to \(u\), \(u^2\) to \(u^3\), and +odd powers to zero. Its square is zero, as is the square of the +\(v\)-part. The two parts commute, so \(d^2=0\). + +Take the homogeneous subspace +\begin{equation}\label{eq:even-complement} + C=\operatorname{span}_{\F_2} + \{u^a v^b:a\in\{0,2\},\ 0\le b<4\}. +\end{equation} +On \(C\), the odd-\(u\) component of \(d\) is multiplication by +\(u\), an isomorphism from \(C\) to the span of the odd-\(u\) +monomials. Thus \(d|_C\) is injective and +\begin{equation}\label{eq:acyclic-frattini} + R=C\oplus dC,\qquad \ker d=\im d=dC. +\end{equation} +Indeed the direct-sum assertion follows by projecting to the +odd-\(u\) monomials; both summands have dimension eight. +It then follows from \(d^2=0\) that the image and kernel, each of +dimension eight, coincide. + +For each of the eight monomials in \eqref{eq:even-complement}, +expand it in the original homogeneous monomials \(\xi^i\eta^\ell\). +Choose \(g\) to be a product of the corresponding +\(u_{i\ell,j}\), each appearing with its coefficient in \(\F_2\), +in any fixed order. Its assigned weight \(w_g\) is the polynomial +degree plus one. Then \(\nu_L(g)\ge w_g\), and +\Cref{lem:shift-basis} gives \(\nu_L(r_g)\ge w_g+1\). +In the graded Frattini quotient the leading vectors of \(g,r_g\) +are respectively the chosen basis of \(C\) and its image under +\(d\). By \eqref{eq:acyclic-frattini} these form a homogeneous +basis. The actual Frattini vectors therefore form a basis as well: +any linear dependence would give one between their lowest nonzero +weight components. Doing this in all three blocks shows that the +48 elements \(g,r_g\), equivalently \(g,\sigma(g)\), generate \(L\). + +Let \(N\) be the closed normal subgroup generated by these 24 +elements \(r_g\). It is invariant under \(\sigma\), because +\[ + \sigma(r_g)=g\sigma(g)^{-1}=r_g^{-1}. +\] +The quotient \(L/N\) is generated by the images of the \(g\), since +there \(\sigma(g)=g\); its induced involution fixes those generators +and hence is the identity. Thus \(N\) is exactly the closed normal +subgroup imposing \(\sigma=\id\). + +Finally, in each block the chosen monomials have weight polynomial +\[ + t(1+t^2)(1+t+t^2+t^3)=t(1+t)(1+t^2)^2. +\] +Adding the three blocks gives \eqref{eq:weight-polynomials}. +Since each \(r_g\) has degree at least \(w_g+1\), its contribution +to the relation bound is at most \(t^{w_g+1}\) for \(01$. +Both conditions hold for $t$ sufficiently close to $1/4$ from below. +We choose the following rational value and check its exact margin +in \Cref{prop:infinite-quotient}: +\begin{equation}\label{eq:t-choice} + t=\frac{49}{200}<\frac14. +\end{equation} +We reserve $1/100$ of the relation bound for deep monodromy and another +$1/100$ for the final boundary powers; the strict inequality below will +justify both allowances. Choose $J$ large enough that \eqref{eq:monodromy-cost} is less than $1/100$. Enlarge $J$, if necessary, until $\Gamma_J$ kills the finite permutation and sign action of \Cref{lem:finite-geometric-monodromy}. This is possible because that action has finite 2-group image. + +Set +\[ + M=\langle\gamma_*^2\rangle\Gamma_J\subset\Gamma. +\] +Normality of $\Gamma_J$ makes this a finite-index subgroup. The action of $\gamma_*^2$ on the pinched sphere group is trivial, by \Cref{lem:finite-geometric-monodromy}. Together with \Cref{lem:deep-invariance}, this proves invariance of $\Pi\to Q_0$ under all of $M$. Moreover $M$ fixes every marking and lies in the parameter sign kernel. It contains $\gamma_*^2$ but not $\gamma_*$. + +Let $B'\to B$ be the connected finite cover corresponding to $M$, and let $C'$ be its smooth projective completion. These are algebraic: Riemann existence applies to finite covers of a complex variety without a properness hypothesis \citep[Expos\'e XII, Th\'eor\`eme 5.1]{SGA1}, and normalization compactifies the curve cover. The selected point $c_*\in C'\setminus B'$ has ramification index exactly two over $s_*$, since $\langle\gamma_*\rangle\cap M=\langle\gamma_*^2\rangle$. + +The $48$ markings split into disjoint algebraic sections over $B'$. Denote their complement by +\[ + U\subset\PP^1\times B'. +\] +Finite-point isotopy gives a locally trivial punctured-sphere bundle. The open base curve is aspherical, and the fixed section at $q(0)$ splits its fundamental-group sequence. Therefore +\[ + \pi_1(U)=\Pi\rtimes M. +\] +The monodromy invariance just proved extends $\Pi\to Q_0$ to a homomorphism +\begin{equation}\label{eq:rho-zero} + \rho_0:\pi_1(U)\longrightarrow Q_0 +\end{equation} +that kills the section subgroup $M$. Its restriction to every punctured fiber has dense image, because the pinched sphere group maps densely onto the pro-2 quotient $Q_0$. + +\subsection{Separating the sections and filling the distinguished fiber} + +Extend the marked sections across $C'\setminus B'$ in $\PP^1\times C'$. Blow up their collision points until their strict transforms are disjoint, obtaining a smooth projective surface +\[ + Y\longrightarrow C' +\] +with $U$ unchanged. This terminates: at any common point of two sections, a blowup decreases their intersection multiplicity by one, so the sum of all pairwise intersection multiplicities strictly decreases. + +All fibers over $C'\setminus B'$ stay reduced with simple normal crossings. Here is the local induction, which also ensures that the full boundary is a simple normal crossing divisor. At a smooth point of a reduced vertical component, take the projection coordinate $u$ and write a section as $w=g(u)$. Blowing up the origin gives the chart $w=uv$, where the transformed section is $v=g(u)/u$ and is still transverse to the new multiplicity-one component $u=0$. In the other chart the projection has the form $u=wb$; its differential vanishes at the vertical node $w=b=0$. A section cannot pass through that node, because its composite with the projection is the identity. Thus every further collision center is a smooth point of a reduced vertical component. When the sections are disjoint, each meets a single vertical component transversely and avoids all vertical nodes. The boundary +\[ + D=Y\setminus U +\] +is consequently a simple normal crossing divisor: locally it is defined +by $u=0$ or $uv=0$ in smooth coordinates. It contains the $48$ +horizontal marked sections and every component of the fibers over +$C'\setminus B'$. + +Over $c_*$ the local marking equation is $w^2=s-s_*$. After the base change of index two it becomes $w^2=u^2$, up to analytic coordinate changes. One blowup separates the two branches $w=\pm u$. Hence the distinguished fiber $D_*$ consists of one main sphere and $24$ tails. Every tail carries two marked sections and meets the main sphere once; the main sphere carries none. + +\begin{figure}[htbp] +\centering +\begin{tikzpicture}[x=1.12cm,y=0.85cm] + \draw[thick] (0,0) -- (8,0); + \foreach \x/\lab in {0.8/1,2.2/2,5.8/23,7.2/24} { + \draw[thick] (\x,-0.12) -- (\x,2); + \draw[fill=white] (\x,0) circle (2.5pt); + \fill (\x,0.7) circle (2pt); + \fill (\x,1.35) circle (2pt); + \node[above] at (\x,2) {tail $\lab$}; + } + \node at (4,1.05) {$\cdots$}; + \node[below] at (4,-0.3) {main $\PP^1$: no markings}; +\end{tikzpicture} +\caption{The distinguished fiber $D_*$ on $Y$, after the base change of index two and +one blowup at each collision. Each of the $24$ rational tails has two +markings (solid dots) and one node on the main component (open circles). +The middle $20$ tails are omitted. The cover constructed in \Cref{sec:surface} +is unramified at the nodes and can branch only at the markings. Its +inverse image may have a different incidence graph.} +\label{fig:distinguished-fiber} +\end{figure} + + +\begin{lemma}\label{lem:distinguished-meridians} +Every meridian around a component of $D_*$ has trivial image under $\rho_0$. +\end{lemma} + +\begin{proof} +A main-component meridian is represented by the circuit along the fixed exterior section, so it is killed by \eqref{eq:rho-zero}. + +For a tail use the blowup chart $w=uv$. A meridian at a sufficiently large fixed slope $v=c$ is represented downstairs by +\[ + u=\varepsilon e^{i\theta},\qquad + w=c\varepsilon e^{i\theta},\qquad 0\le\theta\le2\pi. +\] +Choose $|c|$ larger than the two branch slopes and then choose $\varepsilon$ small. Trivialize the pair motion by an isotopy supported inside the circle containing the two marks but not this meridian. In this trivialization the loop is the base circuit followed by one full pair-boundary loop, up to conjugacy and the choice of multiplication convention. The base circuit is killed by the section splitting; the pair boundary is killed by the pinching map. Both factors therefore have trivial image. +\end{proof} + +\subsection{The remaining boundary relations} + +There are finitely many components of $D$ outside $D_*$. Choose based meridians $\mu_1,\ldots,\mu_b$ for them, and choose lifts $\widetilde\mu_i\in L$ of $\rho_0(\mu_i)\in Q_0$. These lifts exist since $L\to Q_0$ is surjective. Choose an integer $e$ so large that +\begin{equation}\label{eq:boundary-cost} + b\,t^{2^e}<\frac1{100}, +\end{equation} +and impose the additional relators $\widetilde\mu_i^{2^e}$. +This is the high-power method underlying Golod's construction +\citep{Golod1964}, applied here to the finitely many boundary elements. +Each added relator has degree at least $2^e$ by +\eqref{eq:degree-rules}. Denote the resulting pro-2 quotient by $Q$. + +\begin{proposition}\label{prop:infinite-quotient} +The group $Q$ is infinite. The homomorphism +\[ + \rho:\pi_1(U)\longrightarrow Q +\] +induced by $\rho_0$ has dense image on every punctured fiber. Each component meridian of $D$ has finite image, and every component meridian of $D_*$ has trivial image. +\end{proposition} + +\begin{proof} +Only infinitude remains to prove. The generators of $L$ have polynomial $P(t)$ and the involution relators have cost at most $tG(t)$, where \Cref{prop:involution} gives +\[ + P(t)=(1+t)G(t),\qquad + G(t)=3t(1+t)(1+t^2)^2. +\] +At the rational value \eqref{eq:t-choice}, $t^2>3/50$, so +\[ + G(t)> + \frac{3\cdot49\cdot249\cdot53^2}{200^2\cdot50^2} + =\frac{102817827}{100000000} + >\frac{257}{250}. +\] +The complete weighted expression in \Cref{lem:weighted-gs} is therefore less than +\[ + 1-P(t)+tG(t)+\frac1{100}+\frac1{100} + =1-G(t)+\frac1{50} + <-\frac1{125}<0. +\] +The infinite monodromy-relator family is finite in each bounded degree and has convergent cost by \eqref{eq:monodromy-cost}. The remaining families are finite. All hypotheses of \Cref{lem:weighted-gs} hold, proving that $Q$ is infinite. Passage to the quotient preserves all earlier equalities and the density of each fiber image. +\end{proof} + +Notice that $e$ is chosen after the finite cover and compactification. Thus their possibly large number of boundary components introduces no circular parameter choice. The surface construction now uses only the finite local images and the infinite fiber image recorded in \Cref{prop:infinite-quotient}. diff --git a/preprints/A-surface-counterexample-to-Shafarevich-holomorphic-convexity-September-23-2026/build/sections/06-surface.tex b/preprints/A-surface-counterexample-to-Shafarevich-holomorphic-convexity-September-23-2026/build/sections/06-surface.tex new file mode 100644 index 0000000000000000000000000000000000000000..07cf188b7573fdd15ed10e4ee5d7efcfbbcf8f95 --- /dev/null +++ b/preprints/A-surface-counterexample-to-Shafarevich-holomorphic-convexity-September-23-2026/build/sections/06-surface.tex @@ -0,0 +1,310 @@ +\section{The projective surface and its rational fiber} +\label{sec:surface} + +We now turn the quotient of \Cref{prop:infinite-quotient} into a +fundamental-group image on a smooth projective surface. The construction +must retain the distinguished fiber: its rational components will lift +compactly, while the fundamental group of their union will have infinite +image. + +Recall the inputs. The complement $U=Y\setminus D$ lies in a smooth +connected projective surface $Y$, the boundary $D$ has simple normal +crossings, and +\[ + \rho:\pi_1(U)\longrightarrow Q +\] +has dense image in an infinite pro-$2$ group. Its restriction to a +punctured smooth fiber also has dense image. Every component meridian of +$D$ has finite image, and the meridians of the distinguished fiber +$D_*$ have trivial image. This fiber consists of an unmarked main +sphere and $24$ tails, each meeting the main sphere once and carrying +two horizontal markings. + +\subsection{A finite cover that kills all compactification meridians} + +At a crossing, two commuting meridians may have relations that neither +cyclic subgroup detects separately. We therefore choose a finite +quotient that detects the entire local group at every boundary point. + +\begin{lemma}[Finite local detection and descent] +\label{lem:local-detection} +Let $Y$ be a smooth connected projective complex surface, let $D$ be a +simple-normal-crossings divisor, and put $U=Y\setminus D$. Suppose +$\rho:\pi_1(U)\to Q$ has dense image in a profinite group and sends +every component meridian of $D$ to an element of finite order. +There is a finite quotient $Q\twoheadrightarrow P_f$ with the following +properties. +\begin{enumerate} + \item The quotient is injective on the image under $\rho$ of every + sufficiently small local boundary-complement group. + \item The connected finite cover + $\pi:U'\to U$ defined by + $\ker(\pi_1(U)\to P_f)$ extends to a projective morphism + $p:X\to Y$, where $X$ is smooth and connected and + $D_X=X\setminus U'$ is a simple-normal-crossings divisor. + \item Writing $i:U'\hookrightarrow X$ for inclusion, there is a + homomorphism $\rho_X:\pi_1(X)\to Q$ such that + \begin{equation} + \rho_X\circ i_* = \rho\circ\pi_*. + \label{eq:rho-descent} + \end{equation} +\end{enumerate} +Moreover, the resolution used to construct $X$ can be chosen to preserve +every open subset of the normalization on which the surface is smooth +and its reduced boundary has simple normal crossings. +\end{lemma} + +\begin{proof} +We first find a finite quotient detecting all local boundary images, +then compactify and resolve the corresponding cover, and finally +check that the original quotient map kills every new boundary +meridian. + +Choose a small coordinate polydisk $V$ about a point of $D$. Its +boundary complement is +\[ + V\setminus D\simeq(\Delta^*)^r\times\Delta^{2-r}, + \qquad r\in\{1,2\}. +\] +After choosing an access path in $U$, let +$A_V\subset Q$ be the image of its fundamental group. This group is +generated by $r$ commuting finite-order elements, so it is finite. +Only finitely many such images occur up to conjugacy: there is one +meridian type on the smooth part of each boundary component and one +local group at each crossing. The smooth part of an irreducible +component, with its finitely many crossings removed, is connected; +moving the access path along it conjugates the local group. + +For every nonidentity element in representatives of these finite +groups, choose an open normal subgroup of $Q$ that does not contain +it. Their finite intersection $N$ is open and normal. It satisfies +\begin{equation} + N\cap gA_Vg^{-1}=\{1\} + \qquad(g\in Q) + \label{eq:local-detection} +\end{equation} +for every local group. Set $P_f=Q/N$. Density makes the map +$\pi_1(U)\to P_f$ surjective, so its kernel defines a connected +finite Galois cover $\pi:U'\to U$. + +The cover has the complex structure obtained by lifting local charts. +It is algebraic and finite \emph{\'etale} by the Riemann existence +theorem for complex varieties +\citep[Expos\'e XII, Th\'eor\`eme 5.1]{SGA1}. This theorem applies +to the nonproper surface $U$. Normalize $Y$ in the function field of +$U'$ and denote the resulting morphism by $n:\overline Y\to Y$. +It is finite: complex varieties are Nagata, so finiteness of relative +normalization applies to the finite-type morphism $U'\to Y$ +\citep[Tag 03GR]{StacksProject}. +Its restriction over $U$ is $U'$, because the latter is already normal. +Finite morphisms are projective, so $\overline Y$ is projective +\citep[Tag 0B3I]{StacksProject}. + +Resolve $\overline Y$ by projective blowups, leaving its smooth locus +unchanged; strong resolution in characteristic zero has this +prescribed-open property \citep[pp.~822--823]{EncinasHauser2002}. +Next resolve the reduced boundary on the resulting smooth surface. +For curves on a smooth surface this can be done by point blowups: +resolve the singular component germs, decrease tangency intersection +multiplicities, and separate crossings of three or more branches +\citep[Tag 0BIC and its proof]{StacksProject}. +Choose these centers only where the boundary is not already simple +normal crossings. Thus this additional operation preserves every +open subset on which the pair was already smooth with simple normal +crossings. We obtain +\[ + X\xrightarrow{\ r\ }\overline Y + \xrightarrow{\ n\ }Y, + \qquad p=n\circ r, +\] +with $X$ smooth and projective and with $D_X$ a +simple-normal-crossings divisor. The dense open set $U'$ is unchanged, +so $X$ is connected. + +We prove the factorization, including the meridians of resolution +exceptional curves. The inclusion $i$ is surjective on fundamental +groups: a loop can be perturbed off a real-codimension-two divisor. +Its kernel is normally generated by the component meridians of $D_X$. +Indeed, perturb a null-homotopy disk relative to its boundary so that +it avoids the finitely many crossings of $D_X$ and meets its smooth +part transversely. Deleting small disks around the finitely many +intersection points expresses its boundary loop as a product of +conjugates of meridians and their inverses. + +Let $E$ be any component of $D_X$, choose a smooth point $x\in E$ +away from the other components, and take a small transverse disk +$T$ through $x$ with $T\setminus\{x\}\subset U'$. Choose a boundary +polydisk $V$ about $p(x)$ downstairs. By shrinking $T$, we may assume +$p(T)\subset V$. Consequently the projection of a small meridian +in $T\setminus\{x\}$ lies in $V\setminus D$. After including its +access path, its $Q$-image belongs to a conjugate of $A_V$. +This argument applies also when $E$ is exceptional and is contracted +to a boundary point. + +The projected meridian has trivial image in $P_f$, since its lift is +a closed loop in $U'$. Its $Q$-image therefore lies in both $N$ and +a conjugate of $A_V$, and is trivial by +\eqref{eq:local-detection}. All generators of $\ker i_*$ are thus +killed by $\rho\circ\pi_*$. This proves +\eqref{eq:rho-descent}. +\end{proof} + +Apply \Cref{lem:local-detection} to the map supplied by +\Cref{prop:infinite-quotient}. Retain its notation +$P_f,N,U',\overline Y,X,p,\rho_X$. We next verify that the +normalization is already smooth along the distinguished fiber, so +that the resolution can preserve it. + +\subsection{The distinguished fiber remains a union of spheres} + +\begin{lemma} +\label{lem:rational-special-fiber} +Let $h:X\to C'$ be the composite of $p$ and $Y\to C'$. The +resolution can be chosen so that the reduced fiber +\[ + Z=h^{-1}(c_*)_{\mathrm{red}} +\] +has only smooth rational irreducible components and ordinary +transverse crossings. +\end{lemma} + +\begin{proof} +Every vertical meridian along $D_*$ is trivial in $Q$, and hence in +$P_f$, by \Cref{lem:distinguished-meridians}. We describe the +normalization locally at every kind of point of $D_*$. The +restriction of the Galois cover to a boundary-complement polydisk +is classified by its local meridian homomorphism to $P_f$; its +connected components correspond to cosets of the local image. + +At a smooth unmarked point of $D_*$, the sole meridian is trivial. +Every connected local cover therefore extends as a copy of the +polydisk. At a node between the main component and a tail, both +vertical meridians are trivial, and the same assertion holds. + +At a horizontal--vertical intersection choose coordinates $(u,v)$ +with the vertical curve $u=0$ and the horizontal marking $v=0$. +The $u$-meridian is trivial. If the image of the $v$-meridian has +order $e$, the kernel of the local map $\Z^2\to P_f$ is +$\Z\times e\Z$. Each connected local cover thus has the finite +normal extension +\begin{equation} + (u,w)\longmapsto (u,v)=(u,w^e). + \label{eq:distinguished-local-cover} +\end{equation} +This extension is smooth, and its reduced boundary is $uw=0$. +Analytification preserves finiteness and normality +\citep[Expos\'e XII, Propositions~2.1(vi) and 3.2(vi)]{SGA1}. +Thus the local extensions just described agree with $\overline Y$ by +uniqueness of finite normal extension +\citep[Expos\'e XII, Proposition~5.3]{SGA1}. The entire preimage of $D_*$ +lies in the smooth simple-normal-crossings locus of the normalized +pair. The resolution in \Cref{lem:local-detection} can be the +identity on this open neighborhood. + + + +Consider now an irreducible component above the main sphere. +Its map to that sphere is unramified, including at the attaching +nodes, and there are no horizontal markings on the main component. +It is therefore a connected unramified cover of $\PP^1$, hence a +sphere mapping with degree one. A component above a tail can branch +only at its two horizontal markings, and is unramified at the +attaching node. Removing those two points downstairs and all their preimages upstairs +gives a connected finite cover of $\C^*$. Its covering subgroup is +$d\Z\subset\Z$. Thus it is the covering $z\mapsto z^d$ of $\C^*$, +whose smooth compactification is $\PP^1$. + +There are no additional components from resolution over $D_*$. +The local models show that the components just identified are +smooth and meet only in ordinary transverse crossings. This +proves the assertion about $Z$. +\end{proof} + +\subsection{Infinite image on the compact fiber} + +Although $D_*$ is a tree of spheres, a degree-$d$ component above +one of its tails meets $d$ distinct degree-one lifts of the main +sphere: the attaching node has $d$ distinct preimages. The incidence +graph upstairs therefore need not be a tree. For a connected nodal +union of simply connected components, van Kampen identifies its +fundamental group with that of its dual graph, which has one vertex +for each component and one edge for each node. We now prove that +some connected component of $Z$ has infinite fundamental-group image +in $X$. + +\begin{lemma} +\label{lem:infinite-special-image} +Let $Z=h^{-1}(c_*)_{\mathrm{red}}$ be the rational nodal fiber of +\Cref{lem:rational-special-fiber}. Some connected component $Z_0$ of $Z$ satisfies +\[ + \bigl|\im\bigl(\pi_1(Z_0)\longrightarrow\pi_1(X)\bigr)\bigr| + =\infty. +\] +\end{lemma} + +\begin{proof} +Let $F_s\subset U$ be a punctured smooth fiber, with $s\in B'$, +and choose access paths when comparing its group with $\pi_1(U)$. +Its image under $\rho$ is dense in $Q$ by +\Cref{prop:infinite-quotient}. In particular its image in $P_f$ +is all of $P_f$, so its inverse image $F'_s\subset U'$ is +connected. Under the covering inclusion, its group is the +preimage of $N$ in $\pi_1(F_s)$. Thus the image of +$\pi_1(F'_s)$ in $Q$ is +\[ + \rho\bigl(\pi_1(F_s)\bigr)\cap N. +\] +It is dense in $N$, because $N$ is open and the fiber image is +dense in $Q$. The group $N$ is infinite: a finite open subgroup +would make its finite-index overgroup $Q$ finite. Consequently +the image of $\pi_1(F'_s)$ in $Q$ is infinite. By +\eqref{eq:rho-descent}, its image in $\pi_1(X)$ is infinite as +well. + +We transfer this conclusion to $Z$ using an explicit +deformation-retract neighborhood. Regard $X$ as a compact real analytic +manifold and $Z$ as a closed semianalytic subset. This pair admits a +compatible finite triangulation +\citep[Section 3, Theorem 2, p.~464]{Lojasiewicz1964}. +Write it as a finite simplicial pair $(\mathcal K,\mathcal A)$, +and barycentrically subdivide so that $\mathcal A$ is a full +subcomplex: any simplex whose vertices belong to $\mathcal A$ +belongs to $\mathcal A$. For a point of $|\mathcal K|$, let $s_{\mathcal A}$ +be the sum of its barycentric coordinates at vertices of +$\mathcal A$. On the open neighborhood +\[ + V=\{s_{\mathcal A}>0\} +\] +retain those coordinates, divide them by $s_{\mathcal A}$, and +set all other coordinates to zero. Fullness ensures that the +result belongs to $|\mathcal A|$. The straight-line homotopy +inside each simplex gives a strong deformation retraction +$V\to Z$. In particular the inclusion of $V$ into $X$ is +homotopic to a map through $Z$. + +Properness of $h$ supplies the needed containment of nearby +fibers. The set $h(X\setminus V)$ is closed in $C'$ and does +not contain $c_*$. Hence every whole fiber over a sufficiently +small neighborhood of $c_*$ lies in $V$. Choose $s\in B'$ in +this neighborhood. Its connected punctured lifted fiber +$F'_s$ lies in one connected component of $V$; the deformation +retraction takes that component into a connected component +$Z_0$ of $Z$. Therefore its inclusion-induced homomorphism to +$\pi_1(X)$ factors, up to a change-of-basepoint conjugation, +through $\pi_1(Z_0)$. Since the former has infinite image, so +does the latter. +\end{proof} + +\begin{proof}[Proof of \Cref{thm:main}] +The construction above gives a smooth connected projective surface +$X$. By \Cref{lem:rational-special-fiber,lem:infinite-special-image}, +it contains a connected nodal curve $Z_0$ with smooth rational +irreducible components and infinite image in $\pi_1(X)$. + +\Cref{lem:rational-hull} now applies to $(X,Z_0)$. In the simply +connected universal cover $\widetilde X$, every connected component +$W$ above $Z_0$ is a closed noncompact locally finite union of +compact rational curves. For every $w\in W$, the holomorphic +hull of $\{w\}$ contains $W$ and is noncompact. This proves all +the assertions of \Cref{thm:main}. +\end{proof} diff --git a/preprints/A-surface-counterexample-to-Shafarevich-holomorphic-convexity-September-23-2026/paper.pdf b/preprints/A-surface-counterexample-to-Shafarevich-holomorphic-convexity-September-23-2026/paper.pdf new file mode 100644 index 0000000000000000000000000000000000000000..05f84edc15e107f3cbbd98ddfb41c15b9767da93 --- /dev/null +++ b/preprints/A-surface-counterexample-to-Shafarevich-holomorphic-convexity-September-23-2026/paper.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:d99ed633f092375c6b34a026df5d81c56ef9da8b99ca8e9f990146078b71a66a +size 452394