Add preprints files (part 4)

.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
+preprints/A-positive-solution-to-Tingleys-problem-September-23-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/A-power-improvement-in-the-Heilbronn-triangle-lower-bound-September-25-2026/main.pdf filter=lfs diff=lfs merge=lfs -text
+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 filter=lfs diff=lfs merge=lfs -text
+preprints/A-power-saving-for-planar-halving-lines-September-25-2026/main.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/A-power-saving-for-planar-unit-distances-September-23-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/A-Power-Saving-for-Polynomial-Differences-at-Prime-Arguments-October-5-2026/prime-argument-polynomial-differences.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/A-power-saving-for-square-difference-free-sets-September-24-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/A-product-counterexample-to-the-simplex-maximum-for-projection-body-volume-September-24-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/A-projective-fourfold-with-large-fundamental-group-and-non-Stein-universal-cover-October-5-2026/large-fundamental-group-non-stein.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/A-proof-of-Seymours-second-neighborhood-conjecture-September-23-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/A-quadratic-bound-for-Jacobsthals-function-September-25-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/A-radial-continuum-phase-transition-with-algebraic-decay-September-24-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/A-rational-obstruction-to-strong-chromatic-splitting-at-height-three-September-25-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/A-sharp-entropy-bound-and-the-simplex-inequality-for-isotropic-constants-October-5-2026/isotropic-simplex.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/A-sharp-Fourier-certificate-for-planar-circle-packing-September-23-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/A-Sharp-Threshold-Bound-for-Monotone-Graph-Properties-September-25-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/A-single-lattice-covering-bound-of-order-n-log-n-September-23-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/A-singular-normal-affine-surface-with-free-tangent-sheaf-September-23-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/A-Smooth-Metric-with-No-Local-Isometric-Immersion-into-Three-Space-September-24-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/A-smooth-three-torus-diffeomorphism-with-simple-Lebesgue-spectrum-September-23-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/A-spectral-gap-throughout-the-high-temperature-Sherrington-Kirkpatrick-phase-September-24-2026/main.pdf filter=lfs diff=lfs merge=lfs -text
+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

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}
+}
+```
+

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}

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}

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. 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{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}
+}
+
+@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 = {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}
+}
+
+@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}
+}
+
+@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}
+}
+
+@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 = {Also \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}
+}
+
+
+@article{Sarkozy1978III,
+  author = {S{\'a}rk{\"o}zy, A.},
+  title = {On difference sets of sequences of integers. {III}},
+  journal = {Acta Mathematica Academiae Scientiarum Hungaricae},
+  volume = {31},
+  number = {3--4},
+  pages = {355--386},
+  year = {1978}
+}
+
+@article{Lucier2008,
+  author = {Lucier, Jason},
+  title = {Difference sets and shifted primes},
+  journal = {Acta Mathematica Hungarica},
+  volume = {120},
+  number = {1--2},
+  pages = {79--102},
+  year = {2008},
+  doi = {10.1007/s10474-007-7107-1}
+}
+
+@article{RuzsaSanders2008,
+  author = {Ruzsa, Imre Z. and Sanders, Tom},
+  title = {Difference sets and the primes},
+  journal = {Acta Arithmetica},
+  volume = {131},
+  number = {3},
+  pages = {281--301},
+  year = {2008},
+  doi = {10.4064/aa131-3-5}
+}
+
+@article{Wang2020,
+  author = {Wang, Ruoyi},
+  title = {On a theorem of {S{\'a}rk{\"o}zy} for difference sets and shifted primes},
+  journal = {Journal of Number Theory},
+  volume = {211},
+  pages = {220--234},
+  year = {2020},
+  doi = {10.1016/j.jnt.2019.10.009}
+}
+
+@article{Green2024,
+  author = {Green, Ben},
+  title = {On {S{\'a}rk{\"o}zy}'s theorem for shifted primes},
+  journal = {Journal of the American Mathematical Society},
+  volume = {37},
+  number = {4},
+  pages = {1121--1201},
+  year = {2024},
+  doi = {10.1090/jams/1036}
+}
+
+@article{ThornerZaman2024,
+  author = {Thorner, Jesse and Zaman, Asif},
+  title = {An explicit version of {Bombieri}'s log-free density estimate and {S{\'a}rk{\"o}zy}'s theorem for shifted primes},
+  journal = {Forum Mathematicum},
+  volume = {36},
+  number = {4},
+  pages = {1059--1080},
+  year = {2024},
+  doi = {10.1515/forum-2023-0091}
+}
+
+@article{LiPan2009,
+  author = {Li, Hongze and Pan, Hao},
+  title = {Difference sets and polynomials of prime variables},
+  journal = {Acta Arithmetica},
+  volume = {138},
+  number = {1},
+  pages = {25--52},
+  year = {2009},
+  doi = {10.4064/aa138-1-2}
+}
+
+@article{Rice2013,
+  author = {Rice, Alex},
+  title = {{S{\'a}rk{\"o}zy}'s theorem for {$\mathcal P$}-intersective polynomials},
+  journal = {Acta Arithmetica},
+  volume = {157},
+  number = {1},
+  pages = {69--89},
+  year = {2013},
+  doi = {10.4064/aa157-1-4},
+  note = {Corrected version: \href{https://arxiv.org/abs/1111.6559v5}{arXiv:\allowbreak1111.6559v5}, February 1, 2015}
+}
+
+@article{Vaughan1980,
+  author = {Vaughan, R. 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}
+}

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<N,
+ \qquad 1\le\Lambda_B\le1+M_0^kN^{k\tau-1}.
+\end{equation}
+\end{proposition}
+
+\begin{proof}
+The scale bounds follow from $D\le L_D\le D^k$ and $M_0\ge2$; their first
+inequality holds for sufficiently large $N$ because $k\tau<1/2$.
+In particular, with $Q'=n_B^\beta$,
+\begin{equation}\label{eq:shorter-cutoffs}
+ H\le Q'\le Q.
+\end{equation}
+By Proposition~\ref{prop:signed-holder}, it is enough to prove
+\eqref{eq:shorter-scale} for $Z_{l,\mathcal S}(g)$, where $g$ is any one of
+the specialized inputs.
+
+First keep only the frequencies of $g$ with denominator at most $H$,
+and call the resulting real function $g_{\le H}$.  Telescope the diagonal
+integral one slot at a time.  Lemma~\ref{lem:large-denominator}, followed
+by Proposition~\ref{prop:lift-moments} and
+Lemma~\ref{lem:lift-specialization}, bounds the change by
+\begin{equation}\label{eq:specialization-tail}
+ \bigl|Z_{l,\mathcal S}(g)-Z_{l,\mathcal S}(g_{\le H})\bigr|
+ \ll H^{-\gamma}q_BN^{o(1)}=O(N^{-\sigma}).
+\end{equation}
+Indeed specialization costs $q_B^{1/2}$ in the $L^2$ norm of the removed
+part, and $q_B^{1/(2(d-1))}$ in each of the other $d-1$ norms.  Their
+product is $q_B$.  All frequency selections retain whole exact supports
+before specialization, as required by the moment bound.
+Here $N\ge10Q^6$ for large $N$, and the saving follows from
+$\beta\gamma/2-\tau>\sigma$.
+
+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 $0\le i<D$ is supplied by
+\eqref{eq:auxiliary-change}.  We have $Dm-i\ge1$ and
+$T_l(Dm-i)=T_{lD}(m)$, so this would violate avoidance at $l$.
+The nonzero condition is preserved because $\lambda(D)>0$.
+
+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
+$s_{v_{j+1}}-s_{v_j}$, $0\le j<L$, is admissible modulo $M_0$ at $l$.
+Let $\rho_l$ be the proportion of good assignments.  The zero-sum
+admissible sequences of Lemma~\ref{lem:admissible-cycle} give
+\begin{equation}\label{eq:good-proportion}
+ \rho_l\ge\rho_*:=M_0^{1-L}>0.
+\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 $x<y$ because
+$I_1$ precedes $I_2$.  Avoidance therefore gives
+$y-x\notin\mathcal D_l$.  All hypotheses of
+Proposition~\ref{prop:pair-cancellation} hold.
+Its frequency set $\Xi$ is exactly the remaining frequencies on
+$X_{\mathcal S}$ with denominator at most $H$: coprimality with $D$
+excludes the primes below $P$ and the primes in $B$, while every prime
+in a denominator at most $H$ is at most $Q$.
+The coefficient identity becomes
+\[
+ \widehat g_i(\xi)=\frac{DN}{|I_i|}\widehat{J_i}_{[N]}(\xi)
+ \qquad(q(\xi)\le H).
+\]
+Consequently \eqref{eq:pair-cancellation} bounds the low-denominator
+part of \eqref{eq:conditioned-pair} by
+\begin{equation}\label{eq:conditioned-pair-main}
+ O\left(H^{-\gamma/2}D\frac{N^2}{|I_1||I_2|}\right)
+       =O(H^{-\gamma/2}DN^{2\tau}).
+\end{equation}
+
+After multiplication by the expansion bound $O(T^{2(d-2)})$, both
+\eqref{eq:conditioned-pair-tail} and \eqref{eq:conditioned-pair-main}
+are bounded by
+\[
+ N^{-\beta\gamma/4+(3d+2)\tau+o(1)}=O(N^{-\sigma}),
+\]
+since \eqref{eq:parameters} gives
+$\beta\gamma/4>(3d+2)\tau+\sigma$.
+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 $n_B<N$ by
+\eqref{eq:shorter-scale-size}.  The induction hypothesis is available at
+$lD_B$ whether or not that parameter satisfies $lD_B\le n_B^\zeta$.
+Since $n_B\ge N/D_B^k$, it gives
+\[
+ \mathcal Y_{lD_B}(n_B)
+ \le C_0(lD_B)^{b_0}n_B^{-a_0}
+ \le C_0l^{b_0}N^{-a_0}D_B^{s_0}.
+\]
+Use $q_B^{s_0}\le q_B$ and \eqref{eq:exceptional-budget} to bound the
+bracket in \eqref{eq:descent-recurrence} by
+\[
+ \begin{split}
+ C_0l^{b_0}N^{-a_0}M_0^{s_0}
+   \bigl(1-\rho_l+\rho_l/4\bigr)
+ &\le C_0l^{b_0}N^{-a_0}\theta_0,
+ \end{split}
+\]
+where \eqref{eq:good-proportion} was used in the last step.
+Choose a fixed threshold $N_*$, large enough for the recurrence and for
+\[
+ C_1N^{a_0-\sigma}+(1+u_N)\theta_0\le1
+ \qquad(N\ge N_*).
+\]
+Such a threshold exists by \eqref{eq:induction-parameters} and $u_N=o(1)$;
+it is independent of $C_0\ge1$.  Dividing the recurrence by
+$C_0l^{b_0}N^{-a_0}$ now proves the induction step, since
+$C_0l^{b_0}\ge1$.
+Finally choose $C_0$ large enough to cover both the previously treated
+range $l>N^\zeta$ and every $N<N_*$, uniformly in $l$, by
+\eqref{eq:descent-trivial}.  This completes the induction.
+\end{proof}
+
+\begin{proof}[Proof of Theorem~\ref{thm:main}]
+At $l=1$ we have $T_1(m)=m$ and $h_1=h$, so the hypothesis of the theorem
+is exactly avoidance at $l=1$.  Combining \eqref{eq:descent-trivial}
+with \eqref{eq:uniform-descent} gives
+\[
+ \left(\frac{|A|}{N}\right)^d\le C_0N^{-a_0},
+ \qquad |A|\le C_0^{1/d}N^{1-a_0/d}.
+\]
+Thus one may take $c_h=a_0/d$ and $C_h=C_0^{1/d}$, for every $N\ge1$.
+Only strictly ordered pairs, and hence positive differences, were used
+in cancellation.  Restriction to progressions preserved nonzero
+forbidden values and their prime arguments.  A value $h(p)=0$ therefore
+imposes no additional condition at any stage.
+\end{proof}

preprints/A-Power-Saving-for-Polynomial-Differences-at-Prime-Arguments-October-5-2026/build/sections/introduction.tex

@@ -0,0 +1,161 @@
+\section{Introduction}\label{sec:introduction}
+
+A polynomial difference theorem asks how large a set of integers can be
+if the differences of its elements avoid a prescribed polynomial
+sequence. Restricting the polynomial's argument to primes imposes an
+additional arithmetic condition: a root modulo a given modulus must
+be compatible with a prime residue class. We prove a power-saving bound
+when these local conditions hold at every modulus.
+
+Write $\mathbb P$ for the positive rational primes and
+$[N]=\{1,\ldots,N\}$. An integer polynomial $h$ is
+\emph{prime-intersective} if, for every positive integer $q$, there is
+an integer $r$ with
+\[
+ (r,q)=1,\qquad h(r)\equiv0\pmod q.
+\]
+For a set $A$ of integers, write $A-A=\{a-b:a,b\in A\}$.
+
+\begin{theorem}\label{thm:main}
+Let $h\in\Z[x]$ be prime-intersective, of degree at least two and with
+positive leading coefficient. There exist constants $c_h>0$ 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}.

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
+\[
+ J_1(x)J_2(y)\ne0\quad\Longrightarrow\quad
+ x<y\quad\text{and}\quad y-x\notin\mathcal D_l.
+\]
+Set
+\[
+ \Xi=\{\xi\in\Q/\Z:q(\xi)\le H,
+       \ q(\xi)\text{ squarefree},\ (q(\xi),D)=1\}.
+\]
+Then, for the multiplier of any directed cycle edge,
+\begin{equation}\label{eq:pair-cancellation}
+ D\left|\sum_{\xi\in\Xi}\Gamma(\xi)
+       \widehat{J_1}_{[N]}(-\xi)\widehat{J_2}_{[N]}(\xi)\right|
+ \ll H^{-\gamma/2}.
+\end{equation}
+\end{proposition}
+
+Since $M_0$ contains every prime below $P$, the multiplier is defined on all
+of $\Xi$.  To prove the proposition, consider the model kernel
+\begin{equation}\label{eq:model-kernel}
+ k_0(n)=\frac DN\,
+   \mathbf1_{\{1\le n\le N\}}\mathbf1_{\{n\equiv r\ (D)\}}
+   \sum_{\xi\in\Xi}\Gamma(\xi)e(-\xi n).
+\end{equation}
+We will construct a kernel $k_1$ supported on $\mathcal D_l$ such that
+$\sup_\alpha|\sum_n(k_0(n)-k_1(n))e(\alpha n)|\ll H^{-\gamma/2}$.
+The congruence restriction
+in $k_0$ will be imposed on polynomial arguments by a local weight; the
+factors $\Gamma_p$ will be imposed by further weights of mean one.
+
+\subsection{Weights on polynomial arguments}
+
+For a periodic function of the argument $m$, its \emph{unit mean modulo
+$b$} is the uniform average over the classes $m\pmod b$ satisfying
+$(T_l(m),b)=1$.  There are exactly
+\begin{equation}\label{eq:unit-mean-count}
+ n_b=b\prod_{\substack{p\mid b\\p\nmid l}}(1-1/p)
+     =\frac{\varphi(lb)}{\varphi(l)}
+\end{equation}
+such classes.  Indeed, a prime dividing $l$ excludes no argument because
+$(r_l,l)=1$, and any other prime excludes exactly one class modulo that
+prime.  These means factor over prime powers.  They are also consistent
+under replacing $b$ by a multiple: each allowed class has the same number
+of allowed lifts.
+
+At a prime $p\mid D$, choose an argument witnessing admissibility of
+$r\pmod {p^{E_p}}$, and denote the valuation of its derivative by $s_p$.
+Lemma~\ref{lem:local-lifting} supplies a class modulo $p^{E_p-s_p}$ on
+which the unit condition holds and $F$ is constantly $r$ modulo
+$p^{E_p}$.  Uniform argument lifts in this class give uniform higher
+lifts of that value.  Combining the chosen classes gives one class modulo
+\[
+ d_r=\prod_{p\mid D}p^{E_p-s_p}\mid D.
+\]
+Let $\rho_D$ be $n_{d_r}$ times its indicator.  Then $\rho_D$ has unit mean
+one and $\|\rho_D\|_\infty\le D$.
+
+For every prime $p\nmid D$, necessarily $p\ge P$, choose, for each $y$
+with $\pi_p(y)>0$, 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_*<m\le X}&\mathbf1_{\{n=F(m)\}}
+ \frac{F'(m)}{N}\frac{\varphi(l)}{l}\Lambda'(T_l(m))\rho_D(m)\\[-2pt]
+ &\hspace{15mm}\times
+ \sum_{\substack{u\le H\\u\text{ squarefree}\\(u,D)=1}}B_u(m).
+\end{split}
+\end{equation}
+Here and below the argument $m$ is an integer in sums.  Every nonzero
+term has $0<F(m)\le N$ and $T_l(m)$ prime, so $k_1$ is supported on
+$\mathcal D_l\cap[N]$.  No positivity is required of $k_1$.
+
+Choose a fixed constant $C$ so that $Y=ClU$ bounds $T_l(X)$.  The relation
+$a_l=al^k/\lambda(l)$ gives
+\begin{equation}\label{eq:prime-kernel-range}
+ Y\gg N^{1/k},\qquad Y\ll N^{1/k+\zeta},\qquad
+ Y^k\asymp N\lambda(l).
+\end{equation}
+In particular $Y\le N$ for sufficiently large $N$.
+
+\subsection{Uniform Fourier comparison}
+
+For $i=0,1$ use positive-sign transforms
+$S_i(\alpha)=\sum_n k_i(n)e(\alpha n)$ on $\R/\Z$.
+The signs differ from the negative-sign convention for Fourier
+coefficients of functions; this choice makes $S_0$ concentrate at the
+frequencies occurring in \eqref{eq:model-kernel}.
+
+\begin{lemma}[Comparison of the two kernels]\label{lem:kernel-comparison}
+For the data and ranges of Proposition~\ref{prop:pair-cancellation},
+\begin{equation}\label{eq:kernel-comparison}
+ \sup_{\alpha\in\R/\Z}|S_0(\alpha)-S_1(\alpha)|
+ \ll H^{-\gamma/2}.
+\end{equation}
+\end{lemma}
+
+\begin{proof}
+Use major arcs of radius $3N^{-1+\eta}$ around the reduced rationals with
+denominator at most $N^\eta$.  They are disjoint for large $N$, since
+$3\eta<1$.  The remaining points form the minor arcs.  We first bound both
+transforms on the minor arcs; on a major arc we will compare their local
+coefficients.
+
+Write $V_N(y)=N^{-1}\sum_{n\in[N]}e(yn)$.  Expanding the congruence
+indicator in \eqref{eq:model-kernel} gives
+\begin{equation}\label{eq:model-transform}
+ S_0(\alpha)=\sum_{\xi\in\Xi}\sum_{j'\bmod D}
+ \Gamma(\xi)e(-j'r/D)V_N(\alpha-\xi+j'/D).
+\end{equation}
+There are $O(DH^2)$ terms.  Their centers $\xi-j'/D$ are distinct:
+an equality would give a rational with denominator both coprime to $D$
+and dividing $D$, and hence an integer.  Each center has denominator at
+most $DH\le N^{3\beta/2}\le N^\eta$.  On minor arcs, geometric summation
+therefore gives
+\begin{equation}\label{eq:model-minor}
+ |S_0(\alpha)|\ll DH^2N^{-\eta}.
+\end{equation}
+
+To estimate $S_1$, first consider, with any linear twist $y'$, the sums
+\[
+ \sum_{m\in I}\Lambda'(T_l(m))e(\alpha F(m)+y'm)
+\]
+over subintervals $I$ of $(x_*,X]$.  On setting $n=T_l(m)$, expand the
+restriction $n\equiv r_l\pmod l$ as the average of its $l$ additive
+characters.  Each resulting prime sum has a polynomial phase in $n$ whose
+coefficient of $n^k$ is $b_k=\alpha a/\lambda(l)$; the new twists change
+only the linear coefficient because $k\ge2$.
+
+Fix a representative of $\alpha$ and apply Dirichlet approximation to
+$\vartheta=(k!)^2b_k$ with
+\[
+ Q_0=\lfloor Y^kN^{-\eta/2}\rfloor.
+\]
+The resulting reduced fraction $a'/q$ satisfies
+$q\le Q_0$ and $|\vartheta-a'/q|\le1/(qQ_0)\le q^{-2}$.
+By \eqref{eq:prime-kernel-range}, this places $\alpha$ within
+$O(N^{-1+\eta/2})$ of a rational of denominator at most $(k!)^2aq$.
+For a minor-arc point that denominator must exceed $N^\eta$ for large
+$N$.  Hence, using $Y\le N$ and absorbing the fixed factor $(k!)^2a$,
+\[
+ Y^{\eta/4}\le q\le Y^{k-\eta/4}.
+\]
+Proposition~\ref{prop:prime-minor-arcs} applies uniformly in the linear
+twists and in the interval, and gives a bound $O(Y^{1-c_m})$.
+The relation $Y=ClU$ and the lower bound in
+\eqref{eq:prime-kernel-range} imply
+\[
+ Y^{1-c_m}\ll U\,lN^{-c_m/k}
+ \le U N^{\zeta-c_m/k}\ll U N^{-\nu}.
+\]
+This accounts for the dependence on $l$ before any periodic weight is
+expanded.
+
+By \eqref{eq:kernel-polynomial-scale}, the supremum and total variation of
+$F'/N$ on $[x_*,X]$ are $O(U^{-1})$.  Partial summation therefore retains
+an $O(N^{-\nu})$ bound after including this weight.  Use normalized
+Fourier coefficients on each finite period for the other weights.  Since
+$\rho_D$ is a constant times one residue-class indicator, its Fourier
+$\ell^1$ norm is $n_{d_r}\le D$.  For $B_u$ the bound by its period times
+its supremum gives Fourier $\ell^1$ norm at most $u^2$.  Expanding the
+weights in \eqref{eq:prime-kernel}, and using $\varphi(l)/l\le1$, yields
+\begin{equation}\label{eq:prime-minor-kernel}
+ |S_1(\alpha)|\ll D\sum_{u\le H}u^2N^{-\nu}
+ \ll DH^3N^{-\nu}.
+\end{equation}
+The powers in \eqref{eq:model-minor} and
+\eqref{eq:prime-minor-kernel} are $2\beta-\eta$ and
+$5\beta/2-\nu$, respectively.  Both are smaller than $-\beta\gamma/4$,
+so these bounds are $O(H^{-\gamma/2})$.
+
+Now let $\alpha=\alpha_0+y$ be on a major arc, with
+$q=q(\alpha_0)\le N^\eta$ and $|y|\le3N^{-1+\eta}$.
+Decompose $\alpha_0=\alpha_D+\alpha_1$ into the prime-power denominator
+parts supported, respectively, on primes dividing $D$ and on primes not
+dividing $D$.  The coefficient in \eqref{eq:model-transform} belonging
+to the center $\alpha_0$, or zero if that center is absent, is
+\begin{equation}\label{eq:model-local-coefficient}
+ c_0(\alpha_0)=
+ \begin{cases}
+ e(\alpha_Dr)\Gamma(\alpha_1),
+       &q(\alpha_D)\mid D\ \text{and}\ \alpha_1\in\Xi,\\
+ 0,&\text{otherwise}.
+ \end{cases}
+\end{equation}
+Here the sign follows from $\alpha_D=-j'/D$ at that center.  Every other
+center is at circle distance at least $1/(qDH)$ from $\alpha_0$.
+Since $|y|qDH=o(1)$, geometric summation in each of the $O(DH^2)$ terms
+gives
+\begin{equation}\label{eq:model-major}
+ S_0(\alpha)=c_0(\alpha_0)V_N(y)+O(qD^2H^3/N).
+\end{equation}
+
+We next derive the corresponding coefficient for $S_1$, including the
+normalization of the prime weights.  For squarefree $u\le H$ coprime to
+$D$, let $A_u$ be the unit mean of
+\[
+ b_u(m)=\rho_D(m)B_u(m)e(\alpha_0F(m)).
+\]
+Its period divides $b=qDu$, and $\|b_u\|_\infty\le Du$.
+For $x_*\le x'<x''\le X$, Proposition~\ref{prop:prime-progressions}
+gives
+\begin{equation}\label{eq:weighted-prime-progression}
+ \sum_{x'<m\le x''}b_u(m)\frac{\varphi(l)}l\Lambda'(T_l(m))
+ =A_u(x''-x')+
+ O\big(qD^2u^2Y^{31/32}\log^2(2lbY)\big).
+\end{equation}
+Indeed, one class $m\equiv c\pmod b$ corresponds to the class
+$r_l+lc\pmod {lb}$ in a prime interval of length $l(x''-x')$.
+It is reduced exactly when $(T_l(c),b)=1$, since $(r_l,l)=1$.
+For such a class the factor $\varphi(l)/l$ changes the prime main term
+to $(x''-x')/n_b$, by \eqref{eq:unit-mean-count}.
+On a nonreduced class the only possible primes divide $lb$, and their
+total logarithmic weight is $O(\log(lb))$, with zero main term.
+Apply the progression bound at both endpoints, then sum its errors over
+at most $b=qDu$ classes with weights of modulus at most $Du$.
+This proves \eqref{eq:weighted-prime-progression}, including the
+nonreduced classes.
+
+The smooth factor $F'(x)e(yF(x))/N$ has supremum plus total variation
+\[
+ O\big((1+N|y|)/U\big).
+\]
+For example, its derivative is bounded in integral by the contributions
+of $F''/N$ and $2\pi |y|(F')^2/N$, which are $O(U^{-1})$ and
+$O(N|y|/U)$ by \eqref{eq:kernel-polynomial-scale}.
+Partial summation in \eqref{eq:weighted-prime-progression} therefore
+produces main term
+\[
+ \frac{A_u}{N}\int_{x_*}^{X}F'(x)e(yF(x))\,dx
+ =\frac{A_u}{N}\int_{F(x_*)}^{N}e(yt')\,dt'.
+\]
+The normalized integral $N^{-1}\int_{F(x_*)}^{N}e(yt')\,dt'$ differs
+from $V_N(y)$ by
+$O(a_l/N+(1+N|y|)/N)$: the missing initial interval has length
+$F(x_*)\ll a_l$, and comparison of the integral on $[0,N]$ with its
+integer sum costs $O(1+N|y|)$.
+It follows that
+\begin{equation}\label{eq:prime-major}
+ S_1(\alpha)=c(\alpha_0)V_N(y)+O(H^{-\gamma/2}),\qquad
+ c(\alpha_0)=
+ \sum_{\substack{u\le H\\u\text{ squarefree}\\(u,D)=1}}A_u.
+\end{equation}
+Here are the error bounds underlying this assertion.  Since
+$Y^{31/32}/U\ll lN^{-1/(32k)}$, the summed progression error is at most
+\[
+ \frac{1+N|y|}{U}\,qD^2
+ \sum_{u\le H}u^2Y^{31/32}N^{o(1)}
+ \ll N^{2\eta+4\beta+\zeta-1/(32k)+o(1)}.
+\]
+The replacement of the integrals costs, using $|A_u|\le Du$,
+\[
+ O\big(DH^2(U^{-k}+N^{\eta-1})\big)
+ \ll N^{2\beta+k\zeta-1}+N^{2\beta+\eta-1}.
+\]
+The error in \eqref{eq:model-major} is at most
+$N^{\eta+7\beta/2-1}$.  All three estimates are
+$O(N^{-\beta\gamma/4})$: indeed $\beta,\zeta\le\eta/1000$,
+$\eta=1/(10000k)$, and $\gamma<1$.  The first exponent has a strict
+margin, so its $o(1)$ loss is harmless.
+
+It remains to compare $c$ and $c_0$.  This is precisely the reason for
+the unit means and the regular argument classes used above.  At primes
+dividing $D$, the common factor in each $A_u$ is
+\begin{equation}\label{eq:kernel-D-factor}
+ \chi_D(\alpha_0)=
+ \begin{cases}
+ e(\alpha_Dr),&q(\alpha_D)\mid D,\\
+ 0,&\text{otherwise}.
+ \end{cases}
+\end{equation}
+To see both cases, factor $\rho_D$ and the unit mean prime by prime.
+At $p\mid D$ the chosen class has weight of unit mean one and prescribed
+$F$-value modulo $p^{E_p}$.  A character of conductor at most $p^{E_p}$
+therefore gives its value at $r$.  A primitive character of higher
+conductor averages to zero, since Lemma~\ref{lem:local-lifting} gives
+uniform higher lifts of $F$ on that class, all still satisfying the unit
+condition.
+
+Put $q_1=q(\alpha_1)$.  If $u$ has a prime factor outside $q_1$, its
+local $B_p$ factor has unit mean zero and $A_u=0$.
+For $p^j\Vert q_1$, write $\alpha_p$ for the $p^j$-part of $\alpha_1$
+and let
+\[
+ a_{p,j}=\mathbb E_{\substack{m\bmod p^j\\p\nmid T_l(m)}}
+                  e(\alpha_pF(m)),\qquad
+ b_{p,j}=\mathbb E_{\substack{m\bmod p^j\\p\nmid T_l(m)}}
+                  B_p(m)e(\alpha_pF(m)).
+\]
+The surviving $A_u$, for squarefree divisors $u$ of $q_1$, are
+$\chi_D(\alpha_0)$ times products that choose $b_{p,j}$ when $p\mid u$
+and $a_{p,j}$ otherwise.  Moreover,
+\begin{equation}\label{eq:kernel-local-realization}
+ a_{p,j}+b_{p,j}=
+ \begin{cases}
+ \Gamma_p(\alpha_p),&j=1,\\
+ 0,&j\ge2.
+ \end{cases}
+\end{equation}
+For $j=1$, the weight $1+B_p=\psi_p$ pushes the unit mean forward to
+$\pi_p$.  For $j\ge2$, it is supported on allowed regular classes
+modulo $p$, and their uniform argument lifts give uniform lifts of the
+$F$-value.  The primitive character therefore has mean zero.
+
+If $q_1\le H$, every squarefree divisor needed in the product expansion
+occurs in the sum for $c$.  Thus \eqref{eq:kernel-local-realization}
+and \eqref{eq:kernel-D-factor} show that $c=c_0$, including the case of
+a nonsquarefree $q_1$, when both vanish.  If $q_1>H$, 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
+\[
+ |c(\alpha_0)|\le
+ \prod_{p^j\Vert q_1}(|a_{p,j}|+|b_{p,j}|)
+ \le\prod_{p^j\Vert q_1}3p^{-\gamma j}
+ \le q_1^{-\gamma/2}<H^{-\gamma/2}.
+\]
+Indeed $p^{\gamma/2}\ge3$, so each factor is at most
+$p^{-\gamma j/2}$.  Together with \eqref{eq:model-major} and
+\eqref{eq:prime-major}, this proves the comparison on major arcs.
+The minor-arc bounds already proved complete the lemma.
+\end{proof}
+
+\begin{proof}[Proof of Proposition~\ref{prop:pair-cancellation}]
+Extend $J_1,J_2$ by zero to $\Z$ and put
+\[
+ \mathcal B_i=\frac1N\sum_{x,n\in\Z}
+                    k_i(n)J_1(x)J_2(x+n),\qquad i=0,1.
+\]
+To keep track of signs and normalizations, let
+$\widetilde J_i(\alpha)=\sum_xJ_i(x)e(-\alpha x)$, with counting
+measure in this transform.  Character orthogonality gives
+\[
+ \mathcal B_i=\frac1N\int_{\R/\Z}
+ S_i(\alpha)\widetilde J_1(-\alpha)
+                 \widetilde J_2(\alpha)\,d\alpha.
+\]
+Both counting-measure $\ell^2$ norms are at most $\sqrt N$.
+Cauchy--Schwarz, Plancherel, and Lemma~\ref{lem:kernel-comparison}
+therefore imply
+$|\mathcal B_1-\mathcal B_0|\ll H^{-\gamma/2}$.
+The forbidden-support hypothesis gives $\mathcal B_1=0$.
+For every contributing pair in $\mathcal B_0$, the ordering and the
+residue classes imply $1\le n\le N$ and $n\equiv r\pmod D$.
+Substituting \eqref{eq:model-kernel} and setting $y=x+n$ thus yields
+\[
+ \mathcal B_0=\frac{D}{N^2}\sum_{\xi\in\Xi}\Gamma(\xi)
+       \sum_{x,y}J_1(x)J_2(y)e(-\xi(y-x))
+ =D\sum_{\xi\in\Xi}\Gamma(\xi)
+       \widehat{J_1}_{[N]}(-\xi)\widehat{J_2}_{[N]}(\xi).
+\]
+This is exactly \eqref{eq:pair-cancellation}.
+\end{proof}

preprints/A-Power-Saving-for-Polynomial-Differences-at-Prime-Arguments-October-5-2026/build/sections/lifts.tex

@@ -0,0 +1,579 @@
+\section{Multilinear estimates and Fourier lifts}
+\label{sec:lifts}
+
+The tuple laws of Section~\ref{sec:tuples} provide two ways to control
+multilinear averages: reflection positivity compares different inputs
+with a single diagonal quantity, while conditional mixing suppresses
+large Fourier denominators.  We develop both estimates, then prove the
+moment bounds needed to apply them to bounded functions on integer
+intervals.  The signed-form and Fourier-lift arguments follow the corresponding
+section of~\cite{IntersectivePower}.  The signed-form estimates have their
+square-difference counterpart in~\cite[Proposition~4.1 and
+Lemmas~4.2--4.3]{SquarePower}.  We give full proofs of the estimates
+needed here.  The
+change of coordinates in Lemma~\ref{lem:seminorm} carries the auxiliary
+parameter from \(l\) to \(lD\).
+
+Retain the index set \(V=\Pi^w\), where \(\Pi\) consists of the
+permutation words on \([L]\), and put \(d=|V|\); in particular, \(d\) is even.
+Fix a positive integer
+\(l\) and a finite set \(\mathcal P\) of primes at least \(P\).  Set
+\[
+ X_{\mathcal P}=\prod_{p\in\mathcal P}\F_p,
+ \qquad
+ \mu_{l,\mathcal P}=\bigotimes_{p\in\mathcal P}\mu_{l,p}.
+\]
+We collect the prime labels in slot \(v\) into a single variable
+\(z_v\in X_{\mathcal P}\), so \(\mu_{l,\mathcal P}\) is a probability
+law on \(X_{\mathcal P}^V\).  Each single-slot marginal is uniform.
+Norms of functions on \(X_{\mathcal P}\) will always use uniform
+probability measure unless another measure is specified.
+For real functions on this space define
+\[
+ \mathcal Z_{l,\mathcal P}((g_v)_{v\in V})
+ =\int\prod_{v\in V}g_v(z_v)\,d\mu_{l,\mathcal P},
+ \qquad
+ Z_{l,\mathcal P}(g)=\mathcal Z_{l,\mathcal P}((g)_{v\in V}).
+\]
+The empty prime set is allowed: its residue space is a singleton and
+\(Z_{l,\varnothing}(g)=g^d\).
+
+We also fix the Fourier notation on these spaces.  Every frequency has a
+unique additive decomposition
+\[
+ \xi=\sum_{p\in\mathcal P}\frac{a_p}{p}\pmod 1,
+ \qquad 0\le a_p<p,
+\]
+and its character is
+\(e(z\xi)=\prod_{p\in\mathcal P}e(a_pz_p/p)\).
+Equivalently, one may use any integer representative of \(z\) supplied
+by the Chinese remainder theorem.  The \emph{exact support} of this
+character is \(\{p:a_p\ne0\}\); its reduced denominator, denoted
+\(q(\xi)\), is the product of those primes.  For
+\(B\subseteq\mathcal P\) write
+\(q_B=\prod_{p\in B}p\), with \(q_\varnothing=1\).
+
+\subsection{Signed H\"older inequality and seminorm rules}
+
+The finite reflection argument below follows the chessboard method of
+Fr\"ohlich, Israel, Lieb and Simon~\cite[Theorem~4.1]{FILS1978};
+related reflection arguments for graph norms appear in Conlon and
+Lee~\cite[Section~3]{ConlonLee2017}.
+The standard passage from a multilinear H\"older inequality to a seminorm
+is also part of Hatami's criterion for graph norms~\cite[Theorem~2.8 of
+the arXiv version]{Hatami2010}; we prove both steps directly for the
+present tuple laws.
+
+\begin{proposition}[Signed H\"older inequality]
+\label{prop:signed-holder}
+For every positive integer \(l\), finite set \(\mathcal P\) of primes
+at least \(P\), and real functions \(g,g_v:X_{\mathcal P}\to\R\),
+\begin{equation}
+ Z_{l,\mathcal P}(g)\ge0,
+ \qquad
+ \left|\mathcal Z_{l,\mathcal P}((g_v)_{v\in V})\right|
+ \le\prod_{v\in V} Z_{l,\mathcal P}(g_v)^{1/d}.
+ \label{eq:signed-holder}
+\end{equation}
+\end{proposition}
+
+\begin{proof}
+Across any reflection cut of \(V\), the matrix of the product law is
+the tensor product of the corresponding prime-law matrices.  It is
+therefore positive semidefinite.  To explain its consequence for signed
+inputs, fix a cut \(V=V_+\sqcup V_-\) and its reflection \(\theta\).
+Define maps \(\phi_\pm:V\to V_\pm\) that fix the selected half and
+reflect the other half into it.  An assignment of functions is a map
+\(a:v\mapsto g_v\); write its integral as \(\Lambda(a)\).
+The two real functions on the plus-half labels that enter the cut
+bilinear form are
+\[
+ F(x)=\prod_{v\in V_+}a(v)(x_v),
+ \qquad
+ G(x)=\prod_{v\in V_+}a(\theta v)(x_v).
+\]
+Its mixed value is \(\Lambda(a)\), and its two quadratic values are
+\(\Lambda(a\circ\phi_+)\) and \(\Lambda(a\circ\phi_-)\).
+Cauchy--Schwarz for the positive semidefinite matrix gives
+\begin{equation}
+ |\Lambda(a)|^2
+ \le \Lambda(a\circ\phi_+)\Lambda(a\circ\phi_-),
+ \qquad \Lambda(a\circ\phi_\pm)\ge0.
+ \label{eq:fold-cs}
+\end{equation}
+In particular a constant assignment has nonnegative integral, proving
+the first assertion of~\eqref{eq:signed-holder}.
+
+We next give a finite sequence of fold maps that sends every vertex to
+one vertex.  In one block write \(p_i\) for the position of symbol
+\(i\) in its permutation word.  Folding onto the half where \(i\)
+precedes \(i+1\) sorts the pair \((p_i,p_{i+1})\), leaving all other
+entries unchanged.  Apply the comparisons \(i=1,\ldots,L-1\) in that
+order, and repeat this sweep \(L-1\) times.  A sweep moves the largest
+entry to the end.  Once the largest entries occupy their final positions,
+the next sweep leaves them there and places the largest remaining entry
+immediately before them.  Thus every position array becomes
+\((1,\ldots,L)\).  Perform these sweeps in each block.  If their fold
+maps in chronological order are \(\phi_1,\ldots,\phi_m\), then
+\[
+ \phi_m\circ\cdots\circ\phi_1(v)=v_*
+ \quad\text{for every }v\in V,
+\]
+where \(v_*\) has the identity word in every block.
+
+Now fix a finite nonempty list of real functions and let \(M\) be the
+maximum absolute integral over all assignments from that list, allowing
+repetition.  If \(M>0\), 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.

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 $|r_l|<l$ bounds each coefficient by
+$O_h(l^k/\lambda(l))=O_h(a_l)$.
+
+The residues $r_l$ and $r_{lD}$ agree modulo $l$.  Their specified ranges
+therefore give $r_{lD}=r_l-il$ for some $0\le i<D$.  Substitution and
+complete multiplicativity prove \eqref{eq:auxiliary-change}; its last two
+assertions follow from $1\le m_p\le k$.
+
+For the final assertion, factor
+$h=c\prod_i g_i^{d_i}$ over $\Z[x]$, where $c\ne0$ is an integer
+and the $g_i$ are distinct primitive irreducible nonconstant polynomials.
+Exclude primes dividing $c$ or the nonzero integer right-hand sides of
+cleared B\'ezout identities for $g_i,g_j$ with $i\ne j$, and for
+$g_i,g_i'$.  At a root $\mathfrak z_p$ for any remaining prime, exactly one
+factor $g_i$ vanishes, its derivative is a unit, and all the other factors
+have unit values.  Consequently $m_p=d_i$, and the coefficient of
+$X^{m_p}$ in \eqref{eq:root-expansion} is a unit.
+\end{proof}
+
+Define the forbidden values at parameter $l$ by
+\begin{equation}\label{eq:forbidden-family}
+ \mathcal D_l=
+ \{h_l(m):m\in\Z,\ m\ge1,\ T_l(m)\text{ is prime}\}.
+\end{equation}
+A set $A\subseteq[N]$ is \emph{avoiding at $l$} if
+$(A-A)\cap\mathcal D_l\subseteq\{0\}$.  This condition passes to
+progression coordinates: for every $D,n\ge1$ and every integer $c$, the set
+\[
+ A'=\{x\in[n]:\lambda(D)x+c\in A\}
+\]
+is avoiding at $lD$ whenever $A$ is avoiding at $l$.  In fact, a nonzero
+difference $h_{lD}(m)$ in $A'$ would give the nonzero difference
+$\lambda(D)h_{lD}(m)=h_l(Dm-i)$ in $A$.  Formula
+\eqref{eq:auxiliary-change} preserves its prime argument, and
+$Dm-i\ge1$ for $m\ge1$.  This also explains why no restriction at a zero
+polynomial value is needed.
+
+\subsection{Uniform cancellation at large primes}
+
+We next estimate exponential sums over arguments $x$ satisfying
+$p\nmid T_l(x)$, and over the subset on which $h_l'(x)$ is nonzero modulo
+$p$.  Besides controlling local
+Fourier coefficients, the latter estimates will produce zero-sum lists of
+polynomial values: we seek one length $L$, independent of $l$ and $p$,
+for which $L$ selected values sum to zero in each local residue ring.
+At large primes cancellation supplies these lists; at the finitely many
+smaller primes we will repeat a single value.  The selected values must
+also come from argument classes on which values lift uniformly to higher
+prime powers.
+
+Fix
+\begin{equation}\label{eq:local-exponents}
+ \delta=\frac{1}{100k2^k},\qquad \gamma=\frac{\delta}{10}.
+\end{equation}
+For the remainder of this section, $l\ge1$ is arbitrary and $F=h_l$.
+At all sufficiently large primes, $F\bmod p$ is nonconstant, uniformly
+in $l$.  For $p\nmid l$, this follows from $p\nmid a$ and
+\eqref{eq:auxiliary-definition}; for $p\mid l$, formula
+\eqref{eq:auxiliary-padic} reduces to a unit multiple of $Y^{m_p}$.
+When $p>k$, 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
+\[
+ (k-1)(b-1)<n.
+\]
+Since every $F'(x_r)$ is divisible by $p^n$, each interpolating summand
+has all coefficients divisible by $p$.  This contradicts
+$F'\not\equiv0\pmod p$.
+
+Since $b\le n\le j-n$, the class count applies to the full range of
+$x_0$ modulo $p^{j-n}$.  The unit condition retains at least a proportion
+$1/2$ of all arguments.
+Consequently the conditional proportion that can survive $y$-averaging
+is at most
+\[
+ 2(k-1)p^{-b}
+ \le 2(k-1)p^{-j/(3(k-1))},
+\]
+where $\lfloor j/2\rfloor\ge j/3$ for $j\ge2$.  Because
+$8\delta<1/(3(k-1))$, one further increase of $P_1$ makes this at most
+$p^{-8\delta j}$ for every $j\ge2$.  All thresholds were independent of
+$l$.
+\end{proof}
+
+\subsection{Small primes and uniform lifting}
+
+The preceding cancellation estimate will be used only for $p\ge P_1$.
+For the finitely many smaller primes, we instead select a residue class
+on which polynomial values lift uniformly.  The crucial point is that
+its modulus can be fixed independently of $l$.
+
+\begin{lemma}[Uniform lifting classes]\label{lem:local-lifting}
+For every prime $p<P_1$, there is an integer $S_p\ge0$, depending only
+on $h$ and $p$, such that, for every $l\ge1$, some integer $x$ satisfies
+\begin{equation}\label{eq:small-prime-argument}
+ p\nmid T_l(x),\qquad v_p(h_l'(x))\le S_p.
+\end{equation}
+Set
+\begin{equation}\label{eq:local-modulus-exponents}
+ E_p=2S_p+1\quad(p<P_1),\qquad E_p=1\quad(p\ge P_1).
+\end{equation}
+For any prime $p$, let $x$ be an integer satisfying
+\eqref{eq:small-prime-argument} when $p<P_1$, or representing an allowed
+regular argument when $p\ge P_1$.  Write $s=v_p(h_l'(x))$.
+Every integer in the class $x\pmod{p^{E_p-s}}$ satisfies the unit
+condition, and its $h_l$-value is congruent to $h_l(x)$ modulo $p^{E_p}$.
+For each $j>E_p$, 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 $p<P_1$.  For a nonconstant polynomial over $\Z_p$, the
+minimum valuation of its nonconstant coefficients is preserved by an
+affine substitution of unit slope.  One inequality follows by expanding
+the substitution, and the other by applying its inverse.  If $p\nmid l$,
+this observation applied to \eqref{eq:auxiliary-definition} bounds that
+minimum for $F=h_l$ by a constant depending only on $h,p$.  If $p\mid l$,
+use \eqref{eq:auxiliary-padic}: before the unit affine substitution, its
+degree-$m_p$ coefficient is a unit multiple of $b_{m_p,p}$, whose
+valuation is fixed.  The same uniform bound therefore holds in this
+case.  Differentiation increases the minimum valuation by at most
+$\max_{1\le j\le k}v_p(j)$, so the minimum coefficient valuation of
+$F'$ has a uniform upper bound $C_p\ge0$, which we take to be an integer.
+
+There are at least $k$ distinct integers in $[1,kp]$ satisfying
+$p\nmid T_l(x)$.  Interpolate $F'$, of degree at most $k-1$, at any
+$k$ of them.  The interpolation denominators are products of differences
+of integers in this fixed interval.  Let $B_p\ge0$ be a uniform integer
+upper bound for their $p$-adic valuations.  If all $k$ values of $F'$ had
+valuation greater than $C_p+B_p$, Lagrange interpolation would force
+every coefficient of $F'$ to have valuation greater than $C_p$, a
+contradiction.  Thus \eqref{eq:small-prime-argument} holds with
+$S_p=C_p+B_p$.
+
+Now let $p$ be arbitrary and $x,s$ be as in the statement.  In either
+case $E_p\ge2s+1$.  Taylor expansion with integer coefficients gives
+\begin{equation}\label{eq:local-permutation}
+ F(x+p^{E_p-s}y)=F(x)+p^{E_p}\big(c_1y+pC(y)\big),
+ \qquad p\nmid c_1,\quad C\in\Z[y].
+\end{equation}
+Indeed $c_1=p^{-s}F'(x)$, and each term of degree at least two contains,
+after division by $p^{E_p}$, a factor $p^{E_p-2s}$.
+The polynomial $G(y)=c_1y+pC(y)$ permutes the residues modulo every
+power of $p$: for integers $y,z$, the quotient
+$(G(y)-G(z))/(y-z)$ is a unit when $y\ne z$, so
+\[
+ v_p(G(y)-G(z))=v_p(y-z).
+\]
+It follows that $G$ is injective, hence bijective, on every finite
+residue ring.  Formula \eqref{eq:local-permutation} proves constancy
+modulo $p^{E_p}$.  The argument class preserves the unit condition
+because $E_p-s\ge1$.
+
+Finally, uniform arguments in this class modulo $p^j$ correspond to
+uniform $y$ modulo $p^{j-E_p+s}$.  Each residue of $y$ modulo
+$p^{j-E_p}$ occurs exactly $p^s$ times.  Applying the permutation $G$
+modulo $p^{j-E_p}$ therefore gives precisely the asserted uniform
+distribution of polynomial values.
+\end{proof}
+
+Call a residue modulo $p^{E_p}$ \emph{admissible at $l$} if it equals
+$h_l(x)$ for an argument satisfying \eqref{eq:small-prime-argument}
+when $p<P_1$, or for an allowed regular argument when $p\ge P_1$.
+The equality is taken modulo $p^{E_p}$.  Thus every admissible residue
+has an argument class satisfying the unit condition and the uniform lifting property of
+Lemma~\ref{lem:local-lifting}.  For a modulus that is a product of
+distinct factors $p^{E_p}$, admissibility means admissibility at every
+factor.
+
+\begin{lemma}[A fixed admissible cycle length]\label{lem:admissible-cycle}
+Fix an integer $L>2$ such that
+\begin{equation}\label{eq:cycle-length}
+ 4\delta L>2,\qquad p^{E_p}\mid L\quad(p<P_1).
+\end{equation}
+For every $l\ge1$ and every prime $p$, there are $L$ admissible residues
+$y_0,\ldots,y_{L-1}$ modulo $p^{E_p}$, with repetitions allowed, such
+that
+\[
+ y_0+\cdots+y_{L-1}=0\pmod{p^{E_p}}.
+\]
+\end{lemma}
+
+\begin{proof}
+For $p<P_1$, an admissible residue exists by
+Lemma~\ref{lem:local-lifting}.  Repeat it $L$ times and use
+$p^{E_p}\mid L$.  For $p\ge P_1$, take independent uniform allowed
+regular arguments $x_0,\ldots,x_{L-1}$ modulo $p$.  Character
+orthogonality and \eqref{eq:local-sums} give
+\[
+ \left|p\,\mathbb P\left(\sum_{j=0}^{L-1}F(x_j)=0\pmod p\right)-1\right|
+ \le (p-1)p^{-4\delta L}<1.
+\]
+The probability is therefore positive, yielding the required sequence
+of admissible values.
+\end{proof}
+
+The terminology refers to the residue labels
+$z_0=0$ and $z_j=y_0+\cdots+y_{j-1}$ for $1\le j\le L$:
+we have $z_L=z_0$, and the increment on the indexed edge $j\to j+1$
+is the admissible value $y_j$.  Distinct indices need not carry distinct
+residue labels.
+
+We now keep $P_1$, the exponents $E_p$, and $L$ fixed.  For any later
+threshold $P>P_1$, apply Lemma~\ref{lem:admissible-cycle} at each $p<P$
+and combine the lists by the Chinese remainder theorem.  This gives $L$
+admissible residues modulo $\prod_{p<P}p^{E_p}$ with sum zero.  Thus a
+larger prime cutoff never requires a new cycle length.

preprints/A-Power-Saving-for-Polynomial-Differences-at-Prime-Arguments-October-5-2026/build/sections/primes.tex

@@ -0,0 +1,488 @@
+\section{Prime estimates}\label{sec:primes}
+
+The kernel construction requires two uniform estimates: an asymptotic
+formula for primes in residue classes, and cancellation in polynomial
+exponential sums over primes.  We derive the first from a zero-free
+half-plane theorem and prove the second by Vaughan's identity and Weyl
+differencing.  The latter argument also records why the product cutoff
+and the arithmetic coefficients do not obstruct mixed differencing.
+
+For positive integers $n$, put
+\[
+ \Lambda'(n)=(\log n)\mathbf 1_{\{n\text{ is prime}\}},\qquad
+ \Lambda(n)=
+ \begin{cases}\log p,&n=p^j\text{ for a prime }p\text{ and }j\ge1,\\
+ 0,&\text{otherwise}.
+ \end{cases}
+\]
+All sums below run over integers, including sums whose endpoints are
+real.  The constants in this section are independent of the lower
+coefficients of the polynomial phases.
+
+\subsection{Primes in arithmetic progressions}
+
+Here is the precise analytic input from the companion zero-free theorem.
+Only its assertion about Dirichlet $L$-functions is needed.
+
+\begin{theorem}[Zero-free input, {\cite[Theorem~1.1]{ZeroFree}}]
+\label{thm:zero-free-input}
+For every integer $b\ge1$ and every Dirichlet character $\chi$ modulo
+$b$, the Dirichlet $L$-function $L(s,\chi)$ has no zero in
+$\Re s>7/8$.  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
+$2-\sigma_0<r<R<2-7/8$ and use disks of radius $R$ centered at
+$s_t=2+it$.  Theorem~\ref{thm:zero-free-input} makes $G_\chi$
+holomorphic and nonvanishing on each disk, with the principal pole
+removed.  To bound its size there, use periodicity to write
+\[
+ \sum_{n\le u}\chi(n)=\overline\chi\,u+E_\chi(u),\qquad
+ |E_\chi(u)|\le2b,\qquad
+ \overline\chi=\begin{cases}\varphi(b)/b,&\delta_\chi=1,\\0,&\delta_\chi=0.
+ \end{cases}
+\]
+Partial summation continues the Dirichlet series to $\Re s>0$ 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 $0<H<x$, positivity of $\Lambda$ gives
+\[
+ \frac{A_c(x)-A_c(x-H)}{H}
+ \le \sum_{\substack{n\le x\\n\equiv c\pmod b}}\Lambda(n)
+ \le \frac{A_c(x+H)-A_c(x)}{H}.
+\]
+Take $H=x^{31/32}$ for sufficiently large $x$.  The main terms differ
+from $x/\varphi(b)$ by $O(H)$, and the errors are
+\[
+ O\bigl(x^{1+\sigma_0}/H\cdot\log(2b)\bigr)
+   =O\bigl(x^{31/32}\log(2b)\bigr).
+\]
+Bounded $x$ is harmless.  Finally,
+\[
+ \sum_{\substack{p^j\le Y\\j\ge2}}\log p
+ \ll Y^{1/2}\log^2(2Y),
+\]
+so removing higher prime powers proves~\eqref{eq:prime-progression}.
+If a prime lies in a nonreduced residue class modulo $b$, it divides
+$b$; the sum of the logarithms of distinct such primes is at most
+$\log b$.  This proves the last assertion.
+\end{proof}
+
+\subsection{A minor-arc estimate over primes}
+
+\begin{proposition}[Prime polynomial sums]\label{prop:prime-minor-arcs}
+For fixed $k\ge2$ and $0<\rho<1$, there is a constant
+$c_m=c_m(k,\rho)\in(0,1/4)$ with the following property.  If $Y$ is
+sufficiently large, $I\subseteq[1,Y]$ is an interval, and
+$f\in\R[x]$ has degree at most $k$ with coefficient $b_k$ of $x^k$,
+then
+\begin{equation}\label{eq:prime-minor-arc}
+ \left|\sum_{n\in I}\Lambda'(n)e(f(n))\right|
+ \ll_{k,\rho}Y^{1-c_m},
+\end{equation}
+provided $\theta=(k!)^2b_k$ has an approximation $a/q$ satisfying
+\begin{equation}\label{eq:minor-rational-range}
+ (a,q)=1,\qquad |\theta-a/q|\le q^{-2},\qquad
+ Y^\rho\le q\le Y^{k-\rho}.
+\end{equation}
+\end{proposition}
+
+Vaughan's identity will split the prime weight into sums with one short
+variable and bilinear sums.  Weyl differencing turns each into averages
+of linear phases, which rational approximation then bounds.  We first
+prove the needed spacing and differencing estimates.  Uniformity in the
+lower coefficients and in the interval will later permit additive linear
+twists for periodic restrictions and partial summation for smooth weights
+in the kernel construction.
+
+Write $\|u\|$ for the distance of $u\in\R$ to the
+nearest integer, and interpret $\min(M,\|u\|^{-1})$ as $M$ when
+$\|u\|=0$.
+
+\begin{lemma}[Rational spacing]\label{lem:rational-spacing}
+Suppose $(a,q)=1$, $q\ge2$, and $|\theta-a/q|\le q^{-2}$.
+For $R\ge1$ and $M\ge1$,
+\begin{equation}\label{eq:rational-spacing}
+ \sum_{1\le r\le R}\min(M,\|\theta r\|^{-1})
+ \ll (R/q+1)\bigl(M+q\log(2q)\bigr).
+\end{equation}
+\end{lemma}
+
+\begin{proof}
+Within a block of $\lfloor q/2\rfloor$ consecutive integers, two distinct
+indices differ by a nonzero $d$ with $|d|<q/2$.  Thus
+\[
+ \|\theta d\|\ge \|ad/q\|-|d|/q^2\ge1/(2q).
+\]
+The associated points are $1/(2q)$ separated on the circle.  The nearest
+point to zero contributes at most $M$, and summing the others in bands
+of width $1/(2q)$ gives $O(q\log(2q))$, with another $O(M)$ for the
+bounded number of points in the first band.  Partitioning the indices
+into such blocks proves the result, also when $q=2$ or $3$.
+\end{proof}
+
+We will repeatedly group a fixed number of nonzero integer factors by
+their product.  If that product has absolute value at most $Y^{C_k}$,
+the number of such factorizations is $Y^{o(1)}$, uniformly in the
+product.  Indeed, its prime factorization bounds the number by a fixed
+order divisor function, up to a bounded number of sign choices.  For
+every $\epsilon>0$ 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}|<M}
+ \min\bigl(M,\|k!b_km^ku_1\cdots u_{k-1}\|^{-1}\bigr).
+\end{equation}
+
+Next let $A,B$ be positive integers, let $I\subseteq[1,Y]$ be an
+interval, and let $|a_m|,|d_n|\le1$.  Set
+\[
+ S=\sum_{\substack{A\le m<2A\\B\le n<2B}}
+       a_md_n\mathbf 1_{\{mn\in I\}}e(f(mn)),\qquad
+ \theta=(k!)^2b_k.
+\]
+Then
+\begin{equation}\label{eq:type-two-differencing}
+ \left|\frac{S}{AB}\right|^{2^{2k-1}}
+ \ll_k\frac1{A^kB^k}
+ \sum_{\substack{|u_1|,\ldots,|u_k|<B\\
+                  |v_1|,\ldots,|v_{k-1}|<A}}
+ \min\bigl(A,\|\theta u_1\cdots u_kv_1\cdots v_{k-1}\|^{-1}\bigr).
+\end{equation}
+Zero shifts are included in both sums.
+\end{lemma}
+
+\begin{proof}
+We give the mixed argument explicitly.  For a function $g$ of two
+integer variables, define
+\[
+ \Delta^n_u g(m,n)=g(m,n+u)-g(m,n),\qquad
+ \Delta^m_v g(m,n)=g(m+v,n)-g(m,n).
+\]
+Let $D_0$ be the set of pairs in the displayed box with $mn\in I$.
+Extend $a_m$ and $d_n$ by zero outside their respective intervals,
+and extend every summand by zero outside $D_0$.  Cauchy--Schwarz first
+removes $a_m$:
+\[
+ \left|\frac{S}{AB}\right|^2
+ \le\frac1A\sum_{A\le m<2A}
+       \left|\frac1B\sum_n d_n\mathbf 1_{D_0}(m,n)e(f(mn))\right|^2.
+\]
+Expanding the square and grouping by $u=n'-n$ bounds its right side by
+\[
+ \frac1B\sum_{|u|<B}
+ \left|\frac1{AB}\sum_{m,n}
+    d_{n+u}\overline{d_n}\,
+    \mathbf 1_{D_0}(m,n+u)\mathbf 1_{D_0}(m,n)
+    e\bigl(\Delta^n_u f(mn)\bigr)\right|.
+\]
+For fixed $u$ the new weight depends only on $n$ and has modulus at
+most one.  This is exactly the form needed for another $n$-difference.
+In particular, the product cutoff remains inside the sum; it has not
+been replaced by a rectangular domain.
+
+For $j$ fixed shifts $\mathbf u=(u_1,\ldots,u_j)$, let
+\[
+ \begin{split}
+ D_{\mathbf u}&=\{(m,n):
+      (m,n+\varepsilon\cdot\mathbf u)\in D_0
+                   \text{ for every }\varepsilon\in\{0,1\}^j\},\\
+ F_{\mathbf u}(m,n)&=
+    \Delta^n_{u_j}\cdots\Delta^n_{u_1}f(mn).
+ \end{split}
+\]
+The accompanying coefficient $d_{\mathbf u}(n)$ is defined recursively
+by $d_\varnothing(n)=d_n$ and
+\[
+ d_{(\mathbf u,u)}(n)
+   =d_{\mathbf u}(n+u)\overline{d_{\mathbf u}(n)}.
+\]
+It depends only on $n$ and is bounded by one.  At each subsequent step,
+Cauchy--Schwarz in the existing shifts costs only a constant depending
+on $j$, because their total normalized mass is at most $2^j$.
+Cauchy--Schwarz in $m$, followed by the same square expansion, adds one
+$n$-shift.  Induction therefore gives
+\begin{equation}\label{eq:n-difference-stage}
+ \left|\frac{S}{AB}\right|^{2^k}
+ \ll_k\frac1{B^k}\sum_{|u_1|,\ldots,|u_k|<B}
+ \left|\frac1{AB}\sum_{m,n}
+       d_{\mathbf u}(n)\mathbf 1_{D_{\mathbf u}}(m,n)
+       e(F_{\mathbf u}(m,n))\right|.
+\end{equation}
+All summands still have $m$ and $n$ in their original intervals, as the
+zero vertex of every translated cube belongs to $D_0$.
+
+After $k$ differences in $n$, every lower-degree term vanishes, and
+\[
+ F_{\mathbf u}(m,n)=k!b_k u_1\cdots u_k m^k.
+\]
+For $0\le j\le k-1$ and $m$-shifts
+$\mathbf v=(v_1,\ldots,v_j)$, define the domains
+\[
+ \begin{split}
+ D_{\mathbf u,\mathbf v}=\{(m,n):\;&
+ (m+\eta\cdot\mathbf v,n+\varepsilon\cdot\mathbf u)\in D_0\\
+ &\text{for every }\eta\in\{0,1\}^j,
+                         \ \varepsilon\in\{0,1\}^k\}.
+ \end{split}
+\]
+Thus $D_{\mathbf u,\varnothing}=D_{\mathbf u}$.  For $1\le j\le k-1$, let
+\[
+ T_{\mathbf u,\mathbf v}
+ =\frac1{AB}\sum_{m,n}\mathbf 1_{D_{\mathbf u,\mathbf v}}(m,n)
+ e\bigl(\Delta^m_{v_j}\cdots\Delta^m_{v_1}F_{\mathbf u}(m,n)\bigr).
+\]
+To enter this unweighted family, remove $d_{\mathbf u}$ by
+Cauchy--Schwarz in $n$:
+\[
+ \left|\frac1{AB}\sum_{m,n}
+ d_{\mathbf u}(n)\mathbf 1_{D_{\mathbf u}}(m,n)e(F_{\mathbf u}(m,n))
+ \right|^2
+ \le\frac1B\sum_{B\le n<2B}
+ \left|\frac1A\sum_m
+    \mathbf 1_{D_{\mathbf u}}(m,n)e(F_{\mathbf u}(m,n))\right|^2.
+\]
+Expanding the square bounds this by
+$A^{-1}\sum_{|v_1|<A}|T_{\mathbf u,(v_1)}|$.
+This is the first $m$-differencing step, so removing the coefficients
+requires no additional squaring.  For $1\le j\le k-2$, Cauchy--Schwarz
+in $n$ and the same square expansion give
+\[
+ |T_{\mathbf u,\mathbf v}|^2
+ \le\frac1A\sum_{|v|<A}|T_{\mathbf u,(\mathbf v,v)}|.
+\]
+Applying Cauchy--Schwarz also to all existing shift variables, whose
+total normalized mass is bounded in terms of $k$, now yields
+\begin{equation}\label{eq:m-difference-stage}
+ \left|\frac{S}{AB}\right|^{2^{k+j}}
+ \ll_k\frac1{B^kA^j}
+ \sum_{\substack{|u_1|,\ldots,|u_k|<B\\|v_1|,\ldots,|v_j|<A}}
+          |T_{\mathbf u,\mathbf v}|\qquad(1\le j\le k-1).
+\end{equation}
+This accounts for all $k+(k-1)$ squarings.
+
+Take $j=k-1$.  For fixed $n$, the section of $D_{\mathbf u,\mathbf v}$
+is an interval of $m$'s, possibly empty.  In fact each
+condition puts $m+\eta\cdot\mathbf v$ in $[A,2A)$ and its product with
+the positive number $n+\varepsilon\cdot\mathbf u$ in $I$; it therefore
+specifies an interval in $m$.  Their intersection is an interval of
+length at most $A$.  The final phase
+$\Delta^m_{v_{k-1}}\cdots\Delta^m_{v_1}F_{\mathbf u}$ is linear in $m$
+with coefficient
+\[
+ (k!)^2b_k\,u_1\cdots u_kv_1\cdots v_{k-1}.
+\]
+A geometric-sum estimate bounds each $m$-sum by the corresponding
+minimum in~\eqref{eq:type-two-differencing}.  Summing over at most $B$
+values of $n$, the remaining normalized sum is bounded by that minimum
+divided by $A$.  The shifts have normalizing factor
+$B^{-k}A^{-(k-1)}$, which proves~\eqref{eq:type-two-differencing}.
+
+For~\eqref{eq:type-one-differencing}, perform the same square expansion
+$k-1$ times in a single variable.  The translated intersections of $J$
+are intervals of length at most $M$, and the final linear coefficient
+is $k!b_km^ku_1\cdots u_{k-1}$.  The normalizing factor is
+$M^{-(k-1)}$, followed by $M^{-1}$ for the final geometric sum.
+This proves the first bound as well.
+\end{proof}
+
+\begin{proof}[Proof of Proposition~\ref{prop:prime-minor-arcs}]
+We now apply the differencing bounds to Vaughan's decomposition.
+We first prove the estimate with $\Lambda$ in place of $\Lambda'$.
+Put $e_0=\rho/(100k)$ and $Z=Y^{e_0}$.  For an arithmetic function
+$g$, write $g_{\le Z}(n)=g(n)\mathbf 1_{\{n\le Z\}}$ and
+$g_{>Z}=g-g_{\le Z}$.  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 $AB<Y^{1-e_0}$,
+the trivial estimate is already sufficient.  Otherwise put
+$R=A^{k-1}B^k$ and apply~\eqref{eq:type-two-differencing}.
+
+There are $k$ shifts of range $B$ and $k-1$ of range $A$.
+Tuples containing a zero shift contribute
+$O_k(A^{-1}+B^{-1})$.  Group the others by their nonzero absolute
+product, at most $R$.  Lemma~\ref{lem:rational-spacing} and the divisor
+bound imply
+\begin{equation}\label{eq:type-two-saving}
+ \left|\frac{S}{AB}\right|^{2^{2k-1}}
+ \ll Y^{o(1)}\left(
+ \frac1A+\frac1B+\frac1q+\frac1R+\frac q{AR}\right).
+\end{equation}
+Indeed the nonzero-shift bound before expansion is
+\[
+ \frac{Y^{o(1)}}{AR}(R/q+1)(A+q\log(2q));
+\]
+the additional term $O(\log(2q)/A)$ is included in $Y^{o(1)}/A$.
+Now $A,B\ge Y^{e_0}/2$, while
+\[
+ AR=(AB)^k\ge Y^{k(1-e_0)},\qquad
+ \frac q{AR}\le Y^{-\rho+ke_0}.
+\]
+Thus~\eqref{eq:type-two-saving} saves a fixed power.  Restoring the
+coefficients and summing the boxes preserves a positive saving.
+
+The remaining term $\Lambda_{\le Z}$ costs only $O(Z\log(2Z))$.
+This proves the required estimate for $\Lambda$, with some positive
+exponent depending only on $k,\rho$.  Removing higher prime powers
+costs $O(Y^{1/2}\log^2(2Y))$, as in
+Proposition~\ref{prop:prime-progressions}.  Decreasing the exponent
+if necessary gives $c_m\in(0,1/4)$ and proves
+\eqref{eq:prime-minor-arc}.
+\end{proof}

preprints/A-Power-Saving-for-Polynomial-Differences-at-Prime-Arguments-October-5-2026/build/sections/tuples.tex

@@ -0,0 +1,510 @@
+\section{Reflection-positive laws on residue tuples}\label{sec:tuples}
+
+We now construct a probability law on finitely many residue labels at each
+large prime.  The law must serve two purposes: reflection positivity will
+give a multilinear H\"older inequality, while a designated cycle of labels
+will have admissible increments except on a small part of the measure.
+We adapt the reflection-positive construction of
+\cite[Section~3]{IntersectivePower}, imposing the allowed regular argument condition
+from the preceding section.  The proof below includes the positivity
+argument, since the small exceptional measure and the uniformity in the
+auxiliary index $l$ are both needed later.
+
+\subsection{Slots, reflections, and the required properties}
+
+Keep $L$ from Lemma~\ref{lem:admissible-cycle} and the exponents
+$\delta$, $\gamma=\delta/10$ from \eqref{eq:local-exponents} fixed.  Choose an integer
+$t\ge1$ such that $(t+1)\delta/2>5$, 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
+\[
+ v_j=(c^j,\ldots,c^j)\quad(0\le j<L),\qquad v_L=v_0.
+\]
+These slots, in this order, form the designated cycle.
+
+A \emph{cut} is specified by a block $b\in[w]$ and an ordered pair of
+distinct symbols $\alpha,\beta$.  Its plus side $V_+$ consists of slots
+in whose $b$th word $\alpha$ precedes $\beta$; its minus side is
+$V_-=V\setminus V_+$.  The involution $\theta$ exchanges these two
+symbols in block $b$ and fixes all other blocks.  Thus
+$|V_+|=|V_-|=d/2$, and $\theta$ identifies the two halves.
+For a measure $\nu$ on $\F_p^V$, the matrix of masses across this
+cut has rows and columns indexed by $a,a'\in\F_p^{V_+}$ and entry
+\[
+ \nu\{z_v=a_v, z_{\theta v}=a'_v\text{ for every }v\in V_+\}.
+\]
+We call $\nu$ \emph{reflection positive} if every such matrix is symmetric
+positive semidefinite.  Equivalently, each matrix is symmetric and
+\[
+ \int J((z_v)_{v\in V_+})J((z_{\theta v})_{v\in V_+})\,d\nu(z)\ge0
+\]
+for every real function $J$ on $\F_p^{V_+}$.
+
+\begin{proposition}[Tuple laws]\label{prop:tuple-laws}
+There is a threshold $P>P_1$, 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 $z_{v_{j+1}}-z_{v_j}$, $0\le j<L$, is admissible at $l$.
+\item For any integer $D\ge1$ with $p\nmid D$ and any
+$a'\in\F_p$, let $A_{D,a'}$ act on all labels by
+$z_v\mapsto\lambda(D)^{-1}(z_v-a')$.  Then
+\begin{equation}\label{eq:tuple-covariance}
+ (A_{D,a'})_*\mu_{l,p}=\mu_{lD,p},\qquad
+ (A_{D,a'})_*\varepsilon_{l,p}=\varepsilon_{lD,p}.
+\end{equation}
+\item For $\rho_p=\mu_{l,p}$ or $\mu_{l,p}^{\circ}$, every
+$v\in V$, and every $f:\F_p\to\C$ of uniform mean zero,
+\begin{equation}\label{eq:tuple-mixing}
+ \left\|\mathbb E_{\rho_p}
+   \bigl[f(z_v)\mid(z_u)_{u\ne v}\bigr]\right\|_{L^2(\rho_p)}
+ \le p^{-\gamma}\|f\|_{L^2(\F_p)}.
+\end{equation}
+\end{enumerate}
+All thresholds and constants are independent of $l$.
+\end{proposition}
+
+The construction starts from independent labels weighted by admissible
+increments.  Such a law has the cycle property and the required mixing,
+but need not be reflection positive.  We add components in which selected
+blocks are forgotten when labels are assigned.  For cuts in those blocks,
+reflected slots share a list of labels.  We choose bounded interactions
+between their selection indices so that the cut quadratic form controls
+the mean square of its test function.  This lower bound will dominate the possible
+negative contribution from the preceding component.
+The last components need no cycle constraints, and form the exceptional
+submeasure.
+
+\subsection{Two estimates for weighted graphs}
+
+Fix $l$ and $p\ge P_1$, and write $F=h_l$.  Define
+\begin{equation}\label{eq:tuple-edge-weight}
+ W(y)=\frac1k\#\{x\in\F_p:F(x)=y, x\text{ allowed regular}\},
+ \qquad c_p=\mathbb E_{y\in\F_p}W(y).
+\end{equation}
+Since $F\bmod p$ is nonconstant of degree at most $k$, we have
+$0\le W\le1$ and $c_p\ge1/(2k)$.  Lemma~\ref{lem:local-sums}
+for allowed regular arguments gives, with negative-sign Fourier
+coefficients,
+\begin{equation}\label{eq:tuple-edge-fourier}
+ |\widehat W(a)|\le p^{-4\delta}\qquad(a\in\F_p\setminus\{0\}).
+\end{equation}
+Indeed $\widehat W(a)$ is the normalized mean over allowed regular
+arguments multiplied by their proportion divided by $k$.
+
+\begin{lemma}[Weighted graph estimates]\label{lem:graph-estimates}
+Let $G$ be a fixed finite directed graph with no loops and with at most one
+edge on each unordered pair of vertices.  Give its vertices independent
+uniform labels $Y_x\in\F_p$, and put
+\[
+ I_G(Y)=\prod_{x\to y\in E(G)}W(Y_y-Y_x),\qquad e_G=|E(G)|.
+\]
+Then
+\begin{equation}\label{eq:tuple-graph-mean}
+ \mathbb E I_G=c_p^{e_G}+O_G(p^{-4\delta}).
+\end{equation}
+If the vertices are partitioned into two classes and every edge joins
+different classes, regard $I_G$ as a kernel between their uniform label
+spaces.  Its associated operator satisfies
+\begin{equation}\label{eq:tuple-graph-operator}
+ \|I_G-c_p^{e_G}\|_{L^2\to L^2}=O_G(p^{-\delta}).
+\end{equation}
+Both bounds are uniform in $l$.
+\end{lemma}
+
+\begin{proof}
+The convolution kernel $W(y-x)-c_p$ has $L^2$ operator norm at most
+$p^{-4\delta}$ by \eqref{eq:tuple-edge-fourier}.  Telescope the product
+$I_G$ by replacing one edge weight at a time by $c_p$.  In a resulting
+term, hold fixed all labels except those at the endpoints $x,y$ of the
+edge being replaced.  As there is no second edge on this pair, the other
+factors depending on these two labels separate as $a(Y_x)b(Y_y)$,
+with $|a|,|b|\le1$.  The convolution bound controls the conditional
+average by $p^{-4\delta}$.  Summing over the edges proves
+\eqref{eq:tuple-graph-mean}.
+
+For the second assertion write $A,B$ for the label spaces of these two classes and
+$D(a,b)=I_G(a,b)-c_p^{e_G}$.  In uniform $L^2$ spaces, the fourth
+Schatten moment of its operator $T_D$ is
+\[
+ \operatorname{tr}\bigl((T_DT_D^*)^2\bigr)
+ =\mathbb E_{\substack{a,a'\in A\\b,b'\in B}}
+   D(a,b)D(a',b)D(a',b')D(a,b').
+\]
+All kernels here are real.  Expand the four differences.  Each term is
+the graph average of the selected copies of $I_G$, times the corresponding
+power of $-c_p^{e_G}$.  Its formal vertices consist of two separate
+copies of each of the two vertex classes.  No pair of formal vertices receives two
+edges: it determines one corner of the rectangle and one edge of $G$.
+Thus \eqref{eq:tuple-graph-mean} applies to every term.  Replacing all its
+edge weights by $c_p$ makes the expanded sum zero, so the fourth moment
+is $O_G(p^{-4\delta})$.  The fourth power of the largest singular value
+is at most this moment, proving \eqref{eq:tuple-graph-operator}.
+\end{proof}
+
+\subsection{Components obtained by forgetting blocks}
+
+For $S\subseteq[w]$ with $s=|S|\le t+1$, call the blocks in $S$
+marked.  For $v\in V$, its \emph{address} $u_S(v)$ is the tuple of its
+words in the unmarked blocks; the address set is
+$U_S=\Pi^{[w]\setminus S}$.  At every address $u$ take a list
+\[
+ (Y_{u,1},\ldots,Y_{u,\ell_s}),\qquad \ell_s=r_*^s,
+\]
+with all entries independent and uniform in $\F_p$, independently
+over the addresses as well.  A slot of address $u$ will choose one of
+the labels in this list.
+
+For $s\le t$ put a directed edge $u\to u'$ between addresses whenever
+\begin{equation}\label{eq:tuple-address-edge}
+ u'_b=u_bc\quad\text{in at least }t+1\text{ unmarked blocks }b.
+\end{equation}
+Every entry of the first list is joined to every entry of the second,
+with weight
+\[
+ I_S(Y)=\prod_{u\to u'}\prod_{i,i'\in[\ell_s]}
+         W(Y_{u',i'}-Y_{u,i}).
+\]
+There are no loops, since $c\ne1$.  There are also no opposite edges:
+because $c^2\ne1$, the witnessing blocks for $u\to u'$ and for
+$u'\to u$ would be disjoint, requiring $2t+2>w$ 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_{p<P}p^{E_p}\ge2.
+\end{equation}
+The previously chosen $L$ is unchanged when $P$ is enlarged.  For
+$D=M_0$ times a product of distinct primes at least $P$, a residue
+modulo $D$ is called admissible at $l$ if it is admissible at every local
+factor $p^{E_p}$ of $D$.

preprints/A-Power-Saving-for-Polynomial-Differences-at-Prime-Arguments-October-5-2026/prime-argument-polynomial-differences.pdf

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:b42ac864305ade86faecfe7a599249cf07a47681d7ce2008b51f6c8a3591f819
+size 674642

preprints/A-Sharp-Threshold-Bound-for-Monotone-Graph-Properties-September-25-2026/.gitignore

@@ -0,0 +1 @@
+.build-output/

preprints/A-Sharp-Threshold-Bound-for-Monotone-Graph-Properties-September-25-2026/README.md

@@ -0,0 +1,18 @@
+# [A Sharp Threshold Bound for Monotone Graph Properties](paper.pdf)
+
+**Author:** OpenAI
+
+**Date:** September 25, 2026
+
+## Citation
+
+```bibtex
+@misc{OAI:A-Sharp-Threshold-Bound-for-Monotone-Graph-Properties-September-25-2026,
+  author = {{OpenAI}},
+  title = {{A Sharp Threshold Bound for Monotone Graph Properties}},
+  howpublished = {OpenAI Math Release preprint
+                  \href{https://github.com/openai/math/blob/main/preprints/A-Sharp-Threshold-Bound-for-Monotone-Graph-Properties-September-25-2026/paper.pdf}{OAI:A-Sharp-Threshold-Bound-for-Monotone-Graph-Properties-September-25-2026}},
+  year = {2026}
+}
+```
+

preprints/A-Sharp-Threshold-Bound-for-Monotone-Graph-Properties-September-25-2026/build/figures/block.tex

@@ -0,0 +1,82 @@
+\begin{figure}[tb]
+\centering
+\setlength{\unitlength}{0.93pt}
+\begin{picture}(460,190)
+ \put(12,176){\makebox(202,12){\small (a) The free coordinate set $F_B$}}
+ \put(246,176){\makebox(202,12){\small (b) One Fourier support $S$}}
+ \linethickness{0.45pt}
+ \put(40,164){\line(1,0){52}}
+ \qbezier(92,164)(108,164)(108,148)
+ \put(108,56){\line(0,1){92}}
+ \qbezier(108,56)(108,40)(92,40)
+ \put(40,40){\line(1,0){52}}
+ \qbezier(40,40)(24,40)(24,56)
+ \put(24,56){\line(0,1){92}}
+ \qbezier(24,148)(24,164)(40,164)
+ \put(274,164){\line(1,0){52}}
+ \qbezier(326,164)(342,164)(342,148)
+ \put(342,56){\line(0,1){92}}
+ \qbezier(342,56)(342,40)(326,40)
+ \put(274,40){\line(1,0){52}}
+ \qbezier(274,40)(258,40)(258,56)
+ \put(258,56){\line(0,1){92}}
+ \qbezier(258,148)(258,164)(274,164)
+ \put(23,151){\makebox(15,10){$B$}}
+ \put(257,151){\makebox(15,10){$B$}}
+ \thinlines
+ \qbezier(47,133)(65.5,117)(84,101)
+ \qbezier(84,101)(65.5,85)(47,69)
+ \put(47,69){\line(0,1){64}}
+ \qbezier(47,133)(110.5,142)(174,151)
+ \qbezier(47,133)(121,125)(195,117)
+ \qbezier(47,133)(121,103.5)(195,74)
+ \qbezier(47,133)(110.5,86.5)(174,40)
+ \qbezier(84,101)(129,126)(174,151)
+ \qbezier(84,101)(139.5,109)(195,117)
+ \qbezier(84,101)(139.5,87.5)(195,74)
+ \qbezier(84,101)(129,70.5)(174,40)
+ \qbezier(47,69)(110.5,110)(174,151)
+ \qbezier(47,69)(121,93)(195,117)
+ \qbezier(47,69)(121,71.5)(195,74)
+ \qbezier(47,69)(110.5,54.5)(174,40)
+ \put(47,133){\circle*{5}}
+ \put(84,101){\circle*{5}}
+ \put(47,69){\circle*{5}}
+ \put(174,151){\circle*{5}}
+ \put(195,117){\circle*{5}}
+ \put(195,74){\circle*{5}}
+ \put(174,40){\circle*{5}}
+ \put(24,9){\makebox(190,22){\shortstack{\scriptsize All displayed edges are free.\\[-1pt]
+ \scriptsize Edges wholly outside $B$ are fixed.}}}
+ \linethickness{1pt}
+ \qbezier(318,101)(363,126)(408,151)
+ \qbezier(318,101)(373.5,109)(429,117)
+ \qbezier(408,151)(418.5,134)(429,117)
+ \put(281,133){\circle{5}}
+ \put(318,101){\circle*{6}}
+ \put(281,69){\circle{5}}
+ \put(408,151){\circle*{6}}
+ \put(429,117){\circle*{6}}
+ \put(429,74){\circle{5}}
+ \put(408,40){\circle{5}}
+ \put(302,88){\makebox(13,12){$u$}}
+ \put(410,156){\makebox(19,10){\scriptsize $v_1$}}
+ \put(435,115){\makebox(19,10){\scriptsize $v_2$}}
+ \put(435,72){\makebox(19,10){\scriptsize $v_3$}}
+ \put(410,26){\makebox(19,10){\scriptsize $v_4$}}
+ \put(327,143){\makebox(63,10){\scriptsize survives}}
+ \put(430,146){\makebox(28,20){\shortstack{\scriptsize fixed\\[-1pt]\scriptsize by $y$}}}
+ \thinlines
+ \put(440,142){\vector(-1,-1){16}}
+ \put(257,9){\makebox(190,15){\small $|S\cap F_B|=d_S(u)=2$}}
+\end{picture}
+\caption{The block restriction and the one-vertex event.
+In (a), $F_B$ contains every edge incident with $B$, including its
+internal edges; only edges with both endpoints outside $B$ are fixed.
+In (b), only support edges are drawn, and filled vertices are the
+nonisolated vertices $V(S)$.  The event $B\cap V(S)=\{u\}$ leaves
+both support edges incident with $u$ free.  The exterior support
+edge is fixed, and its contribution is retained by the exact
+averaging identity \eqref{eq:restriction-mass}.}
+\label{fig:block}
+\end{figure}

preprints/A-Sharp-Threshold-Bound-for-Monotone-Graph-Properties-September-25-2026/build/main.tex

@@ -0,0 +1,39 @@
+\documentclass[11pt]{article}
+\usepackage[T1]{fontenc}
+\usepackage{lmodern}
+\usepackage[margin=1.03in]{geometry}
+\usepackage{amsmath,amssymb,amsthm,mathtools}
+\usepackage{microtype,graphicx,xcolor}
+\usepackage[hyphens]{url}
+\usepackage[colorlinks=true,linkcolor=blue!50!black,citecolor=blue!50!black,urlcolor=blue!50!black]{hyperref}
+\newcommand{\E}{\mathbb E}
+\DeclareMathOperator{\Var}{Var}
+\newcommand{\1}{\mathbf 1}
+\newcommand{\R}{\mathbb R}
+\newcommand{\N}{\mathbb N}
+\newcommand{\eps}{\varepsilon}
+\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}
+\hypersetup{pdftitle={A Sharp Threshold Bound for Monotone Graph Properties},pdfauthor={OpenAI},pdfsubject={A proof of the Friedgut--Kalai sharp-threshold conjecture}}
+\title{A Sharp Threshold Bound for\\Monotone Graph Properties}
+\author{OpenAI}
+\date{September 25, 2026}
+\begin{document}
+\maketitle
+\begin{abstract}
+We prove the Friedgut--Kalai sharp-threshold conjecture. For every integer $n\ge2$, every nontrivial increasing family of graphs on $n$ vertices invariant under all vertex permutations, and every $0<\varepsilon<1/2$, the edge probabilities at which its probability equals $\varepsilon$ and $1-\varepsilon$ differ by at most $C\log(1/(2\varepsilon))/(\log n)^2$, for a universal constant $C$.
+\end{abstract}
+\input{sections/introduction}
+\input{sections/fourier}
+\input{sections/graph}
+\input{sections/threshold}
+\clearpage
+\input{references}
+\end{document}

preprints/A-Sharp-Threshold-Bound-for-Monotone-Graph-Properties-September-25-2026/build/references.tex

@@ -0,0 +1,74 @@
+\begin{thebibliography}{10}
+
+\bibitem{Bonami1970}
+Aline Bonami,
+\href{https://doi.org/10.5802/aif.357}{\'Etude des coefficients de Fourier
+des fonctions de \(L^p(G)\)},
+\emph{Annales de l'Institut Fourier}
+\textbf{20} (1970), no.~2, 335--402.
+
+\bibitem{BKKKL92}
+Jean Bourgain, Jeff Kahn, Gil Kalai, Yitzhak Katznelson, and Nathan Linial,
+\href{https://doi.org/10.1007/BF02808010}{The influence of variables
+in product spaces},
+\emph{Israel Journal of Mathematics}
+\textbf{77} (1992), 55--64.
+
+\bibitem{BK97}
+Jean Bourgain and Gil Kalai,
+\href{https://doi.org/10.1007/s000390050015}{Influences of variables and
+threshold intervals under group symmetries},
+\emph{Geometric and Functional Analysis}
+\textbf{7} (1997), no.~3, 438--461.
+
+\bibitem{Friedgut22}
+Ehud Friedgut,
+\href{https://doi.org/10.4171/ICM2022/78}{KKL's influence on me},
+in \emph{Proceedings of the International Congress of Mathematicians 2022},
+Vol.~6, Dmitry Beliaev and Stanislav Smirnov (eds.),
+EMS Press, 2023, pp.~4568--4581.
+
+\bibitem{FK96}
+Ehud Friedgut and Gil Kalai,
+\href{https://doi.org/10.1090/S0002-9939-96-03732-X}{Every monotone graph
+property has a sharp threshold},
+\emph{Proceedings of the American Mathematical Society}
+\textbf{124} (1996), no.~10, 2993--3002.
+
+\bibitem{KKL88}
+Jeff Kahn, Gil Kalai, and Nathan Linial,
+\href{https://doi.org/10.1109/SFCS.1988.21923}{The influence of variables
+on Boolean functions},
+in \emph{Proceedings of the 29th Annual Symposium on Foundations of
+Computer Science}, IEEE Computer Society, 1988, pp.~68--80.
+
+\bibitem{KKLMS20}
+Esty Kelman, Guy Kindler, Noam Lifshitz, Dor Minzer, and Muli Safra,
+\href{https://doi.org/10.1007/s00039-020-00544-2}{Towards a proof of the
+Fourier--entropy conjecture?},
+\emph{Geometric and Functional Analysis}
+\textbf{30} (2020), no.~4, 1097--1138.
+Accessible preprint:
+\href{https://arxiv.org/abs/1911.10579v2}{arXiv:1911.10579v2}, 7 May 2020.
+
+\bibitem{ExpectationEquivalence2026}
+OpenAI,
+Integral and fractional expectation thresholds are equivalent,
+OpenAI Math Release preprint
+\href{https://github.com/openai/math/blob/main/preprints/Integral-and-fractional-expectation-thresholds-are-equivalent-September-23-2026/paper.pdf}{OAI:Integral-and-fractional-expectation-thresholds-are-equivalent-September-23-2026},
+2026.
+
+\bibitem{SecondKahnKalai2026}
+OpenAI,
+The second Kahn--Kalai conjecture,
+OpenAI Math Release preprint
+\href{https://github.com/openai/math/blob/main/preprints/The-second-Kahn-Kalai-conjecture-September-24-2026/paper.pdf}{OAI:The-second-Kahn-Kalai-conjecture-September-24-2026},
+2026.
+
+\bibitem{Russo81}
+Lucio Russo,
+\href{https://doi.org/10.1007/BF00535742}{On the critical percolation probabilities},
+\emph{Zeitschrift f\"ur Wahrscheinlichkeitstheorie und verwandte Gebiete}
+\textbf{56} (1981), no.~2, 229--237.
+
+\end{thebibliography}

preprints/A-Sharp-Threshold-Bound-for-Monotone-Graph-Properties-September-25-2026/build/sections/fourier.tex

@@ -0,0 +1,168 @@
+\section{A low-degree estimate on biased products}\label{sec:fourier}
+
+We bound the mass of low-degree Fourier coefficients for Boolean
+functions whose individual influences are small compared with their
+total influence. The bound is linear in total influence, so it can be
+averaged over the conditioned coordinates in Section~\ref{sec:graph}.
+
+Let $J$ be a finite coordinate set, let $0<p<1$, and put
+$\sigma=\sqrt{p(1-p)}$. Expectations and norms in this section are with
+respect to the product Bernoulli-$p$ measure on the indicated coordinates;
+in particular, $\|g\|_q=(\E|g|^q)^{1/q}$. Define
+\[
+  \chi_i(x)=\frac{x_i-p}{\sigma},\qquad
+  \chi_S(x)=\prod_{i\in S}\chi_i(x),\qquad \chi_\varnothing=1.
+\]
+Each $\chi_i$ has mean zero and second moment one. Independence therefore
+makes the $2^{|J|}$ functions $\chi_S$, $S\subseteq J$, orthonormal.
+They form a basis of the $2^{|J|}$-dimensional space of real functions on
+$\{0,1\}^J$. Thus, with $\widehat g(S)=\E[g\chi_S]$,
+\begin{equation}\label{eq:parseval}
+  g=\sum_{S\subseteq J}\widehat g(S)\chi_S,\qquad
+  \E g^2=\sum_{S\subseteq J}\widehat g(S)^2,\qquad
+  \Var(g)=\sum_{\varnothing\ne S\subseteq J}\widehat g(S)^2.
+\end{equation}
+
+For $i\in J$, write
+$\Delta_i g(x)=g(x_{J\setminus\{i\}},1)-g(x_{J\setminus\{i\}},0)$,
+a function on the remaining coordinates. Integrating just coordinate $i$
+gives
+\[
+  \E_i[g\chi_i]
+  =\frac{p(1-p)}{\sigma}
+    \bigl(g(x_{J\setminus\{i\}},1)-g(x_{J\setminus\{i\}},0)\bigr)
+  =\sigma\Delta_i g.
+\]
+Consequently, for $A\subseteq J\setminus\{i\}$,
+$\widehat g(A\cup\{i\})=\sigma\widehat{\Delta_i g}(A)$,
+where the latter Fourier coefficient is taken on $J\setminus\{i\}$.
+For Boolean $h:\{0,1\}^J\to\{0,1\}$, set
+\[
+  I_i(h)=\Pr(\Delta_i h\ne0),\qquad I(h)=\sum_{i\in J}I_i(h).
+\]
+Since $\Delta_i h\in\{-1,0,1\}$, Parseval on the remaining coordinates
+and then summation over $i$ give
+\begin{equation}\label{eq:dirichlet}
+  \sum_{S\ni i}\widehat h(S)^2=\sigma^2 I_i(h),\qquad
+  \sum_{S\subseteq J}|S|\widehat h(S)^2=\sigma^2 I(h).
+\end{equation}
+These identities do not require monotonicity.
+
+The next estimate follows the hypercontractive influence method of
+Kahn, Kalai, and Linial~\cite{KKL88}. We prove the biased two-point
+inequality directly and use the standard product argument of
+Bonami~\cite[Chapter~III, Lemma~1]{Bonami1970}, keeping the dependence
+on the edge probability explicit.
+
+\begin{samepage}
+\begin{lemma}\label{lem:low-degree}
+Let $m\ge2$ be real and let $h:\{0,1\}^J\to\{0,1\}$ satisfy
+$I_i(h)\le I(h)/m$ for every $i\in J$. If
+\[
+  k=\frac{\sigma\log m}{64},
+\]
+then
+\begin{equation}\label{eq:low-degree}
+  \sum_{1\le |S|\le k}\widehat h(S)^2\le m^{-1/4}I(h).
+\end{equation}
+Every degree cutoff is interpreted literally for integer degrees $|S|$;
+in particular, the sum is empty when $k<1$.
+\end{lemma}
+\end{samepage}
+
+\begin{proof}
+We first prove a product norm bound. Put $\rho=\sigma/4$, so
+$0<\rho\le1/8$, and define the linear operator $T_\rho$ by
+$T_\rho\chi_S=\rho^{|S|}\chi_S$. For one coordinate, the bound
+$|\chi_i|\le\sigma^{-1}$ and the second moment identity imply
+\[
+  |\E\chi_i^3|\le\sigma^{-1},\qquad
+  \E\chi_i^4\le\sigma^{-2}.
+\]
+For real $a,b$, expansion and
+$2|ab^3|\le a^2b^2+b^4$ yield
+\begin{align*}
+  \E(a+\rho b\chi_i)^4
+  &\le a^4+6\rho^2a^2b^2
+       +\frac{4\rho^3}{\sigma}|ab^3|
+       +\frac{\rho^4}{\sigma^2}b^4\\
+  &\le a^4+\frac{13\sigma^2}{32}a^2b^2
+       +\frac{9\sigma^2}{256}b^4
+   \le(a^2+b^2)^2,
+\end{align*}
+where the last inequality uses $\sigma^2\le1/4$.
+
+To tensorize this inequality, induct on $|J|$, the empty product being
+immediate. Write $g=a+b\chi_i$ with $a,b$ functions of the other
+coordinates, and set $A=T_\rho a$, $B=T_\rho b$, using the operator on
+those coordinates. Apply the scalar inequality pointwise and then the
+triangle inequality in $L^2$ to obtain
+\[
+  \|T_\rho g\|_4^2
+  \le \|A^2+B^2\|_2
+  \le \|A\|_4^2+\|B\|_4^2
+  \le \|a\|_2^2+\|b\|_2^2
+  =\|g\|_2^2.
+\]
+The penultimate inequality is the induction hypothesis. We have proved
+\begin{equation}\label{eq:noise-norm}
+  \|T_\rho g\|_4\le\|g\|_2
+\end{equation}
+on every finite product.
+
+For any real $d\ge0$, let $P_{\le d}$ be the orthogonal projection onto
+the span of the characters of degree at most $d$. If $u=P_{\le d}u$,
+the operator $T_\rho$ is invertible and Parseval gives
+\[
+  \|T_\rho^{-1}u\|_2^2
+  =\sum_{|S|\le d}\rho^{-2|S|}\widehat u(S)^2
+  \le\rho^{-2d}\|u\|_2^2.
+\]
+Applying \eqref{eq:noise-norm} to $T_\rho^{-1}u$ proves
+$\|u\|_4\le\rho^{-d}\|u\|_2$. Duality in this finite-dimensional
+subspace, followed by H\"older's inequality, now gives for every real
+function $v$
+\begin{equation}\label{eq:projection-norm}
+  \|P_{\le d}v\|_2
+  =\sup_{\substack{u=P_{\le d}u\\\|u\|_2=1}}|\E[vu]|
+  \le\rho^{-d}\|v\|_{4/3}.
+\end{equation}
+
+If $k<1$, \eqref{eq:low-degree} is immediate, so assume $k\ge1$.
+Since $|\Delta_i h|$ is the indicator of a pivotal event,
+$\|\Delta_i h\|_{4/3}=I_i(h)^{3/4}$. Every nonempty Fourier set is
+counted at least once when summing over its coordinates. The derivative
+coefficient identity and \eqref{eq:projection-norm}, applied on
+$J\setminus\{i\}$ with $d=k$, therefore show
+\begin{align*}
+  \sum_{1\le |S|\le k}\widehat h(S)^2
+  &\le\sum_{i\in J}\sum_{\substack{S\ni i\\|S|\le k}}
+       \widehat h(S)^2\\
+  &=\sigma^2\sum_{i\in J}
+       \sum_{\substack{A\subseteq J\setminus\{i\}\\|A|\le k-1}}
+       \widehat{\Delta_i h}(A)^2\\
+  &\le\sigma^2\rho^{-2k}\sum_{i\in J}I_i(h)^{3/2}.
+\end{align*}
+The function $s\mapsto s\log(4/s)$ is increasing on $(0,1/2]$, since
+its derivative is $\log(4/s)-1>0$. 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
+\[
+  \sum_{1\le |S|\le k}\widehat h(S)^2
+  \le\E h^2\le1<m^{-1/4}I(h).
+\]
+This proves the estimate for every $0<p<1$, without a lower bound on
+$\sigma$.
+\end{proof}

preprints/A-Sharp-Threshold-Bound-for-Monotone-Graph-Properties-September-25-2026/build/sections/graph.tex

@@ -0,0 +1,227 @@
+\section{Restricting to the edges incident with a vertex block}
+\label{sec:graph}
+
+Let $E_n=\binom{[n]}2$ be the edge set of the complete graph on $[n]$,
+and let $f\colon\{0,1\}^{E_n}\to\{0,1\}$ be invariant under vertex
+permutations.  Fix $p\in(0,1)$ and put $\sigma=\sqrt{p(1-p)}$.
+All Fourier coefficients, variances, and influences in this section
+use this bias.  Monotonicity is not needed.
+
+The purpose of a vertex block is to connect two different notions of
+degree.  A restriction to the edges incident with the block retains
+enough symmetry to apply Lemma~\ref{lem:low-degree}.  If the block meets
+the nonisolated vertices of a Fourier support at just one vertex $u$,
+then the surviving Fourier degree is the graph degree of $u$.
+An $s$-edge graph with $s\ge1$ has a nonisolated vertex of degree at most
+$\sqrt{2s}$.  Thus a bound for conditional Fourier degrees up to $k$
+will control original degrees up to $k^2/2$.
+
+The restriction identity is the biased-product form of the standard
+Fourier restriction identity; compare
+\cite[arXiv version~2, Lemma~2.7]{KKLMS20}.
+We include its short proof. The next identities hold for every Boolean
+$f$, without a symmetry assumption.
+
+\begin{lemma}[Exact averaging over a restriction]
+\label{lem:restriction}
+Let $F\subseteq E_n$, put $H=E_n\setminus F$, and, for
+$y\in\{0,1\}^{H}$, let $f_y(x)=f(x,y)$ for
+$x\in\{0,1\}^{F}$.  If $y$ has the Bernoulli-$p$ product law, then,
+for every real $k>0$,
+\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 $s\le\binom v2<v^2/2$, its
+average degree over these vertices satisfies
+\[
+ \frac{2s}{v}<\sqrt{2s}.
+\]
+Choose a vertex $u\in V(S)$ whose degree $d_S(u)$ is at most this average.  Then
+\begin{equation}
+ 1\le d_S(u)\le\sqrt{2s}\le k,
+ \qquad v\le2s\le k^2.
+ \label{eq:support-degree}
+\end{equation}
+
+Choose $B$ uniformly among the $m$-element vertex sets.  Conditioned
+on $u\in B$, each other specified vertex belongs to $B$ with
+probability $(m-1)/(n-1)$.  A union bound gives
+\begin{equation}
+ \Pr_B\{B\cap V(S)=\{u\}\}
+ \ge \frac mn
+ \left(1-(v-1)\frac{m-1}{n-1}\right).
+ \label{eq:one-vertex-event}
+\end{equation}
+On this event, $S\cap F_B$ consists of \emph{every} support edge
+incident with $u$, and contains no other support edge.  Its size is
+therefore $d_S(u)$, between $1$ and $k$; it need not be $1$.
+Figure~\ref{fig:block} illustrates this distinction.
+
+\input{figures/block}
+
+To bound the probability uniformly in $p$, note that $k\le t$ and
+\[
+ \frac{m-1}{n-1}\le\frac1{\sqrt n},
+ \qquad
+ (v-1)\frac{m-1}{n-1}
+ \le t^2e^{-t/2}\le256e^{-8}<\frac12.
+\]
+Here $t^2e^{-t/2}$ is decreasing for $t\ge16$.
+Consequently, for every support under consideration,
+\begin{equation}
+ \Pr_B\{1\le|S\cap F_B|\le k\}\ge\frac{m}{2n}.
+ \label{eq:support-detection}
+\end{equation}
+The choice of $u$ may depend on $S$.  The lower bound
+\eqref{eq:support-detection} is applied to each nonnegative summand
+$\widehat f(S)^2$ separately; no block must capture all supports
+simultaneously.  Comparing with \eqref{eq:block-average} and
+cancelling $m/(2n)$ proves \eqref{eq:captured-mass}.
+If $k^2/2<1$, the sum in \eqref{eq:captured-mass} is empty and the
+conclusion holds without choosing a support.
+\end{proof}
+
+\begin{proof}[Proof of Theorem~\ref{thm:variance}]
+First suppose $t=\log n\ge16$, and choose $m$ and $k$ as in
+Lemma~\ref{lem:block-capture}; in particular $m\ge3$ and $k>0$.
+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 $2\le n<e^{16}$, the same Fourier identities immediately
+give
+\[
+ \Var(f)\le\sigma^2I(f)\le\frac14 I(f)
+ \le\frac{64}{(\log n)^2}I(f).
+\]
+Hence the single choice $C_0=2^{17}$ works for every integer $n\ge2$
+and every $p\in(0,1)$, as asserted.
+\end{proof}

preprints/A-Sharp-Threshold-Bound-for-Monotone-Graph-Properties-September-25-2026/build/sections/introduction.tex

@@ -0,0 +1,115 @@
+\section{Introduction}
+\label{sec:introduction}
+
+For an integer $n\ge2$, write $[n]=\{1,\ldots,n\}$ and let $E_n=\binom{[n]}2$ be the edge set of the complete simple graph on $n$ vertices. We identify a graph with an element of $\{0,1\}^{E_n}$. A \emph{graph property} is a family $\mathcal P\subseteq\{0,1\}^{E_n}$ invariant under every permutation of $[n]$. It is \emph{increasing}, or \emph{monotone}, if adding edges preserves membership, and \emph{nontrivial} if it is neither empty nor the whole space.
+
+Write $\mu_p(\mathcal P)$ for the probability that a graph whose edges are present independently with probability $p$ belongs to $\mathcal P$. For $0<a<1$, define
+\[
+ p_a(\mathcal P)=\inf\{p\in[0,1]:\mu_p(\mathcal P)\ge a\}.
+\]
+All logarithms in this paper are natural. In particular, $n$ always counts vertices, not edge coordinates.
+
+\begin{theorem}[Friedgut--Kalai sharp-threshold conjecture]
+\label{thm:threshold}
+For every integer $n\ge2$, every nontrivial increasing family $\mathcal P$ of graphs on $n$ vertices invariant under all vertex permutations, and every $0<\varepsilon<1/2$,
+\begin{equation}
+\label{eq:threshold}
+ p_{1-\varepsilon}(\mathcal P)-p_\varepsilon(\mathcal P)
+ \le \frac{2^{19}}{(\log n)^2}\log\frac1{2\varepsilon}.
+\end{equation}
+\end{theorem}
+
+The influence theorem of Kahn, Kalai, and Linial~\cite{KKL88}
+connected Fourier analysis with the sensitivity of Boolean functions
+to individual coordinates. Bourgain, Kahn, Kalai, Katznelson, and
+Linial extended the influence theorem to product
+spaces~\cite{BKKKL92}. Friedgut and Kalai combined that extension
+with the Margulis--Russo probability-derivative identity to show that
+every increasing family invariant under a transitive coordinate group
+has a narrow transition~\cite[Section~2]{FK96}. For graph properties,
+vertex permutations act transitively on the edge coordinates, and their
+theorem gives the universal width bound
+$C\log(1/(2\varepsilon))/\log n$~\cite[Theorem~1.1]{FK96}.
+They conjectured the denominator $(\log n)^2$ with the same dependence
+on $\varepsilon$~\cite[Conjecture~1.2]{FK96}.
+Theorem~\ref{thm:threshold} resolves that conjecture positively;
+its numerical constant is only a convenient explicit choice.
+
+The squared logarithm is the optimal order in $n$ when
+$\varepsilon$ is fixed. Friedgut and Kalai already pointed to the
+property of containing a clique with order proportional to $\log n$:
+its transition width can be of order $(\log n)^{-2}$
+\cite[p.~2994]{FK96}.
+Bourgain and Kalai went beyond coordinate transitivity by using the
+action of the symmetry group on sets of coordinates~\cite{BK97}.
+For every fixed $\eta>0$ 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 $0<p<1$,
+\begin{equation}
+\label{eq:variance}
+ \Var_p(f)\le\frac{2^{17}}{(\log n)^2}I_p(f).
+\end{equation}
+No monotonicity assumption is needed.
+\end{theorem}
+
+At the unbiased measure, Kelman, Kindler, Lifshitz, Minzer, and Safra
+proved the graph-symmetric estimate
+\[
+ I_{1/2}(f)\ge
+ c\,\frac{(\log n)^2}{(\log\log n)^2}\Var_{1/2}(f)
+\]
+for all sufficiently large $n$ and an absolute $c>0$
+\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 $0<p<1$, including for nonmonotone properties.
+
+Random restrictions and symmetry of Fourier coefficients are central
+to the arguments of Bourgain--Kalai and
+Kelman--Kindler--Lifshitz--Minzer--Safra~\cite{BK97,KKLMS20}.
+Here the restriction is chosen through the vertices: leave free all
+edges touching a random block $B$ of about $\sqrt n$ vertices, while
+conditioning on every other edge. Permutations inside $B$ survive
+every such conditioning. Their edge orbits bound each conditional
+influence, to which an elementary Fourier estimate applies.
+The resulting estimate is linear in total influence, so it can be
+averaged over the conditioned coordinates. Section~\ref{sec:fourier}
+proves this estimate directly, including its dependence on the bias.
+
+The second ingredient regards each edge set indexing a Fourier
+coefficient as a graph. Any nonempty such graph with at most $k^2/2$
+edges has a nonisolated vertex of degree at most $k$. With sufficient
+probability, $B$ meets its nonisolated vertices only at that vertex.
+The restriction then leaves between one and $k$ edges of this graph
+free. This turns control of conditional Fourier degrees up to $k$
+into control of original degrees up to $k^2/2$. Taking $k$ proportional
+to $\sqrt{p(1-p)}\log n$ yields \eqref{eq:variance}; the bias factor
+cancels in the estimate for the remaining Fourier degrees.
+Section~\ref{sec:threshold} then integrates the probability derivative
+to obtain the full quantile bound.

preprints/A-Sharp-Threshold-Bound-for-Monotone-Graph-Properties-September-25-2026/build/sections/threshold.tex

@@ -0,0 +1,77 @@
+\section{From influence to threshold width}
+\label{sec:threshold}
+
+We pass from influence to transition width using the Russo--Margulis
+probability-derivative formula; see Russo~\cite[Lemma~3]{Russo81}
+and Friedgut--Kalai~\cite[Lemma~2.2]{FK96}. We include the finite
+calculation with our pivotal-influence convention and then integrate
+the resulting differential inequality.
+
+\begin{proof}[Proof of Theorem~\ref{thm:threshold}]
+Put $f=\1_{\mathcal P}$, $\mu(p)=\E_p f$, and $C_0=2^{17}$. We first verify the probability-derivative identity with the influence convention used here. If the edge probabilities are allowed to vary separately, $\E f$ is a multilinear polynomial in those probabilities. Its partial derivative in the probability at $e$ is the expectation of
+\[
+ \Delta_e f=f(x_{e\leftarrow1})-f(x_{e\leftarrow0}).
+\]
+For increasing $f$, this difference takes values in $\{0,1\}$, so its expectation is $I_{e,p}(f)$ when all other biases are $p$. Differentiating along the diagonal therefore gives
+\begin{equation}
+\label{eq:russo}
+ \mu'(p)=I_p(f).
+\end{equation}
+This is the usual probability-derivative formula underlying the sharp-threshold method; the calculation above proves it directly in the present finite setting.
+
+Since $f$ is Boolean, $\Var_p(f)=\mu(p)(1-\mu(p))$. Theorem~\ref{thm:variance} and \eqref{eq:russo} imply
+\begin{equation}
+\label{eq:differential}
+ \mu'(p)\ge \frac{(\log n)^2}{C_0}\mu(p)(1-\mu(p))
+ \qquad(0<p<1).
+\end{equation}
+Nontriviality and monotonicity give $f(\varnothing)=0$ and $f(E_n)=1$. Both configurations have positive probability for every interior bias, hence $0<\mu(p)<1$ there, while $\mu(0)=0$ and $\mu(1)=1$. Equation~\eqref{eq:differential} now gives $\mu'(p)>0$ 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.

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

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}
+}
+```
+

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}

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}

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}

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}

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}

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}
+}

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}

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
+0<c_0\le s_y\le C_0.
+\end{equation}
+Indeed the first estimate follows from \(\abs q\le1/100\).
+Differentiating the flow gives
+\[
+y_s=\exp\left(-\int_0^t q_y(r,y(r,s))\dd r\right)>0.
+\]
+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 \(-1/4<b_*\le0\), and assume \eqref{eq:low-bounds}.
+Suppose that, for every compact rectangle
+\[
+R\Subset(-2,2)\times(-2,b_*),
+\]
+the ordinary coordinate derivatives of \(g\) of every order have
+uniform bounds on \(\Phi_z(R)\).
+Then the ordinary coordinate derivatives of \(z\) of every order
+also have uniform bounds on \(\Phi_z(R)\).
+The corresponding assertion holds for the upper cap
+\((-2,2)\times(-b_*,2)\).
+\end{proposition}
+
+Here and below a rectangle has sides parallel to the \(t,s\) axes.
+Its image may depend on the solution, but all margins in coordinate
+labels are chosen independently of the member of the family.
+More precisely, if \(R\Subset R'\) are two such rectangles, the
+uniform bounds for the chart and its inverse give a fixed radius
+of ordinary coordinate balls around points of \(\Phi_z(R)\)
+that stay inside \(\Phi_z(R')\). We can therefore apply local
+Sobolev estimates with uniform constants in the original coordinates.
+No high derivatives of the chart are needed for this step.
+
+\subsection{The differentiated equation}
+
+At the induction step we estimate \(u=\pd_y^lz\) first; the
+Darboux equation then recovers derivatives with more \(x\) indices.
+We differentiate in the original coordinates before passing to
+the flow chart, whose high derivatives are not yet controlled.
+Solving the Darboux equation for \(z_{xx}\) gives
+\begin{equation}\label{eq:P}
+z_{xx}=P(x,y,z_i,z_{xy},z_{yy})
+=\Gamma^r_{xx}z_r+\frac{H_{xy}^2+KE}{H_{yy}}.
+\end{equation}
+The metric and its derivatives are prescribed arguments of \(P\).
+In particular,
+\begin{equation}\label{eq:P-partials}
+P_{z_{xy}}=2q,\qquad P_{z_{yy}}=-q^2-a,\qquad
+P_{z_i}=\Gamma^i_{DD}+O(K).
+\end{equation}
+The notation \(\Gamma^i_{DD}\) contracts the two lower Christoffel
+indices against \(D\), without differentiating \(D\).
+The last identity follows more explicitly from
+\[
+P_{z_i}=\Gamma^i_{DD}
++K\left(\frac{E_{z_i}}{H_{yy}}
++\frac{E\Gamma^i_{yy}}{H_{yy}^2}\right).
+\]
+All coefficients in an \(O(K)\) term in this section are uniformly
+bounded under \eqref{eq:low-bounds}.
+
+\begin{lemma}[Curvature drift]\label{lem:drift}
+The quantities in \eqref{eq:DGa} satisfy
+\begin{equation}\label{eq:drift}
+Dq-P_{z_x}q-P_{z_y}=K_yG+O(K).
+\end{equation}
+\end{lemma}
+
+\begin{proof}
+Put \(Y=\pd_y\). The definitions and the Darboux equation give
+\[
+H(Y,D)=0,\qquad H(D,D)=KE/H_{yy},
+\qquad H(D,\pd_x)=KE/H_{yy}.
+\]
+Thus \(H(D,\cdot)=O(K)\) on bounded coordinate vectors.
+The commutation formula for a covariant Hessian gives
+\begin{equation}\label{eq:hessian-commutation}
+(\nabla_DH)(Y,D)=(\nabla_YH)(D,D)+O(K),
+\end{equation}
+because in dimension two the curvature tensor is a bounded tensor
+multiple of \(K\), and the commutator contracts it against \(dz\).
+
+Differentiate \(H(Y,D)=0\). The term containing \(\nabla_DY\)
+is \(O(K)\), and
+\(\nabla_DD=\Gamma^i_{DD}\pd_i-(Dq)Y\). Hence the left side
+of \eqref{eq:hessian-commutation} equals
+\[
+H_{yy}(Dq-q\Gamma^x_{DD}-\Gamma^y_{DD})+O(K).
+\]
+On the right, differentiating \(H(D,D)=KE/H_{yy}\) gives
+\(K_yE/H_{yy}+O(K)\); the two connection terms are \(O(K)\).
+Division by \(H_{yy}\), followed by \eqref{eq:P-partials},
+proves \eqref{eq:drift}.
+\end{proof}
+
+\begin{lemma}[High derivative equation]\label{lem:high-equation}
+Let \(l\ge8\). Suppose that the ordinary derivatives of \(z\) through
+order \(l\) have uniform \(L^2\) bounds on images of compact rectangles
+in the lower cap, and that the metric satisfies the hypothesis of
+\Cref{prop:cap}. Then \(u=\pd_y^l z\), re-expressed in flow coordinates,
+satisfies
+\begin{equation}\label{eq:linear-cap}
+u_{tt}+Au_{ss}+Bu_t+Cu_s=R_l,
+\end{equation}
+where \(R_l\) is uniformly bounded in \(L^2\) on smaller compact
+rectangles, and
+\begin{equation}\label{eq:linear-coefficients}
+\begin{gathered}
+A=G_1K,\qquad G_1=Gs_y^2,\qquad
+B=-P_{z_x}-2lq_y,\\
+C=(l+1)A_s+O_l(A).
+\end{gathered}
+\end{equation}
+The function \(G_1\) has positive lower and upper bounds and bounded
+first derivatives. The coefficients \(A,A_t,A_s,B,B_s,C\) are
+uniformly bounded.
+\end{lemma}
+
+\begin{proof}
+The inductive \(L^2\) bounds give supremum bounds through order
+\(l-2\) on smaller sets by the local two-dimensional Sobolev
+inequality on the ordinary coordinate balls described above.
+
+Differentiate \eqref{eq:P} \(l\) times in the original \(y\)
+coordinate. In addition to the direct linearization, the terms
+containing derivatives of \(z\) of order \(l+1\) are
+\[
+l\bigl(2q_yu_x-(q^2+a)_yu_y\bigr).
+\]
+These arise when a second-jet argument receives \(l-1\)
+differentiations and the remaining single derivative is placed
+elsewhere. The coefficient derivatives are total derivatives,
+including the explicit metric dependence of \(P\).
+
+Every remaining differentiated solution factor has order at most
+\(l\). A factor of order at least \(l-1\) consumes at least \(l-3\)
+of the \(l\) differentiations. There cannot be two such factors,
+since \(2(l-3)>l\) 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 \(b<b_*\) above the upper edge of \(R_0\).
+Integrations will take place below \(s=b\), with their other three
+edges sufficiently far out that cutoff derivatives lie in
+\[
+3/2<\abs t<2
+\quad\text{or}\quad -2<s<-3/2.
+\]
+Both regions have uniformly positive curvature. At the lateral
+edges, \(t^2-h(y)\ge t^2-1>0\). 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 \(0<K\le c_1d\), since \(d\)
+is bounded above,
+\[
+H_0A\le Cc_1(\lambda d_{\max}+p).
+\]
+Choose \(c_1\) so small that this region lies in the preceding
+neighborhood of the zero set, and that the adverse \(H_0A\)
+term and remaining errors use only a fraction of its positive
+lower bound. This proves \eqref{eq:coercive}. The order of
+choices is \(\epsilon\), then \(\lambda\), then \(c_1\).
+
+\smallskip
+\noindent\emph{The auxiliary elliptic estimate.}
+On the remaining region we claim
+\begin{equation}\label{eq:aux-elliptic}
+\iint_{\{K\ge c_1d\}} d^{p-1}(u_t^2+u_s^2)\dd t\dd s
+\le C\bigl(\norm u_{L^2(R_1)}^2+\norm f_{L^2(R_1)}^2\bigr),
+\end{equation}
+on any of the supports needed for the directed multiplier.
+To prove it, choose a cutoff \(\chi\) farther out at the three
+elliptic edges, equal to one on those supports. Let \(\phi\) be
+a smooth cutoff equal to zero on \((-\infty,c_1/2]\) and one
+on \([c_1,\infty)\), and test the equation against \(-vu\), where
+\[
+v=d^{p-2}\chi^2\phi(K/d)^2.
+\]
+On \(v>0\), \(A\ge c'd\). 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}

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}

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_*<L/4.
+\end{equation}
+
+Choose \(r>0\) 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
+\(\abs s<r/2\) for large \(\tau\), in every solution's flow chart.
+Consequently the image of every compact rectangle in
+\((-2,2)\times(-2,-r/2)\), or in
+\((-2,2)\times(r/2,2)\), misses the pulse support. On all such
+images, \(g_\tau\) differs from the fixed metric \(g_*\) only
+by the approximation error \eqref{eq:approximation}. This verifies
+the metric hypothesis of \Cref{prop:cap} throughout both caps,
+with \(b_*=-r/2\). The proposition gives bounds for every
+derivative of \(z\) on the two cuts
+\begin{equation}\label{eq:cap-cuts}
+\theta=\pm r,\qquad \abs\xi\le Lr.
+\end{equation}
+These cuts lie in compact subsets of the respective flow-coordinate
+caps, uniformly over the solutions.
+\Cref{fig:pulse} shows the cuts and the pulse inside the
+negative-curvature slab.
+
+\input{figures/pulse}
+
+\begin{lemma}[Propagation to the pulse]\label{lem:propagation}
+At \(\theta=\pm\delta/\tau\), on \(\abs\xi\le Lr/2\),
+the functions \(z,z_\theta\) have uniform bounds for every
+tangential derivative.
+\end{lemma}
+
+\begin{proof}
+We record the energy argument, which will also be used on the pulse.
+If \(v\) satisfies
+\[
+v_{\theta\theta}-P_1v_{\theta\xi}-S_1v_{\xi\xi}
+=b^\theta v_\theta+b^\xi v_\xi+R,
+\]
+with the above coefficient bounds and bounded \(b^i\), use the energy
+density and flux
+\[
+e=\frac{v_\theta^2+S_1v_\xi^2}{2},
+\qquad
+j=\frac{P_1v_\theta^2}{2}+S_1v_\xi v_\theta.
+\]
+One has
+\(\abs j\le(\abs{P_1}+\sqrt{S_1})e\).
+Integrate on intervals whose endpoints move inward at speed \(L/4\).
+Multiplication by \(v_\theta\) and integration by parts, with the
+nonpositive boundary loss discarded, gives
+\begin{equation}\label{eq:energy}
+\mathcal E'(\theta)
+\le C\mathcal E(\theta)
++\sqrt{2\mathcal E(\theta)}\,\norm R_{L^2},
+\qquad \mathcal E(\theta)=\int e\dd\xi.
+\end{equation}
+Thus \(\sqrt{\mathcal E}\) is controlled by its initial value and
+the time integral of \(\norm R_2\), with a Gronwall factor.
+One can justify division at zero energy by first using
+\(\sqrt{\mathcal E+\eta}\) and then letting \(\eta\downarrow0\).
+
+Apply this estimate to \(v=\pd_\xi^jz\), \(j\ge8\), obtained by
+differentiating \eqref{eq:Q}. The top second-jet terms give
+the principal coefficients in \eqref{eq:hyperbolic-coefficients}.
+If a second-jet argument receives \(j-1\) differentiations,
+it gives \(v_\xi\) or \(v_\theta\); put these terms, together
+with the direct first-jet linearization, into \(b^i\pd_iv\).
+Their coefficients use only bounded low derivatives.
+Put \(V=(z_\xi,z_\theta)\). Every remaining solution factor is
+a spatial derivative of \(V\) of order at most \(j-1\),
+already controlled by the induction. A factor of order \(j-1\)
+requires at least \(j-2\) of the differentiations, since a
+second-jet argument is already one spatial derivative of \(V\).
+There cannot be two such factors, because \(2(j-2)>j\).
+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 \(C_3\delta<c_3/2\),
+and then let \(\tau\to\infty\). Equations
+\eqref{eq:moment-error} and \eqref{eq:moment-lower} contradict
+\eqref{eq:w-equation}.
+The order of choices is summarized in \Cref{tab:pulse-choices}.
+In particular, no convergence in every smooth seminorm was required
+of the test metrics, and no convergence or common sign across
+different \(\tau\) was required of \(D_\tau\).
+
+\begin{table}[H]
+\centering
+\small
+\renewcommand{\arraystretch}{1.18}
+\begin{tabular}{@{}p{.15\textwidth}p{.36\textwidth}p{.43\textwidth}@{}}
+\hline
+Stage & Quantities fixed & Dependence permitted\\
+\hline
+First & Class, neighborhood, \(g_*\), slab, cutoffs, \(N\)
+& Fixes leading constants \(C_2,C_3,c_3\), independent of
+sufficiently small fixed \(\delta\).\\
+Second & \(\delta>0\), with \(C_3\delta<c_3/2\)
+& Subsidiary derivative bounds may depend on this fixed \(\delta\).\\
+Last & Integers \(\tau\to\infty\)
+& Thresholds and little-oh errors may depend on \(\delta\);
+the leading constants do not.\\
+\hline
+\end{tabular}
+\caption{Parameter order in the pulse contradiction. Every derivative
+order has its own bound; no single bound for all orders is asserted.}
+\label{tab:pulse-choices}
+\end{table}
+
+We have proved that every \(\mathcal A_{M,q_0}\) is nowhere dense.
+Their countable union contains every metric admitting an admissible
+height. Baire's theorem now proves \Cref{prop:patch}.

preprints/A-Smooth-Metric-with-No-Local-Isometric-Immersion-into-Three-Space-September-24-2026/paper.pdf

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:6c2cf05d89a23473c52fecb2aea37bcbda7279db50004cf0b212b000430bd374
+size 435629

preprints/A-Three-Dimensional-Counterexample-to-Integer-Degree-Harmonic-Dimension-Comparison-September-26-2026/build/figures/crossing-cycle.tex

@@ -0,0 +1,25 @@
+\begin{figure}[tbp]
+\centering
+\begin{tikzpicture}[x=1cm,y=1cm,>=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}

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 \(4/9<v<1\) and \(1<c<9v/4\), all sufficiently large integers
+\(k\) admit a complete smooth metric on \(\mathbb R^3\) with nonnegative
+Ricci curvature, asymptotic volume ratio \(v\), and at least \(c(k+1)^2\)
+linearly independent real harmonic functions of pointwise growth at most
+\(k\). This answers Yau's integer-degree dimension comparison question
+negatively in dimension three, with a fixed-factor excess over the Euclidean
+count. For each \(1<c<9/4\), these metrics can be chosen arbitrarily close
+to the Euclidean metric in global bi-Lipschitz distance. The metric may
+depend on \(k\).
+\end{abstract}
+\input{sections/introduction.tex}
+\input{sections/crossings.tex}
+\input{sections/angular-program.tex}
+\input{sections/geometry.tex}
+\input{sections/transfer.tex}
+\input{sections/matching.tex}
+\input{sections/growth.tex}
+\input{sections/bibliography.tex}
+\end{document}

preprints/A-polynomial-time-algorithm-for-three-machine-unit-job-scheduling-September-24-2026/README.md

@@ -0,0 +1,19 @@
+# [A Polynomial-Time Algorithm for Three-Machine Unit-Job Scheduling](paper.pdf)
+
+OpenAI  
+September 24, 2026
+
+## Citation
+
+To cite this manuscript:
+
+```bibtex
+@misc{OAI:A-polynomial-time-algorithm-for-three-machine-unit-job-scheduling-September-24-2026,
+  author = {{OpenAI}},
+  title = {{A Polynomial-Time Algorithm for Three-Machine Unit-Job Scheduling}},
+  howpublished = {OpenAI Math Release preprint
+                  \href{https://github.com/openai/math/blob/main/preprints/A-polynomial-time-algorithm-for-three-machine-unit-job-scheduling-September-24-2026/paper.pdf}{OAI:A-polynomial-time-algorithm-for-three-machine-unit-job-scheduling-September-24-2026}},
+  year = {2026}
+}
+```
+

preprints/A-polynomial-time-algorithm-for-three-machine-unit-job-scheduling-September-24-2026/build/figures/capacity-three.tex

@@ -0,0 +1,26 @@
+\begin{tikzpicture}[x=1cm,y=1cm,font=\small,>=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}

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}

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}

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}

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 \(R(x)<R(y)\), then
+\(M(x)<M(y)\). If \(R(x)=R(y)\) and \(\rk(x)<\rk(y)\),
+then either \(M(y)>M(x)\) 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_-<v_-\le a<b\le b_-.
+ \label{eq:ancestry}
+\end{equation}
+Here \(a\) is an actual ending endpoint of that preceding edge,
+viewed in reverse; it can equal \(v_-\).
+The node \(W\) lies in the open suffix of \(U\) after \(v_-\);
+Figure~\ref{fig:switch} shows this arrangement.
+
+Let \(H_-\) be the low set of the preceding edge in its old
+working frame, regarded as a subset of all of \(U\).
+The extension \(H_-^*\) from \Cref{lem:old-low} is an upset in
+the current, reversed frame. Define
+\begin{equation}
+ G=D_{a_-}\cap(J\setminus\wP a)
+            \cap(J\setminus\wP{v_-})\cap H_-^* .
+ \label{eq:G}
+\end{equation}
+
+\Needspace{3\baselineskip}
+\begin{lemma}\label{lem:G}
+The set \(G\) in \eqref{eq:G} is a global upset satisfying both
+properties in \eqref{eq:G-properties}.
+\end{lemma}
+\begin{proof}
+All four factors in \eqref{eq:G} are global upsets in the current
+frame, so \(G\) is an upset.
+
+Let \(x\in X=W\setminus D_a\).
+If \(x\) had been high at the preceding edge in its old frame,
+\Cref{lem:edge-labels} would put it in the old predecessor cone
+of that edge's ending endpoint. That endpoint is the current \(a\),
+so reversal would give \(x\in D_a\), a contradiction.
+Thus \(x\in H_-\).
+Every job of \(H_-\) precedes the old right boundary of \(U\),
+because all edge cutoffs lie in that cone. Equivalently,
+\(H_-\subseteq D_{a_-}\) in the current frame.
+Since \(x\) is strictly after both \(a\) and \(v_-\), feasibility
+excludes \(x\) from their weak predecessor cones.
+Finally, \(x\in U\), so agreement of \(H_-^*\) on \(U\) gives
+\(x\in H_-^*\). Hence \(X\subseteq G\).
+
+Conversely, suppose \(z\in G\) is scheduled at or before \(a\).
+Membership in \(D_{a_-}\), together with \eqref{eq:ancestry},
+first places it strictly inside \(U\):
+\begin{equation}
+ a_-<\operatorname{time}(z)\le a<b\le b_-.
+ \label{eq:localize-U}
+\end{equation}
+Only now use \(H_-^*\cap U=H_-\).
+The old edge cutoff \(H_-\) was contained in the old center-low
+set. If \(z\) is at or before \(v_-\) in the current frame,
+the old center separator puts it in current \(\wP{v_-}\).
+If it is strictly after \(v_-\) but at or before \(a\), it lies
+in the old normalized side and the old separator at the endpoint
+\(a\) puts it in current \(\wP a\).
+Both possibilities contradict \eqref{eq:G}.
+
+These separator implications remain valid when preparing the current
+node: all intervening reorderings have been confined to \(W=(a,b)\),
+and therefore have not changed any job at or before \(a\) in the
+current frame, the relevant boundary triples, or the fixed old
+cutoff sets.
+\end{proof}
+
+\begin{figure}[t]
+\centering
+\input{figures/switch-localization}
+\caption{The ancestor configuration at a direction change, in the new
+working frame. A candidate early member of \(G\) is first located
+inside \(U\) by \(D_{a_-}\); only there is the earlier cutoff
+extension used. The two weak predecessor exclusions then cover
+the portions at or before \(v_-\) and between \(v_-\) and \(a\).
+The equality \(a=v_-\) is allowed; the intervening segment then
+collapses. The horizontal arrow denotes time.}
+\label{fig:switch}
+\end{figure}
+
+\subsection{Predicate bounds and the induction order}
+
+\begin{lemma}\label{lem:entry-size}
+Every entry constructed above has fewer than \(40\) leaves.
+This bound persists through any number of same-direction updates.
+\end{lemma}
+\begin{proof}
+Root bases, injections, and second switch bases are simple predicates.
+By \Cref{lem:old-low}, the \(H_-^*\) factor in a new \(G\)
+has at most \(21\) leaves. The three other factors of \(G\)
+and its outer rank suffix use four more. Thus a new first switch
+entry has at most \(25\) leaves.
+Further same-direction trimming only tightens that outer rank
+suffix. It does not append a new independent condition.
+The same observation applies to simple entries.
+\end{proof}
+
+For completeness, the construction so far has a well-founded
+dependency order. Assume the parent and ancestor data already satisfy
+their assertions. A same update is defined directly. A switch update
+uses the preceding edge's low set; \Cref{lem:old-low} derives its
+short extension from already existing lists and, when necessary,
+an earlier switch's covering property. \Cref{lem:G} then proves
+the new covering and early-exclusion properties.
+\Cref{lem:update-labels} supplies the high-set identity and
+Invariant~\ref{inv:cones}; \Cref{lem:list-length} supplies the new
+length bound; and \Cref{lem:entry-size} supplies the predicate bound.
+The length argument needs global closure and high-set identities,
+but not the numerical predicate bound. Thus neither bound assumes
+itself at a deeper node.
+
+The root starts this induction. We next prove the remaining boundary
+invariants for every update, using only the parent invariants and
+the current normalizations.

preprints/A-polynomial-time-algorithm-for-three-machine-unit-job-scheduling-September-24-2026/build/paper.tex

@@ -0,0 +1,541 @@
+\documentclass[11pt]{article}
+\usepackage[T1]{fontenc}
+\usepackage[utf8]{inputenc}
+\usepackage{mathpazo}
+\usepackage{courier}
+\usepackage[margin=1.02in]{geometry}
+\usepackage{amsmath,amssymb,amsthm,mathtools}
+\usepackage{microtype}
+\usepackage{needspace}
+\usepackage{enumitem}
+\usepackage{booktabs}
+\usepackage{tikz}
+\usetikzlibrary{arrows.meta,positioning,calc,decorations.pathreplacing}
+\usepackage{xcolor}
+\usepackage[numbers,sort&compress]{natbib}
+\usepackage[colorlinks=true,linkcolor=blue!45!black,citecolor=blue!45!black,urlcolor=blue!45!black]{hyperref}
+\usepackage[nameinlink,capitalise,noabbrev]{cleveref}
+\newcommand{\doi}[1]{doi: \href{https://doi.org/#1}{\nolinkurl{#1}}}
+\pdfinfoomitdate=1
+\pdftrailerid{}
+\pdfsuppressptexinfo=15
+\pdfglyphtounicode{parenleftbig}{0028 FE01}
+\pdfglyphtounicode{parenrightbig}{0029 FE01}
+\pdfglyphtounicode{braceleftBigg}{007B FE04}
+\pdfglyphtounicode{uniondisplay}{22C3 FE02}
+\pdfglyphtounicode{hatwide}{02C6 FE01}
+\pdfglyphtounicode{tildewide}{02DC FE01}
+\pdfgentounicode=1
+\hypersetup{pdftitle={A Polynomial-Time Algorithm for Three-Machine Unit-Job Scheduling},pdfauthor={OpenAI}}
+\usepackage{caption}
+\captionsetup{font=small,labelfont=bf}
+\setlength{\emergencystretch}{2em}
+\setlist[enumerate]{itemsep=3pt,topsep=5pt}
+\setlist[itemize]{itemsep=3pt,topsep=5pt}
+\numberwithin{equation}{section}
+\newtheorem{theorem}{Theorem}[section]
+\newtheorem{lemma}[theorem]{Lemma}
+\newtheorem{proposition}[theorem]{Proposition}
+\newtheorem{corollary}[theorem]{Corollary}
+\theoremstyle{definition}
+\newtheorem{definition}[theorem]{Definition}
+\theoremstyle{remark}
+\newtheorem{remark}[theorem]{Remark}
+\newcommand{\F}{\mathcal F}
+\newcommand{\K}{\mathbf K}
+\newcommand{\Qual}{\mathcal Q}
+\newcommand{\key}{\operatorname{key}}
+\newcommand{\rk}{\lambda}
+\newcommand{\wP}[1]{\widehat P_{#1}}
+\newcommand{\wD}[1]{\widehat D_{#1}}
+\newcommand{\A}{\mathsf A}
+\newcommand{\dotcup}{\mathbin{\dot\cup}}
+\newcommand{\poly}{\operatorname{poly}}
+\newcommand{\YES}{\mathsf{YES}}
+\newcommand{\NO}{\mathsf{NO}}
+\newcommand{\low}{\mathsf{low}}
+\newcommand{\high}{\mathsf{high}}
+\newcommand{\same}{\mathsf{same}}
+\newcommand{\switch}{\mathsf{switch}}
+
+\title{A Polynomial-Time Algorithm for Three-Machine\\
+Unit-Job Scheduling}
+\author{OpenAI}
+\date{September 24, 2026}
+
+\begin{document}
+\maketitle
+\begin{abstract}
+We give a uniform deterministic polynomial-time algorithm for scheduling
+unit-length jobs with arbitrary precedence constraints on three identical
+parallel machines. The algorithm constructs a schedule of minimum makespan
+and decides exactly whether all jobs can finish by a specified deadline.
+The proof reorganizes feasible schedules
+into intervals whose job sets have descriptions of bounded size.
+A dynamic program searches a family containing polynomially many such
+descriptions. Global boundary conditions and simplification of inherited
+information keep the descriptions bounded throughout the decomposition.
+\end{abstract}
+
+\section{Introduction}\label{sec:introduction}
+
+A precedence-constrained schedule must respect two kinds of restrictions:
+jobs related by precedence must run in order, while unrelated jobs compete
+for the available machines. Even when every job takes one unit of time, the
+interaction between these restrictions is subtle. The two-processor case
+was treated by Fujii, Kasami, and Ninomiya
+\citep{FujiiKasamiNinomiya1969,FujiiKasamiNinomiya1971Erratum}, and
+Coffman and Graham gave an optimal list-scheduling algorithm
+\citep{CG1972}. Allowing the number of machines to be part of the input
+yields an NP-complete problem \citep[Theorem~1]{Ullman1975}.
+The fixed-three decision problem appears as OPEN8 in Appendix~A13 of
+Garey and Johnson \citep{GJ1979} and has remained a central complexity
+question in scheduling \citep{NSW2025}.
+
+We consider the following precise decision problem. The input is an
+explicitly listed directed acyclic graph \(G=(V,E)\), with
+\(V=\{1,\ldots,n\}\), and an integer \(T\) satisfying \(1\le T\le n\).
+Every vertex is a nonpreemptive job of processing time one. The question
+is whether there is a function
+\[
+ \tau:V\longrightarrow\{1,\ldots,T\}
+\]
+such that each value is taken at most three times and
+\(\tau(u)<\tau(v)\) for every \((u,v)\in E\).
+There are no release dates, communication delays, eligibility restrictions,
+or additional resources.
+Slot \(t\) means execution during \([t-1,t)\), and the makespan is
+the last completion time. Allowing nonnegative real start times would
+not improve the optimum: fix the machine assignment and within-machine
+order of a feasible schedule. The graph augmented by machine-order arcs
+is acyclic, as witnessed by the original schedule. In a topological order
+of this graph, set each start time to the maximum of zero and its
+predecessors' completion times. Induction gives integer start times,
+and no completion time increases.
+The standard notation for the problem is
+\(P3\mid\mathrm{prec},p_j=1\mid C_{\max}\)
+\citep{GrahamLawlerLenstraRinnooyKan1979}.
+
+\begin{samepage}
+\begin{theorem}\label{thm:main}
+Let an explicitly listed finite directed acyclic graph specify the
+precedence constraints on \(n\ge1\) nonpreemptive unit-length jobs
+on three identical machines. There is a uniform deterministic algorithm
+that constructs a feasible schedule of minimum makespan.
+Given also an integer deadline \(1\le T\le n\), it decides feasibility
+exactly and returns a schedule whenever the answer is affirmative.
+Both tasks can be performed in
+\(O((L+2)^{150020})\) steps on a deterministic multitape Turing machine,
+where \(L\) is the total binary input length.
+\end{theorem}
+\end{samepage}
+
+The theorem resolves the polynomial-time side of this three-machine
+complexity question. Its scope is the explicit model just stated.
+The polynomial supplied by the proof has a very large constant exponent;
+no practical running-time claim is made.
+
+\paragraph{Prior work and the point of departure.}
+Structural restrictions on precedence yield important intermediate
+results. Garey, Johnson, Tarjan, and Yannakakis gave a linear-time
+algorithm on three processors for opposing forests, which are disjoint
+unions of an in-forest and an out-forest
+\citep[Section~4]{GareyJohnsonTarjanYannakakis1983}.
+Dolev and Warmuth obtained a running time
+\(O(n^{h(m-1)+1})\) when the precedence graph has height \(h\), measured
+in arcs, and at most \(m\) processors are available in each slot
+\citep[Theorem~3]{DW1984}. The available number may vary over time.
+This is polynomial when both \(h\) and \(m\) are fixed.
+Their normalized optimal schedules, called zero-adjusted, have a slot
+whose at most \(m-1\) jobs with successors determine the prefix and the
+remaining subproblem, the latter up to isomorphism. When the graph has positive
+height, both recursive pieces have smaller height
+\citep[Theorem~2]{DW1984}.
+
+Nederlof, Swennenhuis, and Węgrzycki developed this structural approach
+into an exact algorithm with running
+time \((1+n/m)^{O(\sqrt{nm})}\) for \(n\) jobs on \(m\) machines, hence
+\(2^{O(\sqrt n\log n)}\) for three machines
+\citep[Theorem~1.1]{NSW2025}.
+Their decomposition at selected slots and memoization of subproblems
+described by source and sink sets already control the number of
+subproblems despite potentially linear recursion depth
+\citep[Sections~5--6]{NSW2025}. Their proper-decomposition
+theorem assumes at most \(m\) sources
+\citep[Definition~1 and Theorem~3.1]{NSW2025}; their full algorithm
+handles arbitrary precedence graphs. The marked intervals below build on
+this decomposition viewpoint. We prove the required separator statements
+and bound the size of each global interval description directly; the
+polynomial bound does not follow by specializing their subexponential bound.
+
+Approximation algorithms also revealed useful recursive structure.
+For fixed machine count and fixed \(\varepsilon>0\), 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
+\[
+ A_0=L_1,\qquad
+ A_i=L_{i+1}\setminus(L_i\cup Z_i)\ (1\le i<k),\qquad
+ A_k=R_k.
+\]
+The inclusions in \eqref{eq:transition} give the ordered disjoint partition
+\begin{equation}
+ W=A_0\dotcup Z_1\dotcup A_1\dotcup\cdots
+       \dotcup Z_k\dotcup A_k.
+ \label{eq:partition}
+\end{equation}
+Every \(A_i\) is smaller than \(W\) and was already accepted. By induction
+it has a full schedule. Every two distinct pieces in \eqref{eq:partition}
+are separated by a selected state, with the earlier piece in its left
+part or triple and the later piece in its triple or right part.
+Compatibility of that state excludes a backward precedence between
+the pieces. Concatenating their schedules with the selected triples is
+therefore a full schedule of \(W\).
+
+To produce that schedule, record a predecessor whenever a state is
+first reached, with initial states distinguished as roots. Trace back
+from a reached final state and use the partition \eqref{eq:partition}.
+Along every transition \(|L|\) increases by at least three, so an
+accepting path has at most \(N/3\) triples. Retrieve the already stored
+schedules of its smaller gap sets and concatenate them with those
+triples. Store this one explicit schedule for \(W\), avoiding any
+repeated recursive expansion.
+
+There are only polynomially many formula instances, sets, states,
+and state pairs because the leaf allowance $K$ is fixed. Sequential scans
+of their bit-vector records therefore give a uniform polynomial-time
+implementation. \Cref{sec:complexity} gives the explicit multitape
+accounting, including the scans and copying needed for reconstruction,
+and proves the stated exponent.
+\end{proof}
+
+The completeness certificate uses only intervals, their marked triples,
+and descriptions; it imposes no bound on the number of levels.
+
+\Needspace{3\baselineskip}
+\begin{definition}\label{def:hierarchy}
+A \emph{bounded-description hierarchy} for a feasible \(J\) has the whole
+schedule as its root interval. At a nonempty node \(W\), one may first
+replace its schedule by a full schedule of the same job set. One then
+marks at least one actual slot; the nonempty open gaps between successive
+marks and the two boundaries become its children. Different children may
+be developed with separate copies of their witnessing exterior schedules.
+The following descriptions are required:
+\begin{enumerate}[label=(\roman*)]
+ \item every node job set belongs to \(\F\);
+ \item for every marked triple, its earlier jobs within the node have a
+ relative description \(W\cap F\), with \(F\in\F\), in original time.
+\end{enumerate}
+\end{definition}
+
+\begin{lemma}\label{lem:hierarchy-dp}
+If a feasible \(J\) has a bounded-description hierarchy, the algorithm
+accepts \(J\).
+\end{lemma}
+\begin{proof}
+Induct on node size. Its marks give compatible states in chronological
+order by property (ii). Their end gaps and transition gaps are empty
+or are its children, belong to \(\F\), and are strictly smaller because
+there is an actual mark. They have already been accepted by induction.
+The states consequently give an initial-to-final path.
+The root belongs to \(\F\) because its predicate is true.
+\end{proof}
+
+We prove the following structural assertion in
+\Cref{sec:separators,sec:construction,sec:lists,sec:boundary,sec:descriptions}.
+The considerably larger allowance \(K=10000\) avoids dependence on tight
+constant bookkeeping.
+
+\begin{theorem}\label{thm:structure}
+Every feasible full three-machine schedule has a bounded-description
+hierarchy. In fact, the global node predicates and relative split
+predicates needed in the hierarchy have fewer than \(500\) leaves.
+\end{theorem}
+
+\begin{proof}[Proof of \Cref{thm:main}, assuming \Cref{thm:structure}]
+Apply \Cref{lem:padding,prop:algorithm,lem:hierarchy-dp,thm:structure}.
+The algorithm is sound on every enumerated set and complete at \(J\).
+For makespan minimization, test all deadlines \(1,\ldots,n\) and take
+the smallest feasible one. A topological ordering gives a feasible
+schedule of length at most \(n\), so this search succeeds.
+Return the stored full schedule for its padded instance and delete the
+isolated padding jobs. Assign the at most three remaining jobs in each
+slot to distinct machines. The schedule has optimum makespan by the
+choice of deadline. The bound in \Cref{prop:algorithm} already allows
+all these deadline trials.
+\end{proof}
+
+\input{separators}
+\input{construction}
+\input{lists}
+\input{boundary}
+\input{descriptions}
+\section{Running time in the explicit input model}\label{sec:complexity}
+
+The preceding structural proof establishes completeness of the finite
+search. We now give the detailed deterministic multitape accounting
+promised in \Cref{prop:algorithm}. Its purpose is to make the
+polynomial-time assertion uniform; the structural argument does not
+require efficient normalization of a schedule or a small recursion depth.
+
+\begin{proof}[Running-time bound in \Cref{prop:algorithm}]
+First specify a conservative implementation using sequential scans.
+Write \(B=C_KN^{3K}\) for an upper bound on the number of enumerated
+formulas, where \(C_K\) depends only on the fixed \(K\).
+There are at most \(B\) distinct sets and at most \(S=BN^3\)
+candidate states for each \(W\).
+Store their bit vectors, reachability marks, and predecessor record
+numbers on work tapes. Each state record uses \(O_K(N)\) bits:
+its indices require only \(O_K(\log N)\) bits.
+Store an explicit full schedule with each previously accepted set.
+This table uses \(O_K(BN^2)\) bits, since a schedule lists at most
+\(N\) job identifiers and their slots.
+
+A complete scan to fetch or update a state record costs
+\(O_K(SN)=O_K(BN^4)\) steps. A lookup in the acceptance table by
+comparing all bit vectors costs \(O_K(BN^2)\) steps.
+Even checking all required precedence relations by repeated scans of
+the \(N\)-by-\(N\) reachability table costs at most \(O(N^4)\).
+Thus one ordered state-pair test and reachability update costs
+\(O_K(BN^4)\) steps, without constant-time random access.
+Starting from the initial states, make at most \(S\) passes over all
+ordered pairs, marking a target when its source is marked and
+\eqref{eq:transition} holds. These passes find every reachable state.
+For all sets together their cost is at most
+\[
+ O_K\bigl(BS^3BN^4\bigr)
+ =O_K(B^5N^{13})
+ =O_K(N^{15K+13}).
+\]
+Generating, evaluating, and deduplicating formulas by scans costs at
+most \(O_K(B^2N^4)\); reachability preprocessing, state construction,
+and table initialization also fit within the displayed bound.
+The subscript \(K\) denotes a constant fixed independently of the input.
+Since \(N\le3n\) and the vertices are explicitly listed, the looser
+bound \(O((L+2)^{150020})\) covers input processing and even all
+\(n\) deadline trials needed for optimization.
+\end{proof}
+
+\begin{remark}\label{rem:compact}
+For the deadline feasibility decision, the same preprocessing can handle
+an encoding that specifies an enormous number of isolated vertices only
+through a binary value of \(n\).
+After checking \(n\le3T\), let \(d\) be the number of vertices appearing
+in edges. If \(T\ge d\), accept: schedule those \(d\) vertices sequentially
+in topological order and fill free positions with the isolated vertices.
+Otherwise \(N=3T<3d\le6|E|\), so expansion is polynomial. This extra
+branch is not needed for the explicit encoding in \Cref{thm:main}.
+\end{remark}
+
+\input{conclusion}
+
+\bibliographystyle{plainnat}
+\bibliography{references}
+\end{document}

preprints/A-polynomial-time-algorithm-for-three-machine-unit-job-scheduling-September-24-2026/build/references.bib

@@ -0,0 +1,135 @@
+@article{FujiiKasamiNinomiya1969,
+  author = {Fujii, M. and Kasami, T. and Ninomiya, K.},
+  title = {Optimal Sequencing of Two Equivalent Processors},
+  journal = {SIAM Journal on Applied Mathematics},
+  volume = {17}, number = {4}, pages = {784--789}, year = {1969},
+  doi = {10.1137/0117070},
+}
+
+@article{FujiiKasamiNinomiya1971Erratum,
+  author = {Fujii, M. and Kasami, T. and Ninomiya, K.},
+  title = {Erratum: {Optimal Sequencing of Two Equivalent Processors}},
+  journal = {SIAM Journal on Applied Mathematics},
+  volume = {20}, number = {1}, pages = {141}, year = {1971},
+  doi = {10.1137/0120018},
+  url = {https://epubs.siam.org/doi/epdf/10.1137/0120018}
+}
+
+@article{CG1972,
+  author  = {Coffman, Jr., Edward G. and Graham, Ronald L.},
+  title   = {Optimal Scheduling for Two-Processor Systems},
+  journal = {Acta Informatica},
+  volume  = {1},
+  number  = {3},
+  pages   = {200--213},
+  year    = {1972},
+  doi     = {10.1007/BF00288685}
+}
+
+@article{Ullman1975,
+  author  = {Ullman, Jeffrey D.},
+  title   = {{NP}-Complete Scheduling Problems},
+  journal = {Journal of Computer and System Sciences},
+  volume  = {10},
+  number  = {3},
+  pages   = {384--393},
+  year    = {1975},
+  doi     = {10.1016/S0022-0000(75)80008-0}
+}
+
+@book{GJ1979,
+  author    = {Garey, Michael R. and Johnson, David S.},
+  title     = {Computers and Intractability: A Guide to the Theory of {NP}-Completeness},
+  publisher = {W. H. Freeman},
+  year      = {1979}
+}
+
+@article{GrahamLawlerLenstraRinnooyKan1979,
+  author = {Graham, R. L. and Lawler, E. L. and Lenstra, J. K. and Rinnooy Kan, A. H. G.},
+  title = {Optimization and Approximation in Deterministic Sequencing and Scheduling: a Survey},
+  journal = {Annals of Discrete Mathematics}, volume = {5}, pages = {287--326}, year = {1979},
+  doi = {10.1016/S0167-5060(08)70356-X},
+  url = {https://ir.cwi.nl/pub/18052},
+}
+
+@article{GareyJohnsonTarjanYannakakis1983,
+  author = {Garey, M. R. and Johnson, D. S. and Tarjan, R. E. and Yannakakis, M.},
+  title = {Scheduling Opposing Forests},
+  journal = {SIAM Journal on Algebraic and Discrete Methods},
+  volume = {4}, number = {1}, pages = {72--93}, year = {1983},
+  doi = {10.1137/0604011},
+}
+
+@article{DW1984,
+  author = {Dolev, Danny and Warmuth, Manfred K.},
+  title = {Scheduling Precedence Graphs of Bounded Height},
+  journal = {Journal of Algorithms},
+  volume = {5},
+  number = {1},
+  pages = {48--59},
+  year = {1984},
+  doi = {10.1016/0196-6774(84)90039-7}
+}
+
+@inproceedings{NSW2025,
+  author    = {Nederlof, Jesper and Swennenhuis, C{\'e}line M. F. and W{\k{e}}grzycki, Karol},
+  title     = {A Subexponential Time Algorithm for Makespan Scheduling of Unit Jobs with Precedence Constraints},
+  booktitle = {Proceedings of the 2025 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA)},
+  year      = {2025},
+  pages     = {535--552},
+  publisher = {Society for Industrial and Applied Mathematics},
+  doi       = {10.1137/1.9781611978322.16}
+}
+
+
+@inproceedings{LeveyRothvoss2016,
+  author = {Levey, Elaine and Rothvo{\ss}, Thomas},
+  title = {A {$(1+\epsilon)$}-Approximation for Makespan Scheduling with Precedence Constraints Using {LP} Hierarchies},
+  booktitle = {Proceedings of the 48th Annual ACM SIGACT Symposium on Theory of Computing},
+  series = {STOC '16},
+  year = {2016},
+  pages = {168--177},
+  publisher = {Association for Computing Machinery},
+  doi = {10.1145/2897518.2897532},
+  url = {https://doi.org/10.1145/2897518.2897532}
+}
+
+@inproceedings{Garg2018,
+  author = {Garg, Shashwat},
+  title = {Quasi-{PTAS} for Scheduling with Precedences Using {LP} Hierarchies},
+  booktitle = {45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)},
+  editor = {Chatzigiannakis, Ioannis and Kaklamanis, Christos and Marx, D{\'a}niel and Sannella, Donald},
+  series = {Leibniz International Proceedings in Informatics (LIPIcs)},
+  volume = {107},
+  year = {2018},
+  pages = {59:1--59:13},
+  publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
+  doi = {10.4230/LIPIcs.ICALP.2018.59},
+  url = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2018.59}
+}
+
+@inproceedings{Li2021,
+  author = {Li, Shi},
+  title = {Towards {PTAS} for Precedence Constrained Scheduling via Combinatorial Algorithms},
+  booktitle = {Proceedings of the 2021 ACM-SIAM Symposium on Discrete Algorithms (SODA)},
+  editor = {Marx, D{\'a}niel},
+  year = {2021},
+  pages = {2991--3010},
+  publisher = {Society for Industrial and Applied Mathematics},
+  doi = {10.1137/1.9781611976465.178},
+  url = {https://doi.org/10.1137/1.9781611976465.178}
+}
+
+@inproceedings{DasWiese2022,
+  author = {Das, Syamantak and Wiese, Andreas},
+  title = {A Simpler {QPTAS} for Scheduling Jobs with Precedence Constraints},
+  booktitle = {30th Annual European Symposium on Algorithms (ESA 2022)},
+  editor = {Chechik, Shiri and Navarro, Gonzalo and Rotenberg, Eva and Herman, Grzegorz},
+  series = {Leibniz International Proceedings in Informatics (LIPIcs)},
+  volume = {244},
+  year = {2022},
+  pages = {40:1--40:11},
+  publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
+  doi = {10.4230/LIPIcs.ESA.2022.40},
+  url = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2022.40}
+}

preprints/A-polynomial-time-algorithm-for-three-machine-unit-job-scheduling-September-24-2026/build/separators.tex

@@ -0,0 +1,162 @@
+\section{Separators with a shared cutoff}\label{sec:separators}
+
+Decomposing a normalized schedule at selected slots is a central device
+in Dolev--Warmuth \citep[Theorem~2]{DW1984} and in the proper-decomposition
+construction of Nederlof, Swennenhuis, and Węgrzycki
+\citep[Section~3]{NSW2025}. Here we prove the shared-cutoff form needed
+for the interval construction: successive boundaries must separate the
+same ideal of low-priority jobs. The global-list invariants in later
+sections control these descriptions inside descendant intervals.
+
+This section concerns one fixed frame and a domain consisting of a
+contiguous interval of a feasible schedule. A set is an \emph{ideal}
+in the domain if it contains every domain predecessor of each of its
+members. Its complement in the domain is an upset.
+
+\begin{definition}\label{def:separator}
+Given an ideal \(S\) of low jobs in a domain, a slot \(z\) is a
+\emph{separator at cutoff \(S\)} if
+\begin{enumerate}[label=(\alph*)]
+ \item every domain job before \(z\) that is outside \(S\) belongs to \(P_z\);
+ \item every domain job after \(z\) that belongs to \(S\) belongs to \(D_z\).
+\end{enumerate}
+The other domain jobs are called high. A left boundary outside the domain
+is also a separator when all low domain jobs descend from it; there are
+then no domain jobs before that boundary.
+\end{definition}
+
+An actual separator has a useful description of its earlier jobs. For
+a global set \(H\subseteq J\), put
+\begin{equation}
+ B_z(H)=P_z\cup\bigl(H\setminus\wD z\bigr).
+ \label{eq:B-definition}
+\end{equation}
+
+\begin{lemma}\label{lem:separator-description}
+If \(z\) is an actual separator for a domain \(U\) at cutoff \(S\),
+and \(H\cap U=S\), then \(B_z(H)\cap U\) is exactly the set of
+domain jobs strictly earlier than \(z\). Reversing order and time,
+and exchanging low with high, preserves the separator property.
+\end{lemma}
+\begin{proof}
+Every job in \(P_z\) is strictly earlier than \(z\). An earlier low
+job is outside \(\wD z\). A low job at \(z\) belongs to \(\wD z\),
+and a later low job belongs to \(D_z\) by separation. Finally, every
+earlier high job belongs to \(P_z\). These observations give both
+inclusions. Reversal exchanges the two conditions in
+\Cref{def:separator}.
+\end{proof}
+
+\subsection{Lexicographic normalization}
+
+A \emph{priority order} is a total order of the domain extending
+precedence. Read a full schedule chronologically, with each triple sorted
+by priority. Among the finitely many feasible full schedules of the
+domain, choose one whose resulting sequence of priority indices is
+lexicographically least. Call it \emph{normalized} for that priority.
+This is an existence choice, not an operation of the decision algorithm.
+
+\begin{lemma}\label{lem:lex-prefix}
+In a normalized schedule, a feasible exchange that puts a better-priority
+job into an earlier slot is impossible. More generally, fix a slot \(r\).
+After any feasible reordering confined strictly after \(r\), a feasible
+exchange of a job at \(r\) with a better-priority job at a later slot
+is still impossible.
+\end{lemma}
+\begin{proof}
+Replacing one member of a sorted triple by a better-priority job
+lexicographically improves that triple. In the second assertion, the
+schedule through \(r\) is unchanged by the intervening reordering.
+The proposed exchange would therefore first change the normalized
+sequence at \(r\), and improve it there. The resulting full schedule
+would contradict minimality, regardless of what happens later.
+\end{proof}
+
+\begin{lemma}[Shared-cutoff walk]\label{lem:shared}
+Fix a normalized full schedule of a domain and let \(S\) and \(S^+\)
+be consecutive priority prefixes, differing by the promotion of one job
+from high to low. Suppose \(s\) is a separator at cutoff \(S\),
+either inside the domain or at its exterior left boundary.
+Then there is a separator \(t\ge s\), found by advancing only to
+domain slots on the right, that separates both \(S\) and \(S^+\).
+If the priority order is specified on a larger job set and the promoted
+job is outside the domain, one may take \(t=s\).
+\end{lemma}
+\begin{proof}
+\textbf{One advance at a fixed cutoff.}
+Keep the normalized schedule and its priority order fixed. Let \(L\)
+be a prefix of that priority order, and suppose the current position
+\(s\) satisfies the earlier-high condition for \(L\).
+If a later low job does not descend from \(s\), choose the earliest
+slot \(t>s\) 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 \(s\le r<t\).
+If \(y\) had no domain successor at or before \(t\), exchanging
+\(x\) and \(y\) would be feasible: all predecessors of \(x\) are
+before \(s\), and delaying \(y\) would violate no successor
+constraint. Moving \(x\) earlier and \(y\) later preserves their
+other precedence constraints. Since \(L\) is a priority prefix,
+\(x\) has better priority than \(y\), contradicting
+\Cref{lem:lex-prefix}. Thus \(y\) has a successor by \(t\).
+The successor is high, since the high set is an upset. If it is
+strictly before \(t\), repeat the argument. Strictly increasing
+times give a chain from \(y\) to a high job at \(t\).
+
+When \(s\) is internal, every earlier high job has a descendant
+at \(s\) by the earlier-high condition. That descendant is high,
+and the preceding tracing extends the chain to \(t\).
+When \(s\) is the exterior left boundary, there are no earlier
+domain jobs. Hence \(t\) satisfies the earlier-high condition.
+We may advance to \(t\) and repeat until there is no later low
+exception. There are finitely many slots, and every advance is
+strict, so the final position also satisfies the later-low condition.
+Throughout this walk the normalized schedule itself stays fixed;
+only the selected separator position changes.
+
+\smallskip\noindent\textbf{Adding one low job.}
+Apply this procedure with \(L=S^+\), starting at the given
+separator for \(S\). The earlier-high condition for \(S^+\)
+holds initially because its high set is smaller. If no advance is
+needed, the same position separates both cutoffs.
+
+Otherwise the first exception is the single promoted job: every
+later job already low at \(S\) descends from the starting position.
+At the first target, the promoted job lies in that target slot,
+so it introduces no earlier high job for the old cutoff \(S\).
+Thus the earlier-high condition holds there for both cutoffs.
+
+In every subsequent advance, the exception is old-low: the promoted
+job stays at the first target, at or before the current position.
+An old-low exception has better priority than every old-high job.
+The same exchange and successor tracing therefore propagates the
+earlier-high condition for \(S\) as well as for \(S^+\).
+At the final position, the later-low condition for \(S^+\) also
+implies the later-low condition for \(S\), since \(S\subseteq S^+\).
+The final position separates both cutoffs.
+
+If the promoted job is outside the domain, the two restricted
+prefixes coincide, so the original position already separates both.
+\end{proof}
+
+\begin{corollary}\label{cor:any-ideal}
+Every ideal of a nonempty set with a full schedule has an actual
+separator in some full schedule of that set.
+\end{corollary}
+\begin{proof}
+Prioritize the ideal before its complement, using a linear extension
+inside each. Normalize. The first actual slot separates the empty
+prefix. Promote the jobs of the ideal one by one and use
+\Cref{lem:shared}. The final position is an actual separator for
+the ideal.
+\end{proof}
+
+The shared cutoff, rather than merely the existence of separate
+separators, will be used for a gap: its low jobs descend from its
+left endpoint, while its high jobs precede its right endpoint.
+Keeping the final position of each promotion is important. Intermediate
+positions assist the proof but need not become marks.

preprints/A-polynomial-time-algorithm-for-three-machine-unit-job-scheduling-September-24-2026/build/tex/runtime/generic/pgf/basiclayer/pgfcore.code.tex

@@ -0,0 +1,33 @@
+
+\ProvidesPackageRCS{pgfcore.code.tex}
+
+\newif\ifpgf@draftmode
+\def\pgf@typeout{\immediate\write0}
+
+\ifdefined\pgfmathloaded\else
+  \input pgfmath.code.tex
+\fi
+
+\ifdefined\pgfintloaded\else
+  \input pgfint.code.tex
+\fi
+
+\input pgfcorepoints.code.tex
+\input pgfcorepathconstruct.code.tex
+\input pgfcorepathusage.code.tex
+\input pgfcorescopes.code.tex
+\input pgfcoregraphicstate.code.tex
+\input pgfcoretransformations.code.tex
+\input pgfcorequick.code.tex
+\input pgfcoreobjects.code.tex
+\input pgfcorepathprocessing.code.tex
+\input pgfcorearrows.code.tex
+\input pgfcoreshade.code.tex
+\input pgfcoreimage.code.tex
+\input pgfcoreexternal.code.tex
+\input pgfcorelayers.code.tex
+\input pgfcoretransparency.code.tex
+\input pgfcorepatterns.code.tex
+\input pgfcorerdf.code.tex
+
+\endinput

preprints/A-polynomial-time-algorithm-for-three-machine-unit-job-scheduling-September-24-2026/build/tex/runtime/generic/pgf/basiclayer/pgfcorearrows.code.tex

@@ -0,0 +1,1011 @@
+
+\ProvidesFileRCS{pgfcorearrows.code.tex}
+
+
+
+
+\def\pgfdeclarearrow#1{%
+  \let\pgf@decl@arrow@defaults\pgfutil@empty%
+  \let\pgf@decl@arrow@means\relax%
+  \let\pgf@decl@arrow@code\relax%
+  \pgfkeys{/pgf/@arrows decl/.cd,
+    name=,
+    setup code=,
+    cache=true,
+    bending mode=orthogonal,
+    parameters=,%
+    #1%
+  }%
+  \ifx\pgf@decl@arrow@name@end\pgfutil@empty%
+    \pgferror{Declaring unnamed arrow}%
+  \else%
+    \pgf@arrow@letter{name@end}%
+    \pgf@arrow@letter{name@start}%
+    \expandafter\let\csname pgf@ar@start@\pgf@decl@arrow@name@start\endcsname\pgf@decl@arrow@name@end%
+    \expandafter\let\expandafter\pgf@old@code\csname pgf@ar@code@\pgf@decl@arrow@name@end\endcsname%
+    \expandafter\let\expandafter\pgf@old@means\csname pgf@ar@means@\pgf@decl@arrow@name@end\endcsname%
+    \ifx\pgf@decl@arrow@code\relax%
+      \ifx\pgf@decl@arrow@means\relax%
+        \pgferror{You must set either the 'means' or the drawing code' keys for arrow tip '\pgf@decl@arrow@name@end'}%
+      \else%
+        \ifx\pgf@old@code\relax%
+          \let\pgf@arrow@tip@sequence\pgfutil@empty%
+          \let\pgf@arrow@translate\relax%
+          \expandafter\let\expandafter\pgf@arrows@direct\csname pgf@arrows@direct@name@end@\pgf@decl@arrow@means\endcsname%
+          \ifx\pgf@arrows@direct\relax%
+            \let\pgf@arrows@options\pgfutil@empty%
+            \expandafter\pgfarrows@initial@parser\pgf@decl@arrow@means[]\pgf@stop%
+          \else%
+            \let\pgf@arrow@tip@sequence\pgf@arrows@direct%
+          \fi%
+          \pgf@arrow@letter@\pgf@arrow@tip@sequence{means}%
+          \expandafter\let\csname pgf@ar@code@\pgf@decl@arrow@name@end\endcsname\relax%
+        \else%
+          \pgferror{Arrow tip '\pgf@decl@arrow@name@end' was already
+            defined with key 'drawing code' previously and you cannot change this}%
+        \fi%
+      \fi%
+    \else%
+      \ifx\pgf@decl@arrow@means\relax%
+        \ifx\pgf@old@means\relax%
+          \pgf@arrow@letter{code}%
+          \pgf@arrow@letter{defaults}%
+          \pgf@arrow@letter{setup}%
+          \pgf@arrow@letter{bending@mode}%
+          \pgf@arrow@letter{par}%
+          \expandafter\let\csname pgf@ar@do@cache@\pgf@decl@arrow@name@end\endcsname\pgf@decl@arrow@cache%
+        \else%
+          \pgferror{Arrow tip '\pgf@decl@arrow@name@end' was already
+            defined with key 'means' previously and you cannot change this}%
+        \fi%
+      \else%
+        \pgferror{You cannot set both the 'means' and also the 'drawing code'
+          keys for arrow tip '\pgf@decl@arrow@name@end'}%
+      \fi%
+    \fi%
+  \fi%
+}
+
+\def\pgf@arrow@letter#1{%
+  \expandafter\pgf@arrow@letter@\expandafter{\csname pgf@decl@arrow@#1\endcsname}{#1}%
+}
+\def\pgf@arrow@letter@#1#2{%
+  \expandafter\let\csname pgf@ar@#2@\pgf@decl@arrow@name@end\endcsname#1%
+}
+
+\pgfkeys{
+  /pgf/@arrows decl/.cd,
+  name/.code={%
+    \pgfutil@in@-{#1}%
+    \ifpgfutil@in@%
+      \pgf@arrows@name@decomp#1\pgf@stop%
+    \else%
+      \def\pgf@decl@arrow@name@end{#1}\let\pgf@decl@arrow@name@start\pgf@decl@arrow@name@end%
+    \fi},
+  means/.store in=\pgf@decl@arrow@means,
+  setup code/.store in=\pgf@decl@arrow@setup,
+  drawing code/.store in=\pgf@decl@arrow@code,
+  cache/.store in=\pgf@decl@arrow@cache,
+  bending mode/.is choice,
+  bending mode/none/.code=\let\pgf@decl@arrow@bending@mode\pgfutil@empty,
+  bending mode/orthogonal/.code=\def\pgf@decl@arrow@bending@mode{\pgfpointcurvilinearbezierorthogonal},
+  bending mode/polar/.code=\def\pgf@decl@arrow@bending@mode{\pgfpointcurvilinearbezierpolar},
+  parameters/.store in=\pgf@decl@arrow@par,
+  defaults/.code={%
+    \let\pgf@arrows@options\pgfutil@empty%
+    \pgfkeys{/pgf/arrow keys/.cd,#1}%
+    \let\pgf@decl@arrow@defaults\pgf@arrows@options%
+  }
+}
+
+\def\pgf@arrows@name@decomp#1-#2\pgf@stop{%
+  \def\pgf@decl@arrow@name@start{#1}%
+  \def\pgf@decl@arrow@name@end{#2}%
+}
+
+
+
+\pgfkeys{/handlers/.tip/.code=%
+  \expandafter\expandafter\expandafter\pgf@arrows@unravel\pgfkeyscurrentpath/\pgf@stop%
+  \expandafter\pgf@arrows@key@call\expandafter{\pgf@arrows@unravelled}{#1}}
+\def\pgf@arrows@unravel#1/{\pgfutil@ifnextchar\pgf@stop{\def\pgf@arrows@unravelled{#1}\pgfutil@gobble}{\pgf@arrows@unravel}}
+\def\pgf@arrows@key@call#1#2{\pgfdeclarearrow{name={#1},means={#2}}}
+
+
+
+
+
+\def\pgfarrowdraw#1{%
+  {%
+    \pgfsetarrowsend{#1}%
+    \pgf@arrow@compute@shortening\pgf@end@tip@sequence%
+    \pgf@arrow@draw@arrow\pgf@end@tip@sequence\pgf@xb%
+  }%
+}
+
+
+
+
+\def\pgfarrowtotallength#1{%
+  {%
+    \pgfsetarrowsend{#1}%
+    \pgf@arrow@compute@shortening\pgf@end@tip@sequence%
+    \expandafter\expandafter\expandafter%
+  }%
+  \expandafter\expandafter\expandafter\pgf@x\expandafter\the\expandafter\pgf@xb\expandafter\pgf@xa\the\pgf@xa%
+}
+
+
+
+
+
+
+\def\pgfsetarrowsend#1{%
+  \let\pgf@arrow@tip@sequence\pgfutil@empty%
+  \pgf@arrows@translatefalse%
+  \expandafter\let\expandafter\pgf@arrows@direct\csname pgf@arrows@direct@name@end@#1\endcsname%
+  \ifx\pgf@arrows@direct\relax%
+    \edef\pgf@marshal{\noexpand\pgfarrows@initial@parser#1[]\noexpand\pgf@stop}%
+    \pgf@marshal%
+  \else%
+    \let\pgf@arrow@tip@sequence\pgf@arrows@direct%
+  \fi%
+  \let\pgf@end@tip@sequence\pgf@arrow@tip@sequence%
+}
+
+\newif\ifpgf@arrows@translate
+
+\def\pgfarrows@initial@parser{%
+  \pgfutil@ifnextchar[\pgfarrows@initial@parser@{\pgfarrows@initial@parser@[]}}%
+
+\def\pgfarrows@initial@parser@[#1]{%
+  \let\pgf@arrows@options\pgfutil@empty%
+  \pgfkeys{/pgf/arrow keys/.cd,#1}%
+  \let\pgf@arrows@options@initial\pgf@arrows@options%
+  \pgfarrows@parser%
+}
+
+\def\pgfarrows@parser#1[{%
+  \pgfkeys@spdef\pgf@temp{#1}%
+  \ifx\pgf@temp\pgfutil@empty%
+    \let\pgf@next\pgf@arrows@empty%
+  \else%
+    \ifx\pgf@temp\pgf@dot@text%
+      \let\pgf@next\pgf@arrows@dot@parsed%
+    \else%
+      \let\pgf@untranslated\pgf@temp%
+      \ifpgf@arrows@translate%
+        \pgf@arrow@swapper%
+      \fi%
+      \expandafter\ifx\csname pgf@ar@code@\pgf@temp\endcsname\relax
+        \expandafter\ifx\csname pgf@ar@means@\pgf@temp\endcsname\relax
+          \let\pgf@next\pgf@arrows@single@char@parser%
+          \let\pgf@temp\pgf@untranslated%
+        \else%
+          \let\pgf@next\pgf@arrows@shorthand@parsed%
+        \fi%
+      \else%
+        \let\pgf@next\pgf@arrows@meta@parsed%
+      \fi%
+    \fi%
+  \fi%
+  \expandafter\pgf@next\pgf@temp[%
+}
+
+\def\pgf@arrows@empty[#1]{%
+  \pgfutil@ifnextchar\pgf@stop\pgf@arrows@parser@done\pgfarrows@parser%
+}
+
+\def\pgf@arrows@single@char@parser#1{%
+  \def\pgf@temp{#1}%
+  \ifx\pgf@temp\pgf@dot@text%
+    \let\pgf@next\pgf@arrows@dot@parsed%
+  \else%
+    \let\pgf@untranslated\pgf@temp%
+    \ifpgf@arrows@translate%
+      \pgf@arrow@swapper%
+    \fi%
+    \expandafter\ifx\csname pgf@ar@code@\pgf@temp\endcsname\relax
+      \expandafter\ifx\csname pgf@ar@means@\pgf@temp\endcsname\relax
+        \let\pgf@temp\pgf@untranslated%
+        \let\pgf@next\pgf@arrows@unknown%
+      \else%
+        \let\pgf@next\pgf@arrows@shorthand@parsed%
+      \fi%
+    \else%
+      \let\pgf@next\pgf@arrows@meta@parsed%
+    \fi
+  \fi
+  \expandafter\pgf@next\pgf@temp[]%
+}
+\def\pgf@arrows@unknown#1[]#2[#3]{%
+  \pgferror{Unknown arrow tip kind '#1#2'}%
+}
+
+\def\pgf@arrows@meta@parsed#1[#2]{%
+  \let\pgf@arrows@options\pgf@arrows@options@initial%
+  \pgfkeys{/pgf/arrow keys/.cd,#2}%
+  \expandafter\pgf@arrows@meta@set\expandafter{\pgf@arrows@options}{#1}{\pgf@arrow@handle}%
+  \pgfutil@ifnextchar\pgf@stop\pgf@arrows@parser@done\pgfarrows@parser%
+}
+
+\def\pgf@arrows@parser@done#1{%
+}
+
+\def\pgf@arrows@dot@parsed#1[#2]{%
+  \pgf@arrows@append@to@tips{\pgf@arrow@handle@dot}%
+  \pgfutil@ifnextchar\pgf@stop\pgf@arrows@parser@done\pgfarrows@parser%
+}
+
+\def\pgf@arrows@meta@set#1#2#3{%
+  \pgf@arrows@append@to@tips{#3{#2}{#1}}%
+}
+\def\pgf@arrows@append@to@tips#1{%
+  \let\pgf@tempa\pgf@arrow@tip@sequence%
+  \def\pgf@tempb{#1}%
+  \ifpgf@arrows@translate%
+    \let\pgf@tempa\pgf@tempb%
+    \let\pgf@tempb\pgf@arrow@tip@sequence%
+  \fi%
+  \expandafter\expandafter\expandafter\def%
+  \expandafter\expandafter\expandafter\pgf@arrow@tip@sequence%
+  \expandafter\expandafter\expandafter{\expandafter\pgf@tempa\pgf@tempb}%
+}
+
+\def\pgf@arrows@shorthand@parsed#1[#2]{%
+  \let\pgf@arrows@options\pgf@arrows@options@initial%
+  \pgfkeys{/pgf/arrow keys/.cd,#2}%
+  \ifx\pgf@arrows@options\pgfutil@empty%
+    \expandafter\pgf@arrows@append@to@tips\expandafter{%
+      \expandafter\pgf@arrow@handle@shorthand@empty\expandafter{\csname pgf@ar@means@#1\endcsname}}%
+  \else%
+    \expandafter\expandafter\expandafter\pgf@arrows@meta@set\expandafter\expandafter\expandafter{%
+      \expandafter\pgf@arrows@options\expandafter}\expandafter{\csname pgf@ar@means@#1\endcsname}{\pgf@arrow@handle@shorthand}%
+  \fi%
+  \pgfutil@ifnextchar\pgf@stop\pgf@arrows@parser@done\pgfarrows@parser%
+}
+
+
+
+\def\pgf@arrow@handle@shorthand@empty#1{#1}
+
+\def\pgf@arrow@handle@shorthand#1#2{%
+  \def\pgf@temp@opt{#2}%
+  \expandafter\expandafter\expandafter\def%
+  \expandafter\expandafter\expandafter\pgf@arrows@nested@options@temp%
+  \expandafter\expandafter\expandafter{\expandafter\pgf@temp@opt\pgf@arrows@nested@options}%
+  \expandafter\let\expandafter\pgf@arrows@nested@options\expandafter\pgf@arrows@nested@options@temp%
+  \expandafter#1%
+  \expandafter\def\expandafter\pgf@arrows@nested@options\expandafter{\pgf@arrows@nested@options}%
+}
+
+\let\pgf@arrows@nested@options\pgfutil@empty
+
+
+
+
+
+\def\pgfsetarrowsstart#1{%
+  \let\pgf@arrow@tip@sequence\pgfutil@empty%
+  \pgf@arrows@translatetrue%
+  \expandafter\let\expandafter\pgf@arrows@direct\csname pgf@arrows@direct@name@start@#1\endcsname%
+  \ifx\pgf@arrows@direct\relax%
+    \edef\pgf@marshal{\noexpand\pgfarrows@initial@parser#1[]\noexpand\pgf@stop}%
+    \pgf@marshal%
+  \else%
+    \let\pgf@arrow@tip@sequence\pgf@arrows@direct%
+  \fi%
+  \let\pgf@start@tip@sequence\pgf@arrow@tip@sequence%
+}
+
+\def\pgf@arrow@swapper{%
+  \expandafter\let\expandafter\pgf@temp\csname pgf@ar@start@\pgf@temp\endcsname%
+  \ifx\pgf@temp\relax%
+    \def\pgf@temp{undefined}%
+  \fi%
+}
+
+\let\pgf@start@tip@sequence\pgfutil@empty
+\let\pgf@end@tip@sequence\pgfutil@empty
+
+
+\expandafter\def\csname pgf@arrows@direct@name@start@\endcsname{}
+\expandafter\def\csname pgf@arrows@direct@name@end@\endcsname{}
+
+
+
+\newdimen\pgfarrowsep
+
+\newif\ifpgfarrowswap
+\newif\ifpgfarrowreversed
+\newif\ifpgfarrowharpoon
+\newif\ifpgfarrowopen
+
+\let\pgf@arrows@stroke@color\pgfutil@empty
+\let\pgf@arrows@fill@color\pgfutil@empty
+
+\def\pgfarrows@length@scale{1}
+\def\pgfarrows@width@scale{1}
+\def\pgfarrows@slant{0}
+
+\let\pgfarrows@scale@list\pgfutil@empty
+\def\pgfarrowsaddtolengthscalelist#1{\expandafter\def\expandafter\pgfarrows@scale@list\expandafter{\pgfarrows@scale@list#1\pgfarrows@length@scale#1}}
+\def\pgfarrowsaddtowidthscalelist#1{\expandafter\def\expandafter\pgfarrows@scale@list\expandafter{\pgfarrows@scale@list#1\pgfarrows@width@scale#1}}
+
+\pgfkeys{
+  /pgf/arrow keys/.cd,
+  scale/.code={%
+    \pgfmathparse{#1}%
+    \expandafter\pgfarrowsaddtooptions\expandafter{\expandafter\def\expandafter\pgfarrows@length@scale\expandafter{\pgfmathresult}}%
+    \expandafter\pgfarrowsaddtooptions\expandafter{\expandafter\def\expandafter\pgfarrows@width@scale\expandafter{\pgfmathresult}}%
+  },
+  scale length/.code={\pgfmathparse{#1}\expandafter\pgfarrowsaddtooptions\expandafter{\expandafter\def\expandafter\pgfarrows@length@scale\expandafter{\pgfmathresult}}},
+  scale width/.code={\pgfmathparse{#1}\expandafter\pgfarrowsaddtooptions\expandafter{\expandafter\def\expandafter\pgfarrows@width@scale\expandafter{\pgfmathresult}}},
+  slant/.code={\pgfmathparse{#1}\expandafter\pgfarrowsaddtooptions\expandafter{\expandafter\def\expandafter\pgfarrows@slant\expandafter{\pgfmathresult}}},
+  sep/.code={
+    \pgfarrowsthreeparameters{#1}%
+    \expandafter\pgfarrowsaddtooptions\expandafter{\expandafter\pgfarrowslinewidthdependent\pgfarrowstheparameters\pgfarrowsep\pgf@x}%
+  },
+  sep/.default=0.88pt .3 1,
+  swap/.code=\pgfarrowsaddtooptions{\ifpgfarrowswap\pgfarrowswapfalse\else\pgfarrowswaptrue\fi},
+  reversed/.code=\pgfarrowsaddtooptions{\ifpgfarrowreversed\pgfarrowreversedfalse\else\pgfarrowreversedtrue\fi},
+  harpoon/.is choice,
+  harpoon/.default=true,
+  harpoon/true/.code=\pgfarrowsaddtooptions{\pgfarrowharpoontrue},
+  harpoon/false/.code=\pgfarrowsaddtooptions{\pgfarrowharpoonfalse},
+  left/.style={harpoon},
+  right/.style={harpoon,swap},
+  color/.code=\pgfarrowsaddtooptions{\def\pgf@arrows@stroke@color{#1}},
+  fill/.code={%
+    \def\pgf@temp{#1}%
+    \ifx\pgf@temp\pgf@nonetext%
+      \pgfarrowsaddtooptions{\pgfarrowopentrue}
+    \else
+      \pgfarrowsaddtooptions{\pgfarrowopenfalse\def\pgf@arrows@fill@color{#1}}
+    \fi
+  },
+  fill/.value required,
+  open/.style={fill=none},
+  .unknown/.code={
+    \expandafter\pgfutil@in@\expandafter!\expandafter{\pgfkeyscurrentname}%
+    \ifpgfutil@in@%
+      \expandafter\pgfarrowsaddtooptions\expandafter{\expandafter\def\expandafter\pgf@arrows@stroke@color\expandafter{\pgfkeyscurrentname}}%
+    \else%
+      \pgfutil@doifcolorelse{\pgfkeyscurrentname}
+      {%
+        \expandafter\pgfarrowsaddtooptions\expandafter{\expandafter\def\expandafter\pgf@arrows@stroke@color\expandafter{\pgfkeyscurrentname}}%
+      }
+      {%
+        \edef\pgf@temp{\pgfkeyscurrentname}%
+        \pgfkeys{/errors/unknown key/.expand once=\expandafter{\pgf@temp}{#1}}%
+      }
+      \fi
+    }
+}
+\def\pgf@nonetext{none}
+
+\def\pgfarrowsaddtooptions#1{\expandafter\def\expandafter\pgf@arrows@options\expandafter{\pgf@arrows@options#1}}
+\def\pgfarrowsaddtolateoptions#1{\expandafter\def\expandafter\pgf@arrows@options\expandafter{\pgf@arrows@options\pgf@arrows@add@to@late@options{#1}}}
+\def\pgf@arrows@add@to@late@options#1{\expandafter\def\expandafter\pgf@arrows@late@options\expandafter{\pgf@arrows@late@options#1}}
+
+
+
+\def\pgfarrowsthreeparameters#1{%
+  \pgfarrowsthreeparameters@#1 +0 +0 +0 \pgf@stop%
+}
+\def\pgfarrowsthreeparameters@#1 #2 #3 #4\pgf@stop{%
+  \pgfmathsetlength\pgf@x{#1}%
+  \pgfmathsetmacro\pgf@temp@a{#2}%
+  \pgfmathsetmacro\pgf@temp@b{#3}%
+  \edef\pgfarrowstheparameters{{\the\pgf@x}{\pgf@temp@a}{\pgf@temp@b}}%
+}
+
+\def\pgfarrowsfourparameters#1{%
+  \pgfarrowsfourparameters@#1 +0 +0 +0 \pgf@stop%
+}
+\def\pgfarrowsfourparameters@#1:#2 #3 #4 #5\pgf@stop{%
+  \pgfmathsetmacro\pgf@temp@z{#1}%
+  \pgfmathsetlength\pgf@x{#2}%
+  \pgfmathsetmacro\pgf@temp@a{#3}%
+  \pgfmathsetmacro\pgf@temp@b{#4}%
+  \edef\pgfarrowstheparameters{{\pgf@temp@z}{\the\pgf@x}{\pgf@temp@a}{\pgf@temp@b}}%
+}
+
+\def\pgfarrows@angle@setup#1#2#3#4{%
+  \pgfarrowslinewidthdependent{#2}{#3}{#4}
+  \pgf@xc\pgf@x
+  \pgf@yc#1 pt%
+  \pgf@yc.5\pgf@yc%
+  \pgfpointpolar{\pgf@yc}{\pgf@xc}
+  \pgfarrowlength\pgf@x%
+  \pgf@y2\pgf@y%
+  \expandafter\pgf@arrows@add@to@late@options\expandafter{\expandafter\pgfarrowwidth\the\pgf@y}%
+}
+
+\def\pgfarrows@angle@setup@prime#1{%
+  \pgf@yc#1 pt %
+  \pgf@yc=0.5\pgf@yc
+  \pgfmathtan@{\pgf@sys@tonumber\pgf@yc}%
+  \pgfarrowwidth\pgfmathresult\pgfarrowlength%
+  \pgfarrowwidth2\pgfarrowwidth%
+}
+
+\def\pgfarrowslinewidthdependent#1#2#3{%
+  \pgf@x#1%
+  \ifdim\pgfinnerlinewidth>0pt%
+    \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

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

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