Add preprints files (part 13)

.gitattributes

@@ -621,3 +621,76 @@ preprints/Termination-for-projective-log-canonical-fourfolds-with-rational-bound
 preprints/Termination-of-generalized-canonical-flips-on-compact-Kahler-fourfolds-October-5-2026/termination-generalized-terminal-flips-compact-kahler-fourfolds.pdf filter=lfs diff=lfs merge=lfs -text
 preprints/Termination-of-generalized-log-canonical-flips-on-compact-Kahler-fourfolds-October-5-2026/termination-generalized-lc-kahler-fourfolds.pdf filter=lfs diff=lfs merge=lfs -text
 preprints/The-Abelian-Zilber-Pink-Conjecture-September-24-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/The-abelianity-conjecture-for-special-compact-Kahler-manifolds-September-23-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/the-additive-indecomposability-of-the-primes-September-24-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/The-affine-Bernstein-theorem-in-dimensions-three-through-nine-September-24-2026/main.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/The-Approximation-Threshold-for-Metric-k-Median-September-24-2026/main.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/The-arithmetic-category-and-stable-recovery-of-lattice-factors-September-23-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/The-Artinian-Lex-Plus-Powers-Betti-Theorem-September-23-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/The-Bass-trace-conjecture-for-complex-group-rings-September-24-2026/The-Bass-trace-conjecture-for-complex-group-rings-September-24-2026.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/The-Blockwise-Alperin-Weight-Conjecture-September-23-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/The-Bogomolov-Pop-reconstruction-theorem-September-23-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/The-Boltzmann-Grad-limit-for-stable-radial-potentials-on-regular-kinetic-intervals-September-23-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/The-Campana-Peternell-conjecture-in-dimension-six-September-25-2026/main.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/The-canonical-massive-continuum-limit-of-the-two-dimensional-O3-model-October-4-2026/massive-continuum-o3.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/The-circulant-Hadamard-conjecture-September-23-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/The-complete-Crouzeix-theorem-September-23-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/The-complexity-of-identifying-a-graph-by-Weisfeiler-Leman-refinement-September-25-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/The-cotype-cotype-conjecture-under-the-approximation-property-September-23-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/The-critical-correlation-exponent-of-the-planar-XY-model-September-24-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/The-critical-dimension-for-one-phase-Bernoulli-minimizers-September-24-2026/The-critical-dimension-for-one-phase-Bernoulli-minimizers-September-24-2026.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/The-critical-Liouville-quantum-sphere-and-geometric-limits-of-FK-maps-at-q-equals-4-September-24-2026/main.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/The-critical-logarithmic-correction-for-the-planar-XY-model-October-5-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/The-Critical-Spin-Field-of-the-Planar-XY-Model-October-5-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/The-crossing-number-of-complete-bipartite-graphs-September-23-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/The-crossing-number-of-complete-graphs-September-23-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/The-curve-scaling-limit-of-half-plane-double-dimers-September-23-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/The-Deligne-Drinfeld-conjecture-September-23-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/The-Dimension-of-the-Two-Adic-Hecke-Algebra-at-Odd-Level-October-5-2026/two-adic-hecke.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/The-dyadic-case-of-the-Erdos-similarity-conjecture-September-25-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/The-E-times-CM-Component-of-Zilber-Pink-for-Curves-in-A2-September-24-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/The-entropy-photon-number-inequality-September-24-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/The-entropy-rate-dimension-formula-for-self-similar-measures-on-the-line-September-24-2026/main.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/The-equivariant-cohomological-Hikita-conjecture-for-arbitrary-quivers-September-24-2026/main.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/The-Euclidean-plane-is-not-five-colorable-September-23-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/The-Euclidean-Steinitz-Bergstrom-theorem-September-24-2026/The-Euclidean-Steinitz-Bergstrom-theorem-September-24-2026.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/The-exact-reconstruction-threshold-for-the-three-state-symmetric-channel-September-25-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/The-Factor-Two-Hardness-Threshold-for-Vertex-Cover-September-23-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/The-Falconer-distance-conjecture-in-all-dimensions-September-23-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/The-first-lattice-correction-to-Blochs-law-October-5-2026/first-lattice-correction-bloch-law.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/The-free-energy-of-the-Ising-random-perceptron-September-24-2026/The-free-energy-of-the-Ising-random-perceptron-September-24-2026.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/The-free-energy-of-the-spherical-random-perceptron-September-24-2026/The-free-energy-of-the-spherical-random-perceptron-September-24-2026.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/The-free-uniform-spanning-forest-is-a-factor-of-IID-September-25-2026/The-free-uniform-spanning-forest-is-a-factor-of-IID-September-25-2026.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/The-Gaussian-free-field-limit-of-integer-Lipschitz-heights-with-two-arc-boundary-data-September-25-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/The-Gaussian-free-field-limit-of-the-balanced-six-vertex-model-with-variance-multiplier-1-over-arcsin-c-over-2-September-23-2026/The-Gaussian-free-field-limit-of-the-balanced-six-vertex-model-with-variance-multiplier-1-over-arcsin-c-over-2-September-23-2026.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/The-Gaussian-Propeller-Bound-in-Every-Dimension-September-24-2026/main.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/The-Generalized-Mukai-Conjecture-September-24-2026/article.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/The-geometric-case-of-the-Erdos-similarity-conjecture-October-5-2026/geometric-erdos-similarity.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/The-Grothendieck-homotopy-hypothesis-via-elementary-expansions-September-24-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/The-height-three-chromatic-overlap-an-explicit-filtration-and-its-attachments-September-27-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/The-higher-dimensional-Erdos-distinct-distances-conjecture-September-23-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/The-Hilbert-Smith-conjecture-in-every-finite-dimension-September-23-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/The-hypercube-Ramsey-number-has-linear-order-September-23-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/The-irrationality-exponent-of-pi-is-2-September-24-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/The-Isoperimetric-Conjecture-for-the-Cubic-Flat-Three-Torus-September-24-2026/article.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/The-joint-Dickman-law-for-consecutive-integers-September-24-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/The-joint-scaling-limit-of-critical-Ashkin-Teller-currents-September-25-2026/main.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/The-Kakeya-maximal-conjecture-in-three-dimensions-September-23-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/The-Kerr-Newman-Penrose-Inequality-for-Axisymmetric-Electrovacuum-Exteriors-October-5-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/The-Kervaire-Invariant-Problem-at-the-Prime-Three-September-24-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/The-Kervaire-Theorem-for-Groups-September-24-2026/The-Kervaire-Theorem-for-Groups-September-24-2026.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/The-Kirchberg-Rordam-character-criterion-September-25-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/The-l1-Bass-Conjecture-for-Discrete-Groups-October-5-2026/l1-bass-conjecture.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/The-logarithmic-Brunn-Minkowski-conjecture-September-23-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/The-low-temperature-Sherrington-Kirkpatrick-fluctuation-scale-September-24-2026/The-low-temperature-Sherrington-Kirkpatrick-fluctuation-scale-September-24-2026.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/The-low-temperature-Sherrington-Kirkpatrick-free-energy-limiting-law-September-24-2026/The-low-temperature-Sherrington-Kirkpatrick-free-energy-limiting-law-September-24-2026.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/The-Mahler-Conjecture-for-General-Convex-Bodies-September-22-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/The-Margulis-Platonov-conjecture-over-global-function-fields-October-5-2026/margulis-platonov-global-function-fields.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/The-Margulis-Platonov-conjecture-over-number-fields-September-23-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/The-maximal-triangular-Hilbert-transform-at-the-symmetric-point-September-24-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/The-maximum-number-of-mutually-unbiased-bases-in-dimension-six-September-24-2026/The-maximum-number-of-mutually-unbiased-bases-in-dimension-six-September-24-2026.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/The-mean-analytic-rank-of-quadratic-twists-of-elliptic-curves-September-23-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/The-metric-Blaschke-theorem-September-23-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/The-Mezard-Parisi-formula-for-diluted-spin-glasses-September-23-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/The-nonmaximal-anti-de-Sitter-Penrose-Inequality-and-original-data-rigidity-October-5-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/The-normalized-volume-gap-in-dimension-four-September-24-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text

preprints/The-Approximation-Threshold-for-Metric-k-Median-September-24-2026/README.md

@@ -0,0 +1,16 @@
+# [The approximation threshold for metric k-median](main.pdf)
+
+OpenAI  
+September 24, 2026
+
+## Citation
+
+```bibtex
+@misc{OAI:The-Approximation-Threshold-for-Metric-k-Median-September-24-2026,
+  author = {{OpenAI}},
+  title = {{The approximation threshold for metric $k$-median}},
+  howpublished = {OpenAI Math Release preprint
+                  \href{https://github.com/openai/math/blob/main/preprints/The-Approximation-Threshold-for-Metric-k-Median-September-24-2026/main.pdf}{OAI:The-Approximation-Threshold-for-Metric-k-Median-September-24-2026}},
+  year = {2026}
+}
+```

preprints/The-Approximation-Threshold-for-Metric-k-Median-September-24-2026/build/figures/survivors.tex

@@ -0,0 +1,27 @@
+\begin{figure}[tbp]
+\centering
+\begin{tikzpicture}[
+  box/.style={draw,rounded corners=2pt,align=center,inner sep=7pt,
+              text width=3.95cm,font=\small},
+  flow/.style={-{Stealth[length=2mm]},thick},
+  note/.style={align=center,font=\small}]
+\node[note] (source) {A full update at an alive source $h\in S$ opens its facility};
+\node[box,below=10mm of source,xshift=-4.8cm] (erased)
+  {$S\cap E_h$\\[3pt]Physically erased\\and removed from $S$};
+\node[box,right=4mm of erased] (kept)
+  {$i\in S\setminus E_h$\\with $ad_i<d_h$\\[3pt]Physically survives\\and remains alive};
+\node[box,right=4mm of kept] (discarded)
+  {$i\in S\setminus E_h$\\with $ad_i\ge d_h$\\[3pt]Physically survives;\\discarded from the analysis};
+\draw[flow] (source.south) -- ++(0,-4mm) -| (erased.north);
+\draw[flow] (source.south) -- (kept.north);
+\draw[flow,dashed] (source.south) -- ++(0,-4mm) -| (discarded.north);
+\node[box,text width=12.95cm,below=8mm of kept] (reset)
+  {The opened facility is reset to $\Delta$ active copies.\\
+   These regenerated copies are not original identities in $S_0$ and do not enter $S$.};
+\end{tikzpicture}
+\caption{One full source update, with no simultaneous relevant action.
+Only the middle group contributes to the continued alive assignment.
+The dashed branch is an analysis discard, not a physical erasure.
+Positions are schematic and represent no metric geometry.}
+\label{fig:survivors}
+\end{figure}

preprints/The-Approximation-Threshold-for-Metric-k-Median-September-24-2026/build/main.tex

@@ -0,0 +1,109 @@
+\documentclass[11pt]{article}
+\usepackage[T1]{fontenc}
+\usepackage{lmodern}
+
+\pdfgentounicode=1
+\pdfglyphtounicode{tfm:lmsy10/mapsto}{007C}
+\pdfglyphtounicode{tfm:lmex10/parenleftbig}{0028}
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+\pdfglyphtounicode{tfm:lmex10/radicalbig}{221A}
+\pdfglyphtounicode{tfm:lmex10/radicalBig}{221A}
+
+\usepackage[margin=1in]{geometry}
+\usepackage{amsmath,amssymb,amsthm,mathtools}
+\usepackage{microtype}
+\usepackage{booktabs,longtable,array}
+\usepackage{enumitem}
+\usepackage{graphicx}
+\usepackage{xcolor}
+\usepackage{tikz}
+\usetikzlibrary{arrows.meta,positioning,calc}
+\usepackage[numbers,sort&compress]{natbib}
+\usepackage[colorlinks=true,linkcolor=blue!50!black,citecolor=blue!50!black,urlcolor=blue!50!black]{hyperref}
+\usepackage[figure]{hypcap}
+\usepackage{bookmark}
+\hypersetup{pdftitle={The approximation threshold for metric k-median},pdfauthor={OpenAI}}
+\renewcommand{\bibsection}{\section*{\refname}\addcontentsline{toc}{section}{\refname}}
+\newtheorem{theorem}{Theorem}[section]
+\newtheorem{lemma}[theorem]{Lemma}
+\newtheorem{proposition}[theorem]{Proposition}
+\newtheorem{corollary}[theorem]{Corollary}
+\theoremstyle{definition}
+\newtheorem{definition}[theorem]{Definition}
+\theoremstyle{remark}
+\newtheorem{remark}[theorem]{Remark}
+\newcommand{\E}{\mathbb E}
+\newcommand{\Prob}{\mathbb P}
+\DeclareMathOperator{\Var}{Var}
+\DeclareMathOperator{\OPT}{OPT}
+\DeclareMathOperator{\cost}{cost}
+\newcommand{\ind}{\mathbf 1}
+\newcommand{\R}{\mathbb R}
+\newcommand{\N}{\mathbb N}
+\newcommand{\eps}{\varepsilon}
+\newcommand{\abs}[1]{\lvert#1\rvert}
+\newcommand{\norm}[1]{\lVert#1\rVert}
+\setlist[itemize]{topsep=4pt,itemsep=3pt}
+\setlist[enumerate]{topsep=4pt,itemsep=3pt}
+\setlength{\emergencystretch}{2em}
+\numberwithin{equation}{section}
+
+\title{The approximation threshold for metric \(k\)-median}
+\author{OpenAI}
+\date{September 24, 2026}
+\begin{document}
+\maketitle
+
+\begin{abstract}
+For every fixed \(\varepsilon>0\), we give a deterministic polynomial-time
+\((1+2/e+\varepsilon)\)-approximation for finite rational metric
+\(k\)-median with specified candidate facilities, opening at most \(k\)
+facilities. Under \(P\ne NP\), the infimum approximation factor in this
+model is therefore \(1+2/e\).
+\end{abstract}
+
+\input{sections/introduction}
+\input{sections/preliminaries}
+\input{sections/preparation}
+\input{sections/inequality}
+\input{sections/rounding}
+\input{sections/derandomization}
+\appendix
+\input{sections/certificates}
+
+\bibliographystyle{plainnat}
+\bibliography{references}
+\end{document}

preprints/The-Approximation-Threshold-for-Metric-k-Median-September-24-2026/build/references.bib

@@ -0,0 +1,217 @@
+@inproceedings{AnandLee2024,
+  author = {Anand, Aditya and Lee, Euiwoong},
+  title = {Separating {$k$}-Median from the Supplier Version},
+  editor = {Vygen, Jens and Byrka, Jaros{\l}aw},
+  booktitle = {Integer Programming and Combinatorial Optimization: 25th International Conference, IPCO 2024},
+  series = {Lecture Notes in Computer Science},
+  volume = {14679},
+  pages = {14--27},
+  publisher = {Springer},
+  address = {Cham},
+  year = {2024},
+  doi = {10.1007/978-3-031-59835-7_2},
+  url = {https://doi.org/10.1007/978-3-031-59835-7_2},
+  note = {Full preprint: arXiv:2401.13819v1, January 24, 2024; \url{https://arxiv.org/abs/2401.13819v1}}
+}
+
+@article{Arya2004,
+  author = {Arya, Vijay and Garg, Naveen and Khandekar, Rohit and Meyerson, Adam and Munagala, Kamesh and Pandit, Vinayaka},
+  title = {Local Search Heuristics for {$k$}-Median and Facility Location Problems},
+  journal = {SIAM Journal on Computing},
+  volume = {33},
+  number = {3},
+  pages = {544--562},
+  year = {2004},
+  doi = {10.1137/S0097539702416402},
+  url = {https://doi.org/10.1137/S0097539702416402}
+}
+
+@misc{BayerTeichmann2005,
+  author = {Bayer, Christian and Teichmann, Josef},
+  title = {The Proof of {Tchakaloff}'s Theorem},
+  year = {2005},
+  eprint = {math/0502473v2},
+  archivePrefix = {arXiv},
+  primaryClass = {math.NA},
+  url = {https://arxiv.org/abs/math/0502473v2},
+  note = {Version 2, 4 May 2005; published in Proceedings of the American Mathematical Society 134(10), 3035--3040 (2006), doi:10.1090/S0002-9939-06-08249-9}
+}
+
+@article{ByrkaEtAl2017,
+  author = {Byrka, Jaros{\l}aw and Pensyl, Thomas and Rybicki, Bartosz and Srinivasan, Aravind and Trinh, Khoa},
+  title = {An Improved Approximation for {$k$}-Median and Positive Correlation in Budgeted Optimization},
+  journal = {ACM Transactions on Algorithms},
+  volume = {13},
+  number = {2},
+  articleno = {23},
+  pages = {23:1--23:31},
+  year = {2017},
+  doi = {10.1145/2981561},
+  url = {https://doi.org/10.1145/2981561},
+  note = {Corrected full manuscript: arXiv:1406.2951v4, 23 April 2016}
+}
+
+@misc{ByrkaEtAl2026,
+  author = {Byrka, Jaros{\l}aw and Guo, Yuhao and Hu, Yang and Li, Shi and Wan, Chengzhang and Wang, Zaixuan},
+  title = {{$k$}-Clustering via Iterative Randomized Rounding},
+  year = {2026},
+  eprint = {2604.06046v1},
+  archivePrefix = {arXiv},
+  primaryClass = {cs.DS},
+  doi = {10.48550/arXiv.2604.06046},
+  url = {https://arxiv.org/abs/2604.06046v1},
+  note = {Version 1, April 7, 2026}
+}
+
+@article{CGLSS2026,
+  author = {Cohen-Addad, Vincent and Grandoni, Fabrizio and Lee, Euiwoong and Schwiegelshohn, Chris and Svensson, Ola},
+  title = {A {$(2+\varepsilon)$}-Approximation Algorithm for Metric {$k$}-Median},
+  journal = {SIAM Journal on Computing},
+  year = {2026},
+  doi = {10.1137/25M181026X},
+  url = {https://doi.org/10.1137/25M181026X},
+  note = {Published online August 24, 2026. Full preprint: arXiv:2503.10972v2, May 19, 2026; \url{https://arxiv.org/abs/2503.10972v2}. First posted March 14, 2025}
+}
+
+@inproceedings{CharikarGuha1999,
+  author = {Charikar, Moses and Guha, Sudipto},
+  title = {Improved Combinatorial Algorithms for the Facility Location and {$k$}-Median Problems},
+  booktitle = {Proceedings of the 40th Annual Symposium on Foundations of Computer Science},
+  pages = {378--388},
+  year = {1999},
+  publisher = {IEEE Computer Society},
+  doi = {10.1109/SFFCS.1999.814609},
+  url = {https://doi.org/10.1109/SFFCS.1999.814609}
+}
+
+@article{CharikarGuhaTardosShmoys2002,
+  author = {Charikar, Moses and Guha, Sudipto and Tardos, {\'E}va and Shmoys, David B.},
+  title = {A Constant-Factor Approximation Algorithm for the {$k$}-Median Problem},
+  journal = {Journal of Computer and System Sciences},
+  volume = {65},
+  number = {1},
+  pages = {129--149},
+  year = {2002},
+  doi = {10.1006/jcss.2002.1882},
+  url = {https://doi.org/10.1006/jcss.2002.1882},
+  note = {Preliminary version in STOC 1999}
+}
+
+@inproceedings{CharikarLi2012,
+  author = {Charikar, Moses and Li, Shi},
+  title = {A Dependent {LP}-Rounding Approach for the {$k$}-Median Problem},
+  booktitle = {Automata, Languages, and Programming},
+  series = {Lecture Notes in Computer Science},
+  volume = {7391},
+  pages = {194--205},
+  publisher = {Springer},
+  year = {2012},
+  doi = {10.1007/978-3-642-31594-7_17},
+  url = {https://doi.org/10.1007/978-3-642-31594-7_17}
+}
+
+@article{Feige1998,
+  author = {Feige, Uriel},
+  title = {A Threshold of {$\ln n$} for Approximating Set Cover},
+  journal = {Journal of the ACM},
+  volume = {45},
+  number = {4},
+  pages = {634--652},
+  month = jul,
+  year = {1998},
+  doi = {10.1145/285055.285059},
+  url = {https://doi.org/10.1145/285055.285059}
+}
+
+@article{GandhiEtAl2006,
+  author = {Gandhi, Rajiv and Khuller, Samir and Parthasarathy, Srinivasan and Srinivasan, Aravind},
+  title = {Dependent Rounding and Its Applications to Approximation Algorithms},
+  journal = {Journal of the ACM},
+  volume = {53},
+  number = {3},
+  pages = {324--360},
+  year = {2006},
+  doi = {10.1145/1147954.1147956},
+  url = {https://doi.org/10.1145/1147954.1147956}
+}
+
+@inproceedings{GowdaEtAl2023,
+  author = {Gowda, Kishen N. and Pensyl, Thomas and Srinivasan, Aravind and Trinh, Khoa},
+  title = {Improved Bi-point Rounding Algorithms and a Golden Barrier for {$k$}-Median},
+  booktitle = {Proceedings of the 2023 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA)},
+  pages = {987--1011},
+  publisher = {Society for Industrial and Applied Mathematics},
+  year = {2023},
+  doi = {10.1137/1.9781611977554.ch38},
+  url = {https://doi.org/10.1137/1.9781611977554.ch38},
+  note = {Preprint: \url{https://arxiv.org/abs/2210.13395v1}}
+}
+
+@article{JainEtAl2003,
+  author = {Jain, Kamal and Mahdian, Mohammad and Markakis, Evangelos and Saberi, Amin and Vazirani, Vijay V.},
+  title = {Greedy Facility Location Algorithms Analyzed Using Dual Fitting with Factor-Revealing {LP}},
+  journal = {Journal of the ACM},
+  volume = {50},
+  number = {6},
+  pages = {795--824},
+  year = {2003},
+  doi = {10.1145/950620.950621},
+  url = {https://doi.org/10.1145/950620.950621},
+  note = {Preprint: \url{https://arxiv.org/abs/cs/0207028v1}}
+}
+
+@article{JainVazirani2001,
+  author = {Jain, Kamal and Vazirani, Vijay V.},
+  title = {Approximation Algorithms for Metric Facility Location and {$k$}-Median Problems Using the Primal-Dual Schema and Lagrangian Relaxation},
+  journal = {Journal of the ACM},
+  volume = {48},
+  number = {2},
+  pages = {274--296},
+  year = {2001},
+  doi = {10.1145/375827.375845},
+  url = {https://doi.org/10.1145/375827.375845}
+}
+
+@article{LiSvensson2016,
+  author = {Li, Shi and Svensson, Ola},
+  title = {Approximating {$k$}-Median via Pseudo-Approximation},
+  journal = {SIAM Journal on Computing},
+  volume = {45},
+  number = {2},
+  pages = {530--547},
+  year = {2016},
+  doi = {10.1137/130938645},
+  url = {https://doi.org/10.1137/130938645},
+  note = {Author manuscript dated September 11, 2014: \url{https://tcs.nju.edu.cn/shili/papers/KM-SIMPCOMP2016.pdf}; the additive-to-ordinary reduction is Theorem 4 in that version}
+}
+
+@misc{OpenAIRecovery2026,
+  author = {{OpenAI}},
+  title = {{Single-exponential recovery and bounded-price strictness for metric $k$-median}},
+  howpublished = {OpenAI Math Release preprint
+                  \href{https://github.com/openai/math/blob/main/preprints/Single-Exponential-Recovery-and-Bounded-Price-Strictness-for-Metric-k-Median-September-24-2026/paper.pdf}{OAI:Single-Exponential-Recovery-and-Bounded-Price-Strictness-for-Metric-k-Median-September-24-2026}},
+  year = {2026}
+}
+
+@inproceedings{Srinivasan2001,
+  author = {Srinivasan, Aravind},
+  title = {Distributions on Level-Sets with Applications to Approximation Algorithms},
+  booktitle = {Proceedings of the 42nd IEEE Symposium on Foundations of Computer Science},
+  pages = {588--597},
+  publisher = {IEEE Computer Society},
+  year = {2001},
+  doi = {10.1109/SFCS.2001.959935},
+  url = {https://doi.org/10.1109/SFCS.2001.959935}
+}
+
+@article{BBLM2005,
+  author = {Boucheron, St{\'e}phane and Bousquet, Olivier and Lugosi, G{\'a}bor and Massart, Pascal},
+  title = {Moment inequalities for functions of independent random variables},
+  journal = {The Annals of Probability},
+  volume = {33},
+  number = {2},
+  pages = {514--560},
+  year = {2005},
+  doi = {10.1214/009117904000000856},
+  url = {https://arxiv.org/abs/math/0503651}
+}

preprints/The-Approximation-Threshold-for-Metric-k-Median-September-24-2026/build/sections/certificates.tex

@@ -0,0 +1,631 @@
+\section{Exact certificates for the scalar bounds}
+\label{app:certificates}
+
+This appendix proves Lemma~\ref{lem:scalar-bounds}.  All terminating
+decimals in this appendix denote exact rational numbers.  The argument
+uses two finite tables of rational polynomial minorants.  We give the
+coefficient rule and the Bernstein conversion formula explicitly, so the
+tables can be verified using only rational arithmetic; the accompanying
+programs are supplementary implementations of these formulas.
+
+Throughout, $X=1-x$, $0\le x\le1$, $0\le w\le X$, $0\le r\le1$, and
+$s\in\{0,1\}$.  We recall the notation
+\begin{align*}
+ P&=\int_0^1t e^{-tx}\,dt,&
+ A&=3-2\int_0^1e^{-tx}\,dt,&
+ m&=A-1-2Px,\\
+ C&=1-sP,& a&=A+2Pw,& b&=(2-s)P(x+w),\qquad z=Pw^2,\\
+ k_s&=(1-s)\frac{1-\sqrt{1-P}}{1+\sqrt{1-P}},&
+ K&=C-k_s-br+1.6z,\\
+ L&=(C-m-b)r+.4z,&
+ M&=(1-r)a+CPr,& G&=M-L.
+\end{align*}
+Also set
+\[
+ d_0=.52,\qquad d_1=.4,\qquad
+ H_0=.394-.42X^2+.04X^3,\qquad H_1=.25,\qquad J_s=CH_s.
+\]
+The elementary integral bounds give $0<P\le.5$ and $1\le A\le2$;
+the latter follows from $1-e^{-tx}\le tx$.  Direct integration gives
+\begin{equation}
+ (1+x)(A-1)=2x(1-P).
+ \label{eq:cert-integral-identity}
+\end{equation}
+
+\subsection{Rational approximations and their errors}
+\label{sec:cert-interpolation}
+
+Use the following polynomials in $X$, with coefficients displayed in
+ascending order:
+\begin{equation}
+\begin{aligned}
+ p(X)={}&.26424112+.16053514X+.05751895X^2\\
+        &\hspace{12mm}+.01313705X^3+.00456774X^4,\\
+ v(X)={}&1.73575888-.52832269X-.16191093X^2\\
+        &\hspace{12mm}-.03442938X^3-.01109588X^4,\\
+ k(X)={}&.0767+.0564X+.0142X^2+.0246X^3.
+\end{aligned}
+\label{eq:cert-interpolants}
+\end{equation}
+We will prove
+\begin{equation}
+ |p-P|\le\epsilon_P:=.0000075,\qquad
+ |v-A|\le2\epsilon_P=.000015,\qquad k_0\le k.
+ \label{eq:cert-approximation-errors}
+\end{equation}
+
+For the first two claims, the integral Taylor series are
+\[
+ P(x)=\sum_{n\ge0}\frac{(-x)^n}{n!(n+2)},\qquad
+ A(x)=3-2\sum_{n\ge0}\frac{(-x)^n}{n!(n+1)}.
+\]
+On $0\le x\le1$ the magnitudes of the summands decrease.  Thus the
+partial sums through orders $12$ and $13$ enclose each defining
+integral.  Evaluating these rational enclosures at
+$X=0,1/4,1/2,3/4,1$ proves that the nodal errors of the displayed
+$p,v$ are, respectively, less than
+$2.343560\cdot10^{-9}$ and $2.344324\cdot10^{-9}$.  In particular,
+each is less than $4\cdot10^{-9}$.  This nodal calculation can be
+performed without evaluating any transcendental number: at each node
+substitute $x=1-X$ into the two finite sums just given and compare
+both rational endpoints with the displayed polynomial value.
+
+Let $I_P,I_A$ be the exact degree-four interpolants at these five nodes.
+The sum of the absolute values of the Lagrange basis polynomials is at
+most
+\[
+ \sum_{i=0}^4\frac{4^4}{i!(4-i)!}=\frac{512}{3},
+\]
+because every numerator factor has absolute value at most one.
+Differentiating with respect to $X$ gives the fifth-derivative bounds
+$1/7$ for $P$ and $1/3$ for $A$.  The absolute node product, with
+$u=|X-1/2|$, is
+\[
+ u(1/4-u^2)|1/16-u^2|.
+\]
+For $u\le1/4$ it is at most $1/256$.  For $1/4\le u\le1/2$, the
+product of the two quadratic factors is at most
+$((1/4-1/16)/2)^2=9/1024$, so the entire expression is at most
+$9/2048$.  Both bounds are less than $.0045$.  The interpolation
+remainder formula therefore gives
+\begin{align*}
+ |p-P|&\le\frac{512}{3}(4\cdot10^{-9})+\frac{.0045}{840}
+       =\frac{31709}{5250000000}<\epsilon_P,\\
+ |v-A|&\le\frac{512}{3}(4\cdot10^{-9})+\frac{.0045}{360}
+       =\frac{9887}{750000000}<2\epsilon_P.
+\end{align*}
+
+The coefficient signs and endpoint values give
+\begin{equation}
+\begin{gathered}
+ .26424112\le p\le.5,\qquad 1\le v\le1.73575888,\\
+ .0767\le k\le.1719,\qquad .014\le H_0\le.394.
+\end{gathered}
+\label{eq:cert-polynomial-ranges}
+\end{equation}
+For $H_0$, use $H_0'(X)=X(-.84+.12X)\le0$.
+The first row of Table~\ref{tab:cert-first} below proves
+\begin{equation}
+ T_k:=4k-(p+\epsilon_P)(1+k)^2>0.
+ \label{eq:cert-k-envelope}
+\end{equation}
+Since $0<k<1$ and $t\mapsto4t/(1+t)^2$ is increasing on $(0,1)$,
+while $4k_0/(1+k_0)^2=P$, inequality~\eqref{eq:cert-k-envelope}
+and $P\le p+\epsilon_P$ imply $k_0\le k$.  This proves all of
+\eqref{eq:cert-approximation-errors}.
+
+Define polynomial versions of the quantities used in the following reductions by
+\begin{equation}
+\begin{gathered}
+ c=1-sp,\qquad N=c-(1-s)k,\qquad l=c+1-v+spx,\\
+ d=d_s,\qquad h=1.6+.4d,\qquad
+ u_0=d/h,\qquad u_1=(2-s)(1+d)/(2h).
+\end{gathered}
+\label{eq:cert-polynomial-notation}
+\end{equation}
+The polynomial families in the reductions are
+\begin{align*}
+ T_1&=1-p-1.12p(1+x)+.4p^2X(1+x),\\
+ T_2&=((1-p)^2-.38pX+px)(1+x)-2x(1-p),\\
+ Q_0(W)&=N-dv+pW(hW-2d),\\
+ Q_1&=N-(2-s)px-d(cp-l)-hp u_1^2,\\
+ Y&=N-p\bigl((2-s)x+(2-s)^2/6.4\bigr),\\
+ Y'&=N+p(1.6X^2-2)\qquad(s=0).
+\end{align*}
+The notation $T(P,A)$ means replacement of $p,v$ by $P,A$, keeping
+$X,k$ and the rational constants fixed.
+
+
+\subsection{Reduction of the first, second, and final scalar bounds}
+
+We first establish the first scalar inequality
+\begin{equation}
+ m+b-.4z/r\le\frac{C(1-P)}{1-Pr}\qquad(0<r\le1).
+ \label{eq:cert-first-scalar}
+\end{equation}
+Put $t=1/r-P>0$.  By~\eqref{eq:cert-integral-identity}, the right
+side of~\eqref{eq:cert-first-scalar} minus its left side equals
+\begin{align*}
+ &C(1-P)-\frac{2x(1-P)}{1+x}+sPx-(2-s)Pw+.4P^2w^2\\
+ &\hspace{35mm}+P\left(\frac{C(1-P)}t+.4w^2t\right).
+\end{align*}
+Applying AM--GM to the last parentheses gives the lower bound
+\begin{equation}
+ C(1-P)-\frac{2x(1-P)}{1+x}+sPx
+ -(2-s-2\sqrt{.4C(1-P)})Pw+.4P^2w^2.
+ \label{eq:cert-amgm}
+\end{equation}
+For $s=0$ the coefficient $2-2\sqrt{.4(1-P)}$ is at most $1.12$.
+The resulting quadratic decreases for $0\le w\le X$: its derivative
+is at most $-1.12P+.8P^2<0$.  Substituting $w=X$ into this lower
+bound gives $X T_1(P)/(1+x)$, where $T_1$ is defined above.  For
+$s=1$, the coefficient $1-2\sqrt{.4}(1-P)$ is at most $.38$.
+Dropping the nonnegative quadratic term and using $w\le X$ gives
+$T_2(P)/(1+x)$.  The numerical coefficient comparisons also follow
+by squaring positive quantities: $.2>.44^2$ and
+$.4>.62^2$.  Table~\ref{tab:cert-first}, with the substitution
+budget proved below, gives $T_1(P),T_2(P)>0$.
+At $X=0$, the extra factor $X$ in the $s=0$ reduction is zero,
+which still proves the required non-strict inequality.
+
+For $r>0$, \eqref{eq:cert-first-scalar} implies
+\[
+ L\ge Cr\left(1-\frac{1-P}{1-Pr}\right)\ge0.
+\]
+For $r=0$, one has $L=.4z\ge0$ directly.
+
+To check $K\ge d_sG$, write $d=d_s$ and $h=1.6+.4d$.
+The difference $K-dG$ is affine in $r$.  With
+$N_P=C-(1-s)k$, replacing $k_s$ by $(1-s)k$ only decreases this
+difference.  Put $\underline K=N_P-br+1.6z\le K$.  The endpoint
+expressions after this replacement are
+\begin{align}
+ (\underline K-dG)|_{r=0}&=N_P-dA+Pw(hw-2d),
+ \label{eq:cert-r0}\\
+ (\underline K-dG)|_{r=1}&=Q_1(P,A)+hP(w-u_1)^2,
+ \label{eq:cert-r1}
+\end{align}
+where
+\[
+ u_0=d/h,\qquad u_1=\frac{(2-s)(1+d)}{2h}
+\]
+and $Q_1$ is defined above.  For $s=1$ we may use the unrestricted
+minimum $w=u_0$ in~\eqref{eq:cert-r0}.  For $s=0$, the constrained
+minimum is at $w=X$ if $X\le u_0$, and at $w=u_0$ otherwise.
+Equation~\eqref{eq:cert-r1} is bounded below by $Q_1(P,A)$ for
+both values of $s$.  These are precisely the $Q_0,Q_1$ checks in
+Table~\ref{tab:cert-first}.
+
+We next reduce the final two scalar bounds.  Recall
+\[
+ \kappa_s(q)=C\frac{1-q}{1-Pq}-(1-q),\qquad 0\le q\le1.
+\]
+For $s=1$, $k_s=0$ and $\kappa_s(q)\le0$.  Thus both relevant
+left sides are bounded below by $C+1.6z-b$.  Its unrestricted
+minimum in $w$ is the true-parameter version of $Y$ above, so the
+rows $Y-H_0$ and $Y-.25$ suffice.
+
+For $s=0$ and $X\ge.5$, use $q\le1$ and $\kappa_0(q)\le k_0$.
+Again a common lower bound is $1-k_0+1.6z-b$, whose unrestricted
+minimum is the corresponding $Y$.  For $s=0$ and $X\le.5$, we
+have $x\ge.5$ and
+\[
+ P(x)\le P(.5)<.47.
+\]
+For example the inequality $e^{-u}\le1-u+u^2/2$ yields
+$P(.5)\le1/2-1/6+1/32=35/96<.47$.
+It follows that $b\ge .52P/(1-P)$, because $x+w\ge.5$ and
+$1-P>.53$.  Since
+\[
+ \kappa_0(q)=\frac{Pq(1-q)}{1-Pq}
+       \le\frac{P}{1-P}(1-q),
+\]
+we obtain $bq+.52\kappa_0(q)\le b$.  The fourth scalar left side
+is consequently at least
+\[
+ 1-.48k_0+1.6z-b.
+\]
+The quadratic $1.6Pw^2-2P(x+w)$ decreases on $0\le w\le X\le.5$,
+so its minimum is $P(1.6X^2-2)$.  Thus the fourth and fifth
+inequalities follow, respectively, from $Y'+.52k-H_0\ge0$ and
+$Y'-.25\ge0$ after substitution.  In the former expression the
+net coefficient of $k$ is $-.48$, so replacing $k_0$ by the upper
+bound $k$ is conservative.
+
+The last scalar bound also gives positivity of $K$: since $b\ge0$,
+\[
+ K=C-k_s-br+1.6z\ge C-k_s-b+1.6z\ge.25>0.
+\]
+
+\subsection{The first coefficient table and its exact verification}
+
+Replacing $p,v$ by $P,A$ in these expressions changes their values
+by less than
+\begin{equation}
+ \sigma=.0002.
+ \label{eq:cert-sigma}
+\end{equation}
+For completeness, the following derivative bounds hold on the line
+segment between $(p,v)$ and $(P,A)$, using $0\le p,P\le.5$,
+$1\le v,A\le2$, and $0\le W\le1$:
+\[
+\begin{array}{c|rr|r}
+ T&|\partial_pT|&|\partial_vT|&
+ 10^6\,|T(p,v)-T(P,A)|\text{ upper bound}\\ \hline
+ T_1&3.64&0&27.3\\
+ T_2&8.76&0&65.7\\
+ Q_0(W)&3.848&.52&36.66\\
+ Q_1&4.848&.52&44.16\\
+ Y&2.625&0&19.6875\\
+ Y'&2&0&15
+\end{array}
+\]
+Here $T_{1,p}=-1-1.12(1+x)+.8pX(1+x)$, with
+$X(1+x)\le1$, and
+$T_{2,p}=[-2(1-p)-.38X+x](1+x)+2x$.
+For $Q_0$, use $h\le1.808$ and $2d\le1.04$.
+For $Q_1$, $(cp-l)_p=1-2sp+sX$ has absolute value at most $2$,
+$(cp-l)_v=1$, and $u_1\le1$.
+The last two derivative bounds follow directly from their displayed
+linear dependence on $p$.  These estimates prove the claimed budget
+even on the full interval, rather than only the listed subintervals.
+Adding $.52k$ or subtracting $H_0,.25$ does not change this budget.
+The row for $T_k$ already includes $\epsilon_P$ explicitly and does
+not need this additional substitution estimate.
+
+Here is the finite verification mechanism used in both coefficient
+tables.  A row with scale $10^a$ and integers
+$\lambda_0,\ldots,\lambda_n$ asserts
+\begin{equation}
+ T(X)\ge t(X):=10^{-a}\sum_{j=0}^n\lambda_jX^j
+       \qquad(0\le X\le1).
+ \label{eq:cert-minorant}
+\end{equation}
+To verify it, expand $T=\sum_jc_jX^j$ and check
+\begin{equation}
+ 10^ac_j\ge\lambda_j\quad(0\le j<n),\qquad
+ 10^a\sum_{j\ge n}\min(0,c_j)\ge\lambda_n.
+ \label{eq:cert-coefficient-rule}
+\end{equation}
+Positive tail terms may be discarded; for every negative tail term,
+$c_jX^j\ge c_jX^n$.  This proves~\eqref{eq:cert-minorant}.
+
+All short minorants have degree at most six.  On any listed interval
+$[\alpha,\beta]$, their degree-six Bernstein coefficients are
+\begin{equation}
+ \mathfrak b_m(t;\alpha,\beta)=10^{-a}
+ \sum_{h=0}^m\frac{\binom mh}{\binom6h}(\beta-\alpha)^h
+       \sum_{j=h}^n\lambda_j\binom jh\alpha^{j-h},
+       \quad 0\le m\le6.
+ \label{eq:cert-bernstein-rule}
+\end{equation}
+Indeed, expand at $X=\alpha+(\beta-\alpha)u$ and convert ordinary
+powers of $u$ to degree-six Bernstein form.  Thus if every coefficient
+in~\eqref{eq:cert-bernstein-rule} is at least $b_*$, then
+$t(X)\ge b_*$ throughout the interval, since the Bernstein basis is
+nonnegative and sums to one.  An endpoints entry with more than two
+numbers prescribes each interval between consecutive numbers.
+
+\begingroup
+\small
+\setlength{\tabcolsep}{3pt}
+\renewcommand{\arraystretch}{1.18}
+\begin{longtable}{@{}lclc>{\raggedright\arraybackslash}p{.39\textwidth}@{}}
+\caption{First coefficient table.  The scale is $10^a$, and the
+ listed vector is $(\lambda_0,\ldots,\lambda_n)$.}
+\label{tab:cert-first}\\
+\toprule
+ $T$&$s$&Endpoints and splits&$a$&$\lambda$\\
+\midrule
+\endfirsthead
+\toprule
+ $T$&$s$&Endpoints and splits&$a$&$\lambda$\\
+\midrule
+\endhead
+\bottomrule
+\endfoot
+ $T_k$&--&$0,.5,1$&4&$(4,73,-384,563,-184,-67)$\\
+ $T_1$&--&$0,1$&2&$(14,-17,3,3,-1)$\\
+ $T_2$&--&$0,1$&2&$(13,10,-22)$\\
+ $Q_0(u_0)$&1&$0,1$&2&$(1,3,-1)$\\
+ $Q_1$&1&$0,1$&2&$(42,-3)$\\
+ $Q_1$&0&$0,1$&2&$(5,-1)$\\
+ $Q_0(u_0)$&0&$u_0,1$&2&$(-2,19,-1)$\\
+ $Q_0(X)$&0&$0,u_0$&2&$(2,-6)$\\
+ $Y-H_0$&1&$0,1$&2&$(3,-9,45,-2)$\\
+ $Y-.25$&1&$0,1$&2&$(17,-9)$\\
+ $Y-H_0$&0&$.5,1$&3&$(-165,50,575,-1)$\\
+ $Y-.25$&0&$.5,1$&2&$(-3,5,15,-1)$\\
+ $Y'+.52k-H_0$&0&$0,.125,.25,.5$&4&$(406,-3483,7208,1786,-3)$\\
+ $Y'-.25$&0&$0,.5$&2&$(14,-38,29,-1)$\\
+\end{longtable}
+\endgroup
+
+Substitution of~\eqref{eq:cert-interpolants} into the definitions
+verifies~\eqref{eq:cert-coefficient-rule} for all fourteen rows of
+Table~\ref{tab:cert-first}.  As additional explicit sign certificates,
+the minima of the finite sets in~\eqref{eq:cert-bernstein-rule}, over
+all prescribed subintervals for each row, are at least $10^{-6}$ times
+the following integers, in row order:
+\begin{equation}
+\begin{gathered}
+ 114, 20000, 10000, 10000, 390000, 40000, 33818,\\
+ 2743, 15000, 80000, 3625, 31250, 356, 21250.
+\end{gathered}
+\label{eq:cert-first-bernstein-lower}
+\end{equation}
+Both~\eqref{eq:cert-coefficient-rule} and these lower bounds are finite
+rational comparisons specified completely by the table and
+\eqref{eq:cert-bernstein-rule}.  For example, the first row has
+smallest Bernstein coefficient $73/640000$, and the thirteenth row
+has smallest coefficient $27341/76800000$.
+Consequently $T_k>0$, and every other short minorant exceeds
+$\sigma=.0002$ on its indicated intervals.  After the substitution
+budget, this completes the first and second groups of scalar bounds
+and the final two scalar bounds.
+
+\subsection{The third scalar bound: perturbation estimate}
+
+It remains to prove
+\begin{equation}
+ ((J_s-b)r+1.6z)M+(C-k_s-J_sr)L\ge0.
+ \label{eq:cert-third-goal}
+\end{equation}
+Since $L\ge0$ is already established, replacing $k_s$ with $(1-s)k$
+only decreases the left side.  We next replace $P,A$ by $p,v$.
+This replacement changes the expression by at most
+\begin{equation}
+ \sigma(r+w^2),\qquad \sigma=.0002.
+ \label{eq:cert-third-error}
+\end{equation}
+Here are full bounds for this assertion.  For either endpoint choice
+$(\pi,\nu)=(P,A)$ or $(p,v)$, put $c_\pi=1-s\pi$, $H=H_s$, and
+write the two products as $F_1F_2+F_3F_4$, where
+\begin{align*}
+ F_1&=[c_\pi H-(2-s)\pi(x+w)]r+1.6\pi w^2,\\
+ F_2&=(1-r)(\nu+2\pi w)+c_\pi\pi r,\\
+ F_3&=c_\pi(1-Hr)-(1-s)k,\\
+ F_4&=[2-\nu-s\pi X-(2-s)\pi w]r+.4\pi w^2.
+\end{align*}
+Writing $\rho=r+w^2$, the ranges already established imply
+\begin{equation}
+ |F_1|\le1.1\rho,\qquad |F_2|\le3,\qquad
+ |F_3|\le1,\qquad |F_4|\le2\rho.
+ \label{eq:cert-factor-bounds}
+\end{equation}
+For $F_1$, both nonnegative terms in its coefficient of $r$ are at
+most one, while $1.6\pi\le.8$.  The expression $F_2$ is a convex
+combination of two quantities between zero and three.  For $F_3$,
+the lower bounds are $1-.394-.1719=.4341$ when $s=0$ and
+$.5(1-.25)=.375$ when $s=1$, and the upper bound is one.
+Finally the coefficient of $r$ in $F_4$ belongs to $[-1,1]$:
+$0\le2-\nu\le1$ and $0\le s\pi X+(2-s)\pi w\le1$.
+
+Let $\delta F_j$ denote the change between the two choices.  With
+$\epsilon_P=.0000075$, direct subtraction gives
+\begin{align*}
+ |\delta F_1|&\le2\epsilon_P\rho,&
+ |\delta F_2|&\le4\epsilon_P,\\
+ |\delta F_3|&\le\epsilon_P,&
+ |\delta F_4|&\le4\epsilon_P\rho.
+\end{align*}
+For the first estimate, the coefficient multiplying
+$\epsilon_Pr$ is $sH+(2-s)(x+w)$, at most $2$ for $s=0$ and at
+most $1.25$ for $s=1$; the remaining coefficient is $1.6$.
+For the second, use the convex combination in $r$, together with
+$|\delta(\nu+2\pi w)|\le4\epsilon_P$ and
+\[
+ |\delta(c_\pi\pi)|
+   =|P-p|\,|1-s(P+p)|\le\epsilon_P.
+\]
+The third estimate is immediate from $0\le1-Hr\le1$.
+For the fourth, use
+$2+sX+(2-s)w\le4$ and $.4\le4$.
+Combining these differences with~\eqref{eq:cert-factor-bounds} and
+the identity $ab-a'b'=(a-a')b+a'(b-b')$ yields
+\begin{align*}
+ |\delta(F_1F_2+F_3F_4)|
+ &\le(3\cdot15+1.1\cdot30+2\cdot7.5+30)10^{-6}\rho\\
+ &=.000123\rho<\sigma\rho\qquad(\rho>0).
+\end{align*}
+If $\rho=0$, both expressions vanish.  Thus
+\eqref{eq:cert-third-error} holds throughout the domain.  The factor
+bounds hold for both choices, so this telescoping estimate already
+accounts for products of perturbations.
+
+\subsection{Bernstein expansion and the second coefficient table}
+
+Use the polynomial quantities in~\eqref{eq:cert-polynomial-notation},
+and set
+\[
+ j=cH_s,\qquad b'_0=(2-s)px,\qquad b'_1=(2-s)pX,
+ \qquad f=2pX,\qquad Z'=pX^2.
+\]
+In the polynomial version of the left side of
+\eqref{eq:cert-third-goal}, subtract $\sigma(r+w^2)$ and put
+$w=X\eta$, $0\le\eta\le1$.  Its degree-three Bernstein
+coefficients in $\eta$, for $i=0,1,2,3$, are
+\begin{equation}
+ B_0+B_1r+B_2r^2,
+ \label{eq:cert-eta-coefficient}
+\end{equation}
+where, writing $e=i/3$, $e_2=i(i-1)/6$, and
+$t=\mathbf{1}_{\{i=3\}}$, the exact formulas are
+\begin{equation}
+\begin{aligned}
+ B_0={}&Z'\bigl((1.6v+.4N)e_2+1.6ft\bigr)-\sigma X^2e_2,\\
+ B_1={}&(j-b'_0)v+Nl\\
+       &+\bigl((j-b'_0)f-b'_1v-Nb'_1\bigr)e-b'_1fe_2\\
+       &-Z'\bigl((1.6(v-cp)+.4j)e_2+1.6ft\bigr)-\sigma,\\
+ B_2={}&(b'_0-j)(v-cp)-jl\\
+       &+\bigl((b'_0-j)f+b'_1(v-cp)+jb'_1\bigr)e+b'_1fe_2.
+\end{aligned}
+\label{eq:cert-bcoeff}
+\end{equation}
+These formulas give all eight expansions, one for each pair $(s,i)$.
+They can also be derived directly by expanding
+\begin{align*}
+ &\bigl([j-(2-s)p(x+w)]r+1.6pw^2\bigr)
+       \bigl((1-r)(v+2pw)+cpr\bigr)\\
+ &\quad +(N-jr)\bigl([l-(2-s)pw]r+.4pw^2\bigr)
+       -\sigma(r+w^2)
+\end{align*}
+in powers of $\eta$ after $w=X\eta$, and using the degree-three
+conversion rule: if $a_h$ is the coefficient of $\eta^h$, the
+$i$th Bernstein coefficient is
+$\sum_{h=0}^i a_h\binom ih/\binom3h$.
+
+In all eight cases $B_0\ge0$.  For $i=0,1$, it is zero.  Otherwise
+\[
+ B_0=X^2\bigl(p(1.6v+.4N)-\sigma\bigr)e_2+1.6pX^2ft.
+\]
+Here $N\ge.5$, $v\ge1$, and $p\ge.26424112$, so the bracket is
+at least $.475434016>0$ and the other term is nonnegative.
+
+Define
+\[
+ D'=4B_0B_2-B_1^2,\qquad S'=B_0+B_1+B_2.
+\]
+For $i\le1$, positivity of $B_1$ and $B_1+B_2$ proves
+nonnegativity of~\eqref{eq:cert-eta-coefficient} on $0\le r\le1$.
+For $i\ge2$, any of the following is sufficient:
+\begin{enumerate}
+ \item $B_1,B_2\ge0$;
+ \item $D'>0$;
+ \item $S'>0$ and $B_0-B_2>0$.
+\end{enumerate}
+For the second criterion, $B_0\ge0$ and $D'>0$ force
+$B_0,B_2>0$; completing the square proves positivity for every
+real $r$.  For the third, use
+\[
+ B_0+B_1r+B_2r^2
+   =rS'+(1-r)(B_0-rB_2),
+\]
+where $B_0-rB_2=(1-r)B_0+r(B_0-B_2)\ge0$.
+
+Table~\ref{tab:cert-third} supplies these signs.  It uses exactly
+the coefficient rule~\eqref{eq:cert-coefficient-rule} and the
+degree-six Bernstein formula~\eqref{eq:cert-bernstein-rule}.
+
+\begingroup
+\small
+\setlength{\tabcolsep}{3pt}
+\renewcommand{\arraystretch}{1.18}
+\begin{longtable}{@{}ccllc>{\raggedright\arraybackslash}p{.38\textwidth}@{}}
+\caption{Second coefficient table.  A row with $i=0,1$ supplies
+ the indicated certificate separately for both values of $i$.}
+\label{tab:cert-third}\\
+\toprule
+ $s$&$i$&Endpoints and splits&$T$&$a$&$\lambda$\\
+\midrule
+\endfirsthead
+\toprule
+ $s$&$i$&Endpoints and splits&$T$&$a$&$\lambda$\\
+\midrule
+\endhead
+\bottomrule
+\endfoot
+ 0&0&$0,1$&$B_1$&2&$(1,90,-39)$\\
+ 0&1&$0,1$&$B_1$&2&$(1,41,-51,28,-1)$\\
+ 0&$0,1$&$0,1$&$B_1+B_2$&2&$(10,-1)$\\
+ 0&2&$0,.05$&$B_1$&2&$(1,-9,-98)$\\
+ 0&2&$0,.05$&$B_2$&2&$(9,-23)$\\
+ 0&2&$.05,1$&$D'$&4&$\begin{gathered}(-2,16,1176,-153,\\-1500,2590,-1000)\end{gathered}$\\
+ 0&3&$0,.015$&$B_1$&2&$(1,-58,-262)$\\
+ 0&3&$0,.015$&$B_2$&2&$(9,-5)$\\
+ 0&3&$.015,.25,.5,1$&$D'$&5&$\begin{gathered}(-12,1180,2070,-32420,\\44210,33740,-14000)\end{gathered}$\\
+ 1&0&$0,1$&$B_1$&2&$(5,30,-7)$\\
+ 1&0&$0,1$&$B_1+B_2$&2&$(12,-4)$\\
+ 1&1&$0,1$&$B_1$&2&$(5,-1)$\\
+ 1&1&$0,1$&$B_1+B_2$&2&$(12,-8)$\\
+ 1&2&$0,.2$&$B_1$&2&$(5,-16,-41)$\\
+ 1&2&$0,.2$&$B_2$&2&$(7,-2)$\\
+ 1&2&$.2,1$&$D'$&3&$(-4,17,92,87,-135)$\\
+ 1&3&$0,.1$&$B_1$&3&$(54,-391,-941,-900)$\\
+ 1&3&$0,.1$&$B_2$&2&$(7,-1)$\\
+ 1&3&$.1,.75$&$D'$&3&$(-4,42,194,478,-995)$\\
+ 1&3&$.75,1$&$S'$&2&$(12,-9)$\\
+ 1&3&$.75,1$&$B_0-B_2$&2&$(-8,-32,67,-1)$\\
+\end{longtable}
+\endgroup
+
+In the printed row order, the Bernstein coefficients of these short
+minorants, over all of each row's subintervals, are at least
+$10^{-6}$ times the following integers:
+\begin{equation}
+\begin{gathered}
+ 10000, 10000, 90000, 3050, 78500, 171, 710,\\
+ 89250, 60, 50000, 80000, 40000, 40000, 1600,\\
+ 66000, 3560, 4590, 69000, 2518, 30000, 52656.
+\end{gathered}
+\label{eq:cert-third-bernstein-lower}
+\end{equation}
+The bound $90000$ in the third position applies to both indices in
+that row.  These are all positive.  The smallest exact Bernstein
+coefficient among the table's checks is
+$193875074979/3200000000000000$, in the ninth printed row.
+Thus all the required strict signs follow from the finite rational
+formula~\eqref{eq:cert-bernstein-rule}.
+
+For clarity, the negative-tail calculations needed for the four
+discriminant rows are detailed next.  Let
+$b_{h,j}=[X^j]B_h$ and write
+\[
+ c_j=[X^j]D'
+     =\sum_u\bigl(4b_{0,u}b_{2,j-u}-b_{1,u}b_{1,j-u}\bigr),
+\]
+where coefficients outside the degree range are zero.  For a short
+minorant of degree $n$, group $\min(0,c_j)$ in consecutive blocks
+of five starting at $j=n$, truncating at degree $28$.  The five block
+sums are bounded below by $.0001$ times the following integers:
+\begin{equation}
+\begin{array}{cc|c|c|rrrrr}
+ s&i&n&\deg D'&1&2&3&4&5\\ \hline
+ 0&2&6&20&-930&-11&-1&0&0\\
+ 0&3&6&22&-1070&-180&-2&-1&0\\
+ 1&2&4&28&-1312&-18&-1&-1&-1\\
+ 1&3&4&28&-8240&-1560&-29&-1&-1
+\end{array}
+\label{eq:cert-tail-blocks}
+\end{equation}
+These twenty rational comparisons follow by substituting
+\eqref{eq:cert-bcoeff} and~\eqref{eq:cert-interpolants} into the
+displayed coefficient formula.  Their row sums are, respectively,
+$-.0942,-.1253,-.1333,-.9831$, which are at least the final scaled
+minorant coefficients $-.1,-.14,-.135,-.995$.
+For all entries before the last, one checks simply
+$10^a[X^j]T\ge\lambda_j$ as in
+\eqref{eq:cert-coefficient-rule}.  The same coefficient rule verifies
+the non-discriminant rows directly.
+
+The tail cutoff is exact: the four discriminant degrees are
+$20,22,28,28$, as recorded in~\eqref{eq:cert-tail-blocks}, so no
+term above degree $28$ has been omitted.  More generally, from
+\eqref{eq:cert-bcoeff}, the respective triples of degrees
+$(\deg B_0,\deg B_1,\deg B_2)$ are
+\[
+\begin{array}{c|rrrr}
+ &i=0&i=1&i=2&i=3\\ \hline
+ s=0&(-\infty,9,9)&(-\infty,10,10)&(10,10,10)&(11,11,9)\\
+ s=1&(-\infty,9,13)&(-\infty,10,13)&(10,14,13)&(11,14,12).
+\end{array}
+\]
+Here $\deg0=-\infty$.  These bounds also justify a uniform cutoff of
+$28$ for every polynomial in the two coefficient tables.
+
+For $(s,i)=(0,2)$, the first criterion covers $X\in[0,.05]$ and
+the discriminant criterion covers $X\in[.05,1]$.
+For $(s,i)=(0,3)$, the corresponding split is $.015$, with the
+listed additional subdivisions used only for Bernstein positivity.
+For $(s,i)=(1,2)$, the split is $.2$.
+For $(s,i)=(1,3)$, the first criterion covers $[0,.1]$, the
+discriminant criterion covers $[.1,.75]$, and the final criterion
+covers $[.75,1]$.  For $i=0,1$, the full interval is covered by
+$B_1$ and $B_1+B_2$.  All endpoints belong to the stated intervals.
+Thus every coefficient~\eqref{eq:cert-eta-coefficient} is nonnegative
+for $0\le X,r\le1$.  Nonnegativity and unit sum of the degree-three
+Bernstein basis in $\eta$ prove the desired polynomial inequality
+for $0\le w\le X$; when $X=0$ one has $w=0$ directly.
+Restoring the error budget~\eqref{eq:cert-third-error} proves
+\eqref{eq:cert-third-goal}.  Together with the preceding reductions,
+this completes the proof of Lemma~\ref{lem:scalar-bounds}.

preprints/The-Approximation-Threshold-for-Metric-k-Median-September-24-2026/build/sections/derandomization.tex

@@ -0,0 +1,585 @@
+\section{Deterministic implementation and completion}
+\label{sec:derandomization}
+
+The preceding constructions yield an additive approximation in expectation.
+We now replace their random choices by finite distributions of polynomial
+support, and then choose the least expensive feasible outcome. Throughout
+this section the desired approximation loss is fixed. In particular, the
+grid denominator, the number of physical rounds, and all parameters chosen
+in Theorems~\ref{thm:preparation} and~\ref{thm:grid-rounding} are constants
+independent of the input size. All input distances are rational, and
+running time includes their encoding lengths.
+
+\subsection{Compression of finite distributions}
+
+A distribution list is a finite list of states, each equipped with a
+nonnegative rational weight, whose weights sum to one. The state records
+the data needed to continue a construction; a feature vector records the
+statistics whose expectations must be retained.
+
+The next lemma is the constructive finite-dimensional Carath\'eodory
+argument for preserving an expectation. Its connection with moment
+matching is classical~\cite{BayerTeichmann2005}; we include the exact
+elimination and its bit bound because both are needed by the algorithm.
+
+\begin{lemma}[Affine support reduction]
+\label{lem:affine-support}
+Let a distribution list have states \(s_1,\ldots,s_M\), weights
+\(\lambda_1,\ldots,\lambda_M\), and feature vectors
+\(v(s_i)\in\mathbb Q^d\). There is an exact algorithm that replaces the
+weights by nonnegative rational weights supported on at most \(d+1\) of
+the same states, preserving
+\[
+  \sum_{i=1}^M\lambda_i v(s_i).
+\]
+Its running time and output encoding length are polynomial in the
+encoding length of the given list and feature vectors.
+\end{lemma}
+
+\begin{proof}
+Discard zero-weight entries. If more than \(d+1\) remain, choose a
+nonzero integral affine dependence \(z\) satisfying
+\[
+ \sum_i z_i=0,
+ \qquad
+ \sum_i z_i v(s_i)=0.
+\]
+Since its coefficients sum to zero, there are both positive and negative
+coefficients. Set
+\[
+ t=\min_{i:z_i>0}\frac{\lambda_i}{z_i},
+ \qquad
+ \lambda'_i=\lambda_i-tz_i.
+\]
+The new weights are nonnegative, have the same sum and feature
+expectations, and include at least one additional zero. Repeating gives
+the support bound.
+
+For completeness, the rational arithmetic has polynomial complexity.
+An affine dependence can be found using at most \(d+2\) columns of the
+matrix with columns \((1,v(s_i))\). Clearing denominators and using
+minors gives an integral dependence of polynomial bit length. If
+\(\lambda_i=b_i/D\) for a common denominator \(D\), and \(j\) attains
+the minimum defining \(t\), then
+\begin{equation}
+ \lambda'_i=
+ \frac{b_i z_j-b_jz_i}{Dz_j}.
+ \label{eq:compression-denominator}
+\end{equation}
+Thus one elimination adds at most the bit length of \(|z_j|\) to a
+common denominator. There are at most \(M\) eliminations, and the
+feature vectors do not change. The resulting bit lengths and the exact
+linear-algebra computations are polynomially bounded.
+\end{proof}
+
+Support reduction preserves every property satisfied by each state in
+the list, whether or not that property appears among the features. It
+does not preserve the entire joint distribution or arbitrary future
+adaptive behavior. We therefore specify the retained features at each
+use and verify the required estimates for the resulting process.
+
+\subsection{A distribution list for the preparation stage}
+
+Let \(M_\star\) denote the number of stars in the preparation
+construction, and write \(X_e\) for the current count of star \(e\).
+Each count remains within its original floor and ceiling. A weighted
+pair update within a strict sign preserves the weighted sum and has
+conditional mean equal to the current count vector. For every pair of
+distinct coordinates \(e,f\), it also satisfies
+\begin{equation}
+ \E[X'_e\mid X]=X_e,
+ \qquad
+ \E[X'_eX'_f\mid X]\le X_eX_f.
+ \label{eq:star-local-moments}
+\end{equation}
+Indeed, unless both coordinates are the selected pair, the product is
+affine in the update parameter. If they are the selected pair, their
+increments have opposite signs and the product is concave. A
+one-coordinate rounding with the correct mean preserves every distinct
+pair-product expectation.
+
+These assertions hold for every admissible state and every permissible
+choice of the next pair. Thus the next pair may be selected adaptively,
+for example as the first two fractional coordinates in a fixed order.
+After branching one update at every state, apply
+Lemma~\ref{lem:affine-support} with the features
+\[
+   X_e \quad\text{and}\quad X_eX_f \qquad(e<f).
+\]
+Induction using~\eqref{eq:star-local-moments} preserves the initial
+first moments and the upper bounds on distinct pair moments. Full
+independence between the positive, negative, and zero-coefficient
+classes is unnecessary for these conclusions.
+
+The weighted sums and floor/ceiling restrictions are properties of each
+retained state. Each productive pair update makes at least one
+fractional count integral, and integral counts are never updated again.
+There are therefore only \(O(M_\star)\) updates, with identity updates
+padding states that finish earlier. Ordinary one-coordinate rounding
+also handles the zero coefficients and the final positive-side
+fraction. We leave the final negative-side fraction unrounded until
+these operations are complete. Each compressed list has
+\(O(M_\star^2)\) entries.
+
+\paragraph{The final negative-side fraction.}
+Condition on a state of this last list. There is at most one fractional
+count, say \(X_e=m+\xi\), where \(0<\xi<1\). The identity of \(e\)
+may depend on the state. Perform the special rounding of the
+preparation construction: use a nominal coin of mean \(\xi\), open the
+outside supplier according to \(m+1\) if its flag is one, and activate
+the extra deficit on a subset of \(D_e\) with marginal \(\xi\) at
+every target.
+
+There is a polynomial-support law for this subset. If
+\(d_e=|D_e|\), choose a size
+\[
+ L\in\{\lfloor\xi d_e\rfloor,\lceil\xi d_e\rceil\},
+ \qquad \E L=\xi d_e,
+\]
+then choose a uniformly random cyclic interval of length \(L\) in a
+fixed cyclic order of the targets. Each target has inclusion
+probability \(\E L/d_e=\xi\), and the support has size at most
+\(2d_e\). Including the nominal coin changes this bound by only a
+constant factor.
+
+For a fixed target, the special star's activation count has conditional
+mean \(X_e\); its nominal count has the same conditional mean. Every
+other star count is integral in the conditioned state. Consequently,
+for any two distinct stars activating this target, the conditional
+expectation of their activation product is the product of their
+current counts. This argument applies separately in every state, so
+it remains valid when the identity of the final fractional star varies.
+After averaging, the activation counts retain the nonpositive
+pair-covariance bounds. Each individual activation count remains
+between the original adjacent integers, giving its variance bound as
+well. No pairwise independence between different targets of this last
+star is required: the deficit estimates are made for one target at a
+time, and the clipping bound sums expectations over targets.
+
+\paragraph{Conditional rounding inside bundles.}
+For each resulting outer state, clip the deficits and rescale the
+bundle masses exactly as in Theorem~\ref{thm:preparation}. Keep this
+outer state fixed while constructing a new list for all the
+within-bundle rounds. Successive pair updates and compression of all
+first and distinct-pair moments of the opening vector preserve its
+marginals and nonpositive covariances conditional on this outer
+state. Updates in different bundles preserve cross-bundle product
+expectations; thus their entire Cartesian product need not be
+enumerated.
+
+In particular, if \(\eta_i\) is the rounded opening minus its
+conditional mean in the fixed outer state, then the noise on a
+client's old assignment obeys
+\[
+ \Var\left(
+   \sum_{i\in F_j\cap\bigcup_sP_s}\eta_i
+   \,\middle|\,\text{outer state}
+ \right)
+ \le
+ g\sum_s\sum_{i\in F_j\cap P_s}\frac{p'_s}{p_s}y_i
+ \le g/p_0\le 2g.
+\]
+Here \(p'_s\le1\) and \(p_s\ge p_0\ge1/2\). This is the conditional
+noise estimate used in the preparation proof. Maintaining a separate
+inner list for every outer state retains this estimate despite
+the varying rescalings.
+
+All remaining preparation estimates use the retained first and second
+moments, or properties holding in every state. In particular, supplier
+availability and the raw opening-count bound hold in each state;
+the clipping estimate and the probability of exceeding its count
+allowance follow from the same deficit moments. The preparation list
+therefore satisfies the same unconditioned cost and budget-success
+bounds as the randomized construction. Discarding budget failures
+and selecting a remaining grid solution of minimum total nearest-unit
+assignment cost gives the corresponding deterministic bound on the
+sum over clients. This single choice need not simultaneously attain
+each client's individual expectation bound.
+
+\subsection{A distribution list for each physical round}
+
+Fix a physical state and a round. For each target copy, first construct
+a small list of source sets of cardinality \(\Delta\) with the
+prescribed edge marginals. Use ordinary fixed-sum pair rounding:
+after each pair update, branch each current state to its at most two
+endpoints and immediately compress the resulting list while preserving
+first moments. Thus no full tree of a target's pair updates is ever
+enumerated, and every intermediate list has polynomial support.
+All sets in this list
+contain the target's own group and respect the edges whose prescribed
+marginal is zero or one.
+
+Combine the target lists by successive independent products. After
+each product, compress while retaining every edge first moment and
+every product of two edge indicators belonging to distinct targets.
+On adjoining a new target independently, its edge products with earlier
+targets have the required expectations; compression then preserves
+them. Induction gives a graph list with all the first and cross-target
+second moments of the independent-target graph construction. Its
+feature dimension and support size are polynomial in the number of
+active copies.
+
+The ideal generator in~\eqref{eq:rnd-ideal-generator} is linear in
+edge indicators except for the terms \(\delta_hY_h\). Expanding these
+terms gives products of edges from a common source to distinct targets,
+and diagonal terms reducible to first moments. The retained statistics
+therefore suffice for its graph average. The heavy-source rule and
+forced openings depend on more than these moments, but this introduces
+no extra averaging requirement: the batch drift
+bound~\eqref{eq:rnd-batch-drift} holds conditional on every realized
+graph, with that graph's own forced openings and chunks. Only after
+that bound is applied do we average the ideal generator, obtaining
+\eqref{eq:rnd-average-drift} for the compressed graph distribution.
+
+\paragraph{Proposal and trigger lists.}
+For every graph, include both round types, each with probability
+\(1/2\). Within either type, construct the proposal or trigger
+indicators by successive independent Bernoulli products, compressing
+after each product while retaining all squarefree monomials of degree
+at most four. These moments coincide exactly with those of the
+independent reference law. There are polynomially many such
+monomials. In the balanced type the variables being compressed are
+the proposals, before conflict filtering.
+
+To see why filtering requires no higher moments, write \(Q_h\) for a
+balanced proposal and \(I_h\) for its acceptance. If \(C_h\) is the
+set of sources conflicting with \(h\), then, in every outcome,
+\[
+ Q_h-\sum_{t\in C_h}Q_hQ_t\le I_h\le Q_h,
+ \qquad
+ I_hI_t\le Q_hQ_t\quad(h\ne t).
+\]
+A balanced source has \(|E_h|=\Delta\), and each target has
+\(\Delta\) possible sources, so \(|C_h|\le\Delta(\Delta-1)\).
+Preservation of first and second proposal moments gives
+\begin{equation}
+ p_*-\Delta(\Delta-1)p_*^2\le\E I_h\le p_*,
+ \qquad
+ \E[I_hI_t]\le p_*^2\quad(h\ne t).
+ \label{eq:list-conflict-moments}
+\end{equation}
+These are the marginal and pair bounds needed for the isolated-action
+and simultaneous-action estimates. Accepted out-sets remain disjoint
+in every outcome.
+
+For an unbalanced round, the count statistic from
+Equation~\eqref{eq:rnd-count-statistic}, conditional on the graph,
+has the form
+\[
+ Z_*=
+ \sum_e c_e I_e+\sum_{e<f}c_{ef}I_eI_f
+\]
+with fixed rational coefficients. It has degree at most two in the
+trigger indicators, and its square has degree at most four after using
+\(I_e^2=I_e\). The compressed law therefore has exactly the same
+\(\E Z_*\), \(\E Z_*^2\), and \(\Var Z_*\) as the independent
+reference law. The variance estimate proved for independent
+indicators transfers to this law, and hence so does the Chebyshev
+count-failure bound. An independent-coordinate variance inequality
+is applied to the reference law before this transfer.
+
+The terminal alive-mass estimate likewise uses only the marginal and
+pair bounds for the \(\Delta\) possible incoming triggers at each
+surviving target. It follows by the same union lower bound as in the
+randomized analysis. Thus all probabilistic estimates in
+Theorem~\ref{thm:grid-rounding} remain valid for these lists.
+
+\paragraph{Retaining adaptive histories.}
+Expand these graph and proposal lists at every reached physical
+history, and retain all resulting physical outcomes over the
+\(T\) rounds. The history tree is not compressed. The alive sets
+used in the analysis can be defined along this tree, but need not be
+computed by the algorithm. Even when two histories reach the same
+physical state with different analysis-only alive sets, the
+next-round estimates apply to either history, since they hold for
+every admissible alive set conditional on the current physical state
+and graph.
+
+The number of active copies is polynomial: there are at most
+\(\Delta\) at each facility, and resetting regenerates copies at an
+existing facility. The number of chunk actions is also polynomial,
+since the total number of source--target incidences is
+\(\Delta\) times the number of target copies. Each history has only
+polynomially many children. Since \(T\) is an input-independent
+constant, the complete history tree has polynomial size. At each
+leaf perform the final deterministic completion, discard outcomes
+exceeding the additive budget, and select one of minimum total
+connection cost.
+
+\subsection{Rational implementation and bit complexity}
+
+We give the encoding details because support size alone does not imply
+a polynomial-time algorithm. Fix rational values of all algorithmic
+parameters. In each of the finitely many rounds, fix in advance a
+rational taper value \(\widehat P\) satisfying its stipulated
+accuracy. The analytic functions of elapsed time, the potential, and
+the analysis-only alive-set decisions need not be evaluated by
+the algorithm.
+
+An optimal basic LP solution has polynomial encoding length. Splitting
+facilities to make assignments use whole pieces introduces only
+polynomially many pieces. Piece masses, bundle masses, and
+intersection masses are sums and differences of the LP coordinates.
+The extra star weights multiply a deficit by the fixed rational
+\(\gamma\), and conversion to grid counts multiplies by the integer
+\(\Delta\). Hence the initial star counts have a common denominator
+\(D_0\) of polynomial bit length.
+
+For weighted pair rounding, write \(a_e=|u_e|\). If a pair update
+makes \(X_e\) integral and leaves \(X_f\) fractional, preservation
+of its weighted sum gives
+\[
+ a_fX'_f=a_fX_f+a_eX_e-a_eX'_e.
+\]
+Throughout these updates, therefore, every weighted count lies in
+\(D_0^{-1}\mathbb Z\). Recovering \(X_f\) only divides by the
+polynomially bounded integer \(a_f\). Counts remain in their original
+integer bounds, so their numerators and their endpoint probabilities
+have polynomial bit length as well. Final single-coordinate rounding
+makes the coordinate integral; zero-coefficient counts retain their
+initial values until rounded directly. Clipping and rescaling give
+fresh rational inputs of polynomial bit length for the within-bundle
+rounds, where the same argument applies with unit coefficients.
+
+The taper radii can also be computed exactly. On an interval where
+\(m\) source copies have saturated marginal one and \(r\) source
+copies have marginal strictly between zero and one, the radius
+equation is
+\[
+ m+\frac{r}{\widehat P}
+ -\frac{\sum_{h\text{ partial}}d(i,h)}{\widehat P N_i}
+ =\Delta.
+\]
+When \(r>0\), an interior solution is
+\begin{equation}
+ N_i=
+ \frac{\sum_{h\text{ partial}}d(i,h)}
+      {r+\widehat P(m-\Delta)}.
+ \label{eq:rational-taper-radius}
+\end{equation}
+The breakpoints are the rational numbers \(d(i,h)\) and
+\(d(i,h)/(1-\widehat P)\). Sorting these values and checking the
+successive intervals finds the smallest solution, including a
+breakpoint solution. If \(r=0\), the equation is constant on the
+interval, and the first point of a solution interval is a breakpoint.
+The separately prescribed case of sufficient
+co-located mass uses \(N_i=0\). All radii, marginals, and resulting
+branch probabilities have polynomial bit length. The physical states
+and their feature values are determined without reference to the
+list weights.
+
+It remains to control the list weights across repeated branching and
+compression. Suppose a feature vector has \(d\) coordinates of bit
+length at most \(L\). In the affine-dependence computation, clearing
+denominators in each row of the matrix on at most \(d+2\) columns
+produces integer entries of bit length \(O(dL)\). A nonzero kernel
+vector can be chosen using minors, with bit length
+\(O(d^2L+d\log(d+1))\) by the determinant bound. These bounds are
+independent of the current list weights.
+
+By~\eqref{eq:compression-denominator}, an elimination multiplies a
+common weight denominator by just one coefficient of such a
+dependence. An exact branch with probability denominator \(q\)
+multiplies a common denominator by at most \(q\). Branching a whole
+list adds the bit lengths of only polynomially many such
+denominators, and taking a product of lists adds their denominator
+bit lengths. There are polynomially many list operations, including
+the entire constant-depth physical history tree. Their common
+denominators and numerators consequently retain polynomial bit
+lengths. Exact linear algebra, comparisons, and cost evaluations
+therefore give a deterministic algorithm whose bit complexity is
+polynomial in the input encoding length for every fixed choice of
+accuracy parameters.
+
+\subsection{Loss allocation and the approximation theorem}
+
+We first record explicitly how conditioning and deterministic
+selection are combined. For a nonnegative cost \(C\) and an event
+\(G\) with \(\Prob(G)\ge1-r\),
+\begin{equation}
+ \E[C\mid G]
+ \le\frac{\E C}{1-r}.
+ \label{eq:conditioning-cost}
+\end{equation}
+Thus a finite distribution list contains a successful outcome whose
+cost is no more than the right-hand side. No independence between
+cost and budget success is required.
+
+\begin{theorem}[Deterministic additive approximation]
+\label{thm:pseudo}
+For every fixed \(\varepsilon>0\), there is an
+instance-independent integer \(c\ge0\) and a deterministic
+polynomial-time algorithm for metric \(k\)-median that opens at
+most \(k+c\) facilities and has total connection cost at most
+\[
+ \left(1+\frac2e+\varepsilon\right)\operatorname{LP},
+\]
+where \(\operatorname{LP}\) is the optimum value of the assignment
+relaxation. In particular, its cost is at most the same factor times
+\(\OPT_k\).
+\end{theorem}
+
+\begin{proof}
+Put \(\alpha_\star=1+2/e\), and choose a fixed positive rational
+number
+\[
+ t\le\min\{1/8,\varepsilon/10\}.
+\]
+Choose the parameters in the constructions underlying
+Theorems~\ref{thm:preparation} and~\ref{thm:grid-rounding} with
+unconditioned cost factors at most \(1+t\) and
+\(\alpha_\star+t\), respectively, and with their respective
+budget-failure probabilities at most \(t\). The proofs of those
+theorems allow the cost and failure tolerances to be made
+arbitrarily small; their stated expectation guarantees absorb this
+same conditioning loss. Here we keep the loss visible.
+
+All parameter choices can be made in a consistent order. After the
+preparation parameters other than the grid are fixed, take an even
+integer \(\Delta\) sufficiently large, set \(g=1/\Delta\), and
+take a rational horizon of order
+\(1+\log(\Delta/t)\). The requirements that the terminal error
+be small and that \(\Lambda g\) be small are compatible because
+\(\log\Delta/\Delta\to0\). Then choose the taper tolerance,
+the selection bias, and the time step successively small. The
+number of rounds is now fixed. Finally choose the heavy-source
+parameter sufficiently small that the sum of the conditional
+count-failure probabilities over all rounds is at most \(t\).
+The resulting additive opening allowances are constants depending
+only on these parameter choices.
+
+Let \(L=\operatorname{LP}\). Apply the preparation list to an
+optimal LP solution. Its total nearest-unit cost has expectation at
+most \((1+t)L\), and its budget event has probability at least
+\(1-t\). Select the least expensive successful grid solution.
+By~\eqref{eq:conditioning-cost}, its total nearest-unit cost
+\(L_{\rm grid}\) satisfies
+\[
+ L_{\rm grid}\le\frac{1+t}{1-t}L.
+\]
+Apply the physical-rounding list to this fixed grid solution. The
+summed cost bound is at most \((\alpha_\star+t)L_{\rm grid}\)
+before discarding count failures, and the budget succeeds with
+probability at least \(1-t\). Selecting the least expensive
+successful integral output therefore gives cost at most
+\[
+ \frac{(1+t)(\alpha_\star+t)}{(1-t)^2}L.
+\]
+Since \(\alpha_\star<2\) and \(t\le1/8\),
+\begin{align*}
+ \frac{(1+t)(\alpha_\star+t)}{(1-t)^2}-\alpha_\star
+ &=
+ \frac{(3\alpha_\star+1)t+(1-\alpha_\star)t^2}
+      {(1-t)^2}\\
+ &\le\frac{7t}{(1-t)^2}
+ <10t
+ \le\varepsilon.
+\end{align*}
+The two additive allowances sum to an instance-independent
+constant, which may be rounded up to an integer \(c\).
+
+Both minimum-cost selections are computable on polynomial-size
+lists, with polynomial bit complexity as proved above. They use
+total costs; the deterministic grid solution is not required to
+attain every client's individual expectation bound. If \(L=0\),
+the nonnegative cost has expectation zero in the preparation list
+and then in the physical-rounding list, so the same argument gives
+zero cost without division by \(L\).
+
+Every opened copy lies at an original candidate facility. Merge
+copies introduced by the construction back to their original
+facility. This preserves connection distances and cannot increase
+the number of opened facilities. Finally,
+\(\operatorname{LP}\le\OPT_k\), proving the last assertion.
+\end{proof}
+
+\subsection{Applying the reduction on rational inputs}
+\label{sec:reduction-implementation}
+
+Before applying Theorem~\ref{thm:li-svensson}, we record its interface
+with the deterministic algorithm just proved. Every call of that reduction
+uses the same clients and distances with a restricted candidate set
+\(F'\subseteq F\); our pseudo-algorithm uses no additional input promise
+or prescribed-open facilities. Its internal forced openings are decisions
+of that call, not constraints passed from another call. Copies are merged
+back to original candidates before returning its output. If
+\(0<|F'|<k\), opening every candidate gives the optimum for that restricted
+instance. If \(F'=\varnothing\) and clients remain, discard the branch.
+The distinguished branch retaining a padded optimal \(k\)-set has
+neither exception. When the additive allowance is zero, the
+pseudo-algorithm already satisfies the required budget.
+
+The zero-optimum case can be recognized before the reduction. Every
+client location must be an available candidate location, and there can
+be at most \(k\) distinct client locations. Equivalently, every
+zero-distance class containing a client must contain a candidate, and
+there are at most \(k\) such client classes. These conditions are necessary
+and sufficient in the original metric; when they hold, open those
+locations. Thus the positive-optimum analysis of the reduction does not
+require division by zero. Co-located pieces occur only inside our
+pseudo-algorithm and do not change this original-metric test.
+
+We also make explicit a rational implementation of the deletion threshold
+in the proof of Li--Svensson's reduction~\cite[Section~2, Algorithm~2]{LiSvensson2016}.
+That proof analyzes a threshold determined by the unknown optimum.
+Its numerical value need not be computed. Suppose a call returns a set
+\(T\) with \(k<|T|\le k+c\). For every set \(T'\subseteq T\) obtainable
+by at most \(c\) deletions with \(|T'|>k\), and every \(i\in T'\), compute
+\[
+ \Delta(T',i)=\cost(T'\setminus\{i\})-\cost(T').
+\]
+These numbers are nonnegative rationals of polynomial bit length.
+There are \(n^{O(c)}\) such numbers. Run the reduction's deletion phase
+for each distinct number as threshold, and for zero, with a fixed order
+for choosing deletions; follow each run by its prescribed enumerative
+completion and retain the cheapest feasible result.
+
+To justify this replacement, let \(B^*\ge0\) be the threshold used in
+the reduction's analysis. Choose the largest enumerated value at most
+\(B^*\), including equality, or zero if there is none. Every test of the form
+\(\Delta(T',i)\le B^*\) has the same answer at this representative
+threshold. Induction through the at most \(c\) deletions therefore
+reproduces the ideal deletion history, including its fixed tie choices.
+Its output is present among the enumerated outputs. The approximation
+analysis still uses \(B^*\); the algorithm uses only exact rational
+comparisons. This adds \(n^{O(c)}\) overhead to the stated reduction.
+All candidate enumeration, distance comparisons and cost evaluations
+likewise use polynomial-bit rational arithmetic.
+
+Finally, the approximation parameter supplied to the reduction can be a
+fixed rational upper bound on our ratio, with its excess charged to the
+allowed accuracy. The auxiliary positive constants in its proof may then
+be chosen rational as well. These observations preserve both determinism
+and polynomial binary-input time; they require no oracle for the optimum.
+
+\begin{proof}[Proof of Theorem~\ref{thm:main}]
+Fix \(\varepsilon>0\). Apply Theorem~\ref{thm:pseudo} with
+loss \(\varepsilon/4\), obtaining a deterministic
+\(c\)-additive approximation, where \(c\) depends only on this fixed
+loss. Choose a fixed rational number \(\alpha\) with
+\[
+ 1+2/e+\varepsilon/4<\alpha<1+2/e+\varepsilon/2.
+\]
+The pseudo-algorithm is a \(c\)-additive \(\alpha\)-approximation.
+The additive-to-ordinary reduction of Theorem~\ref{thm:li-svensson},
+with a fixed positive rational loss at most \(\varepsilon/2\) and the
+implementation in Section~\ref{sec:reduction-implementation},
+gives a deterministic polynomial-time
+algorithm opening at most \(k\) original facilities and having
+cost at most
+\[
+ \left(1+\frac2e+\varepsilon\right)\OPT_k.
+\]
+Copies can be merged back to their original facilities as above.
+If exactly \(k\) facilities are required, add unused facilities
+from \(F\), which is possible because \(k\le|F|\) and can only
+decrease the cost.
+
+\end{proof}
+
+Combining Theorem~\ref{thm:main} with the hardness bound recalled in
+Section~\ref{sec:introduction} proves Corollary~\ref{cor:threshold}:
+under \(P\ne NP\), the infimum of deterministic polynomial-time
+approximation factors in the specified candidate-facility model is
+\(1+2/e\). The upper bound holds for every fixed positive loss and
+does not require attainment of the endpoint.

preprints/The-Approximation-Threshold-for-Metric-k-Median-September-24-2026/build/sections/inequality.tex

@@ -0,0 +1,365 @@
+\section{A finite inequality for the drift}
+\label{sec:inequality}
+
+The cost analysis uses an inequality on a finite probability space.  Its
+comparison matrix need not be symmetric.  The proof allows a positive
+contribution at one index only by charging indices with smaller values;
+the negative contributions at those indices pay the entire charge.
+
+The statement below is independent of the rounding construction.
+Section~\ref{sec:rounding} will apply it to normalized client distances
+and directed edge probabilities; Equation~\ref{eq:rnd-normalization}
+records that substitution. We prove the finite statement first so that
+the geometric analysis can use it as a single input.
+
+For $0\le x\le1$, define
+\begin{equation}
+\label{eq:analytic-coefficients}
+\begin{aligned}
+ A(x)&=1+2\int_0^1(1-e^{-tx})\,dt,&
+ P(x)&=\int_0^1te^{-tx}\,dt,\\
+ B(x)&=2P(x),& m(x)&=A(x)-1-2xP(x).
+\end{aligned}
+\end{equation}
+We suppress the argument $x$ when it is fixed.  Differentiation under the
+integral gives $A'=B$ and
+$A''(x)=-2\int_0^1t^2e^{-tx}\,dt<0$.  In particular,
+$0<P\le1/2$ and $m\ge0$, the latter by concavity and $A(0)=1$.
+Integration by parts also gives
+\begin{equation}
+\label{eq:analytic-identities}
+ (1+x)(A-1)=2x(1-P),\qquad
+ (1+x)A+xB-1=3x,\qquad A(1)=1+\frac2e.
+\end{equation}
+These identities hold at $x=0$ as well, using the integral definitions.
+
+\begin{theorem}[Finite comparison inequality]
+\label{thm:finite-inequality}
+Let $I$ be a finite index set with probability weights $(\pi_i)_{i\in I}$,
+and let $D_i\ge0$ satisfy $\E_iD_i=1$.  Both $\E_i$ and $\E_t$ denote
+expectation with these weights; in a double expectation the two indices
+are sampled independently.  Fix $0\le x\le1$ and $0<y\le1$, and set
+\[
+ \Theta=\frac{P(y-x)^2}{y}.
+\]
+Suppose $q_{ti}\in[0,1]$ and $R_i\ge0$ satisfy
+\begin{equation}
+\label{eq:finite-comparison}
+ R_i(1-Pq_{ti})\le D_i+D_t\qquad\text{whenever }q_{ti}<1.
+\end{equation}
+No symmetry is assumed for $q$.  For any $s_i\in[0,1]$, put
+$r_i=\E_tq_{ti}$ and $C_i=1-Ps_i$.  Then
+\begin{equation}
+\label{eq:finite-inequality}
+\begin{aligned}
+ \E_i\Big[&C_i(1-r_i)R_i
+ +r_i\{mD_i+(2-s_i)Py(D_i+1)\}-C_i(D_i+1)\\
+ &\hspace{12mm}
+ -\E_t(1-q_{ti})\bigl(D_t-(1+m+2Py)D_i\bigr)_+\Big]
+ \le2\Theta.
+\end{aligned}
+\end{equation}
+Here $u_+=\max(u,0)$.
+\end{theorem}
+
+The proof will bound each positive row by charges sent to indices with
+smaller $D$ values, then show that those receiving rows pay all incoming
+charges. The scalar estimates below locate the positive rows, bound the
+largest value they can charge, and certify the required negative
+contributions. Every terminating decimal denotes an exact rational number.
+
+\begin{lemma}[Scalar bounds]
+\label{lem:scalar-bounds}
+Fix $x\in[0,1]$, put $X=1-x$, and take
+$0\le w\le X$ and $s\in\{0,1\}$.  Use the functions in
+Equation~\eqref{eq:analytic-coefficients}, and define
+\begin{equation}
+\label{eq:scalar-parameters}
+\begin{aligned}
+ z&=Pw^2,& b&=(2-s)P(x+w),& a&=A+2Pw,& C&=1-Ps,\\
+ d_0&=.52,& d_1&=.4,& H_0&=.394-.42X^2+.04X^3,& H_1&=.25,\\
+ J_s&=CH_s,&
+ k_s&=(1-s)\frac{1-\sqrt{1-P}}{1+\sqrt{1-P}}.
+\end{aligned}
+\end{equation}
+For $q,r\in[0,1]$, set
+\begin{equation}
+\label{eq:scalar-functions}
+\begin{aligned}
+ \psi(q)&=\frac{1-q}{1-Pq},&
+ \kappa_s(q)&=C\psi(q)-(1-q),\\
+ K(r)&=C-k_s-br+1.6z,&
+ L(r)&=(C-m-b)r+.4z,\\
+ M(r)&=(1-r)a+CPr,& G(r)&=M(r)-L(r).
+\end{aligned}
+\end{equation}
+Then the following five bounds hold on the indicated domains:
+\begin{align}
+ m+b-\frac{.4z}{r}
+ &\le\frac{C(1-P)}{1-Pr}
+ &&(0<r\le1),
+ \label{eq:scalar-first}\\
+ L(r)\ge0,\qquad K(r)>0,\qquad K(r)
+ &\ge d_sG(r)
+ &&(0\le r\le1),
+ \label{eq:scalar-linear}\\
+ \bigl((J_s-b)r+1.6z\bigr)M(r)
+ +(C-k_s-J_sr)L(r)
+ &\ge0
+ &&(0\le r\le1),
+ \label{eq:scalar-charge}\\
+ C-k_s+1.6z-bq+.52\bigl(k_s-\kappa_s(q)\bigr)
+ &\ge H_0
+ &&(0\le q\le1),
+ \label{eq:scalar-type-zero}\\
+ C-k_s+1.6z-b&\ge.25.
+ \label{eq:scalar-type-one}
+\end{align}
+\end{lemma}
+
+\begin{proof}
+The scalar bounds are proved in Appendix~\ref{app:certificates} by
+analytic reductions and exact polynomial certificates.
+\end{proof}
+
+\begin{proof}[Proof of Theorem~\ref{thm:finite-inequality}]
+We may discard indices of probability zero.  The proof treats the
+quantities with a fixed second index $i$ as a row, with $t$ indexing its
+columns.
+
+\paragraph{Reduction of the parameters.}
+For fixed $x,D,q,R,s$, the left side of
+Equation~\eqref{eq:finite-inequality} is nondecreasing in $y$.
+The coefficient of its linear $y$ term is nonnegative, and increasing
+$y$ decreases the positive part that is subtracted.  Replace $y$ there
+by $\max(x,y)=x+w$, where $0\le w\le X=1-x$, and put $z=Pw^2$.
+If $y<x$, then $z=0\le\Theta$; otherwise
+$z=P(y-x)^2\le\Theta$, since $y\le1$.  It is therefore sufficient to
+bound the modified left side by $2z$.
+
+This modified left side is affine separately in every $s_i$, and its
+hypotheses do not constrain $s_i$.  Its maximum over $[0,1]^I$ is
+attained with every $s_i\in\{0,1\}$.  Fix such a choice.  For each row
+use the notation of Lemma~\ref{lem:scalar-bounds}, with $s=s_i$, and
+temporarily suppress the row subscript.  Write $D=D_i$, $R=R_i$,
+$q=q_{ti}$, $Z=D_t$, and $r=\E_tq_{ti}$, and put
+\[
+ U=mD+b(D+1).
+\]
+Notice that $a=1+m+2P(x+w)$.  Define the row contribution
+\begin{equation}
+\label{eq:row-contribution}
+ F=C(1-r)R+Ur-C(D+1)
+       -\E_t(1-q)(Z-aD)_+-(.4D+1.6)z.
+\end{equation}
+Because $\E_iD_i=1$, proving $\E_iF_i\le0$ proves the desired
+bound $2z$.
+
+\paragraph{A centered majorant for each row.}
+For $s=0$, direct maximization gives
+\[
+ \kappa_0(q)=\frac{Pq(1-q)}{1-Pq}
+ \le\frac{1-\sqrt{1-P}}{1+\sqrt{1-P}}=k_0.
+\]
+For $s=1$, one has
+$\kappa_1(q)=-P(1-q)^2/(1-Pq)\le0=k_1$.
+Also
+\begin{equation}
+\label{eq:psi-bound}
+ \psi(q)=1-q+Pq\psi(q)\le1-q+Pq,
+\end{equation}
+since $\psi(q)\le1$.
+
+Equation~\eqref{eq:finite-comparison} implies
+$(1-q)R\le\psi(q)(D+Z)$ when $q<1$.  This last inequality also holds
+when $q=1$, as both sides are zero.  Using
+$(Z-aD)_+\ge Z-aD$ in Equation~\eqref{eq:row-contribution} gives
+\begin{align*}
+ F\le\E_t\big[&Uq+D\{C\psi(q)+a(1-q)-C\}
+              -(.4D+1.6)z-C+\kappa_s(q)Z\big].
+\end{align*}
+The centering identity
+\[
+ \E_t[\kappa_s(q)Z]
+ =k_s-\E_t[(k_s-\kappa_s(q))Z]
+\]
+uses $\E_tZ=1$.  Thus, defining the explicit function
+\begin{equation}
+\label{eq:centered-integrand}
+\begin{aligned}
+ I_s(D;q,Z)={}&[mD+b(D+1)]q
+ +D\{C\psi(q)+a(1-q)-C\}\\
+ &-(.4D+1.6)z-C+k_s-(k_s-\kappa_s(q))Z,
+\end{aligned}
+\end{equation}
+we have
+\begin{equation}
+\label{eq:centered-row-bound}
+ F_i\le\E_t I_{s_i}(D_i;q_{ti},D_t).
+\end{equation}
+The replacement of $k_sZ$ by $k_s$ is made only under expectation.
+Subsequent pointwise bounds will concern the function in
+Equation~\eqref{eq:centered-integrand}.
+
+Since $k_s-\kappa_s(q)\ge0$ and $Z\ge0$, dropping the last term and
+using Equation~\eqref{eq:psi-bound} yields
+\begin{equation}
+\label{eq:pointwise-linear-majorant}
+ I_s(D;q,Z)\le DG(q)-K(q).
+\end{equation}
+Both $G$ and $K$ are affine in their argument, so averaging gives
+\begin{equation}
+\label{eq:row-linear-majorant}
+ F\le DG(r)-K(r).
+\end{equation}
+
+\paragraph{Charges issued by positive rows.}
+If $r=0$, then $q=0$ at every column and averaging
+Equation~\eqref{eq:finite-comparison} gives $R\le D+1$.
+Equation~\eqref{eq:row-contribution} then gives $F\le0$.
+Consequently a row with $F>0$ has $r>0$.  By
+Equations~\eqref{eq:scalar-linear} and
+\eqref{eq:row-linear-majorant}, it also has
+\begin{equation}
+\label{eq:positive-row-location}
+ G(r)>0,\qquad D>\frac{K(r)}{G(r)}\ge d_s.
+\end{equation}
+In this row put
+\[
+ U'=\frac{Ur-(.4D+1.6)z}{Cr}.
+\]
+The identity
+\[
+ U'=\frac{D+1}{C}\left(m+b-\frac{.4z}{r}\right)
+       -\frac mC-\frac{1.2z}{Cr}
+\]
+and Equation~\eqref{eq:scalar-first} imply
+\begin{equation}
+\label{eq:positive-row-uprime}
+ U'\le\frac{(D+1)(1-P)}{1-Pr}.
+\end{equation}
+On the other hand, discarding the nonpositive hinge term in
+Equation~\eqref{eq:row-contribution} gives
+$F/C\le(1-r)R+rU'-(D+1)$.
+If $R\le(D+1)/(1-Pr)$, Equation~\eqref{eq:positive-row-uprime}
+makes this upper bound nonpositive.  Therefore
+\begin{equation}
+\label{eq:positive-row-radius}
+ R>\frac{D+1}{1-Pr},\qquad U'\le R(1-P).
+\end{equation}
+In particular $r=1$ is impossible for a positive row, since
+Equation~\eqref{eq:positive-row-uprime} would then give $U'\le D+1$.
+
+For every column with $q<1$, Equations~\eqref{eq:finite-comparison}
+and \eqref{eq:positive-row-radius} now give
+\[
+ (1-q)R+qU'\le R(1-Pq)\le D+Z.
+\]
+For $q=1$, the excess of the left side over $D+Z$ is exactly
+$U'-D-Z$.  Using $\E_tZ=1$ and again discarding the nonpositive
+hinge term, we obtain
+\begin{equation}
+\label{eq:positive-row-excess}
+ F\le C\E_t\big[\ind_{\{q=1\}}(U'-D-Z)_+\big]
+   \le C\E_t(U'-D-Z)_+.
+\end{equation}
+This treats the unconstrained columns with $q=1$ explicitly.
+
+It remains to bound the largest value to which this row can charge.
+An exact rearrangement gives
+\[
+ U'-D=\frac{br-1.6z-L(r)D}{Cr}.
+\]
+Set $T=(J_s-b)r+1.6z$.  The expression in
+Equation~\eqref{eq:scalar-charge} is
+\begin{equation}
+\label{eq:charge-scalar-rearrangement}
+ TM(r)+(C-k_s-J_sr)L(r)=TG(r)+K(r)L(r)\ge0.
+\end{equation}
+Using $L(r)\ge0$ and Equation~\eqref{eq:positive-row-location},
+we conclude that
+\[
+ br-1.6z-L(r)D
+ \le br-1.6z-\frac{L(r)K(r)}{G(r)}\le J_sr.
+\]
+Thus $U'-D\le H_s$.  Since $C\le1$,
+Equation~\eqref{eq:positive-row-excess} proves
+\begin{equation}
+\label{eq:outgoing-charge}
+ F_i\le\E_t(H_{s_i}-D_t)_+\qquad(F_i>0).
+\end{equation}
+The function $H_0$ decreases from $.394$ to $.014$ on $[0,1]$,
+and $H_1=.25$.  Consequently every nonzero charge in
+Equation~\eqref{eq:outgoing-charge} goes to an index with $D_t<.394$.
+This separates every possible receiving index from all positive rows,
+which have $D_i>d_{s_i}\ge.4$. It remains to prove that the negative
+contribution of each receiving row covers the total charge it receives.
+
+\paragraph{Negative rows cover the incoming charges.}
+Put $c_*=.394$ and define
+\[
+ \mathcal P=\{i:F_i>0\},\qquad
+ \mathcal S=\{i:D_i\le c_*\},\qquad
+ \Gamma_{it}=\ind_{\{i\in\mathcal P\}}(H_{s_i}-D_t)_+.
+\]
+Equation~\eqref{eq:positive-row-location} shows that
+$\mathcal P\cap\mathcal S=\varnothing$, because $d_s\ge.4>c_*$.
+We claim that every $i\in\mathcal S$ satisfies
+\begin{equation}
+\label{eq:incoming-charge}
+ F_i+\E_t\Gamma_{ti}\le0.
+\end{equation}
+
+Fix such an $i$ and a column $t$.  Hold $q=q_{ti}$, $Z=D_t$,
+the two types $s_i,s_t$, and $\chi=\ind_{\{F_t>0\}}$ fixed.
+As a function of a variable $D\in[0,c_*]$,
+\[
+ I_{s_i}(D;q,Z)+\chi(H_{s_t}-D)_+
+\]
+is convex: the first term is affine and the second is a positive part
+multiplied by a nonnegative constant.  It suffices to bound this
+expression at $D=0,c_*$.  This check concerns the explicit majorant;
+it does not require the comparison hypothesis to remain valid when
+the variable $D$ changes.
+
+At $D=c_*$, the incoming charge is zero.  By
+Equation~\eqref{eq:pointwise-linear-majorant}, the remaining term is
+at most $c_*G(q)-K(q)\le0$.  Indeed, if $G(q)<0$, use $K(q)>0$;
+otherwise use $K(q)\ge d_{s_i}G(q)$ and $c_*<d_{s_i}$.
+
+At $D=0$, Equation~\eqref{eq:centered-integrand} reads
+\[
+ I_{s_i}(0;q,Z)
+ =bq-1.6z-C+k_s-(k_s-\kappa_s(q))Z,
+\]
+where the unindexed scalar parameters belong to row $i$.
+If $\chi=1$ and $s_t=0$, the positive donor row $t$ satisfies
+$Z=D_t>d_0=.52$.  Since $k_s-\kappa_s(q)\ge0$,
+Equation~\eqref{eq:scalar-type-zero} gives
+$I_{s_i}(0;q,Z)\le-H_0$, which covers its incoming charge.
+In the other cases the charge is either zero or $H_1=.25$.
+Equation~\eqref{eq:scalar-type-one}, together with $q\le1$,
+gives
+\[
+ I_{s_i}(0;q,Z)\le b-1.6z-C+k_s\le-.25,
+\]
+which covers these cases as well.  Convexity proves the pointwise
+bound throughout $[0,c_*]$.  Apply it at $D=D_i$ and average over
+$t$ in Equation~\eqref{eq:centered-row-bound} to obtain
+Equation~\eqref{eq:incoming-charge}.
+
+Finally, the incoming charge is $\Gamma_{ti}$ by definition, and
+exchanging the two finite sums requires no symmetry of $q$.
+Equation~\eqref{eq:outgoing-charge} bounds all
+positive rows by their outgoing charges;
+Equation~\eqref{eq:incoming-charge} makes all receiving rows pay
+their incoming charges; all other rows have nonpositive contribution.
+Every nonzero $\Gamma_{it}$ has $i\in\mathcal P$ and
+$t\in\mathcal S$.  Exchanging the two indices in the finite sum with
+weights $\pi_i\pi_t$ therefore gives
+\[
+ \E_iF_i\le\E_{i,t}\Gamma_{it}-\E_{i,t}\Gamma_{ti}=0.
+\]
+This proves the stronger bound $2z$ after the parameter reduction and,
+since $z\le\Theta$, proves Equation~\eqref{eq:finite-inequality}.
+\end{proof}

preprints/The-Approximation-Threshold-for-Metric-k-Median-September-24-2026/build/sections/introduction.tex

@@ -0,0 +1,196 @@
+\section{Introduction}\label{sec:introduction}
+
+In metric \(k\)-median, a finite set \(J\) of clients and a finite set \(F\)
+of candidate facilities lie in a common metric space. Given an integer \(k\),
+one chooses at most \(k\) facilities to minimize the sum of the distances
+from clients to their nearest chosen facility. For a nonempty set
+\(S\subseteq F\), write
+\[
+ d(j,S)=\min_{i\in S}d(j,i),\qquad
+ \cost(S)=\sum_{j\in J}d(j,S),\qquad
+ \OPT_k=\min_{\substack{S\subseteq F\\1\le|S|\le k}}\cost(S).
+\]
+The input distances are rational and encoded in binary, and
+\(1\le k\le |F|\). Both \(k\) and the metric are unrestricted.
+Candidate facilities are specified as part of the input; we do not
+require \(F=J\). The empty-client instance has cost zero and can be
+handled immediately.
+
+The facility budget is a central difficulty: a solution with small
+connection cost can be much easier to obtain when extra facilities are
+allowed. Our approach first permits a constant additive surplus, makes
+that approximation deterministic, and then removes the surplus using an
+established reduction.
+
+\begin{theorem}\label{thm:main}
+For every fixed \(\varepsilon>0\), there is a deterministic algorithm which,
+given a finite rational metric \(k\)-median instance with \(1\le k\le |F|\),
+returns \(S\subseteq F\), \(|S|\le k\), such that
+\[
+ \sum_{j\in J}d(j,S)\le
+ \left(1+\frac2e+\varepsilon\right)\OPT_k.
+\]
+Its running time is polynomial in the binary encoding length of the instance.
+The polynomial may depend on \(\varepsilon\).
+\end{theorem}
+
+Hardness below \(1+2/e\) in this facility model follows from the
+Max-\(k\)-Coverage gap: covered clients have distance one and uncovered
+clients have distance at least three. We use the formulation under \(P\ne NP\)
+recorded by Anand and Lee~\cite[Sections~1 and~1.1]{AnandLee2024}, based on
+Feige's perfect-completeness coverage gap~\cite[Theorem~5.3]{Feige1998}.
+For an explicit metric completion, represent sets by facilities and
+universe elements by a disjoint client set. Give distinct points within
+either part distance two, and give a facility--client pair distance one
+for incidence and three otherwise. This is a metric: a same-part edge
+of length two is bounded by \(1+1\), and a cross-part edge of length at
+most three is bounded by \(2+1\). For a universe of size \(m\), the
+connection cost is exactly \(3m-2c\) when the chosen sets cover \(c\)
+elements; a perfect cover has cost \(m\). Thus a gap between
+covering every element and covering at most a
+\(1-1/e+\delta\) fraction gives the lower bound
+\(1+2/e-2\delta\), for every fixed positive \(\delta\).
+Together with Theorem~\ref{thm:main},
+this proves the following consequence.
+
+\begin{corollary}\label{cor:threshold}
+If \(P\ne NP\), the infimum of deterministic polynomial-time approximation
+factors for metric \(k\)-median with specified candidate facilities is \(1+2/e\).
+\end{corollary}
+
+The infimum in Corollary~\ref{cor:threshold} does not assert attainment of the
+endpoint factor. The hardness assumption and candidate-facility model are part
+of the statement. In particular, the corresponding lower-bound question with
+the restriction \(F=J\) is different~\cite{AnandLee2024}.
+
+\subsection{Historical context and method}
+
+Metric \(k\)-median has been a testing ground for LP rounding,
+primal-dual algorithms, and local search. Charikar, Guha, Tardos and
+Shmoys gave the first constant-factor approximation through LP
+rounding~\cite{CharikarGuhaTardosShmoys2002}. Jain and
+Vazirani connected the cardinality constraint with metric facility
+location through Lagrangian relaxation and the primal-dual
+method~\cite{JainVazirani2001}. The combinatorial improvements of
+Charikar and Guha~\cite{CharikarGuha1999} and the greedy dual-fitting
+approach of Jain et al.~\cite{JainEtAl2003} developed this connection
+further. A different route, the bounded-swap local search of Arya et
+al.~\cite{Arya2004}, achieves factor \(3+\varepsilon\).
+
+Li and Svensson~\cite{LiSvensson2016} made constant-additive
+pseudo-approximations a route past the factor-three barrier: a constant
+number of extra facilities can be removed at an arbitrarily small
+additional approximation loss. Subsequent bi-point rounding, including
+the corrected \(2.675+\varepsilon\) guarantee of Byrka et
+al.~\cite{ByrkaEtAl2017}, refined this approach. Gowda et
+al.~\cite{GowdaEtAl2023} obtained \(2.613\) in their SODA 2023 version.
+Cohen-Addad et al.~\cite{CGLSS2026} reached
+\(2+\varepsilon\), first announced in 2025. Byrka et
+al.~\cite{ByrkaEtAl2026} subsequently developed a graph-based iterative
+rounding framework that also gives \(2+\varepsilon\) for metric
+\(k\)-median and extends to other clustering objectives.
+Theorem~\ref{thm:main} reaches every fixed factor above the
+\(1+2/e\) hardness boundary in the general candidate-facility model.
+
+Our construction follows the iterative graph framework, with the
+cost analysis replaced by a directed comparison inequality and a
+potential that retains some surviving assignments after an opening.
+The earlier framework's general transfer theorem uses a parameter at
+least \(2\) in the metric \(k\)-median case
+\cite[Theorem~6.1]{ByrkaEtAl2026}; it does not supply the smaller factor
+proved here. We establish the grid preparation, physical updates,
+comparison, and deterministic implementation together.
+
+A separate companion combines anchor recovery with bounded-price payment
+accounting to give an independent randomized approximation below factor
+\(2\) while respecting the facility budget~\cite[Theorem~1.2]{OpenAIRecovery2026}.
+
+Two other methodological antecedents are useful to distinguish.
+Splitting fractional facilities and organizing assignments around
+separated representatives and bundles are standard LP-rounding
+techniques; Charikar and Li~\cite{CharikarLi2012} provide a close
+predecessor. The pair updates used in our preparation belong to the
+dependent-rounding approach of Srinivasan~\cite{Srinivasan2001} and
+Gandhi et al.~\cite{GandhiEtAl2006}, with weighted variants studied by
+Byrka et al.~\cite{ByrkaEtAl2017}. We prove the particular moment bounds
+and coupled supplier construction needed here. Likewise, reducing the
+support of a distribution while matching moments is a standard
+Carath\'eodory argument; see Bayer and Teichmann~\cite{BayerTeichmann2005}
+for the finite-cubature perspective. Our implementation specifies which
+moments suffice throughout the physical process and proves the required
+rational encoding bounds.
+
+\subsection{Proof overview}
+
+The proof separates the preparation of fractional openings, their
+iterative rounding, and the removal of randomness and surplus. The
+intermediate guarantee is measured against the standard assignment LP:
+its variable \(y_i\) is the amount of opening at facility \(i\), and each
+client receives a unit of this opening mass. The LP has total opening
+mass at most \(k\) and cost at most \(\OPT_k\).
+
+\paragraph{Preparing a constant grid.}
+Section~\ref{sec:preparation} replaces the LP openings by multiples of
+\(g=1/\Delta\), where the even integer \(\Delta\) depends only on the
+accuracy. The total mass increases by at most \(3g\), and each client's
+expected assignment cost increases by an arbitrarily small factor.
+Separated representative clients define disjoint bundles of nearby
+opening mass. A bundle's deficit is the part missing from a unit; its
+suppliers are the facilities or other bundles holding that part of the
+representative's assignment. Rounding deficits together with their
+suppliers controls both the opening budget and the cost of restoring a
+full unit. This coupling must allow one supplier to serve many clients,
+because assignment mass is not a shared capacity.
+
+\paragraph{Rounding the grid.}
+View every positive grid opening as copies of mass \(g\). In each round,
+a directed graph specifies which copies a source erases when that source's
+facility opens. Each target has exactly \(\Delta\) incoming sources,
+chosen with distance-dependent probabilities. Opening and erasing are
+balanced so that only a constant additional number of facilities is
+needed. This source-opening and neighbor-erasure framework, including
+balanced batches and forced large updates, builds on Byrka et
+al.~\cite[Sections~3--6]{ByrkaEtAl2026}.
+
+The connection-cost analysis needs a different comparison below factor
+two. For each client, we track surviving copies from its initial assigned
+unit and a persistent bound on its eventual connection distance. A
+potential combines their remaining assignment cost with that bound. Its
+coefficient depends on elapsed time as well as surviving mass. When one
+of the client's own copies opens, we continue tracking the cheaper
+survivors; this produces a subtraction that pays for other possible
+erasures. The finite inequality in Section~\ref{sec:inequality} captures
+that payment for arbitrary directed comparison coefficients. Its proof
+transfers contributions from positive rows to lower-distance rows, whose
+negative contributions cover them. Appendix~\ref{app:certificates}
+proves every scalar estimate by explicit rational polynomial certificates.
+The geometric substitution and the potential analysis are given in
+Section~\ref{sec:rounding}.
+
+A source erasing many copies is handled by forcing its facility open.
+Because this choice depends on the graph, its potential change is bounded
+for each realized graph first. Only then do we average over graphs. This
+order lets the finite directed inequality control the averaged drift
+without conditioning on a graph-dependent forcing event.
+
+\paragraph{A deterministic solution and the exact budget.}
+The preceding analysis uses randomness, but only specified low-order
+moments of each local choice enter its bounds. Section~\ref{sec:derandomization}
+constructs polynomial-size rational distributions preserving these
+moments. All physical histories are expanded for a constant number of
+rounds, and the cheapest output satisfying the additive budget is chosen.
+The separate rational-arithmetic analysis proves polynomial time in the
+binary input length. The result is a deterministic approximation opening
+at most \(k+c\) facilities for an accuracy-dependent constant \(c\).
+Li and Svensson's reduction then removes this surplus with an arbitrarily
+small additional loss.
+
+\paragraph{Organization.}
+Section~\ref{sec:preliminaries} fixes the model and states the external
+additive-to-ordinary reduction. Section~\ref{sec:preparation} prepares the
+openings. Section~\ref{sec:inequality} proves the finite drift inequality,
+with its scalar certificates in Appendix~\ref{app:certificates}.
+Section~\ref{sec:rounding} gives and analyzes the iterative procedure.
+Section~\ref{sec:derandomization} supplies the deterministic implementation
+and completes the proof of Theorem~\ref{thm:main}.
+

preprints/The-Approximation-Threshold-for-Metric-k-Median-September-24-2026/build/sections/preliminaries.tex

@@ -0,0 +1,104 @@
+\section{Preliminaries and the reduction}\label{sec:preliminaries}
+
+We use the client set \(J\), candidate set \(F\), connection cost
+\(\cost\), and optimum \(\OPT_k\) defined in the introduction.
+Polynomial time includes the binary encodings of all distances.
+We henceforth assume \(J\ne\varnothing\) and \(1\le k\le |F|\).
+This section fixes the fractional representation and states the external
+reduction that will remove the final constant additive surplus.
+
+\subsection{The fractional solution}
+
+We use the standard relaxation
+\begin{equation}\label{eq:lp}
+\begin{aligned}
+ \text{minimize}\quad &\sum_{j\in J}\sum_{i\in F}d(j,i)x_{ij}\\
+ \text{subject to}\quad&
+ \sum_{i\in F}x_{ij}=1 &&(j\in J),\\
+ &0\le x_{ij}\le y_i\le1 &&(i\in F,\ j\in J),\\
+ &\sum_{i\in F}y_i\le k.
+\end{aligned}
+\end{equation}
+Its rational optimum can be found in polynomial time. For fixed openings
+\(y_i\), assign each client to its nearest unit of opening mass, using a fixed
+order to break ties. If \(h_j(t)\) is the distance needed to accumulate mass
+\(t\), put
+\[
+ V_j=\int_0^1h_j(t)\,dt.
+\]
+Then \(L=\sum_jV_j\le\OPT_k\) for an optimal solution of~\eqref{eq:lp}.
+
+As in LP-rounding constructions such as Charikar--Li~\cite[Section~2]{CharikarLi2012},
+it is convenient to split an opening into co-located pieces so that each
+client's nearest-unit assignment uses whole pieces. There are only
+polynomially many needed split points: each client uses at most one partial
+piece in a fixed distance order. Refine each facility simultaneously at all
+the resulting cut points. Denote the whole-piece assignment of client \(j\)
+by \(F_j\). For a set \(U\) of pieces, write \(y(U)=\sum_{i\in U}y_i\);
+thus \(y(F_j)=1\) and \(V_j=\sum_{i\in F_j}y_id(j,i)\).
+
+Internal pieces at one location are allowed to be distinct, so their distance
+can be zero. All geometric arguments use only nonnegativity, symmetry, and
+the triangle inequality. Whenever a procedure returns open pieces, merge
+them by their original facility identity. Merging preserves every client's
+connection cost and cannot increase the number of facilities.
+
+\subsection{Constant-additive pseudo-approximation}
+
+\begin{definition}
+A \(c\)-additive \(\alpha\)-approximation for metric \(k\)-median returns
+at most \(k+c\) facilities of cost at most \(\alpha\OPT_k\).
+The additive allowance \(c\) is independent of the input instance.
+\end{definition}
+
+We use the following established result.
+
+\begin{theorem}[Li--Svensson~\cite{LiSvensson2016}, Theorem~4 of the 2014 manuscript]
+\label{thm:li-svensson}
+Let \(\alpha>1\), and suppose \(\mathcal A\) is a \(c\)-additive
+\(\alpha\)-approximation algorithm for metric \(k\)-median.
+For every \(\eta>0\), there is an ordinary
+\((\alpha+\eta)\)-approximation algorithm with running time
+\(n^{O(c/\eta)}\) times the running time of \(\mathcal A\).
+For fixed \(\alpha,c,\eta\), this is polynomial overhead.
+\end{theorem}
+
+The theorem permits separate client and candidate sets and applies the
+pseudo-algorithm to restricted candidate instances. Our construction works
+on every such instance. We will obtain a deterministic pseudo-algorithm
+before applying Theorem~\ref{thm:li-svensson}; the external reduction is not
+being used to convert an expected-cost statement into determinism.
+
+For each fixed accuracy we permit all grid denominators, numbers of rounds,
+and additive allowances to be constants of any size. They may depend on
+that accuracy, but not on \(k\), the number of input points, their distances,
+or their bit lengths. Thus it suffices to prove, for each fixed
+\(\tau>0\), a deterministic \(c_\tau\)-additive
+\((1+2/e+\tau)\)-approximation. Our analysis in fact bounds the
+pseudo-solution against \(L\), the initial fractional cost. This does not
+assert the same factor for rounding the original relaxation to exactly
+\(k\) facilities: the additive allowance and the subsequent reduction
+are essential.
+
+\subsection{Probability and bookkeeping}
+
+We use \(\E\), \(\Prob\), and \(\Var\) for expectation, probability, and variance.
+Every distribution used below has finite support. Randomized procedures
+first provide the estimates; Section~\ref{sec:derandomization} replaces their
+choices by explicit polynomial-size lists.
+
+Two elementary facts will be used repeatedly. First, for a nonnegative random
+cost \(C\) and an event \(E\) of probability at least \(1-\rho>0\),
+\begin{equation}\label{eq:conditioning}
+ \E[C\mid E]\le\frac{\E C}{1-\rho}.
+\end{equation}
+No cost--event independence is required. Second, a finite distribution
+contains an outcome whose cost is at most its expected cost. These facts
+allow us to discard budget failures and select a single output by total
+cost; we never require all individual client expectation bounds to hold
+simultaneously at that output.
+
+The notation \(O_\Delta(\cdot)\) means that its implicit constant may depend
+on the fixed grid denominator \(\Delta\), and on no input parameter.
+All unqualified constants in the geometric and analytic estimates are
+absolute, unless an explicit previously fixed parameter is indicated.

preprints/The-Approximation-Threshold-for-Metric-k-Median-September-24-2026/build/sections/preparation.tex

@@ -0,0 +1,526 @@
+\section{Preparing the openings}\label{sec:preparation}
+
+We first put the fractional openings on a sufficiently fine constant grid.
+The use of separated representatives and bundles is motivated by the
+discretization approach in~\cite{ByrkaEtAl2026}.  The construction below
+rounds a bundle's missing assignment mass together with the facilities
+that can supply it.  This coupling provides both an additive opening
+bound and a multiplicative connection-cost bound.
+
+Throughout this Section, the client set is nonempty; an instance without
+clients requires no cost analysis.  We start with feasible fractional
+openings $0\le y_i\le1$.  For each client $j$, choose a nearest-unit
+assignment $x_{ij}$, resolving ties in a fixed order, and write
+\[
+ V_j=\sum_i d(i,j)x_{ij}.
+\]
+The feasibility constraints are $\sum_i x_{ij}=1$ and $x_{ij}\le y_i$;
+there is no supply constraint shared by different clients.
+
+\begin{theorem}\label{thm:preparation}
+For every $\eta>0$, there is a constant $\Delta_0$ with the following
+property.  Given a finite rational metric instance and feasible rational
+openings $0\le y_i\le1$ with a nearest-unit assignment of cost $V_j$
+for each client $j$, fix any constant even integer $\Delta\ge\Delta_0$ and let
+$g=1/\Delta$.  There is a randomized procedure, running in expected
+polynomial time on rational inputs, that produces openings
+$\widetilde y_i\in[0,1]\cap g\mathbb Z$, allowing polynomially many
+co-located labeled facility pieces, such that
+\[
+ 1\le\sum_i\widetilde y_i\le\sum_i y_i+3g
+\]
+for every output, and the expected nearest-unit assignment cost of every
+client $j$ is at most $(1+\eta)V_j$.
+In particular the grid can be required to satisfy any additional
+positive instance-independent upper bound on $g$.
+\end{theorem}
+
+The expectation in Theorem~\ref{thm:preparation} is taken after rejecting
+outcomes that exceed the opening budget.  Section~\ref{sec:derandomization}
+gives a deterministic implementation that preserves the corresponding
+bound on the sum of the client costs.
+
+We use the following elementary dependent-rounding observation, in both
+weighted and unweighted form. The mean-preserving pair update and its
+product-supermartingale argument follow the standard
+approach~\cite{Srinivasan2001,GandhiEtAl2006}; weighted rounding with one
+residual fraction is also studied in~\cite[Section~2]{ByrkaEtAl2017}.
+We give the proof to specify exactly the moment guarantees used here.
+
+\begin{lemma}\label{lem:prep-weighted-rounding}
+Let $q_e\ge0$ be rational counts and let $\beta_e>0$ be rational weights.
+One can randomly update the counts within the bounds
+$[\lfloor q_e\rfloor,\lceil q_e\rceil]$, preserving
+$\sum_e\beta_e q_e$ on every update, until at most one count is
+fractional.  Every first moment is preserved, and the expectation of
+every product of two distinct counts is at most the product of their
+initial values.  Rounding the possible remaining fractional count with
+its correct mean preserves these moment assertions.
+For the resulting integer counts $Q_e$,
+\[
+ \Var(Q_e)\le q_e,\qquad
+ \E[Q_eQ_f]\le q_eq_f\quad(e\ne f).
+\]
+The procedure uses linearly many two-coordinate updates.
+\end{lemma}
+
+\begin{proof}
+When two coordinates are fractional, move them along the line
+\[
+ (q_e,q_f)\longmapsto
+ (q_e+t,q_f-(\beta_e/\beta_f)t)
+\]
+to one of the two endpoints allowed by their fixed floor and ceiling
+bounds.  Choose the endpoint probabilities so that the conditional
+mean of $t$ is zero.  At least one coordinate becomes integral, and
+integral coordinates are not updated subsequently.  Each coordinate
+is a martingale.  The product of the two updated coordinates is a
+concave quadratic in $t$; a product involving exactly one updated
+coordinate has unchanged conditional mean.  Thus every distinct-pair
+product is a supermartingale.  Rounding the last fractional coordinate
+with the correct conditional mean leaves all these products unchanged
+in expectation.  Finally, an integer random variable supported on
+$\{m,m+1\}$, with mean $m+\theta$, has variance
+$\theta(1-\theta)\le m+\theta$.  This proves the variance assertion.
+\end{proof}
+
+\begin{proof}[Proof of Theorem~\ref{thm:preparation}]
+We first form disjoint bundles containing most of each representative's
+assigned mass. We then round their deficits jointly with the openings
+that supply them, reserving a small amount of budget for clipping large
+deficits. After rounding inside bundles, every representative can again
+receive one unit. The final part of the proof compares connection costs
+at two representative-distance scales and conditions on the budget event.
+
+We will choose positive rational constants in the order
+\[
+ \delta,\quad M,\quad T_1,\quad\gamma,\quad g,
+\]
+with $\delta$ small, $M$ and then $T_1$ large, and $\gamma$ and then $g$
+small.  All choices depend only on $\eta$; the last choice means that
+every sufficiently fine even grid is permitted.  During the proof we
+require
+\begin{equation}\label{eq:prep-parameter-basics}
+ 0<\delta\le\tfrac14,\qquad 0<\gamma\le1,\qquad
+ \theta:=\frac1{M\delta}\le\tfrac18,\qquad
+ \alpha:=\frac M{\delta T_1}\le\tfrac14.
+\end{equation}
+Set $p_0=1-\theta$ and $\beta=M/T_1=\delta\alpha$.
+
+\paragraph{Splitting and representatives.}
+For each original facility, view its opening $y_i$ as an interval of
+length $y_i$ and its assignment to client $j$ as the initial subinterval
+of length $x_{ij}$.  Split at the endpoints of these subintervals for all
+clients.  There are at most $|J|+1$ positive-mass pieces per original
+facility, and every assignment now uses whole pieces.  Retain the
+notation $y_i$ for the mass of a piece, and let $F_j$ be the set of pieces
+used by client $j$.  Thus
+\[
+ y(F_j)=1,\qquad
+ V_j=\sum_{i\in F_j}y_i d(i,j),\qquad
+ y(W):=\sum_{i\in W}y_i.
+\]
+Pieces retain the distances of their original facility.  They may be
+treated as distinct candidates throughout the rounding; merging
+co-located integral openings at the end preserves all connection
+distances and can only decrease the number of opened facilities.
+
+Process the clients in nondecreasing order of $V_j$.  Select each
+remaining client $s$ as a representative and absorb every remaining
+client $j$ for which $d(j,s)\le MV_j$.  Write $\mathcal S$ for the set of
+representatives.  A client $j$ assigned to $s$ satisfies
+\begin{equation}\label{eq:prep-representative-properties}
+ V_s\le V_j,\qquad d(j,s)\le MV_j.
+\end{equation}
+Distinct representatives satisfy
+\[
+ d(s,t)>M\max(V_s,V_t):
+\]
+indeed, the later one was not absorbed by the earlier one.  Let $L_s$ be
+the distance to a nearest other representative, taking $L_s=\infty$
+when $s$ is the only representative.  Define
+\[
+ P_s=\{i\in F_s:d(s,i)\le\delta L_s\},\quad
+ p_s=y(P_s),\quad c_s=1-p_s,\quad A_s=F_s\setminus P_s,
+\]
+and let $U$ consist of all pieces outside $\bigcup_s P_s$.
+For $L_s=\infty$, interpret $P_s=F_s$ and $c_s=0$.
+
+The bundles $P_s$ are disjoint.  Otherwise a piece in $P_s\cap P_t$
+would imply
+\[
+ d(s,t)\le\delta(L_s+L_t)\le2\delta d(s,t),
+\]
+which is impossible.  For finite $L_s$, the mass in $A_s$ has distance
+at least $\delta L_s$ from $s$, so
+\begin{equation}\label{eq:prep-deficit-small}
+ \delta L_s c_s\le V_s,\qquad
+ 0\le c_s\le\theta,\qquad p_s\ge p_0.
+\end{equation}
+These bounds also cover $V_s=0$: in that case $c_s=0$.
+
+\paragraph{Supplier stars and signed counts.}
+For the raw opening count, give every bundle a base level of one.  A
+supplier star couples planned deficits in its target bundles to the
+mass that will supply them.  One planned count unit subtracts mass $g$
+from each target bundle and adds mass $g$ at the supplier if it lies in $U$.
+A bundle supplier incurs no additional opening in this accounting.
+Thus a star with $d$ targets changes the opening mass by $(1-d)g$ for
+an outside supplier and by $-dg$ for a bundle supplier.
+
+Make a list $\mathcal E$ of stars; a star $e$ has weight $w_e$, a flag
+$a_e\in\{0,1\}$ indicating an outside supplier, a target set
+$D_e\subseteq\mathcal S$, and a specified supplier:
+\begin{enumerate}
+ \item For each $i\in U$, take $w_e=y_i$, $a_e=1$,
+ $D_e=\{s:i\in A_s\}$, and supplier $i$.
+ \item For each pair $s,t$ with $y(A_s\cap P_t)>0$, take
+ $w_e=y(A_s\cap P_t)$, $a_e=0$, $D_e=\{s\}$, and supplier $P_t$.
+ Necessarily $t\ne s$.
+ \item For each $s$ with $c_s>0$, take $w_e=\gamma c_s$, $a_e=0$,
+ $D_e=\{s\}$, and use as supplier the bundle of a nearest other
+ representative.  Such a representative exists because $c_s>0$.
+\end{enumerate}
+The last stars are called extra stars.  The planned count of star $e$
+is $q_e=w_e/g$.  With $u_e=a_e-|D_e|$, the planned raw opening mass is
+\begin{equation}\label{eq:prep-raw-identity}
+ |\mathcal S|+\sum_e u_e w_e
+ =\sum_i y_i-\gamma\sum_s c_s.
+\end{equation}
+To see this, the flag-one weights sum to $y(U)$, while the total weight
+targeting $s$ is $(1+\gamma)c_s$.
+
+Apply Lemma~\ref{lem:prep-weighted-rounding} independently to the
+positive and negative signs of $u_e$, using weights $|u_e|$.  Complete
+the rounding of the positive class by rounding its possible last
+fraction with the correct mean.  Round the zero-coefficient counts
+independently between their floor and ceiling.  Leave the last
+fractional count of the negative class, if any, temporarily unrounded.
+All counts are then integral except possibly this one.
+
+If there is no remaining fraction, the resulting counts are both the
+nominal counts and the actual activation counts.  Otherwise, write the
+remaining count as $m+\xi$, where $0<\xi<1$, and let its star have
+$d=|D_e|$ targets.  Its nominal count is $m+\operatorname{Bernoulli}(\xi)$.
+If its flag is one, open $(m+1)g$ at its supplier, irrespective of the
+nominal coin.  Activate $m$ units at each target, and activate one
+additional unit at every target in a random subset of size
+\[
+ H\in\{\lfloor d\xi\rfloor,\lceil d\xi\rceil\},
+ \qquad \E H=d\xi,
+\]
+whose membership marginal is $\xi$ at every target.  Such a subset is
+obtained by choosing the indicated size with the correct mean and
+then taking a uniformly shifted interval of that size in a fixed
+cyclic ordering of the targets.  The nominal coin can be independent
+of this choice.  All other stars use their integral counts for both
+openings and activations.
+
+This modification bounds the raw opening error independently of the
+number of targets.  For the exceptional star its actual signed
+contribution, in count units, minus its planned contribution is
+\[
+ a_e(m+1)-dm-H-(a_e-d)(m+\xi)
+ =a_e(1-\xi)+d\xi-H<2.
+\]
+A positive coefficient $u_e$ necessarily equals one, so the last
+rounding in the positive class adds less than one further count unit.
+The zero class has no signed contribution.  If $\widetilde y_i$ denotes
+the actual opening on $U$, and $Z_s$ denotes the total activated deficit
+mass at $s$, we therefore have on every outcome
+\begin{equation}\label{eq:prep-raw-bound}
+ \sum_{i\in U}\widetilde y_i+\sum_s(1-Z_s)
+ \le\sum_i y_i-\gamma\sum_s c_s+3g.
+\end{equation}
+Moreover, each actual outside opening dominates both its nominal
+opening and every activation demand placed on it by a single target.
+
+\paragraph{Deficit moments and clipping.}
+The signed count already controls raw opening mass. The remaining
+obstacle is that an activated deficit might leave too little mass in its
+own bundle. We cap that deficit and pay for the added bundle mass with
+the reserve in Equation~\ref{eq:prep-raw-bound}.
+For every $s$, let $A_{es}$ be the integer activation count from a star
+targeting $s$.  Then
+\begin{equation}\label{eq:prep-deficit-moments}
+ \E Z_s=(1+\gamma)c_s,
+ \qquad \Var(Z_s)\le g(1+\gamma)c_s.
+\end{equation}
+Here it matters that the last negative star may depend on the preceding
+random choices.  Condition on the complete count vector immediately
+before its exceptional rounding.  At most one coordinate is
+fractional, so for distinct $e,f$ targeting $s$, at least one of
+$A_{es},A_{fs}$ is deterministic under this conditioning.  Each has
+conditional mean equal to its current count.  Their conditional product
+expectation is therefore the product of those two current counts.
+The distinct-pair product bound in
+Lemma~\ref{lem:prep-weighted-rounding} consequently still holds for
+these activations.  Their individual means are $q_e$, and each is
+supported on $\{\lfloor q_e\rfloor,\lceil q_e\rceil\}$.  Thus
+\[
+ \Var(Z_s)
+ \le g^2\sum_{e:s\in D_e}\Var(A_{es})
+ \le g^2\sum_{e:s\in D_e}q_e
+ =g(1+\gamma)c_s,
+\]
+as asserted.  The nominal counts also retain the same first and
+distinct-pair moment bounds.
+
+Clip the deficit at one half:
+\[
+ Z_s^*=\min(Z_s,\tfrac12),\qquad p'_s=1-Z_s^*.
+\]
+Discard any excess activation units.  This can be done in whole units
+because $\Delta$ is even.  By Equation~\ref{eq:prep-parameter-basics},
+$\mu_s:=\E Z_s\le1/4$.  For any random variable $X$ with mean $\mu<b$,
+\[
+ (X-b)_+\le\frac{(X-\mu)^2}{b-\mu}.
+\]
+Indeed the assertion is immediate when $X\le b$, and otherwise follows
+by expanding $(X-\mu)^2=(X-b+b-\mu)^2$.
+Equation~\ref{eq:prep-deficit-moments} therefore gives
+\[
+ \E(Z_s-\tfrac12)_+\le4g(1+\gamma)c_s.
+\]
+Let $\mathcal G$ be the event that
+\[
+ \sum_s(Z_s-Z_s^*)\le\gamma\sum_s c_s.
+\]
+If some $c_s$ is positive, Markov's inequality gives
+\begin{equation}\label{eq:prep-budget-probability}
+ \Prob(\mathcal G)\ge1-\rho,
+ \qquad \rho:=\frac{4g(1+\gamma)}\gamma.
+\end{equation}
+If all $c_s$ vanish, all activations and discarded amounts are zero,
+and $\mathcal G$ holds surely.  Equations~\ref{eq:prep-raw-bound} and
+\ref{eq:prep-budget-probability} show that a vector with bundle sums
+$p'_s$ has total opening mass at most $\sum_i y_i+3g$ on $\mathcal G$.
+
+\paragraph{Rounding within bundles and supplying one unit.}
+Condition on the complete star-rounding outcome, including the retained
+activations.  In each bundle independently, rescale piece $i\in P_s$ to
+mass
+\[
+ \overline y_i=y_i\frac{p'_s}{p_s}.
+\]
+Since $y_i\le p_s$ and $1/2\le p'_s\le1$, these masses belong to $[0,1]$.
+Their sum $p'_s$ is a grid multiple.  Apply the unweighted version of
+Lemma~\ref{lem:prep-weighted-rounding} to counts $\overline y_i/g$.
+The preserved sum is integral in count units, so there is no final
+fraction.  This produces grid masses $\widetilde y_i$ with conditional
+means $\overline y_i$ and nonpositive conditional pair covariances.
+All masses remain at most one, because one is a grid point.  Likewise
+the outside opening at a piece of original mass $y_i$ is at most
+$g\lceil y_i/g\rceil\le1$, even under the exceptional correction.
+
+Every representative $s$ can now receive one unit: take its entire
+bundle mass $p'_s$ and supply its retained deficit $Z_s^*$ using the
+retained activations.  Each outside supplier has enough mass for its
+activation demand at $s$.  Any bundle supplier is different from $P_s$.
+The total demand of $s$ on any one such bundle is at most its entire
+retained deficit, hence at most $1/2$, while every supplying bundle has
+mass at least $1/2$.  Allocate this demand greedily over its pieces.
+The bundles and $U$ are disjoint, and different clients have separate
+assignment constraints.  This proves feasibility for each client
+individually, even before conditioning on $\mathcal G$.  It also shows
+that the total opening mass is at least one.
+
+\paragraph{Cost comparison at a distant representative scale.}
+Fix a client $j$ and its representative $s$, and put $D=d(j,s)$.
+Let $R_s=\max_{i\in F_s}d(s,i)$, and define the radius
+\[
+ K_s=\begin{cases}L_s,&c_s>0,\\R_s,&c_s=0.\end{cases}
+\]
+If $V_j=0$, Equation~\ref{eq:prep-representative-properties} gives
+$D=V_s=0$.  All pieces of $F_s$ then have distance zero from $j$,
+$c_s=0$, and $Z_s=0$ surely.  The rounded bundle $P_s=F_s$ still has
+one unit at distance zero.  Hence the new cost is identically zero.
+We may therefore assume $V_j>0$ in the remaining cost analysis.
+
+Suppose first that $K_s>T_1V_j$.  We have $R_s\ge\delta K_s$: this is
+immediate if $c_s=0$, and otherwise some assigned mass lies beyond
+$\delta L_s$.  Consequently $D/R_s\le\alpha<1$.  This controls the
+cost of exchanging the portions of $F_j$ and $F_s$ that are not shared.
+Leave the shared pieces matched identically.  The masses of
+$F_s\setminus F_j$ and $F_j\setminus F_s$ are equal.  Each piece in the
+former set is at distance at most $R_s+D$ from $j$.  Each piece in the
+latter set is at distance at least $R_s-D$ from $j$, because the
+nearest-unit assignment at $s$ uses all closer mass before stopping at
+radius $R_s$.  This argument remains valid at ties.  Comparing the two
+residual costs shows that using the old assignment $F_s$ for client $j$
+has cost at most
+\begin{equation}\label{eq:prep-far-exchange}
+ C_j(F_s):=\sum_{i\in F_s}y_i d(j,i)
+ \le\frac{1+\alpha}{1-\alpha}V_j.
+\end{equation}
+
+Serve $j$ with the feasible assignment just constructed for $s$.
+For each $i\in P_s$,
+\[
+ \E\widetilde y_i
+ =y_i\E[p'_s]/p_s\le y_i/p_s.
+\]
+For an ordinary star, the expected retained activation mass is at most
+its original weight.  Thus outside suppliers do not increase their
+expected cost charge.  It remains to compare arbitrary new supply
+inside a bundle $P_t$ with the old pieces in $A_s\cap P_t$.
+This case only arises when $c_s>0$.  Then $D\le\beta L_s$, and for any
+old supplier piece $i\in P_t$,
+\[
+ d(j,i)\ge d(s,t)-D-\delta L_t
+ \ge(1-\beta-\delta)d(s,t).
+\]
+Any two pieces of $P_t$ are at distance at most $2\delta L_t$.
+Because $L_t\le d(s,t)$ and $1-\beta-\delta>0$, every possible new
+supplier piece has distance to $j$ at most
+\[
+ \kappa\,d(j,i),\qquad
+ \kappa:=1+\frac{2\delta}{1-\beta-\delta}.
+\]
+The comparison holds for every old piece, so it also holds relative to
+the weighted mean distance of the old pieces aggregated by that star.
+Multiplying by its expected retained activation mass bounds its
+expected cost by $\kappa$ times its old cost contribution.
+
+Finally, for the extra star at $s$, its supplier $P_t$ satisfies
+$d(s,t)=L_s$ and $L_t\le L_s$.  Every supply position is therefore at
+distance at most $(\beta+1+\delta)L_s$ from $j$.  By
+Equation~\ref{eq:prep-deficit-small}, its expected cost is at most
+\[
+ (\beta+1+\delta)\gamma c_s L_s
+ \le\frac{(\beta+1+\delta)\gamma}{\delta}V_j.
+\]
+There is no extra star when $c_s=0$.
+Combining these bounds with Equation~\ref{eq:prep-far-exchange}, the
+unconditional expected nearest-unit cost in this case is at most
+$B_{\mathrm{far}}V_j$, where
+\begin{equation}\label{eq:prep-far-factor}
+ B_{\mathrm{far}}
+ :=\max\{p_0^{-1},\kappa\}\frac{1+\alpha}{1-\alpha}
+   +\frac{(\beta+1+\delta)\gamma}{\delta}.
+\end{equation}
+
+\paragraph{Cost comparison at a nearby representative scale.}
+Now suppose $K_s\le T_1V_j$.  Consider the mass on the old set $F_j$,
+using nominal rounded masses on $U$ and actual rounded masses in the
+bundles.  This mass is available in the actual opening vector.  Its
+unconditional expected cost is at most $V_j/p_0$: nominal outside
+openings have mean $y_i$, and a bundle piece has expected mass at most
+$y_i/p_s\le y_i/p_0$.
+
+We bound the expected shortfall of this available mass from one.
+Write $M_U$ for its nominal mass on $U\cap F_j$.  The first and pair
+moments of the nominal counts imply
+\begin{equation}\label{eq:prep-outside-noise}
+ \E M_U=y(U\cap F_j),\qquad
+ \Var(M_U)\le g\,y(U\cap F_j)\le g.
+\end{equation}
+For a fixed complete star outcome, put
+\[
+ a_t=y(F_j\cap P_t),\qquad
+ \overline M_P=\sum_t a_t p'_t/p_t,
+\]
+and write $M_P$ for the actual rounded mass on these bundle pieces.
+Conditional on the star outcome, bundle rounding has nonpositive pair
+covariances within each bundle and is independent between bundles.
+The variance of a single grid-rounded mass of conditional mean
+$\overline y_i$ is at most $g\overline y_i$.  Therefore
+\begin{equation}\label{eq:prep-conditional-noise}
+ \E[M_P\mid\text{star outcome}]=\overline M_P,\qquad
+ \Var(M_P\mid\text{star outcome})
+ \le g\overline M_P\le g/p_0.
+\end{equation}
+
+Clipping at $1/2$ cannot increase distance from $c_t\le1/2$, so
+Equation~\ref{eq:prep-deficit-moments} gives
+\begin{align}
+ \E|p'_t-p_t|
+ &=\E|c_t-Z_t^*|\notag\\
+ &\le\E|c_t-Z_t|
+ \le\gamma c_t+\sqrt{g(1+\gamma)c_t}.
+ \label{eq:prep-rescaling-noise}
+\end{align}
+Since $\sum_t a_t\le1$, it follows that
+\[
+ \E\left|\overline M_P-\sum_t a_t\right|
+ \le\sum_t\frac{a_t}{p_t}\E|p'_t-p_t|
+ \le\frac{\gamma+\sqrt{g(1+\gamma)}}{p_0}.
+\]
+Use the identity $y(U\cap F_j)+\sum_t a_t=1$, the triangle inequality,
+and the second-moment bounds in
+Equations~\ref{eq:prep-outside-noise}--\ref{eq:prep-conditional-noise}.
+The expected shortfall is at most
+\begin{equation}\label{eq:prep-shortfall}
+ \E(1-M_U-M_P)_+\le H,
+ \qquad
+ H:=\sqrt g+\sqrt{g/p_0}
+       +\frac{\gamma+\sqrt{g(1+\gamma)}}{p_0}.
+\end{equation}
+In particular no independence between outside counts and rescaled
+bundle masses is being assumed.
+
+Trim any surplus mass on $F_j$, which cannot increase its cost.  To
+fill a remaining shortfall, there is always one unit of available
+opening within distance
+\[
+ QV_j,\qquad Q:=M+(1+\delta)T_1,
+\]
+from $j$.  Indeed, if $c_s=0$, there are no activations at $s$ and $P_s$
+has exactly one unit, all at distance at most
+$D+R_s\le(M+T_1)V_j$.  If $c_s>0$, use $P_s$ and the bundle $P_t$ of a
+nearest other representative.  They are disjoint and each has at
+least half a unit.  Their supply lies within distance
+$D+(1+\delta)L_s\le QV_j$.
+This supply can fill the shortfall even if some of it is already used:
+if the current assignment has mass $1-q$, a ball containing at least
+one opening unit contains at least $q$ unused units.  Allocate those
+units to finish the assignment.  By Equation~\ref{eq:prep-shortfall},
+the unconditional expected nearest-unit cost is at most
+$B_{\mathrm{near}}V_j$, where
+\begin{equation}\label{eq:prep-near-factor}
+ B_{\mathrm{near}}:=p_0^{-1}+QH.
+\end{equation}
+
+\paragraph{Choosing parameters and enforcing the budget.}
+Equations~\ref{eq:prep-far-factor} and~\ref{eq:prep-near-factor} give
+explicit bounds that approach one in the stated order of parameter
+choices.  More precisely, first choose $\delta$ small to make the
+limiting value of $\kappa$ close to one.  Next choose $M$ large so that
+$p_0^{-1}$ is close to one, and then choose $T_1$ large so that $\alpha$
+and $\beta$ are small.  With these constants fixed, choose $\gamma$
+small enough to control both the last term of $B_{\mathrm{far}}$ and
+the term $Q\gamma/p_0$ in $B_{\mathrm{near}}$.  Finally choose $g$ small
+enough to control all square-root terms and $\rho$ in
+Equation~\ref{eq:prep-budget-probability}.  All constants may be
+rational, and all sufficiently small $g=1/\Delta$ with even $\Delta$
+satisfy the resulting strict bounds.  In particular, arrange
+\[
+ \rho<\tfrac12,\qquad
+ \frac{\max\{B_{\mathrm{far}},B_{\mathrm{near}}\}}{1-\rho}
+ \le1+\eta.
+\]
+
+Reject a star-rounding outcome outside $\mathcal G$ and repeat.
+For the purpose of the preceding unconditional analysis, the subsequent
+bundle rounding can be imagined on every outcome.  Let
+$\widetilde V_j$ denote the resulting nearest-unit cost.  Its
+nonnegativity, rather than any independence from $\mathcal G$, gives
+\begin{equation}\label{eq:prep-cost-final}
+ \E[\widetilde V_j\mid\mathcal G]
+ \le\frac{\E\widetilde V_j}{\Prob(\mathcal G)}
+ \le(1+\eta)V_j.
+\end{equation}
+The already established supply and opening bounds hold on every
+accepted outcome.
+
+The split representation, star list, and number of pair updates are
+polynomial in the input size.  Pair updates and subset choices use
+rational probabilities and rational linear operations.  Their
+encodings have polynomial bit length: each weighted update introduces
+only ratios of integer weights bounded by $|\mathcal S|$, and there
+are polynomially many updates; the subsequent rescalings divide by
+the rational masses $p_s$.  Exact rational sampling has expected
+polynomial running time.  The acceptance probability is at least
+one half, so rejection multiplies the expected running time by at
+most two.  This proves the theorem.
+\end{proof}

preprints/The-Approximation-Threshold-for-Metric-k-Median-September-24-2026/build/sections/rounding.tex

@@ -0,0 +1,871 @@
+\section{Tapered iterative rounding}\label{sec:rounding}
+
+This section rounds a sufficiently fine grid solution.  The graph used for
+rounding gives every active copy one unit of possible incoming erasers.
+Its distance-dependent taper makes the opening that accompanies an erasure
+comparable to the client's surviving candidates.  A potential that retains
+some cheap candidates after an opening converts this comparison into an
+application of Theorem~\ref{thm:finite-inequality}.
+
+\begin{theorem}[Rounding a grid solution]\label{thm:grid-rounding}
+For every fixed $\tau>0$ and all sufficiently large constant integers
+$\Delta$, put $g=1/\Delta$.  Suppose a finite rational metric instance has openings
+$\widetilde y_i\in[0,1]\cap g\mathbb Z$ of total mass $M\ge1$.
+For a client $j$, let $V_{j,0}$ be the cost of its nearest unit of this mass.
+There is a constant $C=C(\tau,\Delta)$ and a randomized algorithm of
+expected polynomial running time that outputs a set $O$ of facilities with
+\[
+ |O|\le M+C,
+ \qquad
+ \E d(j,O)\le(1+2/e+\tau)V_{j,0}\quad\text{for every client }j.
+\]
+All parameters and the additive constant are independent of the instance.
+In particular, an initial mass at most $k+c_0$, for a fixed constant
+$c_0$, gives at most $k+c_0+C$ facilities.
+The deterministic implementation of the resulting total-cost guarantee
+is given in Section~\ref{sec:derandomization}.
+\end{theorem}
+
+We prove the theorem using the functions $A,P,B,m$ defined in the preceding
+section.  Thus $B(x)=2P(x)=A'(x)$ and
+$m(x)=A(x)-1-xB(x)$.  We use, in particular,
+\begin{equation}\label{eq:rnd-function-identities}
+ \begin{gathered}
+ 0<B(x)\le1,\qquad B(x)\ge2P(1)>0,\qquad m(x)\ge0,\\
+ (1+x)A(x)+xB(x)-1=3x\qquad(0\le x\le1).
+ \end{gathered}
+\end{equation}
+The identities and derivative bounds used below follow directly from
+the defining integrals.  Constants implicit in $O_\Delta(\cdot)$ may
+depend on $\Delta$, but not on the instance, the round, or the subsequent
+heavy-source threshold.
+
+\subsection{The graph and the physical process}
+
+The source-opening and neighbor-erasure operation follows the framework
+of Byrka et al.~\cite[Definitions~3.2 and~5.1]{ByrkaEtAl2026}.
+We specify the graph, taper, and batches here because their precise
+properties will be used in the sub-two drift analysis.
+
+Represent each opening by active copies of mass $g$, grouped by facility.
+Each group has at most $\Delta$ active copies.  Co-located facilities
+introduced by earlier splitting may remain distinct groups throughout
+the process; they can be merged in the output. A group always means one
+such labeled facility group, rather than an entire zero-distance class.
+An integral group has exactly $\Delta$ copies.
+
+Fix a small time step $\ell>0$ and a large time horizon $\Lambda$.
+There are $T=\lceil\Lambda/\ell\rceil$ rounds.  At the start of round
+$r\in\{0,\ldots,T-1\}$ let $\xi=r\ell$ and $x=e^{-\xi}$.
+For that round choose a fixed rational number $\widehat P$ satisfying
+\begin{equation}\label{eq:rnd-taper-approximation}
+ 0<\widehat P\le\tfrac12,
+ \qquad |\widehat P-P(x)|\le\zeta,
+ \qquad 0<\zeta<\tfrac14.
+\end{equation}
+The finitely many rational approximations may be fixed in advance.
+The analytic value $x$ is used only in the analysis.
+
+For each active target copy $i$, independently of the other targets
+conditional on the current state, choose a set $\mathcal H_i$ of exactly
+$\Delta$ active source copies.  If the colocated active mass is below one,
+choose $N_i>0$ minimally so that
+\begin{equation}\label{eq:rnd-edge-marginals}
+ q_{hi}=\min\left\{1,\frac{(1-d(i,h)/N_i)_+}{\widehat P}\right\},
+ \qquad \sum_h gq_{hi}=1,
+\end{equation}
+and give $\mathcal H_i$ these marginals.  Such a radius exists: the sum
+is continuous and nondecreasing in the positive radius, starts below
+one, and reaches the total active mass at a finite radius.  This last
+observation also handles total active mass exactly one.  The equation
+is piecewise linear in $1/N_i$.  Fixed-sum rounding realizes the
+marginals since their unscaled sum is the integer $\Delta$.
+
+If the colocated active mass is already at least one, set $N_i=0$ and
+choose $\Delta$ colocated copies, including every active copy in the
+target's own group.  Any fixed such choice suffices in this case; denote
+its marginals by $q_{hi}$ as well.  In either case every own-group copy
+belongs to $\mathcal H_i$ with certainty, and
+\begin{equation}\label{eq:rnd-taper-comparison}
+ \begin{aligned}
+  N_i(1-\widehat Pq_{hi})&\le d(i,h)&&\text{if }q_{hi}<1,\\
+  d(i,h)&\le N_i(1-\widehat Pq_{hi})&&\text{if }q_{hi}>0.
+ \end{aligned}
+\end{equation}
+For $N_i>0$ these statements follow from the three cases of the clipped
+linear formula; for $N_i=0$ all sources of positive marginal are
+colocated.  Define the out-set of source $h$ by
+\[
+ E_h=\{i:h\in\mathcal H_i\}.
+\]
+A full update at $h$ erases $E_h$ and opens the facility of $h$.
+
+We use balanced and biased unbalanced batches, with forced openings for
+large updates, as in the approach of Byrka et
+al.~\cite[Algorithms~4--5 and Section~5.4]{ByrkaEtAl2026}.
+The exact batch and chunk rules below are part of our construction.
+To implement many updates while controlling the total opening mass, put
+\[
+ u_h=1-g|E_h|,
+ \qquad
+ \mathcal Z=\{h:u_h=0\},\quad
+ \mathcal P=\{h:u_h>0\},\quad
+ \mathcal N=\{h:u_h<0\}.
+\]
+Every target has $\Delta$ incoming sources, so $\sum_hu_h=0$.  Write
+\begin{equation}\label{eq:rnd-balance-size}
+ A_* =\sum_{h\in\mathcal P}u_h
+     =\sum_{h\in\mathcal N}|u_h|,
+ \qquad
+ |\mathcal P\cup\mathcal N|\le 2A_*/g.
+\end{equation}
+The last inequality uses that each nonzero $u_h$ is a nonzero multiple
+of $g$.  Choose a further small constant $b_*>0$, to be fixed last, and put
+\[
+ B_* =\max\{1,b_*A_*\}.
+\]
+A source is \emph{heavy} if $g|E_h|>B_*$.  Force open the facilities of
+all heavy sources, whether or not any random update is triggered.
+
+All random choices and erasure sets in a round refer to its pre-round
+graph and active copies.  Set $p_*=2\ell g$, and fix a small bias
+$0<\phi\le1$.  Flip an independent fair coin to choose one of the following
+two types of round.
+\begin{enumerate}
+\item In a balanced round, propose each $h\in\mathcal Z$ independently
+with probability $p_*$.  Select a proposed source only if no other
+proposed source has an out-set intersecting its out-set.
+\item In an unbalanced round, select sources of $\mathcal P$
+independently with probability $p_*$, and nonheavy sources of
+$\mathcal N$ independently with probability $p_*(1+\phi)$.
+For each heavy source, partition its out-set into chunks of mass at
+most $B_*+1$ and trigger each chunk independently with probability
+$p_*(1+\phi)$.  A triggered chunk erases just that chunk; its source
+facility is already included among the forced openings.
+\end{enumerate}
+All probabilities are at most one for the parameter choices below.
+Take the union of the selected full out-sets and triggered chunks and
+erase it.  Then reset each forced or selected facility to $\Delta$
+active copies.  Copies removed by either erasure or reset are distinct
+from the regenerated copies.
+
+Two invariants are useful.  First, an integral target group's own
+$\Delta$ copies exhaust each of its incoming sets.  Therefore an
+erasure of an integral group is caused by one of its own sources; that
+source either is selected or is heavy.  The group is reset in either
+case.  Thus integral facilities persist, including when other facilities
+are colocated with them.  Second, every erasure has an associated
+selected or forced opening.  Hence total active mass always stays at
+least one.  The construction of the next round's graph is consequently
+well defined.
+
+\subsection{The opening-count bound}
+
+We first bound the number of facilities independently of connection cost.
+The next lemma controls the total active mass with high probability;
+completion will later turn that mass bound into a facility-count bound.
+
+\begin{lemma}[Constant additive count]\label{lem:rnd-count}
+Fix $g,\phi,\ell,T$ as above, with $\ell$ sufficiently small as a
+function of $\Delta$ and $\phi$.  For every fixed $\eta>0$, one can
+choose $b_*>0$ and a constant $C$ such that the final active mass is at
+most $M+C$ with probability at least $1-\eta$.  The bound holds
+uniformly over all starting states of total mass $M$.
+\end{lemma}
+
+\begin{proof}
+Condition on the entire pre-round state and its realized graph.
+Write $A=A_*$, $B=B_*$, $p=p_*$, and $m_h=g|E_h|$ for this proof.
+If $n$ is the number of active copies, then $\sum_hm_h=n$.
+Each source in the zero class has $m_h=1$, so removing that class gives
+\[
+ \sum_{h\in\mathcal P\cup\mathcal N}m_h
+ =|\mathcal P\cup\mathcal N|.
+\]
+Every heavy source lies in $\mathcal N$ because $B\ge1$.  If $H$ is
+their number, then
+\begin{equation}\label{eq:rnd-heavy-count}
+ H\le\frac{|\mathcal P\cup\mathcal N|}{B}
+ \le\frac{2A}{g\max\{1,b_*A\}}
+ \le\frac{2}{gb_*}.
+\end{equation}
+Thus forced resets add at most a constant mass.  If $A=0$, there are
+no heavy sources and the same conclusions hold directly.
+
+In a balanced round, the selected erasure sets are disjoint and each
+has mass one.  Opening one facility for each selected source adds at
+most one unit per set, so there is no mass increase except the forced
+allowance.
+
+For an unbalanced round index the ordinary selections and heavy
+chunks by actions $e$.  Let $I_e$ be the trigger indicator, $S_e$ its
+erased set, $m_e=g|S_e|$, and $a_e=1$ for an ordinary selection or $0$
+for a chunk.  Resetting facilities adds at most one unit per ordinary
+selection in addition to the forced allowance. More explicitly, the
+change in active mass is at most
+\[
+ H+\sum_e a_e I_e
+   -g\left|\bigcup_{e:I_e=1}S_e\right|.
+\]
+The union is a set of pre-round copy identities. A reset may remove
+additional old copies outside this union, which only decreases the
+change. Counting a selected or forced group more than once likewise
+only enlarges the upper bound. Truncated inclusion--exclusion bounds
+the erased union from below. Consequently the mass increase, apart
+from forcing, is at most
+\begin{equation}\label{eq:rnd-count-statistic}
+ Z_* =\sum_e I_e(a_e-m_e)
+       +\sum_{e<f}g|S_e\cap S_f|I_eI_f.
+\end{equation}
+This remains an upper bound when several selected copies have the
+same facility or when a selected facility was also forced.
+
+Chunking preserves total erasure mass.  There are
+$|\mathcal P\cup\mathcal N|-H$ ordinary actions, and hence
+\begin{equation}\label{eq:rnd-action-total}
+ \sum_e(m_e+a_e)
+ =2|\mathcal P\cup\mathcal N|-H\le4A/g.
+\end{equation}
+The expectation of the linear part of \eqref{eq:rnd-count-statistic}
+is exactly
+\[
+ pA-p(1+\phi)A-p(1+\phi)H\le-p\phi A.
+\]
+The last term pays for the absence of a new opening in a chunk action.
+Each target belongs to at most $\Delta$ action sets, even after
+chunking.  Therefore
+\[
+ \sum_{e<f}g|S_e\cap S_f|
+ \le\frac{\Delta-1}{2}\sum_em_e.
+\]
+Independence of the indicators bounds the overlap expectation by
+$O(p^2\Delta A/g)$.  Since $p=2\ell g$, taking $\ell$ small enough
+relative to $\phi$ and $\Delta$ gives
+\begin{equation}\label{eq:rnd-count-mean}
+ \E Z_*\le-p\phi A/2.
+\end{equation}
+
+Flipping coordinate $I_e$ changes $Z_*$ in absolute value by at most
+$a_e+\Delta m_e$.  The Efron--Stein independent-coordinate variance inequality
+\cite[Section~2.2, Proposition~1]{BBLM2005}, the bound
+$a_e+m_e\le B+2$, and \eqref{eq:rnd-action-total} give
+\begin{equation}\label{eq:rnd-count-variance}
+ \Var Z_*
+ \le O(p\Delta^2)\sum_e(a_e+m_e)^2
+ \le O(p\Delta^2)(B+2)A/g.
+\end{equation}
+If $A>1/b_*$, then $B=b_*A>1$.  Combining
+\eqref{eq:rnd-count-mean} and \eqref{eq:rnd-count-variance} yields
+\begin{equation}\label{eq:rnd-count-failure}
+ \Prob(Z_*>0)
+ \le O\!\left(\frac{\Delta^2b_*}{p\phi^2g}\right).
+\end{equation}
+If instead $A\le1/b_*$, there are at most $2/(gb_*)$ ordinary
+sources in the nonzero classes.  The actual positive mass change is
+then bounded deterministically by this constant plus the forced
+allowance; a deterministic bound on the quadratic statistic is
+unnecessary.
+
+Choose $b_*$ sufficiently small that the union bound for
+\eqref{eq:rnd-count-failure} over the $T$ rounds is at most $\eta$.
+On the complementary event the total increase is at most, for example,
+$4T/(gb_*)$.  This proves the lemma with a constant $C$.
+The proof was conditional on every state and graph, so the same union
+bound applies to the adaptive sequence of rounds.
+\end{proof}
+
+For later use, the mean and variance of $Z_*$ depend only on indicator
+moments through degree four, since $Z_*$ is a quadratic polynomial.
+The probability bounds for balanced selections below need only
+first and second moments of their proposal indicators.  These are
+the moment requirements used in Section~\ref{sec:derandomization}.
+
+\subsection{A persistent backup and final completion}
+
+We next establish a persistent bound on each client's eventual
+connection distance, including trajectories on which none of its assigned
+copies opens. This is the backup-and-completion role in iterative
+rounding~\cite[Section~3.3 and Lemmas~6.3 and~6.5]{ByrkaEtAl2026};
+the following argument proves the bound for our tapered graph and reset
+rule.
+
+Fix a client for the remainder of the analysis.  Let $S_0$ be the
+$\Delta$ copies in its initial nearest unit, chosen using any fixed
+tie-breaking rule.  Write $d_i$ for its distance to copy $i$,
+\[
+ V_0=\sum_{i\in S_0}gd_i,
+ \qquad R=\max_{i\in S_0}d_i.
+\]
+In particular $R\le\Delta V_0$.
+
+\begin{lemma}[Backup and completion]\label{lem:rnd-completion}
+Once a copy of $S_0$ is physically removed, an integral facility within
+distance $5R$ of the client exists and persists to the end.
+At the end of the process one can retain all integral facilities and
+open further facilities, using no more than the total final active
+mass, so that every client is connected within its own distance $5R$.
+\end{lemma}
+
+\begin{proof}
+Consider the round of the first physical removal from $S_0$.  Just
+before that round all of $S_0$ is still active.  For $i,t\in S_0$,
+$d(i,t)\le d_i+d_t\le2R$.  At radius $2R/(1-\widehat P)$, the
+$\Delta$ copies of $S_0$ all have incoming marginal one for target
+$i$.  Minimality of the taper radius gives
+\[
+ N_i\le2R/(1-\widehat P)\le4R.
+\]
+If $R=0$, the zero-radius rule gives the same conclusion directly.
+An erasing source $h$ has $d(i,h)\le N_i$ by
+\eqref{eq:rnd-taper-comparison}, and hence $d_h\le5R$.
+Its facility is either selected or forced open.  A removal by reset
+of $i$'s own group instead opens a facility within $R$.
+The persistence invariant proves the first assertion.
+
+At the end, consider the clients not already served within $5R$ by
+an integral facility.  The first assertion implies that every such
+client still has all its original $S_0$ copies.  Its radius-$R$
+neighborhood of active copies therefore has mass at least one.
+That neighborhood contains no integral group, since such a group
+would itself provide the required connection.
+
+Process these neighborhoods in nondecreasing order of their radii.
+Accept a neighborhood if it is disjoint from all previously accepted
+ones and open any facility in each accepted neighborhood.  These
+disjoint neighborhoods each contain at least one unit of mass and
+contain no integral group.  Their openings, together with the retained
+integral groups, therefore fit within the final active mass.
+If a radius-$R$ neighborhood was skipped, it meets an accepted
+radius-$R'$ neighborhood with $R'\le R$.  Through an intersection
+point, the distance to the accepted neighborhood's chosen facility
+is at most $R+2R'\le3R$.  This proves the asserted $5R$ bound for
+every client, also when some radii are zero.
+\end{proof}
+
+At any intermediate time define $b$ to be the distance from the fixed
+client to its nearest integral facility, capped at $5R$; when no
+integral facility exists use $b=5R$.  By persistence and
+Lemma~\ref{lem:rnd-completion}, the eventual completed solution has
+connection distance at most $b$ at every time on the trajectory.
+The cap need not already be attained by an open facility.  It is a
+valid bound on the eventual connection because completion supplies
+$5R$ on every trajectory.  The quantity $b$ never increases.
+
+\subsection{The potential and its single-action bounds}
+
+For analysis only, track an alive set $S\subseteq S_0$ of original
+copies.  A physically removed copy leaves $S$ permanently, even when
+its facility is reset with regenerated copies.  We also permit the
+additional discards specified below.  Put
+\[
+ y=g|S|,\qquad V=\sum_{i\in S}gd_i.
+\]
+At elapsed time $\xi$, with $x=e^{-\xi}$, define
+\begin{equation}\label{eq:rnd-potential}
+ f=aV+(1-y)b,
+ \qquad
+ a=A(x)+B(x)(y-x)=1+m(x)+B(x)y.
+\end{equation}
+The coefficient $a$ is the value at $y$ of the tangent line to the
+concave function $A$ at $x$; $x$ is a deterministic clock, while $y$
+is the realized alive mass. Its affine dependence on $y$ makes the
+erasure correction quadratic in removed mass and cost, so the graph
+average of the ideal bound below requires only first and second edge
+moments.
+In particular $a\ge1$.  The initial potential is
+\begin{equation}\label{eq:rnd-initial-potential}
+ f_0=A(1)V_0=(1+2/e)V_0.
+\end{equation}
+If $C_j$ denotes the actual connection distance in the completed
+output, the preceding cap property gives
+\begin{equation}\label{eq:rnd-terminal-correction}
+ C_j-f_T\le y_Tb_T-a_TV_T\le5Ry_T.
+\end{equation}
+The potential and every single-action bound below are $O(R)$ uniformly
+over time: $V\le R$, $0\le y\le1$, $b\le5R$, and $a$ is bounded.
+
+We first freeze time and the graph.  Write $B=B(x)$ and
+$b_h=\min\{b,d_h\}$.  For a source $h\notin S$, let
+\[
+ \delta_h=g|E_h\cap S|,
+ \qquad Y_h=\sum_{i\in E_h\cap S}gd_i.
+\]
+If only this full source update affects $S$, its removal changes the
+coefficient to $a-B\delta_h$ and makes the new cap at most $b_h$.
+Its potential change is consequently at most
+\begin{align*}
+ &(a-B\delta_h)(V-Y_h)+(1-y+\delta_h)b_h-f\\
+ &\quad=\delta_hb_h-aY_h-B\delta_h(V-Y_h)
+          +(1-y)(b_h-b)\\
+ &\quad\le\delta_hb_h-aY_h-B\delta_h(V-Y_h).
+\end{align*}
+The same statement holds when $h$ is not alive but its group contains
+alive copies: all those copies belong to $E_h$ by the own-group edges,
+so its reset creates no unaccounted removal.
+
+If an alive source $h\in S$ opens, retain among the physical survivors
+only copies satisfying $ad_i<d_h$.  If their mass and cost are $y',V'$,
+their new coefficient is at most $a$ and their new cap is at most
+$d_h$.  Therefore
+\[
+ f'\le aV'+(1-y')d_h
+ =d_h-\sum_{\substack{i\in S\\h\notin\mathcal H_i}}
+            g(d_h-ad_i)_+.
+\]
+Copies satisfying equality can be discarded without changing this
+bound. Figure~\ref{fig:survivors} separates physical erasure from these
+additional analysis discards. This continuation after an opening
+supplies the subtraction that will enter
+Theorem~\ref{thm:finite-inequality}.
+
+\input{figures/survivors}
+
+The ideal first-order bound, assigning rate $g$ to each full source
+update, is thus
+\begin{equation}\label{eq:rnd-ideal-generator}
+\begin{aligned}
+ \mathcal B={}&
+ \sum_{h\in S}g\left[
+ d_h-\sum_{\substack{i\in S\\h\notin\mathcal H_i}}
+                  g(d_h-ad_i)_+-f\right]\\
+ &+\sum_{h\notin S}g\left[
+ \delta_hb_h-aY_h-B\delta_h(V-Y_h)\right].
+\end{aligned}
+\end{equation}
+At a fixed alive set and cap, the time derivative is
+\begin{equation}\label{eq:rnd-time-derivative}
+ \dot f=-x(y-x)B'(x)V.
+\end{equation}
+Indeed the derivative of $A(x)+B(x)(y-x)$ with respect to $x$ is
+$B'(x)(y-x)$.
+
+\subsection{From the physical batch to the ideal bound}
+
+For the actual analysis update, first account for forced openings by
+removing from $S$ all copies whose groups are forced and updating the
+cap.  Then apply the round's erasures and resets.  If a source that
+was alive after force removals but before these erasures was selected,
+choose one such source $h$ and discard every remaining copy with
+$ad_i\ge d_h$, using the pre-round value of $a$.  This is purely an
+analysis rule and need not be computed by the algorithm.
+
+\begin{lemma}[Conditional batch drift]\label{lem:rnd-batch}
+For every pre-round history and every realized graph, the physical
+round, the above alive-set update, and the time advance satisfy
+\begin{equation}\label{eq:rnd-batch-drift}
+ \E[f_{r+1}-f_r\mid\text{history, graph}]
+ \le\ell(\dot f+\mathcal B)
+       +O_\Delta\bigl((\phi+\ell)\ell R\bigr).
+\end{equation}
+Here $\mathcal B$ is computed from that graph and the original
+pre-round alive set.  The constants and the required upper bound on
+$\ell$ are uniform in $b_*$.
+\end{lemma}
+
+\begin{proof}
+First suppose no group containing an alive copy is forced.  Call an
+action relevant if its ordinary source is in $S$, or its ordinary
+out-set or heavy chunk meets $S$.  There are at most
+$\Delta^2+\Delta$ relevant actions: each of at most $\Delta$ alive
+targets has $\Delta$ incoming sources.  Own-group edges ensure that
+this list also includes every ordinary reset that can affect $S$.
+
+A balanced source has an out-set of exactly $\Delta$ copies.  Every
+one of those copies belongs to at most $\Delta$ source out-sets, so
+the number $c_h$ of potentially conflicting balanced proposals is at
+most $\Delta^2$.  Including the type coin, its selection marginal is
+\[
+ \tfrac12p_*(1-p_*)^{c_h}
+ =\ell g\bigl(1+O_\Delta(\ell)\bigr).
+\]
+An unbalanced ordinary or chunk action has marginal $\ell g$ or
+$(1+\phi)\ell g$.  Any two distinct actions have joint probability
+$O(\ell^2g^2)$; actions of opposite types have joint probability zero.
+These estimates do not assert independence of balanced selections
+after conflict filtering.
+
+If no relevant action occurs, only the cap can improve, so the
+frozen-time potential cannot increase.  If exactly one relevant
+ordinary action occurs, its change is bounded by the appropriate
+single-source expression in \eqref{eq:rnd-ideal-generator}.
+Unrelated selections or forced openings can only lower its cap and
+do not invalidate that upper bound.  The probability of two or more
+relevant actions is $O_\Delta(\ell^2)$, and all possible potentials
+and individual bounds have magnitude $O(R)$.  Replacing isolated-action
+probabilities by marginals and their marginals by $\ell g$ therefore
+costs $O_\Delta((\phi+\ell)\ell R)$.
+
+For a heavy source outside $S$, its forced facility makes the cap at
+most $b_h$ even when only one of its chunks triggers.  Force accounting
+can be done before erasure accounting because no forced group in the
+present case meets $S$; it only improves the cap.  Write
+$\delta_c,Y_c$ for the alive mass and cost removed by chunk $c$ of
+this source, and $\delta=\sum_c\delta_c$, $Y=\sum_cY_c$.  Its summed
+single-chunk bounds satisfy
+\begin{align*}
+ \sum_c[\delta_cb_h-aY_c-B\delta_c(V-Y_c)]
+ &=\delta b_h-aY-B\delta V+B\sum_c\delta_cY_c\\
+ &\le\delta b_h-aY-B\delta(V-Y).
+\end{align*}
+All cross terms omitted in the last step are nonnegative.  Consequently
+the chunk bounds are dominated by the full-source term used in
+$\mathcal B$.
+
+The functions in the potential have uniformly bounded derivatives.
+The chance that a relevant action changes $S$ is $O_\Delta(\ell)$;
+changes of the cap alone do not affect \eqref{eq:rnd-time-derivative}.
+Advancing time thus adds at most
+$\ell\dot f+O_\Delta(\ell^2R)$.  This proves the lemma when no alive
+group is forced.
+
+Now suppose that forcing does remove alive copies.  Let their total
+mass and cost be $\delta\ge g$ and $Y$, and let $b'\le b$ be the
+cap after force accounting.  Every removed copy is in an opened
+facility, so $Y\ge\delta b'$.  At frozen time, the resulting decrease
+is at least
+\begin{align}
+ f-f'
+ &\ge(a-1)Y+B\delta(V-Y)+(1-y)(b-b')\notag\\
+ &\ge BgV+(1-y)(b-b').\label{eq:rnd-force-savings}
+\end{align}
+For the second line, use $a-1=m+By$, $m\ge0$, and
+$y\ge\delta\ge g$.  At least one removed alive copy has distance
+at most $V/g$, so $b'\le V/g$.  With
+\[
+ Q=V+(1-y)b,\qquad D=(1-y)(b-b'),
+\]
+we obtain
+\[
+ \begin{gathered}
+ Q\le(1+1/g)V+D,\qquad BgV+D\ge c_\Delta Q,\\
+ c_\Delta=\min\left\{1,\frac{2P(1)g}{1+1/g}\right\}>0.
+ \end{gathered}
+\]
+Thus forcing supplies a fixed fraction of $Q$ as free savings.
+
+After force accounting, the remaining alive cost is at most $V$ and
+the cap is at most $V/g$.  Any further frozen-time increase has
+magnitude $O_\Delta(V)$, and a change to the alive set still has
+probability $O_\Delta(\ell)$.  If the alive set does not change, the
+cap only improves.  Subsequent actions and time advance therefore
+have expected increase at most $O_\Delta(\ell Q)$.
+
+For comparison with the pre-force ideal expression, we also have
+\begin{equation}\label{eq:rnd-force-generator-size}
+ |\dot f|+|\mathcal B|\le O_\Delta(Q).
+\end{equation}
+For inside sources, $d_h\le V/g$.  If $y<1$, the grid gives
+$1-y\ge g$, so an external cap charge is bounded using $b\le Q/g$.
+If $y=1$, the alive set itself is a full unit; its maximum distance
+is at most $V/g$.  The radius argument from
+Lemma~\ref{lem:rnd-completion} then bounds any source erasing an alive
+copy within $5V/g$ of the client.  Only $O_\Delta(1)$ sources have
+nonzero contributions.  These observations prove
+\eqref{eq:rnd-force-generator-size}.
+
+Choose $\ell$ sufficiently small in terms of these constants and
+$c_\Delta$.  The savings in \eqref{eq:rnd-force-savings} absorb both
+the subsequent expected increase and the possible negative quantity
+$\ell(\dot f+\mathcal B)$.  This proves
+\eqref{eq:rnd-batch-drift} also in the forced case, without an
+additional error term.  If $Q=0$, all the quantities in this
+comparison vanish, so no division by $Q$ is needed.  More explicitly,
+if $V=0$ then $b'=0$, and the force savings remove $(1-y)b$; when
+also $y=1$, relevant source distances and all generator terms are
+zero.  The argument is uniform in the later choice of $b_*$.
+\end{proof}
+
+The order of conditioning in Lemma~\ref{lem:rnd-batch} is essential.
+Whether a group is forced depends on the graph.  We have proved the
+drift comparison for \emph{every} realized graph, in both force cases;
+we will now average that comparison over the original graph law,
+without conditioning the law on a forcing event.
+
+\subsection{Averaging the graph and applying the finite inequality}
+
+The physical batch has now been compared with the ideal generator for
+every realized graph. We average that generator using edge marginals and
+cross-target second moments, then verify the hypotheses of the abstract
+finite comparison inequality. The resulting bound is the cost estimate
+that will be summed over rounds.
+
+\begin{lemma}[Averaged ideal drift]\label{lem:rnd-average}
+Conditional on the pre-round history, the ideal expression satisfies
+\begin{equation}\label{eq:rnd-average-drift}
+ \dot f+\E_{\mathrm{graph}}\mathcal B
+ \le BgV+12\zeta V.
+\end{equation}
+\end{lemma}
+
+\begin{proof}
+The pre-round quantities $S,y,V,b,x$ are fixed throughout this proof.
+Define
+\begin{equation}\label{eq:rnd-graph-statistics}
+ \begin{aligned}
+ w_i&=\sum_{t\in S}gq_{ti},\\
+ K_{it}&=\sum_{h\notin S}gq_{hi}q_{ht},\\
+ s_i&=\frac{\sum_{h\notin S}gq_{hi}^2}{1-w_i}
+       \quad\text{if }w_i<1,
+       \qquad s_i=0\quad\text{if }w_i=1.
+ \end{aligned}
+\end{equation}
+We have $0\le w_i\le y\le1$ and $0\le s_i\le1$.
+The inequality $2uv\le u^2+v^2$ gives
+\begin{equation}\label{eq:rnd-kernel-bound}
+ K_{it}\le\tfrac12\bigl[s_i(1-w_i)+s_t(1-w_t)\bigr].
+\end{equation}
+No symmetry of the directed marginals $q_{hi}$ is assumed.
+
+To average the positive quadratic erasure term, put
+$X_{hi}=\ind_{\{h\in\mathcal H_i\}}$.  Choices for distinct targets
+are independent.  Expanding the two removed quantities consequently
+gives the exact formula
+\begin{align}
+ \E_{\mathrm{graph}}\sum_{h\notin S}g\delta_hY_h
+ &=\sum_{i,t\in S}g^2K_{it}d_t
+   +\sum_{i\in S}g^2d_i\sum_{h\notin S}gq_{hi}(1-q_{hi})\notag\\
+ &\le\sum_{i,t\in S}g^2K_{it}d_t+gV.
+ \label{eq:rnd-diagonal-correction}
+\end{align}
+Here $\sum_hgq_{hi}=1$ bounds the diagonal correction.
+
+For the cap contribution, the external incoming mass at $i$ is
+$1-w_i=(1-y)+(y-w_i)$.  Split it in these proportions.  On its first
+part use $b_h\le b$, and on its second part use, whenever $q_{hi}>0$,
+\[
+ b_h\le d_h\le d_i+d(i,h)
+       \le d_i+(1-\widehat Pq_{hi})N_i.
+\]
+This yields
+\begin{equation}\label{eq:rnd-backup-average}
+ \sum_{h\notin S}gq_{hi}b_h
+ \le(1-y)b+(y-w_i)\bigl[d_i+(1-\widehat Ps_i)N_i\bigr].
+\end{equation}
+If $w_i=1$, then $y=1$ and both sides are zero, so the convention
+for $s_i$ requires no division in that case.
+
+For completeness, averaging \eqref{eq:rnd-ideal-generator} before
+using the cap bound gives, apart from the additive term $BgV$,
+\begin{align*}
+ &[1-(1+y)a-By]V-y(1-y)b\\
+ &\quad+\sum_{i\in S}g\sum_{h\notin S}gq_{hi}b_h
+       +a\sum_{i\in S}gw_id_i+BV\sum_{i\in S}gw_i
+       +B\sum_{i,t\in S}g^2K_{it}d_t\\
+ &\quad-\sum_{i,t\in S}g^2(1-q_{ti})(d_t-ad_i)_+.
+\end{align*}
+The first term of \eqref{eq:rnd-backup-average}, summed over $i$,
+cancels $-y(1-y)b$.  The coefficient of $V$ after adding
+\eqref{eq:rnd-time-derivative} simplifies using
+\begin{equation}\label{eq:rnd-coefficient-identity}
+ (1+y)a+yB+x(y-x)B'-1=3y+B(y-x)^2.
+\end{equation}
+Indeed \eqref{eq:rnd-function-identities} and its derivative give
+\[
+ (1+x)A+xB-1=3x,
+ \qquad A+(2+x)B+xB'=3.
+\]
+Expanding the left side of \eqref{eq:rnd-coefficient-identity} in
+$y-x$ has these as its constant and linear coefficients, and has
+quadratic coefficient $B$.
+
+It follows that $\dot f+\E_{\mathrm{graph}}\mathcal B-BgV$ is at most
+\begin{equation}\label{eq:rnd-expanded-drift}
+\begin{aligned}
+ &-[3y+B(y-x)^2]V\\
+ &\quad+\sum_{i\in S}g\Bigg\{
+ (y-w_i)\bigl[d_i+(1-\widehat Ps_i)N_i\bigr]+aw_id_i
+       +BVw_i+B\sum_{t\in S}gK_{it}d_t\\
+ &\hspace{43mm}
+       -\sum_{t\in S}g(1-q_{ti})(d_t-ad_i)_+\Bigg\}.
+\end{aligned}
+\end{equation}
+
+If $y=0$, the ideal generator and time derivative are zero.  If
+$y>0$ but $V=0$, then every $d_i$ with $i\in S$ is zero.  For any
+$i\in S$ with $N_i>0$, all alive sources are colocated with $i$, so
+$q_{ti}=1$ for every $t\in S$ and $w_i=y$.  If $N_i=0$, its radius
+term already vanishes.  Every term of
+\eqref{eq:rnd-expanded-drift} is then zero.  Thus the lemma holds
+in these degenerate cases without normalization.
+
+Assume now $V>0$, and set
+\begin{equation}\label{eq:rnd-normalization}
+ \mu=V/y,\qquad
+ D_i=d_i/\mu,\qquad
+ r_i=w_i/y,\qquad
+ R_i=(1-2\zeta)N_i/\mu,\qquad C_i=1-Ps_i.
+\end{equation}
+Give each index $i\in S$ probability $g/y$.  Expectations over these
+indices, written $\E_i$ and $\E_t$, satisfy
+$\E_iD_i=1$ and $r_i=\E_tq_{ti}$.
+For $q=q_{ti}<1$, the taper approximation implies
+\[
+ (1-2\zeta)(1-Pq)\le1-\widehat Pq,
+\]
+because the difference is at least
+$2\zeta(1-Pq)-\zeta q\ge2\zeta(1-q)\ge0$.
+The triangle inequality and \eqref{eq:rnd-taper-comparison} give
+\begin{equation}\label{eq:rnd-normalized-comparison}
+ R_i(1-Pq_{ti})\le D_i+D_t\qquad(q_{ti}<1).
+\end{equation}
+These are precisely the radius hypotheses of
+Theorem~\ref{thm:finite-inequality}.
+
+The error in changing the taper coefficient is controlled without
+any bound on individual normalized radii.  Multiply the first
+inequality in \eqref{eq:rnd-taper-comparison} by
+$(g/y)(1-q_{ti})$, sum over $t\in S$, and use
+$1-\widehat Pq_{ti}\ge1/2$ and $d(i,t)\le d_i+d_t$.  This proves
+\begin{equation}\label{eq:rnd-radius-error-control}
+ (1-r_i)N_i/\mu\le2(D_i+1).
+\end{equation}
+Since
+\[
+ 0\le(1-\widehat Ps_i)-(1-2\zeta)(1-Ps_i)\le3\zeta,
+\]
+the replacement of
+$(1-r_i)(1-\widehat Ps_i)N_i/\mu$ by
+$C_i(1-r_i)R_i$ costs at most $6\zeta(D_i+1)$, whose expectation
+is at most $12\zeta$.
+
+Divide \eqref{eq:rnd-expanded-drift} by $yV=y^2\mu$ and put
+$\Theta=P(y-x)^2/y$.  By \eqref{eq:rnd-kernel-bound}, the normalized
+kernel term is at most
+\[
+ B\E_{i,t}K_{it}D_t
+ \le P\E_i\bigl[s_i(1-yr_i)(1+D_i)\bigr].
+\]
+After the radius replacement, the entire normalized expression is
+therefore at most
+\begin{align*}
+ -3-2\Theta+\E_i\Big[&
+ (1-r_i)(D_i+C_iR_i)+ar_iD_i+2Pyr_i\\
+ &+Ps_i(1-yr_i)(1+D_i)
+ -\E_t(1-q_{ti})(D_t-aD_i)_+\Big]+12\zeta.
+\end{align*}
+Using $a=1+m+2Py$ and $\E_iD_i=1$, this is exactly
+\begin{align*}
+ \E_i\Big[&C_i(1-r_i)R_i
+ +r_i\{mD_i+(2-s_i)Py(D_i+1)\}-C_i(D_i+1)\\
+ &-\E_t(1-q_{ti})(D_t-(1+m+2Py)D_i)_+\Big]
+ -2\Theta+12\zeta.
+\end{align*}
+Theorem~\ref{thm:finite-inequality} bounds the expectation by
+$2\Theta$.  Multiplication by $yV$, with $y\le1$, proves
+\[
+ \dot f+\E_{\mathrm{graph}}\mathcal B
+ \le BgV+12\zeta yV\le BgV+12\zeta V,
+\]
+as claimed.
+\end{proof}
+
+\subsection{Survival, accumulated error, and parameter choices}
+
+\begin{lemma}[Decay of the alive mass]\label{lem:rnd-survival}
+For sufficiently small $\ell$ depending only on $\Delta$, every
+client's analysis process satisfies
+\begin{equation}\label{eq:rnd-survival}
+ \E y_T\le(1-\ell/2)^T\le e^{-\Lambda/2}.
+\end{equation}
+\end{lemma}
+
+\begin{proof}
+Condition on a history, a graph, and a currently alive target.
+If its group is forced, the original target identity is removed
+certainly.  Otherwise its $\Delta$ incoming source copies correspond
+to $\Delta$ distinct possible actions: each ordinary source gives
+its selection action, and each heavy source gives the unique chunk
+containing this target.  The marginal and pair estimates in the
+proof of Lemma~\ref{lem:rnd-batch} give an erasure probability at least
+\[
+ \Delta\ell g\bigl(1-O_\Delta(\ell)\bigr)
+       -O(\Delta^2\ell^2g^2)\ge\ell/2.
+\]
+Here the lower bound is the union bound with pair intersections
+subtracted.  A simultaneous reset does not restore the original
+copy identity, and additional analysis discards only reduce survival.
+The conditional expected surviving mass is thus at most
+$(1-\ell/2)y$.  Iteration, starting from $y_0=1$, and $T\ell\ge\Lambda$
+prove the lemma.
+\end{proof}
+
+We finish the proof of Theorem~\ref{thm:grid-rounding}.
+First run the process without rejecting count failures, and complete
+every output by Lemma~\ref{lem:rnd-completion}.  Average
+Lemma~\ref{lem:rnd-batch} over the graph and use
+Lemma~\ref{lem:rnd-average}.  Since $V\le V_0$, $B\le1$, and
+$R\le\Delta V_0$, summing over $T$ rounds gives
+\[
+ \E f_T\le
+ \left[1+2/e+(\Lambda+1)
+       \{g+12\zeta+C_\Delta\Delta(\phi+\ell)\}\right]V_0
+\]
+for a constant $C_\Delta$, provided $\ell\le1$.
+Equations~\eqref{eq:rnd-terminal-correction} and
+\eqref{eq:rnd-survival} consequently imply the unconditional estimate
+\begin{equation}\label{eq:rnd-total-error}
+ \E C_j\le
+ \left[1+2/e+5\Delta e^{-\Lambda/2}
+       +(\Lambda+1)\{g+12\zeta+C_\Delta\Delta(\phi+\ell)\}\right]V_0.
+\end{equation}
+All these estimates also apply when $V_0=0$: then $R=0$, the completed
+connection cost is zero, and no normalization by $V_0$ is used.
+
+Here is an explicit order of choices that avoids circular dependence.
+It suffices to consider $0<\tau\le1$.
+\begin{enumerate}
+\item Choose a sufficiently large constant integer $\Delta$ and put
+$g=1/\Delta$.  For each such choice take, for example,
+\[
+ \Lambda=\left\lceil2\log(40\Delta/\tau)\right\rceil+1.
+\]
+Then $5\Delta e^{-\Lambda/2}\le\tau/8$.  Taking $\Delta$ sufficiently
+large also makes $(\Lambda+1)g\le\tau/8$, since
+$g\log(1/g)\longrightarrow0$.  Any additional upper bound on $g$
+required by the preparation stage may be imposed at the same time.
+\item With $\Delta,\Lambda$ fixed, choose a small positive rational
+$\zeta$ so that $12(\Lambda+1)\zeta\le\tau/8$.
+Next choose a small positive rational $\phi$, and then a small positive
+rational $\ell$, so that
+\[
+ C_\Delta\Delta(\Lambda+1)(\phi+\ell)\le\tau/8.
+\]
+Require in addition every upper bound on $\ell$ used in the count,
+batch, and survival lemmas, and $p_*(1+\phi)\le1$.
+The batch constants are uniform in the remaining threshold $b_*$,
+so these requirements can be imposed before choosing that threshold.
+This fixes the constant number of rounds $T=\lceil\Lambda/\ell\rceil$.
+\item Put $\alpha=1+2/e$ and choose a fixed
+$0<\eta\le\tau/(4(\alpha+\tau))$.
+Finally choose rational $b_*>0$ as in Lemma~\ref{lem:rnd-count} so
+that the final mass is at most $M+C$ with probability at least
+$1-\eta$.
+\end{enumerate}
+The unconditional cost in \eqref{eq:rnd-total-error} is then at most
+$(\alpha+\tau/2)V_0$ for every client.  Repeat the entire process on
+failure of the final mass test.  The accepted output is the
+unconditional output conditioned on that test, whose success
+probability is at least $1-\eta$.  Since connection costs are
+nonnegative,
+\[
+ \E[C_j\mid\text{mass test succeeds}]
+ \le\frac{\alpha+\tau/2}{1-\eta}V_0
+ \le(\alpha+\tau)V_0.
+\]
+Completion uses at most the final active mass, so an accepted output
+has at most $M+C$ facilities.  This proves the count and cost assertions
+of Theorem~\ref{thm:grid-rounding}.
+
+For fixed accuracy all parameters are instance-independent constants.
+The number of current facilities never exceeds the number in the
+split input, and every group has at most $\Delta$ active copies.
+The graph, its marginals, a fixed-sum rounding, the action sets,
+their intersections, and the greedy completion can therefore be
+computed in polynomial time.  The radius equations and all physical
+probabilities are rational; the approximations in
+\eqref{eq:rnd-taper-approximation} are fixed constants.  Only the
+analysis uses the transcendental functions or alive-set discards.
+The expected number of repetitions is at most $1/(1-\eta)$, giving
+expected polynomial running time.  Section~\ref{sec:derandomization}
+replaces the local random choices by polynomial-size lists and obtains
+a deterministic output with the corresponding total-cost bound.

preprints/The-Approximation-Threshold-for-Metric-k-Median-September-24-2026/main.pdf

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:c8176d1671beb83d3a459ff65e5c632d94364ec7e542930bf19fad0b75524d50
+size 562378

preprints/The-Artinian-Lex-Plus-Powers-Betti-Theorem-September-23-2026/README.md

@@ -0,0 +1,18 @@
+# [The Artinian Lex-Plus-Powers Betti Theorem](paper.pdf)
+
+**Author:** OpenAI
+
+**Date:** September 23, 2026
+
+## Citation
+
+```bibtex
+@misc{OAI:The-Artinian-Lex-Plus-Powers-Betti-Theorem-September-23-2026,
+  author = {{OpenAI}},
+  title = {{The Artinian Lex-Plus-Powers Betti Theorem}},
+  howpublished = {OpenAI Math Release preprint
+                  \href{https://github.com/openai/math/blob/main/preprints/The-Artinian-Lex-Plus-Powers-Betti-Theorem-September-23-2026/paper.pdf}{OAI:The-Artinian-Lex-Plus-Powers-Betti-Theorem-September-23-2026}},
+  year = {2026}
+}
+```
+

preprints/The-Artinian-Lex-Plus-Powers-Betti-Theorem-September-23-2026/build/bibliography.tex

@@ -0,0 +1,130 @@
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+2026.
+
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+\href{https://doi.org/10.1016/S0022-4049(03)00130-0}{doi:10.1016/S0022-4049(03)00130-0}.
+\end{thebibliography}

preprints/The-Artinian-Lex-Plus-Powers-Betti-Theorem-September-23-2026/build/figures/endpoint-slices.tex

@@ -0,0 +1,22 @@
+\begin{figure}[htbp]
+\centering
+\begin{tikzpicture}[>=Stealth,font=\small]
+\node (u) at (-3.3,0) {$uz^s\in H$};
+\node (xu) at (3.3,0) {$x_iuz^s\in zH$};
+\node (transfer) at (3.3,-1.5) {$x_iuz^{s-1}\in H$};
+\draw[->] (u) -- node[above] {$\times x_i$} (xu);
+\draw[->] (transfer) -- node[right] {$\times z$} (xu);
+\draw[->,dashed] (u) --
+  node[pos=.47,below,sloped] {transfer $z$ to $x_i$} (transfer);
+\node[align=center] at (-3.2,-1.55)
+  {$x_i[uz^s]=0$ in $H/zH$\\[3pt]
+   $1\le s<a$};
+\end{tikzpicture}
+\caption{A middle slice is annihilated by every earlier variable.
+The dashed arrow is supplied by box stability, or by an earlier
+prescribed power if the monomial leaves the earlier-variable box.
+Consequently all Koszul differentials on a middle slice vanish.
+At $s=0$ there is no factor of $z$ to transfer; at $s=a$ the
+last exponent is outside the box. These are the two endpoints.}
+\label{fig:endpoint-slices}
+\end{figure}

preprints/The-Artinian-Lex-Plus-Powers-Betti-Theorem-September-23-2026/build/main.tex

@@ -0,0 +1,64 @@
+\pdfminorversion=7
+\documentclass[11pt]{article}
+\usepackage[T1]{fontenc}
+\usepackage{lmodern}
+\usepackage[margin=1in]{geometry}
+\usepackage{amsmath,amssymb,amsthm,mathtools}
+\usepackage{microtype}
+\usepackage{enumitem}
+\usepackage{xcolor}
+\usepackage{tikz}
+\usetikzlibrary{arrows.meta,positioning}
+\definecolor{linkblue}{RGB}{26,65,110}
+\PassOptionsToPackage{hyphens}{url}
+\usepackage[colorlinks=true,linkcolor=linkblue,citecolor=linkblue,urlcolor=linkblue]{hyperref}
+\hypersetup{pdftitle={The Artinian Lex-Plus-Powers Betti Theorem},pdfauthor={OpenAI}}
+\pdfinfoomitdate=1
+\pdftrailerid{}
+\pdfsuppressptexinfo=15
+\setlength{\emergencystretch}{2em}
+\setlist{itemsep=3pt,topsep=5pt}
+\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{\C}{\mathbb C}
+\newcommand{\N}{\mathbb Z_{\ge0}}
+\newcommand{\Z}{\mathbb Z}
+\newcommand{\Q}{\mathbb Q}
+\DeclareMathOperator{\Tor}{Tor}
+\DeclareMathOperator{\Hilb}{Hilb}
+\DeclareMathOperator{\Lex}{Lex}
+\DeclareMathOperator{\rank}{rank}
+\DeclareMathOperator{\Sym}{Sym}
+\DeclareMathOperator{\inm}{in}
+\title{The Artinian Lex-Plus-Powers Betti Theorem}
+\author{OpenAI}
+\date{September 23, 2026}
+\begin{document}
+\maketitle
+\begin{abstract}
+We prove the Eisenbud--Green--Harris and lex-plus-powers conjectures over every
+characteristic-zero field for homogeneous regular sequences of any positive
+length, with degrees at least two. For every homogeneous ideal containing such
+a sequence, the corresponding lex-plus-powers ideal has the same Hilbert
+function and at least as large a graded Betti number in every homological and
+internal degree.
+\end{abstract}
+\tableofcontents
+\input{sections/01-introduction}
+\input{sections/01-forms}
+\input{sections/02-filtered}
+\input{sections/03-monomial}
+\input{sections/04-stabilization}
+\input{sections/05-boxes}
+\input{sections/06-tor}
+\input{sections/07-assembly}
+\input{sections/08-characteristic-zero}
+\input{bibliography}
+\end{document}

preprints/The-Artinian-Lex-Plus-Powers-Betti-Theorem-September-23-2026/build/sections/01-forms.tex

@@ -0,0 +1,48 @@
+\section{Commuting forms with division coefficients}\label{sec:forms}
+
+We state the construction theorem from the companion article
+\cite{ArtinianEGH} in the precise form used below. A central
+division algebra over \(k\) is a finite-dimensional
+\(k\)-algebra with center \(k\) in which every nonzero element is invertible.
+In \(\Delta[x]\), the variables \(x_i\) are central and have degree one,
+and elements of \(\Delta\) have degree zero.
+
+\begin{theorem}[Commuting-form construction, {\cite[Theorem~1.1]{ArtinianEGH}}]
+\label{thm:forms}
+Let \(f_1,\ldots,f_n\) be a homogeneous regular sequence in
+\(\C[x_1,\ldots,x_n]\), of degrees \(2\le a_1\le\cdots\le a_n\).
+There are a field extension \(k/\C\), a central division \(k\)-algebra
+\(\Delta\), and a commutative graded \(k\)-subalgebra
+\[
+ k[x_1,\ldots,x_n]\subseteq A\subseteq N=\Delta[x_1,\ldots,x_n]
+\]
+generated by finitely many elements of degree one, containing
+\(t_1,\ldots,t_n\in A_1\), such that
+\begin{equation}\label{eq:triangular-forms}
+ t_h^{a_h}=c_h f_h+\sum_{\ell<h}B_{h\ell}f_\ell
+                  +\sum_{\ell>h}C_{h\ell}t_\ell
+ \qquad(1\le h\le n).
+\end{equation}
+Here \(c_h\in k^\times\), \(B_{h\ell},C_{h\ell}\in A\), and every summand
+has degree \(a_h\). Moreover,
+\[
+ \{t^\alpha=t_1^{\alpha_1}\cdots t_n^{\alpha_n}:
+       \alpha\in\N^n\}
+\]
+is a graded left-\(\Delta\) basis of the \emph{full} algebra \(N\).
+\end{theorem}
+
+The basis assertion in Theorem~\ref{thm:forms} is stronger than a basis of the finite
+complete-intersection quotient: every monomial \(t^\alpha\), without
+an upper bound on its exponents, occurs in the decomposition of \(N\).
+This is what will make \(N\) a free module over a polynomial ring and
+allow us to transfer an entire resolution. The Hilbert-function
+consequence of the foundation theorem alone would not suffice.
+
+\begin{remark}\label{rem:coefficients}
+Commutativity of the forms does not mean that their individual
+\(\Delta\)-coefficients commute. The proof below always places those
+coefficients on the left and multiplies by the \(t_i\) on the right.
+The construction itself has no hypothesis on the additional generators
+of \(I\).
+\end{remark}

preprints/The-Artinian-Lex-Plus-Powers-Betti-Theorem-September-23-2026/build/sections/01-introduction.tex

@@ -0,0 +1,227 @@
+\section{Introduction}\label{sec:introduction}
+
+A Hilbert function records the dimensions of a graded quotient; its graded
+Betti numbers also record the relations among its generators and all higher
+syzygies. The lex-plus-powers conjecture predicts one extremal ideal for
+both kinds of data when the original ideal contains a regular sequence.
+We prove its full Artinian form over \(\C\), and deduce the full
+characteristic-zero statement for regular sequences of arbitrary positive
+length by the established Artinian reduction and coefficient-field descent.
+
+Let
+\[
+ S=\C[x_1,\ldots,x_n],\qquad
+ 2\le a_1\le\cdots\le a_n,\qquad
+ P=(x_1^{a_1},\ldots,x_n^{a_n}),
+\]
+where each variable has degree one. Lexicographic order always has
+\(x_1>\cdots>x_n\), and a lex segment starts with the \emph{greatest}
+monomials. Suppose that a homogeneous ideal \(I\subseteq S\) contains a
+homogeneous regular sequence \(f_1,\ldots,f_n\) with \(\deg f_i=a_i\).
+Here regularity means that multiplication by \(f_i\) is injective on
+\(S/(f_1,\ldots,f_{i-1})\) for each \(i\). Its full length makes
+\(S/I\) Artinian: the complete-intersection Hilbert series is
+\(\prod_i(1+z+\cdots+z^{a_i-1})\).
+For each \(d\ge0\), prescribe
+\begin{equation}
+ J_d=L_d+P_d,\qquad \dim_\C J_d=\dim_\C I_d,
+ \label{eq:prescription}
+\end{equation}
+where \(L_d\) is the largest lex-segment subspace with this property.
+Part of the assertion below is that these dimensions can be attained
+and that the prescribed spaces form an ideal.
+
+For a standard graded polynomial ring \(A\) over a field \(F\), write
+\[
+ \beta^A_{p,j}(B)=\dim_F\Tor^A_p(B,F)_j,
+ \qquad F=A/A_{>0}.
+\]
+The index \(p\) is the homological degree and \(j\) is the internal degree.
+
+\begin{theorem}\label{thm:main}
+For every \(n\ge1\), every ordered degree sequence
+\(2\le a_1\le\cdots\le a_n\), and every homogeneous ideal
+\(I\subseteq\C[x_1,\ldots,x_n]\) containing a regular sequence of these
+degrees, the spaces in \eqref{eq:prescription} exist and
+\(J=\bigoplus_{d\ge0}J_d\) is a homogeneous ideal. For this single ideal,
+\[
+ \beta^S_{p,j}(S/I)\le\beta^S_{p,j}(S/J)
+ \qquad\text{for every \(p\ge0\) and \(j\in\Z\)}.
+\]
+There is no restriction on the degrees or number of the additional
+generators of \(I\).
+\end{theorem}
+
+Thus Theorem~\ref{thm:main} resolves the Artinian lex-plus-powers
+conjecture positively over \(\C\).
+Its Hilbert-function assertion is the Artinian Eisenbud--Green--Harris
+statement; the Betti inequalities are stronger.
+
+The Artinian restriction can be removed from these two conclusions.
+Caviglia and Maclagan proved the Artinian reduction for the
+Eisenbud--Green--Harris conjecture \cite[Proposition~10]{CM08};
+Caviglia and Kummini proved the corresponding reduction for
+lex-plus-powers Betti numbers \cite[Section~4, Theorem~4.1]{CK14}.
+Together with a finite coefficient-field descent, these reductions give
+the following form over every characteristic-zero field.
+
+\begin{corollary}[Eisenbud--Green--Harris and lex-plus-powers in characteristic zero]
+\label{cor:characteristic-zero}
+Let \(F\) be a field of characteristic zero, let
+\[
+ S_F=F[x_1,\ldots,x_n],\qquad
+ 1\le c\le n,\qquad 2\le a_1\le\cdots\le a_c,
+ \qquad P_F=(x_1^{a_1},\ldots,x_c^{a_c}),
+\]
+with the standard grading and lexicographic order \(x_1>\cdots>x_n\).
+Suppose that a homogeneous ideal \(I\subseteq S_F\) contains a homogeneous
+regular sequence \(f_1,\ldots,f_c\) with \(\deg f_i=a_i\).
+For \(d\ge0\), put
+\[
+ q_d=\dim_F I_d-\dim_F(P_F)_d.
+\]
+This integer lies between zero and the number of degree-\(d\) monomials
+outside \(P_F\). Let \(J_d\) be the sum of \((P_F)_d\) and the span of the
+\(q_d\) lexicographically greatest such monomials. Then
+\(J=\bigoplus_{d\ge0}J_d\) is a homogeneous ideal, and for this single ideal
+\[
+ \dim_F(S_F/I)_d=\dim_F(S_F/J)_d\quad(d\ge0),\qquad
+ \beta^{S_F}_{p,j}(S_F/I)\le\beta^{S_F}_{p,j}(S_F/J)
+ \quad(p\ge0,\ j\in\Z).
+\]
+There is no restriction on the degrees or number of the additional
+generators of \(I\).
+\end{corollary}
+
+Corollary~\ref{cor:characteristic-zero} includes non-Artinian quotients
+and fields that are not algebraically closed. The case with no prescribed
+regular sequence is the classical lex-ideal theorem cited below.
+The division-coefficient construction itself remains the full-length
+construction over \(\C\); Section~\ref{sec:characteristic-zero} proves
+the corollary from Theorem~\ref{thm:main}.
+
+\subsection{Context and previous results}
+
+The Eisenbud--Green--Harris (EGH) Hilbert-function conjecture arose in
+their work on higher Castelnuovo theory \cite[Section~4]{EGH}.
+The lex-plus-powers conjecture, generally attributed to Charalambous
+and Evans, strengthens that prediction to graded Betti numbers.
+Francisco and Richert discuss its origins and identify its first
+printed appearance in the work of Evans and Richert
+\cite[Section~5]{FR}; see also the public formulations in
+\cite[Section~1]{Richert} and
+\cite[Conjectures~1.1 and 1.2]{CS}.
+Without a prescribed regular sequence, the extremality of lex ideals
+at fixed Hilbert function is the theorem of Bigatti, Hulett, and
+Pardue \cite{Bigatti,Hulett,Pardue}.
+The additional powers encode the degrees of the regular sequence and
+give the sharper comparison studied here.
+For ideals already containing these powers, Clements--Lindstr\"om
+compression supplies the Hilbert-function comparison \cite{CL}.
+
+The prescribed-powers Betti comparison developed through several stages.
+Mermin, Peeva, and Stillman proved the comparison for ideals containing
+the squares of the variables in characteristic zero
+\cite[Theorem~1.1]{MPS}; Murai established it for strongly stable ideals
+plus the prescribed powers over any field \cite[Theorem~1.1]{Murai}.
+Mermin and Murai then proved the lex-plus-powers comparison for ideals
+containing a monomial regular sequence, again over every field
+\cite[Theorems~3.1 and 8.1]{MM}. They also isolated the reduction to a
+Hilbert-matching monomial ideal with no smaller graded Betti numbers
+\cite[Conjecture~10.2]{MM}. It is this reduction that we obtain from
+division coefficients. Our proof of the subsequent monomial comparison
+is included for completeness; its conclusion is not new.
+
+For nonmonomial regular sequences, the Hilbert-function and Betti problems
+have had different ranges of known cases. Abedelfatah proved EGH for full
+regular sequences whose forms split into linear factors
+\cite[Corollary~4.3]{Abedelfatah15}. Caviglia and Maclagan established
+EGH for fast-growing degrees \cite{CM08}, and Caviglia and De Stefani
+weakened the required growth to
+\(a_i\ge\sum_{j<i}(a_j-1)\) for \(i\ge3\)
+\cite[Theorem~A]{CDeS20}. These Hilbert-function results hold over any
+field. The characteristic-zero Betti theorem of Caviglia and Sammartano
+requires
+\(a_i\ge1+\sum_{j<i}(a_j-1)\) for \(i\ge3\)
+\cite[Theorem~3.8]{CS}. No such growth condition is imposed here.
+
+More recently, Kuzmanovski \cite[Theorem~1.9]{Kuzmanovski} obtained
+asymptotic Betti estimates in characteristic zero in a bounded degree range,
+for fixed regular-sequence degree offsets and generator counts and
+sufficiently large initial degree. That comparison uses the
+reverse-lexicographic generic initial ideal of a lex-plus-powers ideal,
+rather than the lex-plus-powers ideal itself as in Theorem~\ref{thm:main}.
+
+Our treatment of the monomial comparison retains the
+resolution-level mechanism of distractions
+\cite[Theorem~2.19 and Corollary~2.20]{BCR} and the endpoint
+decomposition for stable-plus-powers ideals
+\cite[Lemma~3.7]{CS}, while organizing the last comparison through ranks
+in the Koszul complex.
+
+\subsection{The reduction that carries the Betti numbers}
+
+The essential step is to find a monomial ideal \(M\supseteq P\) such that
+\begin{equation}\label{eq:overview-bound}
+ \Hilb(S/I)=\Hilb(S/M),\qquad
+ \beta^S_{p,j}(S/I)\le\beta^S_{p,j}(S/M)
+ \quad\text{for all \(p,j\)}.
+\end{equation}
+Matching Hilbert functions alone cannot give the second conclusion.
+
+Theorem~1.1 of the companion article \cite{ArtinianEGH} supplies a field
+\(k/\C\), a finite-dimensional central division \(k\)-algebra \(\Delta\), and commuting linear
+forms \(t_1,\ldots,t_n\) in \(N=\Delta[x_1,\ldots,x_n]\). They satisfy
+triangular power identities adapted to the \(f_i\), and their ordered
+monomials form a left-\(\Delta\) basis of \(N\).
+Section~\ref{sec:forms} states exactly the input needed here, including
+the basis assertion for the full algebra rather than only its
+complete-intersection quotient.
+
+Let \(T=k[y_1,\ldots,y_n]\) act on \(N\) on the right by \(y_i\mapsto t_i\).
+Then \(N\) is simultaneously left-\(S\) free and right-\(T\) free, the
+latter of rank \(m=\dim_k\Delta\), with generators in degree zero.
+A minimal \(S\)-resolution of \(S/I\) therefore gives a
+\emph{minimal} \(T\)-resolution of \(N/IN\), multiplying every Betti number
+by the same factor \(m\). With a monomial order adapted to the triangular
+identities, the leading coefficient spaces of \(IN\) in the
+\(t^\alpha\)-expansion are left ideals of \(\Delta\), hence either zero
+or all of \(\Delta\). Their nonzero positions define \(M\); the
+triangular identities ensure that \(M\) contains \(P\). A filtered
+Koszul comparison then gives \eqref{eq:overview-bound} after cancelling
+\(m\).
+This resolution transfer is the additional ingredient beyond the
+Hilbert-function reduction in \cite{ArtinianEGH}.
+
+Starting from \(M\), we work with the monomials in the exponent box
+\(0\le u_i<a_i\). A transfer replaces a factor \(x_\ell\) by \(x_h\),
+where \(h<\ell\), provided the result remains in the box.
+Root-of-unity distractions followed by initial ideals produce an ideal
+\(H\supseteq P\) closed under these transfers, with unchanged Hilbert
+function and no smaller quotient Betti numbers. In each degree, replace
+the box monomials belonging to \(H\) by the same number of greatest
+lexicographic monomials in the box, and retain \(P\). Comparing their bounded
+shadows---the multiples by one variable that remain in the box---shows
+that the resulting space \(G\) is an ideal.
+
+For the Betti comparison, put \(z=x_n\). The modules \(H/zH\) and
+\(G/zG\) over the earlier variables have equal Hilbert functions and
+retain the Betti numbers of the ideals \(H\) and \(G\). They split
+into components according to the exponent of \(z\), from \(0\) to
+\(a_n\). Stability makes every earlier variable annihilate the
+components at exponents \(1,\ldots,a_n-1\). Only the two extreme
+components can therefore contribute nonzero ranks to the Koszul
+differentials. The comparisons on the box slices \(u_n=0\) and
+\(u_n=a_n-1\), induction in the number of variables, and monotonicity
+of differential ranks under submodules and quotients compare these
+endpoint contributions. Since the Koszul terms have equal dimensions,
+the resulting reverse inequality for adjacent rank sums gives the
+required Betti bound.
+
+Sections~\ref{sec:filtered} and \ref{sec:monomial} establish the filtered
+comparison and the transfer to \(M\).
+Sections~\ref{sec:stabilization}--\ref{sec:tor} give the complete monomial
+comparison. Section~\ref{sec:assembly} identifies its output with
+\eqref{eq:prescription} and proves Theorem~\ref{thm:main}.
+Section~\ref{sec:characteristic-zero} then removes the Artinian and
+complex-coefficient restrictions from the Hilbert and Betti conclusions.

preprints/The-Artinian-Lex-Plus-Powers-Betti-Theorem-September-23-2026/build/sections/02-filtered.tex

@@ -0,0 +1,231 @@
+\section{Graded resolutions and leading coefficient spaces}
+\label{sec:filtered}
+
+We first prove a comparison for quotients of a free module whose monomial
+positions have vector-space coefficients.  The application in the next
+section will use a division algebra as that coefficient space.  The
+comparison must retain each internal degree separately, so we work with
+the graded Koszul complex throughout.
+
+Let $F$ be a field and let $A=F[y_1,\ldots,y_r]$, with each variable of
+degree one.  For a finitely generated graded $A$-module $B$, put
+\[
+ \beta^A_{p,j}(B)=\dim_F\Tor^A_p(B,F)_j,
+ \qquad F=A/(y_1,\ldots,y_r).
+\]
+We use the shift convention $B(-s)_d=B_{d-s}$.  All modules considered
+below are bounded below and have finite-dimensional graded components.
+
+\begin{lemma}[Minimal graded resolutions]
+\label{filt:minimal}
+Every finitely generated graded $A$-module $B$ has a graded free resolution
+whose differentials have entries in $A_{>0}$.  In such a resolution, the
+number of free generators of degree $j$ in homological position $p$ is
+$\beta^A_{p,j}(B)$.  If $H\subsetneq A$ is a homogeneous ideal, then
+\begin{equation}
+ \beta^A_{p+1,j}(A/H)=\beta^A_{p,j}(H)
+ \qquad(p\geq 0,\ j\in\Z).
+ \label{filt:ideal-quotient}
+\end{equation}
+If $H$ is monomial, a minimal resolution of $A/H$ can be chosen
+homogeneous for the exponent multigrading, with all generator
+multidegrees in $\Z_{\geq0}^r$.
+\end{lemma}
+
+\begin{proof}
+Lift a homogeneous $F$-basis of $B/A_{>0}B$ to $B$ and map the free module
+on these lifts to $B$.  This map is surjective: in each degree, an element
+not yet expressed in the lifts differs from such an expression by a sum
+of positive-degree multiples of elements of smaller degree, to which
+induction applies.  Its kernel is finitely generated because $A$ is
+Noetherian.  Repeating the construction for successive kernels gives a
+free resolution.  At each stage, reduction modulo $A_{>0}$ identifies the
+chosen free generators with a basis of the corresponding module modulo
+$A_{>0}$, so the next kernel lies in $A_{>0}$ times that free module.
+Thus every differential between free modules has positive-degree
+entries.  Tensoring with $F$ makes these differentials zero and proves
+the assertion about $\Tor$.
+
+A minimal resolution of a proper homogeneous ideal $H$, followed by
+the inclusion $H\hookrightarrow A$ and the quotient $A\twoheadrightarrow
+A/H$, is a minimal resolution of $A/H$: the additional entries also
+have positive degree since $H\subseteq A_{>0}$.  This proves
+\eqref{filt:ideal-quotient}, including $H=0$.
+
+For a monomial quotient, carry out the same construction with
+multihomogeneous lifts.  The initial free module is $A$ in multidegree
+zero.  A free module whose generator multidegrees are nonnegative has
+no component in a multidegree outside $\Z_{\geq0}^r$; neither does its
+kernel or a quotient of that kernel.  Induction therefore gives the
+last assertion.
+\end{proof}
+
+\begin{lemma}[Koszul calculation]
+\label{filt:koszul}
+Let $K_\bullet(B)$ be the Koszul complex on $y_1,\ldots,y_r$ with
+coefficients in $B$.  Its term of internal degree $j$ in position $p$ is
+\[
+ K_p(B)_j=\bigoplus_{\substack{\sigma\subseteq\{1,\ldots,r\}\\
+                              |\sigma|=p}} B_{j-p}e_\sigma,
+\]
+where $e_\sigma$ has degree $p$.  Its homology satisfies
+\[
+ \dim_F H_p(K_\bullet(B)_j)=\beta^A_{p,j}(B).
+\]
+\end{lemma}
+
+\begin{proof}
+For $\sigma=\{i_1<\cdots<i_p\}$ the differential is
+\[
+ d(be_\sigma)=\sum_{q=1}^p(-1)^{q-1}y_{i_q}b\,
+                    e_{\sigma\setminus\{i_q\}}.
+\]
+The complex $K_\bullet(A)$ is the tensor product over $F$ of the
+one-variable complexes
+\[
+ 0\longrightarrow F[y_i](-1)
+   \xrightarrow{\ y_i\ }F[y_i]\longrightarrow0.
+\]
+Each has homology $F$ in position zero and no other homology.  A complex
+of vector spaces splits as its homology with zero differential plus
+contractible two-term summands.  Tensoring these splittings shows that
+$K_\bullet(A)$ is a free resolution of $F$.
+
+To compare this computation with a resolution of $B$, take any graded
+free resolution $F_\bullet\to B$ and form the first-quadrant double
+complex $F_\bullet\otimes_A K_\bullet(A)$.  Augmenting either factor
+gives maps from its total complex to $F_\bullet\otimes_A F$ and to
+$B\otimes_A K_\bullet(A)$, respectively.  The kernels have exact columns
+or exact rows, since the terms of the other factor are free.  Their
+total complexes are exact.  For exact columns, eliminate a cycle's
+component of greatest horizontal index by a vertical boundary; the
+correction introduces terms only in the next column to the left.
+There are finitely many components on each total-degree diagonal,
+so iteration eliminates the cycle.  The argument for exact rows
+interchanges the two indices.  Both augmentation maps consequently induce
+homology isomorphisms.  This identifies the homology of
+$B\otimes_A K_\bullet(A)=K_\bullet(B)$ with $\Tor^A(B,F)$, preserving
+the internal grading.
+\end{proof}
+
+We now order monomial positions without choosing an order on their
+coefficients.  Let $V$ be a finite-dimensional $F$-vector space in
+degree zero, and set
+\[
+ E=V\otimes_F A=\bigoplus_{\alpha\in\Z_{\geq0}^r}Vy^\alpha.
+\]
+Choose a total order $\prec$ on the exponents of each fixed total
+degree, compatible with addition: if $\alpha\prec\beta$, then
+$\alpha+\gamma\prec\beta+\gamma$ for every
+$\gamma\in\Z_{\geq0}^r$.  The first nonzero position of an element is
+its leading position in this order.
+
+For a homogeneous submodule $W\subseteq E$ and an exponent $\alpha$,
+define
+\[
+ V_\alpha=\left\{v\in V:
+   vy^\alpha+\sum_{\substack{|\gamma|=|\alpha|\\\alpha\prec\gamma}}
+        v_\gamma y^\gamma\in W
+        \text{ for some }v_\gamma\in V\right\}.
+\]
+This is an $F$-subspace of $V$.  The resulting leading submodule is
+\[
+ \operatorname{in}_{\prec}W
+       =\bigoplus_{\alpha\in\Z_{\geq0}^r}V_\alpha y^\alpha.
+\]
+Indeed, multiplication of a witnessing element by $y_i$ shows that
+$V_\alpha\subseteq V_{\alpha+e_i}$, so this direct sum is an
+$A$-submodule.
+
+\begin{proposition}[Leading-submodule comparison]
+\label{filt:comparison}
+\label{filt:initial}
+With $A,V,E,W$ and $\prec$ as above, for every $d\in\Z$, every
+$p\geq0$, and every $j\in\Z$ one has
+\begin{align}
+ \dim_F(E/W)_d
+    &=\dim_F(E/\operatorname{in}_{\prec}W)_d,
+       \label{filt:hilbert}\\
+ \beta^A_{p,j}(E/W)
+    &\leq\beta^A_{p,j}(E/\operatorname{in}_{\prec}W).
+       \label{filt:betti}
+\end{align}
+\end{proposition}
+
+\begin{proof}
+Write $Q=E/W$.  In degree $d\geq0$, let $E_{\succeq\alpha}$ be the
+sum of the positions $Vy^\gamma$ with $|\gamma|=d$ and
+$\alpha\preceq\gamma$, and define $E_{\succ\alpha}$ similarly with
+strict inequality.  Their images in $Q_d$ give a finite descending
+filtration.  Its quotient at $\alpha$ is
+\begin{equation}
+ \frac{\operatorname{im}(E_{\succeq\alpha}\to Q_d)}
+      {\operatorname{im}(E_{\succ\alpha}\to Q_d)}
+ \simeq
+ \frac{E_{\succeq\alpha}}
+      {E_{\succ\alpha}+(W_d\cap E_{\succeq\alpha})}
+ \simeq (V/V_\alpha)y^\alpha.
+ \label{filt:slot-quotient}
+\end{equation}
+The last isomorphism takes the coefficient at $\alpha$.  Summing its
+dimensions proves \eqref{filt:hilbert}; both sides vanish for $d<0$.
+
+Fix an internal degree $j\geq0$.  We give all terms of
+$K_\bullet(Q)_j$ a common filtration indexed by the degree-$j$
+exponents.  For a subset $\sigma$, put
+$\mathbf{1}_\sigma=\sum_{i\in\sigma}e_i$.  In the summand with
+exterior label $e_\sigma$, place the coefficient position $\alpha$ at
+\[
+                    \alpha+\mathbf{1}_\sigma.
+\]
+More explicitly, the filtration subspace at $\beta$ in that summand
+is the image of the span of all $Vy^\alpha$ satisfying
+$\beta\preceq\alpha+\mathbf{1}_\sigma$.  The full filtration subspace
+is the direct sum over the labels $\sigma$.  Addition compatibility
+means that the relative order of the positions within one summand
+agrees with their order in $Q_{j-|\sigma|}$.
+
+A differential deleting $i\in\sigma$ multiplies its coefficient by
+$y_i$.  It sends the position with exponent $\alpha$ to that with
+exponent $\alpha+e_i$, and
+\[
+ (\alpha+e_i)+\mathbf{1}_{\sigma\setminus\{i\}}
+       =\alpha+\mathbf{1}_\sigma.
+\]
+Thus the differential preserves the filtration.  By
+\eqref{filt:slot-quotient}, its map on the corresponding associated
+graded positions is, up to the Koszul sign,
+\[
+ V/V_\alpha\longrightarrow V/V_{\alpha+e_i},
+ \qquad [v]\longmapsto[v].
+\]
+This is well defined by $V_\alpha\subseteq V_{\alpha+e_i}$.
+Consequently the associated graded complex is exactly
+$K_\bullet(E/\operatorname{in}_{\prec}W)_j$, decomposed by the total
+exponent $\beta$.
+
+For completeness, if $d:C\to C'$ is a filtration-preserving linear
+map between finite-dimensional filtered spaces, the filtration induced
+on $\ker d$ gives an injection
+\[
+          \operatorname{gr}(\ker d)
+               \hookrightarrow\ker(\operatorname{gr}d).
+\]
+Indeed a leading class represented by an element of $\ker d$ maps to
+zero, and the induced filtration on the kernel makes this inclusion
+injective.  Since taking associated graded preserves dimension, it
+follows that
+$\operatorname{rank}d\geq\operatorname{rank}(\operatorname{gr}d)$.
+At a fixed term of a complex, the homology dimension is the term
+dimension minus the ranks of its incoming and outgoing maps.  Applying
+the rank inequality to these two maps proves that passing to the
+associated graded complex cannot decrease this homology dimension.
+Lemma~\ref{filt:koszul} now gives \eqref{filt:betti}.  For $j<0$ all
+the relevant Koszul terms vanish, so the same assertion holds.
+\end{proof}
+
+Taking $V=F$ recovers the usual comparison with a monomial initial
+ideal; see also \cite[Section~2]{MM}. Keeping $V$ arbitrary is essential
+for the next application:
+the coefficient algebra will enlarge all dimensions by the same
+finite factor, which can then be cancelled.

preprints/The-Artinian-Lex-Plus-Powers-Betti-Theorem-September-23-2026/build/sections/03-monomial.tex

@@ -0,0 +1,218 @@
+\section{A monomial bound containing the prescribed powers}
+\label{sec:monomial}
+
+We use the commuting forms of Theorem~\ref{thm:forms} to construct a
+monomial ideal containing the prescribed powers.  The construction
+preserves the Hilbert function and increases every graded Betti number.
+The first step explains why the original resolution remains minimal
+over the polynomial ring acting through these forms.
+
+\begin{lemma}[Transfer of a minimal resolution]
+\label{mono:transfer}
+Let $S=\C[x_1,\ldots,x_n]$, let $k$ be a field containing $\C$, and
+let $\Delta$ be a finite-dimensional central division $k$-algebra.
+Put $m=\dim_k\Delta$ and
+$N=\Delta[x_1,\ldots,x_n]$, where the $x_i$ are central.
+Suppose that $t_1,\ldots,t_n\in N_1$ commute and that
+\begin{equation}
+          N=\bigoplus_{\alpha\in\Z_{\geq0}^n}\Delta t^\alpha
+ \label{mono:basis}
+\end{equation}
+is a graded direct-sum decomposition.  Give
+$T=k[y_1,\ldots,y_n]$ a right action on $N$ by multiplication,
+$u\cdot y_i=ut_i$.
+For every homogeneous ideal $I\subseteq S$, the right $T$-module
+$Q=N/IN$ satisfies
+\begin{align}
+ \dim_k Q_d&=m\dim_\C(S/I)_d
+          &&(d\in\Z),\label{mono:dimension-transfer}\\
+ \beta^T_{p,j}(Q)&=m\beta^S_{p,j}(S/I)
+          &&(p\geq0,\ j\in\Z).
+          \label{mono:betti-transfer}
+\end{align}
+Here right and left modules over the commutative ring $T$ are
+identified when computing $\Tor$.
+\end{lemma}
+
+\begin{proof}
+Commutativity of the $t_i$ and centrality of $k$ make the stated right
+$T$-action well defined.  The map
+\begin{equation}
+ \Delta\otimes_kT\longrightarrow N,
+ \qquad b\otimes y^\alpha\longmapsto bt^\alpha
+ \label{mono:free-map}
+\end{equation}
+is a graded vector-space isomorphism by \eqref{mono:basis}.  It is
+right $T$-linear because $(bt^\alpha)t_i=bt^{\alpha+e_i}$.
+Thus $N$ is right $T$-free of rank $m$, with a basis in degree zero,
+and
+\begin{equation}
+                         N_{>0}=NT_{>0}.
+ \label{mono:positive-degree}
+\end{equation}
+
+The left $S$-action commutes with the right $T$-action by associativity
+of multiplication in $N$.  Moreover,
+$N=\Delta\otimes_\C S$ as a left $S$-module, so $N$ is left $S$-free
+and therefore flat.  Each element of $I$ is central in $N$, whence
+$IN$ is a two-sided ideal.  In particular it is a right $T$-submodule,
+and
+\[
+               Q\simeq\Delta\otimes_\C(S/I).
+\]
+Taking a complex basis of $(S/I)_d$ proves
+\eqref{mono:dimension-transfer}.
+
+Let $F_\bullet\to S/I$ be a minimal graded $S$-free resolution.
+Since $S$ is commutative, regard its terms as right $S$-modules and
+tensor with the $(S,T)$-bimodule $N$.  Flatness gives a resolution
+\[
+                    F_\bullet\otimes_SN\longrightarrow Q
+\]
+by graded free right $T$-modules.  An entry $f$ of an original
+differential acts on $N$ by left multiplication.  This map is right
+$T$-linear, again by associativity.
+
+Choose a $k$-basis $b_1,\ldots,b_m$ of $\Delta$, which is a right
+$T$-basis of $N$ under \eqref{mono:free-map}.  If $f$ is nonzero and
+has degree $e>0$, then each $fb_u$ belongs to $N_e$.  By the graded
+decomposition \eqref{mono:basis}, its coefficients in the basis
+$b_1,\ldots,b_m$ are homogeneous polynomials in $T$ of degree $e$.
+They therefore lie in $T_{>0}$.  Every nonzero original differential
+entry has positive degree, so the transferred resolution is minimal.
+Each summand $S(-s)$ of $F_p$ becomes $m$ summands $T(-s)$.
+Lemma~\ref{filt:minimal} proves \eqref{mono:betti-transfer} for every
+$p,j$.  If $I=S$, both quotients and their resolutions may be taken
+to be zero, and the same formulas hold.
+\end{proof}
+
+The next proposition supplies the monomial ideal used by the rest of
+the proof.  Its ideal is allowed to depend on $I$, but it is fixed
+simultaneously for all homological and internal degrees.
+
+\begin{proposition}[Monomial reduction]
+\label{mono:bound}
+Let $S=\C[x_1,\ldots,x_n]$, let
+$2\leq a_1\leq\cdots\leq a_n$, and let $I\subseteq S$ be a
+homogeneous ideal containing a homogeneous regular sequence
+$f_1,\ldots,f_n$ with $\deg f_h=a_h$.  Put
+$P=(x_1^{a_1},\ldots,x_n^{a_n})$.
+There is a monomial ideal $M\subseteq S$ containing $P$ such that
+\begin{align}
+ \dim_\C(S/M)_d&=\dim_\C(S/I)_d
+                   &&(d\in\Z),\label{mono:hilbert-bound}\\
+ \beta^S_{p,j}(S/I)&\leq\beta^S_{p,j}(S/M)
+                   &&(p\geq0,\ j\in\Z).
+                   \label{mono:betti-bound}
+\end{align}
+\end{proposition}
+
+\begin{proof}
+If $I=S$, take $M=S$.  Assume henceforth that $I$ is proper.
+Apply Theorem~\ref{thm:forms} to the given regular sequence.  It
+supplies $k$, $\Delta$, and commuting degree-one forms $t_h$ satisfying
+the basis property \eqref{mono:basis} and homogeneous identities
+\begin{equation}
+ t_h^{a_h}=c_hf_h+\sum_{\ell<h}B_{h\ell}f_\ell
+                       +\sum_{\ell>h}C_{h\ell}t_\ell,
+ \qquad c_h\in k^\times,
+ \label{mono:triangular}
+\end{equation}
+where the multipliers belong to $N=\Delta[x_1,\ldots,x_n]$.
+In particular $C_{h\ell}$ is homogeneous of degree $a_h-1$ when
+nonzero.  Define $T=k[y_1,\ldots,y_n]$, $m=\dim_k\Delta$,
+$W=IN$, and $Q=N/W$ as in Lemma~\ref{mono:transfer}.
+
+Under \eqref{mono:free-map}, view $N$ as the free $T$-module with
+coefficient space $V=\Delta$.  In each total degree, order exponents
+by placing the lexicographically smallest tuple
+\[
+                         (\alpha_n,\ldots,\alpha_1)
+\]
+first.  This order is compatible with addition.  Apply
+Proposition~\ref{filt:comparison} to the homogeneous $T$-submodule
+$W$ with this order.
+
+For an exponent $\alpha$, let $V_\alpha\subseteq\Delta$ be its
+leading coefficient space.  It is closed under left multiplication
+by $\Delta$: if $w\in W$ witnesses a coefficient $b$ at $\alpha$
+with no earlier coefficients, then $\lambda w\in W$ witnesses
+$\lambda b$.  Here $W$ is left $\Delta$-linear because it is an ideal
+of $N$.  Hence $V_\alpha$ is a left ideal of $\Delta$.  Since
+$\Delta$ is a division algebra,
+\[
+                         V_\alpha=0\quad\hbox{or}\quad\Delta.
+\]
+Explicitly, if $0\ne b\in V_\alpha$, every $c\in\Delta$ is
+$(cb^{-1})b$ and therefore also belongs to $V_\alpha$.
+
+The set
+\[
+          \mathcal U=\{\alpha\in\Z_{\geq0}^n:V_\alpha=\Delta\}
+\]
+is upward closed under addition of exponent vectors, since the leading
+object is a $T$-submodule.  Define $M\subseteq S$ to be the monomial
+ideal with exponent set $\mathcal U$, and define
+$M_k\subseteq T$ by the same exponent set.  Then
+\begin{equation}
+ N/\operatorname{in}_{\prec}W
+       \simeq\Delta\otimes_k(T/M_k)
+       \simeq(T/M_k)^{\oplus m}
+ \label{mono:initial-quotient}
+\end{equation}
+as graded right $T$-modules.
+
+We next verify the power containment.  Since every $f_\ell$ lies
+in $I$, Equation~\eqref{mono:triangular} gives
+\[
+             t_h^{a_h}-\sum_{\ell>h}C_{h\ell}t_\ell\in W.
+\]
+Expand each $C_{h\ell}$ in the left $\Delta$-basis $t^\alpha$.
+Right multiplication by $t_\ell$ turns a term $bt^\alpha$ into
+$bt^{\alpha+e_\ell}$, using only commutativity of the forms.
+Every resulting exponent has a positive coordinate of index greater
+than $h$.  It consequently lies strictly after $a_he_h$ in the
+chosen order, because $a_he_h$ has zero coordinates at all indices
+greater than $h$.  The displayed element therefore has leading
+position $a_he_h$ with coefficient $1$.  Thus $a_he_h\in\mathcal U$
+for every $h$, proving $P\subseteq M$.
+
+It remains to compare the numerical invariants.  The scalar extension
+of $S/M$ from $\C$ to $k$, with $x_i$ renamed $y_i$, is $T/M_k$.
+Its Hilbert function is unchanged.  Tensoring the Koszul complex over
+$\C$ with the field $k$ also preserves exactness and commutes with
+homology, so
+\begin{align*}
+ \dim_k(T/M_k)_d&=\dim_\C(S/M)_d,\\
+ \beta^T_{p,j}(T/M_k)&=\beta^S_{p,j}(S/M).
+\end{align*}
+Equations \eqref{mono:dimension-transfer} and
+\eqref{mono:initial-quotient}, together with
+\eqref{filt:hilbert}, now give
+\[
+ m\dim_\C(S/I)_d
+   =\dim_k Q_d
+   =\dim_k(N/\operatorname{in}_{\prec}W)_d
+   =m\dim_\C(S/M)_d.
+\]
+Likewise, \eqref{mono:betti-transfer},
+\eqref{filt:betti}, and \eqref{mono:initial-quotient} yield, for
+every $p\geq0$ and every $j\in\Z$,
+\[
+ m\beta^S_{p,j}(S/I)
+  =\beta^T_{p,j}(Q)
+  \leq\beta^T_{p,j}(N/\operatorname{in}_{\prec}W)
+  =m\beta^S_{p,j}(S/M).
+\]
+Cancelling the positive integer $m$ proves both assertions.
+\end{proof}
+
+The right action in Lemma~\ref{mono:transfer} and the placement of the
+$\Delta$-coefficients on the left are deliberate.  The argument uses
+commutativity among the forms $t_i$, but never requires an individual
+coefficient in $\Delta$ to commute with a form.  Proposition~\ref{mono:bound}
+completes the reduction to an ideal containing $P$. The remaining
+comparison is the prescribed-powers theorem of Mermin and Murai
+\cite[Theorem~3.1]{MM}. Sections~\ref{sec:stabilization}--\ref{sec:tor}
+give a direct proof by stabilization, box compression, and a Koszul-rank
+induction.

preprints/The-Artinian-Lex-Plus-Powers-Betti-Theorem-September-23-2026/build/sections/04-stabilization.tex

@@ -0,0 +1,216 @@
+\section{Stabilization inside the exponent box}
+\label{sec:stabilization}
+
+We now start with a monomial ideal containing the prescribed powers.
+The aim of this section is to impose stability inside the finite exponent
+box while preserving the Hilbert function and weakly increasing every
+graded Betti number of the quotient. This is the pure-power
+deformation strategy of Mermin and Murai \cite[Section~3]{MM}.
+
+Let
+\[
+ S=\C[x_1,\ldots,x_n],\qquad
+ 2\le a_1\le\cdots\le a_n,\qquad
+ P=(x_1^{a_1},\ldots,x_n^{a_n}),\qquad b_i=a_i-1.
+\]
+For $1\le r\le n$ and $d\in\Z$, put
+\[
+ \mathcal B_r(d)
+ =\{u\in\Z_{\ge0}^r: |u|=d,\ 0\le u_i\le b_i
+                          \text{ for }1\le i\le r\}.
+\]
+In particular, this set is empty when $d<0$.
+
+\begin{definition}
+\label{stab:definition}
+A subset $X\subseteq\mathcal B_r(d)$ is \emph{stable} if
+\[
+ u\in X,\quad h<\ell,\quad
+ u+e_h-e_\ell\in\mathcal B_r(d)
+ \quad\Longrightarrow\quad u+e_h-e_\ell\in X.
+\]
+A monomial ideal containing $(x_1^{a_1},\ldots,x_r^{a_r})$ is
+\emph{stable in the box} if its included exponent vectors in
+$\mathcal B_r(d)$ form a stable set for every $d$.
+This definition imposes no transfer condition on monomials outside the box.
+\end{definition}
+
+The modification used below acts on one pair of variables.  Fix
+$h<\ell$, write $\xi=x_h$ and $q=x_\ell$, and choose a primitive
+$a_\ell$-th root of unity $\zeta\in\C$.  Define a graded
+$\C$-linear map $D:S\to S$ on monomials by
+\begin{equation}
+\label{stab:distraction-formula}
+ D(x^u)=\left(\prod_{i\ne\ell}x_i^{u_i}\right)
+             \prod_{s=0}^{u_\ell-1}(q-\zeta^s\xi),
+ \qquad u\in\Z_{\ge0}^n,
+\end{equation}
+with the empty product equal to $1$.
+The resolution argument below is the nested-factor mechanism used for
+distractions in~\cite[Theorem~2.19 and Corollary~2.20]{BCR}.
+We give the argument directly because the
+root-of-unity factors here are periodic, rather than eventually constant.
+
+\begin{lemma}
+\label{stab:distraction}
+For every monomial ideal $K\subseteq S$, its vector-space image $D(K)$
+is a homogeneous ideal.  Moreover,
+\begin{gather*}
+ \dim_\C(S/D(K))_d=\dim_\C(S/K)_d
+ \quad(d\in\Z),\\
+ \beta^S_{p,j}(S/D(K))=\beta^S_{p,j}(S/K)
+ \quad(p\ge0,\ j\in\Z).
+\end{gather*}
+The map also satisfies $D(P)=P$.
+\end{lemma}
+
+\begin{proof}
+The expansion of $D(x^u)$ has coefficient $1$ at $x^u$, and every
+other monomial is lexicographically larger than $x^u$.  Thus $D$ is
+unitriangular on each finite-dimensional graded piece and is invertible.
+Multiplication by a variable other than $q$ commutes with $D$, while
+\begin{equation}
+\label{stab:multiplication}
+ qD(x^u)=D(qx^u)+\zeta^{u_\ell}\xi D(x^u).
+\end{equation}
+Since $K$ is spanned by monomials and is an ideal, these identities show
+that $D(K)$ is an ideal.  The invertibility and degree preservation of
+$D$ give the Hilbert-function equality.
+
+To check preservation of $P$, first observe that
+\[
+ \prod_{s=0}^{a_\ell-1}(q-\zeta^s\xi)
+       =q^{a_\ell}-\xi^{a_\ell}\in P.
+\]
+Here the second term belongs to $P$ because $a_h\le a_\ell$.
+If a monomial is divisible by $q^{a_\ell}$, its image under $D$ is
+divisible by this displayed polynomial.  If it is divisible by another
+generator of $P$, that generator remains a factor of its image.
+Consequently $D(P)\subseteq P$.  In each degree this inclusion is an
+inclusion between vector spaces of the same finite dimension, so it is
+an equality.
+
+It remains to compare Betti numbers.  If $K=S$, both quotients are zero.
+Otherwise, choose a minimal exponent-multigraded free resolution
+$F_\bullet\to S/K$ with $F_0=S$, as in
+Lemma~\ref{filt:minimal}.  A nonzero differential entry from a generator
+of multidegree $u$ to one of multidegree $v$ has the form
+$c x^{u-v}$, where $u\ge v$ coordinatewise.  Keeping the ordinary
+graded free modules, replace this entry by
+\begin{equation}
+\label{stab:ratio-entry}
+ c\frac{D(x^u)}{D(x^v)}.
+\end{equation}
+The ratio is a polynomial: all unchanged monomial factors are nested,
+as are the products in~\eqref{stab:distraction-formula}.
+
+For each free summand whose generator has multidegree $v$, define a
+linear change on its coefficient polynomials by
+\begin{equation}
+\label{stab:coefficient-change}
+ C_v(x^{w-v})=\frac{D(x^w)}{D(x^v)},\qquad w\ge v.
+\end{equation}
+More explicitly, the right-hand side is
+\[
+ \left(\prod_{i\ne\ell}x_i^{w_i-v_i}\right)
+       \prod_{s=v_\ell}^{w_\ell-1}(q-\zeta^s\xi).
+\]
+It has the original monomial $x^{w-v}$ with coefficient $1$ and only
+lexicographically larger additional terms.  Hence $C_v$ is an
+invertible graded vector-space map.  These maps need not be
+$S$-linear; the new differential defined by~\eqref{stab:ratio-entry}
+is $S$-linear.
+
+The changes on the free summands intertwine the old and new
+differentials.  Indeed, for a source coefficient $x^{w-u}$ and a
+nonzero entry $c x^{u-v}$, the required equality is precisely
+\[
+ \frac{D(x^w)}{D(x^u)}
+ \left(c\frac{D(x^u)}{D(x^v)}\right)
+      =c\frac{D(x^w)}{D(x^v)}.
+\]
+Thus the new maps form a complex that is exact in every positive
+homological degree.  At $F_0=S$ the change is $C_0=D$, so its image
+from $F_1$ is $D(K)$.  We have obtained a free resolution of $S/D(K)$
+with the same ordinary graded free modules as $F_\bullet$.
+Every replaced nonzero entry still has its original positive degree,
+so this resolution is minimal.  Its free ranks in each degree give
+the asserted equality of all graded Betti numbers.
+\end{proof}
+
+\begin{proposition}
+\label{stab:main}
+Let $S=\C[x_1,\ldots,x_n]$, let $2\le a_1\le\cdots\le a_n$,
+and put $P=(x_1^{a_1},\ldots,x_n^{a_n})$.
+For every monomial ideal $K\supseteq P$, there is a monomial ideal
+$H\supseteq P$ stable in the box such that
+\[
+ \dim_\C(S/H)_d=\dim_\C(S/K)_d\quad(d\in\Z)
+\]
+and
+\[
+ \beta^S_{p,j}(S/K)\le\beta^S_{p,j}(S/H)
+ \qquad(p\ge0,\ j\in\Z).
+\]
+\end{proposition}
+
+\begin{proof}
+If $K$ is stable in the box, take $H=K$.  Otherwise choose a failed
+transfer from $q=x_\ell$ to $\xi=x_h$, with $h<\ell$, and form
+the map $D$ above.  Replace $K$ by
+\[
+ K'=\operatorname{in}_{\mathrm{lex}}D(K),
+\]
+where the leading monomial is the lexicographically largest one.
+By Lemma~\ref{stab:distraction}, $P\subseteq D(K)$, and hence
+$P\subseteq K'$.  That lemma and Proposition~\ref{filt:comparison}
+show that the Hilbert function is preserved and
+\[
+ \beta^S_{p,j}(S/K)\le\beta^S_{p,j}(S/K')
+ \qquad(p\ge0,\ j\in\Z).
+\]
+We prove that repeated replacements terminate by measuring progress
+in every lexicographic prefix.
+
+Fix a degree $d$ and a prefix $E$ of the degree-$d$ monomials, listed
+from greatest to least.  Let $\pi_E$ be the projection onto their span.
+Gaussian elimination with this ordered list of columns gives
+\[
+ \#\{m\in E:m\in K'\}
+       =\dim_\C\pi_E(D(K)_d).
+\]
+For the monomials $m\in K\cap E$, the vectors $\pi_E(D(m))$
+are linearly independent: their coordinates in their original
+monomial positions form a triangular matrix with diagonal entries $1$.
+Consequently
+\begin{equation}
+\label{stab:prefix-growth}
+ \#(K'\cap E)\ge\#(K\cap E)
+\end{equation}
+for every such prefix in every degree.
+
+There is a strict inequality for at least one prefix.  Choose a
+monomial $x^u\in K$ inside the box witnessing the failed transfer, so
+\[
+ m'=\frac{\xi}{q}x^u
+ \quad\text{lies inside the box but does not belong to }K.
+\]
+Its existence implies $1\le u_\ell<a_\ell$.  Let $E$ be the
+degree-$|u|$ prefix ending at $m'$.  Every monomial in $K\cap E$
+is strictly larger than $m'$, so its $D$-image has zero coefficient
+at $m'$.  On the other hand, the coefficient at $m'$ of $D(x^u)$ is
+\[
+ -\sum_{s=0}^{u_\ell-1}\zeta^s
+       =-\frac{1-\zeta^{u_\ell}}{1-\zeta}\ne0.
+\]
+Thus $\pi_E(D(x^u))$ is independent of the projected vectors already
+used for $K\cap E$, proving strictness in~\eqref{stab:prefix-growth}.
+
+Every monomial ideal containing $P$ is determined by the monomials
+it includes in the finite box $0\le u_i<a_i$.  There are therefore
+only finitely many such ideals.  The nondecreasing prefix counts,
+with at least one strict increase at every replacement, prevent a
+repetition.  Continuing whenever stability fails must consequently
+stop at an ideal $H$ stable in the box.  The Hilbert-function equalities
+and all Betti inequalities persist along this finite sequence.
+\end{proof}

preprints/The-Artinian-Lex-Plus-Powers-Betti-Theorem-September-23-2026/build/sections/05-boxes.tex

@@ -0,0 +1,269 @@
+\section{Shadows and end slices in an exponent box}
+\label{sec:boxes}
+
+We now replace the monomials of a stable ideal, degree by degree, by
+lex segments in the exponent box.  The shadow comparison below will
+ensure that the replacement is an ideal.  At the same time we prove
+two comparisons for its first and last slices.
+The bounded-monomial theorem of Clements and Lindstr\"om gives the shadow
+comparison for arbitrary subsets when the bounds are nondecreasing
+\cite{CL}.  Here we assume stability of the subsets, allow the bounds in
+arbitrary order, and prove the two endpoint comparisons together with
+the shadow inequality.
+
+Recall the notation for a box with positive integer bounds
+\(b_1,\ldots,b_r\):
+\[
+ \mathcal B_r(d)=
+ \left\{u\in\Z_{\ge0}^{\,r}:
+       \sum_{h=1}^r u_h=d,\quad u_h\le b_h\text{ for every }h\right\}.
+\]
+Here \(d\) may be any integer; in particular the box is empty when
+\(d<0\) or \(d>\sum_h b_h\).  We also set
+\(\mathcal B_0(0)=\{()\}\) and \(\mathcal B_0(d)=\varnothing\) for
+\(d\ne0\).  No ordering of the bounds is required in this section.
+
+Vectors are ordered lexicographically, with the first coordinate
+taking precedence, and lex segments start with the largest vectors.
+For \(X\subseteq\mathcal B_r(d)\), let
+\(\operatorname{Lex}(X)\) be the lex segment of cardinality \(|X|\).
+Recall that \(X\) is \emph{stable} if
+\[
+ u\in X,\quad h<\ell,\quad
+ u+e_h-e_\ell\in\mathcal B_r(d)
+ \quad\Longrightarrow\quad u+e_h-e_\ell\in X.
+\]
+Every lex segment is stable, since each allowed transfer increases
+the vector in lex order.  Define the bounded shadow and the slices by
+\[
+ \begin{split}
+ \partial X
+   &=\{u+e_h\in\mathcal B_r(d+1):u\in X,\ 1\le h\le r\},\\
+ X(s)
+   &=\{u\in\mathcal B_{r-1}(d-s):(u,s)\in X\}
+       \qquad(0\le s\le b_r).
+ \end{split}
+\]
+The bounds in \(\mathcal B_{r-1}\) are the first \(r-1\) bounds.
+Empty sets have empty shadows; thus these definitions also cover
+negative degrees and empty slices.
+
+\begin{lemma}[Shadows and end slices]
+\label{box:inequalities}
+Let \(r\ge1\), let \(b_1,\ldots,b_r\) be positive integers, and let
+\(d\in\Z\).  If \(X\subseteq\mathcal B_r(d)\) is stable and
+\(Y=\operatorname{Lex}(X)\), then \(\partial Y\) is a lex segment and
+\begin{equation}
+ \label{box:three-inequalities}
+ |\partial X|\ge|\partial Y|,
+ \qquad |X(0)|\ge|Y(0)|,
+ \qquad |X(b_r)|\le|Y(b_r)|.
+\end{equation}
+\end{lemma}
+
+\begin{proof}
+We first prove the assertion about lex shadows, without assuming
+stability of \(X\).  Empty shadows need no argument.  For
+\(v\in\mathcal B_r(d+1)\), with \(d\ge0\), its lex largest
+degree-\(d\) predecessor is
+\[
+ p(v)=v-e_{\max\{h:v_h>0\}}.
+\]
+Every predecessor belongs to the bounded box.  Moreover, the map
+\(p\) preserves weak lex order.  Indeed, suppose \(v>w\), with first
+difference at coordinate \(k\).  Subtraction at the last nonzero
+coordinate cannot change an earlier coordinate of \(v\).  If the
+operation changes a coordinate of \(w\) before \(k\), then
+\(p(v)>p(w)\) already at that coordinate.  Otherwise the strict
+inequality at \(k\) can disappear only if \(v_k=w_k+1\) and the tail
+of \(v\) after \(k\) is zero.  Equality of total degrees then says
+that the tail of \(w\) contains exactly one unit.  Subtracting its
+last nonzero coordinate gives \(p(v)=p(w)\).  In all other cases the
+first strict inequality remains.  Membership of \(v\) in the shadow
+of a lex segment is equivalent to membership of \(p(v)\) in that
+segment, proving that its shadow is lex.
+
+We prove the three inequalities simultaneously by induction on
+\(r\).  If \(\mathcal B_r(d)\) is empty, they are immediate.  For
+\(r=1\), each degree box has at most one vector, so \(X=Y\).
+Assume henceforth that \(r>1\).  A prime will indicate lex
+replacement or shadow in the first \(r-1\) coordinates.
+
+Each slice \(X(s)\) is stable, and stability between slices gives
+\begin{equation}
+ \label{box:original-slices}
+ \partial'X(s)\subseteq X(s-1)
+       \qquad(1\le s\le b_r).
+\end{equation}
+Replace each slice by \(Z(s)=\operatorname{Lex}'(X(s))\), and let
+\(Z\subseteq\mathcal B_r(d)\) be the set with these slices.
+By the induction hypothesis and \eqref{box:original-slices},
+\[
+ |\partial'Z(s)|\le|\partial'X(s)|\le|X(s-1)|=|Z(s-1)|.
+\]
+The shadow on the left and the slice on the right are lex segments
+in the same degree box.  Consequently
+\begin{equation}
+ \label{box:compressed-slices}
+ \partial'Z(s)\subseteq Z(s-1)
+       \qquad(1\le s\le b_r).
+\end{equation}
+Thus a unit in the last coordinate of a member of \(Z\) may be moved
+to any earlier coordinate with available capacity, while remaining
+in \(Z\).
+
+\smallskip\noindent
+\emph{The zero slice.}
+If \(Z(0)=\mathcal B_{r-1}(d)\), then
+\(|Y(0)|\le|Z(0)|=|X(0)|\).  Otherwise let \(v\) be the largest
+vector of \(\mathcal B_{r-1}(d)\) omitted from \(Z(0)\); all members
+of \(Z(0)\) are strictly larger than \(v\).
+
+We claim that every \((u,s)\in Z\) is strictly larger than
+\((v,0)\).  Equality is excluded by the choice of \(v\).  If instead
+\((u,s)<(v,0)\), let \(k<r\) be their first differing coordinate.
+Then \(u_k<v_k\) and earlier coordinates agree. Also \(s>0\),
+since every member of the lex slice \(Z(0)\) is larger than \(v\).
+We will
+move the \(s\) units of the last coordinate into the first \(r-1\)
+coordinates to obtain a vector \(w\le v\).
+
+Let
+\[
+ c=\sum_{k<j<r}(b_j-u_j)
+\]
+be the available capacity strictly after coordinate \(k\).  If
+\(s\le c\), distribute all \(s\) units among those positions; the
+result still has \(w_k=u_k<v_k\).  If \(s>c\), fill all those
+positions and put the remaining \(s-c\) units at coordinate \(k\).
+Since \(|u|+s=|v|=d\) and the prefixes before \(k\) agree,
+\begin{equation}
+ \label{box:tail-filling}
+ s-c
+   =v_k-u_k-\sum_{k<j<r}(b_j-v_j)
+   \le v_k-u_k.
+\end{equation}
+The remaining units therefore fit at \(k\).  If the inequality in
+\eqref{box:tail-filling} is strict, then \(w_k<v_k\).  If equality
+holds, every coordinate of the tail of \(v\) is full, and the
+constructed vector is exactly \(v\).  In either case \(w\le v\).
+All the moves stay within the bounds, so repeated use of
+\eqref{box:compressed-slices} gives \(w\in Z(0)\), a contradiction.
+This proves the claim.
+
+There are thus at least \(|Z|=|X|\) vectors preceding \((v,0)\).
+The lex segment \(Y\) of that cardinality also consists entirely of
+vectors preceding \((v,0)\).  Its zero slice is therefore contained
+in \(Z(0)\), proving
+\begin{equation}
+ \label{box:zero-slice}
+ |Y(0)|\le|Z(0)|=|X(0)|.
+\end{equation}
+
+\smallskip\noindent
+\emph{The top slice.}
+Put \(B=\sum_{h=1}^r b_h\) and let
+\(\rho(u)=(b_1-u_1,\ldots,b_r-u_r)\).  The reflected complement
+\[
+ X^*=\rho\bigl(\mathcal B_r(d)\setminus X\bigr)
+       \subseteq\mathcal B_r(B-d)
+\]
+is stable.  In fact, if an allowed upward transfer took a member
+of \(X^*\) outside \(X^*\), reflection would give an upward transfer
+from a member of \(X\) to a vector outside \(X\), contradicting
+stability.  Reflection reverses lex order, so
+\(Y^*=\rho(\mathcal B_r(d)\setminus Y)\) is the lex segment of
+cardinality \(|X^*|\).
+
+The zero-slice inequality, already proved in \(r\) coordinates,
+applies to \(X^*,Y^*\).  Their zero-slice cardinalities are
+\[
+ \begin{split}
+ |X^*(0)|&=|\mathcal B_{r-1}(d-b_r)|-|X(b_r)|,\\
+ |Y^*(0)|&=|\mathcal B_{r-1}(d-b_r)|-|Y(b_r)|.
+ \end{split}
+\]
+It follows that \(|X(b_r)|\le|Y(b_r)|\).
+
+\smallskip\noindent
+\emph{The shadow.}
+We have established both end-slice comparisons.  To recover the
+shadow bound, we separate its zero slice from the remaining vectors.
+For any stable set \(X\),
+\begin{equation}
+ \label{box:shadow-count}
+ |\partial X|=|\partial'X(0)|+|X|-|X(b_r)|.
+\end{equation}
+Indeed, its shadow vectors with last coordinate zero are precisely
+\(\partial'X(0)\).  Every shadow vector \(v\) with positive last
+coordinate has \(v-e_r\in X\): if it was obtained as \(v=u+e_h\)
+with \(h<r\), transfer one unit of the positive last coordinate of
+\(u\) to coordinate \(h\).  This transfer is within bounds because
+\(v\) is within bounds.  Stability gives \(v-e_r\in X\).
+Thus adding \(e_r\) bijects the members of \(X\) outside its top
+slice with shadow vectors having positive last coordinate,
+proving~\eqref{box:shadow-count}.
+
+The same identity holds for the stable set \(Y\).  Its zero slice
+is a lex segment, and \eqref{box:zero-slice} gives
+\(Y(0)\subseteq\operatorname{Lex}'(X(0))\).  The induction
+hypothesis and monotonicity of shadows under inclusion imply
+\[
+ |\partial'X(0)|
+ \ge\bigl|\partial'\operatorname{Lex}'(X(0))\bigr|
+ \ge|\partial'Y(0)|.
+\]
+Combining this with the top-slice inequality in
+\eqref{box:shadow-count} proves \(|\partial X|\ge|\partial Y|\)
+and completes the induction.
+\end{proof}
+
+We apply the shadow comparison to the degree pieces of a monomial
+ideal.  For \(A=\C[x_1,\ldots,x_r]\) and
+\(P_r=(x_1^{b_1+1},\ldots,x_r^{b_r+1})\), a monomial ideal
+\(H\supseteq P_r\) is stable in the box sense precisely when each
+set
+\[
+ X_d=\{u\in\mathcal B_r(d):x^u\in H\}
+\]
+is stable.  Define its degreewise lex replacement by
+\begin{equation}
+ \label{box:lex-replacement}
+ (\Lambda H)_d
+   =(P_r)_d+
+     \operatorname{span}_{\C}
+        \{x^u:u\in\operatorname{Lex}(X_d)\},
+ \qquad
+ \Lambda H=\bigoplus_{d\ge0}(\Lambda H)_d.
+\end{equation}
+At this point \(\Lambda H\) is a graded vector subspace; the next
+proposition verifies its closure under multiplication.
+
+\begin{proposition}[Lex replacement is an ideal]
+\label{box:ideal}
+Let \(b_1,\ldots,b_r\) be positive integers, and let
+\(H\subseteq\C[x_1,\ldots,x_r]\) be a monomial ideal containing
+\(P_r=(x_1^{b_1+1},\ldots,x_r^{b_r+1})\) and stable in the box
+sense.  Then the subspace \(\Lambda H\) defined in
+\eqref{box:lex-replacement} is a monomial ideal containing \(P_r\).
+It is stable in the box sense and has the same Hilbert function as
+\(H\).
+\end{proposition}
+
+\begin{proof}
+Write \(Y_d=\operatorname{Lex}(X_d)\).  Since \(H\) is an ideal,
+\(\partial X_d\subseteq X_{d+1}\).  Lemma~\ref{box:inequalities}
+therefore gives
+\[
+ |\partial Y_d|\le|\partial X_d|
+                  \le|X_{d+1}|=|Y_{d+1}|.
+\]
+Both \(\partial Y_d\) and \(Y_{d+1}\) are lex segments in
+\(\mathcal B_r(d+1)\), so \(\partial Y_d\subseteq Y_{d+1}\).
+Thus multiplying a monomial of \(\Lambda H\) by a variable either
+gives a monomial of the next lex segment or leaves the box and
+enters \(P_r\).  Products of monomials already in \(P_r\) stay in
+\(P_r\).  This proves the ideal assertion.  Each \(Y_d\) is stable
+and has cardinality \(|X_d|\), which proves stability and equality
+of Hilbert functions.
+\end{proof}

preprints/The-Artinian-Lex-Plus-Powers-Betti-Theorem-September-23-2026/build/sections/06-tor.tex

@@ -0,0 +1,256 @@
+\section{Betti bounds for stable ideals}
+\label{sec:tor}
+
+The preceding section shows that degreewise lex replacement of a
+monomial ideal stable in the box is again an ideal with the same
+Hilbert function.  We prove that this replacement also bounds every
+graded Betti number.  The induction on the number of variables uses
+the first and last box slices.  After reduction modulo the last variable,
+these determine two endpoint modules; the middle modules have zero
+Koszul differentials.
+
+For modules with the same Hilbert function, the Koszul terms have equal
+dimensions in every homological and internal degree.  An upper bound on
+Betti numbers is therefore equivalent to a lower bound on the sum of the
+two adjacent differential ranks.  We first isolate this rank sum, which
+does not increase under taking either submodules or quotients.
+Let $m\ge0$ and $R=\C[x_1,\ldots,x_m]$.  For a graded $R$-module $B$ with
+finite-dimensional graded pieces, write $K^R_\bullet(B)$ for its
+Koszul complex and define its \emph{Koszul rank deficit} by
+\begin{equation}
+\label{tor:deficit-definition}
+ D^R_{p,j}(B)=\dim_\C K^R_p(B)_j-\beta^R_{p,j}(B),
+ \qquad p\ge0,\quad j\in\Z.
+\end{equation}
+Terms of the Koszul complex outside its homological range are zero.
+
+\begin{lemma}
+\label{tor:deficit}
+For every $p\ge0$ and $j\in\Z$, the quantity $D^R_{p,j}(B)$
+is the sum of the ranks of the two Koszul differentials adjacent to
+$K^R_p(B)_j$.  If $U\subseteq B$ is a graded submodule and $V$ is
+a graded quotient of $B$, then
+\[
+ D^R_{p,j}(U)\le D^R_{p,j}(B),\qquad
+ D^R_{p,j}(V)\le D^R_{p,j}(B).
+\]
+The quantity is additive on finite direct sums, satisfies
+\[
+ D^R_{p,j}(B(-s))=D^R_{p,j-s}(B),
+\]
+and vanishes when $R_{>0}B=0$.
+\end{lemma}
+
+\begin{proof}
+Write $d_p(B)_j:K^R_p(B)_j\to K^R_{p-1}(B)_j$ for the Koszul
+differential.  Since its homology computes $\Tor^R(B,\C)$,
+\[
+ \beta^R_{p,j}(B)
+ =\dim_\C K^R_p(B)_j
+       -\operatorname{rank}d_p(B)_j
+       -\operatorname{rank}d_{p+1}(B)_j.
+\]
+This proves the rank formula.  The Koszul complex of $U$ embeds
+termwise into that of $B$, and the image of each of its differentials
+embeds into the corresponding differential image for $B$.
+For the quotient $V$, the termwise surjections give
+\[
+ \operatorname{im}d_p(V)_j
+   =\text{the image of }\operatorname{im}d_p(B)_j
+       \text{ in }K^R_{p-1}(V)_j.
+\]
+Thus each of the two ranks can only decrease under either operation.
+Direct sums and grading shifts commute with the Koszul complex.
+Finally, if all the variables annihilate $B$, every Koszul differential
+is zero.
+\end{proof}
+
+\begin{theorem}
+\label{tor:stable}
+Let $r\ge1$, let $2\le a_1\le\cdots\le a_r$, and let
+$A=\C[x_1,\ldots,x_r]$.  Suppose that $H\subseteq A$ is a
+monomial ideal containing $(x_1^{a_1},\ldots,x_r^{a_r})$ and is
+stable in the box.  Let $G=\Lambda H$ be its degreewise lex
+replacement inside the box, together with the prescribed pure powers.
+Then, for every $p\ge0$ and $j\in\Z$,
+\[
+ \beta^A_{p,j}(H)\le\beta^A_{p,j}(G),
+ \qquad
+ \beta^A_{p,j}(A/H)\le\beta^A_{p,j}(A/G).
+\]
+\end{theorem}
+
+\begin{proof}
+By Proposition~\ref{box:ideal}, $G$ is an ideal stable in the box
+and has the same Hilbert function as $H$.  If $H=A$, then $G=A$
+and both comparisons are equalities.  For proper $H$, the ideal
+comparison implies the quotient comparison in positive homological
+degrees by Lemma~\ref{filt:minimal}:
+\begin{equation}
+\label{tor:ideal-quotient}
+ \beta^A_{p+1,j}(A/H)=\beta^A_{p,j}(H),
+ \qquad
+ \beta^A_{p+1,j}(A/G)=\beta^A_{p,j}(G).
+\end{equation}
+The two quotient Betti numbers in homological degree zero are both
+$1$ when $j=0$ and $0$ otherwise.  We prove the two comparisons
+simultaneously by induction on $r$, establishing the ideal comparison
+and then using~\eqref{tor:ideal-quotient}.  When $r=1$, a monomial
+ideal containing $x_1^{a_1}$ is a power ideal or the full ring, and
+$H=G$.
+
+Assume $r>1$ and $H$ proper, and put
+\[
+ R=\C[x_1,\ldots,x_{r-1}],\qquad z=x_r,\qquad a=a_r.
+\]
+For $s\ge0$, define coefficient ideals
+\[
+ H^{(s)}=\{u\in R:uz^s\in H\},\qquad
+ G^{(s)}=\{u\in R:uz^s\in G\}.
+\]
+These are ascending monomial ideals containing the earlier prescribed
+powers, and they equal $R$ for $s\ge a$.  For $s<a$, their
+stability in the earlier-variable box follows by applying stability
+with the last exponent fixed at $s$.  For $1\le s<a$ we also have
+\begin{equation}
+\label{tor:slice-annihilation}
+ R_{>0}H^{(s)}\subseteq H^{(s-1)},\qquad
+ R_{>0}G^{(s)}\subseteq G^{(s-1)}.
+\end{equation}
+To see the first inclusion, it is enough to multiply a monomial of
+$H^{(s)}$ by an earlier variable $x_i$.  If its earlier-variable
+exponent, or that exponent after multiplication by $x_i$, is outside
+the box, the earlier powers already imply the required membership.
+Otherwise stability moves one factor of $z$ to $x_i$.  The proof
+for $G$ is identical.
+
+We next pass from ideals over $A$ to modules over $R$.  Multiplication
+by $z$ is injective on $H$, since $H$ is an ideal in the domain $A$.
+Set
+\[
+ B_H=H/zH,\qquad B_G=G/zG.
+\]
+Reducing a minimal $A$-free resolution of $H$ modulo $z$ gives a
+minimal $R$-free resolution of $B_H$.  Here is a direct exactness
+check.  Lift a cycle in positive homological degree modulo $z$ to
+an element $y$ of the free resolution, and write $dy=zv$.
+Applying the next differential shows that $v$ is a cycle, using
+injectivity of $z$ on the free modules; in degree zero, apply the
+augmentation and use injectivity of $z$ on $H$ instead.
+Exactness gives $v=du$, so $y-zu$ is a cycle and hence a boundary.
+Its reduction modulo $z$ is the original cycle.  Right exactness
+identifies the cokernel with $B_H$, and all differential entries
+remain of positive degree or become zero, so minimality is preserved.
+The same argument applies to $G$.  Therefore
+\begin{equation}
+\label{tor:reduction}
+ \beta^A_{p,j}(H)=\beta^R_{p,j}(B_H),\qquad
+ \beta^A_{p,j}(G)=\beta^R_{p,j}(B_G).
+\end{equation}
+The injective multiplication maps also give
+\[
+ \dim_\C(B_H)_d=\dim_\C H_d-\dim_\C H_{d-1},\qquad
+ \dim_\C(B_G)_d=\dim_\C G_d-\dim_\C G_{d-1}.
+\]
+Consequently $B_H$ and $B_G$ have equal Hilbert functions.
+
+The monomial decomposition by powers of $z$ describes these modules
+more precisely; this is the decomposition appearing in
+\cite[Lemma~3.7]{CS}.  The coefficient of $z^0$ in $H/zH$ is $H^{(0)}$,
+and its coefficient of $z^s$ for $s\ge1$ is
+$H^{(s)}/H^{(s-1)}$.  Since $H^{(s)}=R$ for $s\ge a$, this gives
+an isomorphism of graded $R$-modules
+\begin{equation}
+\label{tor:slice-decomposition}
+ B_H\cong H^{(0)}
+       \oplus\bigoplus_{s=1}^{a-1}
+             (H^{(s)}/H^{(s-1)})(-s)
+       \oplus(R/H^{(a-1)})(-a).
+\end{equation}
+There are no terms for $s>a$.  The shifts record that multiplication
+by $z^s$ raises ordinary degree by $s$.  By
+\eqref{tor:slice-annihilation}, all middle summands are annihilated
+by the variables of $R$, as illustrated in
+Figure~\ref{fig:endpoint-slices}. The decomposition is $R$-linear
+because multiplying by an earlier variable preserves the exponent of
+$z$. The middle summands still contribute to $\Tor$, but their
+Koszul differentials have rank zero. Thus they contribute nothing to
+the deficits. Lemma~\ref{tor:deficit} yields
+\begin{equation}
+\label{tor:endpoint-deficits}
+ D^R_{p,j}(B_H)
+   =D^R_{p,j}(H^{(0)})
+      +D^R_{p,j-a}(R/H^{(a-1)}),
+\end{equation}
+and the identical formula holds for $G$.
+
+\input{figures/endpoint-slices}
+
+It remains to compare the two endpoint contributions.  Write
+$\Lambda'$ for lex replacement with the earlier prescribed powers
+in $R$.  The induction hypothesis applies to both $H^{(0)}$ and
+$H^{(a-1)}$.  It gives an ideal comparison for the first and a
+quotient comparison for the second.  Their respective lex
+replacements have the same Hilbert functions, and hence the same
+Koszul term dimensions.  Subtracting the Betti inequalities from
+these dimensions gives
+\begin{align}
+ D^R_{p,j}(H^{(0)})
+       &\ge D^R_{p,j}(\Lambda'H^{(0)}),
+       \label{tor:induction-zero}\\
+ D^R_{p,j-a}(R/H^{(a-1)})
+       &\ge D^R_{p,j-a}(R/\Lambda'H^{(a-1)}).
+       \label{tor:induction-top}
+\end{align}
+This also covers any coefficient ideal equal to $R$: its lex
+replacement is $R$, the corresponding ideal comparison is equality,
+and its quotient is zero.
+
+The slice inequalities of Lemma~\ref{box:inequalities} imply
+\begin{equation}
+\label{tor:endpoint-inclusions}
+ G^{(0)}\subseteq\Lambda'H^{(0)},\qquad
+ \Lambda'H^{(a-1)}\subseteq G^{(a-1)}.
+\end{equation}
+Indeed, fix an earlier-variable degree $e\ge0$.  For the first
+inclusion, apply the zero-slice inequality to the included exponent
+sets of $H$ and $G$ in total degree $e$.  For the second, apply
+the top-slice inequality in total degree $e+a-1$.  Each slice of
+$G$ is already a lex segment in the earlier-variable box, so its
+cardinality comparison with the corresponding slice of $H$ becomes
+the stated containment in the appropriate lex replacement.
+All monomials outside the earlier-variable box belong to both ideals.
+This proves the inclusions in every degree.
+
+Apply Lemma~\ref{tor:deficit} to the first inclusion in
+\eqref{tor:endpoint-inclusions} and to the quotient map furnished
+by the second:
+\[
+ R/\Lambda'H^{(a-1)}\longrightarrow R/G^{(a-1)}.
+\]
+Together with~\eqref{tor:induction-zero} and
+\eqref{tor:induction-top}, this gives
+\begin{align*}
+ D^R_{p,j}(H^{(0)})&\ge D^R_{p,j}(G^{(0)}),\\
+ D^R_{p,j-a}(R/H^{(a-1)})
+       &\ge D^R_{p,j-a}(R/G^{(a-1)}).
+\end{align*}
+Adding and using~\eqref{tor:endpoint-deficits}, we obtain
+\[
+ D^R_{p,j}(B_H)\ge D^R_{p,j}(B_G).
+\]
+The equal Hilbert functions of $B_H$ and $B_G$ imply
+\[
+ \dim_\C K^R_p(B_H)_j
+    =\binom{r-1}{p}\dim_\C(B_H)_{j-p}
+    =\dim_\C K^R_p(B_G)_j,
+\]
+with the binomial coefficient interpreted as zero outside its range.
+Subtracting the deficit inequality from these equal dimensions gives
+$\beta^R_{p,j}(B_H)\le\beta^R_{p,j}(B_G)$.
+Equation~\eqref{tor:reduction} proves the ideal comparison over $A$,
+and~\eqref{tor:ideal-quotient} completes the quotient comparison.
+Every step applies to every $p\ge0$ and $j\in\Z$; in particular,
+negative internal degrees and Koszul terms outside their homological
+range introduce no additional cases.
+\end{proof}

preprints/The-Artinian-Lex-Plus-Powers-Betti-Theorem-September-23-2026/build/sections/07-assembly.tex

@@ -0,0 +1,53 @@
+\section{The prescribed lex-plus-powers ideal}\label{sec:assembly}
+
+We now combine the comparisons, keeping the same ideal at every
+homological and internal degree.
+
+\begin{proof}[Proof of Theorem~\ref{thm:main}]
+If \(I=S\), then \(L_d=S_d\) is allowed in every degree and \(J=S\);
+all quotient Betti numbers are zero. Assume \(I\ne S\).
+Proposition~\ref{mono:bound} gives a monomial ideal \(M\supseteq P\)
+with the Hilbert function of \(I\) and
+\[
+ \beta^S_{p,j}(S/I)\le\beta^S_{p,j}(S/M)
+ \qquad(p\ge0,\ j\in\Z).
+\]
+Proposition~\ref{stab:main} gives a box-stable monomial ideal
+\(H\supseteq P\), with the same Hilbert function, and
+\[
+ \beta^S_{p,j}(S/M)\le\beta^S_{p,j}(S/H).
+\]
+By Proposition~\ref{box:ideal}, the degreewise box-lex replacement
+\(G=\Lambda H\) is an ideal, with the same Hilbert function.
+Theorem~\ref{tor:stable} gives
+\[
+ \beta^S_{p,j}(S/H)\le\beta^S_{p,j}(S/G).
+\]
+
+It remains to check that \(G\) is the ideal prescribed in
+\eqref{eq:prescription}.
+In degree \(d\), list all monomials in decreasing lex order and remove
+those belonging to \(P_d\). The remaining list is exactly
+\(\mathcal B_n(d)\) in decreasing lex order.
+The monomials of \(G_d\) outside \(P_d\) are the first
+\[
+ q_d=\dim_\C I_d-\dim_\C P_d
+\]
+positions of this remaining list. This integer lies between zero and
+\(|\mathcal B_n(d)|\), because \(H\supseteq P\) has the same Hilbert
+function as \(I\).
+
+Every box prefix is obtained by intersecting some full-list prefix with
+the box. Thus a lex-segment subspace \(L_d\) with
+\(\dim_\C(L_d+P_d)=\dim_\C I_d\) exists. Enlarging that full-list prefix
+to the largest one of the same resulting dimension adds only monomials
+already in \(P_d\), and does not change the sum. Consequently
+\(L_d+P_d=G_d\) for every \(d\), including the empty and full box cases.
+Hence \(J=G\). The displayed comparisons prove all the asserted Betti
+bounds for this single \(J\).
+\end{proof}
+
+The argument proves idealness and all syzygy bounds together.
+The use of \(\C\) enters the commuting-form construction and supplies the
+roots of unity for stabilization; no conclusion in positive
+characteristic is asserted for an arbitrary regular sequence.

preprints/The-Artinian-Lex-Plus-Powers-Betti-Theorem-September-23-2026/build/sections/08-characteristic-zero.tex

@@ -0,0 +1,85 @@
+\section{Arbitrary regular-sequence length and characteristic-zero fields}
+\label{sec:characteristic-zero}
+
+The new input of this paper is the Artinian theorem over \(\C\). We now
+apply the published Artinian reduction and record the coefficient-field
+argument needed for Corollary~\ref{cor:characteristic-zero}.
+
+\begin{proof}[Proof of Corollary~\ref{cor:characteristic-zero}]
+If \(I=S_F\), the prescription gives \(J=S_F\), and all quotient Betti
+numbers vanish. We may therefore assume that \(I\) is proper.
+
+First let \(F=\C\). After a linear change of coordinates, one may arrange
+that
+\[
+ f_1,\ldots,f_c,x_{c+1},\ldots,x_n
+\]
+is a regular sequence; the list of variables is empty when \(c=n\).
+Write \(\overline f_i\) for the images of the \(f_i\) in
+\(\overline S=\C[x_1,\ldots,x_c]\). They form a full regular sequence
+of degrees \(a_1,\ldots,a_c\). Theorem~\ref{thm:main}, applied in
+\(\overline S\) to every homogeneous ideal containing this sequence,
+gives both the Eisenbud--Green--Harris and lex-plus-powers assertions for
+\(\overline f_1,\ldots,\overline f_c\).
+
+The Artinian reduction of Caviglia--Maclagan
+\cite[Proposition~10]{CM08} transfers the Hilbert-function assertion to
+\(f_1,\ldots,f_c\). Theorem~4.1 of Caviglia--Kummini \cite{CK14}
+transfers the Betti assertion as well. Their formulation uses Betti
+numbers of ideals. For any proper homogeneous ideal \(K\) in a standard
+graded polynomial ring \(A\), the exact sequence
+\(0\to K\to A\to A/K\to0\) gives
+\[
+ \beta^A_{p+1,j}(A/K)=\beta^A_{p,j}(K)
+ \qquad(p\ge0,\ j\in\Z).
+\]
+The remaining quotient Betti number is \(\beta^A_{0,0}(A/K)=1\).
+Thus their convention is equivalent to the quotient convention used
+here. The lexicographic embedding modulo \(P_F\) in that reduction
+selects exactly the prescribed \(q_d\) greatest monomials outside \(P_F\)
+in each degree. A linear change of coordinates preserves Hilbert functions
+and graded Betti numbers, so this proves the corollary over \(\C\).
+
+Now let \(F\) be any characteristic-zero field. Choose homogeneous
+generators \(g_1,\ldots,g_r\) of \(I\), and let \(F_0\subseteq F\) be
+the field generated over \(\Q\) by all coefficients of the \(g_i\) and
+the \(f_i\). Set
+\[
+ S_0=F_0[x_1,\ldots,x_n],\qquad
+ I_0=(g_1,\ldots,g_r,f_1,\ldots,f_c)\subseteq S_0.
+\]
+Then \(I_0S_F=I\). The sequence \(f_1,\ldots,f_c\) is regular in
+\(S_0\): after tensoring each multiplication map with the faithfully
+flat extension \(F/F_0\), it is the corresponding injective map over
+\(S_F\). The finitely generated characteristic-zero field \(F_0\)
+admits an embedding into \(\C\). Extending by this embedding preserves
+regularity, so the case just proved applies to
+\[
+ S_{\C}=\C\otimes_{F_0}S_0,\qquad I_{\C}=I_0S_{\C}.
+\]
+No embedding of \(F\) into \(\C\) is needed.
+
+Let \(J_{\C}\) be the resulting lex-plus-powers ideal. It is a monomial
+ideal, so let \(J_0\subseteq S_0\) be generated by the same monomials and
+put \(J_F=J_0S_F\). For \(E=F\) or \(E=\C\), let
+\(S_E=E\otimes_{F_0}S_0\). Graded field base change preserves the
+dimensions of homogeneous pieces. It also gives, for \(M_0=S_0/I_0\)
+and for \(M_0=S_0/J_0\),
+\[
+ E\otimes_{F_0}\Tor^{S_0}_p(M_0,F_0)_j
+ \;\cong\;
+ \Tor^{S_E}_p(E\otimes_{F_0}M_0,E)_j.
+\]
+For example, this follows by tensoring the Koszul complex of the
+variables with \(E\); its graded homology commutes with this flat
+extension. Consequently the Hilbert equalities and Betti inequalities
+over \(\C\) give the same equalities and inequalities over \(F\).
+Finally, the monomials of \(P_F\) and the lexicographic order are
+determined by their exponents, while the dimensions defining \(q_d\)
+are unchanged by these field extensions. Hence \(J_F\) is precisely
+the degreewise ideal \(J\) prescribed in the corollary.
+\end{proof}
+
+The argument changes the coefficient field only through extensions of
+the common characteristic-zero field \(F_0\). It makes no assertion for
+arbitrary regular sequences in positive characteristic.

preprints/The-Artinian-Lex-Plus-Powers-Betti-Theorem-September-23-2026/paper.pdf

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:81874e1994ff2773543f714989feac78650ddc902f7f36047c6ce8d3288a9c05
+size 447798

preprints/The-Bass-trace-conjecture-for-complex-group-rings-September-24-2026/README.md

@@ -0,0 +1,18 @@
+# [The Bass trace conjecture and the characteristic-zero Kaplansky idempotent conjecture](The-Bass-trace-conjecture-for-complex-group-rings-September-24-2026.pdf)
+
+**Author:** OpenAI
+
+**Date:** September 24, 2026
+
+## Citation
+
+```bibtex
+@misc{OAI:The-Bass-trace-conjecture-for-complex-group-rings-September-24-2026,
+  author = {{OpenAI}},
+  title = {{The Bass trace conjecture and the characteristic-zero Kaplansky idempotent conjecture}},
+  howpublished = {OpenAI Math Release preprint
+                  \href{https://github.com/openai/math/blob/main/preprints/The-Bass-trace-conjecture-for-complex-group-rings-September-24-2026/The-Bass-trace-conjecture-for-complex-group-rings-September-24-2026.pdf}{OAI:The-Bass-trace-conjecture-for-complex-group-rings-September-24-2026}},
+  year = {2026}
+}
+```
+

preprints/The-Bass-trace-conjecture-for-complex-group-rings-September-24-2026/The-Bass-trace-conjecture-for-complex-group-rings-September-24-2026.pdf

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:1c4d90b5ed4de2ef66b4b07b3d5c3e2bfe8822a538177c64ee1d3c3e091120bf
+size 457936

preprints/The-Bass-trace-conjecture-for-complex-group-rings-September-24-2026/build/main.tex

@@ -0,0 +1,63 @@
+\documentclass[11pt]{article}
+\usepackage[T1]{fontenc}
+\usepackage{lmodern}
+\usepackage[margin=1.05in]{geometry}
+\usepackage{amsmath,amssymb,amsthm,mathtools}
+\usepackage{microtype,enumitem,booktabs}
+\usepackage{tikz}
+\usetikzlibrary{arrows.meta,positioning}
+\usepackage[colorlinks=true,linkcolor=blue!45!black,citecolor=blue!45!black,urlcolor=blue!45!black]{hyperref}
+\usepackage[nameinlink,noabbrev]{cleveref}
+\newtheorem{theorem}{Theorem}[section]
+\newtheorem{lemma}[theorem]{Lemma}
+\newtheorem{proposition}[theorem]{Proposition}
+\newtheorem{corollary}[theorem]{Corollary}
+\theoremstyle{definition}
+\newtheorem{definition}[theorem]{Definition}
+\theoremstyle{remark}
+\newtheorem{remark}[theorem]{Remark}
+\newcommand{\C}{\mathbb C}
+\newcommand{\R}{\mathbb R}
+\newcommand{\Z}{\mathbb Z}
+\newcommand{\N}{\mathbb N}
+\DeclareMathOperator{\HS}{HS}
+\DeclareMathOperator{\HH}{HH}
+\DeclareMathOperator{\tr}{tr}
+\DeclareMathOperator{\supp}{supp}
+\DeclareMathOperator{\con}{con}
+\DeclareMathOperator{\id}{id}
+\numberwithin{equation}{section}
+\setlist{itemsep=3pt,topsep=5pt}
+\setlength{\emergencystretch}{1.5em}
+\pdfinfoomitdate=1
+\pdftrailerid{}
+\pdfsuppressptexinfo=15
+\hypersetup{pdftitle={The Bass trace conjecture and the characteristic-zero Kaplansky idempotent conjecture},pdfauthor={OpenAI}}
+\title{The Bass trace conjecture and the characteristic-zero Kaplansky idempotent conjecture}
+\author{OpenAI}
+\date{September 24, 2026}
+\begin{document}
+\maketitle
+\begin{abstract}
+We prove the complex group-ring Bass trace conjecture for every discrete
+group: the Hattori--Stallings trace of a finitely generated projective module
+over its complex group ring is supported on conjugacy classes of finite-order
+elements. As a consequence, for every torsion-free group $G$ and every
+commutative unital domain $R$ of characteristic zero, the only idempotents
+in $RG$ are $0$ and $1$. This proves Kaplansky's idempotent conjecture in
+characteristic zero.
+\end{abstract}
+\tableofcontents
+\input{sections/01-introduction}
+\input{sections/02-framework}
+\input{sections/03-metric}
+\input{sections/04-filtrations}
+\input{sections/05-surgery}
+\input{sections/06-chains}
+\input{sections/07-idempotents}
+\input{sections/08-detector}
+\input{sections/09-completion}
+\input{sections/10-torsion-free}
+\bibliographystyle{plain}
+\bibliography{references}
+\end{document}

preprints/The-Bass-trace-conjecture-for-complex-group-rings-September-24-2026/build/references.bib

@@ -0,0 +1,267 @@
+@article{Bass1976,
+  author = {Bass, Hyman},
+  title = {{Euler} characteristics and characters of discrete groups},
+  journal = {Inventiones Mathematicae},
+  volume = {35},
+  year = {1976},
+  pages = {155--196},
+  doi = {10.1007/BF01390137}
+}
+
+@incollection{Bass1979,
+  author = {Bass, Hyman},
+  title = {Traces and {Euler} characteristics},
+  editor = {Wall, C. T. C.},
+  booktitle = {Homological Group Theory},
+  series = {London Mathematical Society Lecture Note Series},
+  volume = {36},
+  publisher = {Cambridge University Press},
+  address = {Cambridge},
+  year = {1979},
+  pages = {1--26},
+  doi = {10.1017/CBO9781107325449.003}
+}
+
+@article{Hattori1965,
+  author = {Hattori, Akira},
+  title = {Rank element of a projective module},
+  journal = {Nagoya Mathematical Journal},
+  volume = {25},
+  year = {1965},
+  pages = {113--120},
+  doi = {10.1017/S002776300001148X}
+}
+
+@article{Stallings1965,
+  author = {Stallings, John},
+  title = {Centerless groups---an algebraic formulation of {Gottlieb}'s theorem},
+  journal = {Topology},
+  volume = {4},
+  number = {2},
+  year = {1965},
+  pages = {129--134},
+  doi = {10.1016/0040-9383(65)90060-1}
+}
+
+@article{BCM2004,
+  author = {Berrick, A. J. and Chatterji, Indira and Mislin, Guido},
+  title = {From acyclic groups to the {Bass} conjecture for amenable groups},
+  journal = {Mathematische Annalen},
+  volume = {329},
+  number = {4},
+  pages = {597--621},
+  year = {2004},
+  doi = {10.1007/s00208-004-0521-6},
+  url = {https://arxiv.org/abs/1004.1941v1}
+}
+
+@article{BCM2006,
+  author = {Berrick, A. J. and Chatterji, Indira and Mislin, Guido},
+  title = {Homotopy idempotents on manifolds and {Bass}' conjectures},
+  journal = {Geometry \& Topology Monographs},
+  volume = {10},
+  year = {2007},
+  pages = {41--62},
+  doi = {10.2140/gtm.2007.10.41},
+  url = {https://arxiv.org/abs/0903.4341v1}
+}
+
+@incollection{CM2011,
+  author = {Chatterji, Indira and Mislin, Guido},
+  title = {{Hattori--Stallings} trace and {Euler} characteristics for groups},
+  booktitle = {Geometric and Cohomological Methods in Group Theory},
+  editor = {Bridson, Martin R. and Kropholler, Peter H. and Leary, Ian J.},
+  series = {London Mathematical Society Lecture Note Series},
+  volume = {358},
+  pages = {256--271},
+  publisher = {Cambridge University Press},
+  year = {2009},
+  doi = {10.1017/CBO9781139107099.007},
+  url = {https://arxiv.org/abs/0910.4419v1}
+}
+
+@article{Linnell1983,
+  author = {Linnell, P. A.},
+  title = {Decomposition of augmentation ideals and relation modules},
+  journal = {Proceedings of the London Mathematical Society},
+  series = {3},
+  volume = {47},
+  number = {1},
+  year = {1983},
+  pages = {83--127},
+  doi = {10.1112/plms/s3-47.1.83}
+}
+
+@article{BH2015,
+  author = {Berrick, A. J. and Hesselholt, Lars},
+  title = {Topological {Hochschild} homology and the {Bass} trace conjecture},
+  journal   = {J. reine angew. Math.},
+  volume = {704},
+  year = {2015},
+  pages = {169--185},
+  doi = {10.1515/crelle-2013-0051},
+  url = {https://arxiv.org/abs/1305.6438v2}
+}
+
+@article{BV1998,
+  author = {Burger, Marc and Valette, Alain},
+  title = {Idempotents in complex group rings: theorems of {Zalesskii} and {Bass} revisited},
+  journal = {Journal of Lie Theory},
+  volume = {8},
+  number = {2},
+  year = {1998},
+  pages = {219--228},
+  doi = {10.5802/jolt.142},
+  url = {https://jolt.centre-mersenne.org/articles/10.5802/jolt.142/}
+}
+
+@article{OW2023,
+  author = {{\"O}inert, Johan and Wagner, Stefan},
+  title = {Complex group rings and group {$C^*$}-algebras of group extensions},
+  journal = {Journal of Algebraic Combinatorics},
+  volume = {58},
+  year = {2023},
+  pages = {387--397},
+  doi = {10.1007/s10801-022-01183-6}
+}
+
+@article{Connes1985,
+  author = {Connes, Alain},
+  title = {Non-commutative differential geometry},
+  journal = {Publications Math\'ematiques de l'IH\'ES},
+  volume = {62},
+  year = {1985},
+  pages = {41--144},
+  doi = {10.1007/BF02698807}
+}
+
+@article{Burghelea1985,
+  author = {Burghelea, Dan},
+  title = {The cyclic homology of the group rings},
+  journal = {Commentarii Mathematici Helvetici},
+  volume = {60},
+  year = {1985},
+  pages = {354--365},
+  doi = {10.1007/BF02567420}
+}
+
+@article{Eckmann1986,
+  author = {Eckmann, Beno},
+  title = {Cyclic homology of groups and the {Bass} conjecture},
+  journal = {Commentarii Mathematici Helvetici},
+  volume = {61},
+  year = {1986},
+  pages = {193--202},
+  doi = {10.1007/BF02621911}
+}
+
+@misc{Alpert2025,
+  author = {Alpert, Hannah},
+  title = {Simplicial volume and 0-strata of separating filtrations},
+  year = {2024},
+  eprint = {2211.06362},
+  archivePrefix = {arXiv},
+  primaryClass = {math.DG},
+  note = {arXiv:2211.06362v3},
+  url = {https://arxiv.org/abs/2211.06362v3}
+}
+
+@article{Papasoglu2020,
+  author = {Papasoglu, Panos},
+  title = {Uryson width and volume},
+  journal = {Geometric and Functional Analysis},
+  volume = {30},
+  number = {2},
+  pages = {574--587},
+  year = {2020},
+  doi = {10.1007/s00039-020-00533-5}
+}
+
+@article{BraunSauer2021,
+  author = {Braun, Sabine and Sauer, Roman},
+  title = {Volume and macroscopic scalar curvature},
+  journal = {Geometric and Functional Analysis},
+  volume = {31},
+  number = {6},
+  pages = {1321--1376},
+  year = {2021},
+  doi = {10.1007/s00039-021-00588-y}
+}
+
+
+@article{Formanek1973,
+  author = {Formanek, Edward},
+  title = {Idempotents in {Noetherian} group rings},
+  journal = {Canadian Journal of Mathematics},
+  volume = {25},
+  number = {2},
+  year = {1973},
+  pages = {366--369},
+  doi = {10.4153/CJM-1973-037-6}
+}
+
+@article{FarrellLinnell2003,
+  author = {Farrell, F. Thomas and Linnell, Peter A.},
+  title = {Whitehead groups and the {Bass} conjecture},
+  journal = {Mathematische Annalen},
+  volume = {326},
+  year = {2003},
+  pages = {723--757},
+  doi = {10.1007/s00208-003-0424-y}
+}
+
+@article{BartelsLuckReich2008,
+  author = {Bartels, Arthur and L{\"u}ck, Wolfgang and Reich, Holger},
+  title = {On the {Farrell--Jones} Conjecture and its applications},
+  journal = {Journal of Topology},
+  volume = {1},
+  number = {1},
+  year = {2008},
+  pages = {57--86},
+  doi = {10.1112/jtopol/jtm008}
+}
+
+@article{MineyevYu2002,
+  author = {Mineyev, Igor and Yu, Guoliang},
+  title = {The {Baum--Connes} conjecture for hyperbolic groups},
+  journal = {Inventiones Mathematicae},
+  volume = {149},
+  number = {1},
+  year = {2002},
+  pages = {97--122},
+  doi = {10.1007/s002220200214}
+}
+
+@article{Puschnigg2002,
+  author = {Puschnigg, Michael},
+  title = {The {Kadison--Kaplansky} conjecture for word-hyperbolic groups},
+  journal = {Inventiones Mathematicae},
+  volume = {149},
+  number = {1},
+  year = {2002},
+  pages = {153--194},
+  doi = {10.1007/s002220200216}
+}
+
+@article{Kantorovich2006,
+  author = {Kantorovich, L. V.},
+  title = {On the translocation of masses},
+  journal = {Journal of Mathematical Sciences},
+  volume = {133},
+  number = {4},
+  year = {2006},
+  pages = {1381--1382},
+  doi = {10.1007/s10958-006-0049-2},
+  note = {Translation of the 1942 original}
+}
+
+@article{HjorthMolberg2006,
+  author = {Hjorth, Greg and Molberg, Mats},
+  title = {Free continuous actions on zero-dimensional spaces},
+  journal = {Topology and its Applications},
+  volume = {153},
+  number = {7},
+  year = {2006},
+  pages = {1116--1131},
+  doi = {10.1016/j.topol.2005.03.003}
+}

preprints/The-Bass-trace-conjecture-for-complex-group-rings-September-24-2026/build/sections/01-introduction.tex

@@ -0,0 +1,286 @@
+\section{Introduction}
+\label{sec:introduction}
+
+The rank of a projective module over a group ring contains information
+about individual conjugacy classes. This refinement of ordinary rank is
+a basic ingredient of group-ring Euler characteristics: alternating sums
+of projective-module ranks retain information that a numerical Euler
+characteristic forgets \cite{Bass1976,Bass1979}. The Bass trace conjecture
+asks how much of that information can occur at elements of infinite order.
+We first specify the complex version of this question.
+
+For a unital complex algebra $A$, let $K_0(A)$ be the Grothendieck group of
+finitely generated projective right $A$-modules, and set
+\[
+ \HH_0(A)=A/[A,A],
+\]
+where $[A,A]$ is the complex linear span of commutators. An idempotent
+$e=(e_{ij})\in M_q(A)$ represents the projective right module
+$eA^q$, with matrices acting on columns by left multiplication. Its
+Hattori--Stallings trace is
+\[
+ \HS_A([eA^q])=\left[\sum_{i=1}^q e_{ii}\right]\in\HH_0(A),
+\]
+extended additively to virtual projective classes.
+
+Let $G$ be a discrete group and write $\con(G)$ for its set of conjugacy
+classes. The quotient $\HH_0(\C G)$ identifies with
+$\bigoplus_{C\in\con(G)}\C C$ by
+\begin{equation}\label{eq:class-trace}
+ \left[\sum_{h\in G}a_hh\right]
+ \longmapsto
+ \sum_{C\in\con(G)}\left(\sum_{h\in C}a_h\right)C.
+\end{equation}
+Indeed, a commutator has zero sum on each conjugacy class, and the
+differences of conjugate group elements span the kernel of the displayed
+map. Such differences are commutators: $tst^{-1}-s=[t,st^{-1}]$.
+All sums arising from an element of $\C G$ have finite support.
+We write $\HS_G(x)(C)$ for the coefficient at $C$.
+
+The complex group-ring Bass conjecture asks whether these coefficients
+vanish on every infinite-order conjugacy class, for every group and every
+class in $K_0(\C G)$; see
+\cite[Conjecture 2.1]{BCM2004} and \cite[Conjecture 2]{CM2011}.
+
+\begin{theorem}\label{thm:bass}
+Let $G$ be any discrete group. For every $x\in K_0(\C G)$ and every
+infinite-order element $g\in G$,
+\[
+ \HS_G(x)([g]_G)=0.
+\]
+Equivalently, if $\con_{\mathrm{fin}}(G)$ denotes the conjugacy classes of
+finite-order elements, then
+\[
+ \HS_G\bigl(K_0(\C G)\bigr)
+ \subseteq\bigoplus_{C\in\con_{\mathrm{fin}}(G)}\C C.
+\]
+\end{theorem}
+
+This resolves the complex group-ring formulation positively. Finite-order
+coefficients need not vanish away from the identity: for a finite group
+$F$, the idempotent $|F|^{-1}\sum_{h\in F}h$ has coefficient
+$|C|/|F|$ at every conjugacy class $C$. The conclusion therefore differs
+from the identity-only support assertion for integral group rings.
+That integral assertion follows in Section~\ref{sec:completion} by combining
+Theorem~\ref{thm:bass} with the known restriction on its finite-order
+coefficients.
+
+\subsection{Trace rank and the idempotent conjecture}
+
+For a torsion-free group the identity is the only finite-order element,
+so Theorem~\ref{thm:bass} also has numerical and ring-theoretic
+consequences. Define the canonical trace and the augmentation by
+\[
+ \tau_G\!\left(\sum_{g\in G}a_g g\right)=a_1,\qquad
+ \epsilon_G\!\left(\sum_{g\in G}a_g g\right)=\sum_{g\in G}a_g.
+\]
+For a matrix $b=(b_{ij})\in M_q(\C G)$, put
+$\tau_{G,q}(b)=\sum_{i=1}^q\tau_G(b_{ii})$, and let
+$\epsilon_{G,q}(b)\in M_q(\C)$ be the entrywise augmentation.
+The coefficient at $[1]_G$ of the Hattori--Stallings trace defines
+\[
+ \tau_{G,*}:K_0(\C G)\longrightarrow\C,\qquad
+ \tau_{G,*}([e(\C G)^q])=\tau_{G,q}(e).
+\]
+The augmentation also defines the integer-valued augmentation rank
+\[
+ \operatorname{rk}_{\epsilon}:K_0(\C G)
+ \xrightarrow{(\epsilon_G)_*}K_0(\C)
+ \xrightarrow{\dim_{\C}}\Z.
+\]
+
+The idempotent conjecture commonly attributed to Kaplansky asks whether,
+for every torsion-free group $G$ and every field $k$, an element
+$e\in kG$ satisfying $e^2=e$ must be $0$ or $1$; see
+\cite[Problem 3]{OW2023}. It concerns elements of the group ring itself,
+not arbitrary idempotent matrices over that ring.
+
+\begin{corollary}\label{cor:torsion-free}
+Let $G$ be any torsion-free discrete group.
+\begin{enumerate}[label=\textup{(\roman*)}]
+ \item For every $q\geq1$ and every idempotent $e\in M_q(\C G)$,
+ \[
+  \tau_{G,q}(e)=\operatorname{rank}_{\C}\bigl(\epsilon_{G,q}(e)\bigr).
+ \]
+ Consequently $\tau_{G,*}=\operatorname{rk}_{\epsilon}$, with
+ $\Z$ viewed as a subgroup of $\C$, and
+ \[
+  \tau_{G,*}\bigl(K_0(\C G)\bigr)=\Z.
+ \]
+ \item For every commutative unital domain $R$ of characteristic zero,
+ every $e\in RG$ satisfying $e^2=e$ is either $0$ or $1$.
+\end{enumerate}
+\end{corollary}
+
+Part \textup{(ii)} resolves the characteristic-zero Kaplansky idempotent
+conjecture positively, and includes coefficient domains as well as fields.
+Section~\ref{sec:torsion-free} derives both parts from the trace theorem.
+The integer range in part \textup{(i)} concerns algebraic $K_0(\C G)$;
+its relation to the reduced $C^*$-completion is made explicit there.
+
+The integral trace consequence also links the theorem to fixed-point
+theory. A map $f$ is a \emph{homotopy idempotent} if $f\circ f\simeq f$.
+Berrick, Chatterji and Mislin's equivalences \cite[Theorems 1--2]{BCM2006}
+then imply that every homotopy idempotent on a connected closed smooth
+oriented manifold of dimension greater than two is freely homotopic to
+a map with exactly one fixed point. They also identify ordinary and
+$L^2$-Lefschetz numbers for pointed homotopy idempotents of connected
+closed smooth oriented manifolds that induce the identity on the
+fundamental group.
+Corollary~\ref{cor:homotopy-idempotents} states these consequences precisely;
+Section~\ref{sec:completion} explains their connection to projective
+Euler characteristics.
+
+\subsection{The two parts of the proof}
+
+We first describe the algebraic construction to explain the geometric
+statement it needs. The body of the paper begins with that geometric
+statement, formulated independently of group rings, and completes its
+proof before constructing the cycles and cocycles to which it applies.
+
+For a finitely generated group $G$, the algebraic construction starts
+with an idempotent $e\in M_q(\C G)$ and one infinite-order element
+$g\in G$. The finitely many nonzero group coefficients of $e$ determine
+a finite set of steps. Products of matrix
+coefficients along paths from $h$ to $gh$, followed by the ordinary
+matrix trace, give complex weights on finite lists of group elements.
+Idempotency says that summing over any intermediate vertex removes that
+vertex. Cyclicity of matrix trace gives a second identity, a cyclic shift
+whose last vertex is multiplied by $g$.
+
+Project the vertices to the coset set $Y=\langle g\rangle\backslash G$.
+The group $H=C_G(g)/\langle g\rangle$, where $C_G(g)$ is the centralizer
+of $g$, acts freely on $Y$. For every positive even dimension $2m$, the weighted
+lists define a simplicial cycle modulo this action. Its simplices use
+connected sets of vertices in one graph of bounded degree, fixed as $m$
+increases. A Pfaffian cocycle evaluates on this cycle as
+$2^{-m}\HS_G([e(\C G)^q])([g]_G)$. Here a cocycle means an alternating
+function on vertex lists that vanishes on simplicial boundaries. The
+explicit construction and the evaluation occupy
+Sections~\ref{sec:idempotents} and~\ref{sec:detector}.
+
+The geometric part, proved first in Sections~\ref{sec:framework}--\ref{sec:chains},
+shows that every invariant cocycle evaluates to zero on cycles of this
+kind in sufficiently high dimension. We cut the supporting simplicial
+complex successively into pieces of bounded diameter, choosing the
+separating sets nearly to minimize their areas. A volume estimate and
+repeated slicing make the expected number of points in the final
+zero-dimensional set arbitrarily small. A subdivision construction turns
+this point count into a bound for cocycle evaluations. Equivariant chain
+homotopy keeps the evaluation unchanged throughout the construction.
+
+The fixed graph degree makes increasing the dimension useful. Through
+each vertex there are only exponentially many connected vertex sets of a
+given size. On an $n$-simplex, Euclidean structures chosen from spanning
+trees of its vertex set give volume $1/n!$; repeated slicing produces the
+same factorial in the comparison with a point of the final separating
+set. After these factors cancel, radii chosen in terms of the fixed
+degree bound make the remaining volume ratio tend to zero with $n$.
+The local replacement argument turns this strict volume margin into the
+small expected point count. No homological-dimension bound on $H$ is
+assumed.
+
+Two features permit this argument for arbitrary groups. A compact
+probability space with a free action separates overlapping translates of
+local modifications, without any amenability assumption. Also, each top
+simplex has its own averaged Euclidean area functional. The areas need
+not match along common faces; the successive cuts keep the relevant
+dimensional measures away from those faces.
+The cocycles need not be globally bounded: near each original simplex,
+the construction uses only a fixed finite set of possible vertices.
+Bounding the cocycle on those lists suffices for the evaluation estimate.
+
+The final assembly in Section~\ref{sec:completion} chooses a large even
+dimension, forcing the individual trace coefficient to vanish. Passing
+from finitely generated groups to arbitrary groups uses the exact sum
+over subgroup conjugacy classes contained in an ambient conjugacy class.
+Section~\ref{sec:torsion-free} then extracts the augmentation rank from
+the identity coefficient. In the scalar case this rank is $0$ or $1$,
+and the standard faithful-trace argument forces the idempotent itself
+to be $0$ or $1$. A finite-coefficient reduction extends that conclusion
+from complex coefficients to every commutative characteristic-zero domain.
+
+\subsection{Relation to previous work}
+
+The rank construction comes from Hattori and Stallings
+\cite{Hattori1965,Stallings1965}; Bass developed its relation to Euler
+characteristics and formulated the integral conjecture in
+\cite{Bass1976}; see also \cite[Conjecture (4.5)]{Bass1979}.
+Bass proved complex trace vanishing at infinite-order elements for linear
+groups \cite[Theorem (6.1)(f)]{Bass1979}. For the integral trace,
+Linnell established a restriction on the possible nonzero coefficients
+\cite[Lemma 4.1]{Linnell1983}, later recovered through topological
+Hochschild homology by Berrick and Hesselholt \cite[Theorem A]{BH2015}.
+Its finite-order consequence is the input used in
+Section~\ref{sec:completion}.
+
+The scalar idempotent problem also has a long history of trace methods.
+Formanek proved the characteristic-zero conclusion for torsion-free
+groups satisfying the ascending chain condition on subgroups
+\cite[Theorem 1]{Formanek1973}. Burger and Valette revisit the theorems of
+Kaplansky and Zalesskii: the canonical trace of a complex scalar
+idempotent lies in $[0,1]$, its endpoints force the idempotent to be
+trivial, and its value is rational
+\cite[Theorems 1.3 and 2.1]{BV1998}. They also obtain
+$\tau_{G,*}(K_0(\C G))\subseteq\mathbb Q$
+\cite[Corollary 1.2]{BV1998}. Rationality still permits nonintegral
+values strictly between zero and one. For torsion-free groups,
+Theorem~\ref{thm:bass} identifies the trace with augmentation rank,
+which excludes those values. The subsequent faithful-trace argument
+is classical and is included in Section~\ref{sec:torsion-free}.
+
+Broader group classes have been treated by algebraic and analytic
+methods. Farrell and Linnell proved the complex Bass statement for
+elementary amenable groups as part of a result over integral domains
+\cite[Theorem 1.6]{FarrellLinnell2003}. Berrick, Chatterji and Mislin
+obtained it for amenable groups through rational surjectivity of the
+degree-zero Bost assembly map for countable groups
+\cite[Theorems 1.3--1.4 and Lemma 1.5]{BCM2004}. Their result concerns
+the stronger trace assertion on $\ell^1(G)$, with finite support on
+finite-order classes. An algebraic route uses induction from finite
+subgroups: the Farrell--Jones conjecture with complex coefficients
+implies the complex Bass statement
+\cite[Theorems 1.5 and 1.9]{BartelsLuckReich2008}.
+The present proof uses the finite support of group-ring idempotents
+throughout; it requires neither an assembly hypothesis nor a statement
+about a completed group algebra.
+
+For some groups, analytic methods give a stronger idempotent conclusion
+in the reduced group $C^*$-algebra $C_r^*(G)$, the operator-norm closure
+of $\C G$ acting on $\ell^2(G)$. Using Lafforgue's Banach $KK$-theory,
+Mineyev and Yu proved this for torsion-free subgroups of hyperbolic groups
+\cite[Theorems 19--21]{MineyevYu2002}. Puschnigg independently obtained
+the torsion-free hyperbolic case through local cyclic homology
+\cite{Puschnigg2002}; see also \cite[Section 6]{MineyevYu2002}.
+Our conclusion for arbitrary torsion-free groups concerns the group ring
+and the trace on its algebraic $K_0$, as distinguished in
+Section~\ref{sec:torsion-free}.
+
+The higher-dimensional algebraic construction has antecedents in the
+cyclic-homological approach to the conjecture. Connes constructed a pairing
+of even cyclic cocycles with projective classes
+\cite[Part II, \S2, Proposition 14]{Connes1985}. Burghelea's decomposition
+of cyclic homology by conjugacy classes identifies the infinite-order
+components through the reduced centralizers $C_G(g)/\langle g\rangle$
+\cite[Proposition 1.8 and Theorem I]{Burghelea1985}. Eckmann used
+higher-degree vanishing to obtain trace vanishing over rational group
+rings for specified classes of groups of finite rational homological
+dimension \cite[\S3, Theorems 3.1 and 3.3]{Eckmann1986}.
+Sections~\ref{sec:idempotents}--\ref{sec:detector} give direct finite-chain
+constructions in this tradition, including the normalization of the
+Pfaffian detector; no cyclic-homology theorem is needed as an input.
+The sparse-chain theorem replaces a homological-dimension hypothesis
+with connected-support control on the particular cycles.
+
+The geometric proof develops the separating-set method of Papasoglu
+\cite[\S2, especially Lemma 2.5]{Papasoglu2020}, the separating filtrations and simplex-counting
+argument of Alpert \cite[Lemmas 4, 6--7]{Alpert2025}, and the
+Cantor-parameter approach of Braun and Sauer
+\cite[\S\S2, 4--5]{BraunSauer2021}. Alpert's equivariant formulations
+\cite[\S4, especially Lemmas 11--12]{Alpert2025} provide the closest model
+for the local comparisons. Alpert's filtration results concern manifold
+geometry and compatible metrics. We give the full piecewise-linear argument needed
+for simplex-dependent areas and arbitrary cycles in the coinvariant
+chain complex. The separate Pfaffian calculation converts this
+geometric vanishing into a statement about each conjugacy-class
+coefficient.

preprints/The-Bass-trace-conjecture-for-complex-group-rings-September-24-2026/build/sections/02-framework.tex

@@ -0,0 +1,75 @@
+\section{A vanishing theorem for sparse simplicial cycles}
+\label{sec:framework}
+
+We formulate the geometric statement independently of group rings.
+Its input is a graph with a free group action and a simplicial complex
+whose top-dimensional vertex sets are connected in that graph. Its output
+is the vanishing of every invariant cocycle evaluation in sufficiently
+high dimension.
+
+For a set $Y$, let $\C^{(Y)}$ be the vector space with basis
+$\{\delta_y:y\in Y\}$, consisting of finite linear combinations. The
+augmented chains of the full simplex on $Y$ are
+\[
+ E_j=\bigwedge^{j+1}\C^{(Y)}\quad(j\geq0),\qquad E_{-1}=\C.
+\]
+The differential is contraction with the functional
+$\alpha(\sum_y c_y\delta_y)=\sum_y c_y$:
+\begin{equation}\label{eq:boundary}
+ \partial(\delta_{y_0}\wedge\cdots\wedge\delta_{y_j})
+ =\sum_{i=0}^{j}(-1)^i
+ \delta_{y_0}\wedge\cdots\wedge\widehat{\delta_{y_i}}
+ \wedge\cdots\wedge\delta_{y_j}.
+\end{equation}
+In degree zero this is the augmentation to $\C$; the empty wedge is $1$.
+For a simplicial subcomplex $K$, its augmented oriented chains
+$C_*(K)$ form a subcomplex of $E_*$.
+
+If a group $H$ acts on $Y$, it acts on both chain complexes by permuting
+vertices, with the signs prescribed by the exterior product. For an
+$H$-module $V$, its coinvariants are
+\[
+ V_H=V/\operatorname{span}_{\C}\{hv-v:h\in H,\ v\in V\}.
+\]
+The differentials induce differentials on coinvariants. A cycle in
+$C_n(K)_H$ is a class whose boundary is zero there; an arbitrary finite
+chain representing it need not have zero boundary before passage to
+coinvariants. An $H$-invariant linear functional $\psi:E_n\to\C$
+descends to $(E_n)_H$. We call it a cocycle when
+$\psi\partial=0$ on $E_{n+1}$.
+
+\begin{theorem}[Sparse-chain vanishing]\label{thm:sparse}
+For each integer $D\geq2$ there is an integer $n_0(D)$ with the following
+property. Let $Y$ be the vertex set of a countable connected simple graph
+of degree at most $D$, and let a countable group $H$ act by graph
+automorphisms, freely on vertices. Suppose $n\geq n_0(D)$ and $K$ is an
+$H$-invariant subcomplex of the full simplex on $Y$ such that:
+\begin{enumerate}[label=\textup{(\roman*)}]
+ \item $K$ is generated by its $n$-simplices and has finitely many
+ $H$-orbits of simplices;
+ \item the vertices of each $n$-simplex span a connected subgraph of $Y$.
+\end{enumerate}
+Then every $H$-invariant cocycle $\psi:E_n\to\C$ evaluates to zero on
+the image in $(E_n)_H$ of every cycle in $C_n(K)_H$.
+\end{theorem}
+
+The bound $n_0(D)$ depends only on the graph degree, not on $H$, $K$ or
+$\psi$. There is no boundedness hypothesis on $\psi$. We will bound it
+only on a finite set of basis wedges associated with each fixed original
+simplex. This distinction is necessary in the application.
+
+For the proof assume $K\ne\varnothing$; the empty case has no positive
+degree chains. Its realization $|K|$ consists of finitely supported
+probability vectors $p:Y\to[0,1]$ whose supports are simplices of $K$.
+A simplex will also denote its closed affine realization; $\sigma^\circ$
+denotes its relative interior. We call simplices of $K$ \emph{original}
+to distinguish them from the later subdivisions, and call its
+$n$-simplices \emph{top simplices}.
+
+The proof has three steps. First, we choose a metric and simplexwise
+areas for which balls have small volume in high dimension.
+Second, we construct separating filtrations with very few points in
+their final stratum, on average over a free probability parameter.
+Third, we use their subdivisions to construct equivariant chain maps
+homotopic to inclusion whose cocycle evaluations approach zero. Each
+approximation uses a finite filtration; no limiting filtration is needed.

preprints/The-Bass-trace-conjecture-for-complex-group-rings-September-24-2026/build/sections/03-metric.tex

@@ -0,0 +1,262 @@
+\section{A metric and volume bound for connected supports}
+\label{sec:metric}
+
+We work with the graph \(Y\), group \(H\), and nonempty
+\(n\)-dimensional complex \(K\) of Theorem~\ref{thm:sparse}, where
+\(n\geq1\).  We need two kinds of geometric control: a metric with
+finite neighborhoods in which to make cuts, and Euclidean area
+functionals for which those cuts satisfy a slicing inequality.
+We will use a scaled transportation metric on \(|K|\) and,
+separately inside each top simplex, Euclidean structures determined
+by its spanning trees.  Each top simplex will have volume \(1/n!\),
+while every affine branch of a distance function will have gradient
+norm at most one in each of these structures.  Counting the possible
+connected vertex supports will then give the ball-volume estimate
+used in the separating argument.
+
+\subsection{The metric and its finite local geometry}
+
+Represent a point of \(|K|\) by a probability vector \(p\) on \(Y\) whose
+support lies in a simplex of \(K\).  Let \(d_Y\) denote graph distance and
+define
+\begin{equation}
+ \rho(p,q)=\frac1{\sqrt n}
+ \sup_{\operatorname{Lip}_{d_Y}(\ell)\leq1}
+ \sum_{v\in Y}\ell(v)\bigl(p(v)-q(v)\bigr).
+ \label{eq:transport-metric}
+\end{equation}
+This is the scaled Kantorovich--Rubinstein metric on finitely
+supported probability measures, in its Lipschitz-potential form
+\cite[pp.~1381--1382]{Kantorovich2006}. We establish its needed
+properties directly.
+Only finitely many terms occur in the pairing.  To see that the
+supremum is finite and describe the distance function on each simplex,
+let \(V\subset Y\) be a nonempty finite set and put
+\[
+ \mathcal P_V=
+ \left\{\ell\in\R^V:
+   \sum_{v\in V}\ell(v)=0,\quad
+   |\ell(v)-\ell(w)|\leq d_Y(v,w)\quad(v,w\in V)\right\}.
+\]
+This is a nonempty compact polytope.  Indeed, its difference constraints
+and zero mean bound every coordinate by
+\(\max_{v,w\in V}d_Y(v,w)\).  Each \(\ell\in\mathcal P_V\) extends to
+a \(1\)-Lipschitz function on \(Y\) by
+\[
+ \widetilde\ell(y)=\min_{v\in V}\bigl(\ell(v)+d_Y(v,y)\bigr).
+\]
+The difference constraints show that this extension agrees with
+\(\ell\) on \(V\), and the triangle inequality proves that it is
+\(1\)-Lipschitz.  Conversely, a global \(1\)-Lipschitz function restricts,
+after subtraction of its mean, to an element of \(\mathcal P_V\).
+Constants pair to zero with \(p-q\).  Consequently, when \(V\) contains
+both supports, the supremum in \eqref{eq:transport-metric} is the maximum
+over \(\mathcal P_V\).
+
+Symmetry follows by replacing \(\ell\) with \(-\ell\), and the triangle
+inequality follows by applying each potential to
+\(p-q=(p-u)+(u-q)\).  The indicator of a single vertex is
+\(1\)-Lipschitz for \(d_Y\), so
+\begin{equation}
+ |p(v)-q(v)|\leq\sqrt n\,\rho(p,q)
+ \qquad(v\in Y).
+ \label{eq:coordinate-control}
+\end{equation}
+Thus \(\rho\) separates points and is a metric.  It is \(H\)-invariant
+because graph automorphisms permute the admissible potentials.  On
+probability vectors supported in a fixed finite set, it is the
+restriction of a norm on the finite-dimensional space of vectors with
+coordinate sum zero.  It therefore gives the usual topology on every
+finite subcomplex.
+
+For \(c\in|K|\), write
+\[
+ \begin{aligned}
+ f_c(p)&=\rho(c,p),& U_s(c)&=\{p:f_c(p)<s\},\\
+ L_s(c)&=\{p:f_c(p)\leq s\},& S_s(c)&=\{p:f_c(p)=s\}.
+ \end{aligned}
+\]
+On a fixed closed simplex \(\sigma\), the function \(f_c\) is a maximum
+of finitely many affine functions: use the extreme points of
+\(\mathcal P_{V(\sigma)\cup\supp c}\).  Here \(V(\sigma)\) denotes the
+vertex set of \(\sigma\).  Taking all these extreme points makes the
+collection of affine branches equivariant under \(H\).
+
+\begin{lemma}\label{lem:proper}
+Every closed \(\rho\)-ball meets only finitely many simplices of \(K\).
+The complex \(K\) is locally finite, and \(\rho\) gives its usual
+realization topology and is proper.  For each \(c\in|K|\), \(s\geq0\),
+and simplex \(\kappa\), the set
+\[
+ \{h\in H:L_s(hc)\cap\kappa\ne\varnothing\}
+\]
+is finite.  The setwise stabilizer \(H_\kappa\) of a nonempty simplex
+is finite as well.
+\end{lemma}
+
+\begin{proof}
+Let \(A=\supp c\).  The function
+\(d_Y(\,\cdot\,,A)\) is \(1\)-Lipschitz and vanishes on \(A\).
+Using its negative in \eqref{eq:transport-metric} gives
+\begin{equation}
+ \min_{\substack{a\in\supp c\\v\in\supp p}}d_Y(a,v)
+ \leq \sum_v p(v)d_Y(v,A)
+ \leq\sqrt n\,\rho(c,p).
+ \label{eq:support-distance}
+\end{equation}
+If a top simplex \(\sigma\) meets \(L_s(c)\) at \(p\), it therefore
+contains a vertex within distance \(\lfloor s\sqrt n\rfloor\) of \(A\).
+The graph on its \(n+1\) vertices is connected, so every vertex of
+\(\sigma\) is within another \(n\) steps of that vertex.  Thus all
+such simplices use vertices in the finite graph neighborhood
+\[
+ B^Y_{\lfloor s\sqrt n\rfloor+n}(A)
+ :=\{v\in Y:d_Y(v,A)\leq\lfloor s\sqrt n\rfloor+n\}.
+\]
+There are only finitely many possibilities for \(\sigma\) and its
+faces.  Likewise, every top simplex containing a specified vertex
+\(v\) uses vertices in \(B^Y_n(\{v\})\); hence each vertex has a
+finite star.
+
+For completeness, the finite-subcomplex topology statement implies
+the corresponding statement on all of \(|K|\).  Given \(p\), let
+\(\eta=\min\{p(v):v\in\supp p\}>0\).  By
+\eqref{eq:coordinate-control}, the ball of radius
+\(\eta/(2\sqrt n)\) about \(p\) lies in the finite star of
+\(\supp p\).  Within this star the metric and realization topologies
+agree.  The open star of \(\supp p\) is also a neighborhood of \(p\)
+in the locally finite realization, so the agreement is local in
+both topologies.  A closed metric ball is a closed subset of the
+finite union of closed simplices it meets, and is therefore compact.
+
+If \(L_s(hc)\) meets \(\kappa\), \eqref{eq:support-distance} supplies
+\(a\in A\) and \(b\in V(\kappa)\) with
+\(d_Y(ha,b)\leq s\sqrt n\).  For each \(a\), the map \(h\mapsto ha\)
+is injective because \(H\) acts freely on vertices.  The finitely
+many pairs \(a,b\) and finite graph balls therefore allow only
+finitely many \(h\).  Finally, for \(v\in V(\kappa)\), the same
+injection maps \(H_\kappa\) into \(V(\kappa)\).  In particular,
+\(|H_\kappa|\leq|V(\kappa)|\).
+\end{proof}
+
+We will also use the following consequence.  For fixed \(c\) and \(s\),
+only finitely many \(h\in H\) satisfy
+\begin{equation}
+ L_s(hc)\cap L_s(c)\ne\varnothing.
+ \label{eq:finite-ball-overlaps}
+\end{equation}
+Indeed, an intersection gives \(\rho(hc,c)\leq2s\), and then
+\eqref{eq:support-distance} puts some vertex of \(h\supp c\)
+within distance \(2s\sqrt n\) of \(\supp c\).  The same
+injectivity argument applies.
+
+\subsection{Simplex areas and distance gradients}
+
+The metric now provides finite neighborhoods in which we can make
+cuts.  For area comparisons, we use Euclidean structures separately
+inside the top simplices.  They need not agree on shared faces.
+
+Fix a top simplex \(\sigma\).  For each spanning tree \(T\) of the
+connected graph on \(V(\sigma)\), equip its affine span with the
+Euclidean structure in which the tree-edge vectors
+\(\delta_v-\delta_w\), with either orientation for each edge, form
+an orthonormal basis of the translation space.  For \(0\leq j\leq n\),
+let \(a_j^\sigma\) be the average, over these finitely many trees,
+of Euclidean \(j\)-area on polyhedral subsets of \(\sigma^\circ\)
+of dimension at most \(j\).  We allow relatively open faces in
+these sets; pieces of dimension less than \(j\) have zero
+\(j\)-area, and \(a_0^\sigma\) is counting measure.  These area
+functionals are covariant under \(H\), since the action permutes
+the spanning trees.
+
+\begin{lemma}\label{lem:tree}
+In every tree Euclidean structure on a top simplex \(\sigma\), its
+\(n\)-volume is \(1/n!\).  Every affine branch of \(f_c|_\sigma\)
+has gradient of norm at most one, for every \(c\in|K|\).
+The same gradient bound holds after restriction to any affine
+subspace of the span of \(\sigma\).
+\end{lemma}
+
+\begin{proof}
+Root the tree at \(v_0\), orient its edges away from the root, and
+order the remaining vertices so that each parent precedes its
+children.  Put
+\[
+ t_i=\delta_{v_i}-\delta_{\operatorname{parent}(v_i)},
+ \qquad b_i=\delta_{v_i}-\delta_{v_0}
+ \quad(1\leq i\leq n).
+\]
+Each \(b_i\) is the sum of the \(t_j\) along its root path.
+Thus the change-of-basis matrix is triangular with diagonal
+entries one.  The \(t_i\) are an orthonormal basis in the chosen
+metric, so the parallelepiped spanned by the \(b_i\) has volume
+one.  The simplex has volume \(1/n!\).
+
+An affine branch has derivative along \(t_i\) equal, up to sign, to
+\[
+ \frac{\ell(v_i)-\ell(\operatorname{parent}(v_i))}{\sqrt n}
+\]
+for a \(1\)-Lipschitz potential \(\ell\).  Since the two vertices
+are adjacent, this derivative has absolute value at most
+\(1/\sqrt n\).  The squared gradient norm is the sum of the
+squares of these \(n\) derivatives and is at most one.
+Restriction to an affine subspace replaces the gradient by its
+orthogonal projection, which cannot increase its norm.
+\end{proof}
+
+\subsection{The volume margin in high dimension}
+
+We next bound the total top-dimensional area of a metric ball by
+counting its possible connected vertex supports.  Fix
+\[
+ R=4D^2,\qquad r=R/2.
+\]
+For a specified vertex \(v\), there are at most \(D^{2n}\)
+connected sets of \(n+1\) vertices containing \(v\).  To prove
+this, choose a spanning tree of each such set rooted at \(v\).
+A traversal using each tree edge once in each direction is a
+walk of length \(2n\), starting at \(v\), whose visited set is
+exactly the prescribed set.  Choosing one such walk per set
+gives an injection into the length-\(2n\) walks from \(v\),
+of which there are at most \(D^{2n}\).
+
+Set \(k=\lfloor R\sqrt n\rfloor\).  A graph ball of radius \(k\)
+has at most
+\[
+ \sum_{j=0}^kD^j\leq D^{k+1}
+\]
+vertices, since \(D\geq2\).  As \(|\supp c|\leq n+1\), there are
+at most \((n+1)D^{k+1}\) vertices within distance \(k\) of
+\(\supp c\).  By \eqref{eq:support-distance}, each top simplex
+meeting \(L_R(c)\) contains one of them.  The connected-set
+count and Lemma~\ref{lem:tree} therefore give, for every \(c\in|K|\),
+\begin{equation}
+ \sum_{\sigma\ \mathrm{top}}
+ a_n^\sigma\bigl(L_R(c)\cap\sigma^\circ\bigr)
+ \leq V_R:=
+ \frac{(n+1)D^{\lfloor R\sqrt n\rfloor+1}D^{2n}}{n!}.
+ \label{eq:volume-bound}
+\end{equation}
+Counting a simplex more than once only enlarges this upper bound.
+
+With \(D\) fixed, the ratio of this estimate to
+\((R-r)^n/n!\) is
+\[
+ \frac{V_R}{(R-r)^n/n!}
+ =\frac{(n+1)D^{\lfloor4D^2\sqrt n\rfloor+1}}{2^n}
+ \longrightarrow0.
+\]
+Indeed, its logarithm is
+\(-n\log2+4D^2(\log D)\sqrt n+O_D(\log n)\).
+We may therefore choose \(n_0(D)\geq1\) so that, for every
+\(n\geq n_0(D)\),
+\begin{equation}
+ V_R<\frac{(R-r)^n}{n!}.
+ \label{eq:strict-volume}
+\end{equation}
+For the remainder of the geometric proof, fix such an \(n\).
+The graph degree bound \(D\), and hence \(R\) and \(r\), remain
+fixed as dimensions increase.  The strict margin
+\eqref{eq:strict-volume} will convert successive area comparisons
+into a bound on the number of points at the bottom of a
+separating filtration.

preprints/The-Bass-trace-conjecture-for-complex-group-rings-September-24-2026/build/sections/04-filtrations.tex

@@ -0,0 +1,276 @@
+\section{Equivariant separating filtrations}\label{sec:filtrations}
+
+We retain the complex $K$, the metric $\rho$, and the radii $r<R$
+from Section~\ref{sec:metric}.  We will repeatedly cut $K$ into pieces of
+diameter at most $2R$, lowering the dimension of the separating set at
+each step.  The cuts vary over a compact probability space on which $H$
+acts freely.  This allows local modifications to be separated after
+taking account of the parameter, even when their translates overlap in
+$|K|$.
+
+\subsection{A free probability parameter}
+
+The following construction is a finite-color variant of the coinduction
+argument in Braun and Sauer \cite[Theorem 2.1]{BraunSauer2021}, who credit
+the existence theorem to Hjorth and Molberg \cite[Theorem 0.1]{HjorthMolberg2006}.
+We include the construction,
+including its torsion case, so that no free-action theorem is an input.
+
+\begin{lemma}\label{lem:free-parameter}
+Every countable group $H$ admits a continuous free action on a compact
+metrizable zero-dimensional space $X$, preserving a Borel probability
+measure $\mu$.
+\end{lemma}
+
+\begin{proof}
+For $a\in H\setminus\{1\}$, put $k_a=\operatorname{ord}(a)$ if $a$ has
+finite order, and $k_a=2$ otherwise.  Let
+\[
+ X_a=\{\xi:H\longrightarrow\Z/k_a\Z:
+                 \xi(ta)=\xi(t)+1\text{ for every }t\in H\}.
+\]
+This is a closed subspace of the product of finite discrete spaces
+$(\Z/k_a\Z)^H$.  Choose a representative $t_C$ for each left coset
+$C=t_C\langle a\rangle$.  A coloring in $X_a$ is specified uniquely by
+offsets $\omega_C\in\Z/k_a\Z$, through
+$\xi(t_Ca^j)=\omega_C+j$.  This is well defined also when $a$ has finite
+order, because the modulus is then its order.  Thus $X_a$ is nonempty,
+compact and zero-dimensional.
+
+Give the offsets independent uniform distributions.  The left action
+\[
+ (h\xi)(t)=\xi(h^{-1}t)
+\]
+preserves $X_a$ and its probability measure: it permutes the cosets and
+adds a fixed element of $\Z/k_a\Z$ to each offset.  These operations
+preserve every finite cylinder probability.  Take the product over
+$a\ne1$ to obtain $X$ and $\mu$.  Countability gives metrizability, and
+the coordinatewise action is continuous and measure preserving.
+For any $a\ne1$ and any point of $X$, its $a$-coordinate satisfies
+\[
+ (a\xi_a)(1)=\xi_a(a^{-1})=\xi_a(1)-1\ne\xi_a(1).
+\]
+Hence no nonidentity element fixes a point.  If $H$ is trivial, take a
+one-point probability space.
+\end{proof}
+
+Fix such $(X,\mu)$ for the rest of the geometric argument.  In
+particular, no amenability assumption on $H$ is needed.
+
+\subsection{The allowed filtrations}
+
+A triangulation refining $K$ means a finite affine simplicial
+subdivision of each closed original simplex, with agreement on shared
+faces.  A family $T(x)$ of such triangulations, together with specified
+subcomplexes, is \emph{locally constant} if, on each original simplex,
+all its exact affine simplices and subcomplex markings are constant on
+the pieces of a finite clopen partition of $X$.  The partition may
+depend on the original simplex.  We require equivariance:
+$T(hx)=hT(x)$, and likewise for every specified subcomplex.
+Refinement of triangulations always preserves their earlier
+subcomplexes.
+
+Begin with $T_n(x)=K$ and $Z_n(x)=|K|$.  At stage $d$, where
+$n\ge d\ge1$, suppose that $P(x)=Z_d(x)$ is a subcomplex of $T_d(x)$.
+An \emph{admissible} next choice consists of a locally constant equivariant
+refinement $T_{d-1}(x)$ and its subcomplex $Z_{d-1}(x)$ such that
+\begin{enumerate}[label=\textup{(\roman*)}]
+\item $Z_{d-1}(x)\subseteq P(x)$;
+\item for every simplex $F$ of $T_d(x)$ contained in $P(x)$,
+\[
+ \dim\bigl(F\cap Z_{d-1}(x)\bigr)\le\dim F-1;
+\]
+\item every connected component of $P(x)\setminus Z_{d-1}(x)$ has
+$\rho$-diameter at most $2R$.
+\end{enumerate}
+A negative dimension bound means that the intersection is empty.
+Thus the new separator avoids the vertices of $T_d$ and meets each
+higher-dimensional face in lower dimension; its own vertices belong to
+the finer triangulation $T_{d-1}$.
+Connected components use the subspace topology given by $\rho$.
+We call a filtration satisfying these conditions admissible.
+In particular, these conditions imply $\dim Z_i(x)\le i$.
+
+The local operation behind the construction is easiest to describe at a
+fixed parameter. For a separator $Z\subseteq P$, a center $c\in|K|$,
+and $t\in(r,R)$, remove the part of $Z$ inside the ball and insert
+the corresponding slice of $P$:
+\[
+ Z^{\prime}=(Z\setminus U_t(c))\cup(P\cap S_t(c)).
+\]
+The inserted slice prevents a component of $P\setminus Z^{\prime}$
+from passing between the ball and its exterior. Inside the ball its
+diameter is at most $2t$; outside, the old separator remains.
+The next lemma makes this operation compatible with triangulations,
+group translates, and clopen parameter choices. It also constructs the
+level cuts used to start a filtration. Only the affine descriptions of
+distance levels enter the construction; no Euclidean metrics need to
+agree across faces.
+
+\begin{lemma}[Compatible cuts]\label{lem:refinement}
+Fix locally constant equivariant triangulation families $T$ and $S$,
+where $S$ refines $T$, and subcomplex families $P\subseteq T$ and
+$Z\subseteq S$ with $Z\subseteq P$.  Let
+$\{(c_\alpha,B_\alpha)\}_{\alpha\in A}$ be a countable equivariant
+family of centers $c_\alpha\in|K|$ and clopen sets $B_\alpha\subseteq X$.
+Assume that only finitely many of the balls $L_R(c_\alpha)$ meet any
+given original simplex.  Set
+\[
+ \mathcal U_t(x)=\bigcup_{\alpha:\,x\in B_\alpha}U_t(c_\alpha),
+ \qquad
+ \mathcal S_t(x)=\bigcup_{\alpha:\,x\in B_\alpha}S_t(c_\alpha).
+\]
+Outside a countable set of radii $t\in(r,R)$, there is a locally
+constant equivariant triangulation refining $S$ in which both
+\begin{equation}\label{eq:cut-replacement}
+\begin{gathered}
+ P(x)\cap\mathcal S_t(x),\\
+ \bigl(Z(x)\setminus\mathcal U_t(x)\bigr)
+       \cup\bigl(P(x)\cap\mathcal S_t(x)\bigr)
+\end{gathered}
+\end{equation}
+are subcomplexes.  Each inserted level meets every face $F$ of $T$ or
+$S$ in dimension at most $\dim F-1$.  Consequently, if $Z$ satisfies
+the dimension-drop condition relative to $T$, both displayed sets do
+also.  If every component of $P\setminus Z$ has diameter at most $2R$,
+the second displayed set preserves that property.
+\end{lemma}
+
+\begin{proof}
+On a fixed original simplex the function
+$f_c(p)=\rho(c,p)$ is a maximum of finitely many affine functions.
+For a fixed face $F$, a level $\{f_c=t\}$ can have full dimension in
+$F$ only if one of these affine functions is identically $t$ on the
+affine span of $F$.  Otherwise the level is contained in a finite
+union of proper affine hyperplanes.  Thus only finitely many levels
+are exceptional for this face and center.  There are countably many
+original simplices and centers, and only finitely many exact
+triangulation types on each original simplex.  Excluding the
+resulting countable union of exceptional levels gives the stated
+dimension drop simultaneously for all faces of $T$ and $S$.
+
+We give the compatible refinement explicitly.  On each original
+simplex $\sigma$, cut the simplices of $S$ by all hyperplanes $b=t$,
+where $b$ is an affine branch for one of the finitely many relevant
+centers.  Use the
+canonical branch collection from the finite potential polytope in
+Section~\ref{sec:metric}.  On each piece of the finite clopen partition
+recording the old data and active conditions, this gives a finite
+polytopal complex $\mathcal P_\sigma$.  It represents the old
+subcomplexes and the closed conditions defining the two sets in
+\eqref{eq:cut-replacement}.
+
+These polytopal complexes may initially disagree on original faces.
+For each original simplex $\tau$, let $\mathcal Q_\tau$ be the common
+polytopal refinement of
+\[
+ \mathcal P_\sigma|_\tau
+ \quad\text{over all original cofaces }\sigma\supseteq\tau,
+ \quad\text{including }\sigma=\tau.
+\]
+There are finitely many cofaces by Lemma~\ref{lem:proper}.
+The common refinement consists of the nonempty intersections of cells
+and their faces.  If $\tau$ is a face of $\sigma$, then
+$\mathcal Q_\tau$ refines $\mathcal Q_\sigma|_\tau$, since every coface
+of $\sigma$ is also a coface of $\tau$.
+
+Triangulate the original simplices in increasing dimension.  When
+handling $\sigma$, its boundary has already been triangulated by a
+refinement of the boundary restrictions of $\mathcal Q_\sigma$.
+Extend this triangulation over the cells of $\mathcal Q_\sigma$ in
+increasing cell dimension.  Keep cells in the original boundary as
+already triangulated.  For every other positive-dimensional cell,
+cone its triangulated boundary to the arithmetic mean of its vertices,
+which lies in its relative interior.  Zero-dimensional cells need no
+operation.  The cones agree on common faces and give the desired finite
+affine subdivision, preserving the prescribed boundary triangulation.
+
+Every operation commutes with the affine maps induced by $H$:
+restriction, common refinement, and the mean of the full vertex set of
+a polytope are all canonical.  This proves equivariance even for
+nontrivial simplex stabilizers.  On each original simplex the procedure
+uses only finitely many data: those on its faces and their original
+cofaces, together with finitely many relevant cuts.  A common finite
+clopen partition makes all these data constant.  Thus the output has
+the required local constancy.  No manifold condition on $K$ is involved.
+
+The first set in \eqref{eq:cut-replacement} has the dimension bound
+already proved for the levels.  The second is contained in the union
+of those levels with $Z$, so it has the same bound whenever $Z$ does.
+For its separation property, let $C$ be a component of its complement
+in $P(x)$.  If $C$ meets an active $U_t(c_\alpha)$, continuity of
+$f_{c_\alpha}$ and connectedness of $C$ imply that $C$ stays in that
+ball, since it avoids the inserted level.  Its diameter is then at
+most $2t<2R$.  Otherwise $C$ lies outside all active open balls, where
+the old set $Z$ was retained.  It is therefore contained in an old
+component of $P(x)\setminus Z(x)$ and has diameter at most $2R$.
+\end{proof}
+
+The genericity against $S$ is useful when $S$ already contains the old
+separator $Z$: intersections of an inserted level with its faces have
+one less dimension.  The later area comparison will use precisely
+this conclusion.  The lemma itself does not require disjoint active
+balls; disjointness will be imposed when their areas are counted.
+
+\begin{lemma}\label{lem:filtration-exists}
+For any allowed choices through stage $d$, an allowed choice of
+$(T_{d-1},Z_{d-1})$ exists.  Hence a filtration can be continued to
+$Z_0$.
+\end{lemma}
+
+\begin{proof}
+Each original simplex is compact in $\rho$.  Cover representatives of
+the finitely many original simplex orbits by finitely many open balls
+$U_r(c)$, and take all their $H$-translates.  These balls cover $|K|$.
+By Lemma~\ref{lem:proper}, only finitely many translated radius-$R$
+balls meet a given original simplex.  Apply Lemma~\ref{lem:refinement}
+with every activation set equal to $X$, with $T=S=T_d$, and with
+$Z=\varnothing$.  For a common allowed radius $t\in(r,R)$, take
+\[
+ Z_{d-1}(x)=Z_d(x)\cap\bigcup_c S_t(c).
+\]
+This is a subcomplex of the resulting refinement and has the required
+dimension drop.  If $C$ is a component of $Z_d(x)\setminus Z_{d-1}(x)$,
+choose $p\in C$ and a covering ball $U_r(c)$ containing it.
+The connected set $f_c(C)$ avoids $t$ and contains a value below $t$,
+so $C\subseteq U_t(c)$.  Its diameter is at most $2t<2R$.
+The assertion is vacuous for an empty component space.
+\end{proof}
+
+\subsection{Dimension drop along the entire filtration}
+
+We record the consequence that will control chains in
+Section~\ref{sec:chains}.  For any simplex $F$ of $T_i(x)$ contained in
+$Z_i(x)$ and any $0\le j\le i$,
+\begin{equation}\label{eq:iterated-drop}
+ \dim\bigl(F\cap Z_j(x)\bigr)\le\dim F-(i-j).
+\end{equation}
+Indeed, the first step is the defining condition.  Its intersection is
+a subcomplex in the next refinement.  Applying the next dimension
+drop to each simplex of that subcomplex decreases its maximal
+dimension by one.  Induction gives \eqref{eq:iterated-drop}.
+Applied with $i=n$, this also shows that a proper original face meets
+$Z_j(x)$ in dimension strictly less than $j$.
+
+\begin{lemma}[Carriers of the zero-dimensional stratum]\label{lem:carrier}
+Let $(T_i,Z_i)_{i=0}^n$ be an allowed filtration and let $z\in Z_0(x)$.
+For every $i$, there is a unique $i$-simplex of $T_i(x)$ contained in
+$Z_i(x)$ and containing $z$.  The point $z$ lies in its relative
+interior.  In particular, every point of $Z_0(x)$ lies in the interior
+of an original top simplex.
+\end{lemma}
+
+\begin{proof}
+Let $F$ be the unique simplex of $T_i(x)$ whose relative interior
+contains $z$.  Since $Z_i(x)$ is a subcomplex containing $z$, it contains
+$F$, and $\dim F\le i$.  If $\dim F<i$, then
+\eqref{eq:iterated-drop} with $j=0$ would make $F\cap Z_0(x)$ empty.
+Thus $\dim F=i$.  Every other simplex containing $z$ has $F$ as a face,
+so no other $i$-simplex contains it.  Taking $i=n$ proves the last
+assertion.
+\end{proof}
+
+We have obtained a nonempty class of equivariant separating
+filtrations, together with a local cutting operation that stays within
+that class.  The next section chooses separators with area close to
+the infimum and compares them with these cuts.

preprints/The-Bass-trace-conjecture-for-complex-group-rings-September-24-2026/build/sections/05-surgery.tex

@@ -0,0 +1,249 @@
+\section{Local replacement and the number of zero strata}
+\label{sec:surgery}
+
+We now choose the separating filtration so that its zero-dimensional set
+has small expected cardinality in each original simplex.  Throughout this
+section the graph, complex, parameter space, dimension, and radii are fixed
+as in Sections~\ref{sec:metric} and~\ref{sec:filtrations}.  In particular,
+the strict inequality~\eqref{eq:strict-volume} holds.  Only the filtration
+will depend on a positive number $\epsilon$.
+
+\begin{proposition}\label{prop:zero-strata}
+For every $\epsilon>0$ there is an admissible equivariant locally constant
+filtration $(Z_j,T_j)_{j=0}^n$ such that, for every original top simplex
+$\sigma$ of $K$,
+\begin{equation}\label{eq:zero-strata}
+ \int_X\#\bigl(Z_0(x)\cap\sigma\bigr)\,d\mu(x)
+ \le C_\sigma\epsilon.
+\end{equation}
+Here $C_\sigma$ depends on the fixed geometric data but is independent of
+$\epsilon$ and of the complexity of the chosen triangulations.
+\end{proposition}
+
+\subsection{Successive near minimizers}
+
+The comparison with a cut through a ball follows the separating-set
+method of Papasoglu \cite[Section 2, especially Lemma 2.5]{Papasoglu2020}
+and its iterated and equivariant forms in Alpert
+\cite[Lemmas 6--7 and 11--12]{Alpert2025}. We prove the comparisons for our
+simplex-dependent areas, retaining the error from each near minimizer.
+
+\begin{proof}[Proof of Proposition~\ref{prop:zero-strata}]
+For an equivariant family $Z$ of dimension at most $j$, define
+\begin{equation}\label{eq:normalized-area}
+ \mathcal I_j(Z)=
+ \sum_{\sigma\ \mathrm{top}\bmod H}\frac{1}{|H_\sigma|}
+ \int_X a_j^\sigma\bigl(Z(x)\cap\sigma^\circ\bigr)\,d\mu(x).
+\end{equation}
+The sum runs over representatives of the original top simplices, and
+$H_\sigma$ is the finite setwise stabilizer of $\sigma$.  This quantity is
+finite for an admissible family.  There are finitely many representatives,
+and, on each, local constancy and compactness of $X$ give finitely many
+finite polyhedral configurations, each of finite area.
+
+Start with $(Z_n,T_n)=(|K|,K)$.  At step $d=n,\ldots,1$, keep all previously
+chosen data fixed and choose the admissible pair $(Z_{d-1},T_{d-1})$ so that
+$\mathcal I_{d-1}(Z_{d-1})$ is within $\epsilon$ of the infimum over the
+admissible choices at this step.  Lemma~\ref{lem:filtration-exists} ensures
+that this class is nonempty.  Its infimum is a finite nonnegative number,
+so such a choice exists.  Consequently, for every admissible replacement
+$Z'$ at this step,
+\begin{equation}\label{eq:near-minimality}
+ \mathcal I_{d-1}(Z_{d-1})
+ \le \mathcal I_{d-1}(Z')+\epsilon.
+\end{equation}
+Neither an attained minimum nor a limit of filtrations is used.
+
+\subsection{Disjoint replacements in the parameter and spatial variables}
+
+Fix a center $c\in|K|$.  The set
+\[
+ \mathcal F_c=
+ \{h\in H\setminus\{1\}:L_R(hc)\cap L_R(c)\ne\varnothing\}
+\]
+is finite.  Indeed such an intersection implies $\rho(hc,c)\le2R$;
+the support-distance bound, the finite graph balls and freeness on vertices
+then leave only finitely many possibilities for $h$.
+
+There is a finite clopen partition of $X$ into sets $A$ satisfying
+$A\cap hA=\varnothing$ for every $h\in\mathcal F_c$.  To construct one,
+use the free action and the zero-dimensional Hausdorff topology to choose,
+around each $x\in X$, a clopen neighborhood disjoint from its translates
+by this finite set.  A finite subcover exists by compactness; the atoms of
+its common refinement have the required property.  Thus
+\begin{equation}\label{eq:product-disjointness}
+ \{hA\times L_R(hc):h\in H\}
+ \quad\text{is a pairwise disjoint family.}
+\end{equation}
+This partition is fixed independently of $\epsilon$.  The assertion
+concerns the product of parameter and spatial variables; the sets $hA$
+alone need not be disjoint.
+
+Fix one atom $A$ and any clopen subset $B\subseteq A$.  For $r\le s\le R$
+put
+\begin{equation}\label{eq:local-areas}
+ M_j(s)=\int_B\sum_{\sigma\ \mathrm{top}}
+ a_j^\sigma\bigl(Z_j(x)\cap\sigma^\circ\cap L_s(c)\bigr)\,d\mu(x),
+ \qquad 0\le j\le n.
+\end{equation}
+For a generic $t\in(r,R)$, also put
+\[
+ N_{d-1}(t)=\int_B\sum_{\sigma\ \mathrm{top}}
+ a_{d-1}^\sigma
+ \bigl(Z_d(x)\cap\sigma^\circ\cap S_t(c)\bigr)\,d\mu(x).
+\]
+These sums involve only finitely many original simplices.  Unlike
+\eqref{eq:normalized-area}, they run over all such simplices and carry no
+stabilizer weights.
+
+At stage $d$, write $P=Z_d$ and $Z=Z_{d-1}$.  For each $x$, make the
+simultaneous replacement
+\begin{equation}\label{eq:local-replacement}
+ Z'(x)=
+ \left(Z(x)\setminus\bigcup_{h:x\in hB}U_t(hc)\right)
+ \cup
+ \left(P(x)\cap\bigcup_{h:x\in hB}S_t(hc)\right).
+\end{equation}
+The active closed balls are pairwise disjoint by
+\eqref{eq:product-disjointness}.  The families of balls and levels are
+locally finite, and their use is conditioned on clopen sets, so the
+construction is equivariant and locally constant on every original
+simplex.  Apply Lemma~\ref{lem:refinement} with $T=T_d$,
+$S=T_{d-1}$, $P=Z_d$, $Z=Z_{d-1}$, and the centers and activation
+sets $(hc,hB)$.  Outside a countable set of levels, it represents
+$Z'$ by an admissible common refinement: the inserted levels have
+the required dimension drop on faces of $T_d$ and of the old finer
+triangulation $T_{d-1}$, and the lemma preserves the diameter bound
+on components of $P(x)\setminus Z'(x)$.
+
+We next compute the change in~\eqref{eq:normalized-area}.  Genericity
+against the faces of $T_{d-1}$ implies that the old set $Z$ meets any
+inserted level in dimension at most $d-2$.  When $d=1$, that intersection
+is empty.  Hence the removed $(d-1)$-area is unchanged if the open ball is
+replaced by its closure, and the inserted cut has zero $(d-1)$-area
+overlap with the retained old set.
+
+The normalization in~\eqref{eq:normalized-area} makes the removed area
+exactly $M_{d-1}(t)$ and the inserted area exactly $N_{d-1}(t)$.  Here is
+the orbit-counting identity behind this assertion.  Let $Q$ denote the
+relevant equivariant set, and let $W(c)$ denote either the ball or the
+level.  Covariance of area and invariance of $\mu$ give
+\begin{align*}
+ &\sum_{\sigma\ \mathrm{top}\bmod H}\frac1{|H_\sigma|}
+   \sum_{h\in H}\int_{hB}
+   a_j^\sigma\bigl(Q(x)\cap\sigma^\circ\cap W(hc)\bigr)\,d\mu(x)
+ \\
+ &\quad=\sum_{\sigma\ \mathrm{top}\bmod H}\frac1{|H_\sigma|}
+   \sum_{h\in H}\int_B
+   a_j^{h^{-1}\sigma}
+   \bigl(Q(x)\cap(h^{-1}\sigma)^\circ\cap W(c)\bigr)\,d\mu(x)
+ \\
+ &\quad=\int_B\sum_{\tau\ \mathrm{top}}
+   a_j^\tau\bigl(Q(x)\cap\tau^\circ\cap W(c)\bigr)\,d\mu(x).
+\end{align*}
+The map $h\mapsto h^{-1}\sigma$ takes each value in the simplex orbit
+exactly $|H_\sigma|$ times.  All local contributions are nonnegative,
+and only finitely many translates meet any fixed representative simplex.
+The disjointness and zero-overlap statements above therefore yield
+\[
+ \mathcal I_{d-1}(Z')-\mathcal I_{d-1}(Z)
+ =N_{d-1}(t)-M_{d-1}(t).
+\]
+Combining this identity with~\eqref{eq:near-minimality} gives
+\begin{equation}\label{eq:local-comparison}
+ M_{d-1}(t)\le N_{d-1}(t)+\epsilon
+ \quad\text{for almost every }t\in(r,R).
+\end{equation}
+
+\subsection{Slicing and the strict volume margin}
+
+We have compared the area of a separating set inside a ball with that
+of a cut through the preceding set.  Slicing now connects these
+comparisons across the successive dimensions.
+
+Work first in one original top simplex $\sigma$ and one of its tree
+metrics.  Decompose $Z_d(x)\cap\sigma^\circ$ into affine polyhedral
+pieces on which $f_c=\rho(c,\cdot)$ is affine.  On a $d$-dimensional
+piece $Q$, its tangential gradient has norm $\alpha\le1$ by
+Lemma~\ref{lem:tree}.  Euclidean slicing perpendicular to that gradient
+gives
+\[
+ \int_r^b\operatorname{vol}_{d-1}(Q\cap\{f_c=t\})\,dt
+ =\alpha\operatorname{vol}_d(Q\cap\{r<f_c<b\})
+ \le\operatorname{vol}_d(Q\cap L_b(c)).
+\]
+A constant branch contributes at only one level.  Pieces of dimension
+less than $d$, including interfaces of this decomposition, have zero
+$(d-1)$-slice area outside finitely many exceptional levels.  Averaging
+over the tree metrics, summing over original top interiors, and
+integrating over $B$ yields
+\begin{equation}\label{eq:coarea}
+ \int_r^bN_{d-1}(t)\,dt\le M_d(b),\qquad r\le b\le R.
+\end{equation}
+These operations are legitimate by local finiteness and local constancy;
+on the fixed ball only finitely many configurations are involved.
+Every comparison has taken place in a single simplex metric before
+averaging, so no compatibility of metrics across original faces is needed.
+Indeed~\eqref{eq:iterated-drop} bounds $Z_d$ on a proper original face
+by dimension $d-1$, and its generic level slice by dimension $d-2$.
+Such faces contribute no area to the preceding comparison.  The same
+dimension estimate says that $Z_0$ misses every proper original face.
+
+From~\eqref{eq:local-comparison} and~\eqref{eq:coarea},
+\[
+ M_d(b)\ge\int_r^bM_{d-1}(t)\,dt-\epsilon(b-r).
+\]
+Induction, beginning with the monotonicity of $M_0$, gives
+\[
+ M_d(b)\ge
+ \frac{(b-r)^d}{d!}M_0(r)
+ -\epsilon\sum_{i=1}^d\frac{(b-r)^i}{i!}.
+\]
+Set
+\[
+ L=R-r,\qquad A_n=\frac{L^n}{n!},\qquad
+ \mathcal E_n=\sum_{i=1}^n\frac{L^i}{i!}.
+\]
+Since $Z_n=|K|$, the volume bound~\eqref{eq:volume-bound} implies
+\begin{equation}\label{eq:iterated-volume}
+ A_nM_0(r)\le V_R\mu(B)+\epsilon \mathcal E_n.
+\end{equation}
+
+We may now choose
+\[
+ B=A\cap\{x\in X:Z_0(x)\cap L_r(c)\ne\varnothing\}.
+\]
+This set is clopen: finitely many original simplices meet the ball, and
+their exact filtration data are constant on a common finite clopen
+partition.  Although $B$ uses the completed filtration, its use in an
+earlier comparison is legitimate.  Inequality~\eqref{eq:near-minimality}
+holds for every admissible competitor with the data down to stage $d$
+fixed; it imposes no condition on the later stages.  The same $B$ can
+therefore be used in every comparison leading to
+\eqref{eq:iterated-volume}.
+
+Each $x\in B$ contributes at least one zero stratum, and each such point
+lies in a unique original top interior.  Thus $\mu(B)\le M_0(r)$.
+Moreover $M_0(r)$ is the integral of the point count over all of $A$,
+because that count is zero on $A\setminus B$.  The strict inequality
+$V_R<A_n$ now gives the direct expected-count bound
+\begin{equation}\label{eq:ball-zero-strata}
+ \int_A\#\bigl(Z_0(x)\cap L_r(c)\bigr)\,d\mu(x)
+ =M_0(r)\le\frac{\mathcal E_n}{A_n-V_R}\,\epsilon.
+\end{equation}
+
+To pass from the ball estimate to \eqref{eq:zero-strata}, cover a fixed
+original top simplex $\sigma$ by finitely many open
+$r$-balls with fixed centers $c_1,\ldots,c_q$.  Such a cover exists by
+compactness of the simplex.  For each center, use the fixed clopen
+partition constructed above, and write $p_i$ for its number of atoms.
+Sum~\eqref{eq:ball-zero-strata} over these atoms and then over the balls.
+Since the corresponding closed balls also cover $\sigma$, this proves
+\eqref{eq:zero-strata} with
+\[
+ C_\sigma=\frac{\mathcal E_n}{A_n-V_R}\sum_{i=1}^q p_i.
+\]
+All quantities in this constant were fixed independently of $\epsilon$.
+In particular, no bound on the number of subdivision simplices or on
+the cardinality of an individual zero-dimensional fiber is needed.
+\end{proof}

preprints/The-Bass-trace-conjecture-for-complex-group-rings-September-24-2026/build/sections/06-chains.tex

@@ -0,0 +1,324 @@
+\section{From separating filtrations to finite chains}\label{sec:chains}
+
+Fix the dimension $n$ and radii chosen in Section~\ref{sec:metric}.
+For each $\epsilon>0$, use the filtration supplied by
+Proposition~\ref{prop:zero-strata}. We will turn it into an
+$H$-equivariant chain map
+\[
+ F^\epsilon:C_*(K)\longrightarrow E_*
+\]
+that is equivariantly chain homotopic to inclusion. Its evaluation by
+a fixed invariant cocycle will tend to zero. These two properties
+prove Theorem~\ref{thm:sparse}. Throughout this section $n$ remains
+fixed while $\epsilon$ tends to zero.
+
+\subsection{Component averages and their finite support}
+
+For a fixed parameter $x$, take the barycentric subdivision of the
+final triangulation $T_0(x)$. Its vertices are the barycenters $b_F$
+of nonempty simplices $F$ of $T_0(x)$. Define
+\[
+ \ell(F)=\min\{i:b_F\in Z_i(x)\},
+ \qquad Z_{-1}(x)=\varnothing.
+\]
+All $Z_i(x)$ are subcomplexes of $T_0(x)$, so $b_F\in Z_i(x)$
+is equivalent to $F\subseteq Z_i(x)$. Let $U_F$ be the connected
+component of $Z_{\ell(F)}(x)\setminus Z_{\ell(F)-1}(x)$ containing
+$b_F$. For any such component $U$, set
+\begin{equation}\label{eq:component-average}
+ J_U=\bigcup_{p\in U}\supp p,
+ \qquad
+ u_U=\frac{1}{|J_U|}\sum_{v\in J_U}\delta_v.
+\end{equation}
+The next lemma proves that $J_U$ is finite and controls these
+labels as $x$ varies.  The intended image of an oriented barycentric
+simplex is
+\begin{equation}\label{eq:exterior-replacement}
+ [b_{F_0},\ldots,b_{F_j}]
+ \longmapsto u_{U_{F_0}}\wedge\cdots\wedge u_{U_{F_j}}\in E_j.
+\end{equation}
+
+This choice uses the separating filtration in a specific way.
+If nested faces $F\subset F'$ have the same level $i$, the segment
+joining their barycenters lies in $Z_i\setminus Z_{i-1}$.
+Except at its endpoint $b_F$, it lies in the relative interior
+of $F'$, which the subcomplex $Z_{i-1}$ misses; the endpoint also
+avoids $Z_{i-1}$ by its level.  The two barycenters therefore receive
+the same component vector, making any exterior term containing
+both zero.  Along a complete flag $F_0\subset\cdots\subset F_n$,
+with $\dim F_i=i$, the levels are nondecreasing: if
+$F\subset F'\subseteq Z_i$, then $F\subseteq Z_i$.
+A nonzero image must consequently use all levels $0,1,\ldots,n$,
+beginning at a point of $Z_0$.  We will bound the number of these
+flags after establishing that the labels define finite chains.
+
+\begin{lemma}\label{lem:finite-component-labels}
+For every original simplex $\kappa$ of $K$, there is a finite set
+$W_\kappa\subseteq Y$, independent of $\epsilon$ and $x$, such that
+$J_{U_F}\subseteq W_\kappa$ whenever $b_F\in\kappa$. The triangulation
+over $\kappa$, together with all its assigned vectors $u_{U_F}$,
+is locally constant in $x$. These assignments are equivariant.
+\end{lemma}
+
+\begin{proof}
+Each component of $Z_i(x)\setminus Z_{i-1}(x)$ has
+$\rho$-diameter at most $2R$. For $i=0$ the components are
+singletons, since $Z_0(x)$ is a locally finite zero-dimensional
+subcomplex. If $b_F\in\kappa$ and $p\in U_F$, the support-distance
+bound from Lemma~\ref{lem:proper} gives
+\[
+ \min d_Y(\supp b_F,\supp p)
+ \leq\sqrt n\,\rho(b_F,p)\leq 2R\sqrt n.
+\]
+Choose an original top simplex containing $p$. Its $n+1$ vertices
+span a connected graph, so every vertex in $\supp p$ is within
+graph distance $n$ of the vertex furnished by this inequality.
+Consequently we may take
+\[
+ W_\kappa=
+ \left\{v\in Y:
+ d_Y\bigl(v,V(\kappa)\bigr)
+ \leq\left\lceil 2R\sqrt n+n\right\rceil\right\}.
+\]
+The bounded degree of $Y$ makes this set finite. Notice that the
+support of $p$ need not itself span a connected graph; we used
+the containing top simplex.
+
+Let $K[W_\kappa]$ be the finite subcomplex consisting of the simplices
+whose vertices belong to $W_\kappa$. The whole component $U_F$
+lies in its realization. It follows that $U_F$ is exactly the
+component containing $b_F$ in
+\[
+ \bigl(Z_{\ell(F)}(x)\setminus Z_{\ell(F)-1}(x)\bigr)
+ \cap |K[W_\kappa]|.
+\]
+Indeed $U_F$ is a connected subset of this restriction, and every
+connected subset of the restriction containing $b_F$ also lies in
+the original component. Thus restricting to this finite subcomplex
+does not cut off a part of the component needed to compute $J_{U_F}$.
+
+There are finitely many original simplices in $K[W_\kappa]$.
+A common refinement of their finite clopen parameter partitions
+makes all their triangulation and filtration data constant.
+On each resulting piece the components and their sets of support
+vertices are constant as well. This proves local constancy of
+the assignments over $\kappa$. Finally,
+$J_{hU}=hJ_U$, and hence $u_{hU}=h u_U$, proving equivariance.
+\end{proof}
+
+Subdivide each oriented original simplex into its fundamental
+subdivision chain and apply the exterior assignment
+\eqref{eq:exterior-replacement} to every term. This defines a map
+$f_x:C_*(K)\to E_*$. Both operations commute
+with the augmented boundary. For subdivision, interior facets
+cancel in pairs and the boundary facets give the compatible
+subdivision on each original face. For the second operation,
+each assigned vector has coefficient sum one, so contraction
+with the coefficient-sum functional gives
+\[
+ \partial(u_0\wedge\cdots\wedge u_j)
+ =\sum_{i=0}^j(-1)^i
+ u_0\wedge\cdots\wedge\widehat{u_i}\wedge\cdots\wedge u_j.
+\]
+Thus $f_x$ is an augmented chain map.
+
+By Lemma~\ref{lem:finite-component-labels}, its value on any
+original simplex is constant on a finite clopen partition and
+is supported in a fixed finite exterior space. We may therefore
+define
+\begin{equation}\label{eq:averaged-chain-map}
+ F^\epsilon(c)=\int_X f_x(c)\,d\mu(x).
+\end{equation}
+This integral is an actual finite chain: for every $c$ it is a
+finite linear combination of finitely many chain values.
+It commutes with the augmented boundary. Moreover,
+$f_{hx}(hc)=h f_x(c)$, so invariance of $\mu$ shows that
+$F^\epsilon$ is $H$-equivariant.
+
+\subsection{Counting the surviving flags}
+
+We now bound the evaluation of $f_x$ on a top simplex. The number
+of subdivision simplices may depend on $\epsilon$; the following
+bound depends instead on the number of zero-stratum points.
+The component-labeling and flag-counting method is modeled on
+Alpert \cite[Lemmas 4 and 10]{Alpert2025}; the carrier argument below
+supplies the bound without a manifold hypothesis.
+
+\begin{lemma}\label{lem:surviving-flags}
+For every original top simplex $\sigma$ and every $x$, at most
+\[
+ 2^n\#\bigl(Z_0(x)\cap\sigma\bigr)
+\]
+terms in its fundamental barycentric subdivision chain have
+nonzero images under the exterior replacement map.
+\end{lemma}
+
+\begin{proof}
+Suppress $x$. Such a term corresponds to a complete flag
+\[
+ F_0\subset F_1\subset\cdots\subset F_n,
+ \qquad \dim F_i=i,
+\]
+in the restriction of $T_0$ to $\sigma$. As shown when defining
+\eqref{eq:exterior-replacement}, a nonzero term has levels exactly
+$0,1,\ldots,n$. In particular $F_0=\{z\}$ for a point
+$z\in Z_0\cap\sigma$ and $F_i\subseteq Z_i$ for every $i$.
+
+By Lemma~\ref{lem:carrier}, $z$ lies in the relative interior of
+a unique $i$-simplex $S_i$ of $T_i$ contained in $Z_i$. To see why
+this carrier controls the final refinement, let $F_i$ occur in a
+surviving flag through $z$. The $T_i$-carrier of a relative-interior
+point of $F_i$ contains $F_i$ by refinement and belongs to $Z_i$,
+because $Z_i$ is a subcomplex. That carrier has dimension at least
+$i$, since it contains $F_i$, and at most $i$, since $\dim Z_i\leq i$.
+It contains $z$, so it equals $S_i$. Hence $F_i\subseteq S_i$.
+
+For a fixed preceding face $F_{i-1}$, the possible $F_i$ are thus
+full-dimensional simplices in a triangulation of the single affine
+$i$-simplex $S_i$, having $F_{i-1}$ as a facet. There is at most one
+on either side of the affine hyperplane spanned by that facet.
+Otherwise two simplices on the same side would have intersecting
+relative interiors near an interior point of the facet. There
+are therefore at most two choices at each stage and at most $2^n$
+surviving flags through $z$. Summing over $z$ proves the assertion.
+This argument concerns one carrier simplex at a time and uses no
+manifold assumption on $K$ or on the strata.
+\end{proof}
+
+Figure~\ref{fig:flags} illustrates the two choices at each of the two
+stages when $n=2$.
+
+\begin{figure}[htbp]
+\centering
+\begin{tikzpicture}[scale=1.05]
+ \coordinate (A) at (-2,0);
+ \coordinate (B) at (2,0);
+ \coordinate (C) at (0,3);
+ \coordinate (L) at (-1.3333,1);
+ \coordinate (R) at (1.3333,1);
+ \coordinate (z) at (0,1);
+ \fill[blue!9] (L)--(z)--(C)--cycle;
+ \fill[blue!18] (z)--(R)--(C)--cycle;
+ \fill[orange!13] (L)--(z)--(A)--cycle;
+ \fill[orange!24] (z)--(R)--(A)--cycle;
+ \draw[thin,gray] (z)--(A) (A)--(R) (z)--(C);
+ \draw[thick] (A)--(B)--(C)--cycle;
+ \draw[very thick,blue!65!black] (L)--(R);
+ \fill (z) circle (2pt) node[below right=2pt,fill=white,inner sep=1pt] {$z$};
+ \node at (-0.44,1.68) {$F_2$};
+ \node at (0.44,1.68) {$F_2$};
+ \node at (-1.08,0.62) {$F_2$};
+ \node at (-0.25,0.67) {$F_2$};
+ \node[above=2pt] at (-0.66,1) {$F_1$};
+ \node[above=2pt] at (0.66,1) {$F_1$};
+\end{tikzpicture}
+\caption{The flag count in dimension two. The point $z\in Z_0$ has
+two possible incident edges $F_1$ along its one-dimensional carrier.
+Each edge has at most two incident triangles $F_2$ inside the original
+two-dimensional carrier. The four shaded triangles give the four possible
+flags through $z$ in this example. The unshaded triangle has no
+one-dimensional-stratum edge through $z$.}
+\label{fig:flags}
+\end{figure}
+
+Let $\psi:E_n\to\C$ be the invariant cocycle in
+Theorem~\ref{thm:sparse}. For an original top simplex $\sigma$, put
+\[
+ M_\sigma=
+ \max_{v_0,\ldots,v_n\in W_\sigma}
+ \left|\psi(\delta_{v_0}\wedge\cdots\wedge\delta_{v_n})\right|.
+\]
+This is finite, even though $\psi$ need not be bounded on all
+basis simplices of the infinite vertex set. Every vector in
+\eqref{eq:component-average} is a nonnegative convex combination
+of vertices in $W_\sigma$. Multilinearity therefore bounds the
+evaluation of each surviving term by $M_\sigma$. The preceding
+lemma and Proposition~\ref{prop:zero-strata} give
+\begin{align}
+ \left|\psi\bigl(F^\epsilon(\sigma)\bigr)\right|
+ &\leq 2^n M_\sigma
+ \int_X\#\bigl(Z_0(x)\cap\sigma\bigr)\,d\mu(x)
+ \notag\\
+ &\leq 2^n M_\sigma C_\sigma\epsilon
+ \longrightarrow 0.\label{eq:small-chain-evaluation}
+\end{align}
+All constants here are independent of $\epsilon$. Since chains
+have finite support, the same convergence holds on every fixed
+$c\in C_n(K)$.
+
+\subsection{Equivariant comparison and passage to coinvariants}
+
+It remains to show that the small evaluations just obtained equal
+the original cocycle evaluation on every coinvariant cycle.
+We do this by constructing a homotopy before taking coinvariants.
+
+\begin{lemma}\label{lem:equivariant-homotopy}
+Let $K$ be a simplicial complex with an $H$-action and finite
+setwise simplex stabilizers. Let $E_*$ be the augmented exterior
+chain complex on a nonempty $H$-set. Any two $H$-equivariant
+augmented chain maps $A,B:C_*(K)\to E_*$ that agree in degree $-1$
+are $H$-equivariantly chain homotopic.
+\end{lemma}
+
+\begin{proof}
+The target is acyclic as an augmented complex. Indeed, for any
+fixed vertex $v$, the operator $a\mapsto\delta_v\wedge a$ satisfies
+\[
+ \partial(\delta_v\wedge a)+\delta_v\wedge\partial a=a.
+\]
+This contraction need not be equivariant. We use it only to choose
+finite fillers and then impose equivariance by finite averaging.
+
+Set $h_{-1}=0$. Suppose the homotopy has been constructed below
+degree $j$. For an oriented $j$-simplex $e$, the residual
+\[
+ r(e)=A(e)-B(e)-h_{j-1}(\partial e)
+\]
+is a cycle, by the chain-map identities and the previously
+constructed homotopy. Choose a finite chain $t$ with
+$\partial t=r(e)$. Let $S$ be the setwise stabilizer of $e$ and
+let $\chi:S\to\{1,-1\}$ be its orientation character, so that
+$s e=\chi(s)e$. The residual is equivariant, and hence
+$s r(e)=\chi(s)r(e)$. Define
+\[
+ \overline t=\frac1{|S|}\sum_{s\in S}\chi(s)\,s t.
+\]
+Then $\partial\overline t=r(e)$ and
+$s\overline t=\chi(s)\overline t$. Choose such a filler for one
+oriented representative of every simplex orbit, put
+$h_j(e)=\overline t$, and extend equivariantly and linearly.
+The orientation-character identity makes the extension
+well-defined, including when the stabilizer reverses orientation.
+This gives $A-B=\partial h+h\partial$ in degree $j$ and completes
+the induction.
+\end{proof}
+
+\begin{proof}[Proof of Theorem~\ref{thm:sparse}]
+The geometric construction applies for every sufficiently large
+$n$ depending only on $D$, as fixed by \eqref{eq:strict-volume}.
+For nonempty $K$, construct $F^\epsilon$ as above. The stabilizers
+are finite by Lemma~\ref{lem:proper}, and $F^\epsilon$ and inclusion
+$\iota$ both induce the identity in augmented degree $-1$.
+Lemma~\ref{lem:equivariant-homotopy} supplies an equivariant
+homotopy
+\[
+ \iota-F^\epsilon=\partial h+h\partial.
+\]
+
+Let $\overline c$ be a cycle of $C_n(K)_H$ and choose a finite
+chain lift $c\in C_n(K)$. We do not assume that $c$ is a cycle
+before taking coinvariants. Its boundary is a finite sum of
+differences $ha-a$. Equivariance of the homotopy makes
+$h_{n-1}(\partial c)$ another such sum, which the invariant
+functional $\psi$ annihilates. The cocycle identity annihilates
+$\partial h_n(c)$. Thus, for every $\epsilon>0$,
+\[
+ \psi\bigl(\iota(c)\bigr)
+ =\psi\bigl(F^\epsilon(c)\bigr).
+\]
+The right-hand side tends to zero by
+\eqref{eq:small-chain-evaluation}, whereas the left-hand side
+does not depend on $\epsilon$. Hence it is zero. The evaluation
+depends only on the coinvariant class because $\psi$ is invariant.
+For empty $K$ there is no nonzero top chain to consider.
+\end{proof}

preprints/The-Bass-trace-conjecture-for-complex-group-rings-September-24-2026/build/sections/07-idempotents.tex

@@ -0,0 +1,258 @@
+\section{Cycles from an idempotent}
+\label{sec:idempotents}
+
+We now return to group rings.  For an idempotent and one infinite-order
+conjugacy class, we will construct cycles in every positive even degree.
+Their vertices follow steps in one fixed graph of bounded degree. In
+sufficiently high dimension, the supporting complexes will therefore
+satisfy the hypotheses of Theorem~\ref{thm:sparse}. We first record why
+it suffices to perform this construction for finitely generated groups.
+
+The use of cyclic idempotent expressions and reduced centralizers has
+its conceptual background in Connes's pairing with projective classes
+\cite[Part II, Section 2, Proposition 14]{Connes1985} and Burghelea's
+conjugacy-class decomposition \cite[Theorem I]{Burghelea1985}. Our finite
+path identities and the detector's normalization are proved directly;
+no cyclic-homology identification is assumed.
+
+\begin{lemma}[Subgroup locality]\label{lem:locality}
+Let $i:L\hookrightarrow G$ be an inclusion of discrete groups.  Extension
+of scalars defines $i_*:K_0(\C L)\to K_0(\C G)$, and for every
+$y\in K_0(\C L)$ and $C\in\con(G)$,
+\begin{equation}\label{eq:class-aggregation}
+ \HS_G(i_*y)(C)
+ =\sum_{\substack{B\in\con(L)\\ B\subseteq L\cap C}}
+       \HS_L(y)(B).
+\end{equation}
+Only finitely many terms are nonzero.  Moreover, every
+$x\in K_0(\C G)$ is induced from a finitely generated subgroup, which may
+be required to contain any prescribed finite subset of $G$.
+Consequently, the finite-order-support assertion of
+Theorem~\ref{thm:bass} for all finitely generated groups implies that
+assertion for all discrete groups.
+\end{lemma}
+
+\begin{proof}
+We regard free right modules as columns: an idempotent
+$e\in M_r(\C L)$ acts on the left, with image $e(\C L)^r$.
+Tensoring the split decomposition defined by $e$ with $\C G$ identifies
+the extended image with $e(\C G)^r$.  The coefficient sum in an ambient
+class $C$ is the sum of its coefficient sums over all $L$-classes contained
+in $L\cap C$.  This proves \eqref{eq:class-aggregation} for the image of an
+idempotent and then, by subtraction, for every virtual class.  Finiteness
+follows from the finite coefficient support of the matrices representing
+that class.
+
+Write an arbitrary $x\in K_0(\C G)$ as the difference of the classes of
+two finite idempotent matrices.  The group generated by their coefficient
+supports and the prescribed finite subset is finitely generated.  The
+same matrices are idempotent over its group ring, since that ring embeds
+in $\C G$, and their difference induces $x$.  Finally, if an ambient class
+consists of infinite-order elements, so does every subgroup class occurring
+in \eqref{eq:class-aggregation}: subgroup inclusion preserves the order of
+an element.  Vanishing of each subgroup coefficient therefore gives
+vanishing of the ambient coefficient.  This uses all the subgroup classes
+in the sum, since an ambient conjugacy class can split into several of
+them.
+\end{proof}
+
+\subsection{A fixed graph and finite path sums}
+
+For the rest of this section, fix a finitely generated group $G$, an
+idempotent $e\in M_r(\C G)$, and an infinite-order element $g\in G$.
+Write
+\[
+ e=\sum_{s\in G}A_s s,\qquad A_s\in M_r(\C),\qquad
+ S_0=\{s\in G:A_s\ne0\}.
+\]
+The set $S_0$ is finite.  Idempotence says that
+$\sum_{uv=t}A_uA_v=A_t$ for every $t\in G$.  Thus the matrix-valued kernel
+\[
+ a(h,h')=A_{h^{-1}h'}
+\]
+is invariant under simultaneous left translation and satisfies
+\begin{equation}\label{eq:kernel-idempotence}
+ \sum_{k\in G}a(h,k)a(k,h')
+ =\sum_{uv=h^{-1}h'}A_uA_v
+ =a(h,h').
+\end{equation}
+All these sums are finite.  In particular, the order of the matrix
+factors agrees with the column convention for right modules.
+
+Put
+\[
+ J=C_G(g),\qquad H=J/\langle g\rangle,\qquad
+ Y=\langle g\rangle\backslash G,
+ \qquad \bar h=\langle g\rangle h.
+\]
+Choose a finite symmetric generating set $S\subset G\setminus\{1\}$
+containing $S_0\setminus\{1\}$ and its inverses.  Give $Y$ the simple
+undirected graph whose edges join $\bar h$ to $\overline{hs}$ for $s\in S$;
+loops are omitted and repeated edges identified.  The graph is connected
+and has degree at most the fixed integer
+\[
+ D=\max\{2,|S|\}.
+\]
+The group $H$ acts by left multiplication.  This is well-defined because
+$J$ centralizes $g$, and it preserves right-step edges.  It is free on
+vertices: $\overline{ch}=\bar h$ with $c\in J$ implies $ch=g^q h$ for an
+integer $q$, and hence $c=g^q$.  Both $H$ and $Y$ are countable.  We keep
+this graph and this value of $D$ throughout all subsequent choices of
+dimension.
+
+For $k\ge1$, let $Q_k=J\backslash G^k$, with simultaneous left
+translation on tuples.  Define the weight of a tuple by
+\begin{equation}\label{eq:path-weight}
+ w_k(h_0,\ldots,h_{k-1})
+ =\tr\bigl(a(h_0,h_1)\cdots
+ a(h_{k-2},h_{k-1})a(h_{k-1},gh_0)\bigr),
+\end{equation}
+where $w_1(h_0)=\tr(a(h_0,gh_0))$.  These weights are $J$-invariant.
+They define a finite complex linear combination $v_k$ of elements of
+$Q_k$.  Indeed, a nonzero weight has increments
+\[
+ s_i=h_i^{-1}h_{i+1}\quad(0\le i<k-1),\qquad
+ s_{k-1}=h_{k-1}^{-1}gh_0
+\]
+in $S_0$, with
+\[
+ s_0\cdots s_{k-1}=h_0^{-1}gh_0.
+\]
+An increment list determines the tuple from $h_0$.  Any two choices of
+$h_0$ with this conjugate differ by left multiplication in $J$, so the
+list determines at most one element of $Q_k$.  Hence $v_k$ has at most
+$|S_0|^k$ nonzero coefficients.  For a $J$-invariant function $p$ on
+$G^k$, write $v_k(p)$ for the finite weighted sum of its values.  These
+weights need not be real or positive.
+
+\begin{lemma}[Path identities]\label{lem:path-identities}
+For the path sums associated to $e$ and $g$, the twisted cyclic shift
+\[
+ T(h_0,\ldots,h_{k-1})=(h_1,\ldots,h_{k-1},gh_0)
+\]
+preserves $v_k$.  For $k\ge2$, deletion of any entry induces a map
+$d_i:Q_k\to Q_{k-1}$ satisfying
+\[
+ (d_i)_*v_k=v_{k-1}\qquad(0\le i<k).
+\]
+Every $v_k$ has the same total weight
+\begin{equation}\label{eq:trace-coefficient}
+ \lambda:=v_k(1)
+ =\sum_{s\in[g]_G}\tr(A_s)
+ =\HS_G\bigl([e(\C G)^r]\bigr)([g]_G).
+\end{equation}
+\end{lemma}
+
+\begin{proof}
+The shift $T$ is a $J$-equivariant bijection, with inverse
+\[
+ (h_0,\ldots,h_{k-1})\longmapsto
+ (g^{-1}h_{k-1},h_0,\ldots,h_{k-2}).
+\]
+For $k=1$, the shift is the identity on $Q_1$, since $g\in J$.
+For $k\ge2$, the shifted weight has the factors of
+\eqref{eq:path-weight} in cyclic order, since
+$a(gh_0,gh_1)=a(h_0,h_1)$.  Cyclicity of the matrix trace proves
+$T_*v_k=v_k$.
+
+For deletion, fix a representative of the shorter tuple.  Each orbit in
+its fibre has a unique representative with exactly those retained
+entries, because simultaneous left translation acts freely on any
+nonempty tuple.  Thus the fibre sum is over the missing entry in $G$,
+without a stabilizer factor.  Internal deletions follow from
+\eqref{eq:kernel-idempotence}, as does deletion of the last entry using
+\[
+ \sum_u a(h_{k-2},u)a(u,gh_0)=a(h_{k-2},gh_0).
+\]
+For the first deletion, fix $(h_1,\ldots,h_{k-1})$ and put
+$B=a(h_1,h_2)\cdots a(h_{k-2},h_{k-1})$, with $B=I_r$ when $k=2$.
+Then
+\begin{align*}
+ &\sum_{h_0}\tr\bigl(a(h_0,h_1)B a(h_{k-1},gh_0)\bigr)\\
+ &\qquad=\sum_{h_0}\tr\bigl(Ba(h_{k-1},gh_0)a(gh_0,gh_1)\bigr)\\
+ &\qquad=\tr\bigl(Ba(h_{k-1},gh_1)\bigr).
+\end{align*}
+Here the first equality uses trace cyclicity and left invariance; the
+second uses \eqref{eq:kernel-idempotence} with the substitution $u=gh_0$.
+This also proves the assertion for $k=2$.  The kernel has finite row and
+column support, so every fibre sum used in these computations is finite.
+
+The deletion identities show that all total weights equal $v_1(1)$.
+Finally,
+\[
+ J\backslash G\longrightarrow[g]_G,\qquad Jh\longmapsto h^{-1}gh
+\]
+is a bijection: equality of the conjugates is equivalent to
+$h'h^{-1}\in J$.  Therefore
+$v_1(1)=\sum_{s\in[g]_G}\tr(A_s)$, which is precisely the indicated
+Hattori--Stallings coefficient.
+\end{proof}
+
+\subsection{Even-dimensional cycles}
+
+The path identities now turn the finite sums into cycles.  We use the
+exterior complex $E_*$ on the graph $Y$.
+For $k\ge1$, define
+\[
+ \pi_k:\C^{(Q_k)}\longrightarrow(E_{k-1})_H,\qquad
+ [h_0,\ldots,h_{k-1}]\longmapsto
+ [\delta_{\bar h_0}\wedge\cdots\wedge\delta_{\bar h_{k-1}}].
+\]
+The map is well-defined because changing a tuple by an element of $J$
+changes its projected vertices by a simultaneous $H$-translation.
+Set $b_k=\pi_k(v_k)$, and, for $n=2m\ge2$, set
+\[
+ z_n=b_{n+1}\in(E_n)_H.
+\]
+
+\begin{proposition}\label{prop:path-cycle}
+For every positive even integer $n$, the chain $z_n$ is a cycle.
+Moreover, either its defining sum has no nonzero distinct-vertex terms,
+so $z_n=0$, or it is the image of a cycle in $C_n(K)_H$ for an
+$H$-invariant simplicial complex $K$ generated by $n$-simplices, with
+finitely many simplex orbits, whose top vertex sets span connected
+subgraphs of $Y$.  The graph and its degree bound $D$ are independent
+of $n$.
+\end{proposition}
+
+\begin{proof}
+Since $\overline{gh_0}=\bar h_0$, the twisted shift projects to the cyclic
+permutation of the $k$ wedge factors.  Hence
+\[
+ \pi_kT=(-1)^{k-1}\pi_k.
+\]
+Together with $T_*v_k=v_k$, this shows that $b_k=0$ whenever $k$ is even.
+The deletion identities give
+\[
+ \partial b_k=\sum_{i=0}^{k-1}(-1)^i b_{k-1}\qquad(k\ge2).
+\]
+For even $n$, it follows that $\partial z_n=b_n=0$.
+
+A tuple with repeated projected vertices contributes zero to the wedge.
+For every remaining tuple with nonzero weight, consecutive projected
+vertices are joined by edges from the fixed step set $S$.  Their path
+visits all its $n+1$ vertices, so the subgraph spanned by those vertices
+is connected.  Let $K$ be generated by these vertex sets and their
+$H$-translates.  Finite support of $v_{n+1}$ gives finitely many orbits
+of top simplices and of their faces.
+
+The same finite expression for $z_n$ defines a chain in $C_n(K)_H$.
+To see that its boundary is zero there, decompose $E_j$ in every degree
+as the direct sum of the span of the oriented simplices belonging to
+$K$ and the span of all other oriented simplices.  Both summands are
+$H$-invariant, including their orientation signs.  Thus
+\[
+ E_j=C_j(K)\oplus W_j,
+ \qquad
+ (E_j)_H=C_j(K)_H\oplus(W_j)_H.
+\]
+In particular the inclusion remains injective after taking
+coinvariants.  The boundary of the chain in $C_n(K)_H$ maps to
+$\partial z_n=0$, so it is zero already in $C_{n-1}(K)_H$.
+\end{proof}
+
+We have obtained the required sparse cycles in arbitrarily large even
+degrees.  The underlying path sums all have total weight $\lambda$, the trace
+coefficient in \eqref{eq:trace-coefficient}. The next section constructs
+a cocycle whose value on $z_{2m}$ recovers a nonzero scalar multiple of
+that coefficient.

preprints/The-Bass-trace-conjecture-for-complex-group-rings-September-24-2026/build/sections/08-detector.tex

@@ -0,0 +1,203 @@
+\section{A Pfaffian cocycle detecting the trace coefficient}
+\label{sec:detector}
+
+Retain the finitely generated group $G$, idempotent, infinite-order element
+$g$, and notation $J,H,Y,v_k,\lambda,z_{2m}$ of
+Section~\ref{sec:idempotents}.  We construct an $H$-invariant cocycle whose
+value on $z_{2m}$ is $2^{-m}\lambda$.  The construction first records the
+relative powers of $g$ between two lifts of vertices of $Y$.  An alternating
+Pfaffian expression will then remove the dependence on those lifts.
+
+\subsection{Relative exponents between lifts}
+
+\begin{lemma}\label{lem:beta}
+There is a $J$-invariant function $\beta:G\times G\to\R$ such that
+\begin{equation}\label{eq:beta-properties}
+ \beta(h',h)=-\beta(h,h'),\qquad
+ \beta(g^r h,g^s h')=\beta(h,h')+s-r
+ \quad(r,s\in\Z).
+\end{equation}
+\end{lemma}
+
+\begin{proof}
+The diagonal $H$-action on the set of ordered pairs $Y\times Y$ is free,
+because its action on the first coordinate is free.  This also applies to
+diagonal pairs $(y,y)$.  For each orbit choose a representative $(y_o,y'_o)$
+and lifts $(l_o,l'_o)\in G^2$.  For $(h,h')\in G^2$, there is a unique
+$\bar c\in H$ carrying the appropriate representative to the projected
+pair.  Choose a lift $c\in J$ and write
+\[
+ h=g^r c l_o,\qquad h'=g^s c l'_o.
+\]
+For fixed $c$, the integers $r,s$ are unique because $g$ has infinite order.
+Replacing $c$ by $g^t c$ replaces them by $r-t,s-t$.  Thus
+$B(h,h')=s-r$ is well defined.  Commutation with $g$ shows that $B$ is
+$J$-invariant, and independently changing the two lifts gives
+\[
+ B(g^a h,g^b h')=B(h,h')+b-a.
+\]
+The antisymmetrization
+\[
+ \beta(h,h')=\frac{B(h,h')-B(h',h)}2
+\]
+has all the required properties.
+\end{proof}
+
+The function $\beta$ itself need not descend to $Y\times Y$.  In fact,
+\eqref{eq:beta-properties} gives $\beta(h,g^k h)=k$ even though these two
+lifts project to the same vertex.  Nor is there a difficulty if an element
+$\bar c\in H$ exchanges two distinct projected vertices.  If a lift
+$c\in J$ satisfies
+$c h_0=g^a h_1$ and $c h_1=g^b h_0$, its invariance merely requires
+\[
+ 2\beta(h_0,h_1)=b-a,
+\]
+which is precisely what antisymmetrization gives.  Real values are useful
+here: for $G=\langle t\rangle$ and $g=t^2$, the value
+$\beta(1,t)=1/2$ is forced.
+
+\subsection{The cyclic Pfaffian identity}
+
+In degree two, a lift-independent expression is
+\[
+ c(y_0,y_1,y_2)
+ =\beta(h_1,h_2)-\beta(h_0,h_2)+\beta(h_0,h_1),
+ \qquad \bar h_i=y_i.
+\]
+Changing $h_i$ to $g^{r_i}h_i$ adds
+$(r_2-r_1)-(r_2-r_0)+(r_1-r_0)=0$, so $c$ depends only on the
+projected vertices. In higher even degrees we replace $\beta$ by a
+Pfaffian and take the same alternating deletion sum. The Pfaffian
+still depends on the lifts; its deletion sum will define the cocycle.
+We first compute the weighted Pfaffian sum, then prove the descent
+and cocycle properties of its deletion sum.
+
+For a list $\mathbf h=(h_0,\ldots,h_{2m-1})$ with $m\ge1$, let
+\[
+ p_m(\mathbf h)=\operatorname{Pf}
+       \bigl(\beta(h_i,h_j)\bigr)_{0\le i,j<2m},
+ \qquad p_0=1.
+\]
+We use the convention that $p_m(\mathbf h)$ is the coefficient of
+$\varepsilon_0\wedge\cdots\wedge\varepsilon_{2m-1}$ in $b^m/m!$, where
+$\varepsilon_i$ are the coordinate one-forms on the space of list positions
+and
+\[
+ b=\sum_{i<j}\beta(h_i,h_j)\,
+              \varepsilon_i\wedge\varepsilon_j.
+\]
+In particular, $p_1(h_0,h_1)=\beta(h_0,h_1)$, and $p_m$ is alternating in
+its entries and $J$-invariant.  The finite weighted sums
+\[
+ q_m=v_{2m}(p_m)\quad(m\ge1),\qquad q_0=\lambda
+\]
+are therefore defined.  No boundedness of $p_m$ on all lists is needed.
+
+\begin{lemma}\label{lem:pfaffian-recurrence}
+For every $m\ge1$, one has $q_m=\tfrac12q_{m-1}$ and hence
+$q_m=2^{-m}\lambda$.
+\end{lemma}
+
+\begin{proof}
+An ordinary cyclic rotation of a list of length $2m$ changes its Pfaffian
+by the sign $(-1)^{2m-1}=-1$.  In the twisted rotation
+\[
+ T\mathbf h=(h_1,\ldots,h_{2m-1},g h_0),
+\]
+replacing the last entry $h_0$ by $g h_0$ adds $1$ to every matrix entry in
+the last column above the diagonal.  Expansion along that column gives
+\begin{equation}\label{eq:pfaffian-rotation}
+ \begin{split}
+ p_m(T\mathbf h)
+   ={}&-p_m(\mathbf h)\\
+     &+\sum_{i=1}^{2m-1}(-1)^{i+1}
+       p_{m-1}(h_1,\ldots,\widehat{h_i},\ldots,h_{2m-1}).
+ \end{split}
+\end{equation}
+Indeed the last-column cofactor in position $i$, with positions numbered
+$1,\ldots,2m$, has sign $(-1)^{i+1}$; each Pfaffian matching uses exactly
+one entry from that column.
+
+For $m\ge2$, each minor on the second line omits $h_0$ and $h_i$ from the
+original list.  Applying the deletion identity in
+Lemma~\ref{lem:path-identities} twice shows that its weighted sum under
+$v_{2m}$ is $q_{m-1}$.  For $m=1$, the minor is $p_0=1$, whose weighted sum
+is the total weight $v_2(1)=\lambda=q_0$.  Thus this endpoint uses no vector
+$v_0$.  The invariance of $v_{2m}$ under $T$ and
+$\sum_{i=1}^{2m-1}(-1)^{i+1}=1$ now turn
+\eqref{eq:pfaffian-rotation} into
+\[
+ q_m=-q_m+q_{m-1}.
+\]
+The claimed formulas follow.
+\end{proof}
+
+\subsection{Descent to an exterior cocycle}
+
+The preceding calculation detects the coefficient on lists of lifts.
+We next descend it to the exterior complex on $Y$, where the sparse-chain
+theorem applies.
+
+\begin{proposition}\label{prop:detector}
+For every integer $m\ge1$, the formula
+\begin{equation}\label{eq:detector-formula}
+ \psi_m(y_0,\ldots,y_{2m})
+   =\sum_{i=0}^{2m}(-1)^i
+       p_m(h_0,\ldots,\widehat{h_i},\ldots,h_{2m}),
+\end{equation}
+where $h_i\in G$ is any lift of $y_i\in Y$, defines an $H$-invariant
+linear functional $\psi_m:E_{2m}\to\C$ satisfying
+\[
+ \psi_m\partial=0,\qquad
+ \psi_m(z_{2m})=2^{-m}\lambda.
+\]
+Here evaluation on the coinvariant chain $z_{2m}$ is defined by
+$H$-invariance.
+\end{proposition}
+
+\begin{proof}
+On the $2m+1$ list positions put
+\[
+ u=\sum_{i=0}^{2m}\varepsilon_i,
+ \qquad
+ b=\sum_{0\le i<j\le2m}\beta(h_i,h_j)\,
+                         \varepsilon_i\wedge\varepsilon_j.
+\]
+The right side of \eqref{eq:detector-formula} is the coefficient of the
+ordered top form in $u\wedge b^m/m!$.  Moving the factor
+$\varepsilon_i$ past its $i$ preceding positions gives exactly the sign
+$(-1)^i$ in that formula.
+
+Changing the lifts to $g^{r_i}h_i$ replaces $\beta(h_i,h_j)$ by
+$\beta(h_i,h_j)+r_j-r_i$.  Thus it replaces $b$ by
+$b+u\wedge v$, with $v=\sum_i r_i\varepsilon_i$.  Every correction term
+in $u\wedge(b+u\wedge v)^m$ contains two factors $u$ and vanishes.
+This proves independence of the lifts.
+
+The expression is alternating in the lifted list.  After lift independence
+has been established, it is therefore alternating in the projected
+vertices as well.  In particular, if two projected vertices coincide,
+choose equal lifts for them to see that the expression vanishes.  It
+consequently defines a linear functional on
+$E_{2m}=\bigwedge^{2m+1}\C^{(Y)}$.
+For $\bar c\in H$, choose $c\in J$ and use $c h_i$ as the lifts of the
+translated vertices.  The $J$-invariance of $\beta$ proves the
+$H$-invariance of $\psi_m$.
+
+To check the cocycle identity, choose lifts simultaneously for a list of
+$2m+2$ vertices.  On every face, \eqref{eq:detector-formula} is the
+alternating deletion coboundary of $p_m$ on these same lifts.  A second
+deletion cancels each twice-omitted list in its two possible orders.
+Hence $\psi_m\partial=0$.
+
+Finally, $z_{2m}$ is the exterior projection of $v_{2m+1}$.
+Each one-deletion term in \eqref{eq:detector-formula} has weighted sum
+$v_{2m}(p_m)=q_m$, by Lemma~\ref{lem:path-identities}.  Since
+$\sum_{i=0}^{2m}(-1)^i=1$, their signed sum gives
+\[
+ \psi_m(z_{2m})=q_m=2^{-m}\lambda,
+\]
+using Lemma~\ref{lem:pfaffian-recurrence}.  Repeated projected vertices
+already have zero exterior image and zero value under $\psi_m$, so they
+require no separate restriction in this evaluation.
+\end{proof}

preprints/The-Bass-trace-conjecture-for-complex-group-rings-September-24-2026/build/sections/09-completion.tex

@@ -0,0 +1,144 @@
+\section{Completion of the trace theorem}
+\label{sec:completion}
+
+We now combine the geometric vanishing with the detector. The order of
+the choices matters: the graph is fixed by the idempotent before its
+cycle dimension is chosen.
+
+\begin{proof}[Proof of Theorem~\ref{thm:bass}]
+First let $G$ be finitely generated, let $e\in M_q(\C G)$ be an
+idempotent, and fix an infinite-order $g\in G$. Construct the graph on
+$Y=\langle g\rangle\backslash G$ and the free action of
+$H=C_G(g)/\langle g\rangle$ as in Section~\ref{sec:idempotents}.
+Let $D\geq2$ be a degree bound for this fixed graph, and choose an even
+integer $n=2m\geq n_0(D)$ with $m\geq1$.
+
+Proposition~\ref{prop:path-cycle} gives the cycle $z_n$, and
+Proposition~\ref{prop:detector} gives an $H$-invariant cocycle
+$\psi_m:E_n\to\C$ with
+\[
+ \psi_m(z_n)=2^{-m}\lambda,
+ \qquad
+ \lambda=\HS_G([e(\C G)^q])([g]_G).
+\]
+If no nonzero simplices occur, $z_n=0$ and this identity already gives
+$\lambda=0$. Otherwise Proposition~\ref{prop:path-cycle} realizes $z_n$
+as a cycle in $C_n(K)_H$ for an $H$-invariant complex $K$ satisfying all
+hypotheses of Theorem~\ref{thm:sparse}. That theorem gives
+$\psi_m(z_n)=0$, with the same conclusion. Every finitely generated
+projective right $\C G$-module is represented by an idempotent, and every element of
+$K_0(\C G)$ is a difference of two such classes. Additivity proves the
+assertion for all finitely generated groups.
+
+For an arbitrary discrete group $G$, choose idempotents representing
+$x\in K_0(\C G)$ and let $G_0\leq G$ be generated by $g$ and all group
+elements occurring in their entries. Then $G_0$ is finitely generated
+and $x=i_*x_0$ for some $x_0\in K_0(\C G_0)$, where $i:G_0\hookrightarrow G$.
+Lemma~\ref{lem:locality} expresses $\HS_G(x)([g]_G)$ as the sum of the
+coefficients $\HS_{G_0}(x_0)(C)$ over those $G_0$-conjugacy classes
+$C\subseteq G_0\cap[g]_G$. Only finitely many terms are nonzero, and
+every element in every such class has infinite order. Each term
+vanishes by the finitely generated case.
+\end{proof}
+
+\subsection{The integral trace}
+
+For the integral group ring, the same idempotent formula defines
+\[
+ \HS_{\Z G}:K_0(\Z G)\longrightarrow
+ \Z G/[\Z G,\Z G]\cong\bigoplus_{C\in\con(G)}\Z C,
+\]
+where commutators are now taken with their additive span. The integral
+Bass trace conjecture asks for support only at the identity class.
+
+\begin{corollary}\label{cor:integral}
+For every discrete group $G$, the Hattori--Stallings trace of every
+class in $K_0(\Z G)$ is supported on $[1]_G$.
+\end{corollary}
+
+\begin{proof}
+If $P$ is a finitely generated projective right $\Z G$-module, the same
+idempotent matrix over $\C G$ represents $P\otimes_{\Z G}\C G$.
+Consequently its trace coefficients are the images under
+$\Z\hookrightarrow\C$ of those of $P$. Theorem~\ref{thm:bass} and this
+coefficientwise injection give vanishing at all infinite-order elements.
+No injectivity assertion about the map between the two $K_0$ groups is
+required.
+
+For finite-order elements we use Linnell's restriction
+\cite[Lemma 4.1]{Linnell1983}, in the form stated and reproved by
+Berrick and Hesselholt \cite[Theorem A]{BH2015}: if the integral trace of a finitely
+generated projective module is nonzero at $h$, there is an integer
+$a\geq1$ such that $h$ is conjugate to $h^{s^a}$ for every integer
+$s\geq1$. Their convention also uses right modules.
+If $h$ has finite order $t>1$, take $s=t$.
+Then $h^{t^a}=1$, so the restriction would make $h$ conjugate to $1$,
+a contradiction. Thus every nonidentity finite-order coefficient also
+vanishes. Taking differences of projective classes finishes the proof.
+\end{proof}
+
+\subsection{Homotopy idempotents and Lefschetz numbers}
+
+The integral support statement also has fixed-point consequences.
+A self-map $f$ is a \emph{homotopy idempotent} if $f\circ f\simeq f$.
+Such a map on a connected finite CW complex $X$ admits maps
+$u:X\to Y$ and $v:Y\to X$ with
+\[
+ v\circ u\simeq f,\qquad u\circ v\simeq\id_Y
+\]
+\cite[Section 3]{BCM2006}. The second identity says that $Y$ is
+\emph{finitely dominated} by $X$. For a pointed homotopy idempotent
+with $f_\#=\id_G$, where $G=\pi_1(X)$, the induced maps $u_\#$ and
+$v_\#$ identify $\pi_1(Y)$ with $G$. The universal-cover chains of $Y$
+are chain homotopy equivalent to a finite complex $P_*$ of finitely
+generated projective $\Z G$-modules. The sum
+$\sum_i(-1)^i[P_i]\in K_0(\Z G)$ is its projective Euler class. The
+Hattori--Stallings trace of this class records the fixed-point class indices of $f$:
+it is the Reidemeister trace, with conjugacy classes inverted when
+passing from the left-module convention of \cite[Lemma 4.1]{BCM2006}
+to ours. The augmentation trace and the identity coefficient of this projective
+class give the ordinary and $L^2$-Lefschetz numbers, respectively
+\cite[Lemmas 8.1--8.2]{BCM2006}. Berrick, Chatterji and Mislin use these
+relations to deduce the following consequences of the integral support
+statement and its numerical trace consequence.
+
+\begin{corollary}[Berrick--Chatterji--Mislin consequences]
+\label{cor:homotopy-idempotents}
+Let $M$ be a connected closed smooth oriented manifold.
+\begin{enumerate}[label=\textup{(\roman*)}]
+ \item If $\dim M>2$ and $f:M\to M$ satisfies $f\circ f\simeq f$
+ by free homotopy, then some self-map freely homotopic to $f$ has
+ exactly one fixed point.
+ \item Choose $x_0\in M$ and put $G=\pi_1(M,x_0)$. If
+ $f:(M,x_0)\to(M,x_0)$ satisfies $f\circ f\simeq_* f$ by based
+ homotopy and $f_\#=\id_G$, then
+ \[
+  L(f)=L^{(2)}(\widetilde f).
+ \]
+ Here $\widetilde f$ is the lift to the universal cover fixing a chosen
+ point above $x_0$; it is $G$-equivariant. The left side is the ordinary
+ Lefschetz number, and the right side is the alternating sum of von
+ Neumann traces on the induced $L^2$-homology maps, as in
+ \cite[Definition 7.1]{BCM2006}.
+\end{enumerate}
+\end{corollary}
+
+\begin{proof}
+For $x\in K_0(\Z G)$ write $h_C(x)=\HS_{\Z G}(x)(C)$. The
+augmentation trace $\epsilon$ is the sum of these coefficients, while
+the Kaplansky trace $\kappa$ is the identity coefficient. The sum is
+finite, and Corollary~\ref{cor:integral} gives
+\[
+ \epsilon(x)=\sum_{C\in\con(G)}h_C(x)=h_{[1]_G}(x)=\kappa(x).
+\]
+This is the weak Bass conjecture for every group. The cited paper uses
+projective left modules, whereas we use right modules. The anti-involution
+$g\mapsto g^{-1}$, together with transpose on idempotent matrices,
+transports between these conventions and relabels the coefficient at
+$[g]_G$ by $[g^{-1}]_G$. It preserves identity support and both
+$\epsilon$ and $\kappa$. Thus Corollary~\ref{cor:integral} gives the
+classical Bass conjecture in the cited convention, and the displayed
+equality gives weak Bass there. The forward implications of
+\cite[Theorem 1]{BCM2006} and \cite[Theorem 2]{BCM2006} now give
+\textup{(i)} and \textup{(ii)}, respectively.
+\end{proof}

preprints/The-Bass-trace-conjecture-for-complex-group-rings-September-24-2026/build/sections/10-torsion-free.tex

@@ -0,0 +1,121 @@
+\section{Torsion-free groups: trace rank and idempotents}
+\label{sec:torsion-free}
+
+For a torsion-free group, the support theorem leaves only the identity
+conjugacy class. We first identify its coefficient with augmentation
+rank, and then use the two possible scalar ranks to determine scalar
+idempotents.
+
+\begin{proof}[Proof of Corollary~\ref{cor:torsion-free}\textup{(i)}]
+Let $e\in M_q(\C G)$ be idempotent and put
+$x=[e(\C G)^q]\in K_0(\C G)$. Since $G$ is torsion-free,
+Theorem~\ref{thm:bass} says that $\HS_G(x)$ is supported on $[1]_G$.
+Therefore
+\begin{align}
+ \tau_{G,q}(e)
+ &=\HS_G(x)([1]_G)\notag\\
+ &=\sum_{C\in\con(G)}\HS_G(x)(C)\notag\\
+ &=\epsilon_G\!\left(\sum_{i=1}^q e_{ii}\right)
+ =\tr\bigl(\epsilon_{G,q}(e)\bigr)
+ =\operatorname{rank}_{\C}\bigl(\epsilon_{G,q}(e)\bigr).
+ \label{eq:torsion-free-rank}
+\end{align}
+The class sum is finite because the entries of $e$ have finite group
+support. The last
+equality holds because $\epsilon_{G,q}(e)$ is an idempotent complex
+matrix: its image-kernel decomposition makes its trace equal to its
+rank. Taking differences of projective classes gives
+$\tau_{G,*}=\operatorname{rk}_{\epsilon}$. This map has range exactly
+$\Z$, since its values are integers and $[\C G]$ has value $1$.
+\end{proof}
+
+For matrices the conclusion is numerical. For example,
+$\operatorname{diag}(1,0)\in M_2(\C G)$ is an idempotent other than zero
+and the identity, with augmentation rank one. The scalar case is special
+because its augmentation rank can only be zero or one.
+
+\begin{proof}[Proof of Corollary~\ref{cor:torsion-free}\textup{(ii)}]
+First take $R=\C$ and let $e\in\C G$ be idempotent. Equation
+\eqref{eq:torsion-free-rank} with $q=1$ gives
+\[
+ \tau_G(e)=\epsilon_G(e)\in\{0,1\}.
+\]
+We use the standard faithful-trace argument, as in
+\cite[Theorem 2.1 and Remark 2.2]{BV1998}, spelling out a similarity
+to a projection. Embed $\C G$ by left convolution in the group von
+Neumann algebra $\mathcal N(G)$, the commutant of the right regular
+representation on $\ell^2(G)$. Its canonical normalized trace is
+\[
+ \tau_G(T)=\langle T\delta_1,\delta_1\rangle .
+\]
+Its faithfulness can be seen directly: if
+$T\geq0$ and $\tau_G(T)=0$, then $T^{1/2}\delta_1=0$. Since $T^{1/2}$
+commutes with right translations and their translates of $\delta_1$
+span a dense subspace, $T^{1/2}=0$, hence $T=0$.
+
+Set
+\[
+ a=e^*e+(1-e)^*(1-e).
+\]
+For every $\xi\in\ell^2(G)$,
+\[
+ \langle a\xi,\xi\rangle
+ =\|e\xi\|^2+\|(1-e)\xi\|^2
+ \geq \tfrac12\|\xi\|^2.
+\]
+Thus $a$ is positive and invertible in $\mathcal N(G)$, and a direct
+multiplication gives $ae=e^*a$. With $s=a^{1/2}$, the element
+$p=ses^{-1}$ satisfies $p^2=p$ and
+$p^*=s^{-1}e^*s=ses^{-1}=p$. Hence $p$ is a projection, and
+traciality gives $\tau_G(p)=\tau_G(e)$. If this value is zero,
+faithfulness applied to $p=p^*p$ gives $p=0$. If it is one,
+faithfulness applied to $1-p$ gives $p=1$. Similarity then gives
+$e=0$ or $e=1$.
+
+Now let $R$ be any commutative unital domain of characteristic zero
+and write an idempotent $f\in RG$ as
+$f=\sum_{g\in S}r_g g$ with $S$ finite. Put
+\[
+ R_0=\Z[r_g:g\in S]\subseteq R,\qquad F=\operatorname{Frac}(R_0).
+\]
+The field $F$ is finitely generated over $\mathbb Q$, and therefore
+embeds in $\C$. Indeed, send a finite transcendence basis of
+$F/\mathbb Q$ to an algebraically independent tuple of complex
+numbers; the resulting embedding of the rational function field
+extends across the finite algebraic extension $F$ because $\C$ is
+algebraically closed. The coefficientwise maps
+$R_0G\hookrightarrow RG$ and $R_0G\hookrightarrow\C G$ are injective.
+Thus $f^2=f$ already holds in $R_0G$, and its image in $\C G$ is an
+idempotent. The complex case makes that image $0$ or $1$, and
+injectivity gives the same conclusion for $f$. Only the finitely
+generated coefficient field $F$ was embedded; no embedding of all
+of $R$ into $\C$ is required.
+\end{proof}
+
+\begin{remark}[Algebraic and completed trace ranges]
+Continue to assume that $G$ is torsion-free. Let
+$j:\C G\hookrightarrow C_r^*(G)$ be the inclusion into the reduced
+group $C^*$-algebra, and let
+\[
+ j_*:K_0(\C G)\longrightarrow K_0^{\mathrm{top}}(C_r^*(G))
+\]
+be the comparison map to topological $K$-theory. Write
+$\tau_*^{\mathrm{top}}:K_0^{\mathrm{top}}(C_r^*(G))\to\R$ for the
+canonical trace pairing. It agrees with $\tau_{G,*}$ on this image:
+the same positive similarity used above, applied in a matrix algebra
+over $C_r^*(G)$, produces a projection representing the image of a
+matrix idempotent, and traciality preserves its trace. Consequently
+\[
+ \tau_*^{\mathrm{top}}\!\left(j_*K_0(\C G)\right)=\Z.
+\]
+This is the integer range on the algebraic image, not a statement
+about every class of $K_0^{\mathrm{top}}(C_r^*(G))$. In particular,
+the argument does not presume that the algebraic augmentation extends
+to the reduced completion. The stronger completed idempotent question
+is the Kadison--Kaplansky conjecture; see \cite[Section 1]{OW2023}.
+
+The coefficient reduction above uses characteristic zero. An idempotent
+$e\notin\{0,1\}$ would give nonzero zero divisors through $e(1-e)=0$,
+so the zero-divisor conjecture would imply the idempotent conjecture.
+The reverse implication is not established here.
+\end{remark}

preprints/The-Blockwise-Alperin-Weight-Conjecture-September-23-2026/README.md

@@ -0,0 +1,18 @@
+# [The Blockwise Alperin Weight Conjecture](paper.pdf)
+
+**Author:** OpenAI
+
+**Date:** September 23, 2026
+
+## Citation
+
+```bibtex
+@misc{OAI:The-Blockwise-Alperin-Weight-Conjecture-September-23-2026,
+  author = {{OpenAI}},
+  title = {{The Blockwise Alperin Weight Conjecture}},
+  howpublished = {OpenAI Math Release preprint
+                  \href{https://github.com/openai/math/blob/main/preprints/The-Blockwise-Alperin-Weight-Conjecture-September-23-2026/paper.pdf}{OAI:The-Blockwise-Alperin-Weight-Conjecture-September-23-2026}},
+  year = {2026}
+}
+```
+

preprints/The-Blockwise-Alperin-Weight-Conjecture-September-23-2026/build/figures/field-tower.tex

@@ -0,0 +1,40 @@
+\begin{figure}[!ht]
+\centering
+\begin{tikzpicture}[
+  every node/.style={font=\small},
+  field/.style={inner sep=4pt},
+  inclusion/.style={-{Stealth[length=4pt]},semithick}
+]
+\begin{scope}[xshift=-3.45cm]
+  \node[font=\small\bfseries] at (0,4.65) {Characteristic zero};
+  \node[field] (kb) at (0,0) {$K_b$};
+  \node[field] (ks) at (-1.45,1.35) {$K_s$};
+  \node[field] (kbp) at (1.45,1.35) {$K_b'$};
+  \node[field] (l) at (-1.45,2.8) {$L$};
+  \node[field] (lp) at (0,4) {$L'=LK_b'$};
+  \draw[inclusion] (kb)--(ks);
+  \draw[inclusion] (kb)--(kbp);
+  \node[font=\footnotesize,align=left] at (1.85,.35)
+    {unramified\\degree $n$};
+  \draw[inclusion] (ks)--(l);
+  \draw[inclusion] (l)--(lp);
+  \draw[inclusion] (kbp)--(lp);
+\end{scope}
+\begin{scope}[xshift=3.45cm]
+  \node[font=\small\bfseries] at (0,4.65) {Residue fields};
+  \node[field] (f0) at (0,0) {$F_0$};
+  \node[field] (fp) at (1.45,1.35) {$F'$};
+  \node[field] (f) at (-1.45,2.8) {$F$};
+  \node[field] (s) at (0,4) {$S=FF'$};
+  \draw[inclusion] (f0)--(fp);
+  \draw[inclusion] (f0)--(f);
+  \draw[inclusion] (f)--(s);
+  \draw[inclusion] (fp)--(s);
+\end{scope}
+\node[font=\footnotesize,align=center] at (0,-.85)
+ {$\operatorname{res}(K_b)=\operatorname{res}(K_s)=F_0,
+   \qquad [F:F_0]=[L:K_s],\qquad [L:K_s]_p<p^h.$};
+\end{tikzpicture}
+\caption{The auxiliary fields in a descent arising from a small Sylow orbit. Arrows denote field inclusions. The right-hand diagram records the residues of the left-hand fields; in particular $K_b$ and $K_s$ have the same residue. The field $K_b'$ splits the marked sections, and its residue $F'$ splits their reductions. The cyclic extension $K_b'/K_b$ reduces to $F'/F_0$, and the Galois actions of $L'/L$ and $S/F$ induce the same permutations of the marked points.}
+\label{fig:field-tower}
+\end{figure}

preprints/The-Blockwise-Alperin-Weight-Conjecture-September-23-2026/build/figures/proof-interfaces.tex

@@ -0,0 +1,54 @@
+\begin{figure}[!ht]
+\centering
+\begin{tikzpicture}[
+  every node/.style={font=\small,align=center},
+  stage/.style={draw=black!55,rounded corners=2pt,inner sep=7pt},
+  flow/.style={-{Stealth[length=5pt]},semithick}
+]
+\node[stage,text width=12cm] (twist)
+  {Marked genus-one curve: large divisor index\\
+   and independent absolute field differentials\\
+   \Cref{sec:twist}};
+\coordinate (branch-top) at ($(twist.south)+(0,-9mm)$);
+\coordinate (branch-split) at ($(twist.south)+(0,-4mm)$);
+\node[stage,text width=5.5cm,anchor=north] (genus)
+  at ($(branch-top)+(-3.2cm,0)$)
+  {Differential estimate and divisor-degree divisibility\\Sharp inseparable genus bound
+   (\Cref{sec:differentials})};
+\node[stage,text width=5.5cm,anchor=north] (lift)
+  at ($(branch-top)+(3.2cm,0)$)
+  {Valued lift and marked-cover descent\\Small orbit gives an extension
+   with small $p$-part of its degree
+   (\Cref{sec:valuations})};
+\node[fit=(genus)(lift),inner sep=0pt,outer sep=0pt] (branches) {};
+\node[stage,text width=12cm,below=10mm of branches] (specialization)
+  {Section comparison, finite presentation, and connectedness\\
+   No such marked $J$-cover over $L$ with $[L:K_s]_p<p^h$\\
+   when $O_p(J)=1$
+   (\Cref{sec:specialization})};
+\node[stage,text width=12cm,below=7mm of specialization] (count)
+  {Every Sylow orbit has size divisible by $p^h$\\
+   Uniform tuple divisibility (\Cref{thm:tuple-divisibility})};
+\node[stage,text width=12cm,below=7mm of count] (blocks)
+  {Chain inversion and central trace limits (\Cref{sec:group})\\
+   $l(B)=|W_p(B)|$};
+\draw[flow] (twist.south) -- (branch-split) -| (genus.north);
+\draw[flow] (twist.south) -- (branch-split) -| (lift.north);
+\draw[flow] (genus.south) -- (genus.south |- specialization.north);
+\draw[flow] (lift.south) -- (lift.south |- specialization.north);
+\draw[flow] (specialization.south) -- (count.north);
+\draw[flow] (count.south) -- (blocks.north);
+\end{tikzpicture}
+\caption{Mathematical dependencies, not reading order. The covers are geometrically
+connected $J$-Galois covers, with specified $J$-action over their field of
+definition, branched only at the marked points and with inertia order
+divisible by $p$ at every marked point. The two branches provide, respectively, the
+genus estimate and the arithmetic realization of a hypothetical small
+Sylow orbit. For sufficiently large $s$, their combination rules out such a cover over a field whose
+degree has small $p$-part; the geometric cover degree is fixed at $|J|$.
+The resulting uniform
+divisibility supplies the counting hypothesis of the block-algebra
+reduction, which is proved before the geometric construction and applied
+at the end. No weight-counting theorem is used to construct the curves.}
+\label{fig:proof-interfaces}
+\end{figure}

preprints/The-Blockwise-Alperin-Weight-Conjecture-September-23-2026/build/paper.tex

@@ -0,0 +1,103 @@
+\newcommand{\DoNotLoadEpstopdf}{}
+\documentclass[11pt]{article}
+\usepackage[T1]{fontenc}
+\usepackage[utf8]{inputenc}
+\usepackage{lmodern}
+\input{glyphtounicode-cmex}
+\pdfgentounicode=1
+\usepackage[margin=1in]{geometry}
+\usepackage{amsmath,amssymb,amsthm,mathtools}
+\usepackage{enumitem,booktabs,array}
+\usepackage{microtype}
+\usepackage{xcolor}
+\usepackage{etoolbox}
+\usepackage{tikz}
+\usetikzlibrary{arrows.meta,positioning,calc,fit}
+\usepackage[pdftex,unicode,colorlinks=true,linkcolor=blue!45!black,citecolor=green!35!black,urlcolor=blue!55!black]{hyperref}
+\usepackage[nameinlink,capitalise,noabbrev]{cleveref}
+\makeatletter
+\def\paper@thmrefstepcounter[#1]#2{%
+  \let\paper@saved@refstepcounter\cref@old@refstepcounter
+  \let\cref@old@refstepcounter\Hy@theorem@refstepcounter
+  \refstepcounter[#1]{#2}%
+  \let\cref@old@refstepcounter\paper@saved@refstepcounter
+}
+\AddToHook{begindocument/end}{%
+  \patchcmd\cref@thmnoarg{\refstepcounter}{\Hy@theorem@refstepcounter}{}%
+    {\PackageError{paper}{Failed to patch theorem anchors}%
+      {Check the hyperref and cleveref compatibility patch.}}%
+  \patchcmd\cref@thmoptarg{\refstepcounter}{\paper@thmrefstepcounter}{}%
+    {\PackageError{paper}{Failed to patch named theorem anchors}%
+      {Check the hyperref and cleveref compatibility patch.}}%
+}
+\makeatother
+\hypersetup{pdftitle={The Blockwise Alperin Weight Conjecture},pdfauthor={OpenAI},pdfsubject={Modular representation theory, curve covers, and inseparable genus bounds}}
+\numberwithin{equation}{section}
+\newtheorem{theorem}{Theorem}[section]
+\newtheorem{lemma}[theorem]{Lemma}
+\newtheorem{proposition}[theorem]{Proposition}
+\newtheorem{corollary}[theorem]{Corollary}
+\theoremstyle{definition}
+\newtheorem{definition}[theorem]{Definition}
+\newtheorem{notation}[theorem]{Notation}
+\newtheorem{foundation}{Foundation}
+\theoremstyle{remark}
+\newtheorem{remark}[theorem]{Remark}
+\AddToHook{env/theorem/begin}{\crefalias{theorem}{theorem}}
+\AddToHook{env/lemma/begin}{\crefalias{theorem}{lemma}}
+\AddToHook{env/proposition/begin}{\crefalias{theorem}{proposition}}
+\AddToHook{env/corollary/begin}{\crefalias{theorem}{corollary}}
+\AddToHook{env/definition/begin}{\crefalias{theorem}{definition}}
+\AddToHook{env/notation/begin}{\crefalias{theorem}{notation}}
+\AddToHook{env/remark/begin}{\crefalias{theorem}{remark}}
+\crefname{foundation}{Foundation}{Foundations}
+\Crefname{foundation}{Foundation}{Foundations}
+\DeclareMathOperator{\Irr}{Irr}
+\DeclareMathOperator{\IBr}{IBr}
+\DeclareMathOperator{\Br}{Br}
+\DeclareMathOperator{\Gal}{Gal}
+\DeclareMathOperator{\Spec}{Spec}
+\DeclareMathOperator{\Proj}{Proj}
+\DeclareMathOperator{\End}{End}
+\DeclareMathOperator{\Aut}{Aut}
+\DeclareMathOperator{\Hom}{Hom}
+\DeclareMathOperator{\Tr}{Tr}
+\DeclareMathOperator{\im}{im}
+\DeclareMathOperator{\rank}{rank}
+\DeclareMathOperator{\rad}{rad}
+\DeclareMathOperator{\coker}{coker}
+\DeclareMathOperator{\Frac}{Frac}
+\DeclareMathOperator{\ord}{ord}
+\newcommand{\cO}{\mathcal O}
+\newcommand{\Frob}{\operatorname{Frob}}
+\setlist{itemsep=3pt,topsep=5pt}
+\setlength{\emergencystretch}{1.5em}
+\title{The Blockwise Alperin Weight Conjecture}
+\author{OpenAI}
+\date{September 23, 2026}
+
+\begin{document}
+\maketitle
+\begin{abstract}
+We prove the numerical blockwise Alperin weight conjecture for every
+finite group and every prime $p$. For each $p$-block $B$, the number of
+irreducible Brauer characters in $B$ equals the number of conjugacy
+classes of $B$-weights.
+\end{abstract}
+{\fontsize{9}{10}\selectfont
+\tableofcontents
+}
+\clearpage
+
+\input{sections/intro}
+\input{sections/foundations}
+\input{sections/group}
+\input{sections/twist}
+\input{sections/differentials}
+\input{sections/valuations}
+\input{sections/specialization}
+\input{sections/consequences}
+
+\bibliographystyle{amsplain}
+\bibliography{references}
+\end{document}

preprints/The-Blockwise-Alperin-Weight-Conjecture-September-23-2026/build/references.bib

@@ -0,0 +1,448 @@
+@incollection{Alperin1987,
+  author    = {Alperin, J. L.},
+  title     = {\href{https://doi.org/10.1090/pspum/047.1/933373}{Weights for finite groups}},
+  booktitle = {The Arcata Conference on Representations of Finite Groups,
+               Part 1 (Arcata, Calif., 1986)},
+  series    = {Proceedings of Symposia in Pure Mathematics},
+  volume    = {47},
+  pages     = {369--379},
+  publisher = {American Mathematical Society},
+  address   = {Providence, RI},
+  year      = {1987},
+  doi       = {10.1090/pspum/047.1/933373},
+  url       = {https://doi.org/10.1090/pspum/047.1/933373}
+}
+
+@article{Carlson1971,
+  author  = {Carlson, Jon F.},
+  title   = {\href{https://doi.org/10.1017/S0004972700047298}{Block idempotents and the {Brauer} correspondence}},
+  journal = {Bulletin of the Australian Mathematical Society},
+  volume  = {5},
+  number  = {3},
+  pages   = {337--340},
+  year    = {1971},
+  doi     = {10.1017/S0004972700047298},
+  url     = {https://doi.org/10.1017/S0004972700047298}
+}
+
+@article{Carlson1973,
+  author  = {Carlson, Jon F.},
+  title   = {\href{https://doi.org/10.1017/S0004972700042581}{Corrigendum: Block idempotents and the {Brauer} correspondence}},
+  journal = {Bulletin of the Australian Mathematical Society},
+  volume  = {8},
+  number  = {2},
+  pages   = {319},
+  year    = {1973},
+  doi     = {10.1017/S0004972700042581},
+  url     = {https://doi.org/10.1017/S0004972700042581}
+}
+
+@article{KnorrRobinson1989,
+  author  = {Kn{\"o}rr, Reinhard and Robinson, Geoffrey R.},
+  title   = {\href{https://doi.org/10.1112/jlms/s2-39.1.48}{Some remarks on a conjecture of {Alperin}}},
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+  doi     = {10.1112/jlms/s2-39.1.48},
+  url     = {https://doi.org/10.1112/jlms/s2-39.1.48}
+}
+
+@article{NavarroTiep2011,
+  author  = {Navarro, Gabriel and Tiep, Pham Huu},
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+  volume  = {184},
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+  year    = {2011},
+  doi     = {10.1007/s00222-010-0295-2},
+  url     = {https://doi.org/10.1007/s00222-010-0295-2}
+}
+
+@article{Spath2013,
+  author  = {Sp{\"a}th, Britta},
+  title   = {\href{https://doi.org/10.1515/jgt-2012-0032}{A reduction theorem for the blockwise {Alperin} weight conjecture}},
+  journal = {Journal of Group Theory},
+  volume  = {16},
+  number  = {2},
+  pages   = {159--220},
+  year    = {2013},
+  doi     = {10.1515/jgt-2012-0032},
+  url     = {https://doi.org/10.1515/jgt-2012-0032}
+}
+
+@article{FLZ2023,
+  author  = {Feng, Zhicheng and Li, Conghui and Zhang, Jiping},
+  title   = {\href{https://doi.org/10.1016/j.jalgebra.2023.04.025}{On the inductive blockwise {Alperin} weight condition for type {A}}},
+  journal = {Journal of Algebra},
+  volume  = {631},
+  pages   = {287--305},
+  year    = {2023},
+  doi     = {10.1016/j.jalgebra.2023.04.025},
+  url     = {https://doi.org/10.1016/j.jalgebra.2023.04.025}
+}
+
+@article{FMZ2026,
+  author  = {Feng, Zhicheng and Malle, Gunter and Zhang, Jiping},
+  title   = {\href{https://doi.org/10.1007/s00208-026-03415-7}{Generic weights for finite reductive groups}},
+  journal = {Mathematische Annalen},
+  volume  = {394},
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+  pages   = {Article~90},
+  doi     = {10.1007/s00208-026-03415-7},
+  url     = {https://doi.org/10.1007/s00208-026-03415-7}
+}
+
+@misc{Stacks,
+  author       = {{The Stacks Project Authors}},
+  title        = {The {Stacks} Project},
+  year         = {2026},
+  howpublished = {\url{https://stacks.math.columbia.edu}},
+  note         = {Tags cited individually in the text; accessed October 1, 2026}
+}
+
+@misc{MilneAV2022,
+  author       = {Milne, James S.},
+  title        = {Abelian Varieties},
+  year         = {2022},
+  howpublished = {\url{https://www.jmilne.org/math/xnotes/AVs.pdf}},
+  note         = {Author's corrected version, January 2, 2022, of the chapter
+                  in Arithmetic Geometry (1986), pp.\ 103--150}
+}
+
+@misc{MilneLEC2008,
+  author       = {Milne, James S.},
+  title        = {Lectures on {\'E}tale Cohomology},
+  year         = {2008},
+  howpublished = {\url{https://www.jmilne.org/math/CourseNotes/LEC210.pdf}},
+  note         = {Version 2.10, May 20, 2008}
+}
+
+@book{Silverman2009,
+  author    = {Silverman, Joseph H.},
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+  edition   = {Second},
+  series    = {Graduate Texts in Mathematics},
+  volume    = {106},
+  publisher = {Springer},
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+  doi       = {10.1007/978-0-387-09494-6},
+  url       = {https://doi.org/10.1007/978-0-387-09494-6}
+}
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+  author = {Higman, Donald G.},
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+}
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+}
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+}
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+  title = {\href{https://doi.org/10.1007/BF02699128}{Canonical models of surfaces of general type in positive characteristic}},
+  journal = {Publications Math{\'e}matiques de l'IH{\'E}S},
+  volume = {67},
+  year = {1988},
+  pages = {97--144},
+  doi = {10.1007/BF02699128},
+  url = {https://doi.org/10.1007/BF02699128}
+}
+
+@article{Krull1932,
+  author = {Krull, Wolfgang},
+  title = {\href{https://doi.org/10.1515/crll.1932.167.160}{Allgemeine Bewertungstheorie}},
+  journal = {Journal f{\"u}r die reine und angewandte Mathematik},
+  volume = {167},
+  year = {1932},
+  pages = {160--196},
+  doi = {10.1515/crll.1932.167.160},
+  url = {https://doi.org/10.1515/crll.1932.167.160}
+}
+
+@article{Kaplansky1942,
+  author = {Kaplansky, Irving},
+  title = {\href{https://doi.org/10.1215/S0012-7094-42-00922-0}{Maximal fields with valuations}},
+  journal = {Duke Mathematical Journal},
+  volume = {9},
+  number = {2},
+  year = {1942},
+  pages = {303--321},
+  doi = {10.1215/S0012-7094-42-00922-0},
+  url = {https://doi.org/10.1215/S0012-7094-42-00922-0}
+}
+
+@article{Poonen1993,
+  author = {Poonen, Bjorn},
+  title = {\href{https://math.mit.edu/~poonen/papers/amsval.pdf}{Maximally complete fields}},
+  journal = {L'Enseignement Math{\'e}matique},
+  series = {2},
+  volume = {39},
+  number = {1--2},
+  year = {1993},
+  pages = {87--106},
+  url = {https://math.mit.edu/~poonen/papers/amsval.pdf}
+}
+
+@article{DebesDouai1997,
+  author = {D{\`e}bes, Pierre and Douai, Jean-Claude},
+  title = {\href{https://doi.org/10.1016/S0012-9593(97)89922-3}{Algebraic covers: field of moduli versus field of definition}},
+  journal = {Annales scientifiques de l'{\'E}cole Normale Sup{\'e}rieure},
+  series = {4},
+  volume = {30},
+  number = {3},
+  year = {1997},
+  pages = {303--338},
+  doi = {10.1016/S0012-9593(97)89922-3},
+  url = {https://doi.org/10.1016/S0012-9593(97)89922-3}
+}
+
+@incollection{DeligneRapoport1973,
+  author    = {Deligne, P. and Rapoport, M.},
+  title     = {\href{https://doi.org/10.1007/978-3-540-37855-6_4}{Les sch\'emas de modules de courbes elliptiques}},
+  booktitle = {Modular Functions of One Variable II},
+  series    = {Lecture Notes in Mathematics},
+  volume    = {349},
+  pages     = {143--316},
+  publisher = {Springer-Verlag},
+  year      = {1973},
+  doi       = {10.1007/978-3-540-37855-6_4},
+  url       = {https://publications.ias.edu/sites/default/files/Number22.pdf}
+}
+
+@article{Brauer1959,
+  author  = {Brauer, Richard},
+  title   = {\href{https://doi.org/10.1007/BF01162934}{Zur Darstellungstheorie der Gruppen endlicher Ordnung. {II}}},
+  journal = {Mathematische Zeitschrift},
+  volume  = {72},
+  pages   = {25--46},
+  year    = {1959},
+  doi     = {10.1007/BF01162934},
+  url     = {https://doi.org/10.1007/BF01162934}
+}
+
+@article{AlperinFong1990,
+  author  = {Alperin, J. L. and Fong, Paul},
+  title   = {\href{https://doi.org/10.1016/0021-8693(90)90163-I}{Weights for symmetric and general linear groups}},
+  journal = {Journal of Algebra},
+  volume  = {131},
+  number  = {1},
+  pages   = {2--22},
+  year    = {1990},
+  doi     = {10.1016/0021-8693(90)90163-I},
+  url     = {https://doi.org/10.1016/0021-8693(90)90163-I}
+}
+
+@book{AnHissLubeck2024,
+  author  = {An, Jianbei and Hiss, Gerhard and L{\"u}beck, Frank},
+  title   = {\href{https://doi.org/10.1090/memo/1530}{The inductive blockwise {Alperin} weight condition for the {Chevalley} groups {$F_4(q)$}}},
+  series  = {Memoirs of the American Mathematical Society},
+  volume  = {304},
+  note    = {No.~1530},
+  publisher = {American Mathematical Society},
+  address = {Providence, RI},
+  year    = {2024},
+  doi     = {10.1090/memo/1530},
+  url     = {https://doi.org/10.1090/memo/1530}
+}
+
+@article{HuZhou2025,
+  author  = {Hu, Xueqin and Zhou, Yuanyang},
+  title   = {\href{https://doi.org/10.1016/j.jalgebra.2025.01.030}{The blockwise {Alperin} weight conjecture and inertial blocks}},
+  journal = {Journal of Algebra},
+  volume  = {672},
+  pages   = {379--399},
+  year    = {2025},
+  doi     = {10.1016/j.jalgebra.2025.01.030},
+  url     = {https://doi.org/10.1016/j.jalgebra.2025.01.030}
+}
+
+@article{HuangYilmaz2026,
+  author  = {Huang, Xin and Y{\i}lmaz, Deniz},
+  title   = {\href{https://doi.org/10.1016/j.jalgebra.2026.03.045}{{Alperin}'s weight conjecture, {Galois} automorphisms, alternating sums, and functorial equivalences}},
+  journal = {Journal of Algebra},
+  volume  = {701},
+  pages   = {312--327},
+  year    = {2026},
+  doi     = {10.1016/j.jalgebra.2026.03.045},
+  url     = {https://doi.org/10.1016/j.jalgebra.2026.03.045}
+}
+
+@misc{Zhang2026BC,
+  author        = {Zhang, Baoyu},
+  title         = {On the inductive blockwise {Alperin} weight condition for type {$\mathsf B$} and type {$\mathsf C$}},
+  year          = {2026},
+  eprint        = {2609.24774},
+  archivePrefix = {arXiv},
+  primaryClass  = {math.GR},
+  howpublished  = {\url{https://arxiv.org/abs/2609.24774v1}},
+  note          = {Version 1, September 21, 2026},
+  url           = {https://arxiv.org/abs/2609.24774v1}
+}
+
+@article{EGAIII1,
+  author = {Grothendieck, Alexandre and Dieudonn{\'e}, Jean},
+  title = {\href{https://doi.org/10.1007/BF02684273}{{\'E}l{\'e}ments de g{\'e}om{\'e}trie alg{\'e}brique {III}. {\'E}tude cohomologique des faisceaux coh{\'e}rents, Premi{\`e}re partie}},
+  journal = {Publications Math{\'e}matiques de l'IH{\'E}S},
+  volume = {11},
+  year = {1961},
+  pages = {5--167},
+  doi = {10.1007/BF02684273},
+  url = {https://doi.org/10.1007/BF02684273}
+}
+
+@article{EGAIV3,
+  author = {Grothendieck, Alexandre and Dieudonn{\'e}, Jean},
+  title = {\href{https://doi.org/10.1007/BF02684343}{{\'E}l{\'e}ments de g{\'e}om{\'e}trie alg{\'e}brique {IV}. {\'E}tude locale des sch{\'e}mas et des morphismes de sch{\'e}mas, Troisi{\`e}me partie}},
+  journal = {Publications Math{\'e}matiques de l'IH{\'E}S},
+  volume = {28},
+  year = {1966},
+  pages = {5--255},
+  doi = {10.1007/BF02684343},
+  url = {https://doi.org/10.1007/BF02684343}
+}
+
+@article{HST2024,
+  author  = {Hung, Nguyen Ngoc and Sambale, Benjamin and Tiep, Pham Huu},
+  title   = {\href{https://doi.org/10.1007/s11856-024-2613-1}{Bounding {$p$}-{Brauer} characters in finite groups with two conjugacy classes of {$p$}-elements}},
+  journal = {Israel Journal of Mathematics},
+  volume  = {262},
+  pages   = {327--358},
+  year    = {2024},
+  doi     = {10.1007/s11856-024-2613-1},
+  url     = {https://doi.org/10.1007/s11856-024-2613-1}
+}
+
+@article{MNT2023,
+  author  = {Malle, Gunter and Navarro, Gabriel and Tiep, Pham Huu},
+  title   = {\href{https://doi.org/10.1007/s00031-022-09734-8}{On {Alperin}'s lower bound for the number of {Brauer} characters}},
+  journal = {Transformation Groups},
+  volume  = {28},
+  pages   = {1205--1220},
+  year    = {2023},
+  doi     = {10.1007/s00031-022-09734-8},
+  url     = {https://doi.org/10.1007/s00031-022-09734-8}
+}
+
+@article{FMR2025,
+  author        = {Feng, Zhicheng and Mart{\'i}nez, J. Miquel and Rossi, Damiano},
+  title         = {\href{https://doi.org/10.1112/jlms.70470}{{Alperin}'s bound and normal {Sylow} subgroups}},
+  journal       = {Journal of the London Mathematical Society},
+  volume        = {113},
+  number        = {2},
+  pages         = {e70470},
+  year          = {2026},
+  doi           = {10.1112/jlms.70470},
+  url           = {https://doi.org/10.1112/jlms.70470}
+}
+
+@article{BBY2026,
+  author  = {Boltje, Robert and Bouc, Serge and Y{\i}lmaz, Deniz},
+  title   = {\href{https://doi.org/10.1007/s00209-026-03999-5}{On {Alperin}'s conjecture and functorial equivalence of blocks}},
+  journal = {Mathematische Zeitschrift},
+  volume  = {312},
+  pages   = {Article~112},
+  year    = {2026},
+  doi     = {10.1007/s00209-026-03999-5},
+  url     = {https://doi.org/10.1007/s00209-026-03999-5},
+  note    = {Author's version dated February 11, 2026:
+             \url{https://yilmazdeniz.bilkent.edu.tr/papers/alperin-functorial.pdf}}
+}
+
+@article{Chen2024,
+  author = {Chen, William Y.},
+  title = {\href{https://doi.org/10.4007/annals.2024.199.1.5}{Nonabelian level structures, {Nielsen} equivalence, and {Markoff} triples}},
+  journal = {Annals of Mathematics},
+  volume = {199},
+  number = {1},
+  year = {2024},
+  pages = {301--443},
+  doi = {10.4007/annals.2024.199.1.5},
+  url = {https://doi.org/10.4007/annals.2024.199.1.5}
+}

preprints/The-Blockwise-Alperin-Weight-Conjecture-September-23-2026/build/sections/consequences.tex

@@ -0,0 +1,171 @@
+\section{Further consequences}\label{sec:consequences}
+
+The main theorem gives both numerical information about blocks and a
+functorial identity. We first prove the local--global bounds stated in
+\Cref{cor:block-consequences}, then record the more detailed numerical
+classification and the categorical reformulation. These are consequences
+of the numerical theorem through the cited results; none is an input
+to the proof above.
+
+\subsection{Local--global and principal-block bounds}
+
+The comparison with a defect-group normalizer is local to an individual
+block. Summing it over maximal-defect blocks gives the comparison with
+the Sylow normalizer, while the two principal-block bounds relate the
+number of simple modules to the number of classes of $p$-elements.
+
+We use the notation of \Cref{cor:block-consequences}; in particular,
+$P\in\operatorname{Syl}_p(G)$.
+
+\begin{proof}[Proof of \Cref{cor:block-consequences}]
+The local inequality and its abelian-defect equality are Alperin's Consequences~1--2
+\cite{Alperin1987}, as restated in \cite[Section~3, p.~334]{HST2024};
+their numerical weight-conjecture hypotheses follow from \Cref{thm:main}.
+Put $N=N_G(P)$. Since $P$ is a normal Sylow $p$-subgroup of $N$,
+it is a defect group of every block of $N$.
+Brauer's first main theorem therefore matches all $p$-blocks of $N$
+with the maximal-defect $p$-blocks of $G$; this indexing is also used in
+\cite[proof of Lemma~2.14]{FMR2025}. Writing these two block sets as
+$\operatorname{Bl}_p(N)$ and $\operatorname{Bl}_{p,\max}(G)$, respectively,
+the local inequality gives
+\[
+ |\IBr_p(G)|
+ \geq\sum_{B'\in\operatorname{Bl}_{p,\max}(G)}l(B')
+ \geq\sum_{b'\in\operatorname{Bl}_p(N)}l(b')
+ =|\IBr_p(N)|.
+\]
+Finally, \Cref{thm:main} supplies the only conjectural input in
+Hung--Sambale--Tiep's Propositions~3.1--3.2 \cite{HST2024}, which give
+the two principal-block bounds under their respective hypotheses.
+\end{proof}
+
+\subsection{Character counts and defect data}
+
+The next consequence treats blocks whose ordinary-character count
+exceeds their Brauer-character count by one. Besides giving bounds,
+it specifies all possible pairs
+of defect-group order and Brauer-character count, and asserts that
+every listed pair is attained.
+
+\begin{corollary}[Character-count and defect data]\label{cor:character-data}
+Let $B$ be a $p$-block of a finite group with defect group $D$ of order
+$p^d$, and write $k(B)=|\Irr(B)|$. If $k(B)-l(B)=1$, then the possible
+numerical data $(p^d,l(B))$ are exactly the following:
+\begin{enumerate}[label=\textup{(\roman*)}]
+\item There are positive integers $n,m$ such that $d=nm$, every prime divisor of $n$
+      divides $p^m-1$, and $4\mid n$ implies $4\mid p^m-1$, with
+      \[
+       l(B)=\sum_{e\mid n}\frac{p^{em}-1}{ne}J_2(n/e),\qquad
+       J_2(t)=t^2\prod_{\substack{q\mid t\\q\text{ prime}}}(1-q^{-2}).
+      \]
+\item The exceptional pairs are
+      \[
+       \begin{aligned}
+       (p^d,l(B))\in\{&(5^2,7),(7^2,8),(11^2,9),(11^2,35),\\
+                       &(23^2,88),(29^2,63),(59^2,261)\}.
+       \end{aligned}
+      \]
+\end{enumerate}
+All listed data occur, and in every case $k(B)=l(B)+1$.
+\end{corollary}
+
+\begin{proof}
+Hung, Sambale, and Tiep proved this numerical classification assuming
+the numerical weight conjecture for $B$ \cite[Theorem~3.3]{HST2024}.
+\Cref{thm:main} supplies that hypothesis. Their examples establishing
+that every listed value occurs were already unconditional.
+\end{proof}
+
+This classifies only character-count and defect numerical data, not blocks
+up to isomorphism, Morita equivalence, or derived equivalence.
+
+\subsection{Functorial formulation}
+
+The following formulation expresses the numerical theorem as an identity
+of classes of functors. Its alternating sum runs over all strict chains;
+it is not the normal-chain transform used earlier to prove the theorem.
+
+The numerical theorem gives the functorial formulation of
+Boltje, Bouc, and Y\i lmaz \cite[Conjecture~3.3]{BBY2026}.
+Let $k$ and $F$ be algebraically closed fields of characteristics $p$ and
+zero, respectively. We recall the objects used in this optional
+formulation. A finite-dimensional $(kH,kG)$-bimodule is viewed as a
+$k[H\times G]$-module by $(h,g)m=hmg^{-1}$. It is a
+$p$-permutation module if its restriction to a Sylow $p$-subgroup
+has a basis permuted by that subgroup. Such a $p$-permutation bimodule
+is diagonal if it is also
+projective as a left $kH$-module and as a right $kG$-module.
+The notation $T^\Delta(H,G)$ denotes their split Grothendieck group,
+with relations $[M\oplus N]=[M]+[N]$.
+
+The category of group--central-idempotent pairs has objects $(H,c)$,
+including $c=0$. A morphism from $(H,c)$ to $(H',c')$ is an
+$F$-linear combination of diagonal $p$-permutation bimodule classes
+selected by $c'$ on the left and $c$ on the right. Composition is
+induced by tensor product over the intermediate group algebra.
+In the notation of \cite[Section~2]{BBY2026}, the category of
+$F$-linear functors from this category to $F$-vector spaces is
+$\mathcal F\mathcal B\ell_{F,k}$.
+
+This functor category is semisimple in characteristic zero
+\cite[Section~2.5]{BBY2026}. By Yoneda, the endomorphism space of
+the representable $FT^\Delta(-,H)$ is the finite-dimensional
+$F$-space $FT^\Delta(H,H)$. Semisimplicity therefore forces the
+representable to have finite length: each simple summand contributes an
+independent projection. Its direct summands also have finite length.
+Here $K_0(\mathcal F\mathcal B\ell_{F,k})$ denotes the Grothendieck
+group of the finite-length subcategory, the free abelian group on
+simple-functor classes. It is in this sense that classes record the
+simple multiplicities used in \cite[Sections~2.5 and~3]{BBY2026}.
+
+The bimodule $kHc$ acts by precomposition on the representable
+functor $FT^\Delta(-,H)$. Its image is the direct summand
+$FT^\Delta_{H,c}=FT^\Delta(-,H)\circ kHc$ associated to $(H,c)$,
+using the equivalent group and pair descriptions in
+\cite[Sections~2.3--2.4]{BBY2026}. In particular $c=0$ gives the
+zero functor. Write $[[H,c]]_F$ for its class in
+$K_0(\mathcal F\mathcal B\ell_{F,k})$.
+
+Write $S_{1,1,F}$ for the simple
+functor indexed by the trivial pair and its trivial $F$-module.
+Its multiplicity in $FT^\Delta_{H,c}$ is $l(H,c)$: evaluation at
+the trivial group gives the space spanned by indecomposable projective
+$kHc$-modules \cite[proof of Theorem~3.4]{BBY2026}. Thus the
+following identity records all simple-functor multiplicities, with
+the simple-module count as its $S_{1,1,F}$ component.
+
+\begin{corollary}[Functorial Alperin weight identity]\label{cor:functorial-awc}
+Let $b$ be a block idempotent of $kG$, where $G$ is finite. Let
+$\mathcal S_p(G)$ consist of all strict chains
+$\sigma=(1=P_0<P_1<\cdots<P_n)$ of $p$-subgroups, including $(1)$.
+Put
+\[
+ |\sigma|=n,\qquad
+ G_\sigma=\bigcap_{i=0}^nN_G(P_i),\qquad
+ b_\sigma=\operatorname{Br}_{P_n}(b).
+\]
+Since $C_G(P_n)\leq G_\sigma$ and the Brauer image is invariant
+under $G_\sigma$, the element $b_\sigma$ is a central idempotent
+of $kG_\sigma$, possibly zero or a sum of blocks. Thus the pair
+$(G_\sigma,b_\sigma)$ is an object of the category just described.
+Then, summing over representatives of the $G$-conjugacy classes of chains,
+\[
+ \sum_{\sigma\in[G\backslash\mathcal S_p(G)]}
+ (-1)^{|\sigma|}[[G_\sigma,b_\sigma]]_F
+ =\begin{cases}
+ [S_{1,1,F}],&\text{if $b$ has defect zero},\\
+ 0,&\text{if $b$ has positive defect}
+ \end{cases}
+ \quad\text{in }K_0(\mathcal F\mathcal B\ell_{F,k}).
+\]
+\end{corollary}
+
+\begin{proof}
+Boltje--Bouc--Y\i lmaz first prove that, for every group--block pair,
+the alternating sum above is an integer multiple of $[S_{1,1,F}]$:
+every other simple-functor multiplicity cancels
+\cite[Theorem~3.5\textup{(1)}]{BBY2026}. The universal numerical
+blockwise Alperin weight conjecture determines the remaining coefficient
+through its chain formulation. Their Theorem~3.5\textup{(2)} therefore
+gives the stated identity from \Cref{thm:main}.
+\end{proof}

preprints/The-Blockwise-Alperin-Weight-Conjecture-September-23-2026/build/sections/differentials.tex

@@ -0,0 +1,782 @@
+\section{Absolute differentials and the inseparable genus estimate}
+\label{sec:differentials}
+
+The two properties of the marked curve in \Cref{sec:twist} will now
+give an exact lower bound for the arithmetic genus of its purely
+inseparable covers. Throughout this section, a curve over a field $U$
+means a regular integral projective scheme of dimension one over $U$.
+It need not be geometrically regular or geometrically reduced. We set
+\[
+ \chi(C)=\dim_U H^0(C,\mathcal O_C)-\dim_U H^1(C,\mathcal O_C),
+ \qquad \nu(C)=-2\chi(C).
+\]
+Divisor degrees and cohomology dimensions are always taken over the
+stated ground field, and the degree of a vector bundle is the degree
+of its determinant. Thus $\nu(C)$ retains any contribution from an
+inseparable extension of the constant field.
+
+Fix a finite group $J$ as in \Cref{thm:tuple-divisibility} and put
+$m=|J|$. Only this order enters the following estimate, as a bound on
+the cover degrees needed later.
+Put $\kappa=\overline{\mathbb F}_p$. For each $s$, use the fields,
+elliptic curve, and labels constructed in \Cref{prop:twist-index} for
+that value of $s$. In particular,
+$n=p^s$, $F'=\kappa(A_0^n)$, and $E_{F'}\simeq(A_0)_{F'}$.
+Write $\mathcal B$ for the descended finite \'{e}tale divisor of labels;
+over $F'$ it is the sum of the distinct rational points $t_1,\ldots,t_n$.
+For $h\geq1$, $s\geq h$, and a finite extension $F/F_0$ with
+$[F:F_0]_p\leq p^h$, we use two properties of the marked curve.
+Every divisor degree on $E_F$ is divisible by $p^{s-h}$, by
+\Cref{cor:index-base-change}. Over $S=FF'$, any set of the forms
+$t_i^*\eta$ loses at most $h$ dimensions of independence, by
+\Cref{lem:label-rank}, where $\eta$ is a nonzero invariant differential
+on $A_0$.
+
+\begin{theorem}[Exact purely inseparable genus estimate]
+\label{thm:inseparable-genus}
+Fix $h\ge1$ and take $s\ge h$ such that
+\begin{equation}
+ p^{s-h}>m\bigl(h+\log_p m\bigr).
+ \label{eq:genus-rounding-threshold}
+\end{equation}
+Let $F/F_0$ be finite with $[F:F_0]_p\le p^h$, and let $Z$ be the
+normalization of $E_F$ in a finite purely inseparable extension of its
+function field of degree $d\le m$.  Choose a compositum $S=FF'$.
+The curve $Z_S$ is integral and regular.  Let $r_i$ be the ramification
+index of $Z_S\to E_S$ at the unique point over $t_i$.
+With $\nu(Z)$ computed over $F$, one has
+\begin{equation}
+ \frac{\nu(Z)}d\ge\sum_{i=1}^n\left(1-\frac1{r_i}\right).
+ \label{eq:inseparable-genus-bound}
+\end{equation}
+The bound is uniform in $F$, $Z$, and the tuple-class multiset used
+later in the counting argument.
+\end{theorem}
+
+The proof has two parts. Absolute differentials give the same
+inequality with an error of at most $h+\log_p d$. After multiplication
+by $d$, the difference between the two sides is a difference of
+divisor degrees over $F$, hence a multiple of $p^{s-h}$. The strict
+threshold then forces this difference to be nonnegative. We develop
+the differential identities first, together with the separable
+ramification bound needed later in \Cref{sec:specialization}.
+
+\subsection{Divisor degrees, duality, and separable ramification}
+
+The required identities hold over more general fields than the
+twisted-curve construction. Throughout this and the next subsection,
+let $\kappa$ be any perfect field of characteristic $p$, let
+$S/\kappa$ be finitely generated, and put
+$\rho=\operatorname{trdeg}_\kappa S$. Curves and degrees in these two
+subsections are over $S$.
+
+The classical theory of inseparable genus change begins with Tate
+\cite{Tate1952}; Schr\"oer \cite{Schroer2009} gives a modern account
+for geometrically integral curves. Here absolute differentials retain
+arithmetic genus without a geometric-reducedness hypothesis and also
+measure ramification at the marked points.
+
+\begin{lemma}[Degrees over the stated ground field]
+\label{lem:curve-degrees}
+For an invertible sheaf $\mathcal L$ on $C$,
+\begin{equation}
+ \chi(C,\mathcal L)=\deg_S\mathcal L+\chi(C).
+ \label{eq:curve-euler-degree}
+\end{equation}
+A finite dominant morphism $f:C\to C'$ of such curves is finite flat.  If
+its function-field degree is $\delta$, then
+\[
+ \deg_S f^*D=\delta\deg_S D
+ \quad\text{and}\quad
+ \deg_S f_*D'=\deg_S D'
+\]
+for divisors $D$ on $C'$ and $D'$ on $C$.  These assertions, and Euler
+characteristics, add over disjoint unions of regular integral curves.
+After a finite separable extension $S_1/S$, divisor and line-bundle degrees
+and coherent Euler characteristics have the same numerical values over
+$S_1$ as before extension over $S$.
+\end{lemma}
+
+\begin{proof}
+For a closed point $x$ put $\deg_S(x)=[\kappa(x):S]$.  For any divisor $D$
+the quotient in
+\[
+ 0\longrightarrow\mathcal O_C(D)
+ \longrightarrow\mathcal O_C(D+x)
+ \longrightarrow\mathcal Q_x\longrightarrow0
+\]
+has length one over the DVR $\mathcal O_{C,x}$, so its dimension over $S$
+is $[\kappa(x):S]$.  Finite-support sheaves have no higher cohomology.
+Adding and subtracting closed points proves
+$\chi(\mathcal O_C(D))=\chi(C)+\deg_S D$.  A rational trivialization of
+$\mathcal L$ identifies it with $\mathcal O_C(D)$ for a divisor $D$, giving
+\eqref{eq:curve-euler-degree}.  Applied to a principal divisor, the same
+formula gives degree zero, so this degree depends only on $\mathcal L$.
+
+Locally on $C'$, the finite module $f_*\mathcal O_C$ is torsion-free over
+a DVR and therefore free, of generic rank $\delta$.
+For a closed point $x\in C'$ and a uniformizer $t$ there, reduction of
+this free module modulo $t$ gives
+\begin{equation}
+ \sum_{y\mapsto x}e_y[\kappa(y):\kappa(x)]=\delta,
+ \label{eq:local-degree-formula}
+\end{equation}
+where $e_y$ is the valuation of $t$ in $\mathcal O_{C,y}$.
+Indeed the summand at $y$ has a filtration of length $e_y$ with
+successive quotients $\kappa(y)$.  Multiplication by
+$[\kappa(x):S]$ proves the pullback degree formula.  The pushforward
+formula follows from
+$f_*y=[\kappa(y):\kappa(x)]x$ and the tower formula for residue degrees.
+
+Finite separable scalar extension preserves regularity, and a regular
+curve after such extension is a disjoint union of its integral
+components.  Flat base change for coherent cohomology
+in \Cref{foundation:cohomology} gives
+\[
+ H^i(C_{S_1},\mathcal G_{S_1})
+   =H^i(C,\mathcal G)\otimes_S S_1.
+\]
+Thus the dimensions over the respective fields, and hence Euler
+characteristics, agree.  Applying
+\eqref{eq:curve-euler-degree} proves the assertion for line bundles and
+their Cartier divisors.  In particular it also applies to a reduced
+finite divisor: separable scalar extension preserves its reducedness.
+\end{proof}
+
+\begin{proposition}[The absolute determinant computes arithmetic genus]
+\label{prop:absolute-genus}
+The absolute cotangent sheaf $\Omega_{C/\kappa}$ is locally free of rank
+$\rho+1$, and
+\begin{equation}
+ \deg_S\det\Omega_{C/\kappa}=\nu(C).
+ \label{eq:absolute-determinant-genus}
+\end{equation}
+\end{proposition}
+
+\begin{proof}
+An affine coordinate ring of $C$ is a localization of a finitely
+generated integral $\kappa$-algebra: choose a finitely generated domain
+with fraction field $S$, clear the finitely many coefficients in a
+presentation over $S$, and take the contraction of the defining prime
+ideal before localizing.  Its local rings are consequently essentially
+of finite type over $\kappa$.  The regularity criterion over a perfect
+field in \Cref{foundation:regularity} says that these regular local rings
+are local rings at smooth points of the corresponding
+$\kappa$-variety.  Their absolute differentials are free of rank
+\[
+ \operatorname{trdeg}_\kappa\kappa(C)=\rho+1.
+\]
+The sheaf is coherent: the relative differentials over $S$ are coherent,
+and $\Omega_{S/\kappa}$ is finite-dimensional, so the cotangent sequence
+gives a finite presentation locally.  This proves the asserted local
+freeness without making a smoothness assumption over $S$.
+
+Choose a projective embedding $i:C\hookrightarrow\mathbb P^b_S$ and let
+$\mathcal I$ be its ideal sheaf.  This is a regular immersion of
+codimension $b-1$.  Here is a local verification.  Let $R$ be an ambient
+regular local ring, $I$ the local ideal, and $R/I$ the regular local ring
+of $C$.  Their dimensions differ by $b-1$, by the dimension formula at
+the common residue field.  Since both are regular, the kernel of
+\[
+ \mathfrak m_R/\mathfrak m_R^2
+   \longrightarrow
+ \mathfrak m_{R/I}/\mathfrak m_{R/I}^2
+\]
+has dimension $b-1$.  Choose $b-1$ elements of $I$ lifting a basis of this
+kernel.  They form part of a regular system of parameters of $R$ and
+cut out a regular local quotient of dimension $\dim(R/I)$.
+The remaining ideal defining $R/I$ in that quotient is prime and must
+be zero: a nonzero prime in a finite-dimensional local domain strictly
+lowers the quotient dimension.  Thus the chosen elements generate $I$.
+It follows that $\mathcal I/\mathcal I^2$ is locally free of rank $b-1$.
+
+The absolute conormal sequence is in fact short exact:
+\begin{equation}
+ 0\longrightarrow\mathcal I/\mathcal I^2
+ \longrightarrow i^*\Omega_{\mathbb P^b_S/\kappa}
+ \longrightarrow\Omega_{C/\kappa}\longrightarrow0.
+ \label{eq:absolute-conormal}
+\end{equation}
+To check the left end, the kernel of the last arrow is locally free,
+since the last module is locally free.  Its rank is
+$(b+\rho)-(1+\rho)=b-1$.  The conormal sequence surjects
+$\mathcal I/\mathcal I^2$ onto this kernel.  A surjection between locally
+free modules of the same rank is an isomorphism, proving
+\eqref{eq:absolute-conormal}.
+
+The projective space is obtained from $\mathbb P^b_\kappa$ by extending
+constants.  Its absolute cotangents therefore split into relative
+cotangents over $S$ and the constant bundle of $\Omega_{S/\kappa}$.
+Taking determinants in \eqref{eq:absolute-conormal} gives
+\[
+ \det\Omega_{C/\kappa}
+  \simeq\mathcal L_c\otimes_S\det\Omega_{S/\kappa},
+ \qquad
+ \mathcal L_c=
+ \mathcal O_C(-b-1)\otimes
+       \det(\mathcal I/\mathcal I^2)^\vee.
+\]
+The last tensor factor is a constant line and has degree zero.
+
+We compute the degree of $\mathcal L_c$ using the
+projective-space duality in \Cref{foundation:duality}.
+A Koszul resolution for the local regular sequence defining $C$ shows
+that
+\[
+ \mathcal E xt^v_{\mathbb P^b_S}
+       (i_*\mathcal O_C,\mathcal O(-b-1))
+ =\begin{cases}
+   i_*\mathcal L_c,&v=b-1,\\
+   0,&v\ne b-1.
+  \end{cases}
+\]
+After dualizing the Koszul resolution, its top cohomology
+is the dual of the top exterior power of the free module of equations,
+restricted to $C$.  A change of those equations acts on this exterior
+power by the determinant of their change on $\mathcal I/\mathcal I^2$.
+Thus the local top cohomology modules glue to
+$\det(\mathcal I/\mathcal I^2)^\vee\otimes\mathcal O_C(-b-1)$.
+
+The local-to-global Ext spectral sequence has only this one nonzero
+row.  Projective-space duality consequently gives, for $r=0,1$,
+\[
+ H^r(C,\mathcal O_C)^\vee
+  \simeq
+ \operatorname{Ext}^{b-r}_{\mathbb P^b_S}
+       (i_*\mathcal O_C,\mathcal O(-b-1))
+  \simeq H^{1-r}(C,\mathcal L_c).
+\]
+Hence $\chi(C,\mathcal L_c)=-\chi(C)$.  Applying
+\eqref{eq:curve-euler-degree} yields
+$\deg_S\mathcal L_c=-2\chi(C)$, which proves
+\eqref{eq:absolute-determinant-genus}.
+\end{proof}
+
+\begin{lemma}[A separable determinant bound, including the wild term]
+\label{lem:separable-different}
+Let $f:C\to C'$ be a finite dominant, generically separable morphism of
+regular integral projective curves over $S$, of degree $\delta$.
+There is an effective divisor $R_f$ on $C$ such that
+\[
+ \nu(C)=\delta\nu(C')+\deg_S R_f.
+\]
+If $y\in C$ has ramification index $a_y$ over $f(y)$, then
+\begin{equation}
+ \operatorname{ord}_y R_f\ge a_y-1+\mathbf 1_{p\mid a_y}.
+ \label{eq:separable-different-bound}
+\end{equation}
+No separability of $\kappa(y)/\kappa(f(y))$ is required.
+The formula and the inequalities can be added componentwise after a
+finite separable extension of $S$.
+\end{lemma}
+
+\begin{proof}
+The morphism
+$f^*\Omega_{C'/\kappa}\to\Omega_{C/\kappa}$ is generically an isomorphism:
+at the function fields a finite separable extension has no relative
+differentials, and both absolute differential spaces have dimension
+$\rho+1$.  Its determinant is therefore a nonzero regular morphism of
+line bundles.  The divisor of this morphism is effective; call it $R_f$.
+Its degree is the difference of the two determinant degrees, which is
+$\nu(C)-\delta\nu(C')$ by
+\Cref{lem:curve-degrees,prop:absolute-genus}.
+
+Write $x=f(y)$ and choose a uniformizer $t$ of the DVR
+$B=\mathcal O_{C',x}$.  Its absolute differential is primitive in
+$\Omega_{B/\kappa}$, meaning that it is part of a $B$-basis.  Indeed the
+conormal sequence for the residue field is
+\[
+ (t)/(t^2)\longrightarrow
+ \Omega_{B/\kappa}\otimes_B\kappa(x)
+ \longrightarrow\Omega_{\kappa(x)/\kappa}\longrightarrow0.
+\]
+The middle term has dimension $\rho+1$ and the right term dimension
+$\rho$, since $\kappa(x)$ is finite over $S$ and
+\Cref{foundation:fields} computes absolute differential dimension over
+the perfect field $\kappa$.  Thus the first map has nonzero
+one-dimensional image, generated by $dt$.  Consequently $dt$ is
+nonzero modulo the maximal ideal, and it extends to a basis of the
+free module $\Omega_{B/\kappa}$.
+
+In $A=\mathcal O_{C,y}$ write $t=u\tau^{a_y}$, where $u$ is a unit and
+$\tau$ a uniformizer.  Absolute differentiation gives
+\[
+ dt=\tau^{a_y}\,du+
+       a_yu\tau^{a_y-1}\,d\tau.
+\]
+All coordinates of this column of the differential matrix are
+divisible by $\tau^{a_y-1}$; when $p\mid a_y$, the second term is zero,
+so they are divisible by $\tau^{a_y}$.  Expanding the determinant in
+this column proves \eqref{eq:separable-different-bound}.  Every step
+uses absolute differentials over $\kappa$, so an inseparable extension
+of residue fields causes no change to the argument.
+\end{proof}
+
+\subsection{The differential calculation for a height-one map}
+
+We have established the determinant formula for arithmetic genus and
+the separable ramification bound. The remaining ingredient is an
+exact genus identity for a purely inseparable map of exponent one;
+successive such maps will handle arbitrary purely inseparable covers.
+
+Write $T_C=(\Omega_{C/\kappa})^\vee$ for the absolute tangent bundle.
+It has rank $\rho+1$, even when the relative tangent sheaf over $S$ has
+different behavior.
+
+The calculation below is a curve-level form of the canonical-bundle
+formula for a $p$-closed foliation; compare
+\cite[Section~I.1, Proposition~1.1 and Equations~(1.2)--(1.3)]{Ekedahl1988}.
+We prove the version needed here directly from local presentations,
+using absolute differentials over the perfect constants. In this way
+the kernel, the Frobenius-power cokernel, and all inseparable residue
+extensions remain explicit.
+
+\begin{lemma}[Height-one kernels and cokernels]
+\label{lem:height-one}
+Let $\pi:C\to C'$ be a finite dominant morphism of regular integral
+projective curves over $S$ such that
+$\kappa(C)^p\subseteq\kappa(C')$.  Put
+$q=[\kappa(C):\kappa(C')]=p^a$ and
+\[
+ \mathcal F=\ker(T_C\longrightarrow\pi^*T_{C'}),
+ \qquad
+ \mathcal Q=\operatorname{coker}(T_C\longrightarrow\pi^*T_{C'}).
+\]
+Both $\mathcal F$ and $\mathcal Q$ are locally free of rank $a$, and
+\begin{equation}
+ \det\mathcal Q\simeq(\det\mathcal F)^{\otimes p}.
+ \label{eq:height-one-cokernel}
+\end{equation}
+Consequently
+\begin{equation}
+ \nu(C)=q\nu(C')+(p-1)\deg_S\mathcal F.
+ \label{eq:height-one-genus}
+\end{equation}
+\end{lemma}
+
+\begin{proof}
+\textit{The integral presentation.}
+On an affine open of $C'$, every element of the finite upstairs
+coordinate ring has its $p$th power in the downstairs ring: the power
+lies in the downstairs fraction field and is integral, so normality
+puts it in that ring.  There is therefore a unique prime upstairs over
+each downstairs prime, since membership of an element is determined
+by membership of its $p$th power.  At a closed point we may consequently
+write $B\subset A$ for the downstairs and upstairs DVRs, with $A$ still
+finite over $B$.  It is finite free of rank $q$ and satisfies
+$A^p\subset B$.
+
+Let $\tau$ be an upstairs uniformizer, let $\epsilon$ be the
+ramification index, and put $f=[\kappa(A):\kappa(B)]$.
+The element $\tau^p\in B$ has upstairs valuation $p$; all nonzero
+elements of $B$ have upstairs valuation in $\epsilon\mathbb Z$.
+Thus $\epsilon\mid p$, so $\epsilon$ is $1$ or $p$.
+Reduction of the rank-$q$ free module modulo a downstairs uniformizer
+gives $q=\epsilon f$ by \eqref{eq:local-degree-formula}.
+The residue extension has exponent at most one.  Successively
+adjoining a generator outside the preceding residue field gives a
+degree-$p$ extension each time.  Choose residue generators
+$\bar b_1,\ldots,\bar b_r$ in this manner, so their monomials with
+exponents in $\{0,\ldots,p-1\}$ form a $\kappa(B)$-basis of $\kappa(A)$
+and $f=p^r$.  Lift them to $b_1,\ldots,b_r\in A$.  If $\epsilon=p$,
+append $b_{r+1}=\tau$; if $\epsilon=1$, append nothing.  In both cases
+there are exactly $a$ generators, and $c_\ell=b_\ell^p$ belongs to $B$.
+
+For a downstairs uniformizer $t$, the ideal $tA$ is
+$\tau^\epsilon A$.  Filter $A/tA$ by the powers of $\tau$ from $0$ to
+$\epsilon$.  Each successive quotient is $\kappa(A)$, and the lifted
+residue monomials give its basis after multiplication by the
+corresponding power of $\tau$.  It follows that the $q$ monomials
+\[
+ b_1^{i_1}\cdots b_a^{i_a},\qquad 0\le i_\ell<p,
+\]
+are a $\kappa(B)$-basis of $A/tA$.
+By Nakayama they generate $A$ over $B$; since $A$ is free of rank $q$,
+they are a $B$-basis.  The homomorphism from the monic-relation algebra
+therefore gives the actual presentation
+\begin{equation}
+ A\simeq
+ B[X_1,\ldots,X_a]/(X_1^p-c_1,\ldots,X_a^p-c_a),
+ \qquad X_\ell\longmapsto b_\ell.
+ \label{eq:height-one-dvr-presentation}
+\end{equation}
+Both sides have the same free monomial basis over $B$, which verifies
+that there are no further relations.  This includes the cases of a
+nontrivial inseparable residue extension and simultaneous residue
+extension and ramification.
+
+These presentations are available on open neighborhoods.  Shrink an
+open downstairs around the chosen point so that all the $b_\ell$ are
+sections of $\pi_*\mathcal O_C$, all the $c_\ell$ are regular downstairs,
+and the determinant of the displayed monomial basis is invertible.
+Then \eqref{eq:height-one-dvr-presentation} holds there.  Such
+neighborhoods give local presentations throughout the curves.
+
+\textit{Free kernel and free cokernel.}
+Set $N=A\otimes_B\Omega_{B/\kappa}$, and let $W\subset N$ be the span
+of $dc_1,\ldots,dc_a$.  Differentiating the displayed presentation,
+using characteristic $p$, gives
+\begin{equation}
+ \Omega_{A/\kappa}\simeq
+       (N/W)\oplus\bigoplus_{\ell=1}^a A\,db_\ell.
+ \label{eq:height-one-differential-presentation}
+\end{equation}
+Both $N$ and $\Omega_{A/\kappa}$ are free of rank $\rho+1$ by
+\Cref{prop:absolute-genus}.  The summand $N/W$ in
+\eqref{eq:height-one-differential-presentation} is therefore free,
+of rank $\rho+1-a$.  The sequence
+$0\to W\to N\to N/W\to0$ splits, so $W$ is free of rank $a$.
+Its $a$ generators $dc_\ell$ are consequently a basis.
+
+Dualizing now shows that the tangent kernel consists of the
+$\kappa$-derivations $A\to A$ killing $B$, and has free coordinates
+\[
+ D\longmapsto(D(b_1),\ldots,D(b_a)).
+\]
+The tangent cokernel is $W^\vee$, with coordinates
+\[
+ [\delta]\longmapsto
+       (\delta(c_1),\ldots,\delta(c_a)),
+ \qquad \delta\in\operatorname{Der}_\kappa(B,A).
+\]
+Indeed a derivation $B\to A$ extends to $A$ exactly when it kills
+the $c_\ell$; when it does, the values assigned to the $b_\ell$ are
+arbitrary.  Thus both kernel and cokernel are locally free of rank $a$.
+
+\textit{Gluing the determinant lines.}
+On an overlap, write another system of generators as
+$b'_k=P_k(b_1,\ldots,b_a)$ with coefficients in $B$, and put
+\[
+ \mathsf J_{k\ell}=
+ \frac{\partial P_k}{\partial X_\ell}(b_1,\ldots,b_a).
+\]
+A derivation in the kernel kills the coefficients, so its new
+coordinate vector is $\mathsf J$ times its old coordinate vector.
+The matrix $\mathsf J$ is invertible since both coordinate systems are bases.
+Taking $p$th powers of the generator identities gives downstairs
+\[
+ c'_k=(b'_k)^p=P_k^{[p]}(c_1,\ldots,c_a),
+\]
+where $P_k^{[p]}$ means that the coefficients of $P_k$ are raised to
+the $p$th power.  Their absolute differentials vanish.  Differentiating
+this identity by a representative $\delta:B\to A$ of a cokernel class
+therefore gives
+\[
+ \delta(c'_k)=\sum_\ell \mathsf J_{k\ell}^{p}\,\delta(c_\ell).
+\]
+Thus the cokernel coordinate transition is the entrywise Frobenius
+power $\mathsf J^{[p]}$.  Coordinate transitions for determinant lines are
+$\det\mathsf J$ and $\det(\mathsf J^{[p]})=(\det\mathsf J)^p$; if one uses local frames
+instead, both transitions are inverted.  In either convention they
+give exactly \eqref{eq:height-one-cokernel}, with the exponent $+p$.
+
+Finally the exact sequence
+\[
+ 0\longrightarrow\mathcal F\longrightarrow T_C
+ \longrightarrow\pi^*T_{C'}\longrightarrow\mathcal Q
+ \longrightarrow0
+\]
+and \eqref{eq:height-one-cokernel} give
+\[
+ \deg T_C=q\deg T_{C'}-(p-1)\deg\mathcal F.
+\]
+Since $\deg T_C=-\nu(C)$, this is \eqref{eq:height-one-genus}.
+\end{proof}
+
+\subsection{Proof of the exact estimate}
+
+Return to the fields and marked curve in
+\Cref{thm:inseparable-genus}, with $\kappa=\overline{\mathbb F}_p$
+and $S=FF'$. The height-one identity will be applied along a
+Frobenius tower from $Z_S$ to $E_S$. At each stage the label
+covectors bound the degree of the tangent-map image; the resulting
+weighted sum gives the bounded-error estimate before the final
+index argument removes the error.
+
+\begin{proof}[Proof of \Cref{thm:inseparable-genus}]
+\textit{Normalization and the separable splitting field.}
+Normalizations are finite by \Cref{foundation:normalization}, and normal
+curves over fields are regular.  We first record why the finite purely
+inseparable maps in this argument are radicial, that is, universally
+injective.  If $A$ is the integral closure of a normal affine domain
+$B$ in a finite purely inseparable extension, there is an exponent $v$
+such that $a^{p^v}\in B$ for every $a\in A$.  The power belongs to
+$\operatorname{Frac}(B)$ and is integral over $B$, so it belongs to $B$
+by normality.  After tensoring with any $B$-algebra, the $p^v$th power
+of every element still lies in the image of that algebra.  Consequently
+over a prime of the base there is at most one prime upstairs: membership
+of an element in a prime is determined by membership of its $p^v$th
+power downstairs.  The residue extension is purely inseparable for
+the same reason.  This proves the assertion after every base change.
+
+The extension $S/F$ is finite separable, since $F'/F_0$ is finite
+Galois.  Thus $Z_S$ is regular and reduced by
+\Cref{foundation:regularity}.  Its map to the integral curve $E_S$ is
+still finite, surjective, and radicial.  This map is a homeomorphism
+on underlying spaces, so $Z_S$ is irreducible and hence integral.
+It is the normalization of $E_S$ in its function field: it is finite
+and normal with that field, and an element integral over the base is
+also integral over its finite normal coordinate ring.  Its degree
+remains $d$ by finite flatness.  By \Cref{lem:curve-degrees},
+\[
+ \nu(Z_S)=\nu(Z),
+\]
+where the two Euler characteristics are computed over their stated
+fields.  We may therefore do the differential calculation over $S$.
+If $d=1$, this curve is $E_S$, every $r_i=1$, and the theorem holds.
+Assume henceforth that $d>1$.
+
+\textit{The Frobenius tower and its recurrence.}
+Let $M_e=S(E)$ and $M_z=S(Z)$, and set
+\[
+ M_j=M_eM_z^{p^j}\quad(0\le j\le b),\qquad
+ M_0=M_z,\qquad M_b=M_e,
+\]
+where $b$ is the first index for which the last equality holds.
+Let $Y_j$ be the normalization of $E_S$ in $M_j$, and write
+\[
+ \gamma_j:Y_j\longrightarrow E_S,\qquad
+ d_j=[M_j:M_e],\qquad
+ \pi_j:Y_j\longrightarrow Y_{j+1},\qquad
+ q_j=d_j/d_{j+1}=p^{m_j}.
+\]
+In particular $Y_0=Z_S$, $d_0=d$, $Y_b=E_S$, and $d_b=1$.
+The defining fields satisfy
+\begin{equation}
+ M_{j+1}=M_eM_j^p.
+ \label{eq:frobenius-tower-field}
+\end{equation}
+Thus each $\pi_j$ has exponent at most one and
+\Cref{lem:height-one} applies.
+
+Put $T_j=(\Omega_{Y_j/\kappa})^\vee$ and
+\[
+ \mathcal F_j=\ker(T_j\longrightarrow\pi_j^*T_{j+1}).
+\]
+This is also the kernel of the composite tangent map
+$T_j\to\gamma_j^*T_{E_S}$.  To see this, at the function field a
+$\kappa$-derivation kills $M_e$ if and only if it kills
+$M_eM_j^p=M_{j+1}$, since every derivation kills $p$th powers.
+The two kernels are therefore equal generically.  They are equal as
+sheaves: a section whose image vanishes generically has zero image
+everywhere in either of the locally free, hence torsion-free, targets.
+
+The genus identity involves $\deg_S\mathcal F_j$. We will estimate
+this degree through the image of the tangent map to $E_S$.
+Choose a nonzero invariant differential $\eta$ on $A_0$.
+The constant-curve identification of $E_S$ gives the splitting
+\[
+ \Omega_{E_S/\kappa}
+   \simeq\mathcal O_{E_S}\eta
+      \oplus(\mathcal O_{E_S}\otimes_S\Omega_{S/\kappa}).
+\]
+Thus its dual is a trivial bundle of rank $n+1$. In this trivial
+target we can use the marked points to impose linear vanishing
+conditions on the tangent-map image.
+
+Let $\mathcal H_j$ be the image in $\gamma_j^*T_{E_S}$, and put
+$b_j^\circ=-\deg_S\mathcal H_j$.
+It is a torsion-free coherent sheaf on a regular curve, hence locally
+free.  Its rank is $n+1-m_j$, since
+$\operatorname{trdeg}_\kappa S=n$.
+The exact sequence with kernel $\mathcal F_j$ and image $\mathcal H_j$,
+together with \eqref{eq:height-one-genus}, gives
+\[
+ \nu(Y_j)=q_j\nu(Y_{j+1})+(p-1)\deg_S\mathcal F_j,
+ \qquad
+ b_j^\circ=\nu(Y_j)+\deg_S\mathcal F_j.
+\]
+Eliminating $\deg_S\mathcal F_j$ and using $d_j=q_jd_{j+1}$ yields
+\begin{equation}
+ \frac{\nu(Y_j)}{d_j}
+  =\frac1p\frac{\nu(Y_{j+1})}{d_{j+1}}
+     +\left(1-\frac1p\right)\frac{b_j^\circ}{d_j}.
+ \label{eq:genus-recurrence}
+\end{equation}
+The terminal term $\nu(Y_b)$ is zero because $E_S$ has genus one.
+It remains to give a lower bound for
+$b_j^\circ/d_j=-\deg_S\mathcal H_j/d_j$.
+
+\textit{Vanishing at a ramified label.}
+For each $j$ let $S_j$ be the set of labels at which $\gamma_j$ has
+ramification index greater than one.  For $i\in S_j$ write
+$D_{j,i}=\gamma_j^*(t_i)$ and
+\[
+ v_i=(1,-t_i^*\eta)\in S\oplus\Omega_{S/\kappa}.
+\]
+Regard $v_i$ as a constant linear functional on
+$\gamma_j^*T_{E_S}$. We claim that its restriction to $\mathcal H_j$
+vanishes along the entire divisor $D_{j,i}$; equivalently, its values
+lie in $\mathcal O_{Y_j}(-D_{j,i})$.
+
+Let $a$ be a local parameter defining the rational point $t_i$ on
+$E_S$.  Restriction of absolute differentials along $t_i$ sends
+$(\lambda,\beta)$ in the displayed splitting to
+$\lambda t_i^*\eta+\beta$.  Its kernel is the line spanned by $v_i$.
+Since $a$ restricts to zero, $da$ belongs to this kernel modulo $(a)$.
+Moreover its relative component is nonzero: $t_i$ is an $S$-rational
+point on a smooth curve over $S$, and $a$ generates its relative
+cotangent space.  Hence
+\[
+ da\equiv\lambda_i v_i\pmod{a\Omega_{E_S/\kappa}}
+ \quad\text{for some }\lambda_i\in S^\times.
+\]
+At the unique point upstairs write $a=u\tau^r$, with $u$ a unit.
+The nontrivial purely inseparable index $r$ is divisible by $p$;
+therefore
+\[
+ d(\gamma_j^*a)=\tau^r\,du=(\gamma_j^*a)\,u^{-1}du.
+\]
+It lies in the ideal of the \emph{full} divisor $D_{j,i}$ times
+$\Omega_{Y_j/\kappa}$.  Pulling back the preceding congruence and
+dividing by the unit $\lambda_i$ shows that the pullback of $v_i$ lies
+in that same ideal times $\Omega_{Y_j/\kappa}$.
+To relate this differential to the claimed functional, denote the
+tangent map by $d\gamma_j:T_j\to\gamma_j^*T_{E_S}$ and write
+$\gamma_j^*v_i$ for the image of the pulled-back covector in
+$\Omega_{Y_j/\kappa}$. For a local section $\theta$ of $T_j$, duality
+gives
+\[
+ v_i(d\gamma_j(\theta))=(\gamma_j^*v_i)(\theta).
+\]
+The divisibility just proved makes this value a section of
+$\mathcal O_{Y_j}(-D_{j,i})$. Since $\mathcal H_j$ is the image of
+$d\gamma_j$, the claim follows. The vanishing is along
+$\gamma_j^*(t_i)$ with its full multiplicity, not just its reduced
+support.
+
+\textit{From independent vanishing conditions to a degree bound.}
+By \Cref{lem:label-rank}, among the $v_i$ for $i\in S_j$ there are
+$c\ge |S_j|-h$ independent covectors over $S$.
+
+Complete the $c$ chosen covectors to a constant basis of the dual of
+the target trivial tangent bundle.  The full-divisor constraint gives
+an injection
+\begin{equation}
+ \mathcal H_j\hookrightarrow
+ \mathcal O_{Y_j}^{n+1-c}\oplus
+       \bigoplus_{i\ \mathrm{chosen}}\mathcal O_{Y_j}(-D_{j,i}).
+ \label{eq:label-image-injection}
+\end{equation}
+Let $R=n+1-m_j$ be the rank of $\mathcal H_j$.
+The $R$th exterior power of the bundle on the right is a direct sum
+of line bundles, each the tensor product of a choice of $R$ summands.
+The induced
+map from $\det\mathcal H_j$ is nonzero, so it has a nonzero component
+in at least one of these lines. Only $n+1-c$ of the available
+summands are trivial. The chosen line must therefore contain at least
+$R-(n+1-c)=c-m_j$ negative divisor summands when this number is
+positive, and at least zero otherwise.
+Each $D_{j,i}$ has degree $d_j$ by finite flatness and rationality of
+$t_i$. If the nonzero component uses $k$ negative summands, its target
+line has degree $-kd_j$.  A nonzero regular map from one line bundle
+to another has an effective zero divisor, and consequently
+\[
+ \deg_S\mathcal H_j\le-kd_j,
+ \qquad
+ \frac{b_j^\circ}{d_j}\ge c-m_j\ge |S_j|-h-m_j.
+\]
+The argument also covers rank zero, with $\det(0)=\mathcal O$.
+
+\textit{The weighted estimate.}
+Write $r_i=p^{u_i}$.  Every height-one step has local index at most
+$p$, as proved in \Cref{lem:height-one}. The composite
+$Y_0\to Y_j$ consequently has ramification index at most $p^j$ at
+this label. In particular, $u_i\leq b$. Since ramification indices
+multiply, the remaining map
+$Y_j\to E_S$ has index at least $p^{u_i-j}$ whenever $j<u_i$.
+Hence $i\in S_j$ for every $0\le j<u_i$. Iterating
+\eqref{eq:genus-recurrence} with terminal value zero gives
+\[
+ \frac{\nu(Z)}d
+  =\sum_{j=0}^{b-1}
+      p^{-j}\left(1-\frac1p\right)\frac{b_j^\circ}{d_j}.
+\]
+Write $w_j=p^{-j}(1-1/p)$ for the weight of stage $j$. For each
+label, the weights from those first $u_i$ stages sum to
+\[
+ \sum_{j=0}^{u_i-1}p^{-j}\left(1-\frac1p\right)
+      =1-p^{-u_i}=1-\frac1{r_i}.
+\]
+Summing these contributions by label gives
+\[
+ \sum_{j=0}^{b-1}w_j|S_j|
+  =\sum_{i=1}^n\sum_{\substack{0\leq j<b\\i\in S_j}}w_j
+  \geq\sum_{i=1}^n\left(1-\frac1{r_i}\right).
+\]
+The error terms in the degree bound contribute separately. Since
+$m_j\le\log_p d$ and the sum of all the weights is $1-p^{-b}<1$,
+\[
+ \sum_{j=0}^{b-1}w_j(h+m_j)\leq h+\log_p d.
+\]
+Substitute $b_j^\circ/d_j\geq |S_j|-h-m_j$ in the recurrence sum.
+Subtracting the error bound from the label contribution yields
+\begin{equation}
+ \frac{\nu(Z)}d
+   \ge\sum_{i=1}^n\left(1-\frac1{r_i}\right)
+          -\bigl(h+\log_p d\bigr).
+ \label{eq:genus-approximate}
+\end{equation}
+
+\textit{Divisors over $F$ and exact rounding.}
+Multiplying \eqref{eq:genus-approximate} by $d$ bounds the possible
+deficit by $d(h+\log_p d)$. Set $N=p^{s-h}$ and consider the difference
+\[
+ A=\nu(Z)-d\sum_i\left(1-\frac1{r_i}\right).
+\]
+We will realize both terms as degrees of divisors on $Z$, each
+retaining its degree under pushforward to $E_F$. The index property
+\Cref{cor:index-base-change} then makes both degrees divisible by
+$N$. These divisors must be defined over $F$, before the separable
+extension to $S$.
+By \Cref{prop:absolute-genus}, the integer $\nu(Z)$ is the degree over
+$F$ of $\det\Omega_{Z/\kappa}$.  A rational trivialization represents
+this line bundle by a Cartier divisor on the regular integral curve
+$Z$.  Its pushforward to $E_F$ preserves degree over $F$, by
+\Cref{lem:curve-degrees}.  Thus
+\[
+ \nu(Z)\in N\mathbb Z.
+\]
+This uses the absolute determinant on $Z/F$ and its Euler
+characteristic over $F$, including any inseparable constant field.
+
+Let $T$ be the reduced inverse image in $Z$ of $\mathcal B_F$.
+It is an effective divisor on the regular curve $Z$.
+Finite separable scalar extension preserves reducedness, so $T_S$
+is the reduced inverse image of the split labels.  If $z_i$ is the
+unique point above the rational point $t_i$, the local degree formula
+is
+\[
+ r_i[\kappa(z_i):S]=d.
+\]
+It follows that
+\[
+ \deg_F T=\deg_S T_S=\sum_i\frac d{r_i}.
+\]
+In particular this sum counts the full residue degrees of the reduced
+points; those residue fields may be inseparable over $S$.
+The divisor $\gamma^*\mathcal B_F-T$ is defined on $Z$ over $F$, where
+$\gamma:Z\to E_F$, and its degree is
+\[
+ dn-\sum_i\frac d{r_i}
+    =d\sum_i\left(1-\frac1{r_i}\right).
+\]
+Push it to $E_F$ to conclude that this integer also belongs to
+$N\mathbb Z$.
+
+Consequently $A$ is a multiple of $N$. By
+\eqref{eq:genus-approximate} and $d\le m$,
+\[
+ A\ge-d(h+\log_p d)
+      \ge-m(h+\log_p m)>-N.
+\]
+A multiple of $N$ strictly greater than $-N$ is nonnegative.  This
+proves \eqref{eq:inseparable-genus-bound}.
+The sufficient threshold \eqref{eq:genus-rounding-threshold} contains
+no datum from $F$, $Z$, or a tuple-class multiset.  If a threshold
+independent of the prime is desired, the stronger condition
+\[
+ 2^{s-h}>m(h+\log_2m)
+\]
+suffices for every $p\ge2$.  This proves the stated uniformity.
+\end{proof}

preprints/The-Blockwise-Alperin-Weight-Conjecture-September-23-2026/build/sections/foundations.tex

@@ -0,0 +1,167 @@
+\section{Classical foundations used}\label{sec:foundations}
+
+We state the classical inputs to make the scope of the proof explicit. We also use elementary linear algebra, field and group theory, and the basic definitions and functorial formalism of schemes, sheaves, divisors, and K\"ahler differentials. The counting, index, inseparable-genus, valued-field, and specialization assertions particular to this proof are established in the later sections. In particular, no weight-counting theorem or conjectural reduction is an input.
+
+\begin{foundation}[Finite groups]\label{foundation:groups}
+For a finite group $X$, its $p$-subgroups lie in Sylow subgroups of order $|X|_p$. A nontrivial finite $p$-group has nontrivial center and an element of order $p$, and each of its maximal proper subgroups is normal. We use the orbit--stabilizer formula and the usual fixed-point and conjugation actions.
+\end{foundation}
+
+\begin{foundation}[Finite-dimensional algebras and characters]\label{foundation:algebra}
+The Jacobson radical of a finite-dimensional algebra is nilpotent. Over an algebraically closed field, its semisimple quotient is a product of full matrix algebras, one for each simple module up to isomorphism. A finite-dimensional commutative algebra over an algebraically closed field is a product of local algebras with that field as residue, indexed by its primitive idempotents. Finite group algebras in characteristic zero are semisimple. Over a splitting field the simple endomorphism algebras are the scalar field, and an ordinary irreducible character satisfies
+\[
+ \sum_{x\in X}\chi(x)\chi(x^{-1})=|X|.
+\]
+Idempotents lift uniquely from the residue algebra of a finite commutative algebra over a complete discrete valuation ring. Nakayama's lemma applies to finite modules over any local ring.
+\end{foundation}
+
+\begin{foundation}[Commutative algebra]\label{foundation:commalg}
+A finite-type algebra over a Noetherian ring is Noetherian. Integral extensions satisfy lying over. Closed points of a variety over a field are dense and have finite residue fields over the ground field. For an integral variety, the local dimension at a point plus the transcendence degree of its residue field is the transcendence degree of the function field. A regular local ring is a normal domain; a part of a minimal system of generators of its maximal ideal is a regular sequence and its quotient is regular of the correspondingly smaller dimension. A one-dimensional Noetherian normal local domain is a discrete valuation ring, as is a Noetherian local domain with nonzero principal maximal ideal. Finite torsion-free modules over discrete valuation rings are free. Completion of a discrete valuation ring retains its residue field and uniformizer.
+\end{foundation}
+
+\begin{foundation}[Fields and absolute field differentials]\label{foundation:fields}
+If a finite group $D$ acts faithfully by automorphisms on a field $M$, then $M/M^D$ is Galois of degree $|D|$. Finite separable field extensions are simple. A finite field extension has a maximal separable subextension, and the remaining extension is purely inseparable; the number of embeddings in an algebraic closure is the separable degree. Multiplicative groups of finite fields are cyclic. Algebraically closed fields are classified by their characteristic and transcendence degree. If $S$ is a finitely generated field of transcendence degree $\rho$ over a perfect field $\kappa$, then
+\[
+ \dim_S\Omega_{S/\kappa}=\rho;
+ \qquad [S:S^p]=p^\rho\quad\text{in characteristic }p.
+\]
+\end{foundation}
+
+\begin{foundation}[Valuations over a complete discrete base]\label{foundation:valuation}
+A valuation extends to an algebraic field extension. A complete discrete valuation on a field extends uniquely to any finite field extension; its valuation ring is the integral closure of the original ring, is finite over it, and is again a complete discrete valuation ring. The fields to which we apply the latter assertion initially have characteristic zero. No corresponding finiteness assertion for an arbitrary nondiscrete valuation ring is assumed.
+See \cite[Tag 00IA]{Stacks} for prolongation and
+\cite[Tags 032W, 09EM and 0325]{Stacks} for the complete
+discrete-base finiteness assertions.
+\end{foundation}
+
+\begin{foundation}[Descent and finite models]\label{foundation:descent}
+A projective scheme with a semilinear descent datum under a finite Galois field extension descends if equipped with a compatible linearized ample line bundle. The same statement holds for a finite \emph{\'etale} faithfully flat Galois extension of rings and a relatively ample line bundle. Equivariant morphisms and invariant closed subschemes descend along with the scheme. Flatness, smoothness, finiteness, \emph{\'etaleness}, and relative ampleness in these settings are checked after the faithfully flat extension. Geometric integrality of a finite-type scheme over a field can be checked after an extension of that field. A finite diagram of projective schemes and morphisms of finite presentation over a directed union of fields is defined over a finite stage, including its maps to a base scheme already defined there and the identities among those maps. One may realize projective descent by descent of semilinear modules and graded algebras followed by $\Proj$.
+For the projective and module descent assertions, see \cite[Tags 0CCJ, 0CDR and 0D1V]{Stacks}.
+The five locality assertions are respectively
+\cite[Tags 02L2, 02VL, 02LA, 02VN and 0D3C]{Stacks}; geometric
+integrality is covered by \cite[Tags 038K, 0384 and 054P]{Stacks}.
+The finite-stage assertion follows from \cite[Tags 01ZM and 01ZP]{Stacks},
+including chosen projective embeddings among the finite data.
+\end{foundation}
+
+\begin{foundation}[Normalization]\label{foundation:normalization}
+The normalization of an integral variety over a field in a finite extension of its function field is finite. The normalization of a finite-type $\mathbb Z$-domain in its fraction field is finite. In the integral closure of a normal domain in a finite Galois extension of its fraction field, the Galois group acts transitively on the primes above any fixed prime of the base.
+The normalization finiteness assertions follow from the Nagata property in \cite[Tag 0335]{Stacks}.
+\end{foundation}
+
+\begin{foundation}[Regularity, smoothness, and properness]\label{foundation:regularity}
+Finite separable scalar extension preserves regularity and reducedness. Smooth varieties over a field are regular. A regular local ring essentially of finite type over a perfect field $\kappa$ is smooth at the corresponding point of a finite-type $\kappa$-model; its module of absolute differentials is free of rank the transcendence degree of the function field over $\kappa$. We use the valuative criterion for properness, including the unique extension of maps from fraction fields of valuation rings, and that proper maps are closed.
+For regularity and smoothness over a perfect field, see \cite[Tags 00TV and 0B8X]{Stacks}.
+\end{foundation}
+
+\begin{foundation}[Proper cohomology and ample section rings]\label{foundation:cohomology}
+Over a Noetherian base, the higher direct images of coherent sheaves under a proper morphism are coherent. Over an affine base their cohomology commutes with flat affine base change. For a projective scheme over an affine Noetherian base and a relatively ample line bundle, sufficiently high powers give projective embeddings, and sufficiently high twists annihilate higher coherent cohomology. The ample section ring is finitely generated and its $\Proj$ recovers the scheme. Finite pullback preserves ampleness; tensor products of relatively ample line bundles are relatively ample, and a line bundle is relatively ample if a positive power is. Formation of $\Proj$ commutes with base change. On a projective curve over a field, coherent cohomology vanishes in degrees greater than one, and for sheaves of finite support in degrees greater than zero. Global functions on a geometrically integral projective variety over a field equal the ground field.
+For proper coherence, flat cohomology base change, and ample twists, respectively, see \cite[Tags 02O5, 02KH and 0B5T]{Stacks}.
+For Proj recovery and the ampleness rules, see
+\cite[Tags 0EKI, 0892, 0890 and 02NN]{Stacks}.
+\end{foundation}
+
+\begin{foundation}[Differentials, adjunction, and duality]\label{foundation:duality}
+We use the transitivity sequence
+\[
+ \Omega_{V/U}\otimes_VW\longrightarrow\Omega_{W/U}
+ \longrightarrow\Omega_{W/V}\longrightarrow0
+\]
+and the conormal sequence for a closed immersion. Let $S$ be a field. On $\mathbb P^b_S$,
+$\det\Omega_{\mathbb P^b_S/S}=\mathcal O(-b-1)$. For a coherent sheaf $\mathcal G$ there, Serre duality identifies
+\[
+ H^i(\mathcal G)^\vee
+ \simeq\operatorname{Ext}^{b-i}
+       (\mathcal G,\mathcal O(-b-1)).
+\]
+We use the local-to-global Ext spectral sequence and the Koszul resolution of a quotient by a regular sequence. These inputs will yield the required absolute determinant-degree formula even when the curve is not smooth over $S$.
+For the projective-space dualizing complex and the adjunction formalism, see \cite[Tag 0A9W and Section~48.15]{Stacks}.
+The local-to-global Ext sequence is \cite[Tag 0BQP]{Stacks}.
+\end{foundation}
+
+\begin{foundation}[Elliptic curves]\label{foundation:elliptic}
+A smooth Weierstrass cubic with its point at infinity is an elliptic curve. A generalized Weierstrass equation over a discrete valuation ring with unit discriminant gives a smooth elliptic scheme, polarized by three times its origin. Its relative cotangent line is trivial with invariant differential $\eta$, and multiplication by an integer $a$ pulls $\eta$ back to $a\eta$. For a positive integer $a$, multiplication by $a$ on an elliptic curve or elliptic scheme is finite locally free of degree $a^2$.
+For the relative group law, see
+\cite[II, Proposition~2.7]{DeligneRapoport1973}; the unit-discriminant
+Weierstrass family and the invariant cotangent description are given in
+\cite[Tags 072S and 047I]{Stacks}. See also
+\cite[Chapter~III]{Silverman2009} for the Weierstrass formulation.
+Multiplication is finite flat and surjective by
+\cite[Proposition~20.7]{MilneAV2022}; over the Noetherian bases here it
+is finite locally free by \cite[Tag 02KB]{Stacks}.
+\end{foundation}
+
+\begin{foundation}[Complex curves and covers]\label{foundation:complex}
+Finite \'etale algebraic covers of a smooth complex algebraic curve
+correspond to finite unramified topological covers of its analytification,
+including morphisms. For a punctured projective curve, normalization of
+the original projective curve in the resulting algebraic function field
+gives the finite compact cover. Normal complex curves are smooth, and the
+local punctured covering completes by a power map, so monodromy cycle
+lengths give ramification indices. Topological covers are classified by
+fundamental-group actions on finite fibers. The fundamental group of a
+genus-one surface punctured at $n$ points has generators
+$x,y,u_1,\ldots,u_n$ with the single relation
+\[
+ [x,y]u_1\cdots u_n=1.
+\]
+For a finite map of smooth proper curves over an algebraically closed field of characteristic zero, Riemann--Hurwitz states
+\[
+ 2g(C)-2=\deg(f)\bigl(2g(C')-2\bigr)
+              +\sum_{P\in C}(e_P-1).
+\]
+For the algebraic/topological cover correspondence, see \cite[Theorem~3.4]{MilneLEC2008}; for Riemann--Hurwitz, see \cite[Section~53.12, Tag 0C1B]{Stacks}.
+\end{foundation}
+
+\begin{foundation}[Noetherian connectedness]\label{foundation:connectedness}
+A proper morphism over a locally Noetherian base whose structure-sheaf direct image is canonically the structure sheaf of the base has geometrically connected fibers. This is the connected-fiber assertion of Stein factorization. We apply it only after constructing a Noetherian normal model, not directly over the nondiscrete valuation ring.
+See \cite[Tag 03H0]{Stacks}.
+\end{foundation}
+
+\subsection{Coefficient fields and scalar extension}\label{sec:coefficients}
+
+We justify the coefficient choice in \Cref{sec:introduction} without
+requiring a mixed-characteristic lift of every algebraically closed
+field. Put $\kappa=\overline{\mathbb F}_p$ and choose an embedding
+$\kappa\hookrightarrow k$. For any finite group $X$, the ideal
+$\rad(\kappa X)\otimes_\kappa k$ is nilpotent, and the quotient of
+$kX$ by this ideal is the scalar extension of the split matrix product
+$\kappa X/\rad(\kappa X)$. That quotient is again a split semisimple
+algebra. The ideal is therefore exactly $\rad(kX)$, and the simple
+modules correspond under scalar extension.
+
+The center also commutes with this extension: it is the kernel of the
+finitely many linear maps $a\mapsto ax-xa$, $x\in X$, and field
+extension preserves kernels. Each local factor of $Z(\kappa X)$ has
+nilpotent radical and residue $\kappa$; after extension it still has
+nilpotent radical, now with residue $k$, so remains local. Thus
+primitive blocks correspond and their simple-module counts agree.
+Coefficient restriction, restriction to centralizers in the Brauer
+map, and the quotient maps for normal subgroups all commute with
+scalar extension. The residue characters of the local central factors
+extend as well. Consequently the central-character block assignment
+in \eqref{eq:block-assignment} is preserved, for every subgroup and
+subquotient involved in a weight. It suffices to prove the numerical
+identity over $\kappa$.
+
+To construct the coefficient ring, start with $\mathbb Z_p$, lift a
+tower of finite separable extensions of its residue field $\mathbb F_p$
+that exhausts $\kappa$, and complete the union. The construction at the
+start of the proof of \Cref{prop:mixed-lift} shows that this gives a
+complete discrete valuation ring of characteristic zero with residue
+$\kappa$.
+In an algebraic closure of its fraction field, choose matrix
+realizations of every ordinary irreducible representation of every
+subgroup of $G$. There are only finitely many such representations and
+matrix entries. Adjoining their entries gives a finite field extension
+that splits all those subgroups. Its valuation ring is again a complete
+DVR by \Cref{foundation:valuation}; its finite residue extension is
+still $\kappa$, which is algebraically closed. This gives the system
+$(K,\mathcal O,\kappa)$ used in \Cref{sec:group}. Quotient
+representations inflate, so the same coefficients split the required
+subquotients. These coefficient fields are separate from the auxiliary
+geometric fields constructed later.
+
+The remaining sections prove the blockwise chain identity, the marked
+high-index twist, the genus estimate with exact rounding, the required
+valued-field assertions, and the specialization model. The foundations
+above supply their classical inputs, not the desired weight equality.

preprints/The-Blockwise-Alperin-Weight-Conjecture-September-23-2026/build/sections/group.tex

@@ -0,0 +1,839 @@
+\section{From block counts to generating tuples}\label{sec:group}
+
+Fix a prime $p$. Write $O_p(X)$ for the largest normal $p$-subgroup
+of a finite group $X$, and put $[x,y]=xyx^{-1}y^{-1}$. A
+$p$-element includes the identity; a $p'$-element has order prime to
+$p$; and a $p$-singular element has order divisible by $p$.
+The geometric sections will establish the following precise counting
+statement. Its uniformity is part of the assertion.
+
+\begin{theorem}[Uniform divisibility for generating tuples]
+\label{thm:tuple-divisibility}
+Let $J$ be a finite group with $O_p(J)=1$. For $s\geq0$, $n=p^s$,
+and a multiset
+$\mathcal M$ of $n$ conjugacy classes of $p$-singular elements of $J$,
+let $N_{J,s}(\mathcal M)$ be the number, modulo simultaneous
+$J$-conjugation, of ordered tuples
+\[
+ (x,y,u_1,\ldots,u_n)\in J^{n+2}
+\]
+satisfying
+\begin{equation}\label{eq:tuple-relation}
+ [x,y]u_1\cdots u_n=1,\qquad
+ \langle x,y,u_1,\ldots,u_n\rangle=J,
+ \qquad \{[u_1],\ldots,[u_n]\}=\mathcal M.
+\end{equation}
+For every integer $h\geq1$, there is an integer $s_0=s_0(p,J,h)$
+such that
+\[
+ p^h\mid N_{J,s}(\mathcal M)
+ \qquad(s\geq s_0)
+\]
+for every such multiset $\mathcal M$.
+\end{theorem}
+
+The entries $u_i$ remain ordered: prescribing their multiset of classes
+does not quotient by permutations. Empty tuple sets have count zero,
+so the statement includes $J=1$ and groups of order prime to $p$.
+The proof of \Cref{thm:tuple-divisibility} is completed after
+\Cref{prop:no-small-cover}. This section proves that it implies the
+block equality, using \Cref{foundation:groups,foundation:algebra}
+and proving the required counting reductions explicitly.
+
+\subsection{Coefficient projection and the weight transform}
+
+For each ambient finite group $X$, choose a splitting coefficient system
+$(K,\mathcal O,k)$ as in \Cref{sec:introduction,sec:coefficients}, with
+$X$ in place of $G$. Hold this system fixed throughout the calculations
+with $X$ and its subquotients.
+This system also splits all subquotients that occur below: a subgroup of a
+quotient lifts to a subgroup by taking its preimage, and a
+representation of a quotient inflates to that preimage. These
+observations apply after any number of subgroup and quotient steps.
+Ordinary irreducibles may therefore be realized over $K$. Their
+identification with complex irreducibles, when needed, follows by
+first splitting the group algebra over $\overline{\mathbb Q}$ and
+then extending scalars to algebraically closed characteristic-zero
+fields.
+
+For a central idempotent $E\in kX$, define
+\begin{align*}
+ l(X,E)&=\#\{\text{simple $kX$-modules selected by $E$}\},\\
+ k_{\mathrm{ord}}(X,E)&=\#\{\text{ordinary irreducibles selected by the lift of $E$}\},\\
+ z(X,E)&=\#\{\chi\text{ selected by that lift}:\chi(1)_p=|X|_p\}.
+\end{align*}
+Here and throughout, modules and characters are counted up to
+isomorphism. The class sums form a basis of $Z(\mathcal O X)$, and
+their reductions form a basis of $Z(kX)$. Idempotent lifting over the
+complete DVR thus gives a unique central idempotent lifting $E$.
+Every integral central element acts on an ordinary irreducible by
+a scalar in $\mathcal O$: the scalar lies in $K$ by splitting and
+satisfies a monic integral equation by finiteness, and $\mathcal O$
+is integrally closed. Reduction of this scalar defines a character
+of $Z(kX)$. The Artinian local decomposition of that center shows
+that it is precisely $\lambda_b$, where $b$ is the block containing
+the ordinary irreducible.
+
+For $H\leq X$, let $s_H$ denote coefficient restriction from
+$Z(kX)$ to $Z(kH)$, or its integral analogue when appropriate.
+For a $p$-subgroup $Q$, let $\operatorname{Br}_Q$ denote restriction
+of coefficients from the $Q$-invariants $(kX)^Q$ to $kC_X(Q)$.
+
+\begin{lemma}[Normalizer projection]\label{lem:normalizer-projection}
+The map $\operatorname{Br}_Q:(kX)^Q\longrightarrow kC_X(Q)$ is a
+unital algebra homomorphism. Suppose that
+\[
+ Q\leq H\leq N_X(Q),\qquad C_X(Q)\leq H.
+\]
+Writing $\pi_Q:kH\longrightarrow k[H/Q]$ for the quotient map, one
+has, for $v\in Z(kX)$,
+\begin{equation}\label{eq:normalizer-projection}
+ \pi_Q(s_H(v))=\pi_Q(\operatorname{Br}_Q(v)),
+ \qquad
+ \lambda_b(s_H(v))=\lambda_b(\operatorname{Br}_Q(v))
+\end{equation}
+for every block $b$ of $H$. In particular $\lambda_b\circ s_H$ is
+the central character of a unique block of $X$.
+\end{lemma}
+
+\begin{proof}
+To compute the coefficient of $g\in C_X(Q)$ in a product of two
+$Q$-invariant elements, let $Q$ conjugate the pairs $(a,b)$ with
+$ab=g$. Coefficient products are constant on the orbits. Every
+nontrivial orbit has size divisible by $p$, and the fixed pairs are
+exactly those with $a,b\in C_X(Q)$. This proves multiplicativity;
+the identity is preserved.
+
+Each coset $hQ$ is stable under conjugation by $Q$, because
+$h\in N_X(Q)$. Applying the same orbit cancellation to the sum
+of coefficients over that coset proves the first identity in
+\eqref{eq:normalizer-projection}. Also
+$\operatorname{Br}_Q(v)\in Z(kH)$, since its support lies in
+$C_X(Q)\leq H$ and it is invariant under $H$.
+
+Every nonzero module for a finite $p$-group over $k$ has nonzero
+invariants. For completeness, choose a central element $z$ of order
+$p$; the nonzero kernel of $z-1$, whose $p$th power is zero, is a
+module for the smaller group $Q/\langle z\rangle$, and induction
+applies. If the module is simple for $H$, its $Q$-invariants are
+$H$-stable, and hence the whole module. Thus every simple $kH$-module
+factors through $H/Q$, proving the second identity. The composition
+with $\operatorname{Br}_Q$ is a unital algebra character. Characters
+of the finite commutative algebra $Z(kX)$ are its local residue
+characters, one for each block, giving uniqueness.
+\end{proof}
+
+For $H=N_X(Q)$ put
+\[
+ \overline E_Q
+   =\pi_Q(s_H(E))
+   =\pi_Q(\operatorname{Br}_Q(E))\in Z(k[H/Q]).
+\]
+This is an idempotent. Its selected defect-zero quotient characters
+are exactly the weights assigned to the blocks appearing in $E$.
+Indeed, the lift of $\operatorname{Br}_Q(E)$ maps to the lift of
+$\overline E_Q$ by uniqueness of central idempotent lifting. Its
+action on the inflation of a quotient character is therefore one
+exactly when the latter is selected by $\overline E_Q$. If $b$ is
+the block of this inflation, the same test is
+\[
+ \lambda_b(\operatorname{Br}_Q(E))
+   =\lambda_b(s_H(E))=1.
+\]
+By \Cref{lem:normalizer-projection}, this says that the induced block
+is one of the blocks in $E$, with the coefficient-projection
+definition in \eqref{eq:block-assignment}. Once $Q$ is fixed up to $X$-conjugacy,
+there is no further identification of its quotient characters:
+$N_X(Q)$ acts on $N_X(Q)/Q$ by inner automorphisms.
+
+For a conjugation-invariant function $g$ of such pairs $(X,E)$,
+define
+\begin{equation}\label{eq:weight-transform}
+ (Tg)(X,E)=
+ \sum_{\substack{1<Q\leq X\text{ a $p$-subgroup}\\
+                  \text{up to }X\text{-conjugacy}}}
+ g\bigl(N_X(Q)/Q,\overline E_Q\bigr).
+\end{equation}
+Consequently the desired universal equality for block sums is
+$l=(1+T)z$. The identity term includes precisely $Q=1$.
+Every transition in $T$ strictly decreases the $p$-part of the group
+order, so $T$ is nilpotent on each evaluated pair. Thus the formal
+inverse $(1+T)^{-1}=\sum_{r\geq0}(-T)^r$ is evaluation-wise finite.
+
+The next step makes the connection with subgroup-chain formulations of
+the weight conjecture \cite{KnorrRobinson1989}. We derive the particular
+normal-chain identity from this finite inverse, retaining its idempotent
+and orbit data throughout.
+
+\begin{definition}[Chains]\label{def:p-chains}
+A chain in $X$ is a strictly increasing sequence of $p$-subgroups
+\[
+ \mathcal C=(1=Q_0<Q_1<\cdots<Q_r),
+ \qquad |\mathcal C|=r.
+\]
+The length-zero chain $(1)$ is included. Set
+\[
+ H_{\mathcal C}=\bigcap_{i=0}^rN_X(Q_i),
+ \qquad E_{\mathcal C}=\operatorname{Br}_{Q_r}(E).
+\]
+Call $\mathcal C$ \emph{normal} if every $Q_i$ is normal in $Q_r$.
+\end{definition}
+
+For every chain, normal or otherwise,
+$C_X(Q_r)\leq H_{\mathcal C}\leq N_X(Q_r)$, so
+$E_{\mathcal C}$ is an idempotent of $Z(kH_{\mathcal C})$.
+For a normal chain one also has $Q_r\leq H_{\mathcal C}$.
+
+\begin{lemma}[Finite chain inversion]\label{lem:chain-transform}
+The identity $l=(1+T)z$ for all pairs $(X,E)$ is equivalent to
+\begin{equation}\label{eq:chain-inversion}
+ z(X,E)=
+ \sum_{\substack{\mathcal C/X\\\mathcal C\ \mathrm{normal}}}
+ (-1)^{|\mathcal C|}l(H_{\mathcal C},E_{\mathcal C}),
+\end{equation}
+where $\mathcal C/X$ denotes one representative of each
+$X$-conjugacy orbit.
+\end{lemma}
+
+\begin{proof}
+We identify the terms of $T^r l$. After $r$ steps they correspond
+to normal chains of length $r$, with group
+$H_{\mathcal C}/Q_r$ and idempotent
+\[
+ \pi_{Q_r}(s_{H_{\mathcal C}}(E))
+   =\pi_{Q_r}(E_{\mathcal C}).
+\]
+For the induction, the preimage $Q_{r+1}$ of a nontrivial
+$p$-subgroup of $H_{\mathcal C}/Q_r$ is a $p$-group strictly
+containing $Q_r$. It lies in $H_{\mathcal C}$ and therefore
+normalizes every preceding $Q_i$. Conversely every normal one-step
+extension arises in this way. Its normalizer in the quotient has
+preimage
+$H_{\mathcal C}\cap N_X(Q_{r+1})=H_{\mathcal C'}$.
+After fixing $\mathcal C$, conjugation is by its stabilizer
+$H_{\mathcal C}$, which is exactly quotient conjugation at the next
+step. This proves the orbit assertion without an extra multiplicity.
+
+To check the idempotent at the next step, write
+$E=\sum_{x\in X}e_xx$ and put $H'=H_{\mathcal C'}$.
+The new normalizer in $H_{\mathcal C}/Q_r$ is $H'/Q_r$.
+Restriction to this normalizer retains exactly the coefficient sums
+over the $Q_r$-cosets contained in $H'$. Choose a set $A$ of
+representatives for the cosets of $Q_r$ in $Q_{r+1}$. For $h\in H'$,
+the subsequent quotient has coefficient
+\[
+ \sum_{a\in A}\ \sum_{q\in Q_r}e_{haq}
+     =\sum_{t\in Q_{r+1}}e_{ht}
+\]
+at $hQ_{r+1}$: the sets $aQ_r$, $a\in A$, partition $Q_{r+1}$.
+These are precisely the coefficients of
+$\pi_{Q_{r+1}}(s_{H'}(E))$.
+By \Cref{lem:normalizer-projection}, this idempotent is
+$\pi_{Q_{r+1}}(E_{\mathcal C'})$, as required.
+Every simple $kH_{\mathcal C}$-module factors through
+$H_{\mathcal C}/Q_r$ and has the indicated idempotent action.
+Hence the contribution to $T^rl$ is the corresponding term in
+\eqref{eq:chain-inversion}. The finite inverse of $1+T$ now gives
+the equivalence.
+\end{proof}
+
+\subsection{Cocenters and subsections}
+
+By \Cref{lem:chain-transform}, the remaining algebraic objective is
+\eqref{eq:chain-inversion}. We next express ordinary character counts
+as sums of simple-module counts in centralizers. Chain cancellation
+will then replace the alternating sum of $l$ by an alternating sum of
+$k_{\mathrm{ord}}$, which can be computed by central traces.
+
+For an algebra $A$, write $V(A)=A/[A,A]$, where $[A,A]$ is the
+linear span of commutators, and write $\overline a$ for the image of
+$a$. A central element acts on $V(A)$ by multiplication.
+
+Powers modulo commutators underlie K\"ulshammer's generalized Reynolds
+ideals; see \cite[Section~2]{HHKM2005}. We give the finite-dimensional
+argument needed to extract the two block counts.
+
+\begin{lemma}[Cocenter ranks]\label{lem:cocenter-ranks}
+For a finite-dimensional $k$-algebra $A$, the map
+\[
+ F_A:V(A)\longrightarrow V(A),\qquad
+ \overline a\longmapsto\overline{a^p}
+\]
+is well-defined and Frobenius semilinear. Its stable image has
+dimension equal to the number of simple $A$-modules, and its
+summand selected by a central idempotent has dimension equal to
+the corresponding simple-module count.
+For $A=kX$, this stable image is the span of the $p'$-classes.
+The ranks of multiplication by $E$ on that span and on all of
+$V(kX)$ are, respectively, $l(X,E)$ and $k_{\mathrm{ord}}(X,E)$.
+\end{lemma}
+
+\begin{proof}
+In the expansion of $(a+b)^p$, cyclic rotations of any mixed word
+give the same cocenter class. Their orbits have size $p$, leaving
+only $a^p$ and $b^p$. Thus the power operation into the cocenter
+is additive and is Frobenius semilinear. Moreover
+$\overline{(ab)^p}=\overline{(ba)^p}$, by one cyclic rotation;
+additivity shows that this operation kills the linear commutator
+space. It therefore descends to $F_A$.
+
+Let $R$ be the Jacobson radical of $A$. The kernel of
+$V(A)\longrightarrow V(A/R)$ is represented by elements of $R$:
+lift any commutator expression downstairs and subtract it upstairs.
+Since $R$ is nilpotent, a sufficiently high iterate of $F_A$ kills
+this kernel. On the split semisimple algebra $A/R$, the cocenter
+has one matrix-trace coordinate per simple factor, and $F_A$
+raises each coordinate to the $p$th power. It is bijective because
+$k$ is perfect. For example, the trace identity here follows by
+triangularizing a matrix over $k$ and taking the $p$th powers of
+its diagonal entries.
+
+The descending images of $F_A$ stabilize, since they are
+$k$-subspaces of a finite-dimensional space. Denote the stable image
+by $S$ and the quotient map by $q:V(A)\to V(A/R)$.
+Choose $m$ large enough that $S=\operatorname{im}F_A^m$ and
+$F_A^m$ kills $\ker q$. On $S$, the map $F_A$ is surjective
+and hence bijective, because it is semilinear for the automorphism
+$a\mapsto a^p$ of the perfect field $k$. If $x\in S\cap\ker q$,
+then $F_A^m(x)=0$; injectivity on $S$ gives $x=0$.
+To prove surjectivity of $q|_S$, take $y\in V(A/R)$. The
+Frobenius map $F_{A/R}$ is bijective, so choose $y'$ with
+$F_{A/R}^m(y')=y$ and lift $y'$ to $x\in V(A)$. Then
+$F_A^m(x)\in S$ and $q(F_A^m(x))=y$. Thus $q|_S$ is an
+isomorphism. Finally
+$(Ea)^p=Ea^p$ for a central idempotent $E$, so all these statements
+respect its summand.
+
+For a group algebra the cocenter is free on conjugacy classes,
+represented by individual group elements: the commutator
+relations identify exactly conjugate elements. Sufficiently high
+$p$-powers land on $p'$-elements, and taking $p$th powers permutes
+the $p'$-classes. This identifies the stable image. The integral
+cocenter $V(\mathcal O X)$ is free on the same class basis. The
+lift of $E$ projects it onto a free direct summand, whose rank is
+unchanged over $k$ and $K$. Split semisimplicity over $K$ identifies
+this rank with $k_{\mathrm{ord}}(X,E)$.
+\end{proof}
+
+Every group element has a unique commuting factorization into its
+$p$-part and $p'$-part, both powers of that element. A
+\emph{section} is the set of elements whose $p$-parts are conjugate
+to a fixed $p$-element.
+
+The next identity is Brauer's classical subsection count
+\cite{Brauer1959}. The cocenter argument also proves the support
+property needed for the defect-zero detector, without dividing by
+class sizes in characteristic $p$.
+
+\begin{lemma}[Subsection formula and support]\label{lem:subsections}
+For every central idempotent $E\in kX$,
+\begin{equation}\label{eq:subsections}
+ k_{\mathrm{ord}}(X,E)=
+ \sum_{\substack{[u]\text{ an }X\text{-class}\\u\text{ a }p\text{-element}}}
+ l\bigl(C_X(u),\operatorname{Br}_{\langle u\rangle}(E)\bigr).
+\end{equation}
+Multiplication by $E$ preserves the section decomposition of the
+cocenter and the section decomposition of the center into spans
+of class sums.
+\end{lemma}
+
+\begin{proof}
+Fix a $p$-element $u$ and set $C=C_X(u)$. The map
+$\overline t\mapsto\overline{ut}$ identifies the $p'$-class span
+of $V(kC)$ with the section span belonging to $u$ in $V(kX)$.
+It is a bijection of class bases: conjugacy of $ut$ and $ut'$
+forces a conjugating element to centralize their common $p$-part
+$u$, and the converse is immediate.
+
+For $t\in C$ a $p'$-element, conjugation by $\langle u\rangle$
+in the expansion of $Eut$ gives
+\[
+ \overline{Eut}
+   =\overline{\operatorname{Br}_{\langle u\rangle}(E)ut}.
+\]
+Indeed, the coefficients are constant on these orbits and the
+cocenter classes of their products with $ut$ agree; only fixed
+orbits survive. The restricted idempotent is central in $kC$ and
+preserves its $p'$-class span by \Cref{lem:cocenter-ranks}.
+Multiplication by $u$ is central in $kC$, so the displayed
+identification intertwines the two idempotent actions. Their ranks
+are thus the terms in \eqref{eq:subsections}. Summing the ranks
+over the sections proves that identity.
+
+For the center, use the pairing
+\[
+ Z(kX)\times V(kX)\longrightarrow k,
+ \qquad (a,\overline b)\longmapsto\tau(ab),
+\]
+where $\tau$ takes the coefficient of $1$. It is nondegenerate:
+a class sum pairs with an individual representative of its inverse
+class with coefficient one. Multiplication by $E$ is self-adjoint
+for this pairing. Sections on the center pair with the inverse
+sections of the cocenter, so preservation there proves preservation
+here. In particular no class-size division is involved.
+\end{proof}
+
+\subsection{Two cancellations on chain orbits}
+
+\begin{lemma}[Removal of nonnormal chains]\label{lem:nonnormal-cancellation}
+In an alternating sum over chain orbits, every contribution that
+depends only on $H_{\mathcal C}$ and $Q_r$ cancels on the nonnormal
+chains. Thus such a sum may be taken over all chains or only normal
+chains.
+\end{lemma}
+
+\begin{proof}
+For a nonnormal chain, let $i$ be the largest index for which
+$A=Q_i$ is not normal in $P=Q_r$, and let $R$ be the normal closure
+of $A$ in $P$. Then $A<R\leq Q_{i+1}$: every later member is
+normal in $P$. Also $R<P$. To see the latter, put the proper
+subgroup $A$ in a maximal subgroup of $P$, which is normal because
+$P$ is a finite $p$-group; it contains $R$.
+
+Insert $R$ immediately after $A$ if it is absent, and remove it
+if it is present. The last group $P$ and the last nonnormal member
+$A$ are unchanged, so this operation is an involution. Every
+element normalizing $A$ and $P$ normalizes their normal closure
+$R$; hence insertion or removal leaves $H_{\mathcal C}$ unchanged.
+The operation commutes with $X$-conjugation and changes the length
+by one. It induces a fixed-point-free sign reversal on the orbits,
+which proves cancellation.
+\end{proof}
+
+\begin{lemma}[Alternating subsection cancellation]
+\label{lem:alternating-subsections}
+One has
+\begin{equation}\label{eq:alternating-subsections}
+ \sum_{\substack{\mathcal C/X\\\mathcal C\ \mathrm{normal}}}
+ (-1)^{|\mathcal C|}l(H_{\mathcal C},E_{\mathcal C})
+ =
+ \sum_{\substack{\mathcal C/X\\\mathcal C\ \mathrm{normal}}}
+ (-1)^{|\mathcal C|}k_{\mathrm{ord}}(H_{\mathcal C},E_{\mathcal C}).
+\end{equation}
+\end{lemma}
+
+\begin{proof}
+By \Cref{lem:nonnormal-cancellation}, both sums may be taken over
+all chains. Expand the ordinary count by \Cref{lem:subsections}.
+The $u=1$ terms give the simple-module sum. The other terms are
+indexed by $X$-orbits of pairs $(\mathcal C,u)$, with $u\ne1$ a
+$p$-element in $H_{\mathcal C}$. This orbit description follows
+because the stabilizer of a fixed chain is $H_{\mathcal C}$.
+The contribution of such a pair is
+\begin{equation}\label{eq:pair-contribution}
+ l\bigl(H_{\mathcal C}\cap C_X(u),
+         \operatorname{Br}_{Q_r\langle u\rangle}(E)\bigr).
+\end{equation}
+Here $u$ normalizes $Q_r$, so $Q_r\langle u\rangle$ is a
+$p$-group; successive coefficient restrictions have support
+$C_X(Q_r)\cap C_X(u)=C_X(Q_r\langle u\rangle)$, proving the
+idempotent in this expression.
+
+Let $i$ be the largest index, allowing $0$ and $r$, for which
+$u\notin Q_i$. Such an index exists since $Q_0=1$. Toggle
+$R=Q_i\langle u\rangle$ immediately after $Q_i$: insert it
+if absent, and remove it if present. It is a $p$-group, since
+$u$ normalizes $Q_i$, and it is contained in every later member,
+all of which contain $u$. The last member not containing $u$
+therefore remains $Q_i$; this proves involutivity, including an
+insertion after the terminal member and its inverse deletion.
+
+Every element centralizing $u$ and normalizing $Q_i$ normalizes
+$R$. Thus $H_{\mathcal C}\cap C_X(u)$ is unchanged. The product
+of the terminal member with $\langle u\rangle$ is also unchanged,
+even when the terminal member is inserted or deleted. Hence
+\eqref{eq:pair-contribution} is preserved. The operation is
+$X$-equivariant and reverses the sign, so all $u\ne1$ terms cancel.
+\end{proof}
+
+\subsection{Detecting defect zero by central elements}
+
+By \Cref{lem:alternating-subsections}, it remains to prove that the
+alternating ordinary-character sum in
+\eqref{eq:alternating-subsections} equals $z(X,E)$. Their difference
+retains the signed contributions of all characters selected by
+$E_{\mathcal C}$ on positive-length normal chains, but omits the
+selected defect-zero characters at the chain $(1)$. We will realize
+this difference as a trace limit by constructing a central element
+with the corresponding zero-or-one central-character values.
+
+Put
+\[
+ c_X=\sum_{u\in X\,\text{a }p\text{-element}}u,
+ \qquad d_X=\sum_{x,y\in X}[x,y],
+ \qquad I_X(a)=\sum_{x\in X}xax^{-1}.
+\]
+These elements and the image of $I_X$ are central. Call a block
+\emph{semisimple} if its algebra is semisimple; since a block has no
+nontrivial central idempotent, such a block is one full matrix
+algebra over $k$.
+
+The scalar of $d_X$ will identify these semisimple blocks with the
+ordinary defect-zero characters. The complement $1-c_X$ will then remove
+exactly those blocks from the central trace, using the support
+information supplied by $I_X$.
+
+The operator $I_X$ is the group-algebra Higman trace
+\cite[Section~2]{Higman1955}; see also
+\cite[Sections~2.4--2.5]{LorenzTokoly2011}. The use of $c_X$ to
+isolate defect zero goes back to Tsushima \cite[Theorem~1]{Tsushima1971}.
+The following proof records the exact radical-annihilation and
+central-character consequences used in our trace limit.
+
+\begin{lemma}[Averaging and defect zero]\label{lem:defect-zero-detection}
+The image of $I_X$, and also $d_X$, annihilates the Jacobson radical
+of $kX$. On a semisimple block, $d_X$ has nonzero central scalar;
+on every other block its central character is zero. Each semisimple
+block contains exactly one ordinary irreducible, which has defect
+zero, and every ordinary defect-zero irreducible occurs in this
+way. Moreover:
+\begin{enumerate}[label=\textup{(\roman*)}]
+ \item $\operatorname{im}I_X$ is the span of class sums of elements
+ whose centralizers have order prime to $p$;
+ \item each semisimple block idempotent belongs to
+ $\operatorname{im}I_X$ and has zero Brauer image at every nontrivial
+ $p$-subgroup;
+ \item $\lambda_b(c_X)=0$ on every nonsemisimple block $b$.
+\end{enumerate}
+\end{lemma}
+
+\begin{proof}
+\textit{Radical annihilation.}
+The form $(a,b)\mapsto\tau(ab)$ on $kX$ is symmetric and
+nondegenerate. For dual bases $v_i,v_i^*$, the tensor
+$\Omega=\sum_i v_i\otimes v_i^*$ is independent of the chosen
+bases. Apply the linear map $u\otimes v\mapsto uav$ to this
+tensor. In the group basis, whose dual basis consists of the inverses,
+the result is $I_X(a)$. Similarly, apply
+$u\otimes v\otimes w\otimes z\mapsto uwvz$ to
+$\Omega\otimes\Omega$. In the group basis the result is
+$\sum_{x,y\in X}xyx^{-1}y^{-1}=d_X$. Thus in any dual bases
+\begin{equation}\label{eq:averaging-tensor}
+ I_X(a)=\sum_i v_iav_i^*,\qquad
+ d_X=\sum_{i,j}v_iv_jv_i^*v_j^*
+     =\sum_j I_X(v_j)v_j^*.
+\end{equation}
+For $b$ in the radical $R$, the trace identity
+\[
+ \tau(I_X(a)b)=\operatorname{Tr}(L_bR_a)
+\]
+follows by writing the trace in these dual bases; $L$ and $R$
+denote left and right multiplication. Since $L_b$ is nilpotent
+and commutes with $R_a$, this trace is zero. Replace $b$ by $bc$
+for arbitrary $c\in kX$, and use nondegeneracy to obtain
+$I_X(a)b=0$. The final expression in
+\eqref{eq:averaging-tensor} then shows that $d_XR=0$ as well,
+because $v_j^*R\subseteq R$.
+
+\textit{Semisimple blocks and ordinary defect zero.}
+Distinct block ideals are orthogonal for the form. On a
+semisimple block $M_d(k)$ its restriction is $t\operatorname{Tr}$
+for some $t\in k^\times$. Indeed a symmetric functional kills
+commutators, whose quotient in a matrix algebra is the trace line,
+and nondegeneracy forces $t\ne0$. Matrix units $e_{ij}$ have duals
+$t^{-1}e_{ji}$. Since
+$\sum_{i,j}e_{ij}ae_{ji}=\operatorname{Tr}(a)1$, the first
+contraction in \eqref{eq:averaging-tensor} gives
+$I_X(a)=t^{-1}\operatorname{Tr}(a)1$. In particular
+$I_X(e_{ij})=t^{-1}\delta_{ij}1$, so its last expression gives
+\[
+ d_X=\sum_{i,j}I_X(e_{ij})t^{-1}e_{ji}
+     =t^{-2}\sum_i e_{ii}=t^{-2}1.
+\]
+We have therefore proved
+\begin{equation}\label{eq:simple-block-scalars}
+ I_X(a)=t^{-1}\operatorname{Tr}(a)1,
+ \qquad d_X=t^{-2}1
+ \quad\text{on this block}.
+\end{equation}
+In particular its idempotent lies in the image of $I_X$; one can
+take $a=te_{11}$. These formulas do not require $d$ to be nonzero
+in $k$. On a nonsemisimple block, a central element with nonzero
+residue scalar is a unit in its local center. Such a unit cannot
+annihilate the nonzero radical. This proves the assertion about
+the central character of $d_X$.
+
+Now compute the same commutator sum in characteristic zero. If
+$\chi$ has degree $d$, Schur's lemma gives
+\[
+ \sum_{x\in X}\rho(xyx^{-1})
+      =\frac{|X|\chi(y)}d\,1.
+\]
+Multiply by $\rho(y^{-1})$, sum over $y$, and take traces.
+Ordinary character orthogonality gives the scalar
+\begin{equation}\label{eq:ordinary-commutator-scalar}
+ \lambda_\chi(d_X)=\frac{|X|^2}{\chi(1)^2}.
+\end{equation}
+It is integral, by the integrality of ordinary central scalars.
+Its reduction is nonzero exactly when
+$v_p(\chi(1))=v_p(|X|)$, namely in defect zero. The modular
+calculation therefore puts exactly these characters in
+semisimple blocks. Such a block has cocenter dimension one, so
+\Cref{lem:cocenter-ranks} shows that its lifted block selects
+exactly one ordinary irreducible. This also proves existence of
+that character.
+
+\textit{Support and the $p$-element sum.}
+For a group element $g$ one has
+\[
+ I_X(g)=|C_X(g)|\sum_{a\in[g]}a.
+\]
+This proves assertion (i). The support of a semisimple block
+idempotent, already in this image, therefore contains no element
+centralized by a nontrivial $p$-subgroup, proving (ii).
+
+Finally fix a nontrivial $p$-subgroup $P$. Choose
+$z\in Z(P)$ of order $p$. It is central also in $C_X(P)$, and
+multiplication by $z$ preserves the set of $p$-elements of that
+centralizer. With $q=\sum_{j=0}^{p-1}z^j$, the sum of those
+$p$-elements factors as $q$ times a sum of orbit representatives.
+As $q$ is central and $q^2=pq=0$, this proves
+\[
+ \operatorname{Br}_P(c_X)^2=0,
+ \qquad \operatorname{Br}_P(c_X^2)=0.
+\]
+If $p$ divides $|C_X(g)|$, a nontrivial $p$-subgroup
+$P\leq C_X(g)$ makes the coefficient of $g$ in $c_X^2$ vanish
+by this identity. Assertion (i) gives $c_X^2\in\operatorname{im}I_X$.
+On a nonsemisimple block its central character must therefore
+vanish, by the same radical-annihilator argument. Since
+$\lambda_b(c_X)^2=\lambda_b(c_X^2)=0$, assertion (iii) follows.
+\end{proof}
+
+\begin{lemma}[An element with zero-or-one residues]
+\label{lem:detecting-element}
+Let $v=E(1-c_X)$. It is central and supported on $p$-singular
+elements. For a normal chain $\mathcal C$ of positive length, its
+restriction to $H_{\mathcal C}$ satisfies
+\[
+ \lambda_b(s_{H_{\mathcal C}}(v))=\lambda_b(E_{\mathcal C})
+ \in\{0,1\}
+\]
+for every block $b$ of $H_{\mathcal C}$. For the length-zero
+chain, the scalar is one on the nonsemisimple blocks selected by
+$E$ and zero on every other block.
+\end{lemma}
+
+\begin{proof}
+The element $1-c_X$ is the negative sum of the nonidentity
+$p$-elements, hence lies entirely in $p$-singular sections.
+Section preservation from \Cref{lem:subsections} proves the same
+support assertion for $v$. Let $B_0$ be a semisimple block
+idempotent. Since $v$ is central and $B_0kX$ is a full matrix
+algebra, $B_0v$ lies in its one-dimensional center $kB_0$.
+On the other hand, section preservation makes the support of $B_0v$
+$p$-singular. By \Cref{lem:defect-zero-detection}, the support
+of $B_0$ contains only elements whose centralizers have order prime
+to $p$, and therefore no $p$-singular elements. The two support
+conditions force the scalar multiple $B_0v$ to be zero.
+On a nonsemisimple block,
+\Cref{lem:defect-zero-detection} gives
+$\lambda_b(v)=\lambda_b(E)$, proving the length-zero assertion.
+
+For positive length, put $Q=Q_r\ne1$ and $H=H_{\mathcal C}$.
+By \Cref{lem:normalizer-projection}, $\lambda_b\circ s_H$ is a
+central character of $X$ and factors through
+$\operatorname{Br}_Q$. It cannot be the character of a
+semisimple block $B_0$: that would give value one on $B_0$,
+whereas $\operatorname{Br}_Q(B_0)=0$ by
+\Cref{lem:defect-zero-detection}. On its nonsemisimple block,
+$\lambda(v)=\lambda(E)$. Applying
+\Cref{lem:normalizer-projection} to $E$ gives the asserted value
+$\lambda_b(E_{\mathcal C})$.
+\end{proof}
+
+\subsection{Central traces and the final cancellation}
+
+Choose a class-sum lift
+\[
+ \widehat v=\sum_{a\in X}\alpha_a a\in Z(\mathcal O X)
+\]
+of the detecting element $v=E(1-c_X)$ from
+\Cref{lem:detecting-element}, with $\alpha_a=0$ whenever $a$ is
+not $p$-singular.
+For a positive integer $n$, define
+\begin{equation}\label{eq:chain-trace}
+ \Phi_n=
+ \sum_{\mathcal C/X}(-1)^{|\mathcal C|}
+ \operatorname{Tr}\left(
+    s_{H_{\mathcal C}}(\widehat v)^n
+       \mid V(KH_{\mathcal C})\right),
+\end{equation}
+where the operator is central multiplication and all chains are
+included.
+
+\begin{lemma}[The first trace limit]\label{lem:trace-limit}
+In the valuation topology of $K$,
+\begin{equation}\label{eq:trace-limit}
+ \lim_{s\to\infty}\Phi_{p^s}
+ =
+ \sum_{\substack{\mathcal C/X\\\mathcal C\ \mathrm{normal}}}
+ (-1)^{|\mathcal C|}k_{\mathrm{ord}}(H_{\mathcal C},E_{\mathcal C})
+ -z(X,E).
+\end{equation}
+\end{lemma}
+
+\begin{proof}
+The nonnormal chains cancel by
+\Cref{lem:nonnormal-cancellation}, since their trace contribution
+depends only on the stabilizer. For a remaining chain, the
+cocenter of $KH_{\mathcal C}$ has one coordinate per ordinary
+irreducible. The unpowered operator has on these coordinates
+integral central eigenvalues reducing to the scalars in
+\Cref{lem:detecting-element}.
+
+If an eigenvalue reduces to zero, its $p^s$th powers tend to zero.
+If it is $1+a$ with $\operatorname{val}(a)>0$, the binomial
+formula gives
+\[
+ \operatorname{val}((1+a)^p-1)
+ \geq
+ \min\{\operatorname{val}(p)+\operatorname{val}(a),
+                    p\operatorname{val}(a)\}.
+\]
+After normalizing the discrete valuation to integer values, each
+iteration increases the positive valuation by at least one, unless
+the difference has already become zero. These powers tend to one.
+This argument applies also to $p=2$ and requires no finiteness of
+the residue field.
+
+For positive-length chains the limiting trace therefore counts
+all ordinary irreducibles selected by $E_{\mathcal C}$. At length
+zero it omits the semisimple blocks, each of which contributes
+exactly one defect-zero character by
+\Cref{lem:defect-zero-detection}. There are finitely many chains
+and irreducibles, so limits may be summed, proving
+\eqref{eq:trace-limit}.
+\end{proof}
+
+The next identity follows from Frobenius's commutator-counting formula
+and is the weighted genus-one case of the surface formula in
+\cite[Theorem~2.1 and Equation~(3.1)]{Klug2025}. Its single
+denominator is important when chain orbits are replaced by actual
+chains, so we verify the normalization directly.
+
+\begin{lemma}[Trace normalization]\label{lem:tuple-trace}
+For $H\leq X$ and $w=s_H(\widehat v)$,
+\begin{equation}\label{eq:tuple-trace}
+ \operatorname{Tr}(w^n\mid V(KH))
+ =\frac1{|H|}
+   \sum_{\substack{x,y,u_1,\ldots,u_n\in H\\
+                   [x,y]u_1\cdots u_n=1}}
+       \prod_{i=1}^n\alpha_{u_i}.
+\end{equation}
+\end{lemma}
+
+\begin{proof}
+The sum on the right before division is
+$\tau(d_Hw^n)$. The regular trace of any element of $KH$ is
+$|H|$ times its identity coefficient. On the other hand,
+\eqref{eq:ordinary-commutator-scalar} and the regular decomposition
+give
+\begin{align*}
+ |H|\tau(d_Hw^n)
+  &=\sum_{\chi\in\operatorname{Irr}(H)}
+       \chi(1)^2\frac{|H|^2}{\chi(1)^2}\lambda_\chi(w)^n\\
+  &=|H|^2\operatorname{Tr}(w^n\mid V(KH)).
+\end{align*}
+Dividing proves exactly the single denominator in
+\eqref{eq:tuple-trace}.
+\end{proof}
+
+For a subgroup $J\leq X$, define an integer on actual chains,
+without quotienting by conjugation,
+\begin{equation}\label{eq:chain-weight}
+ a_X(J)=
+ \sum_{\substack{\mathcal C\text{ a chain in }X\\
+                  J\leq H_{\mathcal C}}}(-1)^{|\mathcal C|}.
+\end{equation}
+
+\begin{lemma}[Cancellation by a normal $p$-subgroup]
+\label{lem:pcore-cancellation}
+If $O_p(J)\ne1$, then $a_X(J)=0$.
+\end{lemma}
+
+\begin{proof}
+Put $P=O_p(J)$. In any chain normalized by $J$, each $Q_i$ is
+normalized by $P\leq J$, so $Q_iP$ is a $p$-subgroup. Both
+factors are normalized by $J$, making their product
+$J$-normalized. Let $i$ be the last index such that $P\nleq Q_i$;
+it exists since $Q_0=1$ and $P\ne1$. Toggle $Q_iP$ immediately
+after $Q_i$. It strictly contains $Q_i$ and is contained in all
+later members. Insertion and deletion preserve the last member
+not containing $P$, so this is an involution. It is defined also
+for the length-zero chain and for terminal insertion or deletion.
+It preserves the property of being normalized by $J$ and reverses
+the sign. The sum of actual chains in \eqref{eq:chain-weight}
+therefore vanishes.
+\end{proof}
+
+\begin{proposition}[Reduction to uniform tuple divisibility]
+\label{prop:group-reduction}
+If \Cref{thm:tuple-divisibility} holds for every finite group with
+trivial $p$-core, then for every finite group $X$ and central
+idempotent $E\in kX$ one has $l(X,E)=((1+T)z)(X,E)$.
+In particular the block equality in \Cref{thm:main} follows, with
+the stated block assignment and with $Q=1$ included.
+\end{proposition}
+
+\begin{proof}
+The orbit of a chain $\mathcal C$ has size
+$|X|/|H_{\mathcal C}|$. Consequently
+\Cref{lem:tuple-trace} changes the orbit sum
+\eqref{eq:chain-trace} into the following sum over actual tuples
+and actual chains:
+\begin{equation}\label{eq:trace-by-generated-group}
+ \Phi_n=\frac1{|X|}
+ \sum_{\substack{x,y,u_1,\ldots,u_n\in X\\
+                 [x,y]u_1\cdots u_n=1}}
+ a_X\bigl(\langle x,y,u_1,\ldots,u_n\rangle\bigr)
+ \prod_{i=1}^n\alpha_{u_i}.
+\end{equation}
+To verify the factor, each orbit contribution has prefactor
+$1/|H_{\mathcal C}|$; replacing its representative by all
+$|X|/|H_{\mathcal C}|$ conjugates gives the common prefactor
+$1/|X|$. For a fixed tuple the admissible chains are exactly those
+normalized by its generated subgroup. This is precisely
+$a_X(J)$.
+
+By \Cref{lem:pcore-cancellation}, only generated subgroups
+$J$ with $O_p(J)=1$ contribute. By the support of $\widehat v$,
+only tuples with every $u_i$ $p$-singular contribute. Fix such a
+subgroup $J$ and group its generating tuples according to the
+multiset $\mathcal M$ of $J$-conjugacy classes of the $u_i$.
+The product $\prod_i\alpha_{u_i}$ is constant on this multiset:
+the coefficients are constant on $X$-classes, hence on $J$-classes,
+and their product is unchanged by reordering. Denote this integral
+product by $\alpha(\mathcal M)$. A generating tuple has simultaneous
+conjugation stabilizer $Z(J)$, since an element centralizing all
+its entries centralizes the group they generate. Thus its number
+of actual tuples is $|J|/|Z(J)|$ times its orbit count. For
+$n=p^s$, the numerator in \eqref{eq:trace-by-generated-group}
+is therefore
+\begin{equation}\label{eq:uniform-numerator}
+ \sum_{\substack{J\leq X\\O_p(J)=1}}
+ a_X(J)\frac{|J|}{|Z(J)|}
+ \sum_{\mathcal M}
+   \alpha(\mathcal M)N_{J,s}(\mathcal M).
+\end{equation}
+
+Fix any $h\geq1$. By \Cref{thm:tuple-divisibility}, each summand
+of the inner sum belongs to $p^h\mathcal O$ for all sufficiently
+large $s$, with a threshold independent of $\mathcal M$. Although
+the number of multisets grows with $s$, the sum is finite for each
+$s$ and remains in that same ideal. There are only finitely many
+subgroups $J\leq X$, so take the largest of their thresholds.
+The integer factors in \eqref{eq:uniform-numerator} preserve
+divisibility. We conclude that its numerator tends to zero in the
+valuation topology. The fixed denominator $|X|$ does not change
+this conclusion, so $\Phi_{p^s}\longrightarrow0$.
+
+Combine this with \Cref{lem:trace-limit} and
+\Cref{lem:alternating-subsections}. The resulting equality is
+\eqref{eq:chain-inversion}; its integer terms agree in the
+characteristic-zero field $K$, and hence agree as integers.
+The same reasoning applies to every pair and to the splitting
+subquotients arising in the transform. \Cref{lem:chain-transform}
+now gives $l=(1+T)z$. The weight interpretation of
+\eqref{eq:weight-transform} proves the asserted block equality.
+\end{proof}