Add preprints files (part 14)

.gitattributes

@@ -694,3 +694,74 @@ preprints/The-metric-Blaschke-theorem-September-23-2026/paper.pdf filter=lfs dif
 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-ordinary-double-point-gap-in-every-dimension-September-24-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/The-p-adic-section-conjecture-September-24-2026/main.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/The-Partition-Principle-does-not-imply-Choice-September-24-2026/partition-principle-without-choice.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/The-PCP-for-PPAD-conjecture-a-quasilinear-reduction-September-25-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/The-Penrose-inequality-for-maximal-asymptotically-hyperbolic-initial-data-October-5-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/The-periodic-spin-one-Haldane-gap-September-24-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/The-Poisson-Dirichlet-Law-for-Prime-Predecessors-September-24-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/The-Quasi-Riemann-Hypothesis-October-5-2026/paper2.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/The-Quasi-Riemann-Hypothesis-September-30-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/The-rational-Hodge-conjecture-for-CM-abelian-varieties-September-30-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/The-rational-Hodge-conjecture-for-products-of-K3-surfaces-October-4-2026/hodge-conjecture-products-k3.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/The-Reconstruction-Threshold-for-the-Ferromagnetic-Four-State-Potts-Model-October-5-2026/four-state-potts.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/The-Restricted-Geometric-Langlands-Equivalence-in-Positive-Characteristic-September-24-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/The-reverse-logarithmic-Kodaira-inequality-and-additivity-September-26-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/The-second-Kahn-Kalai-conjecture-September-24-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/The-Selmer-converse-for-elliptic-curves-at-every-prime-September-24-2026/main.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/The-sharp-exponential-rate-of-singularity-for-symmetric-Bernoulli-matrices-October-3-2026/symmetric-bernoulli-singularity.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/The-sharp-factor-of-IID-threshold-for-the-free-Ising-model-on-regular-trees-September-26-2026/article.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/The-Sharp-Logarithmic-Exponent-of-r-5-t-September-24-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/The-sharp-singularity-rate-for-biased-symmetric-sign-matrices-October-4-2026/biased-symmetric-sign-singularity.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/The-Sharp-Terminal-Leave-in-Random-Triangle-Removal-September-25-2026/The-Sharp-Terminal-Leave-in-Random-Triangle-Removal-September-25-2026.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/The-Singer-conjecture-in-dimension-four-September-25-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/The-Solomon-Yau-least-volume-theorem-September-23-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/The-spacetime-Penrose-inequality-and-enclosing-area-September-27-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/The-spacetime-Penrose-inequality-with-charge-and-original-data-rigidity-October-5-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/The-spherical-magnetization-law-for-the-three-dimensional-quantum-Heisenberg-ferromagnet-October-5-2026/spherical-magnetization.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/The-spherical-perceptron-with-bi-orthogonally-invariant-disorder-September-24-2026/The-spherical-perceptron-with-bi-orthogonally-invariant-disorder-September-24-2026.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/The-Stable-Hurewicz-Image-of-the-Sphere-at-Two-September-25-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/The-Strong-Hyperkahler-SYZ-Conjecture-September-23-2026/main.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/The-strong-thin-tree-conjecture-September-23-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/The-Subcritical-Henon-Lane-Emden-Conjecture-September-24-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/The-symmetric-Mahler-conjecture-and-its-equality-cases-September-22-2026/build/figures/lens-boundary-law.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/The-symmetric-Mahler-conjecture-and-its-equality-cases-September-22-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/The-trace-cone-classifies-Razak-Jacelon-stabilizations-September-25-2026/The-trace-cone-classifies-Razak-Jacelon-stabilizations-September-25-2026.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/The-two-primary-Birch-Swinnerton-Dyer-formula-in-Selmer-corank-at-most-one-September-24-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/The-Unique-Games-Theorem-September-23-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/The-unique-threshold-bound-state-of-the-SU-N-BFSS-model-September-24-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/The-weak-inhomogeneous-Duffin-Schaeffer-conjecture-September-25-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/The-weak-pinned-planar-distance-theorem-September-23-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/Thermal-FK-Ising-interfaces-and-massive-SLE-October-5-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/Thomason-Model-Structures-in-Every-Strict-Higher-Dimension-September-25-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/Thompsons-group-F-is-nonamenable-September-23-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/Threshold-parallel-repetition-for-finite-dimensional-entangled-games-September-25-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/Topological-triviality-of-mu-constant-families-of-surface-singularities-September-24-2026/main.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/Tracial-projection-methods-and-uniform-property-Gamma-September-23-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/Translative-covering-densities-of-order-n-log-n-September-23-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/Tree-Constructions-for-the-l1-Distortion-of-Binary-Edit-Distance-September-27-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/Triangular-minimality-for-planar-Coulomb-renormalized-energy-September-23-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/Triple-ergodic-averages-with-distinct-integer-slopes-October-4-2026/triple-ergodic-distinct-slopes.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/Turn-Based-Stochastic-Mean-Payoff-Games-in-Deterministic-Quasipolynomial-Time-October-5-2026/stochastic-mean-payoff-games.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/two-limit-cycles-for-quintic-lienard-systems-September-24-2026/two-limit-cycles-for-quintic-lienard-systems-September-24-2026.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/Two-step-monodromy-of-special-quasi-projective-varieties-September-24-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/Ultraflat-real-Littlewood-polynomials-October-5-2026/ultraflat-real-littlewood-polynomials.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/Unbounded-Violations-of-the-Square-Root-Degree-Bound-September-26-2026/Unbounded-Violations-of-the-Square-Root-Degree-Bound-September-26-2026.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/Unconditional-time-lower-bounds-for-Weisfeiler-Leman-equivalence-September-25-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/Uniform-Bi-Holder-Transport-from-Weak-MTW-September-25-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/uniform-bounds-for-planar-polynomial-limit-cycles-September-24-2026/uniform-bounds-for-planar-polynomial-limit-cycles-September-24-2026.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/Uniform-Cartier-sections-for-Fano-type-contractions-September-25-2026/Uniform-Cartier-sections-for-Fano-type-contractions-September-25-2026.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/Uniform-computation-of-the-squared-logarithmic-k-server-bound-September-24-2026/Uniform-computation-of-the-squared-logarithmic-k-server-bound-September-24-2026.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/Uniform-effective-log-Iitaka-fibrations-for-fourfolds-September-26-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/Uniform-excess-charge-for-Coulomb-molecules-and-the-outer-radius-of-neutral-atoms-September-24-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/Uniform-exclusion-of-Landau-Siegel-zeros-October-1-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/Uniform-indices-for-semi-log-canonical-log-Calabi-Yau-pairs-October-5-2026/uniform-slc-index.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/Uniform-Interior-C1alpha-Estimates-for-Infinity-Harmonic-Functions-October-4-2026/interior-c1-infinity-harmonic.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/Uniform-log-Iitaka-fibrations-and-bounded-moduli-denominators-October-4-2026/uniform-log-iitaka.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/Uniform-marked-polygon-estimates-and-sharp-finite-bridge-moments-September-26-2026/main.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/Uniform-Matrix-Hitting-Points-in-Every-Positive-Characteristic-October-4-2026/uniform-matrix-hitting-points-positive-characteristic.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/Uniform-Permanence-in-Weakly-Reversible-Mass-Action-Systems-October-5-2026/permanence.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/Uniform-Pluricanonical-Iitaka-Fibrations-October-3-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/Uniform-quasipolynomial-time-trace-reconstruction-October-5-2026/uniform-trace-reconstruction.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/Uniform-real-Lipschitz-surfaces-on-the-triangular-lattice-September-25-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text

preprints/The-PCP-for-PPAD-conjecture-a-quasilinear-reduction-September-25-2026/README.md

@@ -0,0 +1,18 @@
+# [The PCP-for-PPAD conjecture: a quasilinear reduction](paper.pdf)
+
+**Author:** OpenAI
+
+**Date:** September 25, 2026
+
+## Citation
+
+```bibtex
+@misc{OAI:The-PCP-for-PPAD-conjecture-a-quasilinear-reduction-September-25-2026,
+  author = {{OpenAI}},
+  title = {{The PCP-for-PPAD conjecture: a quasilinear reduction}},
+  howpublished = {OpenAI Math Release preprint
+                  \href{https://github.com/openai/math/blob/main/preprints/The-PCP-for-PPAD-conjecture-a-quasilinear-reduction-September-25-2026/paper.pdf}{OAI:The-PCP-for-PPAD-conjecture-a-quasilinear-reduction-September-25-2026}},
+  year = {2026}
+}
+```
+

preprints/The-PCP-for-PPAD-conjecture-a-quasilinear-reduction-September-25-2026/build/main.tex

@@ -0,0 +1,78 @@
+\documentclass[11pt]{article}
+\makeatletter
+\def\input@path{{vendor/}}
+\makeatother
+\usepackage[utf8]{inputenc}
+\usepackage{amsmath,amssymb,amsthm,mathtools}
+\usepackage{array,booktabs,longtable,tabularx,enumitem}
+\usepackage[margin=1in]{geometry}
+\usepackage{microtype}
+\usepackage[colorlinks=true,linkcolor=blue,citecolor=blue,urlcolor=blue]{hyperref}
+\hypersetup{pdftitle={The PCP-for-PPAD conjecture: a quasilinear reduction},pdfauthor={OpenAI}}
+\setlength{\emergencystretch}{2em}
+\clubpenalty=10000
+\widowpenalty=10000
+\displaywidowpenalty=10000
+\setlist{itemsep=0.25em,topsep=0.5em,beginpenalty=10000}
+\setlist[itemize]{label=\(\bullet\)}
+\renewcommand{\paragraph}[1]{\par\medskip\noindent\textbf{#1}\enspace\ignorespaces}
+\numberwithin{equation}{section}
+\newtheorem{theorem}{Theorem}[section]
+\newtheorem{proposition}[theorem]{Proposition}
+\newtheorem{lemma}[theorem]{Lemma}
+\newtheorem{corollary}[theorem]{Corollary}
+\theoremstyle{definition}
+\newtheorem{definition}[theorem]{Definition}
+\newtheorem{example}[theorem]{Example}
+\theoremstyle{remark}
+\newtheorem{remark}[theorem]{Remark}
+\newcommand{\F}{\mathbb{F}}
+\newcommand{\E}{\mathbb{E}}
+\newcommand{\eps}{\varepsilon}
+\newcommand{\norm}[1]{\lVert #1\rVert}
+\newcommand{\abs}[1]{\lvert #1\rvert}
+\DeclareMathOperator{\dist}{dist}
+\DeclareMathOperator{\clip}{clip}
+\DeclareMathOperator{\supp}{supp}
+\DeclareMathOperator{\rank}{rank}
+\DeclareMathOperator{\poly}{poly}
+\DeclareMathOperator{\id}{id}
+\DeclareMathOperator{\Span}{span}
+\DeclareMathOperator{\im}{im}
+\DeclareMathOperator{\diag}{diag}
+\title{The PCP-for-PPAD conjecture:\\a quasilinear reduction}
+\author{OpenAI}
+\date{September 25, 2026}
+\begin{document}
+\maketitle
+\begin{abstract}
+We prove the quasilinear-size PCP-for-PPAD conjecture of Babichenko,
+Papadimitriou, and Rubinstein. There are fixed positive rational constants
+$\eps$ and $\delta$ and a deterministic polynomial-time reduction that
+transforms an End-of-Line instance of binary length $N$ into a generalized
+circuit of total binary length $N(\log N)^{O(1)}$. From any rational
+assignment of polynomial encoding length that $\eps$-satisfies all but a
+$\delta$ fraction of the gates, a solution to the original End-of-Line
+instance can be recovered in polynomial time, regardless of which gates
+fail. Such assignments always exist, with one fixed polynomial bound on
+their encoding length.
+\end{abstract}
+\tableofcontents
+\input{sections/01-introduction}
+\input{sections/02-macrograph}
+\input{sections/03-outer-codes}
+\input{sections/04-outer-programs}
+\input{sections/05-outer-representation}
+\input{sections/06-outer-composition}
+\input{sections/07-inner-algebra}
+\input{sections/08-inner-binary}
+\input{sections/09-inner-gadget}
+\input{sections/10-geometry}
+\input{sections/11-weak-nash}
+\appendix
+\input{sections/12-extensions}
+\phantomsection
+\addcontentsline{toc}{section}{References}
+\bibliographystyle{plain}
+\bibliography{references}
+\end{document}

preprints/The-PCP-for-PPAD-conjecture-a-quasilinear-reduction-September-25-2026/build/references.bib

@@ -0,0 +1,466 @@
+@inproceedings{BPR2016,
+  author = {Babichenko, Yakov and Papadimitriou, Christos H. and Rubinstein, Aviad},
+  title = {Can Almost Everybody be Almost Happy?},
+  booktitle = {Proceedings of the 2016 ACM Conference on Innovations in Theoretical Computer Science},
+  pages = {1--9},
+  publisher = {ACM},
+  year = {2016},
+  doi = {10.1145/2840728.2840731},
+  eprint = {1504.02411},
+  archivePrefix = {arXiv},
+  primaryClass = {cs.CC},
+  url = {https://arxiv.org/abs/1504.02411v4},
+  note = {Full version: \emph{Can Almost Everybody be Almost Happy? PCP for PPAD and the Inapproximability of Nash}, arXiv:1504.02411v4 (2025), Definitions 1 and 4, Conjecture 2, Theorems 2--4, and Appendix B. \url{https://arxiv.org/abs/1504.02411v4}}
+}
+
+@inproceedings{Fisher2026,
+  author = {Argyrios Deligkas and John Fearnley and Alexandros Hollender and Themistoklis Melissourgos},
+  title = {Fisher Markets with Approximately Optimal Bundles and the Need for a {PCP} Theorem for {PPAD}},
+  booktitle = {Proceedings of the 58th Annual ACM Symposium on Theory of Computing},
+  series = {STOC 2026},
+  year = {2026},
+  pages = {1049--1060},
+  publisher = {Association for Computing Machinery},
+  doi = {10.1145/3798129.3800817},
+  url = {https://arxiv.org/abs/2604.27276v2},
+  eprint = {2604.27276v2},
+  archivePrefix = {arXiv},
+  primaryClass = {cs.GT},
+  note = {Full version, arXiv:2604.27276v2 (18 June 2026), \url{https://arxiv.org/abs/2604.27276v2}. Definition 1, footnote 4, Theorem 2.2 and Appendix A are full-version locators; the conference manuscript numbers the equivalence Theorem 2.3}
+}
+
+@article{DFHM2024,
+  author = {Deligkas, Argyrios and Fearnley, John and Hollender, Alexandros and Melissourgos, Themistoklis},
+  title = {{Pure-Circuit}: Tight Inapproximability for {PPAD}},
+  journal = {Journal of the ACM},
+  volume = {71},
+  number = {5},
+  pages = {31:1--31:48},
+  year = {2024},
+  doi = {10.1145/3678166},
+  eprint = {2209.15149},
+  archivePrefix = {arXiv},
+  primaryClass = {cs.CC},
+  url = {https://arxiv.org/abs/2209.15149v3},
+  note = {Numbered references use arXiv:2209.15149v3 (2024), Section 4.1, Definition 3 and Theorem 4.1. \url{https://doi.org/10.1145/3678166}}
+}
+
+@inproceedings{DEL+2022,
+  author = {Dinur, Irit and Evra, Shai and Livne, Ron and Lubotzky, Alexander and Mozes, Shahar},
+  title = {Locally Testable Codes with Constant Rate, Distance, and Locality},
+  booktitle = {Proceedings of the 54th Annual ACM SIGACT Symposium on Theory of Computing},
+  pages = {357--374},
+  publisher = {ACM},
+  year = {2022},
+  doi = {10.1145/3519935.3520024},
+  eprint = {2111.04808},
+  archivePrefix = {arXiv},
+  primaryClass = {cs.IT},
+  url = {https://arxiv.org/abs/2111.04808v2},
+  note = {Full version arXiv:2111.04808v2 (2021), Theorems 1.1 and 4.5. \url{https://arxiv.org/abs/2111.04808v2}}
+}
+
+@misc{FS1995,
+  author = {Friedl, Katalin and Sudan, Madhu},
+  title = {Some Improvements to Total Degree Tests},
+  howpublished = {\url{https://arxiv.org/abs/1307.3975v1}},
+  year = {2013},
+  eprint = {1307.3975},
+  archivePrefix = {arXiv},
+  primaryClass = {cs.CC},
+  doi = {10.48550/arXiv.1307.3975},
+  url = {https://arxiv.org/abs/1307.3975v1},
+  note = {Corrected version, Theorem 13. Original paper in the Proceedings of the Third Israel Symposium on Theory of Computing and Systems, 1995, pp. 190--198}
+}
+
+@article{EJZ2010,
+  author = {Ershov, Mikhail and Jaikin-Zapirain, Andrei},
+  title = {Property {(T)} for Noncommutative Universal Lattices},
+  journal = {Inventiones Mathematicae},
+  volume = {179},
+  number = {2},
+  pages = {303--347},
+  year = {2010},
+  doi = {10.1007/s00222-009-0218-2},
+  eprint = {0809.4095},
+  archivePrefix = {arXiv},
+  primaryClass = {math.GR},
+  url = {https://arxiv.org/abs/0809.4095v2},
+  note = {Numbered references use arXiv:0809.4095v2 (2009): Theorem 1.1 and Section 6.1, equations (6.1)--(6.2), p. 34. \url{https://arxiv.org/abs/0809.4095v2}}
+}
+
+@article{Shalom1999,
+  author = {Shalom, Yehuda},
+  title = {Bounded Generation and {Kazhdan}'s Property {(T)}},
+  journal = {Publications Math\'ematiques de l'IH\'ES},
+  volume = {90},
+  pages = {145--168},
+  year = {1999},
+  doi = {10.1007/BF02698832},
+  url = {https://numdam.org/item/PMIHES_1999__90__145_0/},
+  note = {Theorem 3.4 and Corollary 3.5, pp. 160--161. \url{https://doi.org/10.1007/BF02698832}}
+}
+
+@misc{Helfgott2013,
+  author = {Helfgott, H. A.},
+  title = {The Ternary {Goldbach} Conjecture Is True},
+  howpublished = {\url{https://arxiv.org/abs/1312.7748v2}},
+  year = {2013},
+  eprint = {1312.7748},
+  archivePrefix = {arXiv},
+  primaryClass = {math.NT},
+  doi = {10.48550/arXiv.1312.7748},
+  url = {https://arxiv.org/abs/1312.7748v2},
+  note = {Version 2 (2014), Main Theorem, Section 1.1, p. 2; conclusion in Section 7.4}
+}
+
+@misc{Xylouris2011,
+  author = {Xylouris, Triantafyllos},
+  title = {On {Linnik}'s Constant},
+  howpublished = {\url{https://arxiv.org/abs/0906.2749v1}},
+  year = {2009},
+  eprint = {0906.2749},
+  archivePrefix = {arXiv},
+  primaryClass = {math.NT},
+  doi = {10.48550/arXiv.0906.2749},
+  url = {https://arxiv.org/abs/0906.2749v1},
+  note = {Diplomarbeit in German, titled \emph{{\"U}ber die Linniksche Konstante}; Theorem 1.1, p.~9}
+}
+
+@article{BG2008,
+  author = {Bourgain, Jean and Gamburd, Alex},
+  title = {Uniform Expansion Bounds for {Cayley} Graphs of {$\mathrm{SL}_2(\mathbb{F}_p)$}},
+  journal = {Annals of Mathematics},
+  volume = {167},
+  number = {2},
+  pages = {625--642},
+  year = {2008},
+  doi = {10.4007/annals.2008.167.625},
+  url = {https://annals.math.princeton.edu/2008/167-2/p07},
+  note = {Theorem 1 and the discussion on p. 626. \url{https://doi.org/10.4007/annals.2008.167.625}}
+}
+
+@article{MGGAFP,
+  author = {Karayel, Emin},
+  title = {Expander Graphs},
+  journal = {Archive of Formal Proofs},
+  year = {2023},
+  month = mar,
+  url = {https://isa-afp.org/entries/Expander_Graphs.html},
+  note = {Isabelle formalization. Theory \path{Expander_Graphs_MGG},
+    function \path{mgg_graph_step} and lemma \path{mgg_numerical_radius};
+    normalization in \path{Expander_Graphs_Definition} and
+    \texttt{expansionD2} in \path{Expander_Graphs_Eigenvalues}.
+    \href{https://isa-afp.org/browser_info/current/AFP/Expander_Graphs/Expander_Graphs_MGG.html}{Online formal development}.
+    Accessed September 25, 2026}
+}
+
+@article{Margulis1973,
+  author = {Margulis, G. A.},
+  title = {Explicit Constructions of Concentrators},
+  journal = {Problemy Peredachi Informatsii},
+  volume = {9},
+  number = {4},
+  pages = {71--80},
+  year = {1973},
+  url = {https://www.mathnet.ru/eng/ppi925},
+  note = {In Russian; English translation in \emph{Problems of Information
+    Transmission} 9(4), 325--332.
+    \href{https://www.mathnet.ru/eng/ppi925}{Journal archive}}
+}
+
+@article{GabberGalil1981,
+  author = {Gabber, O. and Galil, Z.},
+  title = {Explicit Constructions of Linear-Sized Superconcentrators},
+  journal = {Journal of Computer and System Sciences},
+  volume = {22},
+  number = {3},
+  pages = {407--420},
+  year = {1981},
+  doi = {10.1016/0022-0000(81)90040-4},
+  note = {\url{https://doi.org/10.1016/0022-0000(81)90040-4}}
+}
+
+@article{HLW2006,
+  author = {Hoory, Shlomo and Linial, Nathan and Wigderson, Avi},
+  title = {Expander Graphs and Their Applications},
+  journal = {Bulletin of the American Mathematical Society},
+  volume = {43},
+  number = {4},
+  pages = {439--561},
+  year = {2006},
+  doi = {10.1090/S0273-0979-06-01126-8},
+  url = {https://www.math.ias.edu/~avi/PUBLICATIONS/MYPAPERS/HLW06/hlw06.pdf},
+  note = {Survey; Construction 8.1 and Theorem 8.2, p. 503. \url{https://doi.org/10.1090/S0273-0979-06-01126-8}}
+}
+
+@article{Papadimitriou1994,
+  author = {Papadimitriou, Christos H.},
+  title = {On the Complexity of the Parity Argument and Other Inefficient Proofs of Existence},
+  journal = {Journal of Computer and System Sciences},
+  volume = {48},
+  number = {3},
+  pages = {498--532},
+  year = {1994},
+  doi = {10.1016/S0022-0000(05)80063-7},
+  url = {https://www.sciencedirect.com/science/article/pii/S0022000005800637},
+  note = {\url{https://doi.org/10.1016/S0022-0000(05)80063-7}}
+}
+
+@article{Rubinstein2018,
+  author = {Rubinstein, Aviad},
+  title = {Inapproximability of {Nash} Equilibrium},
+  journal = {SIAM Journal on Computing},
+  volume = {47},
+  number = {3},
+  pages = {917--959},
+  year = {2018},
+  doi = {10.1137/15M1039274},
+  eprint = {1405.3322},
+  archivePrefix = {arXiv},
+  primaryClass = {cs.GT},
+  url = {https://arxiv.org/abs/1405.3322v5},
+  note = {Numbered references use arXiv:1405.3322v5 (2016), Theorem 2, Sections 4--7, and Lemma 5. \url{https://arxiv.org/abs/1405.3322v5}}
+}
+
+@inproceedings{Rubinstein2016Settling,
+  author = {Rubinstein, Aviad},
+  title = {Settling the Complexity of Computing Approximate Two-Player {Nash} Equilibria},
+  booktitle = {Proceedings of the 57th IEEE Symposium on Foundations of Computer Science},
+  pages = {258--265},
+  publisher = {IEEE},
+  year = {2016},
+  doi = {10.1109/FOCS.2016.35},
+  eprint = {1606.04550},
+  archivePrefix = {arXiv},
+  primaryClass = {cs.CC},
+  url = {https://arxiv.org/abs/1606.04550v2},
+  note = {Numbered references use the full version, arXiv:1606.04550v2
+    (29 August 2016): Theorem 1.2, Sections 1.1 and 1.3,
+    Lemmas 3.3--3.4, and Sections 5--6.
+    \href{https://arxiv.org/pdf/1606.04550v2}{Full version}}
+}
+
+@misc{Golowich2026,
+  author = {Golowich, Noah},
+  title = {The Fine-Grained Complexity of Approximate {Nash} Equilibrium and Free Games},
+  year = {2026},
+  eprint = {2609.07136},
+  archivePrefix = {arXiv},
+  primaryClass = {cs.GT},
+  doi = {10.48550/arXiv.2609.07136},
+  howpublished = {\url{https://arxiv.org/abs/2609.07136v2}},
+  note = {Version 2, submitted 9 September 2026.
+    Theorem 1.1, Definition 2.1, and Section 2.2}
+}
+
+@article{Brouwer1911,
+  author = {Brouwer, L. E. J.},
+  title = {{\"U}ber {Abbildung} von {Mannigfaltigkeiten}},
+  journal = {Mathematische Annalen},
+  volume = {71},
+  pages = {97--115},
+  year = {1911},
+  doi = {10.1007/BF01456931},
+  url = {https://link.springer.com/article/10.1007/BF01456931},
+  note = {\url{https://doi.org/10.1007/BF01456931}}
+}
+
+@misc{SchuldenzuckerSeuken2019,
+  author = {Schuldenzucker, Steffen and Seuken, Sven},
+  title = {Monotonic and Non-Monotonic Solution Concepts for Generalized Circuits},
+  year = {2019},
+  eprint = {1907.12854},
+  archivePrefix = {arXiv},
+  primaryClass = {cs.CC},
+  doi = {10.48550/arXiv.1907.12854},
+  howpublished = {\url{https://arxiv.org/abs/1907.12854v1}},
+  note = {Version 1, 30 July 2019. Propositions 1--2, Lemma 1, and Sections 4 and 6}
+}
+
+@misc{CDT2007Preprint,
+  author = {Chen, Xi and Deng, Xiaotie and Teng, Shang-Hua},
+  title = {Settling the Complexity of Computing Two-Player {Nash} Equilibria},
+  year = {2007},
+  eprint = {0704.1678},
+  archivePrefix = {arXiv},
+  primaryClass = {cs.GT},
+  doi = {10.48550/arXiv.0704.1678},
+  howpublished = {\url{https://arxiv.org/abs/0704.1678v1}},
+  note = {Version 1. Theorem 1.1, Sections 2 and 4.2, and Lemma 5.6; locators refer to this preprint}
+}
+
+@article{DGP2009,
+  author = {Daskalakis, Constantinos and Goldberg, Paul W. and Papadimitriou, Christos H.},
+  title = {The Complexity of Computing a {Nash} Equilibrium},
+  journal = {SIAM Journal on Computing},
+  volume = {39},
+  number = {1},
+  pages = {195--259},
+  year = {2009},
+  doi = {10.1137/070699652},
+  url = {https://www.cs.ox.ac.uk/people/paul.goldberg/papers/SICOMP-dgp09.pdf},
+  note = {Section 5.1, Lemmas 5.2--5.5. \url{https://doi.org/10.1137/070699652}}
+}
+
+@article{BLR1993,
+  author = {Blum, Manuel and Luby, Michael and Rubinfeld, Ronitt},
+  title = {Self-Testing/Correcting with Applications to Numerical Problems},
+  journal = {Journal of Computer and System Sciences},
+  volume = {47},
+  number = {3},
+  pages = {549--595},
+  year = {1993},
+  doi = {10.1016/0022-0000(93)90044-W},
+  url = {https://people.csail.mit.edu/ronitt/papers/selftestcorrect.pdf},
+  note = {\href{https://people.csail.mit.edu/ronitt/papers/selftestcorrect.pdf}{Author manuscript}, subsection Generic Linear Self-Testing. \url{https://doi.org/10.1016/0022-0000(93)90044-W}}
+}
+
+@article{Benes1964Permutation,
+ author={Bene{\v{s}}, V. E.},
+ title={Permutation Groups, Complexes, and Rearrangeable Connecting Networks},
+ journal={Bell System Technical Journal}, volume={43}, number={4},
+ year={1964}, month=jul, pages={1619--1640},
+ doi={10.1002/j.1538-7305.1964.tb04102.x},
+ url={https://archive.org/download/bstj43-4-1619/bstj43-4-1619_text.pdf},
+ note={Lemma 3 and Theorems 2--3. \url{https://doi.org/10.1002/j.1538-7305.1964.tb04102.x}}
+}
+
+@article{GemmellSudan1992,
+ author={Gemmell, Peter and Sudan, Madhu},
+ title={Highly Resilient Correctors for Polynomials},
+ journal={Information Processing Letters}, volume={43}, number={4},
+ year={1992}, pages={169--174},
+ doi={10.1016/0020-0190(92)90195-2},
+ url={https://people.csail.mit.edu/madhu/papers/1992/gemmell.pdf},
+ note={\href{https://people.csail.mit.edu/madhu/papers/1992/gemmell.pdf}{Author manuscript}, Appendix A, p. 7, presents the Berlekamp--Welch decoder. \url{https://doi.org/10.1016/0020-0190(92)90195-2}}
+}
+
+@article{CooleyTukey1965,
+  author = {Cooley, James W. and Tukey, John W.},
+  title = {An Algorithm for the Machine Calculation of Complex {Fourier} Series},
+  journal = {Mathematics of Computation},
+  volume = {19},
+  number = {90},
+  pages = {297--301},
+  year = {1965},
+  doi = {10.1090/S0025-5718-1965-0178586-1},
+  url = {https://janmr.com/files/papers/cooley-tukey.pdf},
+  note = {Radix-two factorization, pp. 299--300, equations (14)--(22).
+    \href{https://janmr.com/files/papers/cooley-tukey.pdf}{Article scan}}
+}
+
+@article{Bennett1973,
+ author={Bennett, C. H.},
+ title={Logical Reversibility of Computation},
+ journal={IBM Journal of Research and Development}, volume={17},
+ year={1973}, month=nov, pages={525--532},
+ doi={10.1147/rd.176.0525},
+ url={https://www.cs.princeton.edu/courses/archive/fall04/cos576/papers/bennett73.html},
+ note={Reversible cleanup discussion, pp. 525--526. \url{https://doi.org/10.1147/rd.176.0525}}
+}
+
+@inproceedings{DGZ2023,
+  author = {Daskalakis, Constantinos and Golowich, Noah and Zhang, Kaiqing},
+  title = {The Complexity of {Markov} Equilibrium in Stochastic Games},
+  booktitle = {Proceedings of the Thirty Sixth Conference on Learning Theory},
+  series = {Proceedings of Machine Learning Research},
+  volume = {195},
+  pages = {4180--4234},
+  year = {2023},
+  publisher = {PMLR},
+  url = {https://proceedings.mlr.press/v195/daskalakis23a.html},
+  note = {Theorem 4, Conjecture 10 and Appendix D use the published numbering.
+    \href{https://proceedings.mlr.press/v195/daskalakis23a/daskalakis23a.pdf}{Published article}}
+}
+
+@misc{HZ2026,
+  author = {Braverman, Mark and Liu, Jingyi and Xue, Eric and Zhou, Chenghan},
+  title = {Hardness of Approximate {Hylland--Zeckhauser} Equilibria},
+  year = {2026},
+  eprint = {2606.00951},
+  archivePrefix = {arXiv},
+  primaryClass = {cs.GT},
+  howpublished = {\url{https://arxiv.org/abs/2606.00951v2}},
+  note = {Version 2, 9 June 2026. Definition 2.1, Table 1, Theorem 3.2,
+    and Lemmas 3.4 and 3.25}
+}
+
+@inproceedings{PZO2025,
+  author = {Park, Chanwoo and Zhang, Kaiqing and Ozdaglar, Asuman},
+  title = {Multi-Player Zero-Sum {Markov} Games with Networked Separable Interactions},
+  booktitle = {Advances in Neural Information Processing Systems},
+  volume = {36},
+  year = {2023},
+  eprint = {2307.09470v3},
+  archivePrefix = {arXiv},
+  primaryClass = {cs.GT},
+  url = {https://papers.nips.cc/paper_files/paper/2023/hash/75c411b0a06fa9e78f2a516b57b2ce62-Abstract-Conference.html},
+  note = {Numbered references use the full version, arXiv:2307.09470v3
+    (12 July 2025): Definition 1 and Appendix C, Case 1.
+    \href{https://arxiv.org/abs/2307.09470v3}{Full version}}
+}
+
+@misc{FKPP2026,
+  author = {Farina, Gabriele and Kontogiannis, Andreas and Panageas, Ioannis and Pollatos, Vasilis},
+  title = {The Complexity of Computing Coarse Correlated Equilibria in {Markov} Games with a Single Controller},
+  year = {2026},
+  eprint = {2607.10897v1},
+  archivePrefix = {arXiv},
+  primaryClass = {cs.GT},
+  howpublished = {\url{https://arxiv.org/abs/2607.10897v1}},
+  note = {Version 1. Definition 2.6, Theorem 5.5 and Remark 5.7}
+}
+
+@article{Budish2011,
+  author = {Budish, Eric},
+  title = {The Combinatorial Assignment Problem: Approximate Competitive Equilibrium from Equal Incomes},
+  journal = {Journal of Political Economy},
+  volume = {119},
+  number = {6},
+  pages = {1061--1103},
+  year = {2011},
+  doi = {10.1086/664613},
+  url = {https://ericbudish.org/wp-content/uploads/combinatorial_assignment_problem.pdf},
+  note = {Definition 1 and Theorem 1, p.~1069.
+    \href{https://ericbudish.org/wp-content/uploads/combinatorial_assignment_problem.pdf}{Published article}}
+}
+
+@article{OPR2014,
+  author = {Othman, Abraham and Papadimitriou, Christos H. and Rubinstein, Aviad},
+  title = {The Complexity of Fairness through Equilibrium},
+  journal = {ACM Transactions on Economics and Computation},
+  volume = {4},
+  number = {4},
+  pages = {20:1--20:19},
+  year = {2016},
+  eprint = {1312.6249v3},
+  archivePrefix = {arXiv},
+  primaryClass = {cs.GT},
+  note = {The Appendix A locator uses arXiv:1312.6249v3
+    (30 September 2014).
+    \href{https://arxiv.org/pdf/1312.6249v3}{Full version}}
+}
+
+@misc{ABSsw26,
+  author = {Amireddy, Prashanth and Behera, Amik Raj and Srinivasan, Srikanth and Sudan, Madhu and Willumsgaard, Sophus Valentin},
+  title = {A Simple Algebraic Proof of the {PCP} Theorem},
+  year = {2026},
+  eprint = {2608.17429},
+  archivePrefix = {arXiv},
+  url = {https://arxiv.org/abs/2608.17429v1},
+  howpublished = {\url{https://arxiv.org/abs/2608.17429v1}},
+  note = {Section 3.1, Definition 3.1 and Claim 3.2}
+}
+
+@article{Zemor2001,
+  author = {Z{\'e}mor, Gilles},
+  title = {On Expander Codes},
+  journal = {IEEE Transactions on Information Theory},
+  volume = {47},
+  number = {2},
+  pages = {835--837},
+  year = {2001},
+  url = {https://courses.grainger.illinois.edu/cs598sgt/sp2015/zemor.pdf},
+  note = {Section III-A, Lemma 5 and Theorem 6. \url{https://courses.grainger.illinois.edu/cs598sgt/sp2015/zemor.pdf}}
+}

preprints/The-PCP-for-PPAD-conjecture-a-quasilinear-reduction-September-25-2026/build/sections/01-introduction.tex

@@ -0,0 +1,393 @@
+\section{Introduction and the exact search relation}\label{sec:introduction}
+
+The class PPAD organizes total search problems whose existence proofs are
+based on the parity of endpoints in a directed graph~\cite{Papadimitriou1994}. Its canonical problem,
+End-of-Line, asks for an endpoint other than a specified source, when the
+graph is described by predecessor and successor circuits. Generalized
+circuits provide another useful language for this class: a solution assigns
+real values in $[0,1]$ to wires and approximately satisfies local arithmetic,
+comparison, and Boolean constraints. The generalized-circuit intermediary
+was developed by Chen, Deng, and Teng~\cite[Section~4.2]{CDT2007Preprint};
+arithmetic, comparison, and Boolean gadgets also underlie the equilibrium
+reductions of Daskalakis, Goldberg, and
+Papadimitriou~\cite[Section~5.1]{DGP2009}. Rubinstein established
+constant-error hardness with every gate required to be
+satisfied~\cite[Theorem~2]{Rubinstein2018}. These results motivate the
+circuit language used here, with the precise normalization and thresholds
+specified below.
+
+Babichenko, Papadimitriou, and Rubinstein formulated the PCP-for-PPAD
+conjecture as a quasilinear reduction from End-of-Line to generalized
+circuits in which both the allowed numerical error and the allowed fraction
+of violated gates are positive constants~\cite[Conjecture~2]{BPR2016}.
+The failure fraction is the essential robustness requirement: the decoder
+must succeed even if an adversary chooses which constraints to violate.
+The quasilinear size requirement is a separate restriction on the reduction.
+Ordinary polynomial-time PPAD-hardness, even with constant numerical error,
+does not provide it. For example, hardness for every numerical error below $1/10$ is known
+for the standard generalized-circuit problem with every gate required to
+be satisfied~\cite[Theorem~4.1]{DFHM2024}. Subsequent uses of PCP-for-PPAD also
+consider the weaker conjecture asserting ordinary PPAD-hardness with a
+positive failure fraction; the distinction from the original quasilinear
+formulation is explicit in~\cite[Section~2.2.2]{Fisher2026}.
+
+Rubinstein's 2016 work combined encoded-path geometry, a locally
+evolving version of End-of-Line, and a holographic proof system whose
+certificates can be constructed and decoded through restricted
+access~\cite[Sections~3, 5, and~6]{Rubinstein2016Settling}. It established
+quasipolynomial hardness for constant-error two-player Nash under the
+exponential-time hypothesis for PPAD, while leaving the PCP-for-PPAD
+conjecture open and identifying composition as a remaining
+question~\cite[Theorem~1.2 and Section~1.3]{Rubinstein2016Settling}.
+More recently, Golowich derives sharper two-player Nash lower bounds
+for vanishing approximation error under PCP for PPAD together with
+the exponential-time hypothesis for PPAD~\cite[Theorem~1.1]{Golowich2026}.
+These time lower bounds retain their additional exponential-time
+hypothesis; the result here concerns the conjectured reduction itself.
+
+We give a positive resolution of the quasilinear PCP-for-PPAD conjecture.
+Our size bound counts the complete finite output, including connection
+indices, every auxiliary gate, and the bits of all rational parameters.
+The gate conventions below are the standard bounded-rational conventions
+of~\cite{DFHM2024,Fisher2026}; the totality and witness-size statement is
+proved explicitly. When restricted to the nearly linear predecessor and
+successor circuits in the original formulation, the theorem gives the
+conjectured quasilinear reduction there as well.\footnote{The original
+Definition~1 of~\cite{BPR2016} allows $x=0^n$ in its first solution
+disjunct $P(S(x))\ne x$. If $P(S(0^n))\ne0^n$, this is an immediately
+computable solution. Otherwise the instance satisfies the additional
+source promise used below. Thus this normalization requires only circuit
+evaluation and a trivial branch of the reduction: output a fixed one-gate
+Constant circuit and have the decoder return $0^n$.}
+
+\subsection{The source and target problems}
+
+We fix conventional explicit binary encodings of circuit gate lists,
+connection indices, input and output lists, and rational parameters.
+Integers are encoded in binary with delimiters. All size bounds concern
+these complete encodings. Changing between the usual such explicit
+encodings changes only constant or logarithmic factors.
+
+\begin{definition}[End-of-Line]\label{def:eol}
+An instance $I$ consists of two explicitly listed, acyclic, bounded-fanin
+Boolean circuits
+\[
+ S,P:\{0,1\}^n\longrightarrow\{0,1\}^n,\qquad n\geq1,
+\]
+using AND, OR, and NOT gates, with unrestricted fan-out, and satisfying
+\[
+ P(0^n)=0^n,\qquad S(0^n)\ne0^n,\qquad P(S(0^n))=0^n.
+\]
+A solution is a string $v\ne0^n$ such that
+\begin{equation}\label{eq:end-of-line-solution}
+ P(S(v))\ne v\quad\text{or}\quad S(P(v))\ne v.
+\end{equation}
+The input length $N\geq2$ includes the complete descriptions of both
+circuits and their input and output lists.
+\end{definition}
+
+\begin{definition}[Generalized circuit]\label{def:gcircuit}
+A generalized circuit $G=(V,T)$ has an explicitly listed finite set of
+nodes $V$ and a nonempty set of gates $T$. Each gate has one output and at
+most two inputs. A node is the output of at most one gate. Directed cycles
+and unrestricted fan-out are permitted. Every gate parameter $\alpha$ is
+a binary-encoded rational in $[0,1]$.
+
+For an assignment $x:V\to[0,1]$, write $x_a,x_b$ for input values and
+$x_z$ for the output. The five arithmetic gate types are
+$\eps$-satisfied when the corresponding condition holds:
+\begin{align*}
+ \mathrm{Constant}(\alpha;z)&:&\quad \abs{x_z-\alpha}&\leq\eps,\\
+ \mathrm{Scale}(\alpha;a;z)&:& \abs{x_z-\alpha x_a}&\leq\eps,\\
+ \mathrm{Copy}(a;z)&:& \abs{x_z-x_a}&\leq\eps,\\
+ \mathrm{Add}(a,b;z)&:& \abs{x_z-\min\{x_a+x_b,1\}}&\leq\eps,\\
+ \mathrm{Subtract}(a,b;z)&:& \abs{x_z-\max\{x_a-x_b,0\}}&\leq\eps.
+\end{align*}
+The comparison gate $\mathrm{Less}(a,b;z)$ has the two implications
+\[
+ x_a<x_b-\eps\ \Longrightarrow\ x_z\geq1-\eps,
+ \qquad
+ x_a>x_b+\eps\ \Longrightarrow\ x_z\leq\eps.
+\]
+The Boolean gates have the following implications:
+\begin{align*}
+ \mathrm{Or}(a,b;z):\qquad
+ (x_a\geq1-\eps\ \text{or}\ x_b\geq1-\eps)&\Longrightarrow x_z\geq1-\eps,\\
+ (x_a\leq\eps\ \text{and}\ x_b\leq\eps)&\Longrightarrow x_z\leq\eps;\\[2pt]
+ \mathrm{And}(a,b;z):\qquad
+ (x_a\geq1-\eps\ \text{and}\ x_b\geq1-\eps)&\Longrightarrow x_z\geq1-\eps,\\
+ (x_a\leq\eps\ \text{or}\ x_b\leq\eps)&\Longrightarrow x_z\leq\eps;\\[2pt]
+ \mathrm{Not}(a;z):\qquad
+ x_a\leq\eps&\Longrightarrow x_z\geq1-\eps,\\
+ x_a\geq1-\eps&\Longrightarrow x_z\leq\eps.
+\end{align*}
+For these four gate types, every applicable implication must hold. If no
+antecedent holds, any output in $[0,1]$ is permitted. In particular, the
+comparison thresholds are strict and the Boolean thresholds are weak.
+
+Let $M$ be the complete binary encoding length of $G$. For fixed
+$\eps,\delta>0$ and a fixed polynomial $p$, an acceptable assignment is
+rational, has total binary encoding length at most $p(M)$, and
+$\eps$-satisfies at least $(1-\delta)|T|$ gates. Each gate has equal weight;
+at most $\lfloor\delta|T|\rfloor$ gates may fail. Failed gates may violate
+their relations arbitrarily, but all their nodes still have one globally
+consistent value in $[0,1]$.
+\end{definition}
+
+The Boolean predicates make the error parameter nonmonotone: increasing
+$\eps$ can activate an antecedent that was previously vacuous. This issue
+is analyzed by Schuldenzucker and Seuken~\cite[Proposition~1]{SchuldenzuckerSeuken2019}.
+Our gate arguments use one actual tolerance throughout. Later upper bounds
+on that tolerance ensure explicit margins; they do not reclassify a
+satisfied Boolean gate at a larger tolerance.
+
+\subsection{Main theorem}
+
+\begin{samepage}
+\begin{theorem}[Quasilinear PCP for PPAD]\label{thm:main}
+There are fixed rational constants
+\[
+ 0<\eps<\tfrac1{10},\qquad 0<\delta<1,
+\]
+a fixed positive integer $A$, a fixed nonnegative integer $b$, a fixed
+polynomial $p$ with nonnegative integer coefficients, and deterministic
+algorithms $R,D$ with the following properties.
+\begin{enumerate}[label=\textup{(\roman*)}]
+\item Every generalized circuit in Definition~\ref{def:gcircuit} has an
+acceptable rational assignment of encoding length at most $p(M)$.
+\item For every End-of-Line instance $I$ of length $N$, the output $R(I)$
+is a generalized circuit whose complete encoding length satisfies
+\[
+ M\leq A N\bigl\lceil\log_2(N+2)\bigr\rceil^b.
+\]
+\item The construction $R$ runs in time polynomial in $N$, with one fixed
+polynomial bound.
+\item For every acceptable assignment $x$ of $R(I)$, the algorithm $D(I,x)$
+returns a solution of $I$ in time polynomial in $N$ plus the encoding
+length of $x$, with one fixed polynomial bound.
+\end{enumerate}
+All constants, polynomials, and algorithms are chosen once. They use no
+advice depending on the input length and no randomized construction.
+\end{theorem}
+\end{samepage}
+
+The conclusion about $D$ is universal over the locations and values of
+the failed gates. In fact, the soundness argument applies to arbitrary
+real assignments meeting the gate-failure budget; rationality is needed
+for the finite search relation and its computational decoder. The totality
+proof supplies an assignment satisfying every gate on a fixed rational
+mesh. Neither tolerance tends to zero with $N$.
+
+\subsection{Proof strategy: encoded paths and robust local computation}
+
+We begin with the geometric target of the reduction. Replace the
+End-of-Line graph by a locally evolving directed graph with a known
+source, no isolated vertices, and the same endpoint-decoding property.
+Represent each vertex $w$ by a separated binary word $E_w\in\{0,1\}^m$.
+In the box $[-1,2]^{4m}$, viewed as four blocks of $m$ coordinates,
+each edge $u\to v$ traces
+a polygon that first changes a copy of $E_u$ to $E_v$, changes a repeated
+control bit, updates the other copy, and resets the control bit. Rounded
+corners join these polygons into paths and cycles. An additional segment
+enters the known source, and a separate field prescription handles its
+far end. Every other path endpoint encodes an End-of-Line answer.
+
+Section~\ref{sec:geometry} constructs a bounded displacement field $v$
+on this box. Close to a path it uses the directed tangent and the inward
+normal; farther away it passes to a fixed direction. For a sufficiently
+small fixed step $\beta>0$, the clipped move
+$\clip(X+\beta v(X))-X$ has norm at least a fixed positive multiple of
+$\beta$ outside a fixed neighborhood of the desired endpoints. Here
+clipping keeps each coordinate in the box, and the norm is normalized
+by the word length. Thus an accurate enough approximate fixed point
+identifies an endpoint. The proof must also provide local numerical
+access to the field: it cannot inspect the entire encoded graph when
+computing one coordinate of $v$.
+
+This use of encoded paths and restricted access has a close predecessor
+in Rubinstein's holographic proof system~\cite[Sections~3, 5, and~6]{Rubinstein2016Settling}.
+The present reduction needs two further properties simultaneously:
+quasilinear total storage and robustness to an arbitrary fixed fraction
+of failed gates. The two scales of the construction address these
+requirements as follows.
+
+\paragraph{A quasilinear outer word.}
+Sections~\ref{sec:macrograph}--\ref{sec:outer-representation} construct
+canonical representations
+\[
+ R_w=(s_k)_{k\in\Gamma},\qquad
+ s_k\in\{0,1\}^L,\qquad
+ |\Gamma|=N(\log N)^{O(1)},\qquad L=(\log N)^{O(1)}.
+\]
+A bounded number of symbol positions suffices to test local validity,
+propose a neighboring symbol, or perform one correction step. Each such
+operation has Boolean complexity polynomial in $L$, with an exponent
+fixed before the later correction accuracy. Distinct valid words have
+constant symbol distance, and a word within a fixed symbol-error radius
+can be decoded globally in polynomial time.
+
+Begin with the sparse transcript graph of Section~\ref{sec:macrograph},
+whose vertices record partial computations of the predecessor and successor
+circuits and whose edges change one tape position and its occupation bit.
+Transcript validity is expressed by bounded-degree polynomial equations in
+the tape entries. Sampling an individual equation could miss one false
+equation. Sections~\ref{sec:outer-codes} and~\ref{sec:outer-programs}
+instead encode polynomial remainders recording the equation residuals,
+and retain certificates for the intermediate products and linear
+computations that produce them. On exact codewords, a nonzero remainder
+differs from zero at a constant fraction of positions. The code's commuting
+row and column constraints also permit whole-row or whole-column comparisons
+that give a constant gap for an incorrect certificate identity. All stored
+components are tested together on one fixed list of neighboring positions,
+so more certificates enlarge a symbol without diluting the gap.
+
+One tape edit can change all these encoded arrays. Before making an allowed
+edit, Section~\ref{sec:outer-representation} prepares each retained array's
+finite difference: its value on the edited tape minus its value on the old
+tape. Certificates introduced by this preparation need their own increments
+as well. Each new increment has lower assigned polynomial degree in the
+logical tape entries, so this recursion has bounded depth. Once preparation
+is complete, adding the increments updates the whole retained tuple to its
+canonical value on the new tape; the preparation can then be reversed.
+
+The remaining storage issue is the long linear computation needed to prepare
+an increment. It is carried out one small group of Fourier frequencies at a
+time: each such orbit is preserved by the encoder and has only
+polylogarithmically many frequencies. A clock records progress and scratch
+space is reused. Thus the preparation's elapsed time lengthens the
+represented path rather than the data stored at a vertex. The resulting
+words are the locally evolving $R_w$ above.
+
+\paragraph{One fixed binary port.}
+A polynomial-size robust construction applied directly to an $N$-bit
+vertex would not give the required size. Instead we apply it only to
+the $L$-bit outer symbols. The inner construction supplies one injective
+encoding
+\[
+ \mathcal D:\{0,1\}^L\longrightarrow\{0,1\}^{S_0},
+ \qquad S_0=\operatorname{poly}(L),
+ \qquad E_w=(\mathcal D(s_k))_{k\in\Gamma}.
+\]
+Its encoding and positive correction radius are chosen before the
+number of symbols inspected by a local function. From that fixed radius,
+it supports every prescribed fixed average accuracy for requests with
+bounded address density. More outer correction rounds may enlarge the number
+of symbols inspected, and hence the number of input ports in the resulting
+local computation. More inner correction rounds improve its requested
+bit accuracy. These choices may enlarge constant factors, but they do
+not change $\mathcal D$ or its radius. All polynomial factors in $L$
+remain fixed powers of $\log N$.
+
+Sections~\ref{sec:inner-algebra}--\ref{sec:inner-gadget} construct this
+port and its robust decision gadget independently of the outer
+reduction. For a fixed tuple $s=(s_1,\ldots,s_k)$ of $L$-bit inputs
+to a Boolean function $f$, the inner graph is one computation path with
+blank work at its source and output $f(s)$ at its sink. Polynomial charts encode its tapes and
+certificates. Their dependencies are kept separate for each read-only
+input, including throughout the finite-difference closure. Consequently
+the input-dependent parts of the source and its first successor have
+the same formulas in every later invocation. A one-input template at
+these two labels defines $\mathcal D(s_i)$ for one input $s_i$; its bits then supply the
+source data for an actual multi-input computation.
+
+The binary representation in Section~\ref{sec:inner-binary} adds
+point tables, line tables, and tables for the required products.
+Correction keeps both addresses in each product request and preserves
+the required bound on their joint distribution. Its error bound is averaged over requested coordinates;
+arbitrarily accurate correction at every individual coordinate is
+neither asserted nor needed. At the inner scale, the geometric
+construction forces a feedback state near the unique sink, where a
+positive fraction of encoded coordinates reveal $f(s)$. These become the
+many designated outputs of the decision gadget.
+
+\paragraph{From local decisions to an endpoint under gate failures.}
+Section~\ref{sec:outer-composition} uses the outer tester and correction
+routines to supply the field's local information. Two rounded blocks
+serve as possible vertex encodings. On good validation decisions, each
+accepted seed has a unique reference vertex; corrected calls supply its incident
+flags and the requested bits of its own and neighboring encodings.
+The geometric estimator reserves a bounded list of possible pieces and
+estimates their statistics before selecting a numerical winner.
+This order makes average bit accuracy sufficient even when the selected
+piece depends on the output coordinate.
+
+The generalized circuit has many physical replicas of each geometric
+coordinate. Write $X_{ir}$ for replica $r$ of component $i$, and
+$\bar X_i$ for their mean. A fixed-degree expander average $A$ couples
+the replicas. Each output is produced by a circuit that realizes the
+approximate relation
+\[
+ X_{ir}\approx\clip\bigl((AX)_{ir}
+                       +\beta\widehat v_{ir}(\bar X)\bigr),
+\]
+where $\widehat v$ is the bounded local field estimator. First, boundedness
+of the update and the expander gap force the replicas close to their
+means, without assuming that any decision is correct. This synchronization
+then bounds the number of reads far from their means. Randomized
+threshold offsets control ambiguous comparisons, and the gate-failure
+budget controls the circuits containing a failed gate.
+The randomness describes the sampling analysis: the printed circuit assigns
+replicas a deterministically enumerated list of fixed tapes of size polynomial
+in $L$.
+
+These charges apply to the complete wired computation. All hypothetical
+answer branches are allocated before routing, and each physical read
+occurrence is routed separately with bounded load. Expensive Boolean
+decision gadgets are shared by groups of output replicas. Their many
+designated outputs ensure that failures concentrated on extraction
+wires still spoil only a controlled fraction of the group. On reliable
+outputs the update approximates the true field; on unreliable outputs
+whose gates are satisfied, its bounded size still gives an error
+proportional to $\beta$, apart from the numerical gate tolerance. An estimate that keeps clipping inside the
+replica contraction argument then makes the mean an approximate fixed
+point. The displacement lower bound forces $\bar X$ near an endpoint.
+Rounding its first block and applying the global binary and outer
+decoders yields the required End-of-Line answer.
+
+The same synchronization, error charging, and clipped-mean argument is
+used inside the decision gadget. At that scale every output has its own
+bounded estimator circuit; the additional sharing of decision gadgets
+occurs only in the outer composition. In both uses, a fixed hypothetical
+answer branch is analyzed under the original random-tape law, rather
+than by conditioning on that branch being executed.
+
+\paragraph{Organization and imported inputs.}
+Section~\ref{sec:outer-composition} assembles the main theorem from
+the outer representation, the port interface stated there, and the
+independent geometric interface. It proves universal decoding, the
+complete binary size bound, and rational-witness totality.
+Sections~\ref{sec:inner-algebra}--\ref{sec:inner-gadget} prove the port
+interface, and Section~\ref{sec:geometry} proves both the displacement
+bound and finite numerical access. Section~\ref{sec:weak-nash} gives
+the consequences described below. Appendix~\ref{sec:extensions}
+collects the separate full-macrograph and unrestricted-product variants;
+these are not prerequisites for the fixed-port composition.
+
+The outer code uses ternary Goldbach, fixed-generator prime-quotient
+expansion, and an effective least-prime bound
+\cite{Helfgott2013,BG2008,Xylouris2011}. Its simultaneous systematic
+encoders, matrix twists, and joint correction guarantees are proved
+here; a general locally testable code theorem such as~\cite{DEL+2022}
+does not by itself supply this combination. The inner binary construction
+uses the corrected scalar line test and two group-theoretic inputs
+\cite{FS1995,EJZ2010,Shalom1999}; its vector replacement, affine
+expansion, and average correction are proved here. The all-modulus
+Margulis--Gabber--Galil bound supplies the local and replica walks
+\cite{MGGAFP}. The inner fields and groups are introduced afresh and
+are distinct from those of the outer construction.
+
+\subsection{Consequences for games and markets}
+
+Section~\ref{sec:weak-nash} applies the theorem to the conditional
+reductions for weak approximate Nash and relative bimatrix Nash
+\cite{BPR2016}, course allocation~\cite{BPR2016}, markets with
+approximately optimal bundles~\cite{Fisher2026}, stationary equilibria
+in discounted games~\cite{DGZ2023,PZO2025,FKPP2026}, and approximate
+Hylland--Zeckhauser allocation~\cite{HZ2026}. For each target we state
+the precise approximation and encoding conventions before its
+consequence. The cited reductions provide the transfers to these
+problems. We supply the needed gate-basis and fan-out conversions, track
+their fixed error and failure losses, and verify the rational witness
+and decoder bounds. These consequences use fixed-gap circuit hardness;
+the quasilinear complete-output bound is the additional assertion of
+Theorem~\ref{thm:main}.

preprints/The-PCP-for-PPAD-conjecture-a-quasilinear-reduction-September-25-2026/build/sections/02-macrograph.tex

@@ -0,0 +1,436 @@
+\section{A sparse macrograph for End-of-Line}\label{sec:macrograph}
+
+We first replace the circuit transitions by a graph in which one edge
+changes at most one tape bit and its occupation bit.  The graph is
+specified by short labels and local transition rules; its vertices are
+not enumerated.  The use of a clock to expose sparse circuit updates
+has a close precedent in Rubinstein's LocalEndOfALine
+construction~\cite[Sections~5--5.1]{Rubinstein2016Settling}, where a
+counter schedules activation and deactivation of individual wires.
+We give the transcript, reciprocal proposals, and bounded-incidence
+validity description needed here explicitly.
+
+Throughout the outer construction, let $n_0$ be the
+number of input bits of the End-of-Line instance and let $N$ be its
+complete binary encoding length.  We use the promises
+\begin{equation}\label{macro:promises}
+ P(0^{n_0})=0^{n_0},\qquad S(0^{n_0})\ne0^{n_0},\qquad
+ P(S(0^{n_0}))=0^{n_0}.
+\end{equation}
+Thus a nonzero string $y$ is an answer if
+\begin{equation}\label{macro:answer}
+ P(S(y))\ne y\quad\hbox{or}\quad S(P(y))\ne y.
+\end{equation}
+
+\subsection{Sparse transcripts and their two orders}\label{macro:transcripts}
+
+Normalize both circuits to gates of fan-in at most two.  In particular,
+replace any larger AND or OR by a binary tree.  Give every input and
+every gate a distinct tape position.  For each listed output, introduce
+a separate terminal copy gate, even if several listed outputs originally
+name the same wire.  Replace the outgoing uses of a wire by a binary
+tree of copy gates, including its uses as feeds to terminal copies.
+A wire with $d$ uses needs $O(d)$ such gates.  The total number of
+positions is therefore bounded by a constant times the number of gates,
+connections, and input and output list entries in the original instance,
+and hence is $O(N)$.
+
+These changes preserve $S$ and $P$.  They also make every tape position
+an operand of only a bounded number of internal gate rules.  Choose a
+topological order of the nonterminal gates of each circuit, followed by
+all its terminal output copies.  Such an order exists because those
+copies have no internal consumers: any branching needed for other uses
+was placed before them.  Copy gates here are auxiliary transcript rules
+and need not belong to the source circuit's original basis.
+
+Pad the tape with dummy positions to a power of two $h=2^\ell$.
+An arbitrary fixed lower cutoff on $h$ may be imposed, and still
+$h=O(N)$.  Write $[h]=\{0,\ldots,h-1\}$.  On these same positions define
+the two permutations
+\begin{align*}
+ \pi_{\mathrm f}:\;&S\text{ inputs},\ S\text{ gates},\
+ P\text{ inputs},\ P\text{ gates},\ \text{dummies},\\
+ \pi_{\mathrm r}:\;&P\text{ inputs},\ P\text{ gates},\
+ S\text{ inputs},\ S\text{ gates},\ \text{dummies},
+\end{align*}
+using the chosen gate order inside each block.  For a content array
+$D\in\{0,1\}^{[h]}$, let $\operatorname{pref}_{\pi,k}(D)$ agree with
+$D$ on $\pi(0),\ldots,\pi(k-1)$ and be zero elsewhere.  Its occupation
+array is the indicator of these $k$ positions.  Empty and full tapes
+have occupation arrays $\mathbf 0$ and $\mathbf 1$, respectively.
+The content and occupation arrays are separate: an occupied position
+may contain zero.
+
+For a full content array $D$, the notation
+$\operatorname{in}_S(D)$, $\operatorname{out}_S(D)$,
+$\operatorname{in}_P(D)$, and $\operatorname{out}_P(D)$ denotes the
+vectors at the designated input or terminal output positions.  A
+\emph{valid full transcript} $Q_x$ computes $S$ on input $x$, copies
+its output to the inputs of $P$, computes $P$, and satisfies
+\begin{equation}\label{macro:full-transcript}
+ \operatorname{in}_S(Q_x)=x,\qquad
+ \operatorname{in}_P(Q_x)=\operatorname{out}_S(Q_x),\qquad
+ \operatorname{out}_P(Q_x)=x.
+\end{equation}
+Its dummy positions are zero.  Consequently $Q_x$ exists precisely
+when $P(S(x))=x$, and is unique when it exists.
+
+For every $y\in\{0,1\}^{n_0}$, define two complete computation templates,
+without imposing any terminal comparisons.  The template $U_y$ first
+computes $S(y)$ and then $P(S(y))$; the template $V_y$ first computes
+$P(y)$ and then $S(P(y))$.  Both occupy the same tape coordinates and
+have zero dummies.  In $V_y$, the $P$ inputs are $y$ and the $S$ inputs
+are $P(y)$.  Thus $U_y$ is a valid full transcript exactly when
+$P(S(y))=y$, in which case it is $Q_y$; and $V_y$ is a valid full
+transcript exactly when $S(P(y))=y$, in which case it is $Q_{P(y)}$.
+
+\begin{definition}[Computed prefixes]\label{macro:prefixes}
+A \emph{forward prefix of length $k$ with parameter $y$} has the first
+$k$ positions in $\pi_{\mathrm f}$ occupied.  At occupied positions it
+obeys these rules: $S$ inputs copy the corresponding bits of $y$;
+$P$ inputs copy the corresponding designated $S$ outputs; every gate
+obeys its Boolean or copy rule; and dummies are zero.  In addition,
+each occupied designated $P$ output equals the corresponding bit of
+$y$.  Unoccupied contents are zero.
+
+A \emph{reconstruction prefix of length $k$ from $y$} uses
+$\pi_{\mathrm r}$.  Its occupied $P$ inputs copy $y$, its occupied
+$S$ inputs copy the designated $P$ outputs, and its occupied gates
+and dummies obey the same rules as above.  Each occupied designated
+$S$ output must additionally equal the corresponding bit of $y$.
+Unoccupied contents are again zero.
+\end{definition}
+
+Every operand in a prefix rule is either an earlier occupied position
+or a specified bit of $y$.  In particular, a designated comparison is
+an extra test on the newly computed terminal copy, not a substitute
+for its computation rule.
+
+\begin{lemma}[Unique prefixes and their first obstruction]
+\label{macro:prefix-uniqueness}
+For a fixed $y$, the valid forward prefixes exist uniquely for an
+initial interval of lengths $0\le k\le k_{\mathrm f}(y)$ and equal
+$\operatorname{pref}_{\pi_{\mathrm f},k}(U_y)$.  The valid
+reconstruction prefixes likewise exist uniquely for
+$0\le k\le k_{\mathrm r}(y)$ and equal
+$\operatorname{pref}_{\pi_{\mathrm r},k}(V_y)$.  Moreover,
+\begin{align}
+ k_{\mathrm f}(y)=h&\quad\Longleftrightarrow\quad P(S(y))=y,
+ \label{macro:forward-obstruction}\\
+ k_{\mathrm r}(y)=h&\quad\Longleftrightarrow\quad S(P(y))=y.
+ \label{macro:reverse-obstruction}
+\end{align}
+If a maximal length is less than $h$, the next position is a designated
+output at which the additional comparison fails.
+\end{lemma}
+
+\begin{proof}
+Induct along the relevant order.  Input copies, gates, and dummies
+uniquely determine the next content bit from data already determined.
+These are exactly the bits of $U_y$ or $V_y$.  The only possible
+obstruction is an additional terminal comparison.  Shortening a valid
+prefix removes requirements without changing retained bits, so the
+valid lengths form an initial interval.  At length $h$, all terminal
+comparisons have been imposed, giving the two equivalences.
+\end{proof}
+
+\subsection{The cyclic schedule and reciprocal proposals}
+\label{macro:schedule}
+
+A logical tape vector is
+\[
+ b=(A,E^A,B,E^B)\in\{0,1\}^{4h},
+\]
+where $A,B$ are contents and $E^A,E^B$ are occupations.  The schedule
+uses four phases.  To see their effect when all computations succeed,
+suppose $Q_x$ and $Q_y$ exist and $y=S(x)$.  Starting with $A=Q_x$
+and $B$ empty, it fills $B$ with $Q_y$, erases $A$ by reversing
+reconstruction from $y$, copies $B$ into $A$, and erases $B$.
+The resulting tapes are $A=Q_y$ and $B$ empty.  The identity
+$P(y)=x$, which follows from validity of $Q_x$, is what makes the
+reconstruction recover the old tape.
+
+For intermediate prefixes, including those at a failed terminal
+comparison, validity is defined as follows.  The stage $s$ lies in
+$\mathbb Z/(4h)\mathbb Z$.  Write $\mathcal V_s$ for the following
+set of valid logical vectors; the endpoint descriptions in adjacent
+cases are identified.
+\begin{enumerate}[label=\textup{(\roman*)}]
+ \item $0\le s\le h$: $A=Q_x$ is full for some $x$, and $B$ is a
+ forward prefix of length $s$ with parameter
+ $y=\operatorname{out}_S(A)=S(x)$.
+ \item $h\le s\le2h$: $B=Q_y$ is full for some $y$, and $A$ is a
+ reconstruction prefix of length $2h-s$ from
+ $y=\operatorname{in}_S(B)$.
+ \item $2h\le s\le3h$: $B$ is a valid full transcript and
+ $A=\operatorname{pref}_{\pi_{\mathrm f},s-2h}(B)$.
+ This copied prefix has no additional computation requirements.
+ \item $3h\le s\le4h$: $A$ is a valid full transcript and
+ $B=\operatorname{pref}_{\pi_{\mathrm f},4h-s}(A)$.
+\end{enumerate}
+Occupations in every case are the stated full or prefix indicators.
+Stage $4h$ is stage $0$, at which $A$ is full and $B$ is empty.
+
+\begin{lemma}[Boundary agreement]\label{macro:boundaries}
+The two validity descriptions agree at each of $h,2h,3h$, and $4h=0$.
+\end{lemma}
+
+\begin{proof}
+At stage $h$, the first description gives
+\[
+ A=Q_x,\qquad B=Q_y,\qquad y=S(x),\qquad P(y)=x.
+\]
+The reconstruction template from $y$ computes $P(y)=x$ and then
+$S(x)=y$, so it agrees with $A$ at every position and satisfies all
+its comparisons.  This gives the second description.  Conversely,
+the second description at $h$ gives $B=Q_y$ and a full reconstruction
+$A=V_y$ with $S(P(y))=y$.  Put $x=P(y)$.  Then
+$P(S(x))=P(y)=x$, so $A=Q_x$ and $\operatorname{out}_S(A)=y$.
+Since $B=Q_y$, it is the full forward prefix required in the first
+description.  These identities include all copy gates and dummies.
+At $2h$, both descriptions have $A$ empty and $B$ a valid full
+transcript.  At $3h$, both have $A=B$ a valid full transcript.
+At $4h=0$, both have $A$ full and $B$ empty.
+\end{proof}
+
+For each \emph{start frame} $c\in\{0,\ldots,4h-1\}$, define a
+forward proposal $F_c$ from stage $c$ to stage $c+1$, and a reverse
+proposal $R_c$ from stage $c+1$ to stage $c$, with stages reduced
+modulo $4h$.  Thus at current stage $s$, the successor uses $F_s$
+and the predecessor uses $R_{s-1}$, where the latter subscript is
+reduced modulo $4h$.  The following table specifies their content changes.
+Every fill or copy sets the corresponding occupation to one, and
+every erasure sets it to zero.  All other bits remain unchanged.
+\begin{center}
+\begin{tabular}{@{}llll@{}}
+\toprule
+Start frame & Position $j$ & $F_c$ & $R_c$ \\
+\midrule
+$0\le c<h$ & $\pi_{\mathrm f}(c)$ & fill $B_j$ forward & erase $B_j$ \\
+$h\le c<2h$ & $\pi_{\mathrm r}(2h-c-1)$ & erase $A_j$ & fill $A_j$ by reconstruction \\
+$2h\le c<3h$ & $\pi_{\mathrm f}(c-2h)$ & copy $B_j$ to $A_j$ & erase $A_j$ \\
+$3h\le c<4h$ & $\pi_{\mathrm f}(4h-c-1)$ & erase $B_j$ & copy $A_j$ to $B_j$ \\
+\bottomrule
+\end{tabular}
+\end{center}
+A computed fill always writes its Boolean rule value, even if the
+additional terminal comparison fails.  Thus the proposal itself is
+defined separately from the test of whether its result is valid.
+
+The \emph{macrograph} has vertex set
+\[
+ \mathcal V=\{(s,b):s\in\mathbb Z/(4h)\mathbb Z, b\in\mathcal V_s\}.
+\]
+It has an edge $(c,b)\longrightarrow(c+1,b')$ precisely when
+$b'=F_c(b)$ and both endpoints are valid.
+
+\begin{lemma}[Reciprocity and local existence tests]
+\label{macro:local-edges}
+Whenever $b\in\mathcal V_c$ and $b'\in\mathcal V_{c+1}$,
+\begin{equation}\label{macro:reciprocity}
+ b'=F_c(b)\quad\Longleftrightarrow\quad b=R_c(b').
+\end{equation}
+For a valid input to either proposal, validity of its output is
+determined by a bounded number of bits at public, frame-dependent
+addresses.  The forward proposal can fail only when $0\le c<h$ and
+the new forward terminal comparison fails.  The reverse proposal can
+fail only when $h\le c<2h$ and the new reconstruction terminal
+comparison fails.  The graph has indegree and outdegree at most one,
+and no self-edges.
+\end{lemma}
+
+\begin{proof}
+In the first range, erasing the newly filled position recovers its
+previous blank content and occupation.  Conversely, a valid longer
+forward prefix has the uniquely prescribed value at its last
+position, so filling its shortened prefix recovers it.  In the second
+range, erasure shortens a reconstruction prefix.  Its erased value
+is uniquely determined by earlier retained positions and the
+unchanged full tape $B$, so reverse reconstruction recovers it.
+The converse follows from the same uniqueness.  In the last two
+ranges, the unchanged full tape determines the copied value,
+and erasure recovers the blank position.  Boundary agreement applies
+to each use of these statements at a boundary.  In particular,
+at $c=4h-1$, the reverse proposal from stage zero copies
+$A_{\pi_{\mathrm f}(0)}$ into the first forward position of the empty
+$B$ tape, exactly reversing the wraparound erasure.
+
+Shortening a valid computed prefix is always valid, as are extending
+or shortening a copied prefix.  When extending a computed prefix,
+all earlier conditions remain true.  Its next value is obtained
+from at most two earlier operands or a specified full-tape input
+bit, so the only new condition that can fail is the stated terminal
+comparison.  That comparison needs one additional specified bit.
+Thus the relevant operands, old bits, and comparison bits suffice
+to decide both proposals and the corresponding existence flags.
+Equation~\eqref{macro:reciprocity} gives at most one predecessor,
+and the definition gives at most one successor.  Every edge changes
+the stage by one modulo $4h$, and $4h\ge4$, so no edge is a self-edge.
+\end{proof}
+
+\subsection{Endpoints, sparse edits, and uniform size}
+\label{macro:interface}
+
+The following statement is the interface used by the evolving
+representation.
+
+\begin{lemma}[Macrograph interface]\label{lem:macrograph}
+From every promised End-of-Line instance of explicit length $N$, one
+can deterministically construct, in time polynomial in $N$, the
+above tape specification with $h=2^\ell=O(N)$, the stage conditions
+$\mathcal V_c$, and the following data and algorithms.
+\begin{enumerate}[label=\textup{(\roman*)}]
+ \item The macrograph has indegree and outdegree at most one, no
+ self-edges, and no isolated vertices.  It has a distinguished source
+ $v_*$ whose label and actual successor are computable in polynomial
+ time.  Every endpoint other than $v_*$ supplies a nonzero
+ End-of-Line answer by reading a designated vector on a full tape.
+ \item Forward and reverse proposals at the same start frame are
+ reciprocal on valid pairs.  Their edits and existence flags use
+ only a fixed number of logical bits at public addresses, with a
+ fixed finite set of Boolean decision rules.
+ \item There is a fixed finite alphabet
+ $\Sigma=\{-1,0,1\}^4$ and public edit vectors $h_\sigma(c)$ such
+ that each chosen proposal has the form
+ $b\mapsto b+h_\sigma(c)$.  One public address is allotted to each
+ logical array.  On actual edges only the active tape's content
+ position and its occupation position can change.  The same fixed
+ edit formulas are defined over any field, whether or not their
+ arguments represent valid Boolean tapes.
+ \item For each fixed phase, the local tape requirements have
+ bounded arity and bounded operand multiplicity, uniformly in $N$.
+ The full tape specification and all frame-dependent rule and
+ address tables have $O(h)$ records of $O(\log(N+2))$ bits each.
+ A vertex label has $4h+\lceil\log_2(4h)\rceil=O(N)$ bits;
+ validity and endpoint decoding take polynomial time.  These
+ bounds concern descriptions and individual labels and do not
+ require enumeration of the macrograph.
+\end{enumerate}
+\end{lemma}
+
+\begin{proof}
+We first identify all endpoints.  A missing successor can occur
+only in the first range of the schedule.  Write its full tape as
+$A=Q_x$ and put $y=\operatorname{out}_S(A)=S(x)$.  The current
+$B$ tape is its longest valid forward prefix, so
+Lemma~\ref{macro:prefix-uniqueness} gives
+\begin{equation}\label{macro:sink-identity}
+ P(S(y))\ne y,\qquad P(y)=x.
+\end{equation}
+If $y=0^{n_0}$, then $x=P(0^{n_0})=0^{n_0}$ and
+$y=S(0^{n_0})\ne0^{n_0}$, a contradiction.  Thus this designated
+vector is a nonzero End-of-Line answer.  Such a vertex has a
+predecessor: shorten its forward prefix when its length is positive;
+at stage zero use the reverse wraparound copy from the preceding
+frame.
+
+A missing predecessor can occur only at a stage $s=c+1$ with
+$h\le c<2h$.  Its full tape is $B=Q_y$, where
+$y=\operatorname{in}_S(B)$, and its reconstruction prefix has its
+maximal length $k_{\mathrm r}(y)<h$.  Hence
+\begin{equation}\label{macro:source-identity}
+ s=2h-k_{\mathrm r}(y),\qquad S(P(y))\ne y.
+\end{equation}
+For $y\ne0^{n_0}$ this is again a permitted answer.  The vertex
+has a successor obtained by shortening the reconstruction prefix;
+if the prefix is empty, the next copying step supplies a successor.
+Every proposal in the two copying ranges succeeds in the relevant
+direction, except that an incoming reconstruction proposal at stage
+$2h$ is covered by the case just considered.  Together with
+Lemma~\ref{macro:local-edges}, this exhausts the possible missing
+neighbors and proves that no vertex is isolated.
+
+It remains to specify the zero-labelled exception.  The last promise
+in \eqref{macro:promises} supplies the unique full transcript
+$Q_{0^{n_0}}$, whereas the first two promises give
+\[
+ S(P(0^{n_0}))=S(0^{n_0})\ne0^{n_0}.
+\]
+Let $k_*=k_{\mathrm r}(0^{n_0})<h$.  Then $v_*$ is exactly the
+vertex at stage $2h-k_*$ whose $B$ tape is $Q_{0^{n_0}}$ and whose
+$A$ tape is
+$\operatorname{pref}_{\pi_{\mathrm r},k_*}(V_{0^{n_0}})$,
+with their prescribed occupations.  Prefix uniqueness proves that
+this is the only missing-predecessor vertex with designated vector
+zero.  It has no predecessor and has the successor just described.
+Evaluate $U_{0^{n_0}}$ and $V_{0^{n_0}}$, locate the first failed
+reconstruction terminal comparison, and take the preceding prefix.
+This computes $v_*$ and its successor in polynomial time.  This
+proves the endpoint claims, including the existence of the public
+source.  Other directed paths and directed cycles may remain in the
+graph; nothing here requires one connected component.
+
+We next make the local edit interface explicit.  At start frame $c$,
+let $j(c)$ be the position in the transition table.  Allot this
+position to the content and occupation arrays of the active tape,
+and allot the fixed default position $0$ to the two inactive arrays.
+Denote these four addresses by $j_D(c)$, for
+$D\in\{A,E^A,B,E^B\}$.  If $\mathbf e_{D,j}$ is the unit vector
+at position $j$ of array $D$, define, over a field $K$,
+\begin{equation}\label{macro:finite-edits}
+ h_\sigma(c)=\sum_{D\in\{A,E^A,B,E^B\}}
+       \sigma_D\mathbf e_{D,j_D(c)}\in K^{4h},
+ \qquad \sigma\in\{-1,0,1\}^4,
+\end{equation}
+where the integers $-1,0,1$ are mapped into $K$.  For a proposed
+Boolean write, choose its content coefficient as the desired bit
+minus the old bit and its occupation coefficient as the desired
+occupation minus the old occupation.  Set both inactive coefficients
+to zero.  Each coefficient lies in $\{-1,0,1\}$, and the resulting
+additive edit is precisely the proposal, in either direction.
+The vector in \eqref{macro:finite-edits} is public once $c$ and
+$\sigma$ are fixed; addition by it is defined on all of $K^{4h}$.
+The assertion about valid Boolean proposals does not require a
+validity interpretation for other field-valued inputs.
+
+The necessary public data at a frame are its phase, active tape,
+write position, rule type, at most two operand addresses, the old
+content and occupation addresses, and, when applicable, the extra
+comparison address.  A forward fill uses its forward rule; a reverse
+fill uses its reconstruction rule.  Lemma~\ref{macro:local-edges}
+shows that the existence flag is the new comparison result in the
+only direction where failure is possible, and is one otherwise.
+These are fixed Boolean decisions once the finitely many addressed
+bits and public rule data are supplied.  Define the decisions on
+arbitrary Boolean reads by the same formulas, using a fixed default
+rule and address for unused roles.  Their agreement with graph
+neighbors is asserted on valid input vertices.  No global validity
+check or full circuit evaluation is hidden in an existence flag.
+
+For completeness, the bounded-multiplicity assertion includes more
+than internal gate edges.  Each internal value has boundedly many
+uses after copy-tree normalization.  Each designated output is used
+once by the other circuit's input-copy rule and participates in only
+a bounded number of terminal or external-vector comparisons.  Each
+external-vector bit is used once as an input bit and once for its
+corresponding terminal comparison.  Copied-prefix equations use each
+full-tape position once.  Occupation indicators condition only the
+rules at their positions; prefix monotonicity, if expressed by
+adjacent-position tests, reads a position at most twice in each
+order.  Thus all local content, bitness, occupation, gate, copy,
+comparison, and dummy conditions use bounded-arity records, and each
+logical position appears as an operand only boundedly often across
+the fixed list of phases.  The requirement that the occupied prefix
+have its specified length is separate: one may either inspect the
+prescribed occupation pattern directly, or use prefix monotonicity
+together with the occupation sum.  The latter is a single global
+count condition for each tape and will be represented separately in
+the algebraic construction.
+
+All normalization and ordering operations are deterministic.  Their
+records and the two permutations occupy $O(h\log(N+2))$ bits.
+Tabulate the transition data for the $4h$ frames in the same space
+bound, including both directions and defaults.  Scanning the arrays
+and these records tests membership in $\mathcal V_s$ in polynomial
+time.  The designated endpoint vectors and the circuit inequalities
+\eqref{macro:sink-identity} and \eqref{macro:source-identity} give a
+polynomial-time endpoint decoder.  At every fixed stage a valid
+vertex is determined by the input of its full transcript, since
+its other tape is a unique computed or copied prefix.  Consequently
+the graph has at most $4h\,2^{n_0}$ vertices, even though an individual
+label has only $O(N)$ bits.  The construction outputs the normalized
+records, public tables, and algorithms, never this potentially
+exponential vertex list.  All constants in these bounds depend only
+on the fixed Boolean basis and padding cutoff.  This proves the
+remaining uniformity and size assertions.
+\end{proof}

preprints/The-PCP-for-PPAD-conjecture-a-quasilinear-reduction-September-25-2026/build/sections/03-outer-codes.tex

@@ -0,0 +1,1006 @@
+\section{Outer index groups and systematic square codes}\label{sec:outer-codes}
+
+The outer code has two roles.  Its systematic coordinates carry the data
+on which the representation computes, and its redundant coordinates make
+local consistency imply proximity in the metric that counts an entire
+stored symbol as one position.  We construct the code, including its
+matrix twists and simultaneous systematic encoders.  All vector-valued
+statements below use joint support: a coordinate is erroneous when any
+component is erroneous.  Their constants do not depend on the number of
+components or stored slots.  The notation in this section is local to the
+outer construction; in particular, the group constructed here is not the
+affine group used later for the inner construction.
+
+The index group must provide commuting left and right moves, constant
+expansion, and only polylogarithmically many frequencies in each relevant
+orbit. The first two properties support local consistency; the orbit
+bound makes the local filling in Lemma~\ref{lem:outer-fill} possible.
+The code and correction guarantees are summarized in
+Theorem~\ref{thm:outer-codes} and Lemma~\ref{lem:outer-uncertainty}.
+
+\subsection{A group of quasilinear order}
+
+Let $h=2^\ell=O(N)$ be the padded tape length supplied by
+Lemma~\ref{lem:macrograph}.  Fix a sufficiently large absolute integer
+$C$, and choose an even $r$ with
+\begin{equation}\label{oc:outer-size}
+ \ell+C\le r\le\ell+C+2,
+ \qquad M_o=2^r,\qquad G_0=\F_2^r.
+\end{equation}
+We increase $C$ whenever a fixed lower cutoff is needed.  Thus $4h\le M_o$,
+$M_o=O(N)$, and $r=O(\log(N+2))$.  The bits of $x\in G_0$ are ordered
+from least to most significant; $\underline{x}\in[0,M_o)$ denotes the
+integer they represent.
+
+Helfgott's ternary Goldbach theorem says that every odd integer greater
+than five is a sum of three primes \cite[Main Theorem]{Helfgott2013}.
+For our even $r$ we can therefore find primes
+$p_1,p_2,p_3$ with $r-3=p_1+p_2+p_3$, by exhaustive search in $[r]$.
+They need not be distinct or comparable in size.  Partition the $r$ bit
+positions into blocks of lengths $p_a+1$, and let
+\begin{equation}\label{oc:group}
+ H=\prod_{a=1}^3\operatorname{PGL}_2(\F_{p_a}),
+ \qquad \Gamma=G_0\rtimes H.
+\end{equation}
+Each factor acts on its projective line, hence permutes its block of bit
+positions.  Since
+$|\operatorname{PGL}_2(\F_q)|=q(q^2-1)$, we have
+\begin{equation}\label{oc:group-order}
+ |H|\le r^9,\qquad |\Gamma|=M_o|H|\le M_or^9.
+\end{equation}
+A generator of $\F_{p_a^2}^{\times}$, acting on its one-dimensional
+$\F_{p_a}$-subspaces, induces a cycle $R_a$ of length $p_a+1$:
+the scalar subgroup has order $p_a-1$ and its quotient acts freely and
+transitively on these subspaces.  Label each block in this cycle order,
+and denote its first unit vector by $e_a$.  The three finite fields,
+permutation groups, cycles, and their tables have size polynomial in $r$
+and can be found by exhaustive finite-field computations of that size.
+
+The product is $(g,u)(w,v)=(g+uw,uv)$.  It is useful to write a group
+element $k=(g,u)$ instead as $(x,u)$, where $x=u^{-1}g$.
+Left and right multiplication by $(w,v)$ then have the formulas
+\begin{equation}\label{oc:actions}
+ (x,u)\longmapsto (x+u^{-1}v^{-1}w,vu),
+ \qquad
+ (x,u)\longmapsto (v^{-1}(x+w),uv),
+\end{equation}
+respectively.  These formulas also give polynomial-in-$r$ Boolean
+algorithms for multiplication and inversion, using the small tables for
+$H$ rather than a table for $\Gamma$.
+
+\begin{lemma}[Uniform expansion]\label{oc:expansion}
+For every fixed $0<\lambda<1$, the groups $\Gamma$ in
+\eqref{oc:group} admit symmetric labelled Cayley multisets of a fixed
+degree whose averaging operator has norm at most $\lambda$ on mean-zero
+functions.  The same assertion holds for $H$.  Their labels can be
+constructed deterministically in time polynomial in $N$ and represented
+with $\operatorname{polylog}(N)$ bits of public data.  Multiplicities and identity
+labels are permitted.
+\end{lemma}
+
+\begin{proof}
+First consider a factor of $H$.  The reductions of the upper and lower
+unit elementary matrices, with their inverses, have a uniform Cayley
+gap in $\operatorname{SL}_2(\F_p)$.  This is the elementary-generator
+case of the congruence gap discussed in \cite[p.~626]{BG2008}, and also
+follows directly from \cite[Theorem~1]{BG2008}, since the corresponding
+fixed subgroup of $\operatorname{SL}_2(\mathbb Z)$ is non-elementary.
+Passing to the quotient by the center preserves the upper gap.  A
+half-lazy walk has a uniform absolute gap.  The finitely many small
+primes are absorbed by decreasing this fixed gap.
+
+For odd $p$, the subgroup $\operatorname{PSL}_2(\F_p)$ is normal of
+index two in $\operatorname{PGL}_2(\F_p)$.  Add a representative of the
+nonsquare determinant class and its inverse; in the index-one case pad
+with identities.  To check a uniform gap, work in an arbitrary unitary
+representation.  The preceding subgroup gap bounds the distance of a
+vector to its subgroup-invariant projection by a fixed multiple of its
+maximum displacement under the subgroup labels.  Normality makes that
+invariant subspace invariant under the extra label, which controls its
+quotient of order at most two.  Thus distance to the full invariant
+subspace is bounded by a fixed multiple of the maximum displacement
+under all these labels.  The three factor projections commute; applying
+this argument to the three factors gives the same statement for $H$,
+with a fixed-size generating list.  The regular-representation gap used
+here applies to every representation of the finite group, by its
+decomposition into irreducible representations.
+
+We next give a bounded list of translations which, together with these
+$H$ labels, expands $\Gamma$.  Consider a block of $n'$ bits with cycle
+$R$.  For a nonzero frequency $y\in\F_2^{n'}$, let $d'$ be the dimension
+of the span of its cyclic shifts.  Identifying vectors with polynomials
+modulo $Z^{n'}-1$, this span is a cyclic ideal.  Every such ideal of
+dimension $d'$ is specified by a complementary monic divisor of degree
+$d'$.  There are at most $2^{d'}$ such divisors and at most $2^{d'}$
+vectors in each ideal.  In particular, there are at most $2^{2d'}$
+frequencies with this span dimension.  This argument uses only the
+ideal structure of the polynomial quotient and remains valid when
+$Z^{n'}-1$ has repeated factors in characteristic two.
+
+For a uniform seed $a\in\F_2^{n'}$, the correlation vector
+$(y\cdot R^ta)_{t=0}^{n'-1}$ is uniform on a shift-invariant linear
+space of dimension $d'$.  Take $j_0$ independent seeds.  In the product
+of these spaces, the subset with total weight less than $j_0n'/4$ is
+invariant under cyclic shifts and permutation of the seeds.  If this
+subset is nonempty, its uniform distribution has the same marginal at
+every coordinate, with probability of one at most $1/4$.  An information
+set of $j_0d'$ coordinates determines a vector in the product space.
+Entropy subadditivity consequently bounds the size of this subset by
+$2^{j_0d'\mathcal H(1/4)}$, where $\mathcal H$ is binary entropy.
+The subset of weight greater than $3j_0n'/4$ has the same bound, by the
+marginal entropy estimate with probabilities at least $3/4$; it is
+unnecessary for the space to be closed under complementation.
+Thus a fixed sufficiently large $j_0$ makes the probability of any
+failure at any nonzero frequency at most
+\begin{equation}\label{oc:seed-bound}
+ 2\sum_{d'\ge1}
+   2^{-\{j_0(1-\mathcal H(1/4))-2\}d'}<1.
+\end{equation}
+There exist $j_0$ seeds whose cyclic translates have absolute Walsh
+bias at most $1/2$ at every nonzero frequency.  Enumerating all seed
+tuples and checking their biases constructs them in time $2^{O(r)}$,
+which is polynomial in $N$.
+
+Take the Cartesian product of the three block seed lists.  Averaging
+its translations under $\langle R_1\rangle\times\langle R_2\rangle
+\times\langle R_3\rangle$ has absolute Walsh bias at most $1/2$ off
+zero: the three factors multiply, and a nonzero frequency has a nonzero
+block.  Averaging further over cosets in $H$ preserves this bound.
+Denote by $T$ the average of the resulting $H$-conjugates of seed
+translations.  On the orthogonal complement of $G_0$-invariants, $T$
+has all eigenvalues in $[-1/2,1/2]$, by Fourier decomposition for $G_0$.
+
+Let $v$ be a unit vector in a representation of $\Gamma$, and let
+$\eta$ be its maximum displacement under the fixed $H$ labels and the
+fixed product-seed translations.  The $H$-invariant projection $v_H$
+satisfies $\|v-v_H\|\le K\eta$ for an absolute $K$.  Its displacement
+under any seed translation is at most $(1+2K)\eta$.  Since $v_H$ is
+$H$-invariant, the same holds for every $H$-conjugate of a seed.
+Writing $P_{G_0}$ for projection to $G_0$-invariants gives
+\[
+ \|v_H-P_{G_0}v_H\|^2
+ \le 2\langle v_H,(I-T)v_H\rangle
+ \le (1+2K)^2\eta^2.
+\]
+Normality of $G_0$ implies that $P_{G_0}v_H$ is also $H$-invariant.
+Distance to the $\Gamma$-invariant subspace is therefore at most a
+fixed multiple of $\eta$.  The symmetric half-lazy average on our
+fixed generating list has a uniform positive gap: its Dirichlet form
+is the average of squared label displacements divided by a fixed
+constant.  Its spectrum is nonnegative.  A fixed power of this walk
+has mean-zero norm at most the requested $\lambda$.  Reversing words
+shows that the powered list remains symmetric.  Its length is fixed,
+and the same conclusion holds for left and right regular actions.
+\end{proof}
+
+\subsection{Common scalar local codes}
+
+The distance and agreement estimates in this subsection hold over an
+arbitrary field $J$, conditional on the scalar check matrices satisfying
+the stated minor conditions.  We will choose the special outer field
+and those matrices after fixing the constants in these estimates.
+
+The local agreement constant must be fixed before the outer Cayley
+degree is chosen.  To obtain this order, we make the label positions
+the edges of a separate expander.  For every integer $s_b\ge1$, consider
+the Margulis--Gabber--Galil multigraph~\cite{Margulis1973,GabberGalil1981}
+on $\mathbb Z_{s_b}^2$ with the
+eight labelled moves
+\[
+ (x\mathbin\pm2y,y),\quad (x\mathbin\pm(2y+1),y),\quad
+ (x,y\mathbin\pm2x),\quad (x,y\mathbin\pm(2x+1)).
+\]
+Its normalized mean-zero norm is at most $5\sqrt2/8$ for every positive
+integer modulus, including composite moduli.  The primary formal
+development \cite[\texttt{mgg\_graph\_step},
+\texttt{mgg\_numerical\_radius}]{MGGAFP} proves exactly this labelled,
+symmetric, all-modulus statement.  Add eight identity labels for
+half-laziness and take a sufficiently large fixed power.  Viewing the
+result as a bipartite graph between two copies of $\mathbb Z_{s_b}^2$
+gives a graph $B_0$ with fixed degree $d_0$, permutation labels
+$1,\ldots,d_0$, and mean-zero cross-part norm
+\begin{equation}\label{oc:base-parameters}
+ \lambda_0<10^{-4},\qquad
+ c_0=\lfloor d_0/16\rfloor,\qquad \delta_0=1/16.
+\end{equation}
+Take the fixed power large enough that $c_0\ge1$.
+Let $n_b=s_b^2$ and $d=d_0n_b$.  Each of the $d$ edge coordinates
+is named by its left base vertex and permutation label.  Both ends
+of an outer edge use these same coordinate names.
+
+For each of the four types $C_A^0,C_A^1,C_B^0,C_B^1$, choose two
+$c_0\times d_0$ scalar matrices, one for each base side.  Impose the
+corresponding checks at every base vertex, with incident edges in
+permutation-label order.  There are eight check matrices in total;
+their entries are shared by all stars of the same type.  We require
+every $c_0$-column minor to be nonzero.  Section~\ref{oc:encoders-subsec}
+constructs matrices meeting these conditions and the global encoder
+conditions simultaneously.  Until then the following conclusions hold
+for every choice satisfying the stated minors.  For a finite-dimensional
+$J$-space $U$, notation such as $C_A^s(U)$ means that the scalar
+constraints are applied componentwise to a word in $U^d$.
+
+\begin{lemma}[Local distance and tensor agreement]\label{oc:local-agreement}
+Set
+\begin{equation}\label{oc:delta-one}
+ \delta_1=\delta_0(\delta_0-\lambda_0)>0.
+\end{equation}
+Every nonzero word of any of the four local codes, over any
+finite-dimensional $J$-space $U$, has at least $\delta_1d$ nonzero
+coordinates.  There is a constant $C_0<\infty$, depending only on
+$d_0,\delta_0,\lambda_0$, with the following property for every $s_b$,
+every field and check matrices as above, every $U$, and every $s,t$.
+If $F$ is a $d\times d$ table with rows in $C_B^t(U)$ and $D$ is such
+a table with columns in $C_A^s(U)$, and their joint cell-disagreement
+fraction is $z$, there is a table $W$ whose rows and columns obey both
+codes and such that
+\begin{equation}\label{oc:tensor-agreement}
+ \Pr_i[W(i,\cdot)\ne F(i,\cdot)]
+ +\Pr_j[W(\cdot,j)\ne D(\cdot,j)]\le C_0z.
+\end{equation}
+In particular, $C_0$ is independent of $d$, $|J|$, and $\dim_JU$.
+\end{lemma}
+
+\begin{proof}
+The maximal-minor condition gives star distance at least
+$c_0+1\ge\delta_0d_0$: a nonzero word of smaller support would give a
+linear dependence among at most $c_0$ columns, which could be enlarged
+to a dependence among $c_0$ columns.  This also proves the assertion
+for vector entries, either by applying a nonzero scalar component or
+by solving the independent-column equations componentwise.
+
+For a nonzero whole local word, let $l,u$ be the fractions of left and
+right base vertices incident to its support, and let $a=\max(l,u)$.
+Its edge-support fraction $e$ obeys, by star distance and mixing,
+\[
+ \delta_0a\le e\le lu+\lambda_0\sqrt{lu}
+ \le a(a+\lambda_0).
+\]
+Hence $a\ge\delta_0-\lambda_0$ and $e\ge\delta_1$.  Mixing here and
+below follows by subtracting the two constant means and applying the
+cross-part operator norm to the indicator functions.
+
+For agreement, apply the A-base checks to the columns of $F$.  Mark
+every base star at which any check syndrome is nonzero.  Each nonzero
+syndrome, as a row indexed by the $d$ columns, is a scalar linear
+combination of B-code rows, and is therefore a nonzero B-codeword over
+$U$.  Its support has size at least $\delta_1d$.  In each such column
+there must be a discrepancy with $D$ on an edge incident to the marked
+star, since $D$ satisfies that star's check.  A discrepancy cell is
+incident to at most two base stars.  If $w_0$ is the total marked-star
+count divided by $n_b$ (counting both base sides), then
+\begin{equation}\label{oc:marked-size}
+ w_0\le \frac{2zd^2}{\delta_1dn_b}
+       =\frac{2d_0z}{\delta_1}.
+\end{equation}
+
+We now confine every low-discrepancy difference column to edges inside
+a small set of base vertices.  Deleting those row indices, together
+with the columns having large discrepancy, will leave a rectangle on
+which $F$ and $D$ agree.
+
+Close the marked set by adding a vertex whenever at least
+$(\delta_0/2)d_0$ incident edges lead to the set.  If the initial set
+is empty it stays empty.  Otherwise suppose its normalized size first
+reaches $w\ge2w_0$.  A single addition is at most $1/n_b\le w_0$, so
+$w\le3w_0$.  Edges made internal by additions give internal edge
+fraction at least $(\delta_0/2)(w-w_0)$.  Mixing gives the upper bound
+$w^2/4+\lambda_0w/2$.  Since $w\ge2w_0$, these inequalities imply
+$\delta_0\le w+2\lambda_0$.  They are incompatible when
+$3w_0<\delta_0-2\lambda_0$.  For sufficiently small $z$, therefore,
+the final closure $T$ satisfies $|T|/n_b\le2w_0$, with strict
+inequality when $w_0>0$.
+Every star outside $T$ still satisfies all checks on $F$, and fewer
+than $(\delta_0/2)d_0$ of its edges enter $T$.
+
+Fix a constant
+\[
+ 0<z_0<\delta_0(\delta_0/2-\lambda_0).
+\]
+Consider a column whose discrepancy fraction is less than $z_0$.
+The difference between its $F$ and $D$ columns satisfies all checks
+outside $T$.  An outside vertex incident to its support consequently
+has at least $\delta_0d_0$ support edges in total and more than
+$(\delta_0/2)d_0$ support edges to other outside vertices.  The left
+and right fractions of such outside vertices are each at most
+$z_0/\delta_0$, by counting their total support edges.  If $a$ is
+their maximum fraction, mixing on the outside support would give
+\[
+ (\delta_0/2)a\le a(a+\lambda_0),
+ \qquad a\le z_0/\delta_0<\delta_0/2-\lambda_0.
+\]
+Thus $a=0$.  The difference column vanishes on every edge having an
+endpoint outside $T$.
+
+Delete the row indices corresponding to edges internal to $T$, and
+delete columns with discrepancy fraction at least $z_0$.  The former
+fraction is at most $|T|/(2n_b)\le w_0$, since each internal edge consumes
+two incidences in $T$, and the latter fraction is at most $z/z_0$.
+The two tables agree on the complementary rectangle.  Choose a fixed
+$z_*>0$, depending only on the base parameters, so small that for
+$z<z_*$ the closure inequality holds and each deleted fraction is
+less than $\delta_1$.  Puncturing either scalar local code to its
+retained coordinates is then injective by distance.  Its inverse
+on the image is a scalar linear map.  Lift the common rectangle in
+the B direction and then in the A direction.  Applying scalar linear
+maps in one direction preserves the constraints in the other; hence
+the resulting table belongs to the full tensor code.  By injectivity
+it agrees with every retained whole row of $F$ and every retained
+whole column of $D$.  This gives \eqref{oc:tensor-agreement} with
+$2d_0/\delta_1+1/z_0$ for $z<z_*$.  For $z\ge z_*$, use the zero
+tensor table and the bound $2\le(2/z_*)z$.  The maximum of these two
+constants is a suitable $C_0$.  All lifting maps and all support
+counts act on joint vector entries, so no dimension factor occurs.
+\end{proof}
+
+We now fix, once and for all,
+\begin{equation}\label{oc:lambda-choice}
+ 0<\lambda\le\min\{(8C_0)^{-1},\delta_1/16\}.
+\end{equation}
+Apply Lemma~\ref{oc:expansion} to obtain an outer Cayley list of degree
+$D$.  Choose $s_b$ to be a positive multiple of $D$.  Then
+$d=d_0s_b^2$ is divisible by $D$, so label the $d$ local coordinates
+by a balanced repetition of the outer list.  This repetition changes
+neither averaging operator nor its norm.  Use such lists
+$(a_i)_{i=1}^d$ and $(b_j)_{j=1}^d$ for the A and B directions.
+All of $d_0,c_0,\lambda_0,\delta_1,C_0,\lambda,s_b,d$ are now fixed
+constants.  We now specialize $J$; the field choice will not change
+these constants.
+
+Choose an odd characteristic independently of the primes $p_a$
+and independently of the characteristic-two translation group.  Put
+$m=M_o$.  Linnik's theorem, for example the effective bound in
+\cite[Theorem~1.1]{Xylouris2011}, supplies a prime
+\begin{equation}\label{oc:prime}
+ p\equiv1\pmod m,\qquad m<p\le C_Lm^{L}
+\end{equation}
+for absolute constants $C_L,L$.  Searching integers and testing
+primality, even by trial division, takes polynomial time in $m$.
+Search the prime field for an element $\zeta$ of exact order $m$;
+such an element exists because its multiplicative group is cyclic.
+Since $m$ is a power of two, testing $\zeta^m=1$ and
+$\zeta^{m/2}\ne1$ suffices.  We will use
+\begin{equation}\label{oc:field}
+ J=\F_{p^4}.
+\end{equation}
+The proof below verifies that degree four supplies enough room for
+every simultaneous outer-code choice.  One can find an irreducible
+quartic by enumeration and represent $J$ as its polynomial quotient.
+Field operations and the root $\zeta$ then use $O(\log N)$ bits and
+polynomial-in-$\log N$ Boolean computations.  Our fixed lower cutoff
+also ensures $p>m>|H|$, so both $m$ and $|H|$ are invertible in $J$.
+
+\subsection{Twists and simultaneous systematic extension}
+\label{oc:encoders-subsec}
+
+There are four outer parts
+\[
+ V=\Gamma\times\{0,1\}\times\{0,1\}.
+\]
+At $(k,s,t)$ a full slot stores a $d\times d$ matrix $W_k^{st}$ over
+a finite-dimensional $J$-space $U$, with columns in $C_A^s(U)$ and
+rows in $C_B^t(U)$.  The A and B edges impose, respectively,
+\begin{equation}\label{oc:edge-equations}
+ W_{a_ik}^{1t}(i,j)=\alpha_i W_k^{0t}(i,j),
+ \qquad
+ W_{kb_j}^{s1}(i,j)=\beta_j W_k^{s0}(i,j).
+\end{equation}
+The maps $\alpha_i,\beta_j$ are invertible $J$-linear maps on $U$,
+independent of $k,s,t$, and satisfy
+$\alpha_i\beta_j=\beta_j\alpha_i$ for every $i,j$.  An A edge of
+label $i$ transports the entire B-coded row $i$, while a B edge of
+label $j$ transports the entire A-coded column $j$.  A reverse edge
+uses the inverse map.  At their intersection the two transports give
+\[
+\begin{array}{ccc}
+ W_k^{00}(i,j)&\xrightarrow{\ \alpha_i\ }&W_{a_i k}^{10}(i,j)\\
+ {\scriptstyle\beta_j}\big\downarrow&&
+       \big\downarrow{\scriptstyle\beta_j}\\
+ W_{k b_j}^{01}(i,j)&\xrightarrow{\ \alpha_i\ }&W_{a_i k b_j}^{11}(i,j).
+\end{array}
+\]
+Left and right multiplication on $\Gamma$ commute, so the two routes
+reach the same index; commuting entry maps give the same entry value.
+
+The scalar type has $U=J$ and identity twists.  The programs in
+Section~\ref{sec:outer-programs} will also need to multiply scalar
+data $v:\Gamma\to J$ by $(-1)^{e\cdot x}$ for a distinguished bit
+$e=e_a$.  The matrix types below supply edge equations compatible
+with this operation.  The programs will use them to retain one axis
+of code constraints while changing the other.
+For each of the three distinguished bits $e=e_a$, define a
+representation on the basis
+$({\bf e}_{t'})_{t'\in H}$ by
+\begin{equation}\label{oc:rho}
+ \rho_e(g,u){\bf e}_{t'}
+   =(-1)^{(ut'e)\cdot g}{\bf e}_{ut'}.
+\end{equation}
+Its entries are signed permutation entries in the prime field.
+Indeed, for $k=(g,u)$ and $k'=(g',u')$, the exponent in
+$\rho_e(k)\rho_e(k'){\bf e}_{t'}$ is
+\[
+ (u't'e)\cdot g'+(uu't'e)\cdot g
+  =(uu't'e)\cdot(g+ug'),
+\]
+which proves multiplicativity.  In a matrix type let
+$U=\operatorname{Mat}_{|H|}(J)$ and independently choose each axis
+to be untwisted or to use
+\begin{equation}\label{oc:matrix-twists}
+ \alpha_i D=\rho_e(a_i)D,
+ \qquad \beta_j D=D\rho_e(b_j).
+\end{equation}
+Left and right matrix multiplication commute.  Right multiplication
+in \eqref{oc:matrix-twists} specifies edge maps; no convention about
+its order as a group representation is needed.  This gives at most
+$1+3\cdot4=13$ full types.  Finite tuples, with arbitrary repetitions
+of these types, use direct sums of their entry spaces and block
+diagonal twists.
+
+On either local-label graph, let the parity set consist of all edges
+with permutation number in $[4c_0]$.  Its size and complement are
+\begin{equation}\label{oc:parity-set}
+ q_{\rm par}=4c_0s_b^2,\qquad
+ I_A=[d]\setminus\text{A-parity set},\qquad
+ I_B=[d]\setminus\text{B-parity set}.
+\end{equation}
+Both systematic sets are nonempty; in fact they contain at least
+$3d/4$ labels.
+
+\begin{lemma}[Simultaneous systematic encoders]\label{oc:systematic}
+One can choose the eight scalar check matrices over $J$ so that all
+their maximal minors are nonzero and every A and B one-axis system
+of the thirteen types above is systematically invertible.  Precisely,
+arbitrary data at the systematic labels of part zero have a unique
+extension to both parts satisfying that axis's local checks and edge
+equations.  Consequently arbitrary data on
+$\Gamma\times I_A\times I_B$ in part $(0,0)$ have a unique full
+extension satisfying both axes, and the two orders of axis extension
+give the same answer.  The matrices and the full encoders are
+deterministically constructible in polynomial time in $N$.  These
+assertions continue to hold for arbitrary finite direct sums of the
+listed types, without any further field or check-matrix choices.
+\end{lemma}
+
+\begin{proof}
+Consider one A-axis system.  Eliminate part-one variables with the
+A-edge equations.  At each $k$, the two types of base-star checks in
+part zero contribute $2c_0s_b^2$ equations; those in part one contribute
+the same number after substitution.  There are exactly
+$q_{\rm par}=4c_0s_b^2$ parity-label variables at part zero for each
+$k$, so the resulting parity system is square, also after scalarizing
+$U$.  Its entries are linear polynomials in the entries of the four
+A check matrices.
+
+Its determinant polynomial is nonzero in the prime characteristic.
+For this purpose partition the $4c_0$ parity permutation numbers into
+four disjoint sets of $c_0$ numbers, one for each pair consisting of an
+outer part and a base side.  Specialize each associated check matrix
+to selectors of its set.  For a fixed permutation number, every
+label-index edge occurs exactly once at the specified base side.
+At outer part zero its selected coordinate is unshifted; at part one
+it is pulled back through an invertible group shift and an invertible
+entry twist.  Thus every scalarized parity coordinate is selected
+exactly once, up to an invertible block and a permutation.  The
+specialized parity matrix is invertible.  The same proof works for
+the B axis and for every allowed twist type.  It does not require
+this selector specialization itself to have the desired maximal
+minors: it separately witnesses that each determinant is a nonzero
+polynomial.  Each maximal-minor polynomial is also nonzero.  Their
+product is nonzero because a polynomial ring over a field is an
+integral domain.
+
+Here is a degree bound that justifies this field choice without
+reference to any future number of slots.  There are exactly
+\begin{equation}\label{oc:number-variables}
+ 8c_0d_0
+\end{equation}
+scalar indeterminates.  Scalarizing a one-axis parity matrix gives
+dimension at most
+\begin{equation}\label{oc:parity-dimension}
+ q_{\rm par}|\Gamma|\dim_JU
+ \le q_{\rm par}m|H|^3\le q_{\rm par}mr^{27}.
+\end{equation}
+Its determinant has degree at most this number.  At most
+$2(1+3\cdot4)=26$ axis systems cover every required type, with
+redundancies harmless.  The product of the maximal minors of the
+eight check matrices has degree $8c_0\binom{d_0}{c_0}$.  Thus the
+total degree is at most
+\begin{equation}\label{oc:determinant-degree}
+ 26q_{\rm par}mr^{27}+8c_0\binom{d_0}{c_0}<m^3.
+\end{equation}
+For the last inequality increase the fixed cutoff so that
+$r^{27}\le m$, $m\ge2\cdot26q_{\rm par}$, and
+$m^2\ge2\cdot8c_0\binom{d_0}{c_0}$.  The first term is then at
+most $m^3/2$ and the second at most $m^2/2<m^3/2$.
+The exponent three is independent of $s_b$; only the cutoff depends
+on the already fixed local degree.
+
+A nonzero polynomial of total degree less than a finite field's
+cardinality cannot vanish on every tuple over that field.  For
+completeness, view it as a polynomial in the last variable; by
+induction choose the other variables so that a nonzero leading
+coefficient remains nonzero, and then use the univariate root bound.
+Since $|J|=p^4>m^3$, such a tuple exists in $J$.  Enumerate all tuples
+of the fixed number \eqref{oc:number-variables} of field elements,
+testing maximal minors and parity ranks, until one succeeds.  The
+field cardinality and every matrix dimension are polynomial in $N$,
+so this is a deterministic polynomial-time algorithm with a fixed
+exponent.  A direct sum of already invertible parity systems is
+invertible with the direct sum of their inverses.  It introduces no
+new polynomial condition, even when the number of slots grows with
+$N$.
+
+It remains to justify two-axis extension.  Regard an A parity matrix
+as a matrix of operators on $U^\Gamma$: each entry is a scalar
+combination of left translations and A twists.  B entries are scalar
+combinations of right translations and B twists.  They commute
+entrywise across axes.  An invertible A parity matrix $L_A$ therefore
+intertwines the diagonal actions of any B-entry operator $T$ on its
+domain and codomain.  From $L_AT=TL_A$ one obtains
+$L_A^{-1}T=TL_A^{-1}$, with the same indexed domain and codomain
+interpretation.  Thus the operators making up either axis encoder
+commute with every entry operator of the other axis encoder, including
+its inverse parity solve.
+
+Start with the systematic rectangle at part $(0,0)$.  Extend along A
+for every systematic B column and then along B for every resulting
+A row.  The second extension preserves all first-axis constraints
+by this commutation.  Both encoders leave their systematic entries
+unchanged.  Reversing their order gives the same map.  Finally, one-axis
+uniqueness applied in both directions shows that a full word is
+uniquely determined by its systematic rectangle.  Gaussian elimination
+computes these extensions in polynomial time in $N$, or in polynomial
+time in the total explicit input size for an arbitrary direct sum.
+\end{proof}
+
+For $v:\Gamma\to U$, give every systematic cell in part $(0,0)$ the
+same value $v(k)$.  Its full extension is denoted
+\begin{equation}\label{oc:encoder}
+ E_{\alpha,\beta}v,
+\end{equation}
+or $Ev$ when its type is specified.  We call this the
+\emph{duplicated-payload} encoder.  It is linear and injective, and
+its rate in field coordinates is exactly $1/(4d^2)$: its domain has
+$|\Gamma|\dim_JU$ coordinates and its stored word has
+$4d^2|\Gamma|\dim_JU$ coordinates.  In the largest individual type
+the entry dimension is $|H|^2\le r^{18}$.  Therefore one full slot
+occupies $\operatorname{polylog}(N)$ bits at each of $|\Gamma|=O(N\operatorname{polylog} N)$
+sites.  Any fixed power of $\log(N+2)$ slots preserves this size form.
+
+\begin{remark}[Operator invariance]\label{oc:orbit-invariance}
+The preceding inverse argument also justifies restriction to frequency
+orbits, as required by the local filling algorithms.  Expand a function
+of $(x,u)$ in the Walsh characters $(-1)^{\nu\cdot x}$, using $m^{-1}$
+for the inverse transform over $J$.  By \eqref{oc:actions}, a left pull
+preserves each $x$-frequency, with a phase depending on $u$, whereas
+a right pull permutes frequencies by $H$.  Constant entry twists do
+not change the frequency.  For any $H$-orbit $O\subseteq G_0$, both
+the $O$-frequency subspace and its complementary frequency subspace
+are invariant under each parity operator.  Its domain and codomain
+have equal dimensions on this subspace: the parity-label and
+check-label counts are both $q_{\rm par}$.  Global invertibility
+therefore implies invertibility of the restriction.  Its scalar
+dimension is at most $q_{\rm par}|O||H|\dim_JU\le
+q_{\rm par}|H|^4$, polynomial in $r$, so the restriction and its
+inverse can be computed by polynomial-in-$\log N$ field operations.
+This is an algebraic consequence of the explicit parity matrices;
+it does not require reading unspecified global values of a word.
+\end{remark}
+
+\subsection{Proximity and one-axis gaps}
+
+Write $\dist_V$ for relative Hamming distance on the four-part vertex
+set $V$.  When all four parts at the same $k$ are packed into one
+symbol, write $\dist_\Gamma$.  For any two words,
+\begin{equation}\label{oc:packing-metric}
+ \dist_V\le\dist_\Gamma\le4\dist_V.
+\end{equation}
+An \emph{A-only word} retains the A local constraints and A agreements
+at all four parts; any other label indices and any collection of slots
+are regarded as coordinates of a single row symbol.  No B validity
+or B agreement is asserted.  Define B-only words symmetrically.
+
+For a tuple of full slots a local check rejects when any component
+fails its scalar local checks, and an edge is conflicting when any
+slot fails its agreement on the whole row or column carried by that
+edge.  In particular, a growing tuple does not enlarge the set of
+address maps or cause separate per-slot error charges.
+
+\begin{lemma}[Full-code proximity]\label{oc:full-proximity}
+There are fixed $\theta_*>0$ and $K_*<\infty$ such that the following
+holds for every finite tuple of allowed full types and every received
+word.  Check local validity and all A and B agreements at a uniformly
+random packed center, rejecting if any check there fails.  If the
+rejection fraction is $\theta\le\theta_*$, the word is at packed
+distance at most $K_*\theta$ from a word satisfying every local and
+edge constraint.  The constants are independent of the tuple length,
+the field size, and the entry dimensions.  Duplicated-payload
+constraints are addressed separately below.
+\end{lemma}
+
+The proof uses the decreasing-disagreement method of
+\cite[Section~4.2, Algorithm~1 and Propositions~4.7--4.8]{DEL+2022}.
+We prove the version for joint tuples with commuting twists needed here.
+
+\begin{proof}
+First replace every locally invalid whole tuple at a vertex by zero.
+There are at most $\theta|V|$ such vertices.  Initially at most
+$d\theta|V|$ undirected edges conflict, and this replacement creates
+at most $2d\theta|V|$ further conflicts.  Starting with this locally
+valid word, repeatedly replace a whole vertex by a locally valid tuple
+whenever that strictly decreases its number of incident conflicting
+edges.  The total integer conflict count strictly decreases, so at
+most $3d\theta|V|$ replacements occur.  This argument establishes
+existence and termination; it is not asserted to be an efficient
+search through the alphabet.  At most $(1+3d)\theta|V|$ vertices have
+been touched, and the final conflict count is no larger than the
+count before the replacements.
+
+At a stable vertex $v$, let $p_A(v),p_B(v)$ be the fractions of its
+A and B edges that conflict, and put
+$p(v)=p_A(v)+p_B(v)$ and $\mu=\E_Vp$.  Neighbors across A edges
+suggest rows, transported by the appropriate twist or inverse twist.
+They are valid B-code rows.  Neighbors across B edges similarly
+suggest valid A-code columns.  At cell $(i,j)$ these suggestions
+agree unless at least one of the two opposite square edges conflicts:
+the common far corner and commuting twists supply the equality when
+both opposite edges agree.  If $P_A,P_B$ average over the respective
+neighbor maps, their cell-disagreement fraction is at most
+$P_Ap_B(v)+P_Bp_A(v)$.  Lemma~\ref{oc:local-agreement} gives a local
+replacement whose incident conflict fraction is at most $C_0$ times
+this number.  Stability therefore gives pointwise
+\begin{equation}\label{oc:stability}
+ p\le C_0(P_Ap_B+P_Bp_A)\le C_0(P_Ap+P_Bp).
+\end{equation}
+
+All norms and inner products in this proof use uniform normalized
+counting measure on $V$.  Let $\bar p$ be the function equal to the
+mean of $p$ on each of the four parts.  Nonnegativity gives
+$\|\bar p\|_2\le2\mu$.  Each neighbor operator permutes the part
+means and has norm at most $\lambda$ on their orthogonal complement.
+Consequently \eqref{oc:stability} implies
+\[
+ \|p\|_2\le2C_0(2\mu+\lambda\|p\|_2),
+ \qquad \|p\|_2\le C_1\mu,\quad C_1=8C_0,
+\]
+by \eqref{oc:lambda-choice}.  This bound has no dependence on $d$.
+
+Fix an A label $i$, let $S_i$ be the vertices whose incident A edge
+of that label conflicts, and let $p_i=\Pr_V(S_i)$.  At such an edge
+the difference of its two transported rows is a nonzero B-codeword,
+so it has at least $\delta_1d$ nonzero cells.  In each corresponding
+B direction, either a B edge conflicts at one A endpoint or an
+$i$-label A edge conflicts at the opposite B neighbor.  The
+involution exchanging the two A endpoints preserves $S_i$.  Averaging
+this implication, and then subtracting the four part means in the
+mixing term, yields
+\begin{align}
+ \delta_1p_i
+ &\le 2\E_V[1_{S_i}p_B]
+      +\langle1_{S_i},P_B1_{S_i}\rangle \notag\\
+ &\le 2\E_V[1_{S_i}p_B]+4p_i^2+\lambda p_i.
+ \label{oc:conflict-propagation}
+\end{align}
+Indeed, the squared norm of the four part means of $1_{S_i}$ is at
+most $4p_i^2$, and the squared norm of its remaining part is at most
+$p_i$.  Also $p_i\le d\mu$.  For sufficiently small $\mu$, depending
+on the now fixed $d$, we have $4d\mu+\lambda<\delta_1/2$.
+Average \eqref{oc:conflict-propagation} over $i$ and repeat the
+argument with A and B exchanged.  It follows that
+\begin{equation}\label{oc:stable-zero}
+ (\delta_1/2)\mu
+ \le4\E_V[p_Ap_B]\le4\E_V[p^2]\le4C_1^2\mu^2.
+\end{equation}
+If $0<\mu<\delta_1/(8C_1^2)$ this is impossible.  Since the initial
+zeroing left at most $3d\theta|V|$ conflicts, stability has
+$\mu\le6\theta$.  For example, choose $\theta_*$ so that
+$6\theta_*<\min\{\delta_1/(32d),\delta_1/(16C_1^2)\}$.
+Then $\mu=0$.  The final word is exact, and
+\eqref{oc:packing-metric} gives $K_*=4(1+3d)$ as a permissible
+proximity constant for this normalization.  Common scalar local
+checks, block diagonal twists, and joint conflict events justify
+every step for an arbitrary tuple without a slot-count factor.
+\end{proof}
+
+\begin{lemma}[One-axis distance and exact comparisons]\label{oc:one-axis}
+Let $\Delta_A=(\delta_1-\lambda)/4>0$.  Every nonzero A-only or
+B-only word with the specified local constraints and invertible twists
+has relative vertex support at least $\Delta_A$ on $V$.  The same
+holds for differences of two words with the same axis constraints,
+and for arbitrary finite joint tuples.  It also holds for full words.
+
+Among exact full words, the duplicated-payload condition has a fixed
+gap under the following checks: at $s=0$, compare all systematic A
+rows as whole rows, at both values of $t$; at $t=0$, compare all
+systematic B columns as whole columns, at both values of $s$.
+If any payload is not duplicated, the packed-center rejection
+fraction is at least $\Delta_A$.  The tests accept every word $Ev$.
+
+More generally, suppose an expression $X$ and a full target $Y=Ev$
+both obey the same A constraints, and the systematic rectangle of
+$X$ is duplicated with payload $w$.  Equality of their whole
+systematic B columns at $t=0$, including both $s$ parts, holds if
+and only if $v=w$.  If $v\ne w$, the comparison rejects at least a
+$\Delta_A$ fraction of packed centers.  The symmetric assertion
+holds for a shared B axis and comparisons of whole systematic A
+rows at $s=0$.  No constraint on the other axis of $X$ is needed.
+\end{lemma}
+
+\begin{proof}
+For one A slice with fixed $t$, a nonzero vertex has at least
+$\delta_1d$ nonzero row symbols by local distance.  Each is carried
+to a nonzero symbol at its opposite endpoint, because twists are
+invertible.  If $l,u$ are the two vertex-support fractions and
+$a=\max(l,u)$, the fraction $e$ of nonzero carried edges obeys
+\[
+ \delta_1a\le e\le lu+\lambda\sqrt{lu}\le a(a+\lambda).
+\]
+Thus $a\ge\delta_1-\lambda$.  Support on these two parts is at
+least $(\delta_1-\lambda)/2$ in their own normalization and at least
+$\Delta_A$ in the four-part normalization.  At least one slice is
+nonzero.  The argument for B, differences, and joint entries is
+identical.
+
+For two systematic A row labels, their difference at $s=0$, considered
+over both $t$ parts and all $k$, is a B-only word: the scalar B checks
+and B twists are the same for both rows.  If nonzero it has the
+distance just proved.  Differences between systematic B columns are
+A-only words.  A nonduplicated systematic rectangle contains two
+unequal entries connected either by a change of its A label or a
+change of its B label, so one of these comparisons is nonzero.
+Conversely, for a duplicated payload, B-axis uniqueness extends the
+same systematic row data identically at every systematic A label,
+and A-axis uniqueness does the analogous thing for columns.
+
+For the last assertion, if $v=w$, A-axis uniqueness identifies the
+two whole columns at every systematic B label.  If $v\ne w$, a
+systematic payload cell differs somewhere, so one of the compared
+column differences is a nonzero A-only word and has support at least
+$\Delta_A$.  The B-sharing assertion is proved in the same way.
+These comparisons use whole one-axis codewords, which is why an
+arbitrary localized payload disagreement has a constant exact gap.
+\end{proof}
+
+The next estimate will control a growing tuple of A-only scratch
+slots against one exact reference.  It uses the tuple's joint edge
+conflicts, with no factor for its number of components.
+
+\begin{lemma}[One-axis near-code estimate]\label{oc:near-code}
+Let $W$ and $V_0$ be locally A-valid joint tuples on $V$, and suppose
+$V_0$ has exact A agreement with the same twists as $W$.  Let
+$p=\dist_V(W,V_0)$ and let $z_A$ be the A-edge conflict fraction of
+$W$, normalized as the probability that a uniform vertex and a
+uniform A label give a conflict.  Then
+\begin{equation}\label{oc:near-code-inequality}
+ \delta_1p\le z_A+4p^2+\lambda p.
+\end{equation}
+In particular, if $p\le\delta_1/16$, then
+$p\le2z_A/\delta_1$.  Both statements hold for every finite tuple
+of A-only and full slots, uniformly in its size.  They also hold
+with A and B exchanged.
+\end{lemma}
+
+\begin{proof}
+Let $S$ be the joint vertex-disagreement set.  At each vertex in $S$,
+local distance gives at least $\delta_1d$ nonzero difference rows.
+For each corresponding label, either the other endpoint also lies
+in $S$, or agreement of $V_0$ forces an actual conflict of $W$.
+Consequently
+\[
+ \delta_1p\le z_A+\langle1_S,P_A1_S\rangle
+ \le z_A+4p^2+\lambda p,
+\]
+by the four-part mixing bound used above.  For
+$p\le\delta_1/16$ and \eqref{oc:lambda-choice},
+$4p+\lambda<\delta_1/2$, giving the stated rearrangement.
+The proof concerns one fixed received word and one joint reference.
+It can be applied to successively larger prefixes of a tuple without
+modifying the received word or accumulating a number-of-prefixes
+loss.
+\end{proof}
+
+We also allow a clock or control tuple repeated identically at every
+$k$.  For any received tuple, let its distinct-value class proportions
+be $u_j$.  Equality across an expanding symmetric A list has probability
+at most
+\begin{equation}\label{oc:clock-gap}
+ \sum_j u_j^2+\lambda\left(1-\sum_j u_j^2\right).
+\end{equation}
+This follows by applying the mean-zero bound to each class indicator
+and summing.  If the equality rejection is $\eta$, then
+$1-\max_j u_j\le1-\sum_j u_j^2\le\eta/(1-\lambda)$.
+Thus a small rejection fraction implies proximity to one constant
+tuple, irrespective of its bit length.
+
+Use unique bit representations of field elements and canonical zero
+padding.  At a packed center check parsing, ranges, and padding.
+Replace a malformed symbol by a fixed parsable default when reasoning
+from acceptance.  If all checks read from one fixed finite list
+$\Pi$ of permutation addresses, modifying a set of $u|\Gamma|$
+symbols can change their center outcomes on at most
+$|\Pi|u|\Gamma|$ centers, by taking the union of inverse images.
+This applies even when many slots are checked together or branches
+use only subsets of the list.  Combining this observation with
+Lemma~\ref{oc:full-proximity}, \eqref{oc:clock-gap}, and the
+duplicated-payload gap of Lemma~\ref{oc:one-axis} gives proximity
+to exact duplicated full encodings and a repeated clock whenever
+the total packed rejection is sufficiently small.  Indeed, the
+one-time repair changes only $O(\theta)$ centers and hence leaves
+duplication rejection $O(\theta)$; choose its threshold below the
+fixed exact gap.  Any required public range for a corrected clock
+is likewise enforced, since a constant illegal value fails its
+local range test everywhere.  These constants may depend on the
+fixed address-list size, but not on the slot count.
+
+\begin{theorem}[Outer-code interface]\label{thm:outer-codes}
+There exist fixed constants $d$, $\delta_1>0$, $C_0<\infty$,
+$\lambda>0$, $\theta_{\mathrm{out}}>0$, and $K_{\mathrm{out}}<\infty$, with
+$\lambda\le\min\{(8C_0)^{-1},\delta_1/16\}$, having the following
+uniform properties for every input length $N$ and its padded tape
+length $h=O(N)$, after the fixed padding in \eqref{oc:outer-size}.
+\begin{enumerate}[label=(\roman*)]
+\item The group $\Gamma=\F_2^r\rtimes H$ has
+$|\Gamma|\le M_or^9=O(N\operatorname{polylog} N)$, where $M_o=2^r=O(N)$.
+Its A and B label lists are symmetric, of degree $d$, and have
+mean-zero norm at most $\lambda$.  There is a field
+$J=\F_{p^4}$ with $p>M_o>|H|$, $p\equiv1\pmod{M_o}$, and a
+primitive $M_o$-th root $\zeta\in\F_p$.  All public group and field
+data and the common scalar check matrices are deterministically
+constructible in polynomial time in $N$, and the data needed at an
+index have $\operatorname{polylog}(N)$ size and Boolean computation cost.
+\item Each of the scalar and three matrix-representation families,
+with either axis independently twisted or untwisted as specified
+in \eqref{oc:matrix-twists}, has the simultaneous systematic
+encoder \eqref{oc:encoder}.  Arbitrary systematic rectangle data
+extend uniquely; the A and B extensions commute.  Duplicated payload
+encoding is injective, linear, and has rate $1/(4d^2)$ in field
+coordinates.  A full slot has $\operatorname{polylog}(N)$ bits per packed site.
+Arbitrary finite tuples reuse the same checks and inverses.
+\item The local distance and tensor-agreement statement of
+Lemma~\ref{oc:local-agreement} holds uniformly in field size and
+joint vector dimension.  Local validity, edge agreement, and
+duplicated-payload comparisons have perfect completeness.  For
+every finite tuple of full slots, if their joint packed-center
+rejection fraction is $\theta\le\theta_{\mathrm{out}}$, their packed
+distance to a tuple of exact duplicated encodings is at most
+$K_{\mathrm{out}}\theta$.  Canonical parsing and a repeated control
+tuple can be included with the same form of conclusion, changing
+only the fixed constants.
+\item The one-axis distance and shared-axis comparison gaps of
+Lemma~\ref{oc:one-axis} hold with $\Delta_A=(\delta_1-\lambda)/4$.
+For every actual and reference joint tuple satisfying the hypotheses
+of Lemma~\ref{oc:near-code},
+$\delta_1p\le z_A+4p^2+\lambda p$ and, when
+$p\le\delta_1/16$, $p\le2z_A/\delta_1$.  A-only slots require
+no B constraints for these conclusions.  The symmetric B assertions
+hold as well.
+\end{enumerate}
+All constants are independent of $N$ and of the number of later
+program slots.  The field search has only $8c_0d_0$ indeterminates,
+at most $26$ parity determinant factors, and total degree less than
+$M_o^3$; in particular it causes no feedback into $C_0$, $\lambda$,
+or $d$.
+\end{theorem}
+
+\begin{proof}
+Parts (i) and (ii) follow from Lemmas~\ref{oc:expansion} and
+\ref{oc:systematic}, the field construction, and the coordinate
+count after \eqref{oc:encoder}.  Part (iii) is
+Lemma~\ref{oc:local-agreement}, Lemma~\ref{oc:full-proximity}, and
+the parsing and exact-gap argument following \eqref{oc:clock-gap}.
+Decrease $\theta_{\mathrm{out}}$ and increase $K_{\mathrm{out}}$ once to
+cover those fixed costs.  Part (iv) is
+Lemmas~\ref{oc:one-axis} and \ref{oc:near-code}.  The degree and
+dependency assertions are \eqref{oc:number-variables}--
+\eqref{oc:determinant-degree} and the parameter order fixed in
+\eqref{oc:lambda-choice}.  No external local-testability theorem is
+used for these specialized conclusions.
+\end{proof}
+
+\subsection{Correction with value-independent uncertainty supports}
+
+The next guarantee is stronger than correction of a single fixed
+word.  It tracks a set on which values are wholly unspecified, and
+therefore applies to separate bounded-cone simulations with different
+choices on that set.
+
+\begin{lemma}[Outer uncertainty shrinkage]\label{lem:outer-uncertainty}
+Fix the data in Theorem~\ref{thm:outer-codes}, and any finite tuple of
+full or A-only slots with their allowed A twists.  Optionally include
+a control tuple constant over $\Gamma$ and canonical padding.
+There is a deterministic synchronous local correction map $\mathcal C$
+with the following property.  Let $V_0$ be any reference word satisfying
+all A local constraints and A agreements, with constant control and
+canonical parsing.  Let $S\subseteq\Gamma$ and let $W$ be any input
+equal to $V_0$ outside $S$, with arbitrary values or malformed symbols
+on $S$.  Then $\mathcal C W=V_0$ outside
+\begin{equation}\label{oc:uncertainty-set}
+ S'=
+ \left\{k:\frac1d\#\{i:a_ik\in S\}\ge\delta_1/2\right\}
+ \ \cup\
+ \left\{k:\frac1d\#\{i:a_i^{-1}k\in S\}\ge\delta_1/2\right\}.
+\end{equation}
+In particular, $S'$ depends on $S$ alone.  If $u=|S|/|\Gamma|<\delta_1/4$,
+then
+\begin{equation}\label{oc:uncertainty-shrink}
+ \frac{|S'|}{|\Gamma|}
+ \le\frac{2\lambda^2u}{(\delta_1/2-u)^2}\le u/2.
+\end{equation}
+The local map uses a fixed number of permutation addresses and time
+polynomial in the symbol bit length and the public local-parameter
+length, with total default behavior on arbitrary inputs.
+
+For every fixed integer $q\ge0$, the $q$-round map has a bounded
+input cone consisting of fixed permutations of its target; the bound
+may depend on $q$ but not on $N$.  For an arbitrary initial $S$, the
+recursively defined supports contain the errors simultaneously for
+all completions on $S$.  If $|S|/|\Gamma|<\delta_1/4$, their densities
+are at most $2^{-q}|S|/|\Gamma|$ after $q$ rounds.  Different target-cone
+simulations may use different such completions, provided each simulation
+assigns one consistent initial value to every address repeated within
+that cone.  Under this small-initial-support hypothesis,
+$O(\log|\Gamma|)$ rounds correct the whole word exactly in time
+polynomial in its explicit size and $N$.
+\end{lemma}
+
+\begin{proof}
+At a target part $(k,0,t)$, take the row of label $i$ from
+$(a_ik,1,t)$ and apply $\alpha_i^{-1}$.  At $(k,1,t)$, take it from
+$(a_i^{-1}k,0,t)$ and apply $\alpha_i$.  Combine all other indices
+and all slots into one row-symbol vector.  Decode these $d$ row
+suggestions in $C_A^s$ up to strictly fewer than $\delta_1d/2$
+joint row errors.  There is at most one such local word by the
+distance in Lemma~\ref{oc:local-agreement}.  A total deterministic
+implementation enumerates all subsets $T\subseteq[d]$ with
+$|T|>d-\delta_1d/2$, solves the local scalar constraints together
+with the suggested values on $T$ componentwise, and verifies that
+the candidate has fewer than $\delta_1d/2$ joint row errors.
+Such a puncturing is injective by local distance, so each consistent
+system gives at most one candidate.  If the distance promise holds,
+the set of all correct suggestions is one enumerated $T$ and gives
+the required word.  Return zero if no candidate passes.  Malformed
+neighbor symbols are first replaced by the fixed parsable default.
+Restore each control bit by majority on the forward A-neighbor list,
+with fixed tie behavior, and write canonical padding.
+
+If $k\notin S'$, each relevant list contains fewer than
+$\delta_1d/2$ positions of $S$.  Every other row suggestion equals
+the corresponding row of $V_0$, by its exact A agreement.  Thus the
+local decoder recovers all four reference parts and all slots
+jointly.  The same good-neighbor count is strictly below $d/2$, so
+control majorities are also correct.  This proves the support
+containment for every assignment of values on $S$.
+
+For either forward or inverse averaging operator $P$ on $\Gamma$,
+the mean-zero bound gives
+\[
+ \E_k(P1_S(k)-u)^2
+ \le\lambda^2u(1-u)\le\lambda^2u.
+\]
+When $u<\delta_1/4$, Chebyshev's inequality at threshold
+$\delta_1/2-u>0$ and a union bound for the two lists prove the first
+inequality in \eqref{oc:uncertainty-shrink}.  Its right side is then
+at most $32\lambda^2u/\delta_1^2\le u/2$ by
+\eqref{oc:lambda-choice}.
+
+Iterate the deterministic operation on supports in
+\eqref{oc:uncertainty-set}.  The preceding reasoning holds for every
+completion at each step, so the same iterated supports contain all
+possible errors.  A depth-$q$ dependency cone is obtained by
+composing at most $q$ forward or inverse A moves, with the finite
+part and slot bookkeeping.  For fixed $q$ this is a fixed finite
+list of permutations.  A consistent assignment to repeated initial
+addresses in one cone extends to a global initial word agreeing
+with $V_0$ outside $S$.  Applying the global support statement to
+that extension proves the claimed result for that cone.  This
+argument can use a different global extension for each target.
+
+Assume now $|S|/|\Gamma|<\delta_1/4$.  Every iterated support stays
+in this regime, so its integer cardinality decreases by a factor of
+at least two while nonempty.  More than $\log_2|\Gamma|$ rounds make
+it zero.  Running these rounds synchronously on the complete array
+uses $O(|\Gamma|\log|\Gamma|)$ local calls, each of polynomial
+bit complexity as established above.  This proves polynomial-time
+global correction without using the existential greedy repair in
+Lemma~\ref{oc:full-proximity} as an algorithm.
+\end{proof}

preprints/The-PCP-for-PPAD-conjecture-a-quasilinear-reduction-September-25-2026/build/sections/04-outer-programs.tex

@@ -0,0 +1,864 @@
+\section{Certified programs and persistent tape equations}\label{sec:outer-programs}
+
+We turn local code constraints into certificates for computations on the
+tape. This section supplies local reversible filling
+(Lemma~\ref{lem:outer-fill}) and a persistent tuple of tape equations
+(Lemma~\ref{lem:outer-wiring}); Section~\ref{sec:outer-representation}
+will assemble them into the evolving representation.
+
+We retain the outer notation of Sections~\ref{sec:macrograph} and
+\ref{sec:outer-codes}: $h=2^\ell$, $M_o=2^r$, the odd-characteristic
+field $J$, the group $\Gamma=\mathbb F_2^r\rtimes H$, and coordinates
+$k=(g,u)=(ux,u)$.  In particular $r=O(\log N)$,
+$|H|\le r^9$, and a field element has $O(\log N)$ bits.
+All constants and polynomial exponents in this section are fixed
+independently of $N$.  The local star degree $d$ is a constant.
+We pack the four parts $(s,t)\in\{0,1\}^2$ at one $k$ into one symbol;
+``part distance'' instead normalizes over the four parts separately.
+These distances differ by at most a factor of four.
+
+There are two kinds of storage.  A \emph{full slot} has both axis
+constraints and duplicated systematic payload, and thus equals
+$E_{\alpha,\beta}v$ for a unique payload $v:\Gamma\to U$.
+An \emph{A-only scratch slot} has the same local A constraints and
+A-edge agreements, but its other labels and entry coordinates are
+merely vector coordinates: no B condition is imposed.  Keeping all
+intermediate certificates, rather than just the values being computed,
+will let us certify a computation at every point of a reversible schedule.
+
+\subsection{Full operations and a fixed query list}
+
+\begin{definition}[Certified full operation]\label{op:full-operation}
+A full operation specifies parent full slots, a target encoder
+$E_{\mathrm{out}}$, and a local expression $X^{st}_k(i,j)$ in the parents.
+For arbitrary exact duplicated full parents, $X$ must have duplicated
+systematic payload $v$ and satisfy an allowed A-axis constraint system.
+In addition, either its A system is that of the target, or $X$ satisfies
+the target's B system.  The required target is $E_{\mathrm{out}}v$.
+Scalar coefficients in linear combinations are identical at every
+index.  They may depend on an identical clock and on previously
+certified, completed public address data.  The specified covariant
+primitive $W(k)\mapsto W(k)\rho_e(k)$ below may additionally use its
+explicit index-dependent public matrix; it must satisfy the same
+axis-constraint requirements on $X$.
+\end{definition}
+
+In the first case compare $X$ and the target on all systematic B columns
+at $t=0$, retaining every entry in each such column at both A parts.
+In the second case compare all systematic A rows at $s=0$, retaining
+their entire rows at both B parts.  The one-axis uniqueness and distance
+in Theorem~\ref{thm:outer-codes} give a constant $\gamma_{\mathrm{op}}>0$
+with the following alternative: the required payload identity is true
+and all comparisons hold, or these comparisons fail on at least
+$\gamma_{\mathrm{op}}$ of the centers.  Indeed, equality of payloads
+fixes each compared column or row by the shared one-axis extension.
+If a payload differs, at least one compared column or row is a nonzero
+codeword difference for that axis, so its vertex support has constant
+density.  The number of labels and parts is fixed.  A comparison of
+isolated systematic cells would not have this implication.
+
+We use the following full operations, splitting sums into bounded-arity
+stages whenever necessary.
+\begin{enumerate}[label=(\roman*)]
+\item Scalar linear combinations and right pulls
+$R_bF(k)=F(kb)$ share the scalar A system.  Elementary pulls are the
+identity, the three first-bit translations $(e_a,1)$, and one-step
+Singer-cycle rotations of each block and their inverses.
+\item A matrix payload can be reencoded while changing only one axis
+type, using the other as the shared axis.  Constant linear maps and
+contractions of fully untwisted entry alphabets also give full operations.
+\item To multiply a payload by a one-bit Walsh character, we must also
+certify the resulting full encoding. An index-dependent scalar
+multiplier need not preserve the scalar edge equations. The matrix
+representation supplies a shared-axis route through the twisted types.
+Starting from an untwisted scalar slot $W$, form
+\[
+             X^{st}_k(i,j)=W^{st}_k(i,j)\rho_e(k),
+\]
+where $e$ is a distinguished first bit of a block and $\rho_e$ is the
+representation of Section~\ref{sec:outer-codes}.  Multiplicativity gives
+\[
+ X^{1t}_{a_i k}(i,j)=\rho_e(a_i)X^{0t}_k(i,j),\qquad
+ X^{s1}_{k b_j}(i,j)=X^{s0}_k(i,j)\rho_e(b_j).
+\]
+Scalar local checks continue to hold, since the same entry matrix
+multiplies every local scalar.  Reencode first with twisted A and
+untwisted B, sharing A, and then with both axes untwisted, sharing B.
+The column of $\rho_e(g,u)$ indexed by $1_H$ has just one nonzero entry,
+in row $u$, equal to
+$(-1)^{(ue)\cdot g}=(-1)^{e\cdot x}$.
+Summing this entry-alphabet column therefore produces the full scalar
+encoding of $(-1)^{e\cdot x}v_W(k)$.  The contraction is on entry
+coordinates, not on outer star labels.
+\end{enumerate}
+Every intermediate matrix slot in this construction has a specified
+homogeneous linear payload recipe.  An arbitrary multiplier depending
+on one bit, with values $a_0,a_1\in J$, is
+$\frac12(a_0+a_1)I+\frac12(a_0-a_1)M_e$, where
+$M_ef(x,u)=(-1)^{e\cdot x}f(x,u)$ is the corresponding modulation.
+The divisions by two are valid because $J$ has odd characteristic.
+
+Right rotations move these operations to any bit in a block.  For a
+block rotation $v$ the pull convention gives
+\[
+ R_{(0,v)}R_{(e,1)}R_{(0,v^{-1})}=R_{(ve,1)}.
+\]
+Conjugating the modulation in the same way gives the modulation of bit
+$ve$.  A power of a rotation is implemented as a sequence of single
+rotations, returning to the original layout after the operation.
+Thus an arbitrary bit operation costs $O(r)$ elementary full stages,
+without introducing new query maps.
+
+Fix once and for all a permutation list $\Pi$ containing identity,
+the required forward and inverse A and B neighbor maps, left pulls by
+a fixed generating list of $H$, the elementary right pulls above,
+and right pulls by a fixed generating list of $H$ and its inverses.
+Include any bounded compositions used in a local expression.
+Its cardinality $q=|\Pi|$ is independent of $N$.
+All constraints of all slots are checked together at a center using
+the symbols at $\Pi(k)$.  A growing number of slots consequently
+increases local computation and symbol size, but not the number of
+queried positions.  In particular a single false exact full operation
+retains the gap $\gamma_{\mathrm{op}}$, rather than having its gap
+divided by the program length.
+
+\subsection{Scalar transforms with certified stages}
+
+\begin{lemma}[Scalar program library]\label{op:scalar-programs}
+On scalar tables indexed by $x\in\mathbb F_2^r$, the full operations
+above give homogeneous linear programs of $\poly(r)$ stages for:
+bit-conjunction and interval masks; controlled bit flips and bit
+permutations; integer cyclic shifts; discrete Fourier transforms and
+their inverses of lengths $M_o$ and $h$; the Walsh transform; and cyclic
+coefficient multiplication by any systematic Lagrange polynomial
+$L_j$, $0\le j<h$.  The last operation is uniform also when $j$ is
+supplied by already identical address bits.  All intermediate stages
+are full slots with fixed encoder types and query maps in $\Pi$.
+\end{lemma}
+
+\begin{proof}
+A bit-conjunction mask is a product of one-bit masks, applied in
+sequence.  For the integer interval $[0,a)$ with $0\le a\le M_o$,
+represent $a$ by $r+1$ public or already identical control bits.
+Use the zero operator for $a=0$ and the identity for $a=M_o$.
+For $0<a<M_o$, use the disjoint prefix rectangles obtained by
+specifying the most significant differing bit: at a position where
+$a$ has bit one, match its more significant bits and require the
+current bit to be zero.  There are at most $r$ rectangles.
+More explicitly, writing $a_r$ for the endpoint bit, one fixed
+padded program applies the mask
+\[
+ a_r+(1-a_r)\sum_{i=0}^{r-1}a_i\,
+        1[x_i=0]\prod_{j=i+1}^{r-1}1[x_j=a_j].
+\]
+All endpoint decisions and rectangle coefficients are identical at
+every index; the program keeps all candidate stages and uses scalar
+coefficient selection, so its topology and polynomial size do not
+depend on $a$.  Differences of two such masks give arbitrary intervals.
+
+Let $m$ be the indicator of a conjunction not involving bit $i$.
+The pullback of the permutation that flips bit $i$ precisely when
+$m=1$ is
+\begin{equation}\label{op:controlled-flip}
+                    f-mf+R_{(e_i,1)}(mf).
+\end{equation}
+To verify this, if $m(x)=0$ then $m(x+e_i)=0$, and the result is $f(x)$;
+if $m(x)=1$ then $m(x+e_i)=1$, and the result is $f(x+e_i)$.
+The independence of $m$ from the flipped bit is essential here.
+Three controlled XOR operations exchange two index bits, so they
+implement bit reversal and any prescribed bit permutation.
+
+For an integer subtraction of $2^j$ modulo $2^r$, flip bit $i>j$
+when the original bits $j,\ldots,i-1$ are all zero, and flip bit $j$
+unconditionally.  Process the higher bits in descending order;
+then none of the bits used by a condition has yet changed.
+The addition rule uses ones instead of zeros.  Pullbacks reverse the
+chronological composition of index permutations: thus the table
+stages apply these borrow gates in ascending bit order, the reverse
+of their descending index-permutation order.  This implements the
+desired cyclic table shift.  A conjunction uses $O(r)$ bit
+operations, so this procedure is polynomial in $r$.
+
+For the Fourier transform we realize the radix-two Cooley--Tukey
+decomposition~\cite[pp.~299--300]{CooleyTukey1965} by the certified full
+operations above. Let $D=2^v\in\{h,M_o\}$ and let
+$\xi=\zeta^{M_o/D}$ have order $D$.
+The transform acts on the low $v$ integer bits, independently within
+each block of fixed higher bits, and is
+\begin{equation}\label{op:dft}
+       \mathcal F_D(f)(y)=\sum_{z=0}^{D-1}\xi^{yz}f(z).
+\end{equation}
+First reverse these $v$ bits.  At level $j$, pair positions separated
+by XOR of bit $j-1$ in each aligned window of length $2^j$.
+Writing $a<2^{j-1}$ for the offset and $U,V$ for the lower and upper
+entries, replace the pair by
+\begin{equation}\label{op:butterfly}
+          U+\xi^{aD/2^j}V,\qquad U-\xi^{aD/2^j}V.
+\end{equation}
+The factor is a product of one-bit factors, since if
+$a=\sum_{i<j-1}2^ia_i$, then
+$\xi^{aD/2^j}=\prod_{i<j-1}(\xi^{2^iD/2^j})^{a_i}$.
+Half-window masks, the bit pull, these factors, and addition implement
+the displayed rule.  The identity
+\[
+ \sum_{z=0}^{D-1}\xi^{yz}f(z)
+ =\sum_{z=0}^{D/2-1}(\xi^2)^{yz}f(2z)
+  +\xi^y\sum_{z=0}^{D/2-1}(\xi^2)^{yz}f(2z+1)
+\]
+proves the algorithm by induction, with bit reversal placing the
+recursive even and odd subproblems in the required order.  More
+explicitly, if $\operatorname{rev}_a$ reverses $a$ bits, after level
+$j$ the entry at $b2^j+y$, $0\le y<2^j$, is
+\[
+ \sum_{z=0}^{2^j-1}(\xi^{D/2^j})^{yz}
+ f\bigl(\operatorname{rev}_{v-j}(b)+2^{v-j}z\bigr).
+\]
+The level-$j$ butterfly combines the even and odd values of $z$ in
+the level-$(j-1)$ formula, proving this invariant from the bit-reversed
+input at level zero; at level $v$ it is \eqref{op:dft}.
+Using $\xi^{-1}$ and multiplying by $D^{-1}$ gives the inverse:
+$\sum_{y=0}^{D-1}\xi^{y(z-z')}=D\,1[z=z']$.
+Without initial bit reversal, apply the sum/difference butterflies
+directly to the input table, one coordinate bit at a time, and use
+no twiddle factors.  This gives
+\begin{equation}\label{op:walsh}
+                   \mathcal W(f)(x)=\sum_y(-1)^{x\cdot y}f(y).
+\end{equation}
+They can be applied on all $r$ bits or just a specified block.
+The Walsh inverse is $M_o^{-1}\mathcal W$ when all bits are used.
+
+Finally set $\omega=\zeta^{M_o/h}$ and
+\begin{equation}\label{op:lagrange}
+ L_j(Z)=h^{-1}\sum_{a=0}^{h-1}\omega^{-ja}Z^a.
+\end{equation}
+The root sum just used gives $L_j(\omega^i)=1[i=j]$.
+For a length-$M_o$ coefficient table $F$, cyclic multiplication by
+$L_j$ has coefficient at $w$ equal to
+\begin{equation}\label{op:lagrange-convolution}
+ h^{-1}\omega^{-jw}\sum_{a=0}^{h-1}F_{w-a}\omega^{j(w-a)},
+\end{equation}
+with subscripts modulo $M_o$.  The powers are well defined under
+cyclic indexing because $h\mid M_o$.  Each of the two diagonal
+factors splits over the bits of $w$ and has coefficients computable
+from $j$.  If $T_bf(w)=f(w-b\bmod M_o)$, the middle window sum is
+\[
+                    \prod_{i=0}^{\ell-1}(I+T_{2^i}).
+\]
+Every integer in $[0,h)$ has a unique binary expansion, proving this
+identity.  The previously constructed shifts implement it.
+When $\deg(F L_j)<M_o$, cyclic and ordinary multiplication coincide;
+we impose this degree bound at every ordinary-product use.
+
+For clarity, crude stage bounds suffice: a bit operation uses $O(r)$
+elementary stages, a conjunction $O(r^2)$, a carry shift $O(r^3)$,
+and the window product $O(r^4)$.  The Fourier, Walsh, and range-mask
+constructions also fit a fixed polynomial bound.  Coefficient
+calculation, layout descriptions and every elementary local operation
+use polynomially many bit operations in $r$.  Padding unused steps
+with zero or identity operations makes all lengths independent of
+the particular control or address values.
+\end{proof}
+
+\subsection{Triangular A-only certificates}
+
+An A-only program starts from A-coded public constants or a local
+expression $X$ from Definition~\ref{op:full-operation}.  It forms
+identical-coefficient linear combinations and right pulls of earlier
+slots having the same A twist.  Right pulls commute with left
+translations and constant entry twists, and therefore preserve A
+constraints.  A fixed generating list of $H$ suffices to compute all
+desired right-$H$ pulls through stored paths in its Cayley graph.
+Each path has length at most $|H|-1$ after removing repetitions.
+Arbitrary bit pulls use the rotation sequences already described.
+The number of stored stages is polynomial in $r$, while all recurrence
+queries still belong to $\Pi$.
+
+At a fixed clock value, the initialized slots form triangular prefixes
+and satisfy their recurrences; every other scratch slot is zero.
+Filling one next stage evaluates its recurrence locally.  Erasing in
+reverse order replaces that stage by zero while its parents are still
+available.  Each step changes one designated slot or fixed family of
+slots, with the same definition of every retained slot.
+
+\begin{lemma}[Simultaneous scratch restoration]\label{op:scratch-restoration}
+Suppose all scratch recurrences at a legal repeated clock are
+triangular and their references, on arbitrary exact duplicated full
+parents, satisfy their assigned A constraints.  Public address-focus
+recurrences precede every recurrence using the resulting coefficients.
+There exist constants $C_{\mathrm{sc}}$ and $\theta_{\mathrm{sc}}>0$,
+independent of the number of stages, slots and entry coordinates,
+with the following property.  If the packed parsing, clock, full-code,
+duplication, scratch local-code, scratch-edge, recurrence and inactive-zero
+tests reject on a fraction $\theta<\theta_{\mathrm{sc}}$ of centers,
+then after one preprocessing of the parsing, clock and full slots,
+all scratch can be replaced simultaneously by its exact triangular
+reference at total symbol cost at most $C_{\mathrm{sc}}\theta$.
+The references are taken on the corrected full parents; those parents
+need not yet satisfy their advertised payload-operation identities.
+\end{lemma}
+
+\begin{proof}
+Theorem~\ref{thm:outer-codes} first repairs parsing, repetition and
+full codes at $O(\theta)$ symbol cost.  A repeated illegal clock is
+excluded by its range test.  The full-code duplication gap then
+excludes any failure of duplicated payload, for small enough $\theta$.
+Replace all locally A-invalid scratch parts by zero.  This is one
+further packed $O(\theta)$ change.  Call the resulting word $W$, and
+\emph{keep this word fixed throughout the following induction}.
+
+If preprocessing changed at most $C_{\mathrm{pre}}\theta$ packed
+sites, any test using $\Pi$ can change its outcome only on their
+$q$ inverse images.  After harmless fixed conversions of edge and
+part normalizations there are constants $C_z,C_r$ such that
+\begin{equation}\label{op:fixed-word-invariant}
+ z_A(W)\le C_z\theta,\qquad
+ \Pr[\text{some scratch recurrence or inactive-zero test fails on }W]
+       \le C_r\theta.
+\end{equation}
+One can use constant multiples of $1+qC_{\mathrm{pre}}$ for both
+constants.  These bounds concern the \emph{entire} packed scratch
+tuple, so they contain no slot-count factor.
+
+Order all initialized families topologically, putting public focus
+first, and include the inactive-zero families with their zero
+references.  Let $V_j$ be the exact reference of the first $j$ families
+on the corrected full parents and legal clock, and let $p_j$ be the
+joint part distance from the corresponding prefix of $W$ to $V_j$.
+The empty prefix has $p_0=0$.  The full parents may have incorrect
+advertised identities, but are A-coded; every allowed initial $X$
+is A-coded on such parents.  Right pulls and identical coefficients
+preserve this property.  Public focus begins with hardwired correct
+constant encodings and the identical clock.  Its completed reference
+supplies globally identical coefficients to subsequent recurrences.
+Consequently every $V_j$ is exactly A-coded.
+
+An error in the new family occurs only at a recurrence-failure center
+or at a center reading an erroneous earlier part.  A joint part-error
+set of density $p_j$ occupies at most $4p_j$ of the packed sites.
+Taking a union over the $q$ maps, and retaining the old prefix, gives
+\begin{equation}\label{op:prefix-provisional}
+                 p_{j+1}\le C_r\theta+(1+4q)p_j.
+\end{equation}
+This remains true when a recurrence reads all earlier slots or types
+at one of those sites: the error event is already joint.
+Put $K=2C_z/\delta_1$ and choose $\theta_{\mathrm{sc}}$ so small that
+\begin{equation}\label{op:scratch-smallness}
+          (C_r+(1+4q)K)\theta_{\mathrm{sc}}<\delta_1/16.
+\end{equation}
+If $p_j\le K\theta$, the provisional estimate is in the small-error
+range of the A near-code inequality of Theorem~\ref{thm:outer-codes}.
+Apply that inequality to the \emph{same fixed word} $W$, restricted
+to the enlarged tuple, and to $V_{j+1}$:
+\[
+ \delta_1p_{j+1}
+ \le z_A(W|_{j+1})+4p_{j+1}^2+\lambda p_{j+1}.
+\]
+The conflict count of a prefix is at most that of the whole tuple.
+Since $\lambda\le\delta_1/8$ and $p_{j+1}<\delta_1/16$, this implies
+\[
+                 p_{j+1}\le 2z_A(W)/\delta_1\le K\theta.
+\]
+Induction proves this bound for the complete scratch tuple.
+It resets an \emph{estimate}, without making successive word repairs.
+Only now replace the whole scratch tuple by its reference, at symbol
+cost at most $4K\theta$.  The common scalar local checks act on joint
+vectors, and the finitely many twist types act by a block-diagonal
+invertible map; the near-code estimate is therefore valid in exactly
+this joint metric.  Together with preprocessing this proves the lemma.
+\end{proof}
+
+\subsection{Mode-serial filling of a full slot}
+
+The target must remain a full codeword while it is being filled, even
+though its final value can depend on the parents at every index. We use
+a sequence of frequency cutoffs that commute with the encoder. Each
+increment is supported on one $H$-orbit, so its encoded contribution
+can be computed from a polylogarithmic list of coefficients gathered
+locally. A completed certificate for the old cutoff is retained until
+the next cutoff and its increment have both been prepared.
+
+Retaining the completed target while reversing its auxiliary computation
+follows the reversible-cleanup principle of Bennett~\cite[pp.~525--526]{Bennett1973}.
+The two-buffer schedule, local transitions, and certification guarantees
+needed here are established in the following proof.
+
+\begin{lemma}[Local reversible filling]\label{lem:outer-fill}
+For every full operation in Definition~\ref{op:full-operation}, there
+is a canonical reversible schedule from zero target and zero temporary
+scratch to its required full target and zero temporary scratch.
+Parents and all coefficient data remain fixed throughout.  The schedule
+has $O(M_o\poly(r))$ steps, a $\poly(r)$-bit local symbol and local
+algorithm, and an $O(\log M_o+\log\poly(r))$-bit microclock.
+Its checks and both local transition directions use only $\Pi$.
+At each clock location the target is a specified full encoding and
+all scratch consists of specified initialized prefixes.  After
+simultaneous scratch restoration, failure of the required target
+identity has a constant center gap.  Each reference scratch slot
+satisfies its A constraints, even when its full parents have not yet
+been shown to satisfy their advertised payload identities.
+\end{lemma}
+
+\begin{proof}
+Write the operation's local expression as $X$ and its duplicated
+systematic payload as $v$.  For each label, part and entry coordinate
+expand in the $x$ variable:
+\[
+ X(x,u)=\sum_{\nu\in\mathbb F_2^r}\widehat X_\nu(u)(-1)^{x\cdot\nu},
+ \qquad
+ \widehat X_\nu(u)=M_o^{-1}\sum_xX(x,u)(-1)^{x\cdot\nu}.
+\]
+For $0\le\tau\le M_o$ define a Fourier multiplier $P_\tau$ by
+\begin{equation}\label{op:cutoff}
+ w_\tau(\nu)=|H|^{-1}\sum_{a\in H}
+                    1[\underline{a^{-1}\nu}<\tau].
+\end{equation}
+Thus $P_0=0$ and $P_{M_o}=I$.  The inverses of $|H|$ and $M_o$
+exist in $J$.  The multipliers are constant on frequency orbits of
+$H$; they are not asserted to be projections.
+
+Here is the required commutation in detail.  A left pull by $(w,a)$
+sends $(x,u)$ to $(x+u^{-1}a^{-1}w,au)$, so it preserves each
+frequency, changing its coefficient by a known sign and an $H$-index
+permutation.  A right pull sends $(x,u)$ to
+$(a^{-1}(x+w),ua)$, carrying a frequency $\nu$ to $a\nu$, with
+sign $(-1)^{w\cdot a\nu}$.  Hence $P_\tau$ commutes with both pulls.
+The constant entry twists act on a different coordinate and also
+commute.  Every parity operator and every systematic injection thus
+intertwines $P_\tau$ on its domain and codomain.  If an invertible
+parity operator $T$ satisfies $TP_\tau=P_\tau T$, then its inverse
+does too, by multiplication by $T^{-1}$ on both sides.  Applying the
+two one-axis extensions gives
+\begin{equation}\label{op:cutoff-extension}
+              P_\tau E_{\mathrm{out}}v
+                      =E_{\mathrm{out}}(P_\tau v).
+\end{equation}
+No commutation with a preliminary index-dependent operation such as
+$W(k)\mapsto W(k)\rho_e(k)$ is needed: $P_\tau$ is applied to its
+already formed expression $X$.
+
+The mode-bit selector
+\begin{equation}\label{op:mode-selector}
+                   \frac{I+(-1)^bR_{(e_i,1)}}2
+\end{equation}
+retains precisely those frequencies with $\nu_i=b$.
+For each $a\in H$, decompose
+$\underline{a^{-1}\nu}<\tau$ into at most $r$ disjoint prefix
+rectangles and multiply their selectors.  Sum over rectangles and
+average over $a$.  This gives an A-only program computing $P_\tau X$
+with $\poly(r)$ stages.  Include all candidate rectangles and use
+coefficient bits from the microclock, so topology and length are fixed.
+The case $\tau=M_o$ uses the identity with the same padding.
+All selectors use the elementary pull list via stored bit alignments.
+
+The partial target at cutoff $\tau$ is
+$Y_\tau=E_{\mathrm{out}}(P_\tau v)$.
+Compare it with the completed cutoff program using the shared-axis
+test of Definition~\ref{op:full-operation}.  In the shared-B case,
+intermediate scratch is only A-coded.  The test's justification is
+made \emph{after} Lemma~\ref{op:scratch-restoration}: the completed
+reference is then exactly $P_\tau X$, which satisfies the target's
+B constraints by the commutation above.  One may now use B distance
+on its whole systematic A rows.  This argument neither assumes nor
+tests B validity of unfinished scratch.  The shared-A case uses the
+analogous columns.  Thus equality of the required payload suffices,
+and its failure has the same constant gap at every cutoff.
+
+It remains to compute a cutoff increment locally.  Let $a_\tau$ be
+the bit vector for the integer $\tau<M_o$, and set $O=Ha_\tau$.
+The increment multiplier is zero outside $O$ and, for $\nu\in O$, is
+\begin{equation}\label{op:orbit-weight}
+ (w_{\tau+1}-w_\tau)(\nu)
+ =\frac{\#\{a\in H:aa_\tau=\nu\}}{|H|}
+ =\frac1{|O|}.
+\end{equation}
+For each $\nu$ in the fixed-length list $(aa_\tau)_{a\in H}$,
+apply all $r$ mode selectors to $X$.  Repetitions are harmless.
+The resulting pure-mode slot is
+$X_\nu(x,u)=(-1)^{x\cdot\nu}\widehat X_\nu(u)$.
+Also store each right pull $R_{(0,b)}X_\nu$, $b\in H$, through
+generator paths and their intermediate stages.  At the current
+$k=(x,u)$ its systematic payload is
+\begin{equation}\label{op:orbit-focus}
+             (-1)^{(b^{-1}x)\cdot\nu}\widehat v_\nu(ub).
+\end{equation}
+For any desired $u'\in H$, take $b=u^{-1}u'$ and remove the displayed
+known sign.  One symbol now contains every coefficient
+$\widehat v_\nu(u')$ for $\nu\in O$ and every $u'\in H$.
+These are read from locally stored pull slots, not from remote global
+payload positions.
+
+We can apply $E_{\mathrm{out}}$ to this coefficient list locally.
+Remark~\ref{oc:orbit-invariance} supplies the inverse of each one-axis
+parity system on the $O$-frequency subspace: both this subspace and its
+complement are invariant, and the restricted domain and codomain have
+equal dimensions. Here its Fourier coordinates are indexed by frequency,
+$H$, labels, parts and entry coordinates, so the restricted dimension is
+at most
+\begin{equation}\label{op:orbit-dimension}
+         C|O|\,|H|\,\dim_J U\le C|H|^4=\poly(r),
+\end{equation}
+where $C$ depends only on the fixed labels, and
+$\dim_JU\le |H|^2$.  Form its entries using the explicit pulls and
+Walsh signs, solve the two successive axis systems by Gaussian
+elimination, and sum the output modes at $x$.  Together with
+\eqref{op:orbit-weight}, this computes every local entry of
+\[
+                 E_{\mathrm{out}}((P_{\tau+1}-P_\tau)v).
+\]
+Matrix assembly, elimination and field arithmetic take $\poly(r)$
+bit operations.  Canonical parsing and fixed defaults on malformed
+values, illegal clocks or failed pivots make the local algorithm total;
+on the promised data the prescribed matrices are invertible.
+
+Use two cutoff-program buffers and one orbit-focus buffer.
+Initially the target and all three buffers are zero.
+At iteration $\tau$ keep the completed buffer for cutoff $\tau$
+when $\tau>0$, and compare the target with it; at $\tau=0$ compare
+the entire target with zero.  In the other, empty buffer fill the
+cutoff-$\tau+1$ program in dependency order, and fill the orbit focus
+in dependency order.  With both complete, perform one pivot adding
+the local encoded increment to the target.  Its required comparison
+now uses the completed cutoff-$\tau+1$ buffer.  Erase focus in reverse
+order and erase the old cutoff buffer in reverse order when present.
+Swap the buffer roles, determined by the iteration parity.
+At cutoff $M_o$, compare the target directly with $X$ while erasing
+the final cutoff buffer.  This finishes with the full desired target
+and zero temporary scratch.
+
+At every clock value enforce the full-code and duplication constraints
+of the target, all initialized-prefix recurrences, inactive zeros,
+and the stated comparison.  While a new proof is being prepared, a
+completed old proof remains available.  At a pivot both the next
+proof and the increment focus are complete.  In reverse traversal,
+erased prefixes are first reconstructed in forward dependency order,
+so the reverse pivot subtracts the same locally available increment.
+Thus both directions are local, mutually inverse on canonical data,
+and never use a partial certificate as a completed one.
+
+For the size bound, each cutoff program uses at most
+$O(|H|r^3)$ elementary stages up to fixed padding factors.
+The orbit selectors and all right-$H$ paths use at most
+$O(|H|r^2+|H|^3)$ such stages, with a polynomial allowance for
+metadata.  Even a bound $O(r^{30})$ for both programs and their
+descriptions suffices.  Each entry alphabet has at most $r^{18}$
+field coordinates, each of $O(r)$ bits.  The two reused buffers
+therefore have $\poly(r)$ bits per index.  There are $M_o$ iterations
+of a common polynomial sublength, with a special initial omission
+and final erasure; their location is decoded by arithmetic on a
+logarithmic microclock.  No local step loops over all $M_o$ modes.
+The different cutoffs reuse the same zeroed buffers and do not
+contribute a factor $M_o$ to the word length.  Separate buffers for
+the finitely many required A types keep every slot's type fixed.
+These observations prove all bounds and claims of the lemma.
+\end{proof}
+
+\subsection{From sparse tape wiring to row systems}
+
+We next construct a persistent tuple $P(b)$ independent of the clock
+frame, where $b$ comprises the four tape-content and occupation arrays.
+Its canonical formulas are defined for every $b\in J^{4h}$; Boolean
+and phase-validity tests are additional conditions.
+Testing a uniformly chosen tape row would not detect one false row with
+constant probability. Instead we interpolate the row data, certify the
+required products, and encode the remainder of each row-equation
+polynomial modulo $Z^h-1$. A single false row makes that remainder
+nonzero; comparing its full encoding with zero then has the code's
+constant gap. The tuple retains the intermediate linear computations
+needed to perform these checks at every frame.
+
+\begin{lemma}[Persistent wiring and certification]\label{lem:outer-wiring}
+There are $\poly(r)$ persistent full slots $P(b)$, with polynomial
+canonical payload formulas in $b\in J^{4h}$, and packed local tests
+using $\Pi$, with the following properties.
+\begin{enumerate}[label=(\roman*)]
+\item Structural tests uniquely determine $P(b)$ from its four logical
+arrays.  Each variable factor array is a fixed permutation of one
+logical array; a one-address logical edit changes that factor at at
+most one publicly determined address.
+\item Each primary has assigned logical degree at most a fixed $d_2$.
+Every other persistent slot has a homogeneous linear recipe on the
+primaries, including all intermediate matrix reencodings and all
+phase-specific computations.  Its assigned logical degree can be
+the maximum degree of its primary inputs.
+\item There are persistent full encodings of
+\begin{equation}\label{op:address-transform}
+       U_D(x)=\sum_{j=0}^{h-1}(-1)^{x\cdot j}D_j
+\end{equation}
+for every needed logical singleton $D$, where $j$ denotes its padded
+$r$-bit vector.  Its Walsh coefficient at $j<h$ is exactly $D_j$.
+\item Conditional on exact duplicated full codes and a legal repeated
+clock, either all structural formulas hold and $b$ is a valid
+Boolean macrovertex at that clock, or one packed test rejects on
+at least a fixed fraction $\gamma_{\mathrm{per}}>0$ of centers.
+The same alternative without the macro-validity conclusion holds
+if only structural tests are imposed.
+\end{enumerate}
+The canonical tuple and public data are deterministically constructible
+in $\poly(N)$ time.  The number of field coordinates, public local
+parameters and bit operations at one index is $\poly(r)$.
+\end{lemma}
+
+\begin{proof}
+\paragraph{Sparse row equations.}
+Use the normalization in Lemma~\ref{lem:macrograph}.
+The Boolean tests for the four arrays are $z(z-1)=0$ at each row,
+and content must vanish where occupation is zero, by $a(1-e)=0$.
+For occupation in a prescribed order $\pi$, impose
+\[
+        e_{\pi(i+1)}(1-e_{\pi(i)})=0\quad(0\le i<h-1).
+\]
+Together with Booleanity, these equations say that occupied positions
+form a prefix.  Its desired length is imposed separately by its
+integer count.  Full occupation may instead be imposed row by row.
+Fixed gate truth tables have multilinear representing polynomials
+over $J$: for a Boolean input $a\in\{0,1\}^m$ its indicator is
+$\prod_{i:a_i=1}z_i\prod_{i:a_i=0}(1-z_i)$, so summing the indicators
+weighted by the truth-table values gives the desired polynomial.
+The arity is fixed.  Gate, input-copy, designated-comparison, dummy,
+and tape-to-tape-copy equations are conditioned by the occupation
+of their destination where appropriate.  This expresses exactly
+the phase rules in Section~\ref{sec:macrograph}; the phase prescription
+ensures that an occupied computation only depends on earlier occupied
+positions or the fixed full tape.  In a full $Q_x$, its $S$ inputs
+are free, with the terminal and other-input relations imposed as in
+its definition.  At shared phase boundaries fix either equivalent
+description.
+
+For each of the four phases and each unconditional family, organize
+these equations into a fixed number of length-$h$ lists, padding with
+zero equations.  Every operand map from list rows to an original
+logical array has bounded multiplicity: this follows from bounded
+fan-out for internal feeds, separate designated copies, and partial
+permutations for adjacency and same-position copies.
+An operand map of multiplicity at most $B$ splits into $B$ partial
+injections by coloring its preimages separately at each target.
+Each partial injection extends to a permutation of $[h]$, by any
+bijection between the unused domain and range.  Separate logical
+array types and intersect the finitely many operand-color and
+active-row cases.  Because arity and $B$ are constants, this causes
+only a constant number of cases.  Expand the representing polynomials
+into monomials.  Each row list is now a sum of a fixed number of
+bounded-length monomials, whose factors are permuted logical arrays
+or public coefficient and mask arrays.  Empty monomials represent
+constant terms.  Counting coefficient factors in the length still
+gives a fixed bound on the degree.
+
+\paragraph{Routing arbitrary permutations.}
+For completeness a permutation of $2^\ell$ locations has a binary
+switch realization with $2\ell-1$ layers, each layer switching pairs
+along one fixed XOR bit, with a public Boolean mask constant on each
+pair. This is the binary specialization of the Bene\v{s}
+rearrangeability construction~\cite[Lemma~3 and Theorems~2--3]{Benes1964Permutation}.
+Pair sources along the top
+bit and do the same for targets.  The prescribed item routes form a
+bipartite multigraph on these pairs with degree two at every vertex.
+Every component is an even cycle, including a pair of parallel edges;
+alternate its edges with two colors.  An initial top-bit switch and
+a final top-bit switch route each item through the half corresponding
+to its color.  Each half now has a permutation of $2^{\ell-1}$ items.
+Recurse with the same layer schedule in the two halves.  The base case
+$\ell=1$ has one layer and $\ell=0$ none; the recurrence for depth is
+$L(\ell)=L(\ell-1)+2$.  Cycle coloring and recursive routing take
+polynomial time in $h$.
+
+Introduce a variable factor type for every routed intermediate, and
+for each layer introduce the XOR-pulled counterpart $D^*$ of its input
+$D$.  The layer output $D'$ satisfies the row equation
+\begin{equation}\label{op:switch-row}
+                      D'-D-m_{\mathrm{sw}}(D^*-D)=0.
+\end{equation}
+The mask is pair-constant, so this is exactly either the identity or
+a transposition on each pair.  The counterpart identity itself is
+checked by a scalar program below.  These equations are always
+imposed, including in inactive phases.  Induction through the network
+makes each variable factor a known permutation of an original logical
+array, for arbitrary field values of that array.
+There are only $O(r)$ layers among a fixed number of routing networks,
+hence $\poly(r)$ factor and equation types with bounded arity per row.
+For an edit of an original logical array at $j$, every factor derived
+from it changes only at the image of $j$ under its fixed permutation.
+All such addresses for one frame occupy $\poly(r)$ bits.
+Their entire frame-indexed tables need not have a succinct random-access
+formula; they will be stored as public encoded tables and focused
+before use in Section~\ref{sec:outer-representation}.
+
+\paragraph{Primaries and assigned degree.}
+For every factor array $D\in J^h$, define its unique degree-$<h$
+interpolant on the subgroup grid by
+\begin{equation}\label{op:factor-interpolant}
+                  F_D(Z)=\sum_{j=0}^{h-1}D_jL_j(Z).
+\end{equation}
+For each factor-occurrence list of each monomial, include primaries
+for every subset $S$ of its occurrences, with payload the
+length-$M_o$ coefficient table, padded by zeros, of
+\begin{equation}\label{op:subset-primary}
+                         F_S(Z)=\prod_{a\in S}F_{D_a}(Z).
+\end{equation}
+The index set here consists of occurrences: repeated uses of one
+factor remain distinct.  The empty subset gives $1$.
+Identify shared singleton primaries or impose direct full-word
+equalities on copies, and include independent singletons when needed.
+The assigned logical degree $k_S$ counts variable occurrences only;
+public factors have degree zero.  Let $d_2$ bound the length of every
+factor-occurrence list after all switch and row equations have been
+included.  It is a fixed constant, and $k_S\le d_2$.
+The honest ordinary polynomial degree is at most $d_2(h-1)$.
+Choose the earlier size constant so that
+\begin{equation}\label{op:degree-slack}
+                         M_o>(8d_2+8)h.
+\end{equation}
+All primary formulas exist for arbitrary $b\in J^{4h}$.
+
+\paragraph{Index independence and support.}
+Scalar primaries are intended to depend only on $x$, not on $u$.
+Test invariance of their payloads under left $(0,a)$ pulls for the
+fixed generating list of $H$.  Such a pull fixes $x$ and changes
+$u$ to $au$, and shares B with the original untwisted scalar slot.
+Compare whole systematic A rows as above, using the pulled local
+expression without introducing a free encoder witness.
+The shared-B exact gap forces invariance under each generator;
+since these generate $H$, this is precisely independence from $u$.
+No independent invariance test is needed on a derived program stage
+whose exact recipe already determines it, and matrix intermediates
+need not be $u$-independent.
+
+Use range masks from Lemma~\ref{op:scalar-programs} to require
+singleton coefficient support in $[0,h)$ and product-primary support
+in $[0,d_2h+1)$, with sharper public bounds when appropriate.
+Encode the masked complement as a full output and compare that
+entire word with zero.  A nonzero output has full-code distance.
+Compare each public singleton and each empty product with its
+precomputed canonical full encoding.  These are structural conditions.
+
+\paragraph{Product certification.}
+Compute the length-$M_o$ DFT of each relevant primary coefficient
+table by certified full stages.  Its payload at integer $y$ is the
+polynomial's value at $\zeta^y$.  For every nontrivial subset choose
+one occurrence $a\in S$ and check, at one fixed systematic payload
+cell at $s=t=0$,
+\begin{equation}\label{op:primary-product-test}
+                   F_S(\zeta^y)
+                  =F_{S\setminus\{a\}}(\zeta^y)F_{D_a}(\zeta^y).
+\end{equation}
+These are pointwise nonlinear tests; the stored transform stages
+themselves have linear recipes.  Once support, independence and
+transform identities hold, a false equality is the nonzero polynomial
+$F_S-F_{S\setminus\{a\}}F_{D_a}$, of degree less than
+$(d_2+1)h+1<M_o/2$.  A nonzero polynomial of degree $D$ over a field
+has at most $D$ roots, as follows by successively dividing by $Z-z$
+at distinct roots.  Thus the false equality fails at more than half
+the $M_o$ distinct points $\zeta^y$.  Uniform $k$ has uniform $x$,
+and independence from $u$ prevents the extra index from diluting this
+gap.  Induction on subset size forces exactly all products
+\eqref{op:subset-primary} once these tests hold.
+
+\paragraph{Counterparts and the subgroup annihilator.}
+Apply the length-$h$ Fourier transform to the low $\ell$ bits of each
+singleton's padded coefficient table.  Its first block is the value
+table $D$, and every higher block is zero by coefficient support.
+For each XOR counterpart use the certified bit pull on this padded
+table and compare its full output with the counterpart's separately
+computed padded value table.  Their full-operation gap forces the
+exact counterpart identity needed in \eqref{op:switch-row}.
+
+For a row-equation list let $R(Z)$ be the indicated linear combination
+of its product primaries.  Its value at $\omega^j$ is exactly the
+row expression at $j$.  Its remainder modulo $Z^h-1$ has coefficients
+\begin{equation}\label{op:remainder-fold}
+       [Z^i](R\bmod(Z^h-1))
+          =\sum_{z=0}^{M_o/h-1}[Z^{i+zh}]R,
+                \quad 0\le i<h,
+\end{equation}
+and zero padding elsewhere.  This formula follows from
+$Z^{i+zh}\equiv Z^i\pmod{Z^h-1}$, since the coefficient table is
+supported below $M_o$.  It is a short linear program: apply
+$\prod_{i=\ell}^{r-1}(I+R_{(e_i,1)})$ to sum all high-bit blocks,
+then mask to high bits zero.  Require its entire full word to be
+zero when the equation is imposed.
+
+The polynomial $Z^h-1$ is exactly the annihilator of the subgroup grid:
+it has the $h$ distinct roots $\omega^j$, because $h$ is invertible
+in $J$, and its degree is $h$.  The remainder has degree below $h$.
+Therefore it vanishes as a polynomial if and only if $R$ vanishes at
+all grid points.  In one direction divisibility suffices; in the
+other direction a degree-$<h$ polynomial with $h$ distinct roots
+is zero.  This proves that the folded full-word checks impose the
+row equations exactly, without any random sampling loss over rows.
+All structural switch equations and their folded computations are
+always present.  All phase-specific folded computations are also
+stored persistently; the repeated clock selects only which final
+zero conditions are imposed.
+
+\paragraph{Counts and address transforms.}
+The constant coefficient of \eqref{op:factor-interpolant} is
+\begin{equation}\label{op:occupation-count}
+                       [Z^0]F_e=h^{-1}\sum_{j=0}^{h-1}e_j,
+\end{equation}
+since every $L_j$ has constant coefficient $h^{-1}$.
+Mask the occupation coefficient table to index zero, using all bit
+masks.  Compare its full word with the public full encoding of
+$1[\underline x=0]$ times $c h^{-1}$, where the expected count
+$c\in[0,h]$ is computed from the legal repeated stage and phase.
+This scalar is used in the comparison; it is not part of a changing
+persistent value.  On Boolean occupation data, characteristic
+$p>M_o\ge h$ implies that two counts in $[0,h]$ are equal in $J$
+only when they are equal as integers.  Together with monotonicity
+this enforces the prescribed prefix exactly.
+
+For address access, take the padded length-$h$ value table obtained
+above and apply the full $r$-bit Walsh transform.  Its result is
+\eqref{op:address-transform}.  Walsh orthogonality gives
+\[
+ M_o^{-1}\sum_x U_D(x)(-1)^{x\cdot j}
+ =\sum_{i<h}D_i
+        \left(M_o^{-1}\sum_x(-1)^{x\cdot(i+j)}\right)
+ =D_j \qquad(j<h).
+\]
+In particular the high bits of the zero-padded value table cause
+$U_D$ to repeat across the corresponding high output blocks;
+they do not introduce an additional normalization factor.
+
+\paragraph{Exact alternatives and complexity.}
+Every stage of every mask, transform, counterpart comparison and
+remainder computation is stored as a full slot with its specified
+linear recipe, including mixed and untwisted matrix intermediates.
+Give a derived stage the maximum assigned degree of its primary
+ancestors and retain its ancestor subprogram or shared directed
+acyclic recipe.  Public indexed arrays are degree-zero inputs with
+precomputed encoded constants.
+
+On exact duplicated full codewords, a false independence, support,
+public-constant, full-operation, counterpart, folded-zero or count
+identity has a constant gap by the shared-axis or full-code distance.
+If none is false, the polynomial root argument forces all primary
+products.  Singleton support makes the interpolant unique from its
+subgroup values.  The counterpart and switch equations then determine
+all factors as their prescribed permutations of the four original
+logical arrays, and the public factors are fixed.  Thus the structural
+tests force precisely $P(b)$.  The remaining Boolean, prefix, count
+and selected-phase row equations are exactly the macrovertex rules.
+If any is false, its folded or count check has the same exact gap.
+All tests are packed at the center; choosing one false relation in
+this argument entails no division by their number.  The minimum of
+the fixed code gaps and the product gap gives
+$\gamma_{\mathrm{per}}>0$ and proves both alternatives.
+
+There are $O(r)$ routed factor and row families before the scalar
+program expansions, a constant number of occurrence subsets per
+monomial, and polynomially many stages per transform.  Thus the
+number of full slots, recipes and local metadata is $\poly(r)$.
+The entry dimension and field bit length give a further fixed
+polynomial factor.  At one center the checker scans these lists,
+performs field operations and evaluates fixed-length control
+decisions, all in $\poly(r)$ bit operations.  It reads public encoded
+constants only at the publicly specified positions in $\Pi(k)$;
+arbitrary hidden-frame table lookup is never assumed.
+All public permutations, masks, coefficients, encodings and complete
+canonical arrays are constructible globally by polynomial-time
+combinatorial and linear-algebra algorithms.  Since
+$|\Gamma|=M_o|H|=O(N\operatorname{polylog} N)$ and every scalarized encoding system
+has polynomial dimension in $N$, this takes $\poly(N)$ time.
+The persistent word length is $O(N\operatorname{polylog} N)$ field-bit storage,
+with a fixed exponent in the polylogarithmic factor.  This completes
+the construction and proof.
+\end{proof}

preprints/The-PCP-for-PPAD-conjecture-a-quasilinear-reduction-September-25-2026/build/sections/05-outer-representation.tex

@@ -0,0 +1,696 @@
+\section{A locally evolving outer representation}\label{sec:outer-representation}
+
+We now combine the macrograph, the outer codes, and the certified programs
+into one representation. A long preparation is part of the represented
+path: its current position is recorded by a binary clock, while its scratch
+space is reused. Thus preparation time does not multiply the length of a
+represented vertex. The main additional requirement is that every retained
+certificate, including certificates introduced to verify other
+certificates, can change locally when a logical tape entry changes.
+
+All notation in this section belongs to the outer construction. In
+particular, $J$, $H$, and $\Gamma=\F_2^r\rtimes H$ are those of
+Section~\ref{sec:outer-codes}. We retain $M_o=2^r$, the tape length $h$,
+the four logical arrays $b$, and the persistent tuple $P(b)$ of
+Lemma~\ref{lem:outer-wiring}. The assigned degree $k_S$ of a primary
+$F_S$ counts its variable factor occurrences, and $d_2$ is the fixed upper
+bound on the number of factor occurrences. In particular,
+\[
+ r=O(\log N),\qquad M_o=O(N),\qquad |H|\leq r^9,
+ \qquad M_o>(8d_2+8)h.
+\]
+Every persistent program stage has a specified linear payload recipe in
+the primaries. This includes support masks, transforms, routing checks,
+remainder computations for inactive phases, and every matrix-valued
+reencoding intermediate. Its assigned degree is at most the maximum
+assigned degree of the primary inputs to that recipe.
+
+\subsection{Finite differences of the complete certificate tuple}
+
+Fix a start frame $c\in\{0,\ldots,4h-1\}$. The finite edit alphabet
+$\Sigma$ of Lemma~\ref{lem:macrograph} describes additive changes
+$h_\sigma(c)$ of the logical arrays. Enlarge this fixed alphabet by inverse
+edits and a zero edit if necessary. All addresses depend only on
+$(c,\sigma)$ and public routing data. The choice of the edit can depend on
+the critical logical bits; a recipe for a specified edit cannot.
+For a field-valued function $F$ define
+\[
+ \Delta_\sigma F(b)=F(b+h_\sigma(c))-F(b).
+\]
+This definition is made on arbitrary field arrays $b$, independently of
+whether they satisfy the macrograph conditions.
+
+For a variable singleton occurrence $a$, routing gives either no change or
+a change $\epsilon_{a,\sigma}L_{j_{a,\sigma}}(Z)$, where
+$\epsilon_{a,\sigma}\in\{-1,0,1\}$ and $j_{a,\sigma}<h$ are public at
+this frame. A default address is used when $\epsilon_{a,\sigma}=0$.
+Public factors have zero increment. If $S_{\rm var}$ is the set of
+variable \emph{occurrences} in $S$, distributivity gives
+\begin{equation}\label{oe:primary-difference}
+ \Delta_\sigma F_S
+ =\sum_{\varnothing\ne A\subseteq S_{\rm var}}
+     F_{S\setminus A}(b)
+     \prod_{a\in A}\epsilon_{a,\sigma}L_{j_{a,\sigma}}.
+\end{equation}
+Repeated occurrences are retained separately in this formula. Every
+primary on its right has assigned degree at most $k_S-1$. Multiplication
+by a specified Lagrange polynomial is the linear coefficient program of
+Lemma~\ref{op:scalar-programs}. The number of subsets in
+\eqref{oe:primary-difference} is bounded in terms of $d_2$ alone.
+Every ordinary polynomial product appearing here has degree at most
+$d_2(h-1)<M_o$; consequently its implementation by cyclic coefficient
+convolution has no wraparound error. All possible addresses and zero
+coefficients use a common padded program topology.
+
+\begin{lemma}[Closure under finite differences]\label{oe:derivative-closure}
+For every positive-degree persistent full slot $D$ and every
+$\sigma\in\Sigma$, one can allocate a finite certified full program whose
+output, in the same encoder type as $D$, is the increment of $D$ under
+$b\mapsto b+h_\sigma(c)$. One can do this simultaneously for every
+positive-degree full intermediate introduced by these programs, including
+all their verification intermediates. The resulting list has
+$\operatorname{poly}(r)$ slots and description bits. Every new slot has a
+specified linear recipe in persistent primaries of strictly smaller
+assigned degree than its parent target. All identities hold on arbitrary
+field arrays $b$.
+\end{lemma}
+
+\begin{proof}
+For a primary use \eqref{oe:primary-difference}. If $D$ is another
+persistent stage with linear payload recipe $\mathcal L_D$, apply
+$\mathcal L_D$ to the corresponding primary increments, replacing
+degree-zero primary increments by zero. At this fixed frame its
+coefficients, pulls, and encoder types are fixed. Hence
+\[
+ \mathcal L_D\bigl((F_S(b+h_\sigma(c)))_S\bigr)
+ -\mathcal L_D\bigl((F_S(b))_S\bigr)
+ =\mathcal L_D\bigl((\Delta_\sigma F_S(b))_S\bigr).
+\]
+The operations used in Section~\ref{sec:outer-programs} implement this
+identity at the payload level. In particular, multiplying a scalar slot
+by the public matrix $\rho_e(k)$ is linear in that scalar; changing one
+encoder axis or contracting matrix entries is also linear. These
+operations introduce no additional logical degree.
+
+Allocate full slots for every stage of each increment program, with new
+slots for its mixed and untwisted verification stages. An identity or zero
+stage may be inserted to give an output its required type. Order stages
+topologically, so that Lemma~\ref{lem:outer-fill} can initialize one full
+target using already completed parents. For every newly allocated stage,
+record the ancestor directed acyclic graph that expresses its payload in
+the persistent primaries. Its roots have degrees at most one less than
+those of the target whose increment is being computed. If its assigned
+degree is positive, apply the same construction to this new stage for
+every edit. This includes a stage introduced solely for verification: it
+will be retained at full preparation and therefore needs its own
+increment.
+
+Inlining here means substitution in a directed acyclic graph, with
+sharing retained within each target computation. If a primary is used
+many times, substitute its increment program once and fan out its result.
+Separate target computations may use fresh copies. Along any chain of
+newly generated targets the assigned degree drops by at least one.
+Therefore the construction stops after at most $d_2$ derivative tiers.
+At each tier the number of target graphs, their sizes, their ancestor
+descriptions, and their substitutions are bounded by a fixed polynomial
+in the preceding total description size and $r$. A fixed number of such
+polynomial expansions remains polynomial in $r$.
+
+The programs verify the computed increments by their own stage
+relations, starting from the lower-degree primaries. They do not test a
+finite-difference assertion about an otherwise unspecified witness.
+Neither their construction nor their identities use the final validity
+equations. This also ensures that increment recipes remain defined on
+field data produced by an edit that is not a legal macrograph move.
+\end{proof}
+
+\subsection{Public addresses, logical reads, and a reversible preparation}
+
+Public address tables need not admit succinct evaluation at a hidden
+frame $c$. We therefore represent the needed access explicitly. For each
+field entry $a(c)$ of the public data table, form the scalar payload
+\begin{equation}\label{oe:public-walsh-table}
+ V_a(x,u)=\sum_{0\leq c'<4h}a(c')(-1)^{x\cdot c'}.
+\end{equation}
+Integers in this display are identified with their $r$-bit vectors, and
+the coefficient table is zero outside $[0,4h)$. The payload is independent
+of $u$. Hardwire its full encoding at every publicly requested symbol
+position. The table includes operand addresses, rule types,
+terminal-comparison addresses, and choice-rule data for both directions,
+together with the
+permuted edit addresses of every factor type used by
+Lemma~\ref{oe:derivative-closure}. Addresses for inactive phases are
+included too. Strings and indices are split into field entries with fixed
+parsing conventions. There are only $\operatorname{poly}(r)$ entries per
+frame.
+
+The first preparation steps are an A-only public focus. For the bits of
+the repeated frame $c$, apply the mode selectors
+\[
+ \frac{I+(-1)^{c_i}R_{(e_i,1)}}2
+\]
+successively, implementing a nonfirst-bit pull by the fixed generators
+as in Section~\ref{sec:outer-programs}. Applied to
+\eqref{oe:public-walsh-table}, their completed output is
+$a(c)(-1)^{x\cdot c}$ on the systematic payload. Removing that known sign
+recovers $a(c)$ identically at every center. No recurrence using a
+table-dependent coefficient is active before the public focus is
+complete. Retain this focus until preparation has been reversed. On
+malformed data use fixed default field values, indices, and instructions;
+thus every local algorithm is total.
+
+Next, in the topological order of Lemma~\ref{oe:derivative-closure}, fill
+every transient full computation stage by Lemma~\ref{lem:outer-fill}.
+Previously completed targets remain full and their temporary fill scratch
+is empty. At every intermediate location check their direct full-operation
+relations; apply the two-buffer fill checks to the current target; and
+check zero in all still empty full targets and inactive scratch slots.
+The persistent tuple $P(b)$ is checked throughout. All these conditions
+are tested together at the same center.
+
+Finally prepare the critical logical reads. The persistent slot $U_D$
+provided by Lemma~\ref{lem:outer-wiring} has payload
+\[
+ U_D(x)=\sum_{j<h}(-1)^{x\cdot j}D_j.
+\]
+Its Walsh coefficient at address $j$ is $D_j$. Using the completed public
+address focus, build A-only selector stacks for the fixed list of
+critical roles needed by either the forward or reverse macro proposal
+and its existence flag. Unused roles use default addresses. Include
+enough roles for each logical array type; their number is constant.
+Build the identical focus stacks on the full increment outputs for
+$U_D$ for every $\sigma$. The recipe for $U_D$ uses just one degree-one
+singleton, so each increment input has assigned degree zero. At a fixed
+frame its entire focused stack is independent of $b$ as well.
+
+\begin{lemma}[Preparation and local evolution]\label{oe:preparation}
+There is an integer $J_{\max}\geq1$ and a canonical preparation indexed by
+$j=0,\ldots,J_{\max}$ with the following properties.
+\begin{enumerate}[label=\textup{(\roman*)}]
+\item At $j=0$ all frame-dependent full and A-only transient slots are
+zero. At $j=J_{\max}$ all full increment programs and critical focuses are
+complete, and every full-target fill buffer and orbit-focus buffer is
+zero.
+\item For every field array $b$ and frame $c$, every snapshot is uniquely
+specified. Its transition to the next or previous snapshot uses a fixed
+finite list of permutation-neighbors of each requested center.
+\item The number of steps is $J_{\max}=O(M_o\operatorname{poly}(r))$.
+The number of stored slots, the schedule description, and the work to
+interpret one microclock or perform one local transition are
+$\operatorname{poly}(r)$, including entry dimensions and bit lengths.
+\item At full preparation, applying an edit $h_\sigma(c)$ changes the
+entire retained tuple to its canonical value on $b+h_\sigma(c)$ by local
+addition of the stored increment outputs. The public focus is unchanged.
+\end{enumerate}
+\end{lemma}
+
+\begin{proof}
+Concatenate the public focus, all full-target filling schedules, and the
+critical-read focuses. The stage lists for selectors and cutoffs have
+fixed padded lengths independent of their address values. The
+per-target mode schedule consists of $M_o$ iterations of fixed
+polylogarithmic sublength, together with its initial and final buffer
+steps. The exceptional first iteration and final erasure are specified
+separately. Thus a binary microclock can be interpreted by scanning the
+polynomial-in-$r$ list of program boundaries and performing arithmetic
+on $O(\log M_o+\log\operatorname{poly}(r))$ bits. The interpreter selects
+the current iteration, buffer, initialized prefix, and transition; it
+does not execute all preceding iterations. Distinct programs may use
+distinct scratch indices to keep their shapes and twists fixed.
+An idle preparation step supplies $J_{\max}\geq1$ if needed.
+
+For the last assertion, write $D_\sigma(b)$ for the stored full increment
+of a positive-degree full slot $D$. At the top perform, simultaneously,
+\begin{equation}\label{oe:simultaneous-edit}
+ D(b)\longmapsto D(b)+D_\sigma(b)
+             =D(b+h_\sigma(c))
+\end{equation}
+for every persistent and transient full computation slot. Equation
+\eqref{oe:simultaneous-edit} is valid in its own encoder because that
+encoder is linear. The increment slots themselves are included in this
+update whenever their assigned degree is positive; their required
+increments exist by Lemma~\ref{oe:derivative-closure}. Degree-zero slots
+do not change. All additions read the old tuple before writing the new
+one.
+
+The full-target temporary buffers are already empty. Public focus
+depends only on $c$ and remains fixed. Each main critical-read stack is a
+linear selector-and-pull program at fixed public addresses. Adding the
+matching focused increment stack therefore updates all its initialized
+stages to their new canonical values. Those increment stacks have degree
+zero and stay fixed. This verifies the assertion also for the retained
+A-only data. The preparation can now be erased in the reverse order,
+using exactly the same recipes on the new $b$ and unchanged frame.
+\end{proof}
+
+\subsection{The extended graph and its endpoint payload}
+
+Define an extended valid label to be
+\begin{equation}\label{oe:extended-label}
+ v=(c,b,\eta,j),\qquad
+ 0\leq c<4h,\quad \eta\in\{0,1\},\quad 0\leq j\leq J_{\max},
+\end{equation}
+where $b$ is a valid macrograph state at stage $c$ if $\eta=0$, and at
+stage $c+1\pmod{4h}$ if $\eta=1$. We call these modes pre and post,
+respectively. Its raw word $R_v$ contains, at every $k\in\Gamma$, all four
+parts of the persistent tuple $P(b)$, all full and A-only transient data
+in the canonical snapshot $(c,b,j)$, and a canonically encoded repeated
+copy of $(c,\eta,j)$ and its range information. Padding conventions will
+be fixed below.
+
+In pre mode increase $j$ until the top. At the top use the prepared
+critical reads to compute the macrograph's forward choice and existence
+flag. If the edge exists, perform \eqref{oe:simultaneous-edit}, leave
+$c,j$ fixed, and change the mode to post. In post mode decrease $j$ until
+zero, then perform
+\begin{equation}\label{oe:frame-advance}
+ (c,b,1,0)\longmapsto(c+1\bmod4h,b,0,0).
+\end{equation}
+The reverse rules use the reverse preparation transition, the reverse
+macro choice at the same start frame, and the inverse of
+\eqref{oe:frame-advance}. Both directions read their choice and flag
+before the top update. At a missing edge the corresponding proposed word
+is defined to be the current raw word and its flag is zero. This is a
+total proposal convention, not a graph self-edge. At a malformed input
+the local proposal and flag likewise have a fixed total default.
+
+For each valid macro vertex at stage $s$, the extended graph contains the
+directed path
+\begin{equation}\label{oe:replacement-path}
+ \begin{split}
+ (s-1,b,1,J_{\max})&\longrightarrow\cdots
+ \longrightarrow(s-1,b,1,0)\\
+ &\longrightarrow(s,b,0,0)
+ \longrightarrow\cdots\longrightarrow(s,b,0,J_{\max}),
+ \end{split}
+\end{equation}
+with frames taken modulo $4h$. Macro edges join the corresponding ends
+of these paths. Every internal edge changes the clock or mode, and every
+top edge changes mode, so none is a self-edge. The graph has indegree and
+outdegree at most one. Every label belongs to a path of positive length;
+there are no isolated vertices.
+
+Let $(s_*,b_*)$ be the public exceptional macro source of
+Lemma~\ref{lem:macrograph}. The unique exceptional extended source and
+its actual successor are
+\begin{equation}\label{oe:public-source}
+ v_*=(s_*-1,b_*,1,J_{\max}),\qquad
+ v_*^+=(s_*-1,b_*,1,J_{\max}-1).
+\end{equation}
+Their entire raw words are deterministically computable in
+$\operatorname{poly}(N)$ time. Indeed the macro source is public, all
+full encoders and program formulas are computable by global array
+algorithms, and even explicitly following this one preparation takes
+only polynomially many steps.
+
+An extended endpoint can occur only at a top label of
+\eqref{oe:replacement-path}. Map it to its macro vertex by retaining $b$
+and taking stage $c+\eta\pmod{4h}$. If it is not $v_*$, the endpoint
+decoder of Lemma~\ref{lem:macrograph} returns a nonzero string $y$ with
+$P(S(y))\ne y$ or $S(P(y))\ne y$. Thus the endpoint payload consists of
+the logical tapes and the repeated label; it does not require finding a
+history of the transient computation.
+
+\subsection{A packed tester for valid represented vertices}
+
+Let $\Pi$ contain identity, all forward and inverse A- and B-neighbor
+maps needed by the codes, the left $H$ generators used for payload
+independence, the right first-bit shifts and single-cycle rotations and
+their inverses, and the right $H$ generators used by scratch programs.
+Add the finitely many bounded compositions required by a primitive local
+expression. Its cardinality $q=|\Pi|$ is fixed independently of $N$.
+Arbitrary powers of a rotation or paths through $H$ are represented by
+successive stages; they do not add new query maps to this list. At every
+center the tester reads $W|_{\Pi(k)}$ and rejects if \emph{any} applicable
+condition below fails:
+\begin{enumerate}[label=\textup{(\roman*)}]
+\item canonical parsing, legal repeated clock, range checks, and zero
+padding;
+\item all full-slot local codes, A/B agreements, and systematic-payload
+duplication, including targets that are partial or inactive;
+\item all A-only local codes and A agreements, including inactive
+scratch;
+\item all structural persistent relations and the macro-validity
+conditions selected by $(c,\eta)$;
+\item every required current-snapshot relation: completed full-operation
+checks, the current partial-target comparison, initialized scratch-prefix
+recurrences in both buffers and all focuses, and inactive-zero checks.
+\end{enumerate}
+In particular, the tester does not choose one relation uniformly from a
+growing list. All relations share this fixed set of center permutations.
+Write $\operatorname{rej}(W)$ for its rejection probability at uniform
+$k\in\Gamma$, and use normalized whole-symbol distance
+\[
+ \dist(W,W')=\Pr_{k\in\Gamma}[W(k)\ne W'(k)].
+\]
+
+\begin{lemma}[Soundness of the valid-word tester]\label{oe:valid-testing}
+There are fixed constants $C_{\rm rep}<\infty$ and $g_{\rm rep}>0$
+such that, if $\operatorname{rej}(W)<g_{\rm rep}$, some valid extended
+label $v$ satisfies
+\[
+ \dist(W,R_v)\leq C_{\rm rep}\operatorname{rej}(W).
+\]
+The tester has perfect completeness. Conversely,
+$\operatorname{rej}(W)\leq q\dist(W,R_v)$ for every valid label $v$.
+Distinct valid labels have raw distance at least a fixed
+$\delta_{\rm raw}>0$.
+\end{lemma}
+
+\begin{proof}
+Put $\theta=\operatorname{rej}(W)$. The parsing, repetition, and full-code
+proximity conclusions of Theorem~\ref{thm:outer-codes} first change
+$O(\theta)$ centers to give legal constant clock data and exact full
+codewords. A corrected repeated clock of illegal range would cause
+rejection everywhere, so small enough $\theta$ rules it out. Exact
+one-axis distance then forces systematic-payload duplication in every
+full slot. Otherwise that one failed duplication relation already has a
+fixed positive rejection probability. Zeroing locally A-invalid scratch
+parts costs another $O(\theta)$ centers. A change on $z$ centers changes
+any simultaneous test on at most $qz$ centers; this estimate uses the
+common permutation list and does not depend on the number of slots.
+
+We next restore all scratch relative to these corrected full codewords
+and the corrected clock, before using any advertised full-operation
+identity. The reference for the public focus is the actual hardwired
+public word at the corrected frame. Treat its recurrences first. A later
+table-dependent recurrence is active only when this reference focus is
+complete and hence gives identical correct coefficients at every center.
+Each initial expression $X$ in a mode-fill program is A-coded from its
+exact full parents, even if those parents do not yet satisfy their
+advertised payload recipes. The critical and increment focuses also
+start from exact A-coded full words. Thus every required reference
+prefix is exactly A-coded.
+
+These are precisely the reference and ordering hypotheses of
+Lemma~\ref{op:scratch-restoration}. The preprocessing above changed only
+$O(\theta)$ sites, so the common query list bounds both the joint
+A-conflict fraction and the simultaneous recurrence-failure fraction of
+this one preprocessed word by $O(\theta)$. Apply the lemma to its entire
+scratch tuple, including inactive zeros. Its fixed-word induction keeps
+every prefix estimate $O(\theta)$ and makes only one final replacement.
+All scratch is therefore restored at cost $O(\theta)$, independently of
+the number of stages. No full-operation identity has been used to obtain
+this conclusion.
+
+Call the resulting tuple $\widetilde W$. Its remaining tests still fail
+with probability at most $C'\theta$ for a fixed $C'$. By
+Lemma~\ref{lem:outer-wiring}, either its persistent tuple is exactly
+$P(b)$ for logical bits $b$ valid at the selected macro stage, or a
+persistent relation fails with fixed positive probability. Choose
+$\theta$ below that gap divided by $C'$ to obtain the former alternative.
+Now follow the initialization order of the transient full targets.
+Completed targets with canonical parents are forced to their exact
+outputs by the shared-axis gap following Definition~\ref{op:full-operation}.
+For the current partial target, its required cutoff comparison has the
+same gap by Lemma~\ref{lem:outer-fill}. The cutoff scratch has already
+been restored: its completed value is precisely $P_{t'}X$. In the case
+that the target shares only the B axis with $X$, orbit invariance makes
+this completed reference B-coded. Only at this point is B-axis distance
+applied. No intermediate A-only scratch is assumed to be B-valid.
+Inactive full targets are forced to zero by full-word distance.
+
+There are only finitely many kinds of these gaps, uniformly in their
+stage indices, program lengths, and vector dimensions. Their minimum is
+positive. If any relation failed exactly on $\widetilde W$, it alone
+would give at least that gap in the packed test. Decrease $\theta$ once
+more to rule out every such failure. All full slots and restored scratch
+then have the unique canonical values of $(c,b,\eta,j)$. The total
+restoration cost is $C_{\rm rep}\theta$ for a fixed $C_{\rm rep}$.
+
+Canonical snapshots satisfy every test by their construction, proving
+perfect completeness. If a neighborhood avoids the disagreement set
+between $W$ and $R_v$, its test accepts; a union bound over $\Pi$ proves
+the converse. Finally, two labels with different repeated data differ
+at every center. If only their logical arrays differ, one logical
+singleton interpolant differs. Its full encoded difference is nonzero
+and has the fixed distance of Theorem~\ref{thm:outer-codes}, including the
+constant conversion between four-part and whole-symbol metrics. This
+gives $\delta_{\rm raw}>0$.
+\end{proof}
+
+\subsection{Full representation cost and a fixed local Boolean exponent}
+
+We record the size argument before choosing any correction accuracy or
+any number of Boolean ports. This order is needed later. All constants
+in the following count depend only on the fixed gate basis, $d_2$, and
+the already chosen outer code parameters.
+
+One-bit alignment takes $O(r)$ elementary rotations. A conjunction of
+bit conditions takes $O(r^2)$ primitive stages, and a carry or borrow
+shift takes $O(r^3)$. The Lagrange window product takes $O(r^4)$ stages.
+The other transforms and masks in Section~\ref{sec:outer-programs} fit
+within this last bound. There are $O(r)$ initial factor and row families:
+a fixed number of row lists, with $O(r)$ routing layers for each needed
+permutation and a bounded subset list for each monomial. Hence there are
+$O(r^5)$ persistent primitive stages. Storing their ancestor graphs,
+parent indices, types, and coefficient instructions is bounded, with
+ample slack, by $Cr^{20}$ description bits. Public indexed field data
+are counted separately as local constant words below; descriptors refer
+to their entries by indices, rather than copying an entire length-$h$
+array into a descriptor.
+
+Here is an explicit finite-depth bound for derivative expansion. Let $T$
+bound the complete current list of graph descriptions and subgraph
+descriptions, with each slot, parent occurrence, and instruction counted
+at least once. A target graph has at most $T$ nodes and at most $T$
+primary roots. Substitution at one root uses a bounded number of the
+$O(r^4)$ Lagrange programs. There are at most $T$ target graphs. Recording
+an ancestor subgraph for every newly introduced stage, retaining sharing
+within it and allowing fresh copies for separate targets, costs at most
+the square of the size of its target computation. Public address
+instructions and index encodings contribute fixed powers of $r$ and
+logarithms of these list lengths. The following deliberately loose bound
+therefore covers all these operations at one tier:
+\begin{equation}\label{oe:recipe-size-recurrence}
+ T_{a+1}\leq C r^{20}(T_a+1)^6,\qquad T_0\leq Cr^{20}.
+\end{equation}
+For example, the six factors allow a separate factor for choosing a
+target, traversing its graph, substituting its roots, and copying the
+ancestor descriptions, with two additional factors for indexing and
+duplication; the actual substitution keeps more sharing. If
+\begin{equation}\label{oe:recipe-exponent}
+ E_0=20,\qquad E_{a+1}=6E_a+20,\qquad E=E_{d_2},
+\end{equation}
+then all descriptions are $O(r^E)$. This recurrence is composed only
+$d_2$ times, by Lemma~\ref{oe:derivative-closure}.
+
+For each full target, a cutoff program uses at most $|H|$ group elements,
+$r$ prefix rectangles, $r$ selectors per rectangle, and $O(r)$ primitive
+alignment stages per selector. This is $O(r^{12})$ stages. For the orbit
+focus, there are at most $|H|$ frequencies and $|H|$ requested group
+indices. Even finding and storing a separate generator path of length
+$|H|$ for every such pull gives $|H|^3$ stages, in addition to the mode
+selectors. A bound $O(r^{30})$ covers both focus and cutoff work and the
+two buffers. An entry alphabet has dimension at most $|H|^2\leq r^{18}$,
+and a field entry uses $O(r)$ bits. Therefore
+\begin{equation}\label{oe:raw-size-bound}
+ \text{raw data, local public constants, and schedule metadata at one
+ center}\ \leq C r^{E+60}.
+\end{equation}
+This includes public-address focus, critical focus, all slot types and
+all four parts. The long sequence of $M_o$ cutoffs is represented by a
+clock and reusable buffers, not by $M_o$ copies of these data.
+
+Put $\ell_N=\lceil\log_2(N+2)\rceil$. Fix once an integer $C_L>E+100$
+and pad each raw symbol to
+\begin{equation}\label{oe:raw-length}
+ L=(\ell_N+C_{\rm base})^{C_L}.
+\end{equation}
+A fixed lower bound on $C_{\rm base}\geq1$ makes
+\eqref{oe:raw-size-bound} fit for every $N$. The unpadded construction is
+still parameterized by $r$ and the same slot schema. Increasing
+$C_{\rm base}$ later adds zero padding only. The raw word and all
+center-specific public descriptions have total length
+\begin{equation}\label{oe:total-raw-size}
+ |\Gamma|\operatorname{poly}(\ell_N)
+ =M_o|H|\operatorname{poly}(\ell_N)
+ =N\operatorname{poly}(\ell_N).
+\end{equation}
+This is a bound on the listed bits, including index and field
+representations, rather than just the number of slots.
+
+Every local test and proposal can now be implemented by a fixed
+polynomial-time bit algorithm on a bounded number of $L$-bit symbols.
+Parsing, loops over slots and the explicit group $H$, arithmetic in $J$,
+program interpretation, and multiplexing within a supplied
+polylogarithmic array use bounded nests of polynomially bounded loops.
+The largest local computation is the orbit-restricted parity solve of
+Lemma~\ref{lem:outer-fill}. Its dimension is at most
+\[
+ C|O|\,|H|\,\dim_J(U)\leq C|H|^4\leq CL^4.
+\]
+Its explicitly stored matrix has $O(L^9)$ bits if each field entry is
+bounded by $L$ bits. Gaussian elimination uses $O(L^{12})$ field
+operations, and elementary arithmetic and inversion have a schoolbook
+bound $O(L^4)$ in this generous bit-length allowance. Even accessing
+each entry by a linear scan through the matrix, and scanning the full
+slot and metadata lists, adds only fixed polynomial factors. All other
+displayed local operations have the same type of finite loop bound.
+No routine iterates through the $M_o$ cutoffs to answer one query.
+Choose an integer $a$ exceeding the sum of these fixed loop and
+arithmetic exponents. A standard direct time-unrolling of a deterministic
+bit computation of length $O(L^a)$ gives an acyclic bounded-fanin Boolean
+circuit of size $O(L^{2a+2})$: represent each configuration by its used
+tape cells and head indicators, and implement each next configuration
+by Boolean selection. Thus one fixed exponent $c_{\rm raw}$ bounds all
+raw local circuits. Its precise numerical value is immaterial; the
+displayed construction fixes it before correction rounds or port arities
+are selected.
+
+All lists and constants are uniformly constructible. Global algorithms
+enumerate the finite fields, groups, public routing tables, and full
+constant encodings, using the deterministic searches and linear algebra
+of Sections~\ref{sec:outer-codes} and~\ref{sec:outer-programs}. Their
+time is $\operatorname{poly}(N)$, though it need not be quasilinear.
+Input-dependent data without succinct evaluation at $c$ are accessed
+through \eqref{oe:public-walsh-table}; other public constants are
+hardwired only at the bounded, publicly specified neighbors of the
+requested center. There is no hidden access to an unrepresented
+length-$N$ array.
+
+\subsection{Correction and the complete outer interface}
+
+All full and scratch slots of $R_v$ satisfy their A constraints. Apply
+the joint A-axis corrector of Lemma~\ref{lem:outer-uncertainty} to all
+of them, together with majority correction of the repeated clock and
+canonical padding. The local decoder enumerates at most $2^d$ label
+subsets and solves the common scalar constraints componentwise. Since
+$d$ is fixed, this again has a fixed polynomial bound in $L$, independent
+of the number or dimensions of the packed slots. Enlarge $c_{\rm raw}$
+once to cover this one-round routine.
+
+Let $Q\geq2$ bound its neighborhood size and the sizes of the raw
+neighborhoods. For any fixed nonnegative integer $t$, unrolling $t$
+correction rounds followed by a raw local routine uses at most
+\begin{equation}\label{oe:fixed-interface-composition}
+ C(1+Q+\cdots+Q^t)=O(Q^t)
+\end{equation}
+copies of these same $L$-bit circuits. Their intermediate outputs still
+have length $L$. It follows that, for a constant $C_t$ depending on $t$,
+the composite circuit has size at most $C_tL^{c_{\rm raw}}$. Its ports
+are the finitely many compositions of the original permutations to depth
+at most $t$, followed by a raw neighborhood. The exponent is independent
+of $t$. A later increase of $C_{\rm base}$ changes only the linear work
+of checking and writing padding; it changes neither this exponent nor
+the number of ports. In particular, fixed multiplicative constants may
+be absorbed into a subsequent exponent allowance with slack by increasing
+$C_{\rm base}$ after all those constants have been fixed.
+
+\begin{theorem}[Outer representation interface]\label{thm:outer-representation}
+There are fixed positive constants $\delta_{\rm raw},\rho_{\rm raw}$,
+fixed integers $C_L,c_{\rm raw}$, and uniform deterministic algorithms
+with the following properties for every End-of-Line input of length $N$.
+They construct a directed graph of valid extended labels, represented by
+canonical words
+\[
+ R_v\in(\{0,1\}^L)^\Gamma,
+ \qquad |\Gamma|=N\operatorname{poly}(\ell_N),
+ \qquad L=(\ell_N+C_{\rm base})^{C_L},
+ \qquad \ell_N=\lceil\log_2(N+2)\rceil,
+\]
+where $C_{\rm base}$ may be any sufficiently large fixed integer.
+\begin{enumerate}[label=\textup{(\roman*)}]
+\item \textbf{Graph and endpoint data.} The graph has indegree and
+outdegree at most one, no self-edges, and no isolated vertices. It has the
+public distinguished source $v_*$ and actual successor $v_*^+$ in
+\eqref{oe:public-source}; their entire words are computable in
+$\operatorname{poly}(N)$ time. Every other endpoint is mapped in
+$\operatorname{poly}(N)$ time to a nonzero solution of the input
+End-of-Line instance. Every canonical word is computable from its valid
+label in polynomial time.
+\item \textbf{Distance and valid-word testing.} Distinct canonical words
+have normalized symbol distance at least $\delta_{\rm raw}$. A uniform
+center test has perfect completeness and uses only a fixed finite list
+$\Pi$ of permutation-neighbors, of cardinality $q$ independent of $N$.
+For every fixed $\alpha>0$ there is a fixed $\theta_\alpha>0$ such that
+\[
+ \operatorname{rej}(W)<\theta_\alpha
+ \quad\Longrightarrow\quad
+ \dist(W,R_v)<\alpha\quad\hbox{for some valid }v.
+\]
+Also $\operatorname{rej}(W)\leq q\dist(W,R_v)$ for every valid $v$.
+\item \textbf{Local proposals and flags.} At each requested center,
+total deterministic routines on $\Pi(k)$ compute a next or previous
+proposed symbol and an existence bit. On $R_v$ the bit is independent
+of the center and equals the actual edge-existence flag. If it is one,
+the proposed symbols form exactly the canonical neighboring word; if it
+is zero, the default proposal is $R_v$ itself.
+\item \textbf{Value-independent uncertainty correction.} There is a
+total local one-round corrector with a bounded permutation neighborhood.
+If $|U|/|\Gamma|\leq\rho_{\rm raw}$, it maps every word agreeing with
+$R_v$ outside $U$ to a word agreeing with $R_v$ outside a set $U^+$
+determined only by $U$, with $|U^+|\leq|U|/2$. Consequently $t$ rounds
+leave a prescribed uncertainty support of density at most
+$2^{-t}|U|/|\Gamma|$. The guarantee holds for arbitrary values on $U$.
+Different requested-center simulations may use different values there,
+provided repeated occurrences of a position are consistent within each
+individual cone.
+\item \textbf{Global recovery.} From any word within
+$\rho_{\rm raw}$ symbol distance of a valid word, a deterministic
+polynomial-time algorithm recovers its exact extended label and raw
+word. It then recovers the logical bits by systematic coefficient
+extraction and evaluation on the subgroup. In particular it extracts an
+End-of-Line answer whenever the recovered label is a nonexceptional
+endpoint.
+\item \textbf{Uniform cost and order of choices.} The total raw data and
+listed public descriptions have $N\operatorname{poly}(\ell_N)$ bits
+and are constructible in $\operatorname{poly}(N)$ time. Each raw local
+test, proposed symbol, flag, and one-round corrector has a Boolean
+circuit of size at most $CL^{c_{\rm raw}}$ on a fixed number of symbols.
+For each fixed $t$, correction for $t$ rounds followed by any such raw
+routine has size at most $C_tL^{c_{\rm raw}}$ and at most $O(Q^t)$
+symbol ports, with uniform construction for that $t$. The exponents
+$C_L,c_{\rm raw}$ are fixed before $t$ and any later port arity. Further
+fixed additive padding in $C_{\rm base}$ has only linear processing
+cost and does not change these exponents or arities.
+\end{enumerate}
+\end{theorem}
+
+\begin{proof}
+The graph and local evolution were established in
+Lemma~\ref{oe:preparation} and \eqref{oe:replacement-path}. The tester
+and distance are Lemma~\ref{oe:valid-testing}. For example, its constants
+allow
+\[
+ \theta_\alpha=
+ \min\{g_{\rm rep}/2,\alpha/(2C_{\rm rep})\}.
+\]
+All local proposals use the same primitive permutation list as the
+tester, enlarging it by a fixed number of maps if necessary.
+
+For the correction claim, let $d$ and $\delta_1$ be the outer A-code
+degree and distance. Choose $\rho_{\rm raw}$ smaller than both
+$\delta_{\rm raw}/3$ and $\delta_1/4$, and smaller than the fixed
+uncertainty radius in Lemma~\ref{lem:outer-uncertainty}. An admissible
+choice of $U^+$ is the set of centers having at least $(\delta_1/2)d$
+neighbors in $U$ in a forward or inverse A list. Off this set all row
+suggestions have fewer than half the local minimum distance in errors,
+so the joint decoder recovers the reference exactly; the clock majority
+and padding agree there too. If $u=|U|/|\Gamma|$, the spectral estimate
+from that lemma gives
+\[
+ \frac{|U^+|}{|\Gamma|}
+ \leq\frac{2\lambda^2u}{(\delta_1/2-u)^2}
+ \leq\frac u2.
+\]
+Neither this support nor the proof refers to the values inside $U$.
+For a requested-center cone with consistent repeated positions, extend
+its initially supplied values arbitrarily to the whole word, setting
+the reference outside $U$. The same support argument applies. This
+proves the assertion even when separate cones choose different
+extensions.
+
+After $O(\log|\Gamma|)$ synchronous global rounds the uncertainty set
+has cardinality less than one, giving exact recovery in polynomial
+time. From a recovered logical singleton slot, read its coefficient
+payload and evaluate the degree-$<h$ interpolant at $\omega^j$,
+$0\leq j<h$, to obtain the corresponding logical bits. The repeated
+clock gives $(c,\eta,j)$. The distance choice makes the promised
+reference unique. On inputs outside the recovery promise the algorithm
+has a total default; it may check the decoded label and its canonical
+word by the global polynomial-time algorithms. Finally,
+\eqref{oe:recipe-size-recurrence}--\eqref{oe:total-raw-size} and
+\eqref{oe:fixed-interface-composition} give all asserted cost and
+parameter-order statements.
+\end{proof}

preprints/The-PCP-for-PPAD-conjecture-a-quasilinear-reduction-September-25-2026/build/sections/06-outer-composition.tex

@@ -0,0 +1,1183 @@
+\section{Binary composition and the robust outer circuit}\label{sec:outer-composition}
+
+We now convert the symbol representation into a generalized circuit. Two
+features of the conversion require separate arguments. A local Boolean
+function may inspect polynomially many bits of a bounded number of symbols;
+its implementation must tolerate failures among its own gates. Moreover,
+sharing such implementations must not allow one corrupted physical value
+to spoil an unbounded number of decisions. We state the first interface,
+proved in Section~\ref{sec:inner-gadget}, and then give the construction and
+all the counting and synchronization arguments for the second.
+The same sequence is used inside a decision gadget in
+Section~\ref{sec:inner-gadget}: first obtain replica synchronization without
+trusting decisions, then bound unstable comparisons and failed computations,
+and finally use the clipped mean displacement to locate an endpoint.
+At the outer scale, a completed decision is shared by a group of output
+replicas, called a cohort; its internal gates and external read occurrences are charged separately from
+each output's own remaining gates.
+
+Throughout this section, a distance between words is relative Hamming
+distance, unless a norm is displayed. A constant may depend on previously
+fixed accuracies, but never on the End-of-Line input or its length. A
+``fixed branch'' of a randomized answer tree means that the earlier
+hypothetical answers are fixed when the randomized algorithm is described.
+It does not mean conditioning the probability space on receiving those
+answers. This distinction applies also to branches never executed.
+
+\subsection{A fixed port and a robust Boolean decision}
+
+The port family will encode every raw symbol in the same binary format.
+Its tester lets us recognize symbols close enough to use, and its recovery
+radius is fixed before we know how many symbols a corrected local
+computation will inspect. Once that arity is fixed, the robust-decision part of
+the interface realizes one Boolean output of such a computation. It
+provides many designated copies of the answer, so the guarantee remains
+useful even when failed gates concentrate on designated outputs.
+
+\begin{theorem}[Standardized ports and robust decisions]\label{thm:port-gadget}
+For every fixed positive integer exponent allowance $c_{\rm allow}$, there
+are a cutoff $L_{\rm port}$ and a uniform family of injective binary
+encodings, for $L\geq L_{\rm port}$,
+\[
+ \mathcal D:\{0,1\}^{L}\longrightarrow\{0,1\}^{S_0},
+ \qquad S_0=\poly(L),
+\]
+with fixed positive relative distance $d_B$, satisfying \textup{(i)}--
+\textup{(ii)} below. The radius $\nu_B$ in \textup{(iii)} is fixed with
+this port family; the actual-gadget assertions in that part have the
+separate later cutoff specified there.
+\begin{enumerate}[label=\textup{(\roman*)}]
+\item Every coordinate of $\mathcal D(s)$ is a multilinear polynomial in
+the bits of $s$ over $\F_2$, of degree at most $10$. Its coefficient list
+is computable in time polynomial in $L$. The degree bound is independent
+of $c_{\rm allow}$. Encoding and exact recovery from a fixed positive
+relative-error radius are deterministic polynomial-time algorithms.
+\item A perfect-completeness tester makes a bounded number of bit queries
+and performs bounded Boolean answer processing. Its public randomness has
+an enumerable polynomial-size rational support and polynomial bit
+complexity. For every fixed $a>0$, there are a fixed $\theta_B(a)>0$
+and a cutoff $L_B(a)\geq L_{\rm port}$, allowed to depend on $a$, such
+that for every $L\geq L_B(a)$ rejection below $\theta_B(a)$ implies
+distance less than $a$ from $\mathcal D(\{0,1\}^L)$.
+A word at distance $a$ from an encoding has
+rejection at most $C_Ba$, for a fixed $C_B$. Each fixed-branch query has a
+bounded-density marginal relative to the uniform distribution on the
+block. These assertions include arbitrary words and all hypothetical
+branches.
+\item There is a constant $0<\nu_B<d_B/4$ determined with this encoding,
+before any arity is specified. Subsequently fix an arity bound $k$, a
+size factor $C_f$, and a desired output loss $\upsilon>0$. There is a
+further cutoff $L_g$, allowed to depend on these fixed choices, such that
+all actual-gadget assertions below hold for every
+$L\geq\max\{L_{\rm port},L_g\}$. Consider any single-bit Boolean
+function of at most $k$ length-$L$ strings, described
+by an acyclic fan-in-at-most-two circuit over a fixed Boolean basis with
+public constants, of size at most $C_f L^{c_{\rm allow}}$. Some inputs
+may be declared public $0^L$.
+
+There is a uniform generalized subcircuit for this function, with a
+common count scale $G_0=\poly(L)$ for all padded functions under these
+bounds, at most $C_gG_0$ gates, and a set of $W$ designated output wires
+produced by its gates, where
+\begin{equation}\label{out:designated-count}
+ c_gG_0\leq W\leq C_gG_0.
+\end{equation}
+Here $c_g,C_g>0$ are fixed. Each nonpublic external input is a block of
+physical reads $X=3x-1$, with $x\in[0,1]$, compared to a public common
+rational threshold $\xi\in[2/5,3/5]$. Every external comparison occurrence
+in the complete unfolded subcircuit can be routed separately to an
+external replica.
+For each external port position, the total number of these occurrences is
+at most
+\begin{equation}\label{out:port-occurrence}
+ C_{\rm occ}G_0/S_0,
+\end{equation}
+uniformly over $\xi$, the function, and its public parameters. This counts
+all hypothetical branches, including unsuccessful and default branches.
+
+There are fixed $\delta_g,\gamma_g,\eps_g>0$ with the following universal
+soundness property. Fix logical strings $s_1,\ldots,s_k$ and ideal binary
+port words $y_j$ satisfying
+$\dist(y_j,\mathcal D(s_j))\leq\nu_B$ for each nonpublic port.
+For any subsequently chosen fixed physical margin $\omega>0$, use
+\[
+ 0<\eps\leq\eps_g,\qquad \eps<\omega/1000.
+\]
+Suppose at most $\gamma_gG_0$ external comparison occurrences fail to
+have a physical value on the side of $\xi$ specified by the corresponding
+bit of $y_j$, at gap greater than $\omega$. Repeated occurrences need not
+have consistent physical values. Then every assignment of the internal
+nodes in $[0,1]$ with at most $\delta_gG_0$ failed gates makes all but
+$\upsilon W$ designated wires correct near-bits for $f(s_1,\ldots,s_k)$:
+at most $\eps$ for zero and at least $1-\eps$ for one.
+\end{enumerate}
+The port construction and $\nu_B$ are independent of $k,C_f,\upsilon$.
+The later constants, the gate-count bounds, and the topology bounds may
+depend on these fixed choices, but not on $L,\xi$, or the later margin
+$\omega$. The threshold may appear as a rational gate parameter.
+Construction and other parameter computations take polynomial time in
+$L$ and the supplied description lengths, including the bit length of
+$\xi$; all parameter lengths are bounded by a polynomial in these
+quantities. In particular an arbitrarily long externally supplied
+threshold is charged for its bits.
+\end{theorem}
+
+Sections~\ref{sec:inner-algebra} and~\ref{sec:inner-binary} construct the
+algebraic and binary ingredients. Proposition~\ref{prop:dummy-port}
+proves the fixed encoding, tester, and recovery assertions. The remaining
+construction in Section~\ref{sec:inner-gadget} proves the physical occurrence
+bound~\eqref{ig:external-load}, gate count~\eqref{ig:gate-count}, and robust
+output guarantee~\eqref{ig:extraction-loss}, using the independent geometric
+procedures proved in Section~\ref{sec:geometry}. None of these constructions
+uses the outer circuit constructed here.
+
+\subsection{Port substitution, validation, and average correction}
+
+Apply Theorem~\ref{thm:outer-representation}, writing its raw encoding as
+$R_w=(s_k)_{k\in\Gamma}$, with
+\[
+ \ell_N=\lceil\log_2(N+2)\rceil,\qquad
+ L=(\ell_N+C_{\rm base})^{C_L},\qquad
+ |\Gamma|=O(N\ell_N^{O(1)}).
+\]
+Choose $c_{\rm allow}>c_{\rm raw}+12$, with any additional fixed local
+polynomial overhead absorbed in this inequality, and construct
+$\mathcal D$ before choosing correction rounds. Put
+\begin{equation}\label{out:binary-encoding}
+ E_w=(\mathcal D(s_k))_{k\in\Gamma},\qquad m=|\Gamma|S_0.
+\end{equation}
+The product of the two relative distances is a positive Hamming-distance
+bound; write $d_*>0$ for its square root, a lower bound on normalized
+Euclidean separation. The distinguished source and its actual successor are publicly computable in polynomial time.
+
+There is a fixed radius $\rho_{\rm dec}>0$ from which these words can be
+decoded globally. Indeed, let $r_B>0$ be a port recovery radius and
+$r_{\rm raw}>0$ a raw recovery radius. If the whole binary distance is
+less than $r_Br_{\rm raw}/2$, at most an $r_{\rm raw}/2$ fraction of port
+blocks can be at distance at least $r_B$ from their true block. Decode
+all ports, using a fixed default on an invalid or unsuccessful decoder
+output, and then use the raw decoder. An incorrect decoding at a bad
+site need not be detectable; the raw error bound already charges all
+such sites. Fix a smaller radius $\rho_{\rm dec}$ if necessary.
+
+The graph in Theorem~\ref{thm:outer-representation} has indegree and
+outdegree at most one, no self-edges, and no isolated vertices. Write
+$w_*$ for its distinguished source, which has an actual successor, and
+call every other missing-neighbor vertex a desired endpoint. The
+geometric construction applies to this encoded graph. For an edge
+$u\to v$, its four-block polygon successively changes the second, third,
+first, and third blocks:
+\[
+ \begin{aligned}
+ (E_u,E_u,0,0)&\longrightarrow(E_u,E_v,0,0)
+ \longrightarrow(E_u,E_v,1,0)\\
+ &\longrightarrow(E_v,E_v,1,0)
+ \longrightarrow(E_v,E_v,0,0).
+ \end{aligned}
+\]
+Here a scalar in a block denotes its constant repetition. The corners
+are rounded, and a segment in the fourth block enters the distinguished
+source from $(E_{w_*},E_{w_*},0,2)$.
+Theorem~\ref{thm:geometry} constructs a displacement field on
+\[
+ \mathcal B=[-1,2]^{4m},\qquad
+ \norm{U}_{\rm b}^2=\frac1m\sum_i U_i^2,
+\]
+where all distances to subsets of $\mathcal B$ use this norm. Each point
+of its endpoint set $\mathcal E$ has first block exactly $E_w$ for a desired
+endpoint $w$; the top of the extra segment is excluded. For any sufficiently
+small fixed rational $\tau>0$, chosen to meet the validation and
+global-recovery bounds in \eqref{out:tau}, there are constants
+$B_1,c_*,\beta_0>0$ and a field
+$v$ such that, after increasing $B_1$ to an integer,
+\begin{equation}\label{out:geometry-displacement}
+ \norm{v(U)}_\infty\leq B_1-1,\qquad
+ \norm{\clip_{\mathcal B}(U+\beta v(U))-U}_{\rm b}\geq c_*\beta
+\end{equation}
+whenever $\dist(U,\mathcal E)>10\tau$ and $0<\beta\leq\beta_0$.
+These constants precede estimator accuracy and the step $\beta$.
+The circuit will keep many replicas of each physical coordinate and
+force their mean vector close to $\mathcal E$; rounding the first block
+will then permit global recovery.
+
+To approximate the field locally, the numerical procedure of
+Lemma~\ref{lem:geometric-procedures} rounds the first two blocks at one
+common threshold $\xi\in[2/5,3/5]$. Each rounded word is a seed for
+validation. On good decisions an accepted seed must identify one fixed
+reference vertex, independent of the random acceptance decision and of
+the requested confidence or numerical accuracy; a seed in the intermediate
+region may be accepted or rejected. If one of the first two physical
+blocks is within $5\tau$ of a valid word, its seed must be accepted with that
+reference on good decisions for every allowed threshold. The procedure
+then asks for incident-edge flags, bits of each reference and its actual neighbors,
+and bits of the distinguished source and its successor. Each word-bit
+request uses either the coordinate matching the field component being
+estimated or a preallocated uniform coordinate for a bounded scalar
+statistic. Each reference and its neighbors determine a bounded list of
+incident path pieces; a fixed mesh and estimates of their scalar data
+approximate the field near those pieces. Scalar reads use fixed-width
+quantization with a public offset. We next supply these calls. Their
+number is fixed once the geometric mesh, scalar sample counts and
+numerical tolerances have been chosen, before the smaller bit-error
+budgets are assigned.
+
+\begin{lemma}[Protecting functional inputs]\label{out:flags}
+There are constants $0<a_1<a_2<\nu_B$, fixed before the arity of any
+functional invocation, with the following property. A bounded Boolean
+flag routine on an arbitrary binary port block can have arbitrarily
+small fixed failure probability, unflagging every block within $a_1$ of
+an encoding and flagging every block at distance at least $a_2$ from all
+encodings. In a functional invocation replace flagged ports by public
+$0^L$. Except on the union of the completed flag routines' failure
+events, every nonpublic input of every selected invocation is within
+$\nu_B$ of its intended encoding.
+\end{lemma}
+\begin{proof}
+Choose $a_2<\nu_B$ and then $a_1$ so small that
+$C_Ba_1<\theta_B(a_2)/4$. Estimate port-tester rejection by a fixed
+number of independent trials and compare to $\theta_B(a_2)/2$.
+Bounded independent variables give any fixed confidence by increasing
+the fixed sample count. In the interval between the two promises either
+Boolean result is permitted. An unflagged block on the good event is
+within $a_2$ of an encoding, unique because $a_2<d_B/4$; take its logical
+string as the intended input. A flagged input is explicitly a public
+zero, so it requires no external near-port premise. A union bound is
+applied to the finite list of completed flag promises, after that list
+has been fixed. Their internal tester answer trees are not counted as
+new logical promises.
+\end{proof}
+
+Fix a physical vector $U$ and initially fix its rounding threshold and
+scalar offset. A rounded seed $y=(y_k)_{k\in\Gamma}$ is now a fixed
+binary word; use bit one at equality with the threshold. For analysis, define a global nearest-symbol convention
+$s^*(k)$ at every site within $a_2$ of a port encoding, and choose a
+fixed arbitrary string elsewhere. An ideal functional call computes
+its Boolean function on these symbols at unflagged inputs and on
+$0^L$ at flagged inputs. Within one correction cone, deduplicate repeated
+raw sites and share their flag and initial logical value. Different
+cones may make different choices at uncertain sites.
+
+\begin{lemma}[Binary validation]\label{out:validation}
+There are fixed $0<\rho_{\rm comp}<\rho_{\rm acc}$, with
+$\rho_{\rm acc}$ smaller than the uniqueness radius of
+\eqref{out:binary-encoding}, and a finite ideal Boolean validation
+procedure such that, with arbitrarily high fixed probability, acceptance
+implies distance less than $\rho_{\rm acc}$ from a unique valid $E_w$,
+and distance less than $\rho_{\rm comp}$ implies acceptance with the
+true reference. The procedure uses port flags, port tests, and bounded
+raw-checker functional calls. It remains valid with averaged stationary
+expander-walk sampling as described below.
+\end{lemma}
+\begin{proof}
+Choose $\rho_{\rm acc}$ small enough that
+$\rho_{\rm acc}/a_1$ is within the raw uncertainty-correction radius.
+Choose $\alpha<\rho_{\rm acc}/4$ and take the raw gap
+$0<\theta_\alpha\leq1$ from Theorem~\ref{thm:outer-representation}.
+Writing $q=|\Pi|$ for its bounded checker-neighborhood size, choose
+$0<a<a_1$ and $b_0>0$ satisfying
+\begin{equation}\label{out:validation-constants}
+ \alpha+a+b_0<\rho_{\rm acc},\qquad
+ qb_0<\theta_\alpha/16,
+ \qquad p_{\max}=b_0\theta_B(a)>0.
+\end{equation}
+Decrease $p_{\max}$ if necessary. Let $p$ be the average port-tester
+rejection over all sites. Estimate $p$, and independently simulate the
+raw checker at sampled centers using the flagged/substituted inputs.
+Accept when the first estimate is at most $p_{\max}/2$ and the simulated
+raw rejection estimate is at most $\theta_\alpha/2$.
+
+On statistics of error less than $p_{\max}/4$, acceptance gives
+$p<p_{\max}$. Consequently the fraction of sites not within $a$ of any
+port encoding is less than $b_0$. At a center whose entire neighborhood
+avoids these sites, good flag promises leave all inputs unflagged and
+make the simulated checker exactly the checker on the global word
+$s^*$. Exceptional centers occupy at most $qb_0$. In the analysis,
+also estimate the true checker result on $s^*$ to accuracy
+$\theta_\alpha/8$ and the exceptional-center fraction to accuracy
+$\theta_\alpha/16$. These are fixed bounded statistics; the algorithm
+need not compute the hypothetical true statistic. On their good events
+and the completed flags' good events,
+\[
+ \Pr[\text{raw checker rejects }s^*]
+ \leq\theta_\alpha/2+\theta_\alpha/8+qb_0+\theta_\alpha/16
+ <\theta_\alpha.
+\]
+Raw soundness makes $s^*$ $\alpha$-close to a valid $R_w$. Charging all
+raw-disagreement sites and all non-$a$-close sites at cost one yields
+\[
+ \dist(y,E_w)<\alpha+a+b_0<\rho_{\rm acc}.
+\]
+The uniqueness radius makes this reference independent of subsequent
+random acceptance decisions for the fixed $y$.
+
+Conversely, if $\dist(y,E_w)<\rho_{\rm comp}$, then
+$p\leq C_B\rho_{\rm comp}$. By Markov's inequality at most
+$\rho_{\rm comp}/a$ of the ports are at distance at least $a$ from their
+true block. Outside their $\Pi$-neighborhoods the simulated checker is
+the perfectly accepting raw checker. Choose $\rho_{\rm comp}$ so small
+that both means are strictly below one quarter of their acceptance
+thresholds; choose the sampling errors with the remaining slack. Flag
+error probabilities are set after the sample counts. This proves both
+assertions. The proof requires concentration of the sampled exceptional
+fraction when $p$ is small. It never requires an arbitrarily long batch
+to avoid every exceptional site.
+\end{proof}
+
+Choose the geometric radius so that
+\begin{equation}\label{out:tau}
+ 4(10\tau)^2<\rho_{\rm dec},\qquad
+ \frac{(5\tau)^2}{(2/5)^2}<\rho_{\rm comp}.
+\end{equation}
+The second inequality supplies physical validation completeness:
+every incorrectly rounded coordinate of a binary word costs at least
+$(2/5)^2$ in squared distance for every allowed threshold. Fix the
+resulting $B_1,c_*,\beta_0$ and fix $\kappa=1/100$, the replica gap used
+below. Set the unscaled estimator error target with slack, smaller than
+$c_*\kappa/100$ in normalized root-mean-square norm.
+
+\begin{lemma}[Average correction of every relevant word role]
+\label{out:average-correction}
+After fixing the high-level mesh and statistical call counts, boundedly
+many raw correction rounds suffice to supply every required incident
+flag and every requested encoding bit, with any prescribed fixed average
+error. The average is over a uniform output component or its
+uniform-marginal statistical requests. The conclusion holds
+unconditionally for every true reference that can be accepted on good
+validation decisions, before selecting pieces or numerical winners.
+All resulting single-bit functions fit the fixed exponent allowance of
+Theorem~\ref{thm:port-gadget}.
+\end{lemma}
+\begin{proof}
+Fix a rounded seed that can be accepted on good validation, and its
+unique reference $R_w=(s_k)$. Put
+\[
+ U_0=\{k:\dist(y_k,\mathcal D(s_k))>a_1\},
+ \qquad |U_0|/|\Gamma|\leq\rho_{\rm acc}/a_1.
+\]
+Outside $U_0$, good flags unflag the port and its unique nearest logical
+string is $s_k$. Within $U_0$, substitution and intermediate flag choices
+may supply arbitrary values. In each deduplicated cone extend those
+initial values to a whole raw word agreeing with $R_w$ outside $U_0$.
+The raw uncertainty theorem gives a residual set $U_t$, depending only
+on $U_0$, with arbitrarily small fixed density after sufficiently many
+fixed rounds $t$. Its guarantee holds simultaneously for every such
+extension. Thus different cones and correlated intermediate flags cause
+no additional exceptional support. A corrected symbol is right off
+$U_t$; a local proposal and edge flag are right off a fixed permutation
+expansion of $U_t$.
+
+Compute an edge flag at a uniform-marginal center. For a requested own
+or neighboring encoding bit, compute the appropriate raw output symbol
+and then the requested coordinate polynomial of $\mathcal D$. Matching
+coordinates are uniform in the output average; the statistical requests
+have the same center marginal. Each fixed permutation expansion
+therefore costs at most its size times $|U_t|/|\Gamma|$. Set $t$ for
+the already fixed list of high-level true roles and calls. No union over
+all possible intermediate flag patterns is necessary.
+
+The raw simulation uses at most $C_tL^{c_{\rm raw}}$ Boolean gates, with
+exponent independent of $t$. A degree-at-most-ten Boolean polynomial has
+at most $\sum_{j=0}^{10}\binom Lj=O((L+1)^{10})$ monomials. Compute each
+monomial by a bounded-fanin AND tree and their sum by XOR gates expanded
+in the fixed basis. Coefficients and the requested bit's description are
+publicly generated. The extra bit-index and padding work has a fixed
+polynomial exponent. Our initial choice of $c_{\rm allow}$ leaves
+slack for this work. The number of raw ports and the multiplicative
+factor may increase with $t$, but are now fixed. The port encoding and
+its radius do not change.
+\end{proof}
+
+The completed port flags are now amplified for the finite list of
+logical promises in validation and these correction cones. Conditional
+on a sampled site, they use fresh tester randomness. Make the probability
+that any selected invocation has a nonpublic input outside $\nu_B$
+arbitrarily small. This protection concerns arbitrary rounded seeds,
+including seeds undergoing validation, and does not require exclusion
+of the geometric endpoint neighborhood. Uncertain flag results only
+choose which already uncertain inputs are substituted.
+
+\subsection{Small-support ideal tapes and complete branch counts}
+
+\begin{lemma}[Averaged expander sampling]\label{out:walk}
+The statistics needed above can be sampled with polynomial-in-$L$
+random support per output coordinate, with any desired fixed confidence
+in the average over output coordinates.
+\end{lemma}
+\begin{proof}
+Write an output coordinate as $(b,k_0,z_0)$ with $b\in[4]$,
+$k_0\in\Gamma$, and $z_0\in[S_0]$. Use fixed-length labeled walks in the
+outer Cayley expander starting at $k_0$. Fresh bounded segments and
+auxiliary uniform positions in $[S_0]$ supply all center and word
+statistics. Averaging over uniform $k_0$ makes every walk position
+stationary. If the mean-zero operator norm is at most $\lambda<1$,
+bounded centered functions at times $s,t$ have covariance at most
+$C\lambda^{|s-t|}$. Independent auxiliary randomness only changes the
+bounded diagonal terms and replaces off-diagonal functions by their
+conditional means. Therefore a batch of $q_s$ samples satisfies
+\[
+ \operatorname{Var}\left(\frac1{q_s}\sum_{j=1}^{q_s} Z_j\right)
+ \leq\frac{C}{q_s^2}\sum_{s,t=1}^{q_s}\lambda^{|s-t|}
+ \leq\frac{C(1+\lambda)}{(1-\lambda)q_s}.
+\]
+Chebyshev's inequality and a finite union give any fixed confidence by
+choosing a sufficiently large fixed $q_s$. The argument permits distinct
+bounded functions at the different slots and separate batches for
+physical blocks.
+
+Apply it to port rejection, the fixed global raw-checker and exceptional
+indicators in validation, and each actual reference candidate's
+statistics in the geometric procedure. At fixed rounded seeds those
+reference vertices are fixed whenever good acceptance is possible.
+Analyze all their roles before a numerical minimum or uncertain
+acceptance chooses which ones to use. The concentration is not
+conditioned on the realized answer path. Each walk has a fixed number
+of fixed-degree label choices. Each remaining choice has polynomial-in-
+$L$ support, and only a fixed number of choices is used. Their joint
+support per output coordinate is consequently polynomial in $L$.
+\end{proof}
+
+The preceding routines now meet the local-access contract of
+Definition~\ref{geo:access-contract}. Validation supplies fixed unique
+references, average correction supplies the matching and statistical bit
+requests, and the walk lemma supplies concentration for each underlying
+true candidate. The geometric procedure reserves at most eight true word
+roles and thirty-two piece roles, as enumerated in
+Section~\ref{geo:candidates}. Missing roles are omitted; duplicate roles
+may be retained.
+For fixed $U$ outside the endpoint neighborhood and each fixed admissible
+binary and scalar-quantization offset, it gives arbitrarily small
+prescribed $4\E_{i,\mathrm{trial}}|\widehat v_i-v_i(U)|^2$, with $i$
+uniform among the $4m$ components. All bit and flag errors and all
+candidate statistics are analyzed before a numerical winner is selected.
+On the resulting good event, Lemma~\ref{geo:stability} controls every
+sufficiently near-minimizing candidate in each component before selection, so a winner correlated with
+the requested component is harmless. The needed bit-error bounds remain
+average bounds over the allocated requests.
+
+Apart from the supplied calls, the procedure uses bounded comparison and
+Boolean-answer processing, public rational numerical tables, and a
+signed dyadic output in $[-B_1,B_1]$, including on every bad branch.
+A fixed-width scalar-quantization grid with an independently randomized
+offset suffices. Refining the finite grids of offsets changes comparison
+thresholds, not the number of bins, answers, or slots. All these
+statements concern the unscaled field.
+
+This outer application uses the accuracy and finite-role clauses
+\textup{(i)}--\textup{(iii)} of
+Lemma~\ref{lem:geometric-procedures} with completed ideal Boolean
+functional calls. Its optional physical-read clause \textup{(iv)} is
+not applied to the full outer gadget expansion. The fixed count
+$H_{\rm out}$ below counts direct outer read slots, and $B_{\rm out}$
+counts ideal functional invocations. Theorem~\ref{thm:port-gadget} and
+Lemma~\ref{out:routing} separately bound every physical read created by
+substituting those invocations.
+
+Applying parts \textup{(i)}--\textup{(iii)} of
+Lemma~\ref{lem:geometric-procedures} at this ideal-oracle level, with
+the preceding accuracy choices, we obtain ideal dyadic outputs satisfying,
+for every eligible fixed $U$ and every fixed admissible offset,
+\begin{equation}\label{out:ideal-accuracy}
+ 4\E_{i,\mathrm{tape}}\abs{\widehat v_i-v_i(U)}^2
+       <(c_*\kappa/100)^2.
+\end{equation}
+All outputs remain in $[-B_1,B_1]$ on bad trials. The good-port failure
+probability can independently be made arbitrarily small on this
+high-level call list. It is useful to record the order explicitly:
+\begin{enumerate}[label=\textup{(\arabic*)}]
+\item Fix geometric roles, mesh, scalar counts and numerical tolerances.
+\item Choose validation confidence and raw correction rounds for this
+finite logical-call list; fix the number of ports in every cone.
+\item Amplify each completed flag for its prescribed failure probability.
+Amplification adds tester reads, not new logical promises.
+\item Only now unfold the complete outer answer tree, leaving functional
+calls unexpanded and obtaining a fixed number $H_{\rm out}$ of direct
+read slots and $B_{\rm out}$ of hypothetical functional invocations.
+\item Choose $\upsilon$ for these $B_{\rm out}$ invocations and apply
+Theorem~\ref{thm:port-gadget} with the now fixed arity and size factor.
+\end{enumerate}
+This order does not define a bit accuracy by a union bound over the very
+answer tree created by amplifying that accuracy.
+
+At the ideal-oracle level, unfold every flag pattern, substitution
+pattern, quantized bin and hypothetical answer. Functional-call nodes
+remain unexpanded: each contributes one completed Boolean oracle answer,
+and its physical implementation is separately accounted for below.
+Thus $H_{\rm out}$ counts only direct read slots and $B_{\rm out}$ counts
+functional invocations. Supply randomness to unused nodes too. Earlier
+answers that determine an invocation's function or addresses are fixed
+publicly in that copy; they are not additional fragile input wires.
+Thus all port functions remain Boolean functions on their declared raw
+ports. A wrong numerical answer selects another reserved role or a
+finite table case. It cannot become an arbitrary hidden word address.
+Each hypothetical raw site is $\pi(k_0)$ for $\pi$ in a fixed finite set
+$\mathcal P$: compose walk-label moves, raw-neighborhood permutations
+and correction-cone moves. This set is independent of later offset-grid
+refinement. Direct scalar and bit slots use matching $z_0$, independently
+sampled uniform within-block positions, or the port tester's smooth
+queries. For fixed $k_0$ and fixed offsets, summing their unconditioned
+position densities over the finite complete tree gives a constant.
+
+The tape law in the next lemma includes both late offset grids---the
+binary threshold and the independent scalar-quantization offset---together
+with all internal choices at the hypothetical nodes. The grids are fixed
+before this final law is rationalized. The preceding accuracy estimates
+hold for each fixed offset, and hence for this joint law; the finite grids
+multiply its support bound only by a fixed factor.
+
+\begin{lemma}[Rational tapes and their read loads]\label{out:tapes}
+The true tape law can be replaced deterministically, at each output
+coordinate, by a list of a common square length $T_{\rm tap}=\poly(L)$,
+with total variation at most $\eta_0/S_0$, for any prescribed fixed
+$\eta_0>0$. The lists use only choices in the original support. They
+preserve \eqref{out:ideal-accuracy} and any bounded rare-event estimate
+with arbitrarily small fixed slack. Across these lists, every base
+physical coordinate is read by at most $C T_{\rm tap}$ direct
+comparison slots in the complete unfolded outer trees.
+\end{lemma}
+\begin{proof}
+Let $Q(L)$ bound the enumerable rational support of every tape law.
+For a common square $T\geq Q(L)S_0/\eta_0$, round each mass $Tp_j$
+down to an integer and distribute the remaining copies among nonzero
+support entries. Each mass error is at most $1/T$, and total variation
+is at most $Q(L)/T\leq\eta_0/S_0$. Floors, remainders and the ordering
+of entries are computed with exact rational arithmetic in polynomial
+time. The probability descriptions have polynomial bit complexity.
+A function in $[0,A]$ changes expectation by at most $A$ times total
+variation, proving the bounded-statistic assertions.
+
+For the load assertion, explicitly list all potential direct slots
+$\mathcal A$, including their fixed earlier-answer paths; set
+$H=|\mathcal A|$. If a slot is omitted or supplied a public value, its
+read subdistribution is dominated by that of its potential slot; it is
+never renormalized on being selected. Fix a target $(b',k',z')$ and a
+possible $\pi\in\mathcal P$. Its preimage $k_0=\pi^{-1}(k')$ is fixed.
+For every fixed offset, the true law gives
+\begin{equation}\label{out:direct-density}
+ \sum_{z_0=1}^{S_0}
+ \E[\text{number of direct reads at }(b',k',z')]
+ \leq C,
+\end{equation}
+where $C$ is the sum of the unconditioned fixed-slot density constants.
+Matching coordinates contribute one, uniform sampled coordinates
+contribute their bounded counts, and tester queries contribute their
+smoothness constants. Branches and seed roles add only a fixed factor.
+A position-count random variable lies in $[0,H]$. Changing the law for
+one output by total variation $\eta_0/S_0$ adds at most
+$H\eta_0/S_0$ to its expectation; summing over $z_0$ adds only
+$H\eta_0$. Sum the fixed preimages and physical roles and multiply by
+$T_{\rm tap}$. This is the asserted $O(T_{\rm tap})$ load.
+A constant total-variation tolerance without the $S_0$ denominator
+would not give this bound.
+\end{proof}
+
+The occurrence bound \eqref{out:port-occurrence} for each substituted
+gadget uses the same full-tree principle at its own scale. Its tape
+approximation must control loads both on its internal word, of length
+$m_{\rm in}$, and on an external port, of length $S_0$. Section~\ref{sec:inner-gadget}
+therefore uses total variation $\eta/\max\{4m_{\rm in},S_0\}$; the
+calculations \eqref{ig:external-load}--\eqref{ig:internal-load} give
+$O(G_0/S_0)$ external occurrences per port position and $O(R_{\rm in})$
+internal reads per base coordinate, where $R_{\rm in}$ is the gadget's
+replica count and $G_0=4m_{\rm in}R_{\rm in}$.
+This accounts for unequal polynomial block lengths without conditioning
+on an executed branch.
+
+\subsection{Cohorts, individual read routing, and gate realization}
+
+At each base component $i\in[4m]$, allocate one cohort for every tape in
+its list. A cohort has $K_c$ output replicas, where $K_c$ is a square
+satisfying
+\begin{equation}\label{out:cohort-size}
+ \max\{1,C_gG_0\}\leq K_c\leq C_KG_0
+\end{equation}
+for a fixed $C_K$; increasing $C_g$ if necessary also covers all
+designated-wire counts. The total number of replicas per component is
+$R=T_{\rm tap}K_c$, again a square. Each cohort contains its complete
+bounded list of functional invocations, sharing these invocations
+among its $K_c$ outputs. For each invocation, assign its $W$ designated
+wires to these outputs by round robin. Since $W\leq K_c$,
+\begin{equation}\label{out:round-robin}
+ \#\{\text{uses of any designated wire}\}
+ \leq\lceil K_c/W\rceil\leq 2K_c/W.
+\end{equation}
+A fraction $\upsilon$ of wrong designated wires therefore affects at
+most $2\upsilon$ of the cohort outputs. The remaining gates, called a
+direct cone, are separately allocated for each output replica.
+
+\begin{lemma}[Bounded physical read multiplicity]\label{out:routing}
+Every physical replica can be made the input of at most a fixed number
+of comparison occurrences, counting all outer and inner hypothetical
+branches. The construction is deterministic and uses no shared rounded
+bit between different comparison occurrences.
+\end{lemma}
+\begin{proof}
+Fix a physical block and a raw site $k'$. For each of the fixed possible
+address permutations, a functional port requesting $k'$ has a fixed
+$k_0$ preimage. There are $S_0$ output bit positions and $T_{\rm tap}$
+tapes at that preimage, and only a bounded number of invocation/port
+roles. Thus at most $C S_0T_{\rm tap}$ invocation ports request $k'$.
+By \eqref{out:port-occurrence}, any position in that site is used at most
+\begin{equation}\label{out:gadget-load}
+ C S_0T_{\rm tap}\,\frac{C_{\rm occ}G_0}{S_0}
+       \leq C' T_{\rm tap}K_c=C'R
+\end{equation}
+times. Lemma~\ref{out:tapes}, followed by replication over the cohort,
+gives $O(T_{\rm tap}K_c)=O(R)$ direct comparison occurrences at each
+base coordinate. Enumerate all these occurrences and assign successive
+ones round robin to its $R$ physical replicas. Each actual replica has
+bounded resulting multiplicity. Each comparison is performed in its
+own gate cone directly on that replica's $x$ value. In particular, an
+error in one comparison wire is not broadcast to other inner cones.
+\end{proof}
+
+Write the physical replica values and their means as
+\[
+ X_{ir}=3x_{ir}-1,\qquad \bar X_i=R^{-1}\sum_rX_{ir},\qquad
+ \norm{Y}_{\rm rep}^2=\frac1{mR}\sum_{i=1}^{4m}\sum_{r=1}^R Y_{ir}^2.
+\]
+A vector constant over replicas has replica norm equal to its base norm.
+Let $J$ be orthogonal projection onto these constant-over-replica
+vectors, so $JX=\bar X$ with the usual identification.
+
+For each $i$, average the replica labels using the half-lazy
+Margulis--Gabber--Galil operator $A_{\rm rep}$ on
+$(\mathbb Z/q\mathbb Z)^2$, $q=\sqrt R$. Its sixteen equally weighted
+labels consist of eight identity labels and the eight moves
+\[
+ (a\pm2b,b),\quad(a\pm(2b+1),b),\quad
+ (a,b\pm2a),\quad(a,b\pm(2a+1)),
+\]
+with multiplicities retained. For every positive integer modulus the
+normalized eight-move operator is symmetric with nonconstant numerical
+radius at most $5\sqrt2/8$; see the primary formal proof
+\cite[\texttt{mgg\_numerical\_radius}]{MGGAFP}. Thus
+\begin{equation}\label{out:replica-gap}
+ A_{\rm rep}J=J,\qquad
+ \norm{A_{\rm rep}(I-J)Y}_{\rm rep}
+       \leq(1-\kappa)\norm{(I-J)Y}_{\rm rep},
+ \qquad \kappa=1/100.
+\end{equation}
+The modulus need not be prime. Loops and parallel labels are kept as
+individual averaging inputs. The mean-zero space for modulus one is
+zero. Every averaging connection has bounded multiplicity as well.
+
+We specify the direct cones using exactly the gate types in
+Definition~\ref{def:gcircuit}. This also specifies how public constants
+and finite numerical processing are charged. Every needed constant
+node is the fresh output of a Constant gate in the cone using it, unless
+its value is incorporated as a legal Scale parameter or into a Boolean
+truth table. A physical comparison at threshold $t\in[-1,2]$ is a
+Constant gate with parameter $(t+1)/3$ and a Less gate on the physical
+replica's $x$ node and that constant, with orientation chosen for the
+required bit. Outside the physical range its answer is a public
+constant. On a satisfied comparison, an ideal mean gap greater than
+$4\omega$, an actual read deviation at most $\omega$, and
+$\eps<\omega/1000$ leave a strict Less antecedent even after the
+constant error: the node-scale gap is at least $\omega-\eps>\eps$.
+
+Boolean finite tables, flag processing, and quantization-bin processing
+use And, Or and Not with fan-in at most two. If their inputs are correct
+near-bits, each gate produces the same correct near-bit interval,
+without accumulating Boolean error. In particular the weak inequalities
+at $\eps$ and $1-\eps$ are sufficient. A selection among hypothetical
+invocations is implemented as a Boolean multiplexer: And each candidate
+answer with its selector and Or the resulting terms. An inactive
+selector is a low near-bit and forces that term low even if the unused
+candidate answer is arbitrary in $[0,1]$. Thus unsuccessful unused
+branches are shielded, without assuming that their outputs are bits.
+All finite answer tables have fixed size, although their public entries
+may depend on the tape and threshold.
+
+Represent the ideal dyadic field value by
+\begin{equation}\label{out:signed-value}
+ \widehat v=B_1(p^*-n^*),\qquad
+ p^*,n^*\in[0,1],\quad p^*n^*=0,
+\end{equation}
+using a fixed number of binary digits; enlarge the dyadic precision
+within the earlier estimator slack. Scale the near-bit digits by their
+dyadic weights and combine them by clipped Add gates to obtain $p,n$.
+Compute the $x$-average by sixteen Scale gates of weight $1/16$ and a
+bounded Add chain. All exact partial sums are in $[0,1]$. Choose a
+positive rational step with
+$t_\beta=\beta B_1/3\leq1$. Scale $p,n$ by $t_\beta$, add the positive
+term to the computed average, then subtract the negative term. The last
+output, or a final Copy output, is $x_{ir}$ itself. When the ideal signs
+are exclusive this expression is exactly
+\[
+ \frac{1+\clip_{[-1,2]}((A_{\rm rep}X)_{ir}
+                         +\beta\widehat v)}3.
+\]
+Indeed if the negative term vanishes the only possible clipping is at
+the upper end; if the positive term vanishes it is at the lower end.
+The affine change of coordinates commutes with the stochastic average.
+
+\begin{lemma}[Coarse and fine direct-cone bounds]\label{out:direct-cone}
+There is a fixed $C_{\rm ar}$, determined by the finite direct-cone
+shape before $\beta$ is filled in, such that every entirely
+$\eps$-satisfied direct cone obeys
+\begin{equation}\label{out:coarse-pointwise}
+ |X_{ir}-(A_{\rm rep}X)_{ir}|
+       \leq2\beta B_1+C_{\rm ar}\eps.
+\end{equation}
+No correctness of its Boolean or functional inputs is required.
+If its direct comparisons and selected invocation answers are correct
+near-bits for the ideal run on $\bar X$, then also
+\begin{equation}\label{out:fine-pointwise}
+ |X_{ir}-\clip_{[-1,2]}((A_{\rm rep}X)_{ir}
+                +\beta\widehat v_{ir}(\bar X))|
+       \leq C_{\rm ar}\eps.
+\end{equation}
+\end{lemma}
+\begin{proof}
+All hidden nodes, including $p,n$, remain in $[0,1]$ even when their
+logical values are wrong. Satisfied final Scale gates therefore make
+each signed term at most $t_\beta+\eps$. The absolute change from the
+computed average is at most their sum plus the final arithmetic errors.
+The averaging chain is within $C\eps$ of its exact average. Returning
+to physical units gives \eqref{out:coarse-pointwise}, after enlarging
+$C_{\rm ar}$. For the fine bound, induction through the shielded
+Boolean selections supplies the correct digits in
+\eqref{out:signed-value}. Scale is nonexpansive and clipped Add and
+Subtract are Lipschitz with respect to the sum of their input errors.
+For an explicit bound, let $J_d$ be the number of dyadic digits in
+one magnitude. A magnitude uses $J_d$ Scale and $J_d-1$ Add gates;
+its error is at most $(3J_d-1)\eps$. The sixteen Scale and fifteen Add
+gates for the average contribute at most $31\eps$. The two final
+Scale gates and the final Add/Subtract pair therefore contribute a
+total error at most $(6J_d+33)\eps$ in node units, or
+$(18J_d+99)\eps$ in physical units. An optional final Copy adds
+$3\eps$. The coarse bound's arithmetic error is at most $108\eps$
+including that Copy. Thus $C_{\rm ar}=18J_d+108$ suffices for this
+implementation, uniformly in all Scale parameters in $[0,1]$.
+The exclusive-sign identity gives \eqref{out:fine-pointwise}.
+\end{proof}
+
+Every invocation owns a disjoint gate set, and every direct cone owns a
+disjoint gate set. Internal outputs are fresh; designated and external
+wires are read-only fan-outs. The cyclic physical replica output is
+produced by the final gate of its own unique cone and by no other gate.
+There is no promised free constant node. Let
+\begin{equation}\label{out:gate-count}
+ K_{\rm out}=4mR,
+ \qquad |T|\leq C_{\rm tot}K_{\rm out}.
+\end{equation}
+The second assertion follows because a cohort has a bounded number of
+invocations, each with $O(G_0)=O(K_c)$ gates, and $K_c$ direct cones of
+bounded size. Each designated output is included in its invocation's
+gate count. In particular, concentrating failures on extraction gates
+receives the same charge as any other failures.
+
+\subsection{All failure and spoilage charges}
+
+We next choose the late offsets. Use a uniform rational grid in
+$[2/5,3/5]$ for $\xi$ and, independently, a uniform rational grid modulo
+the fixed scalar-bin width for the quantization offset. For every fixed
+physical value, the probability of lying within $4\omega$ of a binary
+threshold or any relevant shifted scalar boundary is at most
+\begin{equation}\label{out:anticoncentration}
+ C_{\rm bin}(D_{\rm grid}^{-1}+\omega),
+\end{equation}
+where $D_{\rm grid}$ is a lower bound on the two grid sizes and
+$C_{\rm bin}$ depends only on the already fixed widths and bin count.
+This follows by counting grid points in intervals of total length
+$O(\omega)$, adding at most a fixed number of boundary grid points.
+The number of comparison slots is unchanged when the grids are refined.
+The ideal accuracy estimate is conditional on every allowed offset, so
+refinement cannot invalidate it.
+
+Call a routed comparison occurrence \emph{spoiled} if its mean value is
+within $4\omega$ of its prescribed boundary or if its physical replica
+value differs from its mean by more than $\omega$. Otherwise its input
+has the prescribed ideal side with physical gap greater than $\omega$.
+For direct cones, spoiled comparisons are charged to their output. For
+functional invocations, count the occurrences themselves, including
+all hypothetical branches.
+
+\begin{lemma}[Deterministic charges for arbitrary assignments]
+\label{out:charges}
+There are fixed counting constants with the following properties.
+For an assignment with at most $\delta|T|$ failed gates:
+\begin{enumerate}[label=\textup{(\roman*)}]
+\item At most a $C_{\rm tot}\delta$ fraction of direct cones contains a
+failed gate.
+\item Cohorts containing an invocation with more than $\delta_gG_0$
+failed gates account for at most $C\delta/\delta_g$ of all outputs.
+\item The total number of excessive-deviation comparison occurrences,
+divided by $K_{\rm out}$, is at most
+$C\norm{X-\bar X}_{\rm rep}^2/\omega^2$.
+\item The averaged number of ambiguous gadget comparison occurrences,
+divided by $K_{\rm out}$, is at most
+$C(D_{\rm grid}^{-1}+\omega+\eta_0)$.
+The fraction of direct cones with an ambiguous comparison has the same
+form of bound. Here the averaging over tapes is the actual empirical
+frequency of the circuit, and the bounds hold for every fixed mean
+vector.
+\item Cohorts containing an invocation with more than $\gamma_gG_0$
+spoiled external occurrences account for at most $C/\gamma_g$ times
+the total normalized gadget-spoil count.
+\end{enumerate}
+\end{lemma}
+\begin{proof}
+Disjointness of direct gate sets proves (i). The number of invocations
+violating the threshold in (ii) is at most
+$\delta|T|/(\delta_gG_0)$. Charging $K_c$ outputs for each and using
+\eqref{out:cohort-size} and \eqref{out:gate-count} proves (ii), even if
+the same cohort is charged repeatedly. For (iii), Markov's inequality
+gives at most
+$mR\norm{X-\bar X}_{\rm rep}^2/\omega^2$ excessive physical replicas.
+Lemma~\ref{out:routing} bounds their total comparison multiplicity by
+a constant. The normalization $K_{\rm out}=4mR$ only improves the bound.
+
+For (iv), fix a port of an invocation at site $k$ and threshold $\xi$.
+Equation~\eqref{out:port-occurrence} bounds its ambiguous occurrences by
+\[
+ C_{\rm occ}G_0\,
+ \frac{\#\{z:|\bar X_{kz}-\xi|\leq4\omega\}}{S_0}.
+\]
+Every site is an image of a uniform $k_0$ under a member of the fixed
+permutation list. Sum over that list and all invocation ports under the
+unconditioned tape law, before selecting any answer branch. Averaging the offset and using
+\eqref{out:anticoncentration} bounds the expected number per trial by
+$C G_0(D_{\rm grid}^{-1}+\omega)$. The full occurrence bound is uniform
+in threshold-dependent function data. Tape rationalization changes a
+count bounded by $CG_0$ by at most $CG_0\eta_0/S_0$. Dividing each
+trial's charge by its $K_c$ output replicas gives (iv). Direct slots use
+the full-tree density argument in \eqref{out:direct-density}, and the
+same grid count; their count per trial is bounded by $H_{\rm out}$.
+This reasoning applies to every fixed eventual mean, rather than a
+mean selected before construction. Finally, (v) is the same threshold
+counting proof as (ii), with spoiled occurrences replacing failed gates.
+\end{proof}
+
+\subsection{Parameter order and synchronization before decisions}
+
+We give an acyclic choice of all remaining constants. Reserve enough
+slack in \eqref{out:ideal-accuracy} that after tape rationalization the
+ideal error norm is at most $c_*\kappa/20$. Choose $q_0>0$ so small that
+\begin{equation}\label{out:q0}
+ 20(B_1+1)\sqrt{q_0}<c_*\kappa/20.
+\end{equation}
+The geometric and statistical accuracies, correction rounds, and
+completed flags are chosen as above to put the ideal good-port failure
+probability below, for example, $q_0/20$. These are logical-call budgets.
+After the complete outer tree is fixed, choose
+$2B_{\rm out}\upsilon<q_0/20$ and obtain the gadget constants. Increasing
+$K_c$ for these gadgets weights every trial equally and does not
+magnify its ideal failure probability.
+
+Now choose fine fixed offset grids, then a small positive rational
+$\omega$, then a small tape tolerance $\eta_0$, and then a desired
+synchronization radius $d_{\rm sync}>0$. By
+Lemmas~\ref{out:tapes} and~\ref{out:charges}, they can be chosen so that,
+whenever $\norm{X-\bar X}_{\rm rep}\leq d_{\rm sync}$, the sum of the
+following empirical output fractions is less than $q_0/2$: failure of
+the ideal good-port event, direct ambiguous/excessive comparisons,
+and cohorts having an invocation with too many spoiled occurrences.
+For example assign $q_0/20$ to each grouped term, with the
+$C/\gamma_g$ factors included before choosing the responsible parameter.
+Ideal squared error is bounded, so make $\eta_0$ small enough for its
+reserved norm slack as well. This order is possible because all counts
+and density constants are already fixed and independent of these late
+grids and margins.
+
+Choose a positive rational $\beta\leq\beta_0$ with
+$\beta B_1/3\leq1$ and small enough that
+$4\beta B_1/\kappa<d_{\rm sync}/2$. Finally choose positive rational
+$\eps<1/10$ and $\delta<1$ so small that
+\begin{gather}
+ \eps\leq\eps_g,\qquad\eps<\omega/1000,
+ \qquad C\delta/\delta_g<q_0/20,\label{out:epsilon-delta}\\
+ \frac{4\beta B_1+2C_{\rm ar}\eps
+             +6\sqrt{C_{\rm tot}\delta}}{\kappa}<d_{\rm sync},
+ \label{out:coarse-budget}\\
+ 2C_{\rm ar}\eps+6\sqrt{C_{\rm tot}\delta}
+                <\beta c_*\kappa/20.
+ \label{out:fine-budget}
+\end{gather}
+There is positive slack in each inequality at $\eps=\delta=0$, so
+fixed positive rational choices exist. If necessary decrease them for
+any remaining direct numerical slack. All unstable events other than
+failed direct cones, including the mapped extraction errors, now total
+less than $q_0$.
+
+The finitely many lower-size cutoffs are imposed after these choices by
+increasing $C_{\rm base}$. These include $L_{\rm port}$, the finitely
+many $L_B(a)$ for the chosen tester accuracies, and $L_g$ for the fixed
+arity, size factor, and output loss. This does not create arity feedback. The raw
+algorithm's exponent is independent of correction rounds; the chosen
+port allowance has slack above that exponent and the degree-ten output
+polynomial evaluation. For the fixed cones, increasing the additive
+padding absorbs remaining multiplicative factors and cutoffs into a
+single fixed size factor, even factor one if desired. Padding checks
+and zero construction are $O(L)$ scans with uniform coefficients; their
+implementation does not perform hidden work growing with
+$C_{\rm base}$. Public computation of bit-polynomial coefficients may
+have a larger fixed polynomial exponent than the circuit evaluating
+the resulting monomials. Only the latter exponent is charged to the
+functional size allowance. No parameter varies with $N$ after this
+single choice.
+
+\begin{lemma}[Coarse synchronization]\label{out:coarse-sync}
+Every assignment satisfying all but $\delta|T|$ gates has
+$\norm{X-\bar X}_{\rm rep}<d_{\rm sync}$, without any assumption that
+its functional decisions are correct.
+\end{lemma}
+\begin{proof}
+Apply \eqref{out:coarse-pointwise} to nonfailed direct cones. At failed
+ones, both $X_{ir}$ and $(A_{\rm rep}X)_{ir}$ belong to $[-1,2]$, so
+their difference is at most three. The factor four in the norm and
+Lemma~\ref{out:charges}(i) give
+\[
+ \norm{X-A_{\rm rep}X}_{\rm rep}
+ \leq4\beta B_1+2C_{\rm ar}\eps
+                  +6\sqrt{C_{\rm tot}\delta}.
+\]
+On the other hand \eqref{out:replica-gap} and the reverse triangle
+inequality give
+$\kappa\norm{X-\bar X}_{\rm rep}
+\leq\norm{X-A_{\rm rep}X}_{\rm rep}$.
+Equation~\eqref{out:coarse-budget} proves the assertion. In particular,
+failures at physical feedback outputs, threshold constants, or the
+numeric averaging gates are already included; no decision is trusted
+in establishing this estimate.
+\end{proof}
+
+\subsection{The clipped mean contradiction and universal decoding}
+
+\begin{lemma}[Fine residual]\label{out:fine-residual}
+If an assignment meets the gate-failure budget and
+$\dist(\bar X,\mathcal E)>10\tau$, then
+\begin{equation}\label{out:residual}
+ E_{\rm res}:=
+ \norm{X-\clip_{\mathcal B}(A_{\rm rep}X+\beta v(\bar X))}_{\rm rep}
+                  <\beta c_*\kappa/2.
+\end{equation}
+Here clipping is componentwise on the replicas, and $v(\bar X)$ is
+repeated identically over them.
+\end{lemma}
+\begin{proof}
+Fix the actual mean $\bar X$ in all ideal tape experiments. Coarse
+synchronization permits all spoilage charges already budgeted. Discard
+cohorts with a high-failure or high-spoil invocation. In every remaining
+invocation whose ideal branch-pattern inputs are all near ports or
+public substitutions, Theorem~\ref{thm:port-gadget} applies. Mark all
+cohort outputs receiving an incorrect designated wire from any such
+invocation. Equations~\eqref{out:designated-count} and
+\eqref{out:round-robin} bound their union by
+$2B_{\rm out}\upsilon$ of the outputs. Do this marking for every
+near-input hypothetical invocation before following the selected path.
+Non-near-input invocations are irrelevant unless selected; on the ideal
+good-port event a selected invocation necessarily is a near-input one.
+
+Outside these marks and the other unstable events, every direct
+comparison in the complete tree has the needed margin and every
+selected invocation supplies its correct near-bit. Induct on the
+ideal answer sequence: earlier reliable bits select its actual
+substitution/function branch, the selected gadget is reliable, and
+Boolean shielding removes arbitrary unused outputs. Thus the whole
+selected direct computation equals its ideal Boolean run, and
+\eqref{out:fine-pointwise} holds.
+
+For an unstable but entirely satisfied direct cone, use only
+\eqref{out:coarse-pointwise} and $|v_i(\bar X)|\leq B_1$:
+its discrepancy from the clipped true-field update is at most
+$3\beta B_1+C_{\rm ar}\eps$. This is proportional to $\beta$ apart
+from numeric error. Failed direct cones use only the range bound three.
+The empirical ideal error, clipping nonexpansiveness, and the total
+unstable fraction at most $q_0$ therefore give
+\[
+ E_{\rm res}
+ \leq\beta\bigl(c_*\kappa/20+6B_1\sqrt{q_0}\bigr)
+      +2C_{\rm ar}\eps+6\sqrt{C_{\rm tot}\delta}
+ <\beta c_*\kappa/2
+\]
+by \eqref{out:q0} and \eqref{out:fine-budget}. This includes arbitrary
+concentration on extraction gates: above-budget invocations lose their
+whole cohort; below-budget invocations lose their guaranteed fraction
+of designated wires, and the lower bound in
+\eqref{out:designated-count} prevents a concentration factor.
+\end{proof}
+
+\begin{lemma}[Clipping and replica means]\label{out:clip-mean}
+Let $A$ be a stochastic replica operator satisfying
+\eqref{out:replica-gap}, and let $c$ be constant across the replicas of
+each component. Put $F=\clip_{\mathcal B}(AX+c)$ and
+$E=\norm{X-F}_{\rm rep}$. Then
+\[
+ \norm{X-JX}_{\rm rep}\leq E/\kappa,\qquad
+ \norm{JX-\clip_{\mathcal B}(JX+c)}_{\rm b}\leq E/\kappa.
+\]
+\end{lemma}
+\begin{proof}
+Write $Q=I-J$ and $g=\clip_{\mathcal B}(JX+c)$, a repeated vector.
+The mean projection minimizes distance to repeated vectors, and
+coordinate clipping is nonexpansive. Hence
+\[
+ \norm{QF}_{\rm rep}
+ \leq\norm{F-g}_{\rm rep}
+ \leq\norm{A(X-JX)}_{\rm rep}
+ \leq(1-\kappa)\norm{QX}_{\rm rep}.
+\]
+Projecting $X-F$ gives
+$\norm{QX}_{\rm rep}\leq E+(1-\kappa)\norm{QX}_{\rm rep}$,
+which proves the first inequality. Taking means and then comparing to
+$g$ gives
+\[
+ \norm{JX-g}_{\rm b}
+ \leq\norm{J(X-F)}_{\rm rep}+\norm{JF-g}_{\rm rep}
+ \leq E+(1-\kappa)\norm{QX}_{\rm rep}\leq E/\kappa.
+\]
+This argument does not exchange averaging and clipping.
+\end{proof}
+
+Apply Lemma~\ref{out:clip-mean} with $c=\beta v(\bar X)$ and the fine
+residual. If the mean were farther than $10\tau$ from $\mathcal E$, it
+would give
+\[
+ \norm{\bar X-\clip_{\mathcal B}(\bar X+\beta v(\bar X))}_{\rm b}
+ <\beta c_* ,
+\]
+contrary to \eqref{out:geometry-displacement}. Thus every assignment
+meeting the gate-failure budget satisfies
+\begin{equation}\label{out:endpoint-proximity}
+ \dist(\bar X,\mathcal E)\leq10\tau.
+\end{equation}
+Choose an endpoint witnessing this bound. Its first block is exactly
+$E_w$ for a desired endpoint. Round the first block of $\bar X$ at
+$1/2$. Each erroneous bit contributes at least $1/4$ to its squared
+coordinate distance from $E_w$, and so the resulting word $y$ obeys
+\[
+ \dist(y,E_w)\leq4(10\tau)^2<\rho_{\rm dec}.
+\]
+The global decoder following \eqref{out:binary-encoding} recovers $w$.
+The endpoint decoder in Theorem~\ref{thm:outer-representation} and
+Lemma~\ref{lem:macrograph} returns a nonzero End-of-Line solution. It
+does not require knowing which gates failed or which endpoint witnessed
+\eqref{out:endpoint-proximity}.
+
+This defines a deterministic decoder on every acceptable rational
+assignment. Means and comparisons can be computed exactly: forming a
+common denominator within a group increases its bit length by at most
+the sum of its input denominator bit lengths, followed by polynomial
+integer-arithmetic overhead. The port and raw algorithms are polynomial
+time, with arbitrary fixed defaults on out-of-promise inputs. The
+proved proximity places this actual input inside their guaranteed
+radii. Hence decoding is polynomial in $N$ plus the total assignment
+encoding length. The soundness argument itself applied to all real
+assignments satisfying the failure budget.
+
+\subsection{Complete output length, uniformity, and totality}
+
+Every constituent scale is a fixed polynomial in $L$, and
+$|\Gamma|=O(N\ell_N^{O(1)})$. Equations~\eqref{out:gate-count} and
+\eqref{out:cohort-size} give, for some fixed exponent $b_1$,
+\begin{equation}\label{out:size}
+ |V|+|T|=O(mT_{\rm tap}K_c)
+       =O(|\Gamma|S_0T_{\rm tap}G_0)
+       =O(N\ell_N^{b_1}).
+\end{equation}
+This count includes every copied direct cone, every entire shared
+invocation, its extraction gates, every Constant gate, and every
+comparison in every hypothetical branch. Routing creates one connection
+record per counted input occurrence; it is not an implicit description
+of a larger circuit. Explicit node and gate identifiers require
+$O(\log(N\ell_N^{b_1}))=O(\ell_N)$ bits. Field/index metadata, locally
+generated function coefficients, and every rational parameter have
+fixed polynomial-in-$L$ bit bounds. The offsets, bin widths,
+$\beta,\eps,\delta$ and numerical precisions are fixed rational
+constants. In particular, the threshold-input length qualification in
+Theorem~\ref{thm:port-gadget} contributes only a fixed bound here.
+If a representation repeats local parameter descriptions instead of
+sharing them, multiplying \eqref{out:size} by their polynomial-in-$L$
+length still changes only a fixed logarithmic exponent. Input/output
+lists and delimiters obey the same bound. Thus there are a fixed
+positive integer $A$ and fixed nonnegative integer $b$ for which the
+complete output encoding has length at most $AN\ell_N^b$.
+
+All tape masses and their integer multiplicities are deterministically
+enumerable. So are every raw-neighborhood and correction-cone address,
+every padded function, every inner invocation gate, every replica move,
+and every round-robin routing record. Public outer encodings and the
+source data are constructed by the polynomial-time algorithms already
+proved. Polynomially many operations on polynomial-bit integers suffice
+for all these tasks. Products and compositions of the finitely many
+fixed polynomial bounds give one deterministic polynomial-time
+reduction. It never lists the graph of all logical vertex labels. The
+gate set is nonempty, all parameters lie in $[0,1]$, fan-in is at most
+two, and the fresh-output construction enforces the required unique
+output ownership despite the feedback cycles.
+
+It remains to establish the finite totality assertion for arbitrary
+generalized circuits, independently of this reduction.
+
+\begin{lemma}[A fixed rational mesh satisfies every gate]
+\label{out:totality}
+For every fixed rational $0<\eps<1/10$, every generalized circuit in
+Definition~\ref{def:gcircuit} has a rational assignment satisfying all
+its gates, each coordinate having denominator
+$q=\lceil20/\eps\rceil$. The complete assignment length is bounded by
+one fixed polynomial with nonnegative integer coefficients in the
+circuit encoding length.
+\end{lemma}
+\begin{proof}
+Construct a continuous self-map of $[0,1]^V$. At an arithmetic gate's
+unique output use its exact numerical operation. At a Less gate with
+inputs $a,b$, use the clipped linear ramp that is zero when
+$x_b-x_a\leq-\eps/2$ and one when
+$x_b-x_a\geq\eps/2$. For Boolean gates define a continuous signal
+$h:[0,1]\to[0,1]$ equal to zero on $[0,2\eps]$, equal to one on
+$[1-2\eps,1]$, and linear between them. Use respectively
+\[
+ \max\{h(x_a),h(x_b)\},\qquad
+ \min\{h(x_a),h(x_b)\},\qquad 1-h(x_a)
+\]
+for Or, And, and Not. Set coordinates not produced by gates to zero.
+Unique output ownership makes these prescriptions a well-defined
+continuous cube map. Brouwer's fixed-point theorem~\cite{Brouwer1911} gives a fixed point
+$x^*$; no efficient computation of it is asserted.
+
+Round every coordinate to the nearest point of
+$\{0,1/q,\ldots,1\}$, obtaining $\widetilde x$ with
+$|\widetilde x_v-x_v^*|\leq1/(2q)\leq\eps/40$.
+Constant errors are at most $\eps/40$, Scale and Copy errors at most
+$2\eps/40$, and Add and Subtract errors at most $3\eps/40$, using
+parameter bounds and nonexpansive clipping. For a triggered strict
+Less antecedent after rounding, its signed input gap before rounding
+has the same sign and magnitude strictly greater than
+$\eps-2\eps/40>\eps/2$. Its fixed-point output is therefore the
+required endpoint zero or one, which rounding preserves.
+
+If a weak Boolean low antecedent
+$\widetilde x_a\leq\eps$ holds, then
+$x_a^*\leq\eps+\eps/40<2\eps$, so $h(x_a^*)=0$.
+Similarly $\widetilde x_a\geq1-\eps$ implies
+$h(x_a^*)=1$. Every applicable Or, And, or Not conclusion follows
+from its displayed exact formula, including equality at the weak
+threshold. The corresponding fixed-point output is zero or one and
+is unchanged by rounding. This verifies every gate implication.
+
+The denominator $q$ is a fixed integer depending only on $\eps$.
+There are at most a fixed multiple of $M+1$ explicitly listed nodes
+in a length-$M$ input, and each output rational has a fixed bit bound.
+Accounting also for order, indices if used, and delimiters, a
+sufficiently large fixed integer $C$ makes
+$p(M)=C(M+1)^2$ an upper bound for the entire assignment. This
+polynomial has nonnegative integer coefficients and is independent of
+the particular circuit and its parameter denominators.
+\end{proof}
+
+\begin{proof}[Proof of Theorem~\ref{thm:main}]
+The preceding argument uses Theorem~\ref{thm:port-gadget}, whose construction
+is proved in Sections~\ref{sec:inner-algebra}--\ref{sec:inner-gadget}, and
+Lemma~\ref{lem:geometric-procedures}, proved in Section~\ref{sec:geometry}.
+Use the fixed rational $\eps,\delta$ chosen in
+\eqref{out:epsilon-delta}--\eqref{out:fine-budget} and the polynomial
+$p$ from Lemma~\ref{out:totality}. That lemma proves totality, with all
+gates satisfied. The explicit construction and \eqref{out:size} give
+the claimed complete quasilinear encoding bound and a deterministic
+polynomial-time reduction. Equations~\eqref{out:endpoint-proximity}
+and~\eqref{out:tau}, followed by the global endpoint decoder, give the
+required solution from every acceptable assignment, with the stated
+polynomial running time. Every constant and algorithm has been fixed
+independently of the input. This proves all four assertions.
+\end{proof}

preprints/The-PCP-for-PPAD-conjecture-a-quasilinear-reduction-September-25-2026/build/sections/07-inner-algebra.tex

@@ -0,0 +1,1552 @@
+\section{The inner path and its algebraic representation}\label{sec:inner-algebra}
+
+We construct the algebraic part of the decision gadget promised in
+Theorem~\ref{thm:port-gadget}.  Its graph consists of one directed path for
+each tuple of read-only inputs.  Besides encoding these paths, the
+construction arranges that every chart involving one read port has exactly
+the same native layout and formula in a common one-port template.  This
+literal correspondence, including all verification auxiliaries, will supply
+the distinguished source in Section~\ref{sec:inner-gadget}.
+
+\emph{All field, clock, dimension, and size notation is reset in this
+section.}  In particular, the fields and the polynomial-size affine domain
+below are independent of the outer index group.  The construction and its
+proof do not use the outer circuit realization.  All graphs in this section
+are described implicitly; we never enumerate their exponentially many
+Boolean input tuples.
+
+\subsection{Parameters and the extended path}\label{ia:path-parameters}
+
+Fix the exponent allowance $c_{\rm allow}$ in
+Theorem~\ref{thm:port-gadget}, and choose a fixed integer
+$c_0>c_{\rm allow}+5$.  For a port length $L$, set
+\begin{equation}
+ n'=(L+2)^{c_0},\qquad
+ h=\Theta\bigl((\log(n'+2))^{1/4}\bigr),\qquad
+ H=\mathbb F_h,\qquad q=h^v,                         \label{ia:parameters}
+\end{equation}
+where $h$ is a deterministically chosen power of two and $v$ is a fixed
+sufficiently large integer.  Let $\ell$ be the least positive integer
+coprime to $v$ for which $h^\ell\ge(n')^3+3$, and put
+\begin{equation}
+ K_1=\mathbb F_{h^\ell},\quad K=\mathbb F_{q^\ell},\quad
+ C=K_1^\times,\quad M=h^\ell-1,\quad D=\ell^2-1.
+                                                               \label{ia:fields}
+\end{equation}
+Fix a generator $\gamma$ of $C$.  Since $\gcd(\ell,v)=1$, the subfields
+$K_1$ and $\mathbb F_q$ of $K$ intersect in $H$ and generate $K$.
+An $H$-basis of $K_1$ is therefore an $\mathbb F_q$-basis of $K$.
+We use it throughout: one $K$ variable has $\ell$ coordinates over
+$\mathbb F_q$, and its $H^\ell$ digit grid is precisely $K_1$.
+
+An interval of $v$ consecutive integers contains an integer congruent to
+$1$ modulo $v$.  Thus the coprimality restriction changes the first size
+threshold for $\ell$ by at most $v$, and
+\begin{equation}
+ \ell=\Theta(\log n'/\log\log n'),\qquad
+ h=o(\ell),\qquad |K|=q^\ell=(n')^{O(1)}.          \label{ia:size-bounds}
+\end{equation}
+Choose $v$ large enough that $q$ dominates the fixed power of $D$ needed
+by the scalar line--point input in Section~\ref{sec:inner-binary}, with
+positive power slack.  In particular, $D/q\longrightarrow0$.  After $v$,
+fix an integer $a>v$ and set
+\begin{equation}
+ B=\lceil q^{1/a}\rceil\le\ell                     \label{ia:radix-size}
+\end{equation}
+for all sufficiently large $L$.  Every fixed multiple of $\ell h$ is then
+eventually below $D$.  All these choices precede the number of ports and
+the particular Boolean function.
+
+All finite sets used below have polynomial cardinality in $n'$.  Field
+presentations, compatible embeddings, bases, generators, interpolation
+matrices, and deterministic orderings can consequently be found by
+exhaustive finite-field algebra in polynomial time in these cardinalities.
+For instance, irreducible presentations and suitable generators can be
+found by enumeration and their defining properties checked by finite
+arithmetic.  No claim of polynomial time in the logarithm of a field's
+cardinality is required.  Fix these conventions once for the common
+template.  Finitely many smaller sizes can be handled by enlarging the
+fixed lower cutoff as in the gadget statement.
+
+Now fix an arbitrary constant number $k\ge1$ of ports and an arbitrary
+constant size factor for the Boolean circuit of
+$f:(\{0,1\}^L)^k\to\{0,1\}$.  Invocations with fewer ports are padded
+by unused public-zero ports.  Normalize the circuit to a topological list
+of gates with Boolean rules of degree at most two over characteristic two,
+and finish with a separate output copy gate.  This costs only a constant
+factor in circuit size.  The size cutoff may depend on $k$ and on the
+fixed size factor.
+
+The logical tape has read-only groups $s_i\in\{0,1\}^L$ for $i\in[k]$,
+a work group $w$ indexed by $K_1$, and legacy bits $e_z$ indexed by
+$z\in C$.  Each $s_i$ is embedded by the same fixed injection of $[L]$
+into $K_1$; other grid entries are specified zeros.  Stages
+$r=0,\ldots,M-1$ have labels $c=\gamma^r$.  Work is blank at stage zero.
+During step $r=j-1\to j$, for $1\le j\le L$, copy $s_i[j]$ to a
+distinct work location $w_{i,j}$ for every $i$ in parallel.  Thereafter
+assign one new gate location per step, using earlier work locations or
+public constants.  Reserve a fixed grid position $y_{\rm out}$ solely
+for the final output assignment.  Pad remaining steps by idle work rules.
+The number of work positions and nonidle steps is at most
+\[
+ kL+O(1)+O(L^{c_{\rm allow}}),
+\]
+with constants allowed to depend on the circuit size factor.  The choices
+above make both counts fit below the available grid and stage sizes.
+
+At stage $c$, require $e$ to be one-hot at $c^{-1}$.  Require every
+unassigned work position to be zero, every assigned copy to equal its
+specified input, and every assigned gate to satisfy its rule.  Consequently
+each Boolean tuple $s=(s_i)_i$ has exactly one valid tape
+$b=(w,e,(s_i)_i)$ at every stage.  Join consecutive stages by their
+assignment steps, and disable the wrap step $\gamma^{-1}\to1$.
+
+\begin{lemma}[Validity and a constant edit alphabet]\label{ia:validity}
+The union of the stage-valid tapes is described by polynomially many
+Boolean equations of degree at most $d=3$.  For fixed $k$, every macro
+step and its inverse use a fixed finite alphabet of flips, whose
+coefficients are determined by a bounded number of logical reads and
+public rule bits.  Equations involving $s_i$ and other groups can be
+written as the sum of an $(e,s_i)$ contribution and an $(e,w)$
+contribution.  No equation requires a product involving two distinct
+positive port groups.
+\end{lemma}
+
+\begin{proof}
+Impose bitness on all logical coordinates, and
+\begin{equation}
+ e_z e_{z'}=0\quad(z\ne z'),\qquad
+ \sum_{z\in C}e_z=1.                              \label{ia:one-hot}
+\end{equation}
+For each stage $c$, multiply all its work requirements by
+$e_{c^{-1}}$.  Gate equations have degree at most two before this
+multiplication, so the resulting degree is at most three.  A copy
+requirement at an occupied position is specifically
+\begin{equation}
+ e_{c^{-1}}w_{i,j}-e_{c^{-1}}s_i[j]=0.             \label{ia:copy-equation}
+\end{equation}
+The requirements are vacuous at unselected stages.  At the one selected
+stage, the prefix equations determine all assigned work positions in
+topological order and force every other position to zero.  This proves
+the characterization and the dependency assertion.
+
+For start frame $c$, list $k$ work write slots and the two legacy slots
+$c^{-1},(\gamma c)^{-1}$.  During a gate step only the first work slot
+is active; use a zero unit vector for every inactive slot.  Write
+$v_1(c),\ldots,v_{k+2}(c)$ for these public vectors in the logical tape
+space, and for $\sigma\in\{0,1\}^{k+2}$ put
+\begin{equation}
+ h_\sigma(c)=\sum_{r=1}^{k+2}\sigma_r v_r(c).
+                                                               \label{ia:flip-mask}
+\end{equation}
+At a valid pre tape, choose a work coefficient to be the old bit plus
+its required new assignment; choose both legacy coefficients to be one.
+The required assignment is a read input, a degree-at-most-two gate of
+earlier work bits, or the unchanged value for an inactive slot.  At a
+valid post tape, the inverse clears precisely the newly assigned
+positions and flips the same legacy positions.  Earlier operands have
+not changed, so the two operations are reciprocal.  There is a bounded
+list of old-value, operand, and input reads for fixed $k$.  Public phase,
+gate, activity, and wrap bits select their roles.  Define the same
+coefficient rules on arbitrary Boolean read answers; at the disabled
+wrap use an idle work proposal.  This gives a finite branch system
+without any assumption that those arbitrary answers describe a valid
+tape.
+\end{proof}
+
+\begin{definition}[Extended inner graph]\label{ia:extended-path}
+An extended label is $(c,b,\eta,t)$, where $c\in C$,
+$\eta\in\{0,1\}$, and $0\le t\le q-1$.  In pre mode $\eta=0$,
+$b$ must be valid at stage $c$; in post mode $\eta=1$, it must be valid
+at stage $\gamma c$.  Edges are the following:
+\begin{enumerate}
+\item in pre mode, increase $t$ by one until $q-1$;
+\item at full pre progress, perform the enabled macro edit and change
+to post mode, retaining the frame and full progress;
+\item in post mode, decrease $t$ by one until zero;
+\item at empty post progress, change to empty pre mode in frame
+$\gamma c$, leaving $b$ unchanged.
+\end{enumerate}
+The fourth operation is included also for $c=\gamma^{-1}$; only the
+second operation is disabled at that frame.
+\end{definition}
+
+\begin{lemma}[Endpoints and output]\label{ia:path-endpoints}
+For each $s$, the extended labels with that read-only tuple form a
+single directed path, without isolated vertices or self edges.  Its
+unique source and immediate successor are respectively
+\begin{equation}
+ (\gamma^{-1},b_0(s),1,q-1),\qquad
+ (\gamma^{-1},b_0(s),1,q-2),                      \label{ia:source-successor}
+\end{equation}
+where $b_0(s)$ has blank work and legacy bit $e_1=1$.  Its other
+endpoint is the full pre label at stage $M-1$, and there
+$w[y_{\rm out}]=f(s)$.
+\end{lemma}
+
+\begin{proof}
+Replace the unique macro tape at stage $x$ by its post labels in frame
+$\gamma^{-1}x$, in decreasing order of $t$, followed by its pre labels
+in frame $x$, in increasing order of $t$.  The frame-advance edge joins
+the two halves.  Each enabled macro edit joins the full pre end of one
+such segment to the full post start of the next.  The only absent join
+is the wrap join from the final stage to stage zero.  This gives exactly
+the stated path and endpoints.  Every segment has more than one vertex,
+so there are no isolated vertices; each edge changes a label component,
+so there are no self edges.  The final work assignment and all its
+ancestors have been completed at the last stage, proving the output
+claim.
+\end{proof}
+
+\subsection{Chart conventions and lifted cyclic interpolation}
+\label{ia:lifts-and-masks}
+
+A native chart is an $\mathbb F_q$-coefficient polynomial on
+$O(\ell)$ coordinates over $\mathbb F_q$, possibly with slots.  A
+persistent chart has at most one slot axis of length $O(\ell)$; stacks
+will have a bounded number of such axes.  A full rotating or projection
+variable always occupies one whole $K$ label.  Native charts will later
+be embedded in one common $K^u$, with unused coordinates ignored.
+
+For slot lengths $t_1,\ldots,t_r$ and masks
+$L_b\in\mathbb F_q^{t_b}$, write
+\begin{equation}
+ L\cdot F(Y)=\sum_{z_1,\ldots,z_r}
+             \left(\prod_{b=1}^r L_b[z_b]\right)
+             F_{z_1,\ldots,z_r}(Y).                \label{ia:contraction}
+\end{equation}
+There are $q^{O(\ell)}$ mask tuples.  To query any specified mask tuple
+smoothly, choose independent uniform $A_b$, substitute
+$L_b=A_b+(L_b-A_b)$, and expand the $2^r$ terms.  Each term has uniform
+mask marginals, even conditional on the point and the requested tuple.
+A nonzero tensor remains nonzero after independent random contractions
+with probability at least $(1-1/q)^r$: at each step a nonzero linear
+map has a kernel of relative size at most $1/q$.
+
+We repeatedly use the elementary polynomial zero bound: a nonzero
+polynomial of total degree $d_*$ on $\mathbb F_q^m$ vanishes on at most
+a $d_*/q$ fraction of the grid.  For completeness, expand in the last
+variable with leading coefficient of degree at most $d_*-r$ and last
+variable degree $r$.  Inductively that coefficient vanishes on at most
+$(d_*-r)/q$ of the earlier coordinates; elsewhere there are at most
+$r$ roots in the last variable.  Induction on $m$ proves the bound.
+We apply it also after adjoining the finitely many mask groups.
+Thus any nonzero degree-$O(D+\ell h)$ chart identity, including a
+whole slot list, is detected with constant probability.  An ignored
+coordinate group is checked by comparing two independent replacements
+of that group with common uniform other coordinates.  Its difference
+is a polynomial; the same bound gives an exact alternative between
+independence and constant rejection probability.  Throughout, smoothness
+means bounded marginal density relative to uniform point and mask
+coordinates.  Ignored coordinates are independently filled uniformly.
+
+Ordinary interpolation on the $H$ digits extends a function on $K_1$
+with individual degrees below $h$ and total degree at most
+$\ell(h-1)$.  The punctured cyclic grid requires a different extension.
+
+\begin{lemma}[Cyclic lift, including descent at zero]\label{ia:cyclic-lift}
+For $0\le j<M$, write $j=\sum_{t=0}^{\ell-1}j_t h^t$ and define
+\begin{equation}
+ \widetilde j=\sum_{b=0}^{\ell-1}j_{vb\bmod\ell}q^b.
+                                                               \label{ia:lift-exponent}
+\end{equation}
+Replacing $Z^j$ by $Z^{\widetilde j}$ in interpolation on $C$ gives a
+linear extension to $K$, of coordinate degree at most $\ell(h-1)$.
+For $\mathbb F_q$-valued data this extension has coefficients in
+$\mathbb F_q$ in the chosen coordinates, including its value at zero.
+It commutes with $Z\mapsto xZ$ for every $x\in C$.
+\end{lemma}
+
+\begin{proof}
+Modulo $M$ we have $q^b=h^{vb}$ and $h^\ell=1$, hence
+\[
+ \widetilde j\equiv
+ \sum_b j_{vb\bmod\ell}h^{vb\bmod\ell}=j\pmod M.
+\]
+The two monomials therefore agree on $C$, where the powers
+$1,Z,\ldots,Z^{M-1}$ form the usual interpolation basis.  Each map
+$Z\mapsto Z^{q^b}$ is linear over $\mathbb F_q$, so the coordinate
+degree of the lifted monomial is at most $\sum_t j_t\le\ell(h-1)$.
+
+Multiplication of $\widetilde j$ by $q$ modulo $q^\ell-1$ cyclically
+rotates its base-$q$ digits.  Under \eqref{ia:lift-exponent}, this is
+the permutation $j\mapsto qj\bmod M$.  The digit list all of whose
+entries equal $h-1$ is excluded by $j<M$; exponent zero is consequently
+the unique representative of residue zero on $C$.  If the original
+interpolant is $\sum_j a_jZ^j$ and its values belong to $\mathbb F_q$,
+uniqueness of interpolation on $C$ gives
+\[
+ a_{qj\bmod M}=a_j^q.
+\]
+The digit rotation then proves that its lift $F$ satisfies
+$F(z)^q=F(z)$ for every $z\in K^\times$.  At zero, $F(0)=a_0$ and
+$a_0^q=a_0$, so the same assertion holds.
+
+The coordinate degree is below $q$ after our cutoff.  Conjugating the
+coordinate coefficients by the $q$-power map leaves the values on
+$\mathbb F_q^\ell$ unchanged.  Uniqueness of interpolation with
+individual degrees below $q$ forces each coefficient to be fixed by
+that conjugation, and thus to lie in $\mathbb F_q$.  Finally
+$x^{\widetilde j}=x^j$ for $x\in C$, which proves the scaling assertion
+on every basis monomial.  These arguments apply coefficientwise in any
+additional independent variables.
+\end{proof}
+
+\subsection{Canonical certificates for linear conditions}
+\label{ia:linear-certificates}
+
+Partial-sum polynomial certificates also appear in Rubinstein's
+construction~\cite[Sections~6.2--6.3]{Rubinstein2016Settling}.
+Here each input chart has its own canonical certificates, and contributions
+from different charts meet only in a tested identity.  We require these
+recipes to remain linear in their input chart and, in rotating calls, to
+respect cyclic scaling of the designated variables even when the final
+condition fails.
+
+\begin{lemma}[Smooth linear verification]\label{ia:linear-tool}
+Let a fixed finite list of primary native charts, each scalar or with
+one $O(\ell)$ slot axis, be subject to an explicit
+$\mathbb F_q$-linear condition $\mathcal S$ on their full evaluation
+arrays.  Its matrix has polynomially many rows and columns, with
+logarithms $O(\ell\log q)$.  Coefficient and special-grid conditions
+may be expressed by public linear matrices.  A nonrotating call may
+adjoin a separately enforced primary $1$ for affine terms.
+
+In a rotating call, at most one full variable in each primary is
+designated to scale by a common $x\in C$.  Assume $\mathcal S$ is
+invariant under this simultaneous pullback and imposes the lifted
+shape of Lemma~\ref{ia:cyclic-lift} in every rotating variable.  No
+arbitrary noninvariant affine term is allowed in such a call.
+
+The condition has a canonical verification construction with the
+following properties.
+\begin{enumerate}
+\item A fixed number of auxiliary families is stored \emph{separately
+for each primary block}.  Each has native dimension $O(\ell)$ and at
+most one slot axis of length $O(\ell)$.  Its canonical arrays are
+linear in that primary's arrays, also when $\mathcal S$ fails.
+\item Honest input coordinate degree $d_{\rm in}$ produces auxiliary
+degree $O(d_{\rm in}+\ell h)$.  A fixed mixture of bounded, smooth
+linear field probes is perfectly complete on legal canonical tuples
+within the degree budgets.  On genuine degree-at-most-$D$ polynomial
+tuples, either $\mathcal S$ holds and every auxiliary is canonical, or
+the mixture rejects with a fixed positive probability.
+\item In a rotating call each auxiliary has a new designated full
+variable $R$, and rotating its primary by $x$ changes the auxiliary by
+$R\mapsto xR$.  Its canonical recipe has the lifted shape in $R$
+whenever its primary has the required shape, even if final rows fail.
+All canonical arrays and public maps are polynomial-time constructible.
+\end{enumerate}
+\end{lemma}
+
+\begin{proof}
+We give the construction, uniqueness, degree, and probe arguments.
+
+\paragraph{Why arbitrary scaling factors preserve the linear condition.}
+In a rotating call substitute $RZ$ for the designated variable of a
+primary and denote the resulting chart by $f_R$.  An unchanged primary
+stays unchanged.  We first show that legal input arrays satisfy
+$\mathcal S$ for every $R\in K$, rather than only for $R\in C$.
+Extend scalars in the evaluation space and its linear subspace
+$\mathcal S$ from $\mathbb F_q$ to $K$.  Since $|C|=M$ is odd, it is
+invertible in characteristic two.  The character projections
+\[
+ \Pi_j=\frac1M\sum_{x\in C}x^{-j}\rho(x)
+ \qquad(0\le j<M)
+\]
+for the simultaneous scaling action $\rho$ decompose the space and
+preserve every invariant subspace.  On the imposed lift shape, the
+character $j$ is represented by the single exponent $\widetilde j$
+in every rotating primary.  Nonrotating components occur in character
+zero.  Substitution by $R\ne0$ therefore acts by the scalar
+$R^{\widetilde j}$ on each character space, and preserves the extended
+subspace.  At $R=0$ it projects onto character zero, which also preserves
+the subspace.  Scaled evaluations of our $\mathbb F_q$-valued chart
+are still $\mathbb F_q$-valued; intersecting with the original
+evaluation space, or using the original defining rows, proves the
+assertion over $\mathbb F_q$.
+
+A failed original row fails at $R=1$, so it remains a nonzero polynomial
+condition.  Tests requiring the substitution sample $R$ uniformly from
+$K^\times$; this changes a uniform density by at most
+$(1-|K|^{-1})^{-1}$ and keeps primary point marginals uniform.  The
+substitution is bilinear in coordinate variables and at most doubles
+total degree.  In a nonrotating call omit $R$ everywhere below.
+
+\paragraph{Separate row contributions and partial sums.}
+Pad the row set by zero rows to $H^\rho$, with $\rho=O(\ell)$.
+For one primary block with $m_0$ native $q$-coordinates, put $m'=vm_0$.
+An $H$-basis of $\mathbb F_q$ gives a packing map
+\[
+ w_{\rm pack}:\mathbb F_q^{m'}\longrightarrow\mathbb F_q^{m_0}.
+\]
+This is a surjective $\mathbb F_q$-linear map whose restriction to
+$H^{m'}$ is a bijection onto $\mathbb F_q^{m_0}$.  In particular it
+sends uniform inputs to uniform outputs.  For each primary slot $s_0$,
+interpolate that block's public matrix entries as
+$A_{s_0}(I,X)$ on $H^\rho\times H^{m'}$, with individual degrees below
+$h$.  Define, for this block only,
+\begin{align}
+ p_{m'}(R,I,X)
+   &=\sum_{s_0}A_{s_0}(I,X)
+                 f_{s_0,R}(w_{\rm pack}(X)),       \label{ia:linear-top}\\
+ p_j(R,I,X_1,\ldots,X_j)
+   &=\sum_{x_{j+1},\ldots,x_{m'}\in H}
+      p_{m'}(R,I,X_1,\ldots,X_j,x_{j+1},\ldots,x_{m'})
+      \quad(0\le j<m').                           \label{ia:partial-sums}
+\end{align}
+The top identity requests the primary with the public slot mask
+$(A_{s_0}(I,X))_{s_0}$, made smooth by additive splitting.  This is a
+linear probe; the matrix entries are not unknown operands.
+
+The final condition is the tested identity
+\begin{equation}
+ \sum_{\text{primary blocks }b} p_{0,b}(R,I)=0.
+                                                               \label{ia:final-row-sum}
+\end{equation}
+\emph{No chart for this mixed sum is stored.}  On legal data it
+vanishes on every $I\in H^\rho$ and every $R\in K$.  Its individual
+$I$ degrees are below $h$, and its honest $R$ coordinate degrees are
+below $q$, so this is a polynomial identity.  Conversely, its
+vanishing at $R=1$ and at the row grid, with correct partial sums,
+is exactly the original matrix condition.
+
+\paragraph{Division enforces all partial sums uniquely.}
+For $1\le j\le m'$ introduce quotient and remainder charts and impose
+\begin{align}
+ p_j&=(X_j^h-X_j)Q_j+X_j^{h-1}p_{j-1}+E_j,
+                                                        \label{ia:division}\\
+ p_{j-1}&\text{ is independent of }X_j,
+ \qquad \deg_{X_j}E_j\le h-2.                       \label{ia:division-bounds}
+\end{align}
+In characteristic two,
+\[
+ \sum_{x\in H}x^r=0\quad(0\le r<h-1),\qquad
+ \sum_{x\in H}x^{h-1}=1.
+\]
+Here the $r=0$ identity is $h=0$ in the field.  The coefficient of
+$X_j^{h-1}$ in the remainder modulo $X_j^h-X_j$ is thus precisely the
+sum of the original polynomial over $X_j\in H$.  Euclidean division
+by this monic polynomial uniquely fixes $Q_j$, the top remainder
+coefficient $p_{j-1}$, and $E_j$.  Starting at
+\eqref{ia:linear-top}, downward induction fixes all certificates and
+gives \eqref{ia:partial-sums}.  Independence from later trailing
+coordinates follows in the same induction.
+
+To test the degree restriction without concentrated reads, write
+\begin{equation}
+ G_j(R,I,X_{\ne j},U)
+   =\sum_{b=0}^{h-2}E_{jb}(R,I,X_{\ne j})U_b,
+ \qquad U\in\mathbb F_q^{h-1},                     \label{ia:coefficient-chart}
+\end{equation}
+where $E_{jb}$ is the coefficient of $X_j^b$ in $E_j$.  Check
+independence from $X_j$, addition in $U$, and homogeneity under a
+uniform nonzero scalar $\lambda\in\mathbb F_q$.  Sampling a nonzero
+scalar makes $\lambda U$ uniform; zero homogeneity already follows
+from addition.  If these identities are exact, $G_j$ is pointwise
+$\mathbb F_q$-linear in $U$, and uniqueness of reduced interpolation
+makes its polynomial linear in $U$.  Now check
+\begin{equation}
+ E_j(R,I,X)=G_j(R,I,X_{\ne j},(1,X_j,\ldots,X_j^{h-2})).
+                                                               \label{ia:coefficient-evaluation}
+\end{equation}
+The special $U$ tuple is queried as two uniform additive summands.
+Only \emph{after} linearity has been established in the exact-soundness
+case analysis do we use the $O(D+h)$ degree bound for this substitution.
+The earlier linearity discrepancies have degree $O(D)$ and themselves
+have constant gap.  Equation~\eqref{ia:coefficient-evaluation} then
+enforces \eqref{ia:division-bounds} and fixes $G_j$ uniquely.
+
+\paragraph{A whole list is checked at once.}
+Sampling a single recurrence index would lose a factor $m'$.  Instead
+choose a cyclic permutation $\omega$ of the $m'$ scalar coordinates
+so that coordinate $j$ of $\omega^jX$ is one fixed coordinate of $X$.
+Store
+\begin{equation}
+ \overline p_j(R,I,X)=p_j(R,I,\omega^jX),
+ \qquad
+ p_{j-1}(R,I,\omega^jX)
+      =\overline p_{j-1}(R,I,\omega X),             \label{ia:recurrence-frames}
+\end{equation}
+and store $Q_j,E_j,G_j$ in the equation-$j$ frame, with $U$ unchanged.
+Both public powers in \eqref{ia:division}, and the predecessor
+independence axis in its corresponding frame, are now common across
+all $j$.  A uniform mask on the recurrence-index list checks all its
+equations with boundedly many point probes.  Endpoint slices and
+predecessor shifts only change the requested linear mask, which is
+split smoothly.  The coefficient families use the same $U$ group in
+every slot.  If $m'=0$, retain only the unsummed top contribution and
+introduce no division family.
+
+\paragraph{Degree, covariance, and exact gap.}
+The top degree is at most
+\begin{equation}
+ 2d_{\rm in}+(\rho+m')(h-1).                       \label{ia:tool-degree}
+\end{equation}
+Monic division by $X_j^h-X_j$, reduction, coefficient extraction, and
+finite summation do not increase total degree.  Multiplication by a
+linear $U_b$ costs at most one.  This proves the honest degree bound.
+There is a fixed number of families per primary; the growing recurrence
+index is one $O(\ell)$ slot axis, and the additional $U$ domain has
+$h-1\le\ell$ coordinates.
+
+First test ignored coordinates and the independence and linearity
+devices.  Unless they hold identically, one has a nonzero polynomial
+discrepancy of degree $O(D)$ and constant rejection probability.
+Once they hold, every remaining displayed discrepancy has degree
+$O(D+\ell h)=o(q)$ and a failed slot identity survives a random
+contraction with constant probability.  Exact identities give the
+uniqueness just proved and then \eqref{ia:final-row-sum}.  A fixed
+mixture of test types therefore has a uniform positive gap.  Perfect
+completeness concerns canonical tuples within the budgets; it does not
+claim that every legal primary of degree exactly $D$ has auxiliaries
+of degree at most $D$.
+
+For covariance even on invalid inputs, expand a lifted primary after
+substitution over $K$ as
+\[
+ f_R=\sum_{j=0}^{M-1}R^{\widetilde j}v_j
+\]
+in its evaluation arrays.  All subsequent operations in
+\eqref{ia:linear-top}--\eqref{ia:coefficient-chart} are the same fixed
+linear maps on that primary, independent of $R$.  This includes
+division and coefficient extraction, since honest-degree evaluation
+arrays determine the required polynomials by interpolation.  These
+maps preserve the exponent list, and rotating the input by $x\in C$
+replaces $R$ by $xR$.  No final zero-row identity is used.  In particular
+every canonical certificate is defined even when its primary belongs
+to data failing the final condition.
+
+Finally, all point substitutions used in these tests are coordinate
+permutations, surjective packing, nonzero scaling, fresh independent
+coordinate replacement, or additive splitting.  They have bounded
+point marginals.  Requested masks are split.  All arrays have
+polynomial size, and every construction operation is finite linear
+algebra on them.  This proves the remaining claims.
+\end{proof}
+\subsection{Persistent charts and their dependency tags}
+\label{ia:persistent}
+
+Every persistent chart has a canonical recipe $F(c,b;Y)$, possibly
+with one slot axis.  At each fixed frame $c$, both its values and its
+coordinate coefficients have an assigned multilinear Boolean degree
+$\kappa_F\le d=3$ in $b$, after reducing by $b_j^2=b_j$.  This
+logical degree is different from the coordinate degree bounded by
+$D$.  All recipes are defined for every Boolean tape, including tapes
+that fail validity.  A recipe either is independent of $c$, or has
+one designated full variable whose scaling realizes $c\mapsto xc$.
+
+Give each family a tag in $\{0,1,\ldots,k\}$.  A tag-$0$ recipe
+depends only on $c,w,e$ and public data; that public data may include
+the circuit, work addresses, and the port index.  A tag-$i$ recipe
+depends only on $c,s_i,e$ and public data fixed by the standardized
+one-port template.  It may ignore $s_i$ if it is a certificate created
+inside that template.  Call tags compatible if they are equal or one
+of them is zero.  After fixing a finite Boolean transition branch,
+tag-$0$ update operands will all have tag $0$, and tag-$i$ update
+operands will have compatible tags.
+
+\paragraph{Monomials and the separated validity calls.}
+Use the ordinary full-grid extension for $W_w$ and each $W_{s_i}$,
+and the cyclic lift for $W_e$.  Include the constant chart $1$,
+tested against its prescribed value.  For all ordered lists of length
+at most $d$ from $\{w,e\}$, and separately from $\{s_i,e\}$ for
+each $i$, include
+\begin{equation}
+ W_{g_1,\ldots,g_j}(b;Y_1,\ldots,Y_j)
+    =\prod_{r=1}^j W_{g_r}(b;Y_r).                 \label{ia:monomial-charts}
+\end{equation}
+Repeated types use independent variables.  Share the legacy-only
+types.  A monomial containing $s_i$ has tag $i$; all other monomials
+have tag $0$.  Enforce each nontrivial product by factorization into
+its shorter prefix and last factor on independent native variables,
+together with all ignored-coordinate requirements.
+
+Apply the nonrotating linear tool as follows.
+\begin{enumerate}
+\item One tag-$0$ validity call uses the monomials from $\{w,e\}$
+and $1$.  It imposes the order-one extension shapes, bitness on the
+grids, the one-hot requirements \eqref{ia:one-hot}, and all blank and
+gate requirements of Lemma~\ref{ia:validity}.  Use one work-requirement
+row for each pair (stage, work grid position), with a zero row when
+that position is an occupied copy.  Other conditions have fixed
+lists of rows.  Circuit data affect matrix entries only.
+\item For each $i$, a separate copy/port call imposes the shape,
+padding, and bitness of $W_{s_i}$, and the copy rows indexed by
+$(c,j)$ for $c\in C$ and $1\le j\le L$.  The row is zero when
+the stage index of $c$ is less than $j$.  Otherwise it compares
+\begin{equation}
+ W_{e,s_i}(c^{-1},\iota(j))
+     -W_{e,w}(c^{-1},w_{i,j})=0,                  \label{ia:separate-copy-row}
+\end{equation}
+where $\iota$ is the common input injection.  Primary-list, row-order,
+and padding conventions are identical in every such call.  The
+matrix blocks for port-dependent primaries depend only on these
+common stage and source-bit indices.  Work destinations occur only
+in the separate $W_{e,w}$ matrix block.  Auxiliaries for port-dependent
+blocks receive tag $i$; auxiliaries for the work-side block receive
+tag $0$.
+\end{enumerate}
+For example, bitness is the linear row
+\[
+ W_{g,g}(y,y)-W_g(y)=0
+\]
+on represented grid entries.  Although its two grid arguments
+coincide, it is verified as a matrix row, not by a concentrated
+unknown-product query.  The same representation handles every
+degree-at-most-three Boolean validity term.  Extension shapes are
+linear evaluation/interpolation conditions; the cyclic shape is
+$\mathbb F_q$-linear by Lemma~\ref{ia:cyclic-lift}.
+
+If these conditions and the factorization identities hold exactly
+on genuine polynomials, order-one grid entries are bits, their
+shapes fix their full extensions, products fix higher monomials,
+and the rows give exactly a valid union-of-stages tape.  The linear
+tool fixes each certificate separately.  The initial auxiliary
+assigned degrees can all be bounded by $d$, or by the smaller
+degree of their primary when useful.  Since every certificate map
+is linear in one primary, the same degree statement holds on
+arbitrary Boolean tapes even when a final validity row is nonzero.
+
+Include also a tag-$0$, point-constant output chart
+\begin{equation}
+ F_{\rm out}(b)=w[y_{\rm out}].                    \label{ia:output-chart}
+\end{equation}
+A separate nonrotating linear call ties it to the designated grid
+entry of $W_w$ and enforces its ignored coordinates.  Give it and
+the associated certificates assigned degree one.  This output chart
+does not need a critical-address projection for transitions.
+
+\paragraph{Frame and clock-times-monomial charts.}
+Let $J_c(X)$ be the cyclic lift of the indicator of $c^{-1}$ on
+$C$.  It has tag $0$, assigned degree zero, and
+\begin{equation}
+ J_{xc}(X)=J_c(xX).                                \label{ia:clock-covariance}
+\end{equation}
+Tie it to the legacy chart by
+\begin{equation}
+ W_e(X)=J_c(\gamma^\eta X).                        \label{ia:clock-legacy}
+\end{equation}
+Below, $\eta$ is enforced as a point-constant bit.  Equation
+\eqref{ia:clock-legacy} is then a finite choice of two smooth
+polynomial identities.  One-hot legacy validity determines a unique
+$c$ and the required pre/post relation, and fixes $J_c$.  For every
+monomial $W$, including $1$, include
+\begin{equation}
+ T_W(c,b;X,Y)=J_c(X)W(b;Y),                         \label{ia:clock-monomial}
+\end{equation}
+with independent arguments.  It has the same tag and degree tier as
+$W$, and rotates only $X$.  Enforce it by factorization.
+
+\begin{lemma}[Clock-indexed linear formulas]\label{ia:clock-formula}
+Suppose for each $y\in C$ that $L_y$ is an explicitly given
+$\mathbb F_q$-linear map from selected monomial evaluation arrays
+of logical degree at most $k_0$ to the full native array of a scalar
+or one-slot output.  There is a canonical chart, lifted in $Z$, with
+\begin{equation}
+ G(c,b;z,\cdot)=L_{cz}(W(b))\quad(z\in C).         \label{ia:clock-formula-value}
+\end{equation}
+It rotates only $Z$, and it and its verification auxiliaries have
+assigned degree at most $k_0$.  A rotating linear-tool call verifies
+them.  The recipes and their covariance hold for all Boolean $b$.
+\end{lemma}
+
+\begin{proof}
+Use the needed $T_W$ inputs and the output $G$ as primaries.  Impose
+their lifted shapes in the designated variables and the linear rows
+\begin{equation}
+ G(z,\cdot)=\sum_{x\in C}L_{x^{-1}z}(T(x,\cdot))
+ \quad(z\in C).                                  \label{ia:clock-convolution}
+\end{equation}
+Rows also range over every output evaluation position and slot.
+On canonical $T$ inputs, only $x=c^{-1}$ contributes, giving
+\eqref{ia:clock-formula-value}.  The lift shape then determines the
+whole chart uniquely.  The row condition is invariant under the
+simultaneous replacement $G(z)\mapsto G(u_0z)$ and
+$T(x)\mapsto T(u_0x)$: setting $x'=u_0x$ changes the map index to
+$(x')^{-1}u_0z$.  The shape conditions are invariant too, so
+Lemma~\ref{ia:linear-tool} applies.  Every input has logical degree
+at most $k_0$; interpolation and all per-primary certificate operations
+are linear and preserve that degree.  Covariance follows first on
+$C$ and then by uniqueness of the lift, and for the auxiliaries by
+the last part of the linear tool.  No validity row on $b$ was used.
+Public clock-dependent terms are supplied through $T_1=J_c$, keeping
+the rotating condition linear.
+\end{proof}
+
+Apply this lemma to a fixed list, for fixed $k$, of critical addresses
+needed by either transition direction.  A critical output chart has
+only the variable $Z$ and gives at $z\in C$ the required logical read
+or rule bit for absolute frame $cz$.  For each port, make a separate
+tag-$i$ read chart that reads $s_i[j]$ during initial copy step $j$
+and is identically zero on every other step.  This is a standardized
+degree-one call independent of all subsequent computation.  Work
+reads have separate tag-$0$ charts, defined as zero when inactive;
+there are a bounded number per write slot for old and operand reads.
+Phase, gate, activity, and wrap-disable bits use tag-$0$ degree-zero
+calls.  Fix their numbers and tiers by the rule basis and write-slot
+bound, padding inactive roles.  Their public values and work address
+maps may depend on the circuit.
+
+Every auxiliary of a tag-$0$ clock call has tag $0$.  Every auxiliary
+of a standardized tag-$i$ clock call has tag $i$, even an auxiliary
+for a shared legacy-only or public $T$ primary in that call.  The
+shared primary itself retains its base tag and formula.  This
+distinction between an input's tag and the ancestry of its
+per-call certificate is part of the construction.
+
+A fixed-frame tape edit must update every persistent chart, including the
+certificates introduced to verify other charts.  Its effect on a chart
+$F$ is the difference
+\[
+ F(c,b+h_\sigma(c);Y)-F(c,b;Y).
+\]
+For a positive-tier chart we store this difference in a new chart with a
+clock variable $Z$, whose restriction at $Z=1$ gives the required change.
+Taking a Boolean difference lowers the logical degree.  We can also store the
+changes of this new chart and its certificates, stopping after at most
+three levels.  The construction below supplies these polynomial charts;
+the projection stacks in Section~\ref{ia:projection-stacks} will realize
+the restriction at $Z=1$ without a concentrated point read.
+
+\begin{lemma}[Derivative closure for all persistent families]
+\label{ia:derivative-closure}
+The persistent system can be closed under every required fixed-frame
+Boolean edit, including edits to verification auxiliaries, with a
+fixed number of families for fixed $k$.  Each has at most one
+$O(\ell)$ slot axis, native dimension $O(\ell)$, honest coordinate
+degree $O(\ell h)$, and assigned logical degree at most three.  The
+closure and its polynomial-time canonical recipes are defined on
+invalid Boolean tapes as well.
+\end{lemma}
+
+\begin{proof}
+Let $F$ have positive assigned tier $k_0$.  For every required flip
+mask, retain its native variables $Y$ and slots, adjoin a new full
+variable $Z$, and prescribe on the cyclic grid
+\begin{equation}
+ G_F^\sigma(c,b;z,Y)
+   =F(cz,b+h_\sigma(cz);Y)-F(cz,b;Y),\qquad z\in C.
+                                                               \label{ia:derivative}
+\end{equation}
+Extend in $Z$ by the cyclic lift.  Both parent terms are evaluated
+in frame $cz$; the parent's existing argument coordinates remain
+exactly $Y$.  A tag-$0$ parent uses the full flip alphabet
+\eqref{ia:flip-mask}.  For a tag-$i$ parent, only the two legacy
+coefficients matter: the input bits $s_i$ are never written and
+work bits are absent from its recipe.  Use the fixed four masks
+\begin{equation}
+ h_\chi(y)=\chi_1\mathbf 1_{e_{y^{-1}}}
+            +\chi_2\mathbf 1_{e_{(\gamma y)^{-1}}},
+ \qquad \chi\in\{0,1\}^2,                        \label{ia:port-flip-mask}
+\end{equation}
+independent of $k$ and $i$.
+
+For a squarefree monomial, a fixed Boolean translation gives
+\begin{equation}
+ \prod_{j\in S}(b_j+h_j)-\prod_{j\in S}b_j
+   =\sum_{T\subsetneq S}\left(\prod_{j\in T}b_j\right)
+                         \left(\prod_{j\in S\setminus T}h_j\right).
+                                                               \label{ia:degree-drop}
+\end{equation}
+Thus \eqref{ia:derivative} has logical degree at most $k_0-1$.
+Lifting its new variable adds at most $\ell(h-1)$ to its coordinate
+degree.  For each absolute frame $y$, output evaluation point, and
+inherited slot, the derivative is a known linear combination of
+grid monomials of degree at most $k_0-1$: from $\{w,e\}$ for tag
+$0$, or from $\{s_i,e\}$ for tag $i$.
+
+These coefficients are computed from the \emph{canonical parent
+recipe}, not from the presently supplied parent table.  An explicit
+method is bounded-degree Boolean interpolation.  For a multilinear
+function $P$ of degree at most $r$ on a universe of $N$ bits, its
+coefficient on $S$, $|S|\le r$, is
+\begin{equation}
+ [b_S]P=\sum_{T\subseteq S}(-1)^{|S|-|T|}P(\mathbf 1_T).
+                                                               \label{ia:boolean-inversion}
+\end{equation}
+In characteristic two the signs are all one.  Use this on the
+right side of \eqref{ia:derivative}, which is computable on every
+Boolean argument, including translated invalid tapes.  There are
+only $\sum_{j\le r}\binom Nj=\poly(n')$ required subsets, because
+$r\le3$ and $N=\poly(n')$.  For tag $i$ the interpolation universe
+is the common ordered list of its $L$ source bits and $M$ legacy
+bits, with no work coordinates.  Interpret the resulting coefficients
+as public maps $L_y$ on the permitted monomial arrays.
+
+Apply Lemma~\ref{ia:clock-formula} using only inputs through tier
+$k_0-1$.  The output derivative and every newly introduced auxiliary
+receive that tier and the parent's tag.  In particular a certificate
+for a shared tag-$0$ $T$ input in this positive call receives the
+positive ancestry tag.  The derivative rotates only its new variable,
+since on the grid
+\[
+ G_F^\sigma(u_0c,b;z,Y)=G_F^\sigma(c,b;u_0z,Y),
+\]
+and lift uniqueness extends this to $K$.  Its older parent variables
+are not additionally scaled under a frame change.  Each verification
+auxiliary instead rotates its own new $R$, as proved by the linear
+tool.
+
+Repeat this operation for every added positive-tier family, including
+derivatives and all their proof families.  Every descendant call
+strictly lowers its assigned tier, whether or not its actual degree
+was already smaller.  The recursion ends after at most three levels.
+A family has only finitely many children for fixed $k$.  A derivative
+inherits its parent's single slot axis; a verification auxiliary uses
+its own recurrence-index axis rather than tensoring it with primary
+slots.  Consequently the number of families is fixed, their native
+dimensions and slot lengths remain $O(\ell)$, and their coordinate
+degrees remain $O(\ell h)$.  Tier-zero charts require no derivative
+for a fixed-frame Boolean edit.
+
+All maps have polynomial construction cost: there are polynomially
+many frames, evaluations, slots, monomials, and rows, and only a
+fixed number of recursive levels.  Finite sums, monic division,
+coefficient extraction, and interpolation are fixed linear operations
+on each primary array.  They neither create a further logical product
+nor import another primary's dependency.  This remains true when the
+final row sum fails, by Lemma~\ref{ia:linear-tool}.  The asserted
+invalid-tape recipes and their degree bounds follow as well.
+\end{proof}
+
+\begin{lemma}[Literal positive-port ancestry]\label{ia:port-ancestry}
+There are deterministic local call and ancestry conventions such that
+the entire tag-$i$ construction in an actual $k$-port graph is
+identical, after renaming $s_i$ to $s$, to the positive-tag construction
+in the one-port template with the same $L,n'$ and field parameters.
+The agreement includes primary lists, row indexing and zero padding,
+packing maps, all auxiliary and derivative families, native dimensions,
+slot orders, and canonical arrays whenever $(c,e,s_i)=(c,e,s)$.
+It holds independently of the actual work destinations, $k$, and $f$.
+\end{lemma}
+
+\begin{proof}
+Choose row lists, their order, and zero padding separately for each
+call's local template.  Do not choose a row dimension by maximizing
+over all calls in the actual graph.  In each copy/port call use the
+same input-shape rows, bitness rows, and $(c,j)$ copy rows, and the
+same full primary-list conventions.  Although the work-side matrix
+depends on $w_{i,j}$, its dimensions do not.  This makes all
+port-dependent primary blocks literal copies of their dummy blocks.
+
+For a concrete example, consider $W_{e,s_i}$ in
+\eqref{ia:separate-copy-row}.  Its matrix $A_{e,s}(I,X)$ knows the
+stage and the source position $\iota(j)$, and has no reference to
+$w_{i,j}$.  That destination appears only in the separate
+$A_{e,w}$ block.  The top formula for its positive certificate is
+\[
+ p_{m',e,s}(I,X)
+       =A_{e,s}(I,X)W_{e,s_i}(w_{\rm pack}(X)),
+\]
+with the evident slot contraction when needed.  Its row padding,
+packing, recurrence permutation, recurrence-index order, and
+coefficient-vector domain are therefore exactly those of the
+one-port construction.  Each partial sum, quotient, remainder, and
+coefficient chart uses that block alone.  The equality with the
+work-side contribution is tested only in \eqref{ia:final-row-sum};
+there is no mixed stored chart from which work addresses could
+enter this positive family.
+
+The critical source-bit calls agree directly, since they use the
+same initial-copy address rule and the same zero rule thereafter.
+For the remaining families, induct on positive ancestry.  If a
+parent's recipe and native layout agree, its derivative uses exactly
+the same restricted mask \eqref{ia:port-flip-mask}.  Its values on
+every subset in \eqref{ia:boolean-inversion}, including translated
+invalid inputs, are identical in the two templates.  Hence its maps
+$L_y$ are identical.  Make each derivative call separately by parent
+type and mask, with the fixed allowed lower-tier monomial list, full
+output-equality rows, shape rows, deterministic order, and local
+padding.  Its output and every per-primary auxiliary then agree
+literally by the linear-tool formulas.  This includes auxiliaries
+whose primary is a shared legacy-only or constant $T$ chart: the
+input formula is common and the auxiliary's own call matrix is the
+standardized positive one.  These auxiliary families are themselves
+parents in the induction.  Strictly decreasing tiers terminate it.
+
+Keep descendants from distinct standardized calls separate, without
+coalescing them differently with circuit-dependent tag-$0$ families.
+Additional full work masks and additional tag-$0$ calls then create
+separate branches and do not change any positive branch.  The
+construction does not use validity to establish this correspondence.
+\end{proof}
+
+\begin{lemma}[A common dimension and inherited full labels]
+\label{ia:placements}
+The common point domain $K^u$ and all per-chart dimension bounds can
+be fixed before $k$.  Persistent charts can be placed in it so that
+every derivative preserves all its parent's labels and slot order
+and adds an unused whole label for its new variable.  Corresponding
+positive families have identical placements.  For each fixed $k,L$,
+all actual graph family and slot layouts can be fixed independently
+of the size-bounded circuit's values and wiring.
+\end{lemma}
+
+\begin{proof}
+Initially a monomial uses at most $d$ full variables and a $T$ chart
+one more.  Base linear calls have native dimension $O_d(\ell)$.
+Their row counts are bounded by $q^{C\ell}$ for an absolute fixed
+$C$ independent of $k$: the work grid, clock grid, input length, and
+stage count are at most $h^\ell$, and shape conditions are polynomial
+in the corresponding full evaluation sizes.  Occupied-copy calls
+are separate, so their number does not enter an individual row
+dimension.  They use fixed stage/source-index lists.
+
+For a derivative call on a parent of $m_0$ native coordinates and
+one possible slot axis, its output, shape, and clock-equality rows
+have size polynomial in $q^{m_0+C\ell}$, with a fixed exponent.
+Its permitted monomial input list is fixed by its tag and tier.
+Different masks have separate calls.  Therefore $\rho/\ell$, the
+new native dimension divided by $\ell$, and the new slot length
+divided by $\ell$ are bounded by a fixed multiple of one plus the
+corresponding parent bounds.  Digit unpacking contributes the fixed
+factor $v$; the coefficient-vector domain contributes at most
+$h-1\le\ell$ coordinates.  Iterating through the at most three
+tier drops gives an absolute bound $C_{\rm dim}\ell$, independent
+of the number of calls or ports.
+
+Choose $u$ to accommodate this bound and at most $d+1$ additional
+whole labels along an inherited derivative-axis chain.  Put every
+full $K$ variable in a whole label, and pack remaining native
+$q$-coordinates into additional labels, ignoring their padding.
+An initial chart or a new verification auxiliary is a placement
+root; unrelated roots may reuse labels according to their native
+templates.  A derivative retains exactly its parent's placements
+and chooses a deterministically first unused whole label for its
+new variable.  Repeat along that chain.  No labels have to be
+globally disjoint across all ports: tests between unrelated charts
+assemble their native arguments separately and fill ignored
+coordinates uniformly.  Enforce all ignored-coordinate conditions.
+These deterministic choices depend only on local templates and
+ancestry, proving positive-family agreement.
+
+For tag-$0$ calls at fixed $k,L$, use the full uniform
+(stage, work-position) validity list, fixed critical-read roles and
+tiers, the full fixed flip alphabet for every parent, and the
+allowed monomial, shape, and output lists fixed by domain and tier.
+Only matrix entries and rule values vary with the circuit.  This
+makes the entire actual layout independent of circuit wiring, as
+claimed.
+\end{proof}
+\subsection{Projection stacks and their exact tests}\label{ia:projection-stacks}
+
+Critical reads and tape edits require algebraic restrictions of charts
+with one or more full labels set to $1$.  Reading these special points
+directly would not preserve a uniform point marginal.  Instead we
+express each restriction as a sum of invertible affine pullbacks and
+store the affine-pulled values in slots.  The progress coordinate $t$
+specifies which of these slots are active.  At full progress, summing
+over the new stack axes will give the required restriction; changing
+progress will use bounded slot shifts and affine pulls.
+
+Give each critical address output its clock-formula variable as a
+projection axis.  Other initial persistent charts, including all
+verification auxiliaries, start with no projection axes.  A derivative
+inherits its parent's axes, with their exact labels, and adds its new
+outer variable.  New verification auxiliaries again start with no
+axes.  Denote the resulting set by $A_F$; its size is at most $d+1$.
+
+Fix a generator $g$ of $\mathbb F_q^\times$.  For a full label
+$\alpha$, let $D_\alpha$ be pullback by
+\begin{equation}
+ Z_\alpha\longmapsto 1+g(Z_\alpha-1).              \label{ia:dilation}
+\end{equation}
+This is an invertible $K$-affine map on the common domain.
+
+\begin{lemma}[Projection by full-label dilations]\label{ia:projection}
+If the coordinate degree in label $\alpha$ is below $q-1$, then
+\begin{equation}
+ \sum_{r=0}^{q-2}D_\alpha^r F
+       =F\big|_{Z_\alpha=1},                      \label{ia:projection-identity}
+\end{equation}
+where the right side ignores that label.  Dilations on distinct
+full labels commute.
+\end{lemma}
+
+\begin{proof}
+Center the $\mathbb F_q$ coordinates of $Z_\alpha$ at the coordinate
+vector of $1\in K$.  Decompose $F$ into parts homogeneous of degree
+$r$ in the centered coordinates.  Their sums under all nonzero
+scalar dilations have multiplier
+\[
+ \sum_{j=0}^{q-2}g^{jr}
+   =\begin{cases}1,&r=0,\\0,&0<r<q-1.\end{cases}
+\]
+For $r=0$ this is $q-1=1$ in characteristic two, and for positive
+$r$ it is a geometric sum with ratio different from one.  Thus
+only the constant homogeneous part remains.  Pullbacks on disjoint
+labels commute by direct substitution.
+\end{proof}
+
+\paragraph{Mode and progress marker.}
+Represent $\eta$ by a point-constant scalar chart.  Enforce constancy,
+then membership in $\mathbb F_2\cdot1$ by random additive binary
+forms annihilating $1$.  A value outside this subspace is detected
+with probability $1/2$; on the subspace, a fixed form sending $1$
+to $1$ reads the bit.
+
+Let $N=B^a$.  Represent progress by a point-constant marker tensor
+$M^{\rm mark}_j$, with $a$ radix axes of length $B$, indexing
+$0\le j<N$.  At $t=0$ it is zero; at $t>0$ it is one-hot at $t-1$.
+After constancy is enforced, test with independent uniform separable
+masks $L,L'$ the identity
+\begin{equation}
+ (L\cdot M^{\rm mark})(L'\cdot M^{\rm mark})
+                  =(LL')\cdot M^{\rm mark},       \label{ia:marker-identity}
+\end{equation}
+where each axis of $LL'$ is an entrywise product.  Expanding in the
+separate mask variables, exactness says
+$M_j^{\rm mark}M_{j'}^{\rm mark}=0$ for $j\ne j'$ and
+$(M_j^{\rm mark})^2=M_j^{\rm mark}$ for each $j$.  Hence the tensor
+is zero or has exactly one coefficient equal to one.  A nonzero
+discrepancy is a polynomial of bounded mask degree and is detected
+with constant probability.  Enforce zero support for
+\begin{equation}
+ q-1\le j<N.                                      \label{ia:marker-padding}
+\end{equation}
+This range is a union of at most $a+1$ radix rectangles: compare
+digits from the most significant end, separating the first larger
+digit and the boundary itself.  Masked restrictions check every
+rectangle with boundedly many probes.  The remaining possibilities
+are exactly $t\in\{0,\ldots,q-1\}$.  Both mode and marker have
+tag $0$.
+
+For each persistent $F$ and each nonempty $S\subseteq A_F$, include
+a stack with the same tag, point placements, and inherited slots as
+$F$, and $a$ additional radix slot axes for each $\alpha\in S$.
+Its intended formula is
+\begin{equation}
+ C^S_{F,\mathbf j}(c,b,\eta,t;Y)=
+ \begin{cases}
+ \displaystyle\left(\prod_{\alpha\in S}
+           D_\alpha^{t-1-j_\alpha}\right)F(c,b;Y),
+             &j_\alpha<t\text{ for every }\alpha\in S,\\[2pt]
+ 0,           &\text{otherwise},
+ \end{cases}
+ \qquad C_F^\varnothing=F.                         \label{ia:stack-formula}
+\end{equation}
+Indices run through $0,\ldots,N-1$ on each integer axis.
+
+\begin{lemma}[Canonical stacks and bounded radix probes]\label{ia:stacks}
+For a legal marker, the stacks are uniquely characterized by a fixed
+number of types of bounded-probe polynomial identities.  These tests
+are perfectly complete and have constant exact gap on
+degree-at-most-$D$ polynomials.  Their only unknown products are
+marker times smaller-stack products, with independent full operand
+point marginals and compatible tags.  All slot shifts, restrictions,
+and boundary slices have bounded separable-mask expansions.
+\end{lemma}
+
+\begin{proof}
+For each nonempty $S$, fix one $\alpha\in S$ and require
+\begin{equation}
+ C^S_{F,(j_\alpha,\mathbf j_-)}
+  =D_\alpha C^S_{F,(j_\alpha+1,\mathbf j_-)}
+      +M^{\rm mark}_{j_\alpha}
+                 C^{S\setminus\{\alpha\}}_{F,\mathbf j_-},
+                                                               \label{ia:stack-recurrence}
+\end{equation}
+with the upper boundary at $j_\alpha=N$ defined to be zero.
+For fixed $\mathbf j_-$ this is triangular downward in $j_\alpha$.
+If the marker is zero its only solution is zero.  If it is one-hot
+at $t-1$, its solution is
+\[
+ C^S_{F,(j_\alpha,\mathbf j_-)}
+   =\begin{cases}
+     D_\alpha^{t-1-j_\alpha}
+             C^{S\setminus\{\alpha\}}_{F,\mathbf j_-},
+                       &j_\alpha<t,\\
+     0,                 &j_\alpha\ge t.
+    \end{cases}
+\]
+Induction on $|S|$ gives exactly \eqref{ia:stack-formula}.
+
+Here are explicit slot formulas for the shift, including boundary
+handling.  Use little-endian radix digits
+$j=\sum_{s=0}^{a-1}j_sB^s$ and let
+$\lambda_L(j)=\prod_{s=0}^{a-1}L_s[j_s]$.  With out-of-range
+coefficients zero,
+\begin{align}
+ \sum_{j=0}^{N-1}\lambda_L(j)C_{j+1}
+   &=\sum_{r=0}^{a-1}L^{(r,+)}\cdot C,
+                                                        \label{ia:radix-plus}\\
+ L_s^{(r,+)}[d]
+   &=\begin{cases}
+      L_s[B-1]\mathbf1_{d=0},&s<r,\\
+      \mathbf1_{d\ge1}L_r[d-1],&s=r,\\
+      L_s[d],&s>r,
+     \end{cases}                                      \nonumber\\
+ \sum_{j=0}^{N-1}\lambda_L(j)C_{j-1}
+   &=\sum_{r=0}^{a-1}L^{(r,-)}\cdot C,
+                                                        \label{ia:radix-minus}\\
+ L_s^{(r,-)}[d]
+   &=\begin{cases}
+      L_s[0]\mathbf1_{d=B-1},&s<r,\\
+      \mathbf1_{d\le B-2}L_r[d+1],&s=r,\\
+      L_s[d],&s>r.
+     \end{cases}                                      \nonumber
+\end{align}
+Indeed the first pullback mask on index $k>0$ is
+$\lambda_L(k-1)$.  Partition $k$ by its least digit $r$ which is
+nonzero; precisely the lower digits borrow.  The second pullback
+is $\lambda_L(k+1)$ for $k<N-1$; partition by the least digit below
+$B-1$.  Overflow is absent in both formulas.  Slicing at zero is
+a single separable restriction, and interval restrictions use the
+bounded rectangle decomposition used in \eqref{ia:marker-padding}.
+Tensoring these expansions over the bounded number of stack axes
+still gives a bounded number of separable masks.  Split every
+requested mask additively as in \eqref{ia:contraction}.
+
+Contract the entire identity \eqref{ia:stack-recurrence} over all
+slots.  The last term factors into a contraction of the marker on
+axis $\alpha$ and a contraction of the smaller stack on its other
+axes and inherited slots.  In the exact-soundness analysis marker
+constancy has already been imposed, so its full point can be sampled
+independently of the smaller stack point.  This retains perfect
+completeness on canonical data.  If the recurrence is not an
+identity, its discrepancy has point degree at most $2D$ and bounded
+mask degree, giving constant rejection probability.  Honest stacks
+are affine pullbacks of persistent charts or zero, so they have the
+same honest coordinate degree.  Marker tag $0$ is compatible with
+every parent tag.  This proves all claims.
+\end{proof}
+
+\subsection{Exact validation and locally encoded evolution}
+\label{ia:validation-and-evolution}
+
+\begin{lemma}[Exact algebraic validity]\label{ia:exact-validator}
+For each fixed $k$ there is a fixed finite mixture of bounded-probe
+tests with perfect completeness on canonical extended labels and a
+uniform constant gap on genuine degree-at-most-$D$ polynomial
+families.  Passing all identities is equivalent to being the unique
+canonical tuple of a valid extended label.  The gap is independent
+of $L$ past the fixed cutoff and uniform over the allowed circuit
+coefficients.  Linear probes have bounded point and mask density.
+Every unknown product has compatible tags and can be queried with
+independent full point marginals for its two operands.
+\end{lemma}
+
+\begin{proof}
+The tests just constructed constitute a fixed number of family
+types for fixed $k$; growing slot lists are checked by whole
+contractions.  We prove exact soundness in an order that licenses
+all substitutions.  First consider ignored-coordinate requirements,
+mode and marker constancy, and the independence and coefficient
+linearity devices of every linear tool.  If one fails, its direct
+test has constant gap.  Otherwise these are exact polynomial
+identities.  Monomial factorization and the nonrotating base calls
+then force a Boolean tape, all its monomial extensions, its
+union-of-stages validity, its output chart, and the canonical
+per-primary certificates.  Mode and marker membership give unique
+$\eta$ and $t$.  The legacy identity \eqref{ia:clock-legacy}
+forces the unique frame $c$ and fixes $J_c$; clock-monomial
+factorizations fix all $T_W$.
+
+Now apply the clock-formula calls in construction order.  Their
+uniqueness fixes each critical output, each derivative, and every
+auxiliary.  A derivative's rows were computed from a prescribed
+parent recipe; the proof never treats an unverified parent table
+as an additional oracle for those rows.  The strictly decreasing
+tier construction is therefore acyclic.  Finally the triangular
+stack identities give \eqref{ia:stack-formula}.  Hence exactness
+of all identities gives precisely a valid label's canonical tuple.
+Conversely every valid label satisfies every identity for every
+random choice in the support.
+
+If exactness fails, the first failed identity in this ordering has
+point degree $O(D+\ell h)$ after the stated preliminary devices
+and bounded degree in its mask variables.  The polynomial zero and
+contraction bounds give a uniform positive rejection probability.
+Taking the minimum over the fixed mixture gives the asserted gap.
+Honest degrees are $O(\ell h)<D$ eventually, also after any later
+fixed-factor enlargement of the point domain.  The constants in
+these degree bounds and in the finite family list depend on fixed
+templates, not on the values of public circuit coefficients.
+
+For the product claim, the only possibilities are monomial
+factorizations, $J_cW$ factorizations, marker membership, and marker
+times smaller stack.  In a factorization, choose the two native
+operand arguments independently, assemble the output's relevant
+groups from them, and independently fill all ignored coordinates.
+Even repeated operand types then have independent full operand
+points.  The relevant output groups are independent uniform too.
+For marker products use independent points, justified by the
+earlier constancy step.  Every such pair has compatible tags by
+construction.  Bitness, diagonal grid entries, copy equalities,
+and all other correlated grid uses are linear matrix rows, so they
+require no concentrated unknown-product query.  The linear tool's
+packing, permutation, and splitting probes have bounded individual
+point density.  Dependent or prescribed masks, including those
+inside products, are smoothed by fresh independent axis splittings.
+These observations prove the stated probe interface.
+\end{proof}
+
+\begin{lemma}[All neighbor coordinates evolve by bounded affine pulls]
+\label{ia:evolution}
+For canonical valid data, edge existence and the transition branch
+in either direction are determined using only mode, empty/full
+marker comparisons, and boundedly many critical projected bits at
+full progress.  Equality decisions can have any prescribed fixed
+error probability by a fixed number of repetitions.  Once the
+finite Boolean branch is fixed, every requested contraction in the
+neighbor tuple is a bounded public linear sum of contractions in
+the current tuple at invertible $K$-affine pullbacks of the common
+point.  Its coefficients and slot pulls depend on the branch,
+requested masks, and public data, but not on the point, row index,
+or further hidden values.  Tag-$0$ updates use only tag $0$;
+tag-$i$ updates use only compatible types.
+\end{lemma}
+
+\begin{proof}
+We give formulas for each transition, including the boundaries.
+Write $C^{S,t}_{F,\mathbf j}$ for the stack at progress $t$.
+
+\paragraph{Progress increase and decrease.}
+On increasing from $0$ to $1$, initialize every nonempty stack by
+\begin{equation}
+ C^{S,1}_{F,\mathbf j}
+       =\mathbf1_{\mathbf j=\mathbf0}F,
+ \qquad M^{\rm mark,new}_j=\mathbf1_{j=0}.         \label{ia:increase-zero}
+\end{equation}
+For $0<t<q-1$, and $\mathbf j$ in the padded radix range, let
+$Z(\mathbf j)=\{\alpha\in S:j_\alpha=0\}$ and put
+$k_\alpha=0$ for $\alpha\in Z(\mathbf j)$ and
+$k_\alpha=j_\alpha-1$ otherwise.  Then
+\begin{equation}
+ C^{S,t+1}_{F,\mathbf j}
+    =\left(\prod_{\alpha\in Z(\mathbf j)}D_\alpha\right)
+           C^{S,t}_{F,\mathbf k},\qquad
+ M^{\rm mark,new}_j=M^{\rm mark,old}_{j-1},       \label{ia:increase-positive}
+\end{equation}
+with zero padding.  For $j_\alpha>0$, the old exponent is
+$t-1-(j_\alpha-1)=t-j_\alpha$, as required; for $j_\alpha=0$,
+one extra dilation gives exponent $t$.  Old inactive coefficients
+remain zero.  Partitioning by $Z(\mathbf j)$ and using the radix
+pull formulas implements this with boundedly many mask operations.
+For a decrease from $t>0$, the formula is simply
+\begin{equation}
+ C^{S,t-1}_{F,\mathbf j}
+     =C^{S,t}_{F,(j_\alpha+1)_{\alpha\in S}},\qquad
+ M^{\rm mark,new}_j=M^{\rm mark,old}_{j+1}.       \label{ia:decrease}
+\end{equation}
+Its exponent is $t-1-(j_\alpha+1)=t-2-j_\alpha$ and its support
+is $j_\alpha<t-1$.  In particular $t=1$ produces empty stacks
+and a zero marker.  Persistent data remain fixed during these
+progress transitions.
+
+\paragraph{Critical readings and full-progress edits.}
+At $t=q-1$, for a critical chart $F$ with projection axis $\alpha$,
+\begin{equation}
+ \sum_{j=0}^{N-1}C^{\{\alpha\}}_{F,j}
+     =\sum_{j=0}^{q-2}D_\alpha^{q-2-j}F
+     =F\big|_{Z_\alpha=1}.                       \label{ia:critical-projection}
+\end{equation}
+All-ones radix masks perform this sum.  A critical output ignores
+every other native group, so the result is a constant-in-point
+logical or public rule bit.  Lemma~\ref{ia:validity} therefore
+determines the edit mask from finitely many such bits.
+
+Fix that mask and direction.  For every positive-tier parent $F$
+and every $S\subseteq A_F$, including $S=\varnothing$, add the
+increment
+\begin{equation}
+ \Delta C^S_{F,\mathbf j}
+    =\sum_{r=0}^{N-1}
+      C^{S\cup\{\alpha_{\rm new}\}}_{G_F^\sigma,(\mathbf j,r)}.
+                                                               \label{ia:edit-increment}
+\end{equation}
+For a positive port parent use its corresponding legacy mask
+$\chi$.  For active old indices, commuting all inherited-label
+dilations with projection on the new label gives
+\begin{align}
+ \Delta C^S_{F,\mathbf j}
+   &=\left(\prod_{\alpha\in S}D_\alpha^{q-2-j_\alpha}\right)
+          G_F^\sigma(c,b;1,Y)                    \label{ia:edit-calculation}\\
+   &=\left(\prod_{\alpha\in S}D_\alpha^{q-2-j_\alpha}\right)
+          \bigl(F(c,b+h_\sigma(c);Y)-F(c,b;Y)\bigr).
+                                                        \nonumber
+\end{align}
+If any inherited index is inactive, every summand is zero.  This
+is precisely the new stack minus the old one, including the empty
+stack, which is the persistent chart itself.  The recursion supplied
+the necessary lower-tier derivative for \emph{every} positive-tier
+parent, including proof and derivative charts, so all coordinates
+update simultaneously from old data.  Tier-zero charts stay fixed
+under the Boolean change.  The marker is unchanged and the mode
+changes as prescribed.  Reverse edits obey the same calculation
+with the inverse branch's coefficients.
+
+\paragraph{Frame change and edge flags.}
+At empty progress all stacks are zero.  For frame advance, a
+rotating persistent chart is pulled back on its one designated
+variable by multiplication by $\gamma$; for reverse advance use
+$\gamma^{-1}$.  A nonrotating persistent chart stays fixed.  The
+empty stacks remain zero, and the mode changes.  In particular,
+an auxiliary with native variables $(R,I,X,U)$ rotates only $R$;
+its packing, division, and coefficient formulas are not recomputed
+by the update.
+
+The successor rules are: pre increases unless full; at full pre
+perform the enabled edit, or report the absent wrap edge; post
+decreases unless empty; at empty post advance the frame.  The
+predecessor rules are: full post performs the inverse enabled edit,
+or reports the absent wrap edge; other post reverses a decrease;
+positive-progress pre reverses an increase; empty pre reverses the
+frame advance.  Empty must also be distinguished for initialization
+in \eqref{ia:increase-zero}.  These are all finite cases.  Comparing
+the marker with the public zero tensor or the public full tensor
+uses random separable masks and binary forms, with a constant
+inequality-detection gap.  A fixed number of repetitions makes
+either comparison as reliable as prescribed.  Mode and critical
+constant bits are read with forms taking $1$ to $1$.  No progress
+index scan or proposed-target validity check is needed for edge
+flags on the valid inner path.
+
+Finally, all point maps in these formulas are identities, full-label
+dilations \eqref{ia:dilation}, or full-label multiplications by
+$\gamma^{\pm1}$, and their bounded compositions.  They are
+invertible $K$-affine maps.  In \eqref{ia:edit-increment}, the child
+retains the parent's placements and has an already available extra
+whole label; the sum projects that label to one and makes it
+irrelevant to the target coordinate.  Thus a requested update has
+the form
+\begin{equation}
+ L\cdot F^{\rm new}(Y)
+    =P_L(Y)+\sum_{r=1}^{R_*}a_r
+                         (L_r\cdot F_r)(T_rY),    \label{ia:affine-update}
+\end{equation}
+where $R_*$ is bounded, $P_L$ is publicly computable, $a_r$ and
+$L_r$ are fixed by the requested masks and branch, and $T_r$ is
+an invertible $K$-affine map.  No $a_r$ or slot pull depends on
+$Y$ or on a later row index.  The inherited tags in
+Lemma~\ref{ia:derivative-closure}, and the public fixed branch,
+give the tag assertions.
+\end{proof}
+
+\begin{remark}[Native tests and encoded updates have different maps]
+\label{ia:testing-versus-evolution}
+The arbitrary $\mathbb F_q$ coordinate permutations and surjective
+digit packing in Lemma~\ref{ia:linear-tool} need not be $K$-affine.
+They are used to construct certificates and to test systematic
+target-point values with bounded density.  They are not maps for
+propagating an encoded parity row or a line-coefficient array.
+Equation~\eqref{ia:affine-update} is the separate evolution
+interface for those encodings.  For example, the proof chart
+$F(c,b;R,I,X,U)$ changes frame only through its whole $R$ label,
+and changes Boolean tape through its stored child's full-label
+projection.  Its canonical division is not rerun on the requested
+encoded symbol.  This distinction permits the binary construction
+to use ordinary systematic representations for all native tests
+while transporting neighbor updates through its common affine
+action and linear row encoding.
+\end{remark}
+\begin{lemma}[Source data, original indices, and Boolean degree]
+\label{ia:source-data}
+At each of the two labels in \eqref{ia:source-successor}, every
+tag-$0$ chart and stack is publicly computable.  Every tag-$i$
+chart and stack has exactly the value of its corresponding
+positive-tag one-port-template chart at the same native point,
+slot, and mask indices, evaluated on input $s_i$.  These statements
+hold for the source and its first successor separately, with
+their indicated progresses $q-1$ and $q-2$.
+
+At either label, every prime-field coordinate function of every
+chart value or coordinate coefficient has multilinear degree at
+most three in the read inputs.  Any common public linear
+representation of these arrays preserves this bound and their
+original-index correspondence.  A binary bilinear form of two
+such represented values has degree at most six.  Its coefficients
+are computable in polynomial time by bounded-degree Boolean
+interpolation.
+\end{lemma}
+
+\begin{proof}
+At both labels, $c=\gamma^{-1}$, work is blank, $e_1=1$ and other
+legacy bits are zero, the mode is post, and progress is the public
+value specified in \eqref{ia:source-successor}.  Tag-$0$ persistent
+recipes depend only on these values and public circuit data, so
+they are computable.  Their derivatives might evaluate a translated
+tape with nonblank work, but every flip vector is then public and
+the translated work is public too.  Such a tape need not be valid:
+the canonical recipe is evaluated by its finite-array operations,
+and never by solving for a valid circuit computation.  In
+particular a work-side copy certificate uses its own work primary
+and public matrix, not a stored sum involving $s_i$.  Recursively,
+all its derivative and proof arrays remain public.  Stack
+dilations at either fixed progress preserve this conclusion.
+
+Positive persistent recipes and layouts agree by
+Lemma~\ref{ia:port-ancestry}, and their placements agree by
+Lemma~\ref{ia:placements}.  The projection-axis choices, inherited
+labels, $q,a,B$, and radix slot conventions also agree.  Formula
+\eqref{ia:stack-formula} therefore gives identical stacks at each
+of the two progresses, including every descendant proof stack.
+This proves a literal map of original algebraic indices, not
+merely equality up to unspecified isomorphisms or comparable sizes.
+Applying the same public row extension, restriction to an ordered
+line, extraction of its coefficients, slot contraction, or basis
+map to corresponding arrays preserves both values and indices.
+Later balancing by repeated copies may use different multiplicities;
+it does not alter this statement about original indices.
+
+At a fixed frame every persistent value and coefficient has
+assigned degree at most three in the Boolean tape.  At the present
+labels all tape bits except the read inputs are fixed public bits.
+Fixed-progress stacks are fixed affine pullbacks or zero.  Their
+degree is thus at most three in the read inputs as well.  A public
+linear operation over $\mathbb F_q$, or over any fixed public
+extension field followed by prime-field coordinates, is
+$\mathbb F_2$-linear on these functions.  It cannot increase
+multilinear degree.  For two degree-three coordinate vectors, a
+binary bilinear form is a sum of products of two such coordinate
+functions and hence has degree at most six.  Copies do not alter
+degree.  Evaluate the specified canonical bit on supports of size
+at most six and apply \eqref{ia:boolean-inversion}; the number of
+evaluations and each canonical finite-array computation are
+polynomial.  This proves the coefficient-computation assertion.
+\end{proof}
+
+\subsection{The complete inner algebra interface}\label{ia:interface}
+
+\begin{lemma}[Inner algebra interface]\label{lem:inner-algebra-interface}
+Fix $c_{\rm allow}$.  The field and grid parameters
+\eqref{ia:parameters}--\eqref{ia:radix-size}, the common domain
+$K^u$, and the one-port positive family templates can be chosen
+before the port count and the function.  For every fixed $k$ and
+fixed circuit size factor, and all sufficiently large $L$, a
+uniform polynomial-time construction has the following properties.
+\begin{enumerate}
+\item \textbf{Paths and endpoints.}
+The graph on all port tuples is the graph of
+Definition~\ref{ia:extended-path}.  On each fixed tuple $s$ it is
+one directed path with source and first successor
+\eqref{ia:source-successor}, no isolated vertices or self edges,
+and a sink at full pre stage $M-1$.  The point-constant output
+chart \eqref{ia:output-chart} equals $f(s)$ at the sink.
+
+\item \textbf{Canonical polynomial arrays and their size.}
+Each valid extended label has a unique canonical tuple
+$\mathcal A(c,b,\eta,t)$ of $\mathbb F_q$-coefficient polynomials
+on $K^u$.  Persistent native dimensions are at most
+$C_{\rm dim}\ell$, with $C_{\rm dim}$ independent of $k$, and
+each persistent family has at most one $O(\ell)$ slot axis.
+Stack families have a bounded number of slot axes, each of length
+$O(\ell)$.  Their honest coordinate degrees are $O(\ell h)<D$.
+For fixed $k$, the number of families is fixed and every full
+evaluation table, fully masked table, public matrix, and indexing
+map has polynomial size and is polynomial-time constructible.
+The layouts for fixed $k,L$ are independent of the circuit's
+values and wiring.  The construction does not enumerate all
+Boolean tapes or graph vertices.
+
+\item \textbf{Exact validity with smooth binary-form probes.}
+There is a fixed finite mixture of bounded-probe algebraic tests,
+perfectly complete on canonical valid tuples.  For every tuple
+of genuine degree-at-most-$D$ polynomials, either it is a
+canonical valid tuple or the rejection probability is at least
+$\theta_{\rm alg}>0$.  For fixed $k$ this gap and the probe density
+bounds are uniform in $L$ and in the allowed circuit coefficients.
+Each linear probe has bounded point and mask marginals.  Every
+unknown product has compatible tags and independent full point
+marginals for its two operands, also for repeated types.
+The tests require only a bounded number of additive binary forms
+and binary bilinear forms of hidden field values, followed by a
+bounded number of Boolean operations.  They do not require reading
+the full binary expansion of a hidden field element.
+
+\item \textbf{Invalid tapes and finite derivative closure.}
+Canonical recipes exist also for arbitrary Boolean $b$ at every
+specified $c,\eta,t$, whether or not that label is valid.  They
+retain their coordinate bounds, logical degree at most three,
+per-primary dependencies, and designated frame covariance.
+Every positive-tier persistent family, including every
+verification auxiliary, has all its required lower-tier
+derivatives.  The recursion terminates in at most three tier
+drops.  A nonvalid tuple so obtained has the same exact rejection
+gap; no alternative certificate can hide a failed row condition.
+
+\item \textbf{Neighbor evolution.}
+On valid data, both edge flags and finite transition branches use
+only mode, empty/full marker comparisons, and boundedly many
+critical projected bits at full progress.  At any desired fixed
+reliability, these distinctions use boundedly many binary probes.
+After a branch is fixed, every neighbor contraction has the
+form \eqref{ia:affine-update} with boundedly many invertible
+$K$-affine pulls of the common point.  Coefficients and slot
+pulls are independent of point and row position.  Tag-$0$
+updates use tag $0$ only; positive updates use compatible tags.
+The formulas update all proof data simultaneously.  Their
+fixed-branch algebraic identities remain valid on canonical
+invalid Boolean inputs, without asserting that the resulting
+proposed label is valid.
+
+\item \textbf{Standardized ports and source access.}
+Every positive-tag family has the literal per-call ancestry and
+original-index correspondence of Lemma~\ref{ia:port-ancestry},
+including all descendants, full-label placements, projection
+axes, and stacks.  At the source and first successor, the
+tag-$0$ part is public, and tag $i$ agrees with the one-port
+template on $s_i$.  The source Boolean-degree bounds are three
+under public linear representations and six after one binary
+bilinear pairing, as in Lemma~\ref{ia:source-data}.  These
+templates and bounds are fixed before actual $k$ or any later
+correction accuracy is chosen.
+\end{enumerate}
+\end{lemma}
+
+\begin{proof}
+The path and endpoint assertions are
+Lemmas~\ref{ia:validity} and~\ref{ia:path-endpoints}.  The linear
+tool, clock-formula construction, derivative closure, and common
+placements give the persistent arrays.  Lemma~\ref{ia:stacks}
+gives their stacks.  At fixed $k$, all depths and family counts
+are fixed, while all dimensions and total mask lengths are
+$O(\ell)$.  Consequently every domain and mask space has size
+$q^{O(\ell)}=\poly(n')=\poly(L)$.  Row and coefficient arrays
+have the same polynomial bounds.  Their construction consists
+of the explicit finite-array operations and bounded-degree
+interpolation above.  In particular, increasing the number of
+separate calls does not change the already chosen common $u$.
+
+Lemma~\ref{ia:exact-validator} proves the exact gap and all point
+and tag restrictions.  Here is its conversion to the asserted
+binary-form interface.  A nonzero element of $\mathbb F_q$ is
+detected by a uniformly random additive form
+$\mu:\mathbb F_q\to\mathbb F_2$ with probability $1/2$.
+Applying $\mu$ to a field equality turns its public linear terms
+into additive forms, since
+\[
+ \mu\left(\sum_r a_r x_r\right)
+      =\bigoplus_r(\mu\circ[a_r])(x_r),
+ \qquad [a_r](x)=a_rx.
+\]
+For a product, the requested bit is the binary bilinear form
+$(x,y)\mapsto\mu(xy)$.  Random annihilating forms handle mode
+membership as already specified.  A bounded field test therefore
+uses a bounded number of such bits and Boolean operations,
+losing at most a fixed factor in its gap.  Prescribed masks and
+forms can be split by fresh additive randomness.  Their public
+descriptions have polynomial size; even the space of all binary
+bilinear forms on $\mathbb F_q$ has cardinality
+\[
+ 2^{(\log_2 q)^2}=\poly(n'),
+\]
+since $\log q=O(\log\log n')$.  A bounded sequence of all the
+displayed random choices consequently has polynomial support.
+
+The invalid-tape assertion follows from the per-primary
+canonical formulas and the finite-difference construction, not
+from successful final rows.  If a label is invalid, a validity
+or frame/legacy condition fails and exact soundness rejects;
+uniqueness of the division certificates prevents a different
+proof table from canceling the failed condition.
+Lemma~\ref{ia:evolution} proves all fixed-branch update formulas,
+including for canonical invalid tapes.  Frame changes occur
+only when stacks are empty, so no commutation between centered
+dilations and frame scaling is needed.  Finally
+Lemmas~\ref{ia:port-ancestry}, \ref{ia:placements}, and
+\ref{ia:source-data} give the exact standardized source
+correspondence and its Boolean-degree bounds.  This proves
+every part of the interface.
+\end{proof}

preprints/The-PCP-for-PPAD-conjecture-a-quasilinear-reduction-September-25-2026/build/sections/08-inner-binary.tex

@@ -0,0 +1,1379 @@
+\section{Binary testing and average correction at a fixed radius}
+\label{sec:inner-binary}
+
+We convert the polynomial arrays of Section~\ref{sec:inner-algebra} into
+binary words.  The correction guarantee is averaged over the requested
+encoded coordinates.  Its radius is fixed before the requested accuracy
+and the density of those coordinates are specified.  This quantifier
+order is essential when the one-port construction is subsequently used
+many times.
+
+All notation in this section is inner notation.  In particular, the
+groups below are polynomial-size affine groups and are unrelated to the
+outer index group.  Fix one of the padded templates of
+Lemma~\ref{lem:inner-algebra-interface}.  We call each polynomial family
+in its canonical tuple, together with its slot axes, a \emph{track}.
+We use its fields
+\(\F_q\subset K=\F_{q^\ell}\), common target domain \(K^u\), degree
+bound \(D=\ell^2-1\), and fixed number of slot-mask axes per track.
+The integer \(u\) and the field conventions are common to the actual
+and one-port templates.  Honest chart degrees, including the constant
+factor enlargement below, are \(O(\ell h)<D\) for sufficiently large
+size.  The field-size choice already made in
+Section~\ref{ia:path-parameters} ensures \(q/D\longrightarrow\infty\).
+Constants in this section may depend on the fixed template, but are uniform
+in the public circuit within that template.
+
+\subsection{Rectangular coordinates and the common row code}
+\label{ib:rectangular-code}
+
+The algebraic neighbor formulas use invertible affine maps of the target
+\(K^u\). We will add parity rows so that local row decoding can reduce
+errors, and use a group action to move a requested value among columns
+without changing it. The rectangular coordinates below make this action
+transitive on point and ordered-line columns. Target affine maps commute
+with it, so neighbor updates remain column permutations after row encoding.
+
+Choose a constant integer \(r>2u\), \(r\ge3\).  A rectangular point
+is \(I=(P,z)\), where \(P(x)=Ax+b\) maps \(K^r\) affinely to
+\(K^u\) and \(z\in K^r\).  Retain points with \(\rank A=u\).
+For an ordered \(\F_q\)-line
+\[
+ I_t=(P_0+tP_1,z_0+tz_1),\qquad t\in\F_q,
+\]
+retain it precisely when the matrix obtained by stacking \(A_0,A_1\)
+has rank \(2u\) and \(z_1\ne0\).  Every point on such a line is
+retained, since a dependence in \(A_0+tA_1\) would give a dependence
+in the stacked matrix.  A sequential row-exposure bound shows that
+the omitted point and line fractions are \(o(1)\) as \(\abs K\to
+\infty\); for example rank failure for an \(s\)-by-\(r\) uniform
+matrix is at most \(\sum_{j=0}^{s-1}\abs K^{j-r}\).
+
+Write
+\[
+ U(K)=K^r\rtimes\mathrm{SL}_r(K),\qquad
+ \mathcal H=U(K)\times U(K),\qquad
+ (g_1,g_2)(P,z)=(P\circ g_1^{-1},g_2z).
+\]
+This is an affine action on the rectangular \(\F_q\)-coordinates,
+and acts on ordered lines without changing their parameter.  There
+is one orbit of retained points and one orbit of retained ordered
+lines.  Indeed, right special-linear transformations are transitive
+on full-rank matrices with \(u\) rows, and on those with \(2u\)
+rows: first align by a general-linear transformation and then adjust
+its determinant on an unused complementary dimension.  Translation
+in the argument of \(P\) adjusts the one or two offsets because the
+corresponding matrix has full row rank.  Independently, the second
+affine factor is transitive on points and on ordered pairs of a base
+point and a nonzero direction.  All retention conditions are
+invariant.  In particular the point at each fixed parameter of a
+uniform retained ordered line is uniform on retained points.
+
+An invertible affine target map \(d:K^u\to K^u\) induces
+\((P,z)\mapsto(d\circ P,z)\).  It commutes with \(\mathcal H\),
+preserves both retained domains, and preserves line parameters.  A
+target chart \(f\) is represented by \(f(P(z))\); this substitution
+increases its coordinate degree by at most a factor two.
+
+\begin{lemma}[Uniform affine-group expansion]
+\label{ib:affine-expansion}
+There is a fixed symmetric labelled multiset
+\(u_1,\ldots,u_{d_e}\) in each \(\mathcal H\), with \(d_e\ge8\),
+whose averaging operator has norm at most \(1/64\) on mean-zero
+functions.  The labels, group operations, and complete group tables
+are deterministically constructible in polynomial time in the inner
+size.  The degree and gap are independent of the fields.
+\end{lemma}
+
+\begin{proof}
+Here are the two primary group-theoretic inputs and their exact use.
+Ershov--Jaikin-Zapirain~\cite[Theorem~1.1]{EJZ2010} prove property
+\((T)\) for \(\mathrm{EL}_r(R)\) for every finitely generated
+associative unital ring \(R\) and \(r\ge3\).  Their generating-set
+bound in Section~6.1 applies to the elementary matrices with entries
+in a fixed ring-generating set containing 1.  In particular, their
+equations (6.1)--(6.2) give the positive Kazhdan constant
+\[
+ \kappa_r=\bigl[8(48\sqrt2+2\sqrt{3r})\bigr]^{-1}
+\]
+for
+\(\{e_{ij}(1),e_{ij}(t):i\ne j\}\) in
+\(\mathrm{SL}_r(\F_2[t])\); the equality with the elementary group
+holds because \(\F_2[t]\) is Euclidean.  No restriction excluding
+characteristic two occurs in this theorem.
+
+Shalom~\cite[Theorem~3.4 and Corollary~3.5]{Shalom1999}, applied to the discrete
+commutative ring \(R=\F_2[t]\), bounds the displacement of every
+translation in \(R^2\rtimes\mathrm{SL}_2(R)\) by 484 times the
+maximum displacement on the basis translations and
+\(e_{12}(1),e_{12}(t),e_{21}(1),e_{21}(t)\).  Signs coincide in
+characteristic two.  This is the homogeneous displacement form of
+the corollary's tolerance statement, obtained by scaling and taking
+limits.  It bounds the translation sector; it alone does not assert
+expansion of the full affine group.
+
+Set \(\Lambda=R^r\rtimes\mathrm{SL}_r(R)\) and let \(S\) consist
+of all basis translations and the elementary matrices just listed
+for all ordered coordinate pairs.  Embed Shalom's rank-two group on
+each such pair.  Since a translation is a product of \(r\)
+coordinate translations, every unitary representation satisfies
+\[
+ \sup_{a\in R^r}\norm{av-v}
+       \le 484r\max_{s\in S}\norm{sv-v}.
+ \tag{\ref{ib:affine-expansion}.1}
+\]
+Let \(V_N\) be the translation-invariant subspace.  Its orthogonal
+projection commutes with \(\Lambda\), since the translations are
+normal.  On \(V_N^\perp\), the minimum-norm point of the closed
+convex translation-orbit hull is zero.  The displayed bound therefore
+gives
+\(\norm v\le484r\max_{s\in S}\norm{sv-v}\) on this subspace.
+On \(V_N\), a representation without group-invariant vectors is a
+representation of \(\mathrm{SL}_r(R)\) without invariant vectors,
+so the preceding Kazhdan bound applies.  Orthogonally combining the
+two bounds gives a positive Kazhdan constant \(\kappa\) for
+\((\Lambda,S)\).  Explicitly, if \(\kappa_r\) is the quoted
+ linear-group constant, one may take
+\(\kappa=((484r)^2+\kappa_r^{-2})^{-1/2}\).
+
+The set \(S\) generates \(\Lambda\): the third-index elementary
+commutator multiplies entries, entries in a root subgroup add, and
+conjugating basis translations gives arbitrary coordinate
+translations.  Choose a field presentation
+\(K=R/(p(t))\) by a monic irreducible polynomial.  Elementary
+generation over a field shows that the quotient map surjects onto
+\(U(K)\).  Pulling back its mean-zero regular representation
+preserves the Kazhdan bound.  If \(d=\abs S\), the half-lazy walk
+has Dirichlet form
+\[
+ \langle v,(1-M)v\rangle
+   =\frac{1}{4d}\sum_{s\in S}\norm{sv-v}^2
+   \ge\frac{\kappa^2}{4d}\norm v^2
+\]
+on mean-zero vectors, and nonnegative spectrum.  On the product
+group, choose a factor uniformly and take this step there.  The
+commuting tensor-factor operators give another uniform positive
+gap and nonnegative spectrum.  A sufficiently large fixed power has
+mean-zero norm at most \(1/64\).  Keep all walk labels, including
+identities and coincident group elements, as a symmetric multiset;
+repeat labels if necessary to ensure \(d_e\ge8\).  All matrices
+have fixed dimensions over polynomial-size fields, so enumeration
+and every stated computation are polynomial-time finite-array
+operations.
+\end{proof}
+
+Form a bipartite graph with both vertex sets \(\mathcal H\), and
+edges
+\(\mathcal R=[d_e]\times\mathcal H\), where \((i,g)\) joins
+left \(g\) to right \(u_i g\).  Its cross-part normalized second
+singular norm is \(\lambda\le1/64\).  Put
+\(c_e=\lfloor d_e/4\rfloor\).
+
+\begin{lemma}[Systematic row extension]
+\label{ib:row-extension}
+There is a polynomial-size extension field \(J/\F_q\) and two
+\(c_e\)-by-\(d_e\) matrices \(H_L,H_R\) over \(J\) whose every
+\(c_e\)-column minor is nonzero, such that the edge code
+\begin{align}
+ \sum_i(H_L)_{bi}U_{i,g}&=0,\label{ib:left-row}\\
+ \sum_i(H_R)_{bi}U_{i,u_i^{-1}x}&=0\label{ib:right-row}
+\end{align}
+has a unique linear extension \(E\) from labels \(i>2c_e\) to all
+labels.  The same holds with symbols in any finite-dimensional
+\(J\)-space.  Each local code has distance at least \(c_e+1\),
+and \(E\) commutes with every right shift
+\((R_hU)_{i,g}=U_{i,gh}\).
+\end{lemma}
+
+\begin{proof}
+Treat the \(2c_ed_e\) matrix entries as indeterminates.  The square
+system matrix on parity labels \(1,\ldots,2c_e\) has nonzero
+determinant polynomial: specialize left row \(b\) to select label
+\(b\), and right row \(b\) to select label \(c_e+b\).  This gives
+a permutation matrix, including when some generators coincide.
+Multiply this determinant by all maximal minors of both local
+matrices.  Each factor is a nonzero polynomial; their product is
+nonzero and has total degree at most
+\[
+ 2c_e\abs{\mathcal H}
+     +2c_e\binom{d_e}{c_e}.
+\]
+Take the least extension \(J/\F_q\) of cardinality exceeding this
+bound.  It has polynomial cardinality.  A nonzero polynomial of
+smaller total degree than \(\abs J\) cannot vanish everywhere on
+\(J^{2c_ed_e}\).  Exhaustive search over this constant number of
+entries, testing the determinant and minors, therefore finds a
+specialization in polynomial time.  Inverting the parity matrix
+constructs \(E\).
+
+Column independence proves local distance \(c_e+1\), also for
+vector symbols: a nonzero vector-valued local word supported on at
+most \(c_e\) columns would contradict that independence after any
+nonzero coordinate projection.  Right shifts permute left stars
+and right stars in \eqref{ib:left-row}--\eqref{ib:right-row} and
+preserve the systematic labels.  Uniqueness proves \(ER_h=R_hE\).
+\end{proof}
+
+Choose this field, these matrices, and their deterministic ordering
+from the common point and field parameters alone, independently of
+the number or identities of tracks.  Every track uses this one
+encoder.  On systematic rows prescribe, slotwise,
+\[
+ Y_{i,g}(I)=f(P'(z')),\qquad (P',z')=gI,
+\]
+and extend columnwise by \(E\).  Equivariance yields
+\begin{equation}
+ Y_{i,gh}(I)=Y_{i,g}(hI)
+ \quad\hbox{for all rows, including parity rows.}
+ \label{ib:covariance}
+\end{equation}
+Every parity-slot function is a degree-in-bound polynomial over
+\(J\).  At an ordered retained line store its actual restriction,
+using coefficients through degree \(D\) and padding by zero.
+These coefficients obey the same row constraints and covariance.
+The commuting target maps act by the same column permutations on
+point tables, line tables, and their encoded parity rows.
+In particular, covariance gives
+\(Y_{i,gh}(h^{-1}I)=Y_{i,g}(I)\) for a point or an ordered line \(I\).
+Thus changing the row and making the inverse change to the column
+preserves the requested symbol. This is the movement among columns that
+will accompany local row decoding.
+
+\subsection{Binary blocks and smooth probes}
+\label{ib:binary-blocks}
+
+For each track there are point and line blocks, both denoted \(Y\)
+with their type understood.  A point position specifies a row edge,
+a retained point, one \(\F_q\)-vector mask per slot axis, and an
+additive binary form \(\mu:J\to\F_2\).  Its bit is \(\mu\) of
+the slot contraction.  A line position additionally specifies masks
+\(A,B\in\F_q^\ell\) for the coefficient index.  If \(Y(t)\)
+is already contracted on the original slots, the bit is
+\begin{equation}
+ \mu\left(\sum_{j,l=0}^{\ell-1}A[j]B[l]
+                     [t^{j+\ell l}]Y(t)\right).
+ \label{ib:line-bit}
+\end{equation}
+This uses \(D+1=\ell^2\).  Evaluation at a specified parameter
+uses the separable masks \((t^j)_j\) and \((t^{\ell l})_l\),
+with the usual convention \(t^0=1\).
+
+The short two-mask coefficient representation in \eqref{ib:line-bit}
+is adapted, through the inherited polynomial-size construction, from
+Amireddy--Behera--Srinivasan--Sudan--Willumsgaard
+\cite[Section~3.1, Definition~3.1 and Claim~3.2]{ABSsw26}.
+This attribution concerns the coefficient encoding and its evaluation
+masks, not the correction and evolution machinery developed here.
+
+Fix an \(\F_q\)-linear map \(\pi:J\to\F_q\) restricting to
+the identity on \(\F_q\).  For each compatible ordered pair of
+point-track types, include a product block \(Z\).  Compatibility
+means that one tag is zero or both positive tags agree.  A type may
+be paired with itself, with two independent positions.  An original
+\(Z\) position consists of the two complete edge/point/slot-mask
+positions, \(b_1,b_2\in J\), and a binary bilinear form \(Q\)
+on the \(\F_2\)-coordinates of \(\F_q\).  For the two contracted
+symbols the bit is
+\begin{equation}
+ Q\bigl(\pi(b_1Y_1),\pi(b_2Y_2)\bigr).
+ \label{ib:product-bit}
+\end{equation}
+There are no line-product blocks and no products of distinct
+positive tags.
+
+Give every block equal weight.  If its original length is \(L_b\),
+repeat each original bit uniformly until every block has length
+\(\prod_b L_b\).  Every raw query chooses a uniform repetition
+copy.  Let \(b_\#\) be the block count, \(m_{\rm in}\) the total
+length, and \(\mathcal W(v)\) the canonical word for an extended
+label \(v\).  These lengths are polynomial in the inner size:
+there are constantly many blocks, \(q^{O(\ell)}\) slot masks,
+fixed-dimensional matrix domains over \(K\), polynomial-size
+\(J\), and \(2^{(\log_2q)^2}\) bilinear forms \(Q\).
+Here \(\log q=O(\log\log n')\), so the last quantity is also
+polynomial in \(n'\).  The product used for balancing has constantly
+many polynomial factors.  Lengths are fixed within a padded template
+independently of the circuit coefficients.  Positive-tag original
+indices and canonical formulas coincide with the corresponding
+one-port indices; repetition counts need not coincide.
+
+We will repeatedly use the following explicit smoothing operation.
+For a specified additive argument \(a\), choose uniform \(r\)
+and write \(a=r+(a+r)\).  Multilinearity expands a requested value
+into a bounded XOR of terms, each uniform in that argument.
+Split different axes independently.  This applies to slot masks,
+coefficient masks, bit forms, \(Q\), and, when needed, the two
+multiplier axes.  Each term has uniform marginals jointly across
+its independently split axes, conditional on the entire specified
+request.  Pull back all public scalars and other linear maps before
+splitting.  This remains true for zero coefficients; a zero term
+may instead be omitted and supplied as public zero.
+
+To probe a target value at \(y\in K^u\), choose a uniform row
+group \(g\), full-rank \(A\), and \(z\), set
+\(P(x)=Ax+y-Az\), and use the systematic position
+\begin{equation}
+ (i,g,g^{-1}(P,z)).
+ \label{ib:systematic-probe}
+\end{equation}
+For uniform \(y\), this is jointly uniform in row and retained
+point within its systematic label; every fiber has the same number
+of \((A,z)\) choices.  A density bound for \(y\) therefore gives
+the same bound for this joint distribution.  Independently chosen
+target arguments can use independent representations.  An additive
+form on a target \(\F_q\)-value extends to \(J\), for example by
+composition with \(\pi\).  These procedures are defined on every
+raw word and are smooth before any replacement or soundness claim.
+
+\subsection{Dimension-independent replacement lemmas}
+\label{ib:replacement-tools}
+
+The tester's soundness proof replaces arbitrary binary data by symbol
+tables and then by polynomial arrays. The two replacement lemmas below
+make the error cost independent of the number of field coordinates and
+slot entries.
+
+We state precisely the low-degree input being used.  The corrected
+Theorem~13 of Friedl--Sudan~\cite{FS1995} says that for every
+\(\eta>0\) there is \(C_\eta<\infty\) such that, if
+\(\abs{\F}\ge C_\eta D\), \(D\ge1\), and a function
+\(f:\F^m\to\F\) and degree-at-most-\(D\) univariate claims
+\(p_{x,h}\) satisfy
+\[
+ e=\Pr_{x,h\in\F^m,\,t\in\F}
+       [p_{x,h}(t)\ne f(x+th)]\le \tfrac18-\eta,
+\]
+then a polynomial \(p:\F^m\to\F\) of total degree at most \(D\)
+satisfies \(\dist(f,p)\le2e\).  The variables in this probability
+are independent and uniform, including the possible direction \(h=0\);
+the field-size constant does not depend on \(m\).  We use
+\(\eta=1/16\).  Neither a vector-valued conclusion nor a restriction
+on the line domain is part of this cited statement.
+
+\begin{lemma}[Vector and deleted-domain low-degree replacement]
+\label{ib:vector-ld}
+Let \(V\) be any finite-dimensional \(\F_q\)-space.  In the preceding
+field regime, line--point disagreement tending to zero for
+\(V\)-valued point functions and degree-in-bound line claims implies
+point distance tending to zero from a single \(V\)-valued polynomial
+of total degree at most \(D\), uniformly in both dimensions.
+The conclusion remains true if the point and ordered-line domains
+omit fractions tending to zero, every retained line lies entirely in
+the retained point domain, and the point at a uniform parameter on a
+uniform retained line is uniform on retained points.  In this case the
+mean fraction of line coefficient vectors differing from the
+polynomial's restrictions also tends to zero.
+\end{lemma}
+
+\begin{proof}
+First use the full domains.  For every \(\alpha\in V^*\), scalar
+projection and the cited theorem give a polynomial \(p_\alpha\) with
+\(\dist(\alpha f,p_\alpha)\le2e\).  Distinct scalar polynomials in
+this degree range differ at a fraction at least \(1-D/q\) of points.
+Consequently, when \(6e<1-D/q\), comparison with
+\((\alpha+\beta)f\) forces
+\[
+ p_{\alpha+\beta}=p_\alpha+p_\beta,
+ \qquad p_{a\alpha}=a p_\alpha \quad(a\in\F_q).
+\]
+The second identity uses at most \(4e\) disagreement; the assertions
+with a zero form or scalar follow in the same way.  Coefficientwise
+finite-dimensional duality now gives one \(V\)-valued polynomial
+\(p\) for which \(\alpha p=p_\alpha\) for all \(\alpha\).
+A uniform \(\alpha\) detects each nonzero vector with probability
+\(1-1/q\), so
+\[
+ (1-1/q)\dist(f,p)
+ =\E_\alpha\dist(\alpha f,p_\alpha)\le2e.
+\]
+There is no factor depending on \(\dim V\).
+
+For deleted domains, extend the point function arbitrarily and give
+every added line any degree-in-bound claim, for example zero.  The
+full incidence disagreement is at most the retained-line disagreement
+plus the omitted-line fraction.  The full-domain conclusion therefore
+gives vanishing error also on retained points.  On any retained line
+whose claim differs from the restriction of \(p\), a nonzero scalar
+projection of their difference is a univariate polynomial of degree
+at most \(D\).  The two claims disagree at a fraction at least
+\(1-D/q\) of its parameters.  Averaging the original test error and
+the point replacement error, using the asserted uniform point
+marginal, bounds the fraction of such lines by their sum divided by
+\(1-D/q\).  This tends to zero and proves the coefficient claim.
+\end{proof}
+
+The addition-test argument below uses the majority self-correction method
+of Blum, Luby, and Rubinfeld~\cite[Generic Linear Self-Testing]{BLR1993}.
+We include it together with the field-homogeneity, multiple-axis, and
+slice-averaging extensions needed for our vector alphabets.
+
+\begin{lemma}[Axis replacement]
+\label{ib:axis-replacement}
+For elementary binary vector groups, the addition test has a
+dimension-independent high-agreement replacement by an additive map,
+also with a vector-group alphabet.  For \(\F_q\)-spaces of
+characteristic two, addition and scalar-homogeneity tests give
+replacement by an \(\F_q\)-linear map.  For any fixed number of axes,
+testing these identities in each axis at uniform other arguments gives
+replacement by a multilinear map as the failure probability tends to
+zero, uniformly in every domain and alphabet dimension.  These
+conclusions also hold in mean over additional slice parameters.
+\end{lemma}
+
+\begin{proof}
+Suppose first that \(f:G\to A\) fails addition with probability
+\(e\), where both groups have exponent two.  For fixed \(x\) put
+\(D_x(y)=f(x+y)+f(y)\).  The addition identities on the pairs
+\((x+y,y+z)\) and \((y,y+z)\) imply \(D_x(y)=D_x(z)\).
+Each pair is uniform when \(y,z\) are independent uniform elements,
+so the collision probability is at least \(1-2e\), for every \(x\).
+The most frequent value \(L(x)\) therefore has mass at least
+\(1-2e\).  The identity
+\[
+ D_{x+y}(z)=D_x(y+z)+D_y(z)
+\]
+and three majority bounds imply \(L(x+y)=L(x)+L(y)\) when
+\(6e<1\): at least one \(z\) realizes all three majority values.
+If \(f(x)\ne L(x)\), the conditional addition failure at \(x\)
+is at least \(1-2e\).  Thus
+\(\dist(f,L)\le e/(1-2e)\).
+
+For the field version, transfer the homogeneity test to \(L\).
+The scalars \(a\) satisfying \(L(ax)=aL(x)\) identically form an
+additive subgroup of \(\F_q\).  If it is proper, at least half the
+nonzero scalars fail; for each failing scalar its nonzero additive
+discrepancy is nonzero on at least half the inputs.  The homogeneity
+test with uniform nonzero scalar would then have failure at least
+\(1/4\).  This contradicts failure tending to zero after a vanishing
+replacement cost.  Hence \(L\) is field-linear.
+
+Induct on the number \(j\) of axes.  Markov's inequality, with a
+threshold tending slowly to zero, makes all but a vanishing fraction
+of first-axis slices good for the remaining \(j-1\) tests.  Replace
+good slices by multilinear maps and the remaining slices by zero.
+This changes a vanishing fraction of values.  Regard the resulting
+slices as a map into the vector space of multilinear maps on
+\(j-1\) axes.  A nonzero such map is nonzero on at least
+\(2^{-(j-1)}\) of uniform inputs: successively contract each axis,
+using that a nonzero linear map has a kernel of relative size at most
+\(1/2\).  Thus vanishing first-axis test failure on values gives
+vanishing failure as equalities in this vector-space alphabet.
+The one-axis result finishes the induction.  The same Markov cutoff
+over any additional parameters proves the mean version.
+\end{proof}
+
+\subsection{Testing the binary representation}
+\label{ib:testing}
+
+\begin{proposition}[Smooth tester and distance]
+\label{ib:tester}
+For each fixed template, the canonical encodings of valid extended
+labels have constant relative distance.  They admit a perfectly
+complete bounded-bit tester with the following uniform gap: for every
+fixed \(a>0\), there are \(\theta(a)>0\) and a size cutoff such that
+rejection below \(\theta(a)\) implies distance less than \(a\) from
+one canonical valid word.  Each potential raw read on each fixed
+answer branch has bounded density relative to the balanced word,
+without conditioning on execution of that branch.  Consequently
+rejection is at most a fixed constant times distance from a canonical
+valid word.
+\end{proposition}
+
+\begin{proof}
+Take a positive fixed mixture of the test families described below,
+including each present block type.  Their query counts are bounded,
+and all requested nonuniform masks are implemented by the splitting
+operation above.  We prove the gap by a sequential replacement
+argument.  Consider any sequence of increasing sizes and words whose
+rejection tends to zero, allowing the public circuits to vary in the
+fixed template.  We show distance tending to zero from a canonical
+valid word.  A bounded smooth test retains vanishing rejection after
+a vanishing-distance change, by the union bound over its potential
+reads; for a branching test use its entire bounded answer tree.
+
+\paragraph{Copies, symbols, and slot tensors.}
+Compare two independent random copies of a uniform original bit.
+If the fraction of ones among its copies is \(p\), the test rejects
+with probability \(2p(1-p)\), which is at least
+\(\min(p,1-p)\).  Replacing all copies by their majority costs
+vanishing distance.  In each point or line \(Y\) block test
+additivity in \(\mu\).  Lemma~\ref{ib:axis-replacement} and binary
+double duality replace the block, in mean over other indices, by
+Hadamard tables of genuine \(J\)-symbols.  Next test addition and
+\(\F_q\)-homogeneity in every slot-mask axis, including the two
+coefficient-mask axes for a line.  Test symbol equality by uniform
+\(\mu\), which detects any inequality with probability \(1/2\);
+pull field scalars onto this form before smoothing its raw accesses.
+The same replacement lemma makes all these symbols honest
+multilinear slot tensors at each edge and point or line, at vanishing
+mean cost.  No global polynomial property has yet been asserted.
+
+\paragraph{Point and line polynomials.}
+At a uniform edge, retained ordered line, and parameter, compare the
+point symbol with line evaluation, using uniform original slot masks
+and a uniform bit comparison.  Split the two specified evaluation
+masks in \eqref{ib:line-bit}.  A nonzero symbol discrepancy has a
+constant binary detection probability, so the line--point
+disagreement tends to zero in mean.  For most edge and original-mask
+tuples apply Lemma~\ref{ib:vector-ld}, with \(V=J\) as an
+\(\F_q\)-space, to obtain a degree-at-most-\(D\) rectangular
+polynomial.  Use zero on the exceptional tuples.  The lemma's
+line-coefficient conclusion also replaces the line tables by the
+restrictions of these polynomials at vanishing cost.
+
+The selected polynomials need not yet be coherent between original
+mask tuples.  Transfer the previous multilinearity identities to
+them using their mean point error.  A false identity between
+degree-in-bound polynomials is nonzero on at least
+\(1-D/q-o(1)\) of retained points.  Thus the identities fail with
+vanishing probability as equalities in the vector space of
+degree-in-bound polynomials.  Apply
+Lemma~\ref{ib:axis-replacement} in that vector-space alphabet, in
+mean over edges, and use zero on exceptional edges.  Changing a
+vanishing fraction of mask tuples changes a vanishing fraction of
+point and line bits.  We now have genuine polynomial slot families
+at every row, and their exact line restrictions.
+
+\paragraph{Covariance and common systematic data.}
+Test \eqref{ib:covariance} on point symbols with independent uniform
+\(g,h,I\), uniform masks, and a bit form.  Undo each row's chart:
+\[
+ F_{i,g}(I)=Y_{i,g}(g^{-1}I).
+\]
+If two such polynomial slot tuples differ, the discrepancy is seen
+with constant probability, first on a retained point, then by
+successive nonzero mask contractions, then by the bit form.  Since
+\(g\) and \(gh\) are independent uniform elements, for each fixed
+label and track the squared frequencies of the uncharted tuples
+sum to \(1-o(1)\).  Their maximum frequency is therefore
+\(1-o(1)\).  Replace other rows by this predominant tuple, charted
+back.  This changes vanishing point and line distance and enforces
+covariance exactly, including on lines.
+
+Compare uncharted tuples of the finitely many systematic labels,
+using a shared uniform retained point and masks and independent row
+groups.  The same constant gap forces an exactly common polynomial
+tuple \(F\) on all systematic labels of a track.  Test that its
+contracted values lie in \(\F_q\) by a random binary form on
+\(J/\F_q\), pulled back to \(J\) and split.  The quotient is an
+\(\F_q\)-space.  If a polynomial coefficient were outside
+\(\F_q\), its quotient would be a nonzero vector polynomial and
+would have a constant detection gap.  Consequently every systematic
+coefficient is in \(\F_q\).
+
+\paragraph{Factoring through the target and enforcing parity.}
+Choose uniform \(y\in K^u\) and two conditionally independent
+generic representations as in \eqref{ib:systematic-probe}.  Compare
+their values, with independent row groups.  Each individual marginal
+is smooth; this test is a linear comparison and makes no product
+query on the correlated pair.  For a slot coefficient the uncharted
+identity tested is
+\begin{equation}
+ F(A,y-Az,z)=F(A',y-A'z',z').
+ \label{ib:factor-target}
+\end{equation}
+The difference has coordinate degree at most \(2D\), and the
+full-rank restrictions omit an \(o(1)\) fraction.  A failed
+polynomial identity therefore has constant detection probability,
+also after the bounded slot contractions.  Hence
+\eqref{ib:factor-target} holds identically.  Fix any full-rank
+\(A'\) and any \(z'\).  Its right side defines a target polynomial
+\(f(y)\) of degree at most \(D\) with \(\F_q\) coefficients.
+Substitute \(y=Az+b\) to obtain the exact factorization
+\(F(P,z)=f(P(z))\).
+
+Test the left and right star equations at uniform stars and retained
+points, with slot masks and random \(J\)-bit forms; pull row
+coefficients onto these forms.  In an exactly covariant family, a
+failed polynomial star relation fails on every right-shift translate
+of that star, with an invertible affine change of argument.
+Polynomial, mask, and bit detection thus give a constant rejection
+gap for any failure.  All star identities hold once the rejection is
+sufficiently small.  Lemma~\ref{ib:row-extension} now forces the
+exact parity extension of the systematic target data and its line
+restrictions.
+
+\paragraph{Product blocks.}
+In each present \(Z\) block test additivity in \(Q\) and replace
+at vanishing cost by exact \(Q\)-linear maps, in mean over all its
+other indices.  Sample a uniform independent pair of
+edge/point/slot-mask indices and uniform multipliers.  Test the
+product identity for
+\(Q(x,x')=\alpha(x)\alpha'(x')\), with independent uniform
+binary forms \(\alpha,\alpha'\) on \(\F_q\).
+On the \(Z\) side split only \(Q\), keeping these already uniform
+other indices fixed.  On each \(Y\) side pull the multiplier and
+\(\pi\) into the requested bit form and split that form.  Compare
+with the Boolean product of the two resulting bits.
+
+The \(Y\)'s are already genuine symbols.  The difference between
+a wrong \(Q\)-linear map and their true product is a nonzero binary
+matrix.  A uniform outer-product form detects it with probability
+at least \(1/4\), by two successive nonzero contractions.  The
+fraction of wrong maps therefore tends to zero.  Replace them by
+\eqref{ib:product-bit}.  This works separately for every compatible
+pair and never invokes a missing product block.
+
+\paragraph{Algebraic validity.}
+Finally run the algebraic validator of
+Lemma~\ref{lem:inner-algebra-interface} on the recovered target
+polynomials, using systematic representations.  Linear target
+probes need only their individual bounded-density marginal.  For
+each product probe that lemma gives compatible tags and independent
+uniform full target arguments.  Represent these independently in
+rectangular coordinates.  The corresponding \(Z\) query therefore
+has the required joint density.  Identity multipliers on
+\(\F_q\)-symbols and the demanded bilinear form implement the
+field-level product; split the necessary slot, form, and multiplier
+axes for its raw reads.  The exact algebraic gap forces canonical
+extended validity.  The native linear maps used by the validator,
+including coordinate packing, need not commute with the row action:
+they are used here only to specify systematic target arguments.
+
+All replacements have vanishing total binary cost, so the original
+word is at vanishing distance from a canonical valid word.  If the
+stated fixed-gap conclusion failed, for each integer \(j\) one
+could choose a size at least \(j\) and a word of distance at least
+\(a\) with rejection below \(1/j\), contradicting this sequence
+argument.  Each constituent test is perfectly complete for every
+random choice in its support.  All its raw marginals are smooth by
+the displayed representations and splitting; this also proves the
+claimed linear upper bound on rejection near a canonical word.
+
+For distance, different logical tapes differ in an order-one grid
+polynomial.  Its difference is nonzero on a constant fraction of
+uniform target arguments, hence on a constant fraction of systematic
+row/point positions; a bit form detects the symbol difference with
+probability \(1/2\).  A concerned block has fixed positive weight.
+If the logical tape agrees, different valid extended labels differ
+in mode or in the progress marker, because legacy and mode fix the
+frame.  These constant slot tensors likewise have a fixed
+nonzero-contraction gap.  This gives a constant lower bound on
+relative distance.
+\end{proof}
+
+\subsection{Value-independent uncertainty supports}
+\label{ib:uncertainty}
+
+The correction argument uses uncertainty sets rather than the actual
+wrong symbols at intermediate stages.  This permits separate
+consistent choices in each bounded dependency cone.
+
+The alternating local decoding strategy is classical; see
+Z\'emor~\cite[Section~III-A, Lemma~5 and Theorem~6]{Zemor2001}.
+The proofs below supply the vector-space, repeated-edge,
+value-independent-support, and finite-cone refinements needed here,
+rather than invoking that result verbatim.
+
+For a \(Y\) block, a column fixes a retained point or line and all slot
+masks, including line coefficient masks. At that column, varying the
+edge gives a row codeword of \(J\)-symbols in the reference encoding.
+The binary data for one such symbol form its Hadamard table, including
+all repetition copies; we call this its \emph{inner table}. The first
+lemma controls uncertainty in these row-code positions without examining
+the uncertain symbols' values.
+
+\begin{lemma}[One-axis support contraction]
+\label{ib:one-axis-support}
+For the row code above, set
+\[
+ t_* =\lfloor c_e/2\rfloor+1,\qquad
+ \vartheta=t_*/d_e\ge1/8,\qquad \zeta_*=1/256.
+\]
+Starting with an uncertain edge set of density
+\(\zeta\le\zeta_*\), a constant number of alternating local
+decoding half-rounds leaves an arbitrarily small fixed uncertain
+edge density.  The resulting support depends only on the starting
+uncertainty set and contains every possible output error for every
+assignment equal to the reference codeword off that set.  This holds
+for symbols in any finite-dimensional \(J\)-space.
+\end{lemma}
+
+\begin{proof}
+At a star decode to the unique local codeword within
+\(t_*-1=\lfloor c_e/2\rfloor\) errors, if it exists; otherwise
+output zero on that star.  Distance \(c_e+1\) gives uniqueness.
+This decoder uses a bounded number of linear symbol operations and
+symbol-equality decisions: enumerate the information sets of size
+\(d_e-c_e\), solve using the complementary invertible minor,
+and count agreements with each candidate.  There are constantly
+many such choices.
+
+On a half-round mark a star bad if it meets at least \(t_*\)
+currently uncertain edges, and declare all edges incident to bad
+stars possibly uncertain afterward.  Every other star decodes
+correctly, regardless of the values on uncertain edges.  The first
+bad-star fraction is at most \(\zeta/\vartheta\le8\zeta\).
+If \(x\) and \(y\) are consecutive bad-star fractions on opposite
+sides, the new bad stars receive at least a \(\vartheta\) fraction
+of their incident edges from the preceding bad stars.  Expander
+mixing with multiplicities gives
+\begin{equation}
+ \vartheta y\le xy+\lambda\sqrt{xy}.
+ \label{ib:mixing-contraction}
+\end{equation}
+When \(x\le\vartheta/2\), either \(y=0\), or division and
+squaring imply
+\[
+ y\le(2\lambda/\vartheta)^2x\le x/16.
+\]
+Since \(8\zeta_*\le\vartheta/2\), induction applies at every
+subsequent half-round.  The current edge-support fraction equals
+the latest bad-star fraction.  A constant number of half-rounds
+therefore gives any prescribed fixed residual density.  Every
+support operation used only sets of uncertain positions, proving
+the value-independent assertion.
+\end{proof}
+
+For a \(Z\) block, a column fixes both retained points, their slot
+masks, and \(Q\). Its row position is a joint pair of edges, and its
+inner table consists of the multiplier pairs and repetition copies.
+The reference multiplier tables obey row constraints along each edge
+axis, but the two axes use different scalar actions. We verify these
+actions before applying a two-axis contraction.
+
+For \(Z\), the relevant symbol alphabet is the additive group
+\(\mathcal B_{\rm mul}\) of all binary bilinear functions
+\(F:J\times J\to\F_2\).  It carries two commuting \(J\)-space
+structures
+\begin{equation}
+ (a\cdot_1F)(b_1,b_2)=F(ab_1,b_2),\qquad
+ (a\cdot_2F)(b_1,b_2)=F(b_1,ab_2).
+ \label{ib:two-actions}
+\end{equation}
+Additivity in each argument and the field laws verify the vector
+space axioms and commutation.  Distinct such functions differ on
+at least \(1/4\) of uniform multiplier pairs.  Fixing the two
+outer points, the original slot masks, and \(Q\) makes the true
+multiplier tables across edge pairs a two-axis codeword.  For
+example the first-axis relation with coefficients \(a_j\) is
+\[
+ \sum_j Q\bigl(\pi(b_1a_jY_{1,j}),\pi(b_2Y_2)\bigr)
+ =Q\bigl(\pi(b_1\sum_j a_jY_{1,j}),\pi(b_2Y_2)\bigr)=0.
+\]
+The second-axis relation is identical with the roles exchanged.
+The next lemma permits precisely these two possibly different space
+structures.
+
+\begin{lemma}[Two-axis support contraction and cone consistency]
+\label{ib:two-axis-support}
+Consider an array on two edge axes whose reference obeys the row
+code in each slice, with possibly different \(J\)-space structures
+on the same additive symbol group for the two axes.  Put
+\(\zeta_{**}=\zeta_*^2/4\).  From initial uncertainty density at
+most \(\zeta_{**}\), a constant number of first-axis half-rounds
+followed by a constant number of second-axis half-rounds leaves any
+prescribed fixed residual density.  The uncertainty support is
+value-independent.  For one output position, the same guarantee
+holds when arbitrary values are chosen on initially uncertain
+positions specifically for its finite dependency cone, provided
+each repeated initial position receives one consistent value inside
+that cone.
+\end{lemma}
+
+\begin{proof}
+First-axis slices with uncertainty density greater than
+\(\zeta_*\) occupy at most a \(\zeta_*/4\) fraction.  Keep these
+whole stripes uncertain.  On all other slices use
+Lemma~\ref{ib:one-axis-support} to make residual density at most
+\(\delta_1\).  In all but \(2\delta_1/\zeta_*\) of the
+second-axis slices, the additional residual outside these stripes
+has density at most \(\zeta_*/2\).  Such a slice starts with
+uncertainty at most \(3\zeta_*/4<\zeta_*\).  Contract it to
+\(\delta_2\), leaving all other second-axis slices uncertain.
+The final density is at most
+\[
+ 2\delta_1/\zeta_*+\delta_2,
+\]
+which is arbitrarily small.  Intermediate arrays need not satisfy
+both slice constraints: each phase is compared to the fixed
+reference and uses its appropriate space structure only.
+
+A fixed number of rounds and constant degree give a finite
+dependency cone.  Extend any consistent assignment in that cone
+arbitrarily to the other uncertain initial positions, retaining the
+reference elsewhere.  The global support statement applies to this
+extension and determines the same output.  Separate cones may use
+different values on their uncertain inputs; no common assignment
+between cones is needed.
+\end{proof}
+
+\subsection{Bit simulation of a finite decoding cone}
+\label{ib:cone-simulation}
+
+\begin{lemma}[Inner-table flags and arbitrary-form prediction]
+\label{ib:inner-table-prediction}
+There are constants \(0<\nu_1<\nu_2<1/64\), independent of the
+number of row rounds, with the following properties.  For a table
+of bits, including its actual repetition copies, one can distinguish
+distance at most \(\nu_1\) from a legal inner table from distance
+greater than \(\nu_2\) from every legal inner table, with any
+prescribed fixed reliability and bounded smooth queries.  The legal
+inner tables are either Hadamard tables of \(J\)-symbols or binary
+bilinear functions on \(J\times J\).  Inside radius \(\nu_2\)
+the legal symbol is unique, and any specified bit of it can be
+predicted with arbitrarily high fixed reliability by bounded queries
+with uniform inner-index and copy marginals.  The specified form or
+multiplier pair may be any function of previously exposed data;
+the assertion holds conditional on those data, using fresh randomness.
+\end{lemma}
+
+\begin{proof}
+Random-copy agreement and binary addition, or the two-axis binary
+tests, are perfectly complete.  Lemma~\ref{ib:axis-replacement},
+together with the majority replacement of copies, gives a
+dimension- and copy-count-independent positive rejection lower
+bound \(\gamma_2\) at distance greater than a fixed \(\nu_2\)
+from every legal inner table.  If this failed, a sequence of such
+tables with rejection tending to zero would contradict that lemma.
+Near a legal table, smoothness bounds rejection by \(C_2\) times
+distance, with an absolute \(C_2\).  Choose
+\(\nu_1<\min(\nu_2,\gamma_2/(4C_2))\).  Estimating rejection
+from independent trials and comparing with \(\gamma_2/2\)
+distinguishes the two promised cases with any fixed reliability;
+Chebyshev's inequality suffices to choose a bounded sample count.
+The intermediate annulus may be assigned either flag.
+
+Distinct Hadamard tables differ on half their forms.  Distinct
+bilinear functions differ on at least a quarter of multiplier
+pairs.  Repetition preserves these distances, proving uniqueness
+within \(\nu_2\).  For a specified Hadamard form \(\mu\), choose
+uniform \(\alpha\), query random copies at \(\alpha\) and
+\(\mu+\alpha\), and XOR.  Each query is uniform in its inner
+index, so one trial errs with probability at most \(2\nu_2\).
+For a bilinear table at specified \((b_1,b_2)\), choose independent
+uniform \(r_1,r_2\) and use
+\begin{align*}
+ F(b_1,b_2)={}&F(r_1,r_2)+F(b_1+r_1,r_2)\\
+             &+F(r_1,b_2+r_2)+F(b_1+r_1,b_2+r_2).
+\end{align*}
+Each pair is jointly uniform, with a fresh random copy, and the
+error probability is at most \(4\nu_2<1/16\).  Independent
+repetition and majority give any desired fixed error.  These bounds
+hold for every specified bit, so remain valid after conditioning
+on all prior data if the splits and copies are fresh.
+\end{proof}
+
+\begin{lemma}[Consistent bit simulation]
+\label{ib:bit-simulation}
+Fix a finite cone of one- or two-axis row decoding, a reference satisfying
+the corresponding row constraints, and a requested bit of the output
+symbol. For analysis, declare an initial position uncertain when its
+received inner table is more than \(\nu_1\) from its reference table,
+and propagate this set by the support rules above. With bounded bit
+queries the requested bit can be simulated to any fixed error probability
+\(\sigma>0\), on a consistent assignment of initially uncertain
+symbols that agrees with the reference at every initial table
+within \(\nu_1\) of its true inner table. If the output position lies
+outside the value-independent support, the predicted bit equals its
+reference bit except with probability \(\sigma\).  All hidden operations are bit equalities
+and requested bits; field coefficients and address computations
+are public.  The routine is bounded and defined on arbitrary words.
+\end{lemma}
+
+\begin{proof}
+List the cone's initial positions and deduplicate repeated positions
+within that cone.  Flag each distinct table once by
+Lemma~\ref{ib:inner-table-prediction}, calling the far outcome
+flagged, and assign flagged positions the zero symbol.  At an
+unflagged table within \(\nu_2\) of a legal table, assign its
+unique nearby symbol.  Choose the flag reliability so that, except
+with a prescribed small probability, every table within \(\nu_1\)
+of its true symbol is unflagged and no table farther than \(\nu_2\)
+from all legal symbols is unflagged.  On this event the assignment
+is well-defined and consistent inside the cone.  Certain inputs
+receive their true symbols: a competing symbol within \(\nu_2\)
+would contradict inner distance.  The remaining assignments,
+including arbitrary choices in the gap annulus, are permitted by
+Lemmas~\ref{ib:one-axis-support} and~\ref{ib:two-axis-support}.
+
+There is no need to recover the assigned symbols in full.  Unfold
+the finite symbol-level decision tree for information-set decoding,
+agreement counts, and fallback-zero outcomes.  On each fixed branch
+every candidate, comparison, and output is a linear expression in
+the assigned initial symbols.  For two axes these expressions are
+sums of compositions of the two scalar actions
+\eqref{ib:two-actions}.  Their coefficients are obtained publicly
+from the fixed local matrices and the finite branch choices.
+
+A false symbol equality is detected by a fresh uniform Hadamard
+form with probability \(1/2\), or by a fresh uniform multiplier
+pair with probability at least \(1/4\).  Each resulting bit of a
+linear expression reduces, by additivity and scalar pullback, to
+a bounded number of specified bits of the initial assigned tables.
+Use Lemma~\ref{ib:inner-table-prediction} on unflagged inputs and
+public zero on flagged inputs.  Independent equality trials make
+the chance of missing a false equality arbitrarily small; an exact
+equality fails only if one of its input-bit predictions fails.
+The requested final output bit is handled in the same manner.
+
+The order of choices prevents circular error bookkeeping.  First
+fix the row rounds and the complete finite symbol-level tree.
+Next choose flag and equality-detection budgets for its bounded
+number of possible decisions.  This fixes the number of
+initial-bit predictions needed in each tested expression.  Then
+choose their smaller error budgets and their repetitions.  Given
+successful flags, every later specified bit has the uniform
+worst-request guarantee of the previous lemma conditional on prior
+answers, using fresh splits after all scalar pullbacks.  A union
+bound, or conditioning successively on the preceding correct
+symbol-level decisions, bounds total simulation error by
+\(\sigma\).  We do not choose the same budgets by union-bounding
+over the later amplified bit tree.  After all choices that full
+tree is nevertheless finite and bounded.  When a promise fails,
+the same finite comparisons and fallback-zero rules still define
+an output; only correctness is relinquished.
+\end{proof}
+
+\subsection{Arbitrarily accurate average correction}
+\label{ib:average-correction-section}
+
+For a point \(Y\) request its \emph{outer index} means its row
+edge together with its retained point; for a line \(Y\) request
+it means its edge and retained ordered line.  For \(Z\) it means
+the \emph{joint pair} of edge/point indices.  A request law has
+density at most \(C_{\rm req}\) within its block when its outer
+index probability is at most \(C_{\rm req}\) times the corresponding
+uniform probability.  Every other requested mask, form, or multiplier
+may be an arbitrary public function of the request.
+
+Call a binary word \emph{structurally canonical} if it is obtained from
+the canonical recipes of Lemma~\ref{lem:inner-algebra-interface} at a
+specified frame, mode, progress, and arbitrary Boolean tape, followed by
+the point, line, row-extension, product, and repetition constructions
+above. The tape need not satisfy the final validity conditions. Such a
+word nevertheless satisfies the row-code, covariance, and multilinear
+mask identities and has exactly the prescribed product tables.
+
+\begin{proposition}[Average correction from one fixed radius]
+\label{ib:average-correction}
+For every fixed template there is \(\rho>0\) such that the following
+holds for all sufficiently large sizes.  Let a received balanced
+word \(w\) satisfy \(\dist(w,\mathcal W)\le\rho\), where
+\(\mathcal W\) is structurally canonical; validity of its label
+is not needed here.  For every fixed \(C_{\rm req}<\infty\) and
+\(\eta>0\), a bounded-bit algorithm predicts requests with outer
+density at most \(C_{\rm req}\) with average error at most \(\eta\).
+It has smooth potential raw reads on every fixed hypothetical answer
+branch.  The radius is independent of \(C_{\rm req}\), \(\eta\),
+the row-round count, and the subsequent amplification count.
+\end{proposition}
+
+\begin{proof}
+Use the constants \(\nu_1,\nu_2\) already chosen, and put
+\(\zeta=\min(\zeta_*,\zeta_{**})\). Use the column and inner-table
+descriptions in Section~\ref{ib:uncertainty}: rows are edges for \(Y\) and edge pairs
+for \(Z\). Call an initial row position uncertain if its received inner
+table is more than \(\nu_1\) from its reference table.
+
+Each block has error fraction at most \(b_\#\rho\).  If
+\(u_{\rm unc}\) is the uncertain edge fraction in a uniform
+column, or pair fraction for \(Z\), then
+\begin{equation}
+ \E u_{\rm unc}\le\frac{b_\#\rho}{\nu_1}.
+ \label{ib:uncertain-average}
+\end{equation}
+Call columns with \(u_{\rm unc}>\zeta\) bad.  Their fraction is
+at most \(b_\#\rho/(\nu_1\zeta)\).  On every other column,
+choose enough rounds in the support lemmas to leave uncertainty
+fraction at most \(\delta'>0\), whose value will be fixed later.
+No residual-support claim is made on bad columns.
+
+For a \(Y\) request, independently split all its slot masks,
+including the line coefficient masks, and expand.  Keep its
+specified Hadamard form as the inner bit request.  For each expanded
+term choose independent uniform \(h\in\mathcal H\) and replace
+the row/outer pair by
+\begin{equation}
+ (i,g,I)\longmapsto(i,gh,h^{-1}I).
+ \label{ib:random-shift}
+\end{equation}
+Covariance preserves the true value.  For a \(Z\) request split
+both slot-mask lists and \(Q\), expand, and shift its two row/outer
+indices independently by \eqref{ib:random-shift}.  Keep the
+specified multipliers as its inner bit request.  Let \(T_0\) be
+a fixed common upper bound on the expansion count, determined by
+the template, before any accuracy choices.
+
+For each fixed original request, every term has a uniform column:
+the relevant \(\mathcal H\)-action is transitive and all its
+column-mask axes have been independently randomized.  In \(Z\)
+the two independent shifts give the uniform pair of outer points.
+Choose the radius once and for all so that
+\begin{equation}
+ \frac{T_0b_\#\rho}{\nu_1\zeta}\le\frac1{64}.
+ \label{ib:fixed-radius}
+\end{equation}
+Then, for every individual request, the chance that any term uses
+a bad column is at most \(1/64\).  This conclusion does not assume
+that the shifted row is independent of the shifted outer point.
+
+For every fixed shift, \eqref{ib:random-shift} is a permutation
+of the complete row/outer index; the independent pair of shifts
+is a permutation of the complete \(Z\) index.  Thus averaging
+over an outer-density-\(C_{\rm req}\) request law preserves that
+density bound.  Expanded column masks are uniform conditional on
+the indices.  The mean chance of hitting the good-column residual
+support in any of the terms is consequently at most
+\(T_0C_{\rm req}\delta'\).  By Markov's inequality, outside an
+exceptional fraction at most \(8T_0C_{\rm req}\delta'\) of
+original requests, the conditional sum of those residual-hit
+probabilities is at most \(1/8\).
+
+Bit-simulate every term's row cone by
+Lemma~\ref{ib:bit-simulation}, with total simulation error at most
+\(1/64\).  On any nonexceptional request, the full randomized
+XOR trial therefore errs with probability at most
+\[
+ \frac1{64}+\frac18+\frac1{64}=\frac5{32}<\frac14.
+\]
+Cone-specific assignments are sufficient because the support bounds
+cover every consistent assignment in each cone.  Choose
+\(\delta'\) so that \(8T_0C_{\rm req}\delta'\le\eta/2\).
+This determines a constant row-round count.  Repeat the entire
+trial independently at the fixed request, including shifts, flags,
+and all predictions, and take majority.  A bounded number of
+repetitions makes its error at each nonexceptional request at most
+\(\eta/2\).  Adding the exceptional fraction proves average error
+at most \(\eta\) from the fixed radius \eqref{ib:fixed-radius}.
+This proves an average guarantee, and makes no assertion of
+arbitrarily accurate correction at every prescribed outer index.
+Smoothness is proved in the next lemma.
+\end{proof}
+
+\begin{lemma}[All hypothetical read slots are smooth]
+\label{ib:all-branch-density}
+For the tester, the corrector of
+Proposition~\ref{ib:average-correction}, and the finite linear
+compositions used below, use their stated bounded outer-index request
+laws, with joint pair density for \(Z\). Freeze any hypothetical sequence
+of hidden answer bits.  Under the original, unconditioned public random tape
+and request law, the raw-read subdistribution of every potential
+slot has bounded density relative to its balanced input block.
+After all accuracy counts have been fixed, the sum of these bounds
+over the complete finite answer tree is a constant independent of
+the size.  Defaults, wrong flags, and wrong equality answers satisfy
+the same assertion.  Public numerical thresholds that only select
+among these fixed schedules do not change the bound.
+\end{lemma}
+
+\begin{proof}
+First enumerate every potential initial row-cone \emph{path slot},
+including all candidate uses, before deduplication.  A term's
+initial covariance shift is the joint permutation
+\eqref{ib:random-shift}.  A fixed row-star path subsequently changes
+the edge label among finitely many possibilities and multiplies
+\(g\) on the left by a fixed group element.  A right-star move
+from label \(i\) to \(j\), for example, uses
+\(g\mapsto u_j^{-1}u_i g\).  Each such map is a permutation
+within a fixed edge-label class, so costs only a fixed label factor
+in a density bound.  In a two-axis cone the same reasoning applies
+separately to each axis and preserves the joint pair bound.
+Consequently each enumerated path slot has a fixed density bound.
+
+Deduplication, representative selection, and omission of unused
+terms only decide whether an already enumerated slot is read.
+Its resulting unconditioned subdistribution is dominated by the
+original slot distribution.  We never condition and renormalize
+on the event that this slot was chosen as a representative or that
+its flag had a specified outcome.  Summing the bounds of all
+potential slots, including unexecuted ones, bounds their complete
+occurrence count.  A flagged, failed, or unused zero value is public
+zero and requires no placeholder query at a special address.
+
+On a frozen symbolic branch, all information-set interpolation
+matrices depend only on \(H_L,H_R\) and public graph indices.
+The relevant inverses exist by their maximal-minor property.
+The coefficients of all selected linear expressions, including
+the compositions of \eqref{ib:two-actions}, can therefore be
+computed publicly.  Hidden work is only the bounded equality and
+output-bit calls described in Lemma~\ref{ib:bit-simulation}.
+Their specified forms and multiplier pairs may depend on every
+frozen answer, public coefficient, and preceding random choice.
+Fresh splits \emph{after} these pullbacks give uniform raw inner
+forms or jointly uniform multiplier pairs conditional on all that
+information.  Zero coefficients can be split as well, or omitted.
+Fresh uniform copies give the required copy marginal.  Wrong
+answers change the selected expression, but introduce no new type
+of row-cone path.
+
+The tester's schedules were explicitly smooth before replacements.
+The neighbor compositions below use finite lists of commuting
+affine column permutations and changes between fixed block types,
+so the same argument adds only fixed type factors.  Dropping a
+public operand takes a marginal of a joint law, which also
+preserves a density bound.  Finally freeze all simulation and
+amplification counts, unfold the full raw bit tree, and sum these
+finitely many bounds.  This tree may be large, but is constant at
+the chosen accuracy.  A later threshold offset that selects its
+branches changes neither its slot count nor these public address
+maps.  The assertion concerns the unconditioned schedules for
+all branches, not correctness on unsuccessful branches.
+\end{proof}
+
+The request statement also applies to a subdistribution within a
+type: pad its missing mass by the uniform law, which increases a
+density bound by at most 1, and charge the original subdistribution
+by the resulting error bound.  A bounded list of type or branch
+choices is handled by adding their bounds and error budgets.
+This does not normalize on a rare branch.  Likewise, if a mixed
+product has a public operand, a joint density
+\(p(x,y)\le C/(\abs X\abs Y)\) implies
+\begin{equation}
+ \sum_y p(x,y)\le C/\abs X.
+ \label{ib:public-marginal}
+\end{equation}
+The surviving \(x\) is the complete edge/point index, not only
+its point.  The requested form
+\(u\mapsto Q(\pi(b_1u),\pi(b_2v_{\rm public}(y)))\) may depend
+on the discarded index; it does not select a new surviving index.
+One may forget an actual repetition-copy index and choose a fresh
+uniform copy in the corresponding one-port block.  This changes
+only the fixed block-weight factor, not the original-index bound.
+
+All public supports used here have polynomial cardinality for fixed
+accuracy.  Groups, retained points and lines, representation fibers,
+fields, masks, forms, and copy sets are polynomial-listable; only a
+constant number of independent draws is used.  Their sampling laws
+are rational.  Public indices, coefficient operations, and the
+bounded Boolean answer tables can be enumerated in polynomial time.
+Neither a hidden field symbol nor a hidden array is processed as one
+unit-cost query.
+
+\subsection{Neighbor bits and global port recovery}
+\label{ib:neighbors-recovery}
+
+\begin{proposition}[Local flags and encoded neighbors]
+\label{ib:neighbors}
+Within a sufficiently small fixed radius of a valid canonical word,
+one can decide its incident-edge flags with arbitrarily high fixed
+reliability, and supply its own bits and those of every existing
+neighbor with arbitrarily small average error for bounded-density
+requests.  The number of bit probes is bounded at each accuracy.
+The complete hypothetical answer tree, including wrong decisions
+and missing-edge defaults, satisfies
+Lemma~\ref{ib:all-branch-density}.
+\end{proposition}
+
+\begin{proof}
+Reduce \(\rho\) if needed to lie below half the distance of
+Proposition~\ref{ib:tester}; a nearby valid label is then unique.
+Use average correction at uniform systematic edge/point positions
+to read the mode and the needed critical-bit projections of
+Lemma~\ref{lem:inner-algebra-interface}.  At full progress the
+all-ones contraction of the relevant singleton stack is a constant
+polynomial equal to its critical Boolean read.  Use a specified
+additive form taking 1 to 1; its actual raw queries are smoothed by
+the corrector.  Such uniform systematic requests have bounded
+outer density even though the logical read itself has a sparse
+public address.
+
+Compare the progress marker with its public empty and full tensors
+by uniform separable masks and bit forms at systematic points.
+On the true constant tensors a false equality has a fixed gap by
+successive nonzero contractions.  Corrected probes and sufficiently
+many repetitions distinguish these cases with arbitrary fixed
+reliability.  Together with mode and the full-progress critical
+bits, including the wrap predicate, these are exactly the finite
+decisions required by the path transition rules.  Explicitly, a
+pre successor increases progress unless full; at full it performs
+the enabled edit or has a missing wrap edge.  A post successor
+decreases progress unless empty; at empty it advances the frame.
+For predecessors, full post reverses the edit or has the missing
+wrap edge, other post reverses a decrease, positive-progress pre
+reverses an increase, and empty pre reverses a frame advance.
+An increase also distinguishes empty progress for initialization.
+There is no scan of the full progress index and no proposed-tape
+validity test.
+
+Fix the true Boolean transition branch of an existing edge.  The
+algebraic interface expresses every updated contraction as a
+bounded sum of public terms and public scalar multiples of old
+contractions at invertible \(K\)-affine pullbacks of the common
+target.  Slot pulls and scalars are independent of the target point
+and row.  The pullbacks commute with the rectangular action, so
+they commute through the common row encoder as column permutations.
+Public initialization terms have publicly computable actual
+parity values.  Thus each requested point \(Y\) bit is an XOR
+of boundedly many input \(Y\) requests with specified masks and
+forms at bounded-density indices.  For line \(Y\), the affine
+map preserves the line parameter, so the identical argument applies
+to the separable coefficient contractions
+\eqref{ib:line-bit}.  The derivative update uses the separately
+stored child and its summed new-axis stack; it does not recompute
+native packing, polynomial division, or coefficient extraction
+from a requested line or parity symbol.
+
+For a \(Z\) bit insert the two \(Y\) update sums into
+\eqref{ib:product-bit} and expand by bilinearity.  Two variable
+terms give an available input \(Z\) block by tag compatibility;
+absorb public coefficients into multipliers or masks.  The two
+row/point coordinates undergo separate affine column permutations,
+with their rows fixed.  This preserves their joint density even
+when the two input types coincide: their two positions are still
+distinct coordinates of the product domain, never identified into
+a diagonal.  A term with one public operand is an arbitrary
+specified binary linear form of the other \(J\)-symbol, and
+\eqref{ib:public-marginal} gives the necessary complete surviving
+row/point bound.  Here the public operand is its actual encoded
+\(J\)-symbol, including parity values, not an unencoded target
+value.  Two public operands give a public bit.
+
+Apply Proposition~\ref{ib:average-correction} to every needed term
+with a suitably smaller accuracy.  Each output type and fixed
+branch has a bounded list of transformations, including radix
+pulls and zero terms.  Charge their error and density bounds by
+their sum before selecting a term or branch.  Charge the finite
+decision errors separately, and analyze each relevant correct
+branch with its true decisions fixed before intersecting the good
+events.  Wrong decisions still select from the same public lists;
+missing or unused values are public zero.  Therefore the
+all-hypothetical-branch smoothness assertion follows from
+Lemma~\ref{ib:all-branch-density}.
+\end{proof}
+
+\begin{proposition}[Deterministic global read-port recovery]
+\label{ib:global-recovery}
+At a sufficiently small fixed balanced distance from a canonical
+word, every order-one logical read port can be recovered exactly
+in deterministic polynomial time.  In particular this applies to
+the one-port template.  No constant-query claim is made for this
+global decoder.
+\end{proposition}
+
+\begin{proof}
+At every systematic edge/point position in the port's order-one
+point block, read the entire Hadamard table including copies and
+choose a nearest \(J\)-symbol, breaking ties publicly.  Whenever
+the table error is less than \(1/4\), the selected symbol is true.
+Markov's inequality bounds the fraction of incorrect systematic
+symbols by a fixed multiple of balanced word error, accounting for
+the block weight and the systematic row fraction.
+
+Map these symbols to their represented target arguments using the
+known row action.  All fibers have equal size: for a fixed target
+\(y\), any full-rank \(A\) and any \(z\) determine the unique
+offset \(y-Az\), and the number of choices is the same for every
+row group.  Take a plurality in each fiber.  Every wrong plurality
+requires at least half its fiber's symbols to be wrong, so this
+loses at most a factor two.  Project outputs outside the subfield
+to any fixed \(\F_q\)-estimate, which cannot spoil a correct
+symbol, and take another plurality over ignored target coordinates.
+After reducing the fixed radius, we have an estimate
+\(\widehat f:\F_q^\ell\to\F_q\) of the native order-one
+polynomial with symbol error at most a sufficiently small fixed
+\(\alpha>0\).
+
+For each desired grid point \(x\), enumerate all directions
+\(z\in\F_q^\ell\).  Along the \(q\)-word
+\((\widehat f(x+tz))_{t\in\F_q}\), the expected number of
+errors over \(z\) is at most
+\(1+(q-1)\alpha\): every nonzero \(t\) makes \(x+tz\) uniform.
+Let
+\[
+ t_{\rm RS}=\left\lfloor\frac{q-D-1}{2}\right\rfloor.
+\]
+For sufficiently small fixed \(\alpha\) and large size, Markov's
+inequality shows that a strict majority of directions have at most
+\(t_{\rm RS}\) errors, since \(D/q\to0\).
+
+Use the Berlekamp--Welch reconstruction method in the linear-system
+presentation of Gemmell and Sudan~\cite[Appendix~A]{GemmellSudan1992}.
+Specifically, uniquely decode each line by solving the linear equations
+\[
+ N(t)=E(t)\widehat f(x+tz)\quad(t\in\F_q),\qquad
+ E\text{ monic of degree }t_{\rm RS},\quad
+ \deg N\le D+t_{\rm RS}.
+\]
+On a good direction, the error locator, padded by any monic factor
+to degree \(t_{\rm RS}\), provides a solution with
+\(N=Ef_{\rm line}\).  Every solution has this equality: the
+difference \(N-Ef_{\rm line}\) has degree at most
+\(D+t_{\rm RS}\) and vanishes at at least
+\(q-t_{\rm RS}>D+t_{\rm RS}\) correct positions.
+Divide, verify degree and agreement, and use public zero on a
+failure.  The strict majority of returned values at parameter zero
+is exactly \(f(x)\).  Repeat for all port grid indices.
+All domains, tables, directions, fields, and linear systems here
+are polynomial-size, so the complete algorithm is deterministic
+polynomial time.
+\end{proof}
+
+\subsection{The binary interface}
+
+\begin{lemma}[Inner binary interface]
+\label{lem:inner-binary-interface}
+For the inner algebraic families of
+Lemma~\ref{lem:inner-algebra-interface}, there is a deterministic
+polynomial-time binary encoding with the following properties,
+uniform over the circuits in each fixed padded template.
+\begin{enumerate}[label=(\roman*)]
+\item Its balanced blocks are the point and line \(Y\)'s and the
+compatible point-product \(Z\)'s above.  Canonical valid extended
+labels have constant distance, and admit the perfectly complete
+bounded-bit tester and fixed-gap proximity conclusion of
+Proposition~\ref{ib:tester}.
+\item One radius \(\rho>0\), fixed by the template before any
+request accuracy or density, permits every fixed average error
+\(\eta>0\) for requests of any fixed bounded outer-index density.
+For \(Z\), the requirement is a joint pair-density bound.  Other
+masks, multipliers, and forms may be arbitrary request-dependent
+ones.  Finite type subdistributions are charged without
+renormalization.  Mixed products with a public operand may be
+reduced to arbitrary linear forms using
+\eqref{ib:public-marginal}.
+\item In a small fixed neighborhood of a valid label, incident
+flags and current or existing-neighbor encoded bits have the
+bounded-query reliability and averaged accuracy in
+Proposition~\ref{ib:neighbors}.  Every routine is total with
+bounded decisions and public-zero defaults.
+\item After all fixed accuracy counts are chosen, all hypothetical
+raw read slots in the full answer tree have a finite summed
+density bound as in Lemma~\ref{ib:all-branch-density}.  This
+includes potential cone paths before deduplication, unsuccessful
+flags and comparisons, and arbitrary pulled-back forms.  All
+coefficient and address computations are public, all remaining
+hidden processing uses boundedly many bits, and all random
+supports and rational sampling tables are polynomial-time
+constructible.
+\item The common row encoder, field presentations, positive-tag
+original layouts, and fixed-label formulas are identical in
+corresponding actual and one-port families.  Repetition copies
+may differ and are sampled afresh in the receiving family.
+At a fixed distinguished source or its first successor, a
+positive \(Y\) bit has Boolean degree at most three in that
+port's read input, and a same-positive-tag \(Z\) bit has degree
+at most six.  An order-one port has the deterministic polynomial
+global recovery of Proposition~\ref{ib:global-recovery}.
+\end{enumerate}
+In particular, the radius for the one-port template can be fixed
+before a later actual port-count bound is chosen.  The actual
+template's own radius may depend on that bound through its number
+of blocks.
+\end{lemma}
+
+\begin{proof}
+The construction and the preceding propositions prove
+(i)--(iv) and global recovery.  For (v), field and row-code choices
+were made solely from the common point parameters.  The positive
+charts, including per-primary verification auxiliaries and their
+tagged descendants, have the literal common-index ancestry of
+Lemma~\ref{lem:inner-algebra-interface}.  Columnwise linear
+extension and fixed-parameter line restriction preserve that
+identity.  Passing through an original index and choosing a fresh
+copy therefore suffices even when balanced multiplicities differ.
+
+At either of the two specified fixed labels, legacy, work, frame,
+mode, and progress are public, and every positive persistent recipe
+has Boolean degree at most three in its port input.  Public matrix
+operations, division by fixed monic polynomials, coefficient
+extraction, slot contraction, fixed affine substitutions, line
+restriction, \(J\)-linear row extension, coordinate changes, and
+additive bit forms are all \(\F_2\)-linear in these hidden-data
+functions.  None raises that degree.  The one additional binary
+bilinear operation in \eqref{ib:product-bit} gives degree at most
+six.  Finally \eqref{ib:fixed-radius} uses only the template's
+\(T_0,b_\#,\nu_1,\zeta\); the later requested density and
+accuracy affect only row rounds and repetitions.  Applying it
+first to the fixed one-port template proves the last assertion.
+\end{proof}

preprints/The-PCP-for-PPAD-conjecture-a-quasilinear-reduction-September-25-2026/build/sections/09-inner-gadget.tex

@@ -0,0 +1,817 @@
+\section{The fixed port and its robust decision gadget}\label{sec:inner-gadget}
+
+We now prove Theorem~\ref{thm:port-gadget}. Throughout this section all
+fields, groups, block lengths, and path labels are those of the inner
+construction of Sections~\ref{sec:inner-algebra} and~\ref{sec:inner-binary}.
+In particular, its polynomial-size affine group is unrelated to the outer
+index group. We first construct the port independently of the eventual
+arity. We then fix an actual arity and construct a generalized subcircuit
+for one Boolean decision. The subcircuit will use the geometric procedures
+of Lemma~\ref{lem:geometric-procedures}, whose proof is independent of this
+section. Word distances in this section are relative Hamming distances;
+vector distances use the displayed normalized block norm.
+
+\subsection{A port fixed before the arity}
+
+Fix the exponent allowance $c_{\rm allow}$ and the common field, point,
+slot, and row-code parameters supplied by
+Lemmas~\ref{lem:inner-algebra-interface} and
+\ref{lem:inner-binary-interface}. In particular these parameters are
+chosen from $n'=(L+2)^{c_0}$, with fixed $c_0>c_{\rm allow}+5$, before
+specifying the number of input ports or the actual Boolean circuit.
+Construct the one-port instance of the inner path with the prescribed
+initial copy phase and, afterwards, a fixed trivial computation. Include
+all its positive-tag validity, critical-address, derivative, and
+verification-auxiliary families, as well as its tag-zero families. This is
+the \emph{dummy template}. Write $a_s$ for its distinguished source and
+$a_s^+$ for its immediate successor. Their common tape has blank work,
+stage-zero legacy, input $s$, frame $\gamma^{-1}$, and post mode. Their
+progress values are respectively $q-1$ and $q-2$. Define
+\begin{equation}\label{ig:port-definition}
+ \mathcal D(s)=\bigl(\mathcal W_{\rm dum}(a_s),
+                         \mathcal W_{\rm dum}(a_s^+)\bigr),
+ \qquad S_0=2m_{\rm dum}.
+\end{equation}
+Both halves use the entire balanced binary word of the same dummy template.
+
+\begin{proposition}[Fixed dummy port]\label{prop:dummy-port}
+The map in~\eqref{ig:port-definition} has polynomial length, constant
+relative Hamming distance $d_{\mathcal D}>0$, and the following properties.
+\begin{enumerate}[label=\textup{(\roman*)}]
+\item Every output bit is a multilinear polynomial in $s$ over $\F_2$ of
+ degree at most six. Its coefficients, and the whole canonical encoding,
+ are computable deterministically in time polynomial in $L$.
+\item There is a perfectly complete tester using a bounded number of bit
+ reads and bounded Boolean processing. Its public randomness is uniformly
+ enumerable with polynomial-size support and rational probabilities of
+ polynomial bit length. Every fixed branch has unconditioned read
+ subdistributions of bounded density relative to $[S_0]$. There is a
+ constant $C_B$ such that proximity $a$ to a port word implies rejection
+ at most $C_Ba$. For every fixed $a>0$, there is a fixed
+ $\theta_B(a)>0$ such that rejection less than $\theta_B(a)$ implies
+ proximity less than $a$ to some $\mathcal D(s)$, for all sufficiently
+ large $L$.
+\item There are a deterministic polynomial-time exact decoder at a fixed
+ positive Hamming radius and a fixed
+ $0<\nu_B<d_{\mathcal D}/4$. From any word within $\nu_B$ of
+ $\mathcal D(s)$, either half supplies bits of its canonical word with
+ any fixed requested average accuracy under any fixed bounded-density
+ law on the original edge and outer indices. For a product block this
+ law is on the joint pair of indices. Requested masks and additive
+ forms may depend arbitrarily on the public request. The read procedures
+ have unconditioned fixed-branch smoothness, including their unsuccessful
+ branches.
+\end{enumerate}
+The dummy template, its decoder radius, and $\nu_B$ are all fixed before
+any later actual arity, circuit-size factor, or correction-round count.
+They are not decreased when those constants increase.
+\end{proposition}
+
+\begin{proof}
+The inner distance assertion applies separately to each half. Distinct
+inputs give distinct labels already in the order-one input track, so it
+gives the claimed distance for their concatenation.
+
+For testing, mix the ordinary valid-label testers on the two halves with
+the following additional checks. At systematic point probes compare the
+legacy chart with the public stage-zero polynomial, the mode with post,
+and the progress marker with its specified one-hot tensor, using the
+source or successor progress value as appropriate. Uniform slot masks
+and additive binary forms give smooth bit versions of these checks; a
+specified form can also be split into fresh uniform summands. On exact
+valid labels, an incorrect location condition has a fixed positive
+rejection gap by the low-degree or constant-tensor distance. Correct
+conditions force the blank-work tape, the specified progress and mode,
+and frame $\gamma^{-1}$. Finally compare the two order-one input point
+tracks at identical original systematic edge/point indices and identical
+uniform bit forms, choosing copies independently in the two halves.
+Different inputs give a fixed positive discrepancy gap. The two reads
+have smooth marginals individually; they do not require a product block.
+Each mixture weight is a fixed positive constant.
+
+Here is the fixed-gap implication explicitly. Let $g>0$ be a common lower
+bound for the rejection of any violated location or equality requirement
+on exact pairs of valid labels, with the mixture weights included. Let
+$C$ bound the change in these tests' rejection under changes to a fraction
+of the whole port. Given $a>0$, choose
+\[
+ 0<a_0<\min\{a,g/(4C)\}.
+\]
+Ordinary soundness supplies a positive rejection threshold forcing each
+half within $a_0$ of an exact valid label. Take the total tester threshold
+small enough to imply both of these assertions and also smaller than
+$g/2$. The pair of nearby exact labels is at distance less than $a_0$
+from the received port. If any required exact condition were wrong,
+smoothness would make the received rejection at least
+$g-Ca_0>g/2$, a contradiction. Thus the exact pair is precisely
+$\mathcal D(s)$ and the received distance is less than $a$. This defines
+$\theta_B(a)$. Perfect completeness and the upper bound $C_Ba$ follow by
+charging every read whose answer differs from the exact port word. All
+choices and computations have the bounded, polynomially enumerable
+properties asserted in the two inner interfaces.
+
+At the two fixed labels all logical inputs except $s$ are public.
+Canonical persistent values and their coordinate coefficients have
+logical degree at most three, even on invalid Boolean tapes. Their
+verification certificates use public linear maps on one primary array:
+multiplication by the public test matrix, summation, division by a public
+monic polynomial, coefficient extraction, and interpolation. These
+operations preserve degree. Fixed-progress stack formation, restriction
+to a point or a line, line-coefficient extraction, slot contraction, row
+extension, and additive bit forms are linear over the prime field. Thus a
+$Y$ bit has degree at most three. A $Z$ bit is a binary bilinear form of
+two such operands and has degree at most six. Repeating bits for balance
+changes nothing. The degree bound is independent of the allowance and
+of all future template choices.
+
+For completeness, the coefficient algorithm need not manipulate a
+large symbolic expression. For a specified bit function $b$, evaluate
+the canonical dummy word on $1_T$ for all $T\subseteq[L]$ with
+$|T|\le6$. Its coefficient at the squarefree monomial indexed by $A$ is
+\[
+ c_A=\sum_{T\subseteq A}b(1_T)\quad\text{in }\F_2,
+ \qquad |A|\le6.
+\]
+Boolean multilinear inversion and the degree bound give every
+coefficient; there are $O((L+1)^6)$ evaluations, each polynomial-time.
+
+Use the deterministic order-one recovery from
+Lemma~\ref{lem:inner-binary-interface} on the first half. A port error
+fraction $r$ causes an error fraction at most $2r$ there, so a fixed
+sufficiently small $r$ gives exact recovery of $s$. Finally choose
+$\nu_B<d_{\mathcal D}/4$ with $2\nu_B$ inside the dummy average-correction
+radius and, if necessary, the just established recovery radius.
+A port within $\nu_B$ corrupts either half by at most $2\nu_B$.
+The dummy corrector therefore answers the stated requests from that
+half; its read density relative to the whole port loses only a factor
+two. Its radius is independent of the requested accuracy and density
+constant. Every constant used here belongs to the one-port template.
+\end{proof}
+
+\subsection{Exact original-index correspondence and virtual source bits}
+
+Now fix an actual arity $k$, a circuit-size factor, and a Boolean circuit
+$f$ covered by Theorem~\ref{thm:port-gadget}, padding unused inputs with
+public zero ports. Fix words $y_i$ with
+\begin{equation}\label{ig:fixed-ports}
+ \dist\bigl(y_i,\mathcal D(s_i)\bigr)\le\nu_B.
+\end{equation}
+For a public zero port all bits of $\mathcal D(0^L)$ are computed publicly.
+The true graph used below is the single actual-template path for the
+fixed tuple $s=(s_1,\ldots,s_k)$, rather than the union over possible
+input tuples. Write $w_*(s)$ and $w_*^+(s)$ for its source and immediate
+successor. These have the same public frame, tape locations, mode, and
+progress as their dummy counterparts, but their input-dependent encoding
+bits must be supplied from the ports.
+
+\begin{lemma}[Source correspondence and virtual access]\label{ig:virtual-source}
+For either $w_*(s)$ or $w_*^+(s)$, every tag-$i$ point or line $Y$
+operand agrees with its dummy counterpart for $s_i$ at the same original
+indices, after renaming the port tag. The same is true of a $Z$ block
+whose two operands both have tag $i$. Tag-zero data at either label are
+publicly computable. Consequently source and successor encoded-bit
+requests with bounded original-index density admit arbitrarily accurate
+average answers from~\eqref{ig:fixed-ports}, with smooth external reads.
+For mixed products the requisite density is joint density on the complete
+edge/point indices before marginalization. Actual and dummy balanced-copy
+multiplicities need not agree.
+\end{lemma}
+
+\begin{proof}
+Apply Lemma~\ref{ia:source-data} at the two labels in
+\eqref{ia:source-successor}. Both have frame $\gamma^{-1}$, blank work,
+stage-zero legacy, and post mode; their progresses are respectively
+$q-1$ and $q-2$. At either progress the lemma identifies each positive
+algebraic array with its one-port counterpart at the same native point,
+slot, and mask indices. This includes verification auxiliaries and all
+derivative descendants: Lemma~\ref{ia:port-ancestry} fixes their separate
+call layouts independently of work destinations, the number of ports,
+and the actual circuit. The placements and fixed-progress stack formulas
+are common as well.
+
+The common row field, row matrices, systematic extension, and point/line
+domains were fixed before $k$. Applying these same linear operations
+preserves the equality at the original encoded indices, for the source
+and its first decrement separately. The actual template may balance its
+block types with different copy counts. Forgetting its copy index and
+choosing a fresh uniformly random dummy copy preserves the original index
+and costs only the fixed actual block-weight factor.
+
+The same source-data lemma makes every tag-zero array public at these
+labels. Its conclusion includes derivatives evaluated on public shifted
+work, even when the shifted Boolean tape is invalid: canonical certificate
+recipes are defined there and do not solve a final validity equation.
+Thus no work-side certificate needs to evaluate $f(s)$ or a stored sum
+with a port-dependent primary. The actual encoded $J$-symbols of these
+public arrays, including their parity values, are computed with the
+actual template's common row encoder.
+
+The only remaining case is a product with tags $(i,0)$ or $(0,i)$.
+Write $x$ for the positive operand's entire original edge/point index and
+$z$ for the public operand's entire index. After the indicated contractions
+and public scalar multipliers its bit is, in one orientation,
+\begin{equation}\label{ig:mixed-form}
+ Q\bigl(\pi(b_i u(x)),\pi(b_0 v_{\rm pub}(z))\bigr)
+       =\mu_{z,b_i,b_0,Q}\bigl(u(x)\bigr),
+\end{equation}
+where $\mu_{z,b_i,b_0,Q}$ is a specified additive binary form. The dummy
+$Y$ corrector permits precisely such arbitrary forms, including dependence
+on both requested positions and all the public coefficients. It retains
+$x$ at the same original index. If the request law satisfies
+\[
+ p(x,z)\le\frac{C}{|\mathcal X|\,|\mathcal Z|},
+ \qquad\text{then}\qquad
+ \sum_zp(x,z)\le\frac{C}{|\mathcal X|}.
+\]
+This proves the required density for the dummy $Y$ request on both its
+edge and its point. A same-positive-tag $Z$ request retains both original
+indices and their joint density; even when its types coincide, no pair
+is replaced by a diagonal. A $(0,0)$ product is public. There are no
+products of two distinct positive tags. Proposition~\ref{prop:dummy-port}
+now supplies all the answers. Fixing a source/successor choice, orientation,
+or block type costs a fixed factor. All error and density bounds are
+summed before any later answer-dependent selection; no density bound is
+conditioned on that selection.
+\end{proof}
+
+\subsection{Validation of the true path and numerical access}
+
+Let $m=m_{\rm in}$ be the length of one actual-template word, and let
+$b_\#$ be its number of equally weighted original binary block types.
+The path labels have fixed separation $d_*>0$ in the normalized Euclidean
+norm. Constants from this point may depend on the actual arity,
+circuit-size factor, and desired output-loss fraction $\upsilon>0$.
+They will not alter $\mathcal D$ or $\nu_B$.
+
+The ordinary binary tester recognizes the union of paths over all input
+tuples. Before using a seed for geometry, we must ensure that its nearby
+label belongs to the path for the externally supplied tuple $s$.
+We do this by comparing each input track with its corresponding port.
+
+\begin{lemma}[Validation at the fixed external tuple]\label{ig:validation}
+There are fixed $\rho_{\rm in},\rho_{\rm in,comp}>0$ and bounded-read
+validation routines for an internal binary seed $z\in\{0,1\}^m$ such
+that, with arbitrarily small prescribed failure probability, acceptance
+implies distance less than $\rho_{\rm in}$ to a unique label on the
+true-$s$ path, and distance less than $\rho_{\rm in,comp}$ to a true-path
+label implies acceptance. These radii are chosen before the reliability
+of the routines. For each fixed seed that can be accepted on good
+decisions its reference label is fixed independently of the decision
+randomness. Current and actual incident-neighbor bits and existence flags
+can subsequently be supplied with arbitrary fixed average accuracy for
+the request laws of Lemma~\ref{lem:geometric-procedures}.
+\end{lemma}
+
+\begin{proof}
+Choose $\rho_{\rm in}$ within the whole-word uniqueness and actual-template
+correction radii. Estimate rejection of the ordinary tester for the union
+of all port tuples. For each port $i$, also sample the order-one point
+track $W_{s_i}$ of $z$ at uniform systematic original edge/point indices
+and uniform forms, and compare it with the bit at the identical original
+index of the corrected dummy source half of $y_i$. Choose a random actual
+copy for the direct read. This input track is unrotating and uses the
+same extension at every actual path label.
+
+On an exact valid label the ideal mismatch is zero if its $i$th port is
+$s_i$, and at least a constant $p_s>0$ otherwise: the nonzero order-one
+polynomial differs on a constant fraction of represented arguments, and
+a uniform additive form detects a nonzero difference with probability
+$1/2$. We can choose one $p_s$ for the fixed templates. A balanced error
+fraction $a'$ in $z$ changes a comparison rate by at most $C'_k a'$,
+where $C'_k$ includes the systematic and block-weight density factors.
+Choose
+\[
+ 0<a'<\rho_{\rm in},\qquad C'_k a'<p_s/8.
+\]
+Use half the ordinary soundness threshold for $a'$ as the rejection
+cutoff, and $p_s/2$ as each mismatch cutoff. Good statistics with fixed
+positive slack imply that an accepted seed is $a'$-close to an exact
+valid label. The mismatch gap then forces every port of that label to
+be $s_i$. Conversely, smoothness of the ordinary tester and of the
+comparison probes makes all rates lie strictly below their cutoffs when
+the seed is within a sufficiently small fixed
+$\rho_{\rm in,comp}>0$ of a true-path label.
+
+First fix sample counts that concentrate the underlying ideal rates at
+the required slack; independent samples and Chebyshev's inequality
+suffice. Next choose the dummy correction accuracy so that the bounded
+number of supplied comparison bits fail with the desired total
+probability. This order leaves $a'$, $\rho_{\rm in,comp}$, and $\nu_B$
+unchanged. On good acceptance, the ordinary correction and neighbor
+procedures of Lemma~\ref{lem:inner-binary-interface} apply to the fixed
+unique reference. They return the true incident flags, including the
+disabled wrap flag, and true encoding bits at existing neighbors with
+arbitrary average accuracy. Analyze each possible reference of the fixed
+seed before intersecting with acceptance; one never conditions a
+smoothness estimate on that event. On unsuccessful branches every routine
+still has a bounded Boolean output or a fixed default.
+\end{proof}
+
+The point-vector space for the gadget is
+\begin{equation}\label{ig:point-space}
+ \mathcal B=[-1,2]^{4m},\qquad
+ \|U\|_{\rm b}^2=\frac1m\sum_{i=1}^{4m}U_i^2.
+\end{equation}
+Among its first-block coordinates select the systematic point $Y$ bits
+of the constant output track $F_{\rm out}$, with forms $\mu$ satisfying
+$\mu(1)=1$, including all their balanced copies. At the unique sink all
+these bits equal $f(s)$. At least half the row labels are systematic,
+half the forms take $1$ to $1$, and the track has weight $1/b_\#$.
+Consequently the selected fraction among all $4m$ coordinates is at least
+\begin{equation}\label{ig:selection}
+ c_{\rm sel}=\frac1{16b_\#}.
+\end{equation}
+Choose the geometric parameter $\tau>0$ so small that
+\begin{equation}\label{ig:tau}
+ \frac{(5\tau)^2}{(2/5)^2}<\rho_{\rm in,comp},
+ \qquad
+ \frac{16(10\tau)^2}{c_{\rm sel}}<\frac{\upsilon}{4}.
+\end{equation}
+A binary disagreement after rounding a physical block at any threshold
+in $[2/5,3/5]$ incurs squared error at least $(2/5)^2$ from the true bit.
+Thus the first inequality gives the geometric completeness condition.
+Obtain fixed $B_1,c_*,\beta_0>0$ from
+Lemma~\ref{lem:geometric-procedures} and its geometric lower bound,
+increasing $B_1$ to an integer if necessary. Write $\mathcal E$ for the
+endpoint set of the true path. It excludes the top of the distinguished
+vertical segment; here every member of $\mathcal E$ has first block equal
+to the unique sink encoding. In particular,
+\begin{equation}\label{ig:geometric-bound}
+ \|v(U)\|_\infty\le B_1-1,
+ \qquad
+ \|\clip_{\mathcal B}(U+\beta v(U))-U\|_{\rm b}\ge c_*\beta
+\end{equation}
+whenever $\dist(U,\mathcal E)>10\tau$ and $0<\beta\le\beta_0$.
+
+We verify the numerical access contract rather than presupposing access
+to hidden circuit values. There are at most eight true word roles: each
+of two seed roles and its two possible neighbors, and the special source
+and successor. Four incident edge roles contribute four straight pieces
+and three internal fillets each; with two possible seed junctions and
+the special vertical and its bottom fillet there are at most 32 piece
+roles. Lemma~\ref{ig:virtual-source} always supplies the special roles,
+even when neither seed validates. The other roles use
+Lemma~\ref{ig:validation} and the actual neighbor routines. Their requests
+are full-word uniform samples for statistics or the coordinate matching
+the requested field component. Condition on a fixed output block type,
+or sum over the finite type list, to obtain the required edge/outer laws,
+including joint laws for $Z$ blocks. Mode, progress, and critical decisions
+use their own uniform systematic marginals.
+
+Fix the geometric mesh, numerical tolerances, scalar sample counts, and
+high-level role list first. Allocate calls for the possible roles and
+matching components before any numerical minimum or branch is selected.
+For a fixed input and rounding threshold, each seed has at most one
+reference that good validation could accept, so there are only boundedly
+many underlying actual candidates. Independent uniform-coordinate
+samples concentrate all their bounded statistics, including physical
+block statistics, without conditioning on which candidate wins. Set the
+required bit and decision error budgets for these high-level calls next,
+and only then fix the lower-level correction and repetition counts.
+A numerical winner chooses a reserved role and mesh case; it cannot turn
+a hidden answer into a new arbitrary word address.
+
+Raw scalar samples can be quantized on fixed-width grids with public
+random offsets. Their finite range and fixed widths give finitely many
+answers. All public field, address, coefficient, mesh, reciprocal,
+square-root, and trigonometric calculations are made during the
+reduction, to the fixed required precision. A finite table indexed by
+the bounded answers supplies the required numerical operations. The
+procedures return bounded values on arbitrary unsuccessful trials as
+well. Refining an offset grid later changes the thresholds and public
+table entries, but not the bin count, word-address primitives, or number
+of answer slots.
+
+It follows from Lemma~\ref{lem:geometric-procedures}, with these explicit
+oracles, that for $\kappa=1/100$ we can fix a finite unscaled estimator
+whose signed output has the form
+\begin{equation}\label{ig:ideal-estimator}
+ \widehat v_i(U,y)=B_1(p_i^*-n_i^*),\qquad
+ p_i^*,n_i^*\in[0,1]\text{ fixed-precision dyadics},\quad p_i^*n_i^*=0,
+\end{equation}
+and, whenever $\dist(U,\mathcal E)>10\tau$,
+\begin{equation}\label{ig:ideal-error}
+ 4\E_{i,t}\bigl|\widehat v_i(U,y)-v_i(U)\bigr|^2
+        <(c_*\kappa/100)^2.
+\end{equation}
+Here $i$ is uniform on $[4m]$, $t$ is the true public tape, and the
+inequality holds conditional on each fixed admissible internal rounding
+threshold and scalar offset. Rounding the signed output to its fixed
+dyadic grid is included in the accuracy. All choices to this point are
+independent of $\beta$ and of any external comparison margin.
+
+\subsection{Full tapes, deterministic copies, and read routing}
+
+We require bounds for every read actually wired into the circuit. Unfold
+all hypothetical Boolean, validation, equality, and quantization answers,
+assigning fresh public randomness to unexecuted nodes as well. After the
+accuracy choices above this is a fixed finite tree. Let $H$ bound the
+number of its raw comparison occurrences. A branch may select an
+unsuccessful seed, a wrong flag, a wrong transition type, or a default;
+these choices do not license a new addressing primitive.
+
+More explicitly, list each potential initial row-cone path before
+removing repeated positions. On a frozen symbolic branch the covariance
+map $(g,I)\mapsto(gh,h^{-1}I)$ is a joint permutation; a further row-star
+step is a fixed left multiplication of the edge coordinate, with a change
+among boundedly many edge labels. On product blocks the two axes undergo
+these transformations separately. Neighbor proposals use a finite list
+of the affine pullbacks from the inner algebra. Omissions, deduplication,
+and representative choices only select subdistributions of the listed
+paths, without renormalization. A flagged zero is a public value and
+needs no artificial dummy read at a concentrated address. Information-set
+inverses and linear-expression coefficients are public functions of the
+frozen branch and indices. Fresh additive splitting is performed after
+all form or multiplier pullbacks, so arbitrary answer-dependent requested
+forms still give the uniform inner marginals. These facts are exactly
+the unconditioned guarantees of
+Lemma~\ref{lem:inner-binary-interface}; they apply whether the answers
+on the frozen branch are correct or not.
+
+These inner routines satisfy the additional physical-read
+hypothesis~\ref{geo:access-finite} of
+Definition~\ref{geo:access-contract}. We therefore invoke part~(iv) of
+Lemma~\ref{lem:geometric-procedures} at the physical-read level: each
+completed inner bit or decision already has a bounded physical answer
+tree with the stated unconditioned branch marginals.
+
+For a fixed internal offset choice, sum these bounds over all potential
+slots and geometric roles. With $i$ uniform on $[4m]$, the expected number
+of reads of a specified internal base coordinate is at most $C/(4m)$,
+and of a specified position of any one external port is at most $C/S_0$,
+for a fixed constant $C$. Internal direct reads use uniform statistic
+positions or matching output positions. Virtual external reads use the
+original-index correspondence and the marginal calculation
+in~\eqref{ig:mixed-form}. The constants $H,C$ are uniform over the
+function and over the external threshold $\xi$ for the fixed bounds,
+and remain unchanged by later offset-grid refinement. These are counts
+on the whole tree, not smoothness conditioned on execution.
+
+Choose finite uniform rational grids for the internal common binary
+threshold $\xi_{\rm in}\in[2/5,3/5]$ and the scalar offsets modulo the
+already fixed quantization widths. If $D_{\rm grid}$ is a lower bound
+on their sizes, then for a fixed scalar value the probability of being
+within $4\omega_{\rm in}$ of a binary threshold or one of its relevant
+shifted boundaries is at most
+\begin{equation}\label{ig:ambiguity}
+ O\bigl(D_{\rm grid}^{-1}+C_{\rm width}\omega_{\rm in}\bigr),
+\end{equation}
+where $C_{\rm width}$ is fixed by the widths and finite range. Sum the
+conditional slot-density bounds over all base positions at every offset,
+and then over the bounded slot list. Thus the true-law fraction of cones
+having any such ambiguous internal comparison is uniformly as small as
+desired, for every fixed mean array, by refining the grids and decreasing
+the positive rational $\omega_{\rm in}$. External comparisons are not
+included here; their spoil budget is a separate hypothesis.
+
+For each output coordinate $i$, the full tape law, including the offsets
+and conditional random choices at all hypothetical nodes, has enumerable
+polynomial support and rational probabilities of polynomial bit length.
+A fixed product or finite tree of polynomial-size choices still has
+these bounds. Choose a common square $R$ and replace each law by an equal
+list of $R$ tapes with total variation at most
+\begin{equation}\label{ig:tv}
+ t_{\rm tv}=\frac{\eta_{\rm tv}}{\max\{4m,S_0\}}.
+\end{equation}
+Indeed floor $R$ times each probability and distribute the remaining
+integer counts within the support. If $P(L)$ is a common support bound,
+choosing a square $R\ge P(L)\max\{4m,S_0\}/\eta_{\rm tv}$ suffices.
+A square within a fixed factor of this polynomial bound is available,
+and the integer and rational operations cost polynomial time. The bound
+and $R$ can be common to all padded functions in the stated class.
+Use the $r$th listed tape at physical replica $(i,r)$ and put
+\begin{equation}\label{ig:gscale}
+ G_0=4mR.
+\end{equation}
+By bounded outputs, sufficiently small fixed $\eta_{\rm tv}$ preserves
+from~\eqref{ig:ideal-error} the empirical bound
+\begin{equation}\label{ig:empirical-error}
+ \left(\frac1{mR}\sum_{i,r}
+       |\widehat v_{ir}(U,y)-v_i(U)|^2\right)^{1/2}
+       \le c_*\kappa/20,
+\end{equation}
+as well as the desired rare-event slack in~\eqref{ig:ambiguity}.
+This estimate holds for every fixed $U$: the tape lists depend only on
+the public laws, and total variation controls every bounded function of
+the tape, including its error on that $U$.
+
+For clarity, the unequal-size load calculation is
+\begin{align}
+ \#\{\text{reads of external position }j\}
+ &\le G_0\left(\frac C{S_0}
+            +\frac{H\eta_{\rm tv}}{\max\{4m,S_0\}}\right)
+ \le (C+H\eta_{\rm tv})\frac{G_0}{S_0},
+ \label{ig:external-load}\\
+ \#\{\text{reads of internal base coordinate }j\}
+ &\le G_0\left(\frac C{4m}
+            +\frac{H\eta_{\rm tv}}{\max\{4m,S_0\}}\right)
+ \le (C+H\eta_{\rm tv})R.
+ \label{ig:internal-load}
+\end{align}
+For one position the count on a tape lies in $[0,H]$, so changing its
+law by total variation $t_{\rm tv}$ changes its expectation by at most
+$Ht_{\rm tv}$, which proves these inequalities. They hold even when
+$m/S_0$ grows as an arbitrarily large fixed power of $L$.
+Route the internal occurrences round-robin to the $R$ replicas of their
+base coordinate. Every replica is then read a bounded number of times.
+Every external occurrence is exposed separately for the caller to route;
+\eqref{ig:external-load} proves the constant $C_{\rm occ}$ in
+Theorem~\ref{thm:port-gadget}.
+
+\subsection{An explicit generalized subcircuit}
+
+Let $R=b^2$ and identify replica labels with $(\mathbb Z/b\mathbb Z)^2$.
+On each base coordinate use the average $A$ of the eight labeled moves
+\[
+ (u\pm2v,v),\quad (u\pm(2v+1),v),\quad
+ (u,v\pm2u),\quad (u,v\pm(2u+1))
+\]
+and eight identity labels, each with weight $1/16$. The all-modulus
+Margulis--Gabber--Galil bound, in the primary formal development
+\cite[\texttt{mgg\_numerical\_radius}]{MGGAFP}, bounds the normalized
+upper mean-zero eigenvalue of the eight-move operator by $5\sqrt2/8$.
+The operator is symmetric with spectrum in $[-1,1]$. Half-laziness
+therefore yields
+\begin{equation}\label{ig:replica-gap}
+ \|AD\|\le(1-\kappa)\|D\|,\qquad \kappa=1/100,
+\end{equation}
+on replica deviations, since $(1+5\sqrt2/8)/2<1-\kappa$.
+The statement includes every positive integer $b$ and retains loops
+and repeated labels. No prime-modulus choice or randomized graph search
+is needed.
+
+Each physical replica is a node $x_{ir}\in[0,1]$, interpreted as
+$X_{ir}=3x_{ir}-1\in[-1,2]$. Its own \emph{direct cone} produces it
+using the prescribed 16 averaging inputs and its fixed estimator tape.
+All gates of different direct cones are distinct; they may read the
+same existing node as input. The estimator uses only the nine gate types
+of Definition~\ref{def:gcircuit}, as follows.
+
+For every raw comparison occurrence create its own Constant threshold
+node and a Less gate, oriented to give the desired high bit. Internal
+binary thresholds and quantization boundaries are translated from
+physical coordinates by $a\mapsto(a+1)/3$. Out-of-range constant cases
+use Boolean constants. An external occurrence reads its individually
+routed external node and compares it with a Constant at $(\xi+1)/3$.
+Threshold gates are inside the cone and are never shared between cones.
+All public zero bits are produced by Constant gates.
+
+Unfold finite answer-dependent choices and implement their Boolean
+answers and finite lookup outputs by truth-table circuits over And, Or,
+and Not. A multiplexer, for example, is
+$(c\mathbin{\mathrm{And}}a)\mathbin{\mathrm{Or}}
+ ((\mathrm{Not}\,c)\mathbin{\mathrm{And}}b)$.
+Copy gates may provide fan-out, although unrestricted input fan-out
+already suffices. Induction through these circuits preserves a correct
+near-bit with the same tolerance: weak Boolean antecedents apply to
+inputs at most $\eps$ or at least $1-\eps$. This costs a fixed number
+of gates because the complete answer tree and output precision have
+already been fixed. In particular, polynomial-size field elements and
+indices are computed publicly to choose wires, coefficients, and tables;
+they are not hidden strings being processed by a bounded Boolean cone.
+
+Apply Scale with parameter $1/16$ to each averaging input and accumulate
+with Add to obtain their mean. Use Scale and Add to sum the fixed dyadic
+weights of the signed-output bits, obtaining nodes $p,n\in[0,1]$ for
+$p^*,n^*$ in~\eqref{ig:ideal-estimator}. For a positive rational $\beta$
+chosen below, put $t_\beta=\beta B_1/3\le1$. Scale $p,n$ by
+$t_\beta$, add the positive term to the averaging node, and subtract
+the negative term, in that order, to produce $x_{ir}$. All parameters
+lie in $[0,1]$. Exact partial sums for averages and dyadic expansions
+are at most one; hence their clipped Add realization is exact on ideal
+inputs. Clipped sums and differences are Lipschitz in their inputs, so
+there is a fixed arithmetic constant $C_{\rm ar}$ with the following
+two implications whenever the entire direct cone is $\eps$-satisfied.
+First,
+\begin{equation}
+ |X_{ir}-(AX)_{ir}|
+   \le2\beta B_1+C_{\rm ar}\eps.
+       \label{ig:coarse-direct}
+\end{equation}
+Under the same hypothesis, if all its raw comparisons give the ideal
+near-bits, then
+\begin{equation}
+ \left|X_{ir}-\clip_{[-1,2]}
+       \bigl((AX)_{ir}+\beta\widehat v_{ir}\bigr)\right|
+   \le C_{\rm ar}\eps.
+       \label{ig:fine-direct}
+\end{equation}
+Here are explicit error counts. Sixteen Scale gates and fifteen Add
+gates produce an averaging node $\widetilde a$ with
+$|\widetilde a-a|\le31\eps$, where $a=(Ax)_{ir}$.
+For
+\[
+ H(a,p,n)=\max\{\min\{a+t_\beta p,1\}-t_\beta n,0\},
+\]
+the two final Scale gates, Add, and Subtract give
+$|x_{ir}-H(a,p,n)|\le35\eps$. Since
+$|H(a,p,n)-a|\le2t_\beta$, this proves
+\eqref{ig:coarse-direct} with arithmetic error $105\eps$ and requires
+no estimator correctness. If the raw comparisons give the ideal near-bits,
+pad each signed dyadic expansion to $J$ bits. Its $J$ Scale gates and
+$J-1$ Add gates reconstruct each of $p^*,n^*$ within
+$(3J-1)\eps$: at most $J\eps$ comes from its input bits and
+$(2J-1)\eps$ from the arithmetic gates. Every exact partial sum is
+at most one. The map $H$ is 1-Lipschitz in $a$ and $t_\beta$-Lipschitz
+in each of $p,n$. Since $p^*n^*=0$,
+\[
+ 3H(a,p^*,n^*)-1
+   =\clip_{[-1,2]}\bigl((AX)_{ir}
+                         +\beta B_1(p^*-n^*)\bigr).
+\]
+Thus both estimates hold with $C_{\rm ar}=18J+99$. This calculation
+also covers intermediate saturations.
+
+For each selected first-block coordinate and each replica add a separate
+extraction pair: a Constant at $1/2$ in $x$ coordinates and a Less gate
+comparing that constant with $x_{ir}$. Its output is the designated
+near-bit. These gates are not used in the feedback. Let $W$ be the number
+of designated outputs. The construction has
+\begin{equation}\label{ig:gate-count}
+ c_{\rm sel}G_0\le W\le G_0,
+ \qquad |T_{\rm invocation}|\le C_gG_0
+\end{equation}
+for a fixed $C_g\ge1$ including every extraction gate and every hypothetical
+branch. Each node is the output of at most one gate, and cycles arise only
+through the prescribed physical input nodes. Gate listing, tape
+construction, address calculations, and all parameters other than the
+supplied threshold cost polynomial time and polynomial bit length in
+$L$, with fixed exponents for the chosen bounds. Uses of an arbitrary
+rational $\xi$ additionally cost its supplied bit length (and polynomial
+time in that length). The gate topology and count do not depend on that
+length or on the external physical margin.
+
+\subsection{Robustness under arbitrary failed gates}
+
+We give the choice order and prove the universal assignment guarantee.
+For a replica array put
+\begin{equation}\label{ig:replica-norm}
+ \|Y\|_{\rm rep}^2=\frac1{mR}\sum_{i,r}Y_{ir}^2,
+ \qquad \bar X_i=\frac1R\sum_rX_{ir}.
+\end{equation}
+Repeated mean vectors have precisely the norm~\eqref{ig:point-space}.
+Projection onto means and onto deviations are orthogonal contractions.
+The maximum norm of an array bounded coordinatewise by three is six;
+this explains the constants in the failure charges below.
+
+\paragraph{Parameter choices.}
+Choose $q_{\rm bad}>0$ so that
+\begin{equation}\label{ig:qbad}
+ 6B_1\sqrt{q_{\rm bad}}<c_*\kappa/10,
+\end{equation}
+and choose $\gamma_g>0$ much smaller than $q_{\rm bad}$. Choose the
+internal grids, the positive $\omega_{\rm in}$, and the tape-TV tolerance
+so that, for every fixed means, the empirical fraction of cones with an
+ambiguous internal comparison is less than $q_{\rm bad}/4$.
+The full-tree constants $H,C$ are already fixed. By
+\eqref{ig:internal-load}, let $D_{\rm read}$ bound the number of routed
+internal comparison inputs using any actual replica. Choose
+$d_{\rm sync}>0$ sufficiently small that
+\begin{equation}\label{ig:sync-choice}
+ \frac{D_{\rm read}d_{\rm sync}^2}{4\omega_{\rm in}^2}
+       <q_{\rm bad}/4,
+ \qquad
+ \frac{64d_{\rm sync}^2}{c_{\rm sel}}<\upsilon/4.
+\end{equation}
+If $\|X-\bar X\|_{\rm rep}\le d_{\rm sync}$, the first bound and
+Markov's inequality charge fewer than $q_{\rm bad}/4$ of the cones for
+any internal routed read deviating more than $\omega_{\rm in}$ from its
+base mean. Choose grouped budgets with positive slack, so that this
+charge, the ambiguity charge, and the $\gamma_g$ external charge sum
+to less than $q_{\rm bad}$.
+
+Now fix a positive rational $\beta\le\beta_0$ such that
+$t_\beta\le1$ and $4\beta B_1/\kappa<d_{\rm sync}/2$.
+Finally choose positive rational $\eps_g,\delta_g$ with
+\begin{gather}
+ \eps_g<\min\{1/100,\omega_{\rm in}/1000\},\qquad
+ \delta_g/c_{\rm sel}<\upsilon/4,
+ \label{ig:last-small}\\
+ \frac{4\beta B_1+2C_{\rm ar}\eps_g+6\sqrt{\delta_g}}
+         {\kappa}<d_{\rm sync},\qquad
+ 2C_{\rm ar}\eps_g+6\sqrt{\delta_g}
+         <\beta c_*\kappa/10.
+ \label{ig:epsilon-delta}
+\end{gather}
+All inequalities can hold because each right side has positive slack
+when the later constants are decreased. None of these choices uses the
+subsequently supplied external margin $\omega>0$. That margin imposes
+only the additional restriction $\eps<\omega/1000$ on the actual gate
+tolerance, alongside $\eps\le\eps_g$.
+
+Fix any assignment of values in $[0,1]$ to all nodes with at most
+$\delta_gG_0$ failed invocation gates. Assume that at most
+$\gamma_gG_0$ of the external comparison occurrences are spoiled:
+every other occurrence lies on the side of $\xi$ prescribed by the
+fixed word $y_i$, with physical gap greater than $\omega$.
+Different occurrences are not required to have equal physical values.
+A direct cone is called failed if any of its own gates fails. Since its
+gates are disjoint from those of other direct cones, their failed fraction
+is at most $\delta_g$.
+
+\paragraph{Coarse synchronization.}
+First obtain synchronization without assuming any correct estimator
+answer. Apply~\eqref{ig:coarse-direct} on the unfailed cones and the
+physical range bound on the others. The triangle inequality, the
+factor-four norm normalization, and~\eqref{ig:replica-gap} give
+\begin{equation}\label{ig:coarse-sync}
+ \kappa\|X-\bar X\|_{\rm rep}
+ \le\|X-AX\|_{\rm rep}
+ \le4\beta B_1+2C_{\rm ar}\eps+6\sqrt{\delta_g}
+ <\kappa d_{\rm sync}.
+\end{equation}
+For the first inequality write $D=X-\bar X$ and use
+$\|D\|\le\|D-AD\|+\|AD\|$. Mark as unstable a cone with an
+ambiguous internal comparison at these fixed means, an excessive routed
+internal deviation, or any spoiled external occurrence. The fraction of
+unstable cones is less than $q_{\rm bad}$ by the choices above. An
+external spoiled occurrence charges only its own cone; no repeated-value
+consistency was used.
+
+\paragraph{Fine residual and endpoint proximity.}
+Suppose for a contradiction that
+$\dist(\bar X,\mathcal E)>10\tau$. In a stable, unfailed cone,
+each internal comparison has physical mean gap greater than
+$4\omega_{\rm in}$ and input deviation at most $\omega_{\rm in}$.
+Its transformed $x$ gap is therefore at least $\omega_{\rm in}$.
+The Constant error and the strict Less tolerance are dominated by
+$\eps\le\eps_g<\omega_{\rm in}/1000$, forcing the ideal near-bit.
+Every external comparison likewise has $x$ gap greater than $\omega/3$;
+its Constant and Less errors are dominated by $\eps<\omega/1000$.
+Thus~\eqref{ig:fine-direct} applies with the empirical ideal estimator
+on $(\bar X,y)$. On unstable but unfailed cones,
+\eqref{ig:coarse-direct} and $\|v\|_\infty\le B_1$ give discrepancy
+at most $3\beta B_1+C_{\rm ar}\eps$ from the clipped mean-field
+update at $AX$. On failed cones that discrepancy is at most three.
+Using~\eqref{ig:empirical-error}, obtain
+\begin{align}
+ E_{\rm in}
+ &:=\bigl\|X-\clip_{\mathcal B}
+              (AX+\beta v(\bar X))\bigr\|_{\rm rep}\notag\\
+ &\le\beta\bigl(c_*\kappa/20+6B_1\sqrt{q_{\rm bad}}\bigr)
+       +2C_{\rm ar}\eps+6\sqrt{\delta_g}
+ <\beta c_*\kappa/2.
+ \label{ig:refined-error}
+\end{align}
+The notation $\clip_{\mathcal B}$ here means coordinatewise clipping
+on every replica, with $v(\bar X)$ repeated over replicas.
+
+We keep the clipping inside the contraction argument. Put
+\[
+ F=\clip_{\mathcal B}(AX+\beta v(\bar X)),\qquad
+ C_0=\clip_{\mathcal B}(\bar X+\beta v(\bar X)),
+\]
+where $C_0$ is constant on replicas. Nonexpansiveness and
+\eqref{ig:replica-gap} show
+\[
+ \|F-C_0\|_{\rm rep}\le(1-\kappa)\|X-\bar X\|_{\rm rep},
+ \qquad
+ \|F-\bar F\|_{\rm rep}\le(1-\kappa)\|X-\bar X\|_{\rm rep}.
+\]
+Projecting $X-F$ onto deviations now gives
+$\|X-\bar X\|_{\rm rep}\le E_{\rm in}/\kappa$.
+Projecting onto means and comparing $\bar F$ with $C_0$ gives
+\begin{align*}
+ \|\bar X-C_0\|_{\rm b}
+ &\le E_{\rm in}+(1-\kappa)\|X-\bar X\|_{\rm rep}\\
+ &\le E_{\rm in}/\kappa<c_*\beta.
+\end{align*}
+This contradicts~\eqref{ig:geometric-bound}. No commutation of clipping
+with taking means was asserted. Hence
+$\dist(\bar X,\mathcal E)\le10\tau$, and its first block has
+mean squared distance at most $(10\tau)^2$ from the sink word.
+
+\paragraph{Designated outputs.}
+It remains to account for adversarial concentration on designated
+outputs. Among all $G_0$ replica indices, discard those in the first
+block with mean discrepancy at least $1/4$ from the corresponding sink
+bit, and those with replica deviation greater than $1/8$. The squared
+bounds just proved and~\eqref{ig:coarse-sync} bound their respective
+counts, with harmless slack for normalization, by
+\[
+ 16(10\tau)^2G_0\quad\text{and}\quad64d_{\rm sync}^2G_0.
+\]
+At each remaining selected position, the physical replica differs by
+less than $3/8$ from the bit $f(s)$, so its distance from the physical
+threshold $1/2$ is greater than $1/8$. In $x$ coordinates the gap is
+greater than $1/24$. Since $\eps\le\eps_g<1/100$, a satisfied
+extraction pair returns the correct near-bit. Every extraction pair has
+its own two gates, so at most $\delta_gG_0$ designated outputs are
+charged for their own failed gates, even if every failed gate is
+concentrated there. By~\eqref{ig:gate-count}, the incorrect fraction is
+at most
+\begin{equation}\label{ig:extraction-loss}
+ \frac{16(10\tau)^2+64d_{\rm sync}^2+\delta_g}{c_{\rm sel}}
+ <\upsilon.
+\end{equation}
+The choice of $\tau$, synchronization radius, and failure budget reserved
+this divisor explicitly.
+
+We have proved the assertion for every assignment satisfying the stated
+failure and spoil budgets, with all failed gates otherwise arbitrary.
+Taking $c_g=c_{\rm sel}$, together with
+Proposition~\ref{prop:dummy-port},~\eqref{ig:external-load}, and
+\eqref{ig:gate-count}, gives every assertion of
+Theorem~\ref{thm:port-gadget}. The proof uses no part of the outer
+circuitization: its only remaining independent input is the geometric
+lemma proved next.

preprints/The-PCP-for-PPAD-conjecture-a-quasilinear-reduction-September-25-2026/build/sections/10-geometry.tex

@@ -0,0 +1,1167 @@
+\section{Encoded paths and finite local displacement procedures}
+\label{sec:geometry}
+
+The encoded-path and displacement-field method follows the construction
+in Rubinstein~\cite[Sections~3.1--3.2]{Rubinstein2016Settling}: current and successor
+blocks trace a copy--toggle polygon, and radial interpolation moves from
+the tangent through the inward and reverse-tangent directions to a fixed
+vertical direction. Here we prove the geometric interface at the normalized
+codeword scale, including rounded corners, displacement after clipping,
+and finite local access with average errors. This section is independent
+of both circuit realizations. Its hypotheses concern entire binary words and local access
+to their bits; they do not depend on the algebraic interpretation of an
+individual coordinate. In particular, in the inner application the graph
+is the path for the fixed true input tuple, and the distinguished source
+is accessed through the virtual procedures of Section~\ref{sec:inner-gadget}.
+
+\subsection{The geometric statement and its parameters}
+\label{geo:setup}
+
+Let $G$ be a finite directed graph of indegree and outdegree at most one,
+with no self-edge and no isolated vertex. Fix a source $w_*$ with an actual
+successor. All other vertices missing an incoming or an outgoing edge are
+called desired endpoints. Suppose that distinct vertices have encodings
+$E_w\in\{0,1\}^m$ satisfying
+\begin{equation}\label{geo:distance}
+ \norm{E_u-E_v}_{\rm b}\ge d_*>0\qquad(u\ne v).
+\end{equation}
+We decrease $d_*$ to a positive rational at most one. On any number of
+blocks of length $m$ use the inner product and norm
+\begin{equation}\label{geo:norm}
+ \langle U,V\rangle_{\rm b}=m^{-1}\sum_j U_jV_j,
+ \qquad \norm{U}_{\rm b}^2=m^{-1}\sum_j U_j^2.
+\end{equation}
+An unqualified norm in this section means this norm. The infinity norm
+is unnormalized. Write $\mathcal B=[-1,2]^{4m}$, and let
+$\clip_{\mathcal B}$ denote coordinatewise clipping to this cube.
+Clipping is nonexpansive for~\eqref{geo:norm}. Scalars occupying a block
+mean constant repetitions.
+
+\begin{theorem}[Encoded-path geometry]\label{thm:geometry}
+Under these hypotheses, there is a union $\mathcal L\subset\mathcal B$
+of directed, rounded paths and cycles, with desired endpoint set
+$\mathcal E$, having the following properties. For every sufficiently
+small positive rational $\tau$, there are a positive integer $B_1$ and
+positive rational constants $c_*,\beta_0$, depending only on $d_*$ and
+$\tau$, and a field $v:\mathcal B\to\mathbb R^{4m}$ such that
+\begin{align}
+ \norm{v(X)}_\infty&\le B_1-1,\label{geo:bounded-field}\\
+ \norm{\clip_{\mathcal B}(X+\beta v(X))-X}
+   &\ge c_*\beta
+ \quad\text{if }\dist(X,\mathcal E)>10\tau,
+       \quad 0<\beta\le\beta_0.\label{geo:displacement}
+\end{align}
+The field is continuous on the relative open set
+$\{X\in\mathcal B:\dist(X,\mathcal E)>10\tau\}$.
+Every $P\in\mathcal E$ has first block exactly $E_w$ for a desired
+endpoint $w$. Consequently, if $\norm{X-P}\le10\tau$, rounding $X_1$
+at $1/2$ gives relative Hamming distance at most $4(10\tau)^2$ from
+$E_w$. The exceptional top of the distinguished vertical is excluded
+from $\mathcal E$ and has field $-\mathbf e$, where
+$\mathbf e=(0,0,0,1)$.
+
+The initial choice of $\tau$ may impose any further fixed positive upper
+bounds, including validation and global-recovery radii. All these choices,
+and $B_1,c_*,\beta_0$, precede every numerical accuracy choice and the
+choice of a step size $\beta$.
+\end{theorem}
+
+We prove the theorem before formulating the precise access contract for
+its finite numerical implementation. In particular no global regularity
+claim at desired endpoints will be needed.
+
+\subsection{Polygons, fillets, and local projection}
+\label{geo:curves}
+
+For an edge $u\to v$, draw, with fourth block zero,
+\begin{equation}\label{geo:polygon}
+ \begin{aligned}
+ &(E_u,E_u,0)\xrightarrow{\;2\;}(E_u,E_v,0)
+                 \xrightarrow{\;3\uparrow\;}(E_u,E_v,1)\\
+ &\qquad\xrightarrow{\;1\;}(E_v,E_v,1)
+                 \xrightarrow{\;3\downarrow\;}(E_v,E_v,0).
+ \end{aligned}
+\end{equation}
+An arrow label names the changing block; the up and down arrows record
+the direction of the third block. This is a schematic of block changes,
+not a distance-preserving planar drawing.
+Concatenate these polygons at graph-vertex junctions. Put
+$H_*=(E_{w_*},E_{w_*},0)$ and prepend the segment
+$(H_*,2)\longrightarrow(H_*,0)$ to the path out of $w_*$.
+We call the four segment types in~\eqref{geo:polygon} \emph{horizontal},
+meaning that their fourth block is zero; this includes the two third-block
+toggles. Every horizontal segment has length in $[d_*,1]$, and consecutive unit
+directions are orthogonal: the successive varying blocks in a polygon
+are $2,3,1,3$, a graph junction changes from block $3$ to block $2$,
+and the special bottom changes from block $4$ to block $2$.
+
+\begin{lemma}[Separation of original segments]\label{geo:separation}
+There is $d_{\rm sep}>0$, depending only on $d_*$, such that distinct
+nonconsecutive original segments are at distance at least $d_{\rm sep}$.
+Their adjacency graph, with adjacency meaning consecutiveness, has no
+triangle.
+\end{lemma}
+\begin{proof}
+For binary words $a,b,c$ and $0\le s\le1$,
+\begin{equation}\label{geo:binary-chord}
+ \norm{(1-s)a+sb-c}\ge\min(s,1-s)\norm{a-b}.
+\end{equation}
+Indeed on a coordinate where $a,b$ differ, the chord coordinate is $s$
+or $1-s$, whose distance from a binary coordinate is at least
+$\min(s,1-s)$. Summing squares proves the claim. In particular, a
+point on this chord within distance $z$ of a binary word is within
+distance $z$ of one of the chord endpoints.
+
+Consider first segments varying the same block. If their distance is
+less than $d_*$, all fixed encoding blocks and repeated binary levels
+agree. For block $2$, the tail agrees and determines the outgoing
+edge. For block $1$, the head agrees and determines the incoming edge.
+For block $3$, an upward segment has a fixed unequal pair $(E_u,E_v)$,
+whereas a downward segment has an equal pair $(E_v,E_v)$; the two types
+cannot have the same fixed pair. Within either type that pair determines
+the segment, using uniqueness of the incoming edge for the downward
+type. There is only one block-$4$ segment.
+
+For two segments varying different blocks, choose points at distance
+$z$. Each varying block of a horizontal segment is within distance $z$
+of a fixed binary block of the other. By~\eqref{geo:binary-chord}, its
+point is within distance $z$ of an endpoint of its original segment.
+For the special vertical, its distance from a horizontal segment forces
+its height to be at most $z$, so it is within $z$ of its bottom. The
+two selected endpoints are therefore within $3z$. Distinct original
+endpoints are separated by at least $d_*$: one of their encoding blocks,
+binary third levels, or vertical levels differs. Taking $3z<d_*$ forces
+the endpoints to coincide.
+
+An unequal ordered pair $(E_u,E_v)$ occurs at the two internal toggle
+corners of its own edge only, with the third level distinguishing them.
+An endpoint $(E_v,E_v,1,0)$ belongs to the last internal corner of the
+unique incoming edge. An endpoint $(E_v,E_v,0,0)$ belongs to at most
+the incoming final segment and the outgoing first segment. If both
+exist, they are consecutive. At $w_*$ there is no incoming graph edge,
+and the special bottom takes exactly that role. The top belongs only
+to the special vertical. Thus equality of endpoints of distinct
+segments implies consecutiveness. For example $d_{\rm sep}=d_*/4$
+is sufficient for the asserted strict-distance argument. The adjacency
+components are paths and cycles; any cycle contains at least eight
+segments because the graph has no self-edge. Adding a segment at a
+path beginning creates no triangle.
+\end{proof}
+
+Choose a positive rational $a_0$ such that
+\begin{equation}\label{geo:trim-choice}
+ 2a_0<d_{\rm sep}/2,\qquad4a_0<d_*,\qquad a_0<1/10.
+\end{equation}
+Trim arclength $a_0$ from both ends of every horizontal segment, whether
+or not its outer graph junction is present. At a shared corner $C$ with
+incoming and outgoing unit directions $u_0,v_0$, join the trim points by
+\begin{align}
+ Z(s)&=C-a_0u_0+a_0v_0
+             +a_0(\sin s\,u_0-\cos s\,v_0),
+             &0\le s\le\pi/2,\label{geo:fillet}\\
+ T(s)&=\cos s\,u_0+\sin s\,v_0.\label{geo:fillet-tangent}
+\end{align}
+The arc has speed $a_0$ in the parameter $s$, and $T$ is its directed
+unit tangent. Figure~\ref{fig:geo-fillet} shows this replacement in the
+plane of its two orthogonal directions. Use the same fillet at the special
+bottom, trimming the vertical there only. Its top is not trimmed. Unmatched horizontal trim
+points remain endpoints. Let $\mathcal L$ be the resulting compact union
+and set
+\begin{equation}\label{geo:endpoint-set}
+ \mathcal E=\{\text{unmatched endpoints of }\mathcal L\}
+                \setminus\{(H_*,2)\}.
+\end{equation}
+
+The first three blocks stay in $[0,1]$: on a fillet the separate varied
+block coordinates stay in the intervals traversed by their original
+segments. The fourth block is zero except on the vertical and its bottom
+fillet; on that fillet it is a repeated scalar in $[0,a_0]$.
+Consequently $\langle T,\mathbf e\rangle\le0$ everywhere. Fix an
+integer $C_t\ge2/d_*$. For arclength $\lambda$,
+\begin{equation}\label{geo:tangent-bounds}
+ \norm T_\infty\le C_t,\qquad
+ \norm{dT/d\lambda}\le1/a_0,\qquad
+ \norm{dT/d\lambda}_\infty\le2C_t/a_0.
+\end{equation}
+These derivative bounds hold on piece interiors and give the same
+Lipschitz bounds across joins because the tangent is continuous there.
+
+\begin{figure}[htbp]
+\centering
+\setlength{\unitlength}{1pt}
+\begin{picture}(290,160)
+\thinlines
+\multiput(180,30)(-5,0){10}{\line(-1,0){2}}
+\multiput(180,30)(0,5){10}{\line(0,1){2}}
+\put(180,30){\circle*{3}}
+\put(187,24){\makebox(0,0)[l]{$C$}}
+\put(130,80){\circle*{3}}
+\put(122,87){\makebox(0,0)[r]{$O$}}
+\thicklines
+\put(20,30){\line(1,0){110}}
+\put(62,30){\vector(1,0){25}}
+\put(74,18){\makebox(0,0)[t]{$u_0$}}
+\qbezier(130,30)(139.946,30)(149.134,33.806)
+\qbezier(149.134,33.806)(158.323,37.612)(165.355,44.645)
+\qbezier(165.355,44.645)(172.388,51.677)(176.194,60.866)
+\qbezier(176.194,60.866)(180,70.054)(180,80)
+\put(180,80){\line(0,1){65}}
+\put(180,103){\vector(0,1){24}}
+\put(170,116){\makebox(0,0)[r]{$v_0$}}
+\put(130,30){\circle*{3}}
+\put(180,80){\circle*{3}}
+\put(130,9){\makebox(0,0)[t]{$C-a_0u_0$}}
+\put(188,80){\makebox(0,0)[l]{$C+a_0v_0$}}
+\end{picture}
+\caption{Rounding one orthogonal corner. The dashed legs are removed;
+the directed path follows the retained straight portions and the fillet
+from $C-a_0u_0$ to $C+a_0v_0$. Its center is
+$O=C-a_0u_0+a_0v_0$ and its radius is $a_0$. This local geometry lies
+in the orthonormal $(u_0,v_0)$ plane.}
+\label{fig:geo-fillet}
+\end{figure}
+
+The numerical procedure will choose a curve point by minimizing estimated
+distances. We therefore need more than a unique projection: a curve point
+with small squared-distance excess must be close to the true projection,
+and its tangent close to the projection's tangent, in every coordinate.
+The next lemma gives these bounds before any candidate is selected.
+
+\begin{lemma}[Local chains and strong excess]\label{geo:projection}
+There is $\rho_{\rm loc}>0$, depending only on $d_*,a_0$, such that
+two points $Z,Z'\in\mathcal L$ with $\norm{Z-Z'}<\rho_{\rm loc}$
+are joined along one local straight or single-corner chain. Its length
+$\Delta$ satisfies $\Delta\le\sqrt2\norm{Z-Z'}$.
+For every sufficiently small $\tau>0$, if
+\begin{equation}\label{geo:tube-hypothesis}
+ r=\dist(X,\mathcal L)\le4\tau,\qquad
+ \dist(X,\mathcal E\cup\{(H_*,2)\})>10\tau,
+\end{equation}
+there is a unique closest point $Z$, which is interior and satisfies
+$\langle X-Z,T(Z)\rangle=0$. If $Z'\in\mathcal L$ has squared excess
+$g=\norm{X-Z'}^2-r^2\le\tau^2$ and signed arclength difference $s'$
+from $Z$ on that chain, then
+\begin{equation}\label{geo:strong-excess}
+ g\ge (s')^2/2.
+\end{equation}
+In particular,
+\begin{align}
+ \norm{Z'-Z}&\le\sqrt{2g},&
+ \norm{Z'-Z}_\infty&\le C_t\sqrt{2g},\label{geo:point-error}\\
+ \norm{T(Z')-T(Z)}&\le\sqrt{2g}/a_0,&
+ \norm{T(Z')-T(Z)}_\infty&\le2C_t\sqrt{2g}/a_0.
+ \label{geo:tangent-error}
+\end{align}
+The tangent at an endpoint competitor is its one-sided tangent.
+\end{lemma}
+\begin{proof}
+Support a straight point by its original segment, and an arc point by
+the two segments of its corner. An arc point is within $a_0$ of each
+supporting segment. Choose $\rho_{\rm loc}+2a_0<d_{\rm sep}$.
+Every across-point pair of supports is then identical or consecutive.
+Each point's support set is itself a clique, so their union is a clique
+and contains at most two segments by Lemma~\ref{geo:separation}. In the
+one-segment case both points lie on its retained straight. In the
+two-segment case they lie on the two retained straights and their
+shared fillet. Opposite-end trims are disjoint by~\eqref{geo:trim-choice}.
+This also rules out unintended identifications or intersections.
+
+Along a two-direction chain, all tangent coefficients in its orthogonal
+directions $u_0,v_0$ are nonnegative, and
+$\langle T,u_0+v_0\rangle\ge1$. Integrating in the directed order gives
+$\Delta\le\langle Z'-Z,u_0+v_0\rangle\le\sqrt2\norm{Z'-Z}$.
+Reverse the order if necessary. A straight chain has equality between
+length and endpoint distance.
+
+Compactness gives a minimizer. The endpoint assumption makes every
+minimizer interior, so differentiating squared distance along its
+$C^1$ path gives the asserted perpendicularity. A competitor of the
+stated excess satisfies
+\[
+ \norm{Z'-Z}\le4\tau+\sqrt{17}\tau<9\tau.
+\]
+Require $9\tau<\rho_{\rm loc}$. The local chain then has
+$|s'|<13\tau$, and integration of~\eqref{geo:tangent-bounds} gives
+\[
+ Z'-Z=s'T(Z)+R,\qquad \norm R\le(s')^2/(2a_0).
+\]
+For $|s'|/(2a_0)<1$, expansion of squared distance and perpendicularity
+therefore imply
+\begin{align*}
+ g&=\norm{Z'-Z}^2-2\langle X-Z,Z'-Z\rangle\\
+  &\ge(s')^2\left[
+      \left(1-\frac{|s'|}{2a_0}\right)^2-\frac r{a_0}\right]
+   \ge(s')^2/2,
+\end{align*}
+where, for example, $\tau/a_0\le1/64$ makes the last inequality hold.
+Applying this to a second minimizer proves uniqueness. Integrating the
+unit tangent, its infinity bound, and its two Lipschitz bounds over
+length $|s'|\le\sqrt{2g}$ proves~\eqref{geo:point-error} and
+\eqref{geo:tangent-error}. All estimates are independent of $m$.
+\end{proof}
+
+At an unmatched beginning, the trimmed segment varies only block $2$;
+at an unmatched ending it varies only block $3$. Their first blocks
+are therefore exactly the encoding of the missing-neighbor vertex.
+There are no isolated vertices requiring a separate convention. The
+source's horizontal beginning is joined to the special vertical and
+is not an endpoint of $\mathcal L$. This proves the endpoint assertion
+in Theorem~\ref{thm:geometry}. If $P\in\mathcal E$ and $\norm{X-P}\le10\tau$, every wrongly
+rounded first coordinate costs at least $1/4$ in squared error, proving
+the Hamming bound in that theorem.
+
+\begin{remark}[Local Lipschitz continuity of projection]
+\label{geo:projection-lipschitz}
+Suppose both $X$ and $Y$ satisfy~\eqref{geo:tube-hypothesis}, and
+$\norm{X-Y}\le\tau/20$. Write $P,Q$ for their respective projections
+onto $\mathcal L$ and $r_X,r_Y$ for their distances to $\mathcal L$.
+With the same sufficiently small choice of $\tau$, one has
+\begin{equation}\label{geo:projection-lipschitz-bound}
+ \left(1-\frac{r_X+r_Y}{a_0}\right)\norm{P-Q}^2
+ \le \langle X-Y,P-Q\rangle
+ \le \norm{X-Y}\norm{P-Q}.
+\end{equation}
+In particular, the projection is locally $2$-Lipschitz on this tube.
+This is a local assertion away from the specified endpoints.
+
+Indeed, the conclusion is immediate if $P=Q$. Otherwise,
+$\norm{P-Q}\le r_X+\norm{X-Y}+r_Y\le(8+1/20)\tau<\rho_{\rm loc}$.
+Lemma~\ref{geo:projection} therefore puts $P,Q$ on one local chain,
+whose arclength $s$ satisfies $s^2\le2\norm{P-Q}^2$.
+Integrating~\eqref{geo:tangent-bounds} from either end expresses its
+chord as the signed length times the tangent there, with remainder
+of norm at most $s^2/(2a_0)$. Perpendicularity at $P$ and $Q$ gives
+\[
+ \big|\langle X-P,P-Q\rangle\big|
+       \le\frac{r_Xs^2}{2a_0},\qquad
+ \big|\langle Y-Q,P-Q\rangle\big|
+       \le\frac{r_Ys^2}{2a_0}.
+\]
+Expanding $X-Y=(P-Q)+(X-P)-(Y-Q)$ proves the first inequality
+in~\eqref{geo:projection-lipschitz-bound}; Cauchy--Schwarz proves
+the second. Finally $r_X+r_Y\le8\tau$ and $\tau/a_0\le1/64$
+make its left coefficient at least $7/8$, and in particular at least
+$1/2$, proving the stated factor $2$.
+\end{remark}
+
+\subsection{A field with uniform displacement after clipping}
+
+For the inside prescription, away from the desired endpoint neighborhood,
+successive radial intervals interpolate from the directed tangent through
+the inward normal and reverse tangent to the fixed fourth-block direction.
+The four directional stages in~\eqref{geo:Phi} parallel
+\cite[Equation~(1)]{Rubinstein2016Settling}; the projection and clipping estimates
+below establish the bounds needed for the present curve.
+A separate prescription above fourth-block mean level $1/2$ handles the
+distinguished vertical. We check that the prescriptions agree and that
+clipping preserves positive displacement away from the desired endpoints.
+The normalized tube bound permits individual coordinates of $X$ to be
+far from the curve, so raw field length alone does not prove this last
+claim. We will bound the fraction of coordinates where clipping can
+remove motion, and then handle the upper face separately.
+\label{geo:field}
+
+Fix $\tau$ satisfying Lemma~\ref{geo:projection}, and impose also
+\begin{equation}\label{geo:tau-choice}
+ \tau<1/100,\qquad6(C_t+1)\tau\le1/4.
+\end{equation}
+All further fixed validation or recovery upper bounds on $\tau$ are
+imposed now. For $X=(X_1,X_2,X_3,X_4)$ put
+$\mu=m^{-1}\sum_j(X_4)_j$. For a point $Z$ and a unit tangent $T$ let
+$r=\norm{X-Z}$ and define
+\begin{equation}\label{geo:Phi}
+ \Phi(X;Z,T)=
+ \begin{cases}
+ (1-r/\tau)T+(Z-X)/\tau,&0\le r\le\tau,\\[2pt]
+ -(r/\tau-1)T+\dfrac{2\tau-r}{\tau r}(Z-X),
+                       &\tau\le r\le2\tau,\\[5pt]
+ -\dfrac{3\tau-r}{\tau}T+\dfrac{r-2\tau}{\tau}\mathbf e,
+                       &2\tau\le r\le3\tau,\\[5pt]
+ \mathbf e,&r\ge3\tau.
+ \end{cases}
+\end{equation}
+The values at shared boundaries agree. There is no reciprocal radius
+in the formula used near zero.
+
+For $\mu\le1/2$ and $\dist(X,\mathcal E)>10\tau$, use
+$v(X)=\Phi(X;Z,T(Z))$ when $\dist(X,\mathcal L)\le3\tau$, and
+$v(X)=\mathbf e$ otherwise. The top is at distance at least $3/2$
+from this region, by the difference of fourth-block means; hence the
+projection lemma applies. At the remaining inside points put $v(X)=0$.
+For $\mu\ge1/2$, the default upward motion can be lost at the upper
+face. We add motion toward the distinguished vertical, with coefficient
+zero at the mean interface. Close to that vertical the tangent direction
+is downward, including at its top. Set
+\begin{equation}\label{geo:outside}
+ \begin{aligned}
+ Z_\mu&=(H_*,\mu),\qquad r_\mu=\norm{X-Z_\mu},\qquad
+ h_\mu=\min\{1,2(\mu-1/2)\},\\
+ v(X)&=\Phi(X;Z_\mu,-\mathbf e)+h_\mu(Z_\mu-X).
+ \end{aligned}
+\end{equation}
+
+\begin{lemma}[Agreement at the mean interface]\label{geo:interface}
+In $2/5<\mu<3/5$, the inside prescription extended to this slab equals
+$\Phi(X;Z_\mu,-\mathbf e)$. The outside extension, using
+$h_\mu=\max\{0,\min\{1,2(\mu-1/2)\}\}$, differs from it in norm by
+at most $12|\mu-1/2|$ and in each component by at most $6|\mu-1/2|$.
+Thus the definitions agree at $\mu=1/2$.
+\end{lemma}
+\begin{proof}
+Every nonvertical piece, including the bottom fillet, has fourth height
+at most $a_0<1/10$. Its distance to a point in the slab exceeds $3/10$,
+by the mean inequality $\norm{X_4-z\mathbf1}\ge|\mu-z|$.
+It cannot be in the $4\tau$ tube. The retained vertical contains all
+heights of this slab, and
+\[
+ m^{-1}\sum_j((X_4)_j-t)^2
+ =m^{-1}\sum_j((X_4)_j-\mu)^2+(t-\mu)^2
+\]
+shows that its nearest line point is $Z_\mu$. For $T=-\mathbf e$,
+the last annulus of~\eqref{geo:Phi} is already identically
+$\mathbf e$ at every $r\ge2\tau$. Consequently the same equality
+holds even where the closest curve piece is far away and the inside
+field is default. The desired endpoints have fourth height zero and
+the top has height two, so neither endpoint exception occurs in the
+slab. Each coordinate of $Z_\mu-X$ has magnitude at most three and
+its norm is at most six. Multiplication by
+$h_\mu\le2|\mu-1/2|$ proves both bounds.
+\end{proof}
+
+The projection is continuous wherever it is unique in the tube:
+if $X_n\to X$, compactness makes every subsequential limit of their
+projections a minimizer for $X$, hence the unique minimizer. Its tangent
+is continuous along the local chains. Formula~\eqref{geo:Phi} matches
+the default at $3\tau$, and Lemma~\ref{geo:interface} matches the mean
+interface. This proves precisely the continuity claimed in
+Theorem~\ref{thm:geometry}.
+
+\begin{proof}[Proof of the displacement assertion in Theorem~\ref{thm:geometry}]
+Choose $\beta_0>0$ satisfying all the upper bounds on $\beta$ used below.
+\paragraph{Inside the tube and in the inside default region.}
+First suppose $\mu\le1/2$ and $r=\dist(X,\mathcal L)\le3\tau$.
+For $r\le\tau$ the raw field is the convex combination of $T$ and
+the inward unit normal with coefficients $1-r/\tau,r/\tau$.
+For $\tau\le r\le2\tau$ it is the convex combination of $-T$ and
+the inward unit normal with coefficients $r/\tau-1,2-r/\tau$.
+The two directions in each combination are orthogonal. At $r=0$
+the field is $T$. In the third annulus the two unit directions are
+$-T,\mathbf e$, whose inner product is nonnegative. Therefore in all
+three annuli $\norm{v(X)}\ge1/\sqrt2$.
+
+Write $v=b+w(Z-X)$, where $\norm b_\infty\le C_t+1$ and
+$0\le w\le1/\tau$. All coordinates of $Z$ lie in $[0,1]$: for a
+vertical projection its height is at most $\mu+3\tau<1$, and all
+other pieces already have this property. Require
+$\beta/\tau\le1/2$ and $\beta(C_t+1)\le1/4$.
+The preliminary point $Y=X+\beta w(Z-X)$ lies in the box. If
+$|X_i-Z_i|<1/2$, then $-1/2<Y_i<3/2$, and the additional displacement
+$\beta b_i$ causes no clipping. The number of remaining coordinates,
+divided by $m$, is at most $4r^2\le36\tau^2$. At each, the clipping
+loss is at most $\beta(C_t+1)$ because $Y$ was in the box. The norm of
+all loss is at most $6\tau\beta(C_t+1)\le\beta/4$. Subtracting it
+from the raw displacement leaves at least $\beta/4$.
+
+In the inside default case, let $p$ be the fraction of fourth coordinates
+at most $7/4$. Since all entries are at least $-1$,
+\[
+ 1/2\ge\mu\ge-p+(7/4)(1-p),\qquad p\ge5/11.
+\]
+For $\beta\le1/4$ these coordinates retain the full upward displacement,
+giving norm at least $\beta\sqrt{5/11}$.
+
+\paragraph{Above the mean interface.}
+Now suppose $\mu\ge1/2$, and abbreviate $Z=Z_\mu$, $r=r_\mu$.
+The fourth coordinates of $Z-X$ have mean zero, so they are orthogonal
+to $\mathbf e$. Write $v=b\mathbf e+w(Z-X)$, with the scalar table
+\begin{equation}\label{geo:outside-coefficients}
+\begin{array}{c|cc}
+\text{radius}&b&w\\\hline
+0\le r\le\tau&-1+r/\tau&1/\tau+h_\mu\\
+\tau\le r\le2\tau&r/\tau-1&2/r-1/\tau+h_\mu\\
+r\ge2\tau&1&h_\mu
+\end{array}
+\end{equation}
+and hence
+\begin{equation}\label{geo:outside-norm}
+ |b|\le1,\quad0\le w\le w_{\max}:=1/\tau+1,
+ \quad b^2+w^2r^2\ge1/2.
+\end{equation}
+The last assertion follows from the two orthogonal components of the
+first two radial regimes; adding $h_\mu$ increases the nonnegative
+normal coefficient. In the last regime $b=1$. Require
+$\beta w_{\max}\le1/2$. The move toward $Z$ is a convex move inside
+the box, and horizontal coordinates undergo no further move or clipping.
+
+If $b<0$, then $r<\tau$. The subsequent downward step cannot clip at
+the upper boundary. Where $|(X_4)_j-\mu|<1/2$, the preliminary fourth
+coordinate is nonnegative since $\mu\ge1/2$, and the following step
+is at least $-\beta\ge-1$. Thus lower clipping can occur only on a
+fraction at most $4\tau^2$ of the fourth coordinates. Its loss on each
+is at most $\beta$, giving norm loss at most $2\tau\beta$. By
+\eqref{geo:outside-norm} this leaves at least $\beta/4$.
+
+It remains to treat $b\ge0$. Set
+\[
+ \sigma^2=m^{-1}\sum_j((X_4)_j-\mu)^2,\qquad
+ H=\norm{(X-Z)_{1,2,3}},\qquad r^2=H^2+\sigma^2,
+\]
+and choose a positive rational
+\begin{equation}\label{geo:eta-choice}
+ \eta\le\min\left\{\frac\tau2,\frac1{4w_{\max}},
+           \frac12\min\left(\frac12,\frac1{4w_{\max}}\right)\right\}.
+\end{equation}
+If $\sigma\ge\eta$, a uniform fourth coordinate satisfies
+\[
+ \sigma^2\le3\E|(X_4)_j-\mu|
+        =6\E(\mu-(X_4)_j)_+.
+\]
+Put $c=\eta^2/12$. Since the positive part is at most three,
+$\E(\mu-(X_4)_j)_+\ge\eta^2/6$ implies that a fraction at least
+$\eta^2/36$ have $\mu-(X_4)_j\ge c$: otherwise its expectation
+would be less than $c+3\eta^2/36=\eta^2/6$.
+On those coordinates the room to the upper boundary is at least $c$.
+Also $b+wc\ge c/12$. Indeed~\eqref{geo:outside-norm} and $r\le6$
+imply that either $b\ge1/2$ or $w\ge1/12$; $c<1$ handles the first
+alternative. For $\beta\le1$, clipping therefore retains displacement
+at least $\beta c/12$ on that fraction. Its norm is at least
+$\beta\eta^3/864$.
+
+Suppose instead $\sigma<\eta$. If $wH\ge\tau/4$, the horizontal
+displacement, which is not clipped, is at least $\beta\tau/4$.
+Otherwise~\eqref{geo:outside-norm} gives
+\[
+ b^2\ge\frac12-w^2H^2-w^2\sigma^2
+       \ge\frac12-\frac{\tau^2}{16}-w_{\max}^2\eta^2
+       \ge\frac38,
+\]
+so $b\ge1/2$. If $\mu\le1$, let
+$d_0=\min\{1/2,1/(4w_{\max})\}$. By~\eqref{geo:eta-choice}, at
+least three quarters of the fourth coordinates have deviation from
+$\mu$ at most $d_0$. Their upper room is at least $1/2$, and their
+velocity is at least $b-w_{\max}d_0\ge1/4$. They retain at least
+$\beta/4$ for $\beta\le1$, giving norm at least
+$\sqrt3\beta/8>\beta/8$. If $\mu>1$, then $h_\mu=1$, $w\ge1$,
+and $b\ge0$ in~\eqref{geo:outside-coefficients} requires $r\ge\tau$.
+It follows that
+\[
+ wH\ge H\ge\sqrt{\tau^2-\eta^2}\ge\sqrt3\tau/2,
+\]
+contrary to the remaining case assumption. This exhausts all outside
+points, including the upper face. At $(H_*,2)$, $r=0$ and the formula
+gives $v=-\mathbf e$ exactly.
+
+Take positive rational
+$c_*\le\min\{1/8,\tau/4,\eta^3/864\}$ and positive rational
+$\beta_0$ satisfying all restrictions just made. All coordinate
+differences used in the formulas have magnitude at most three, so any
+integer $B_1\ge C_t+6+3/\tau$ satisfies~\eqref{geo:bounded-field},
+including the zero convention near inside endpoints. These choices
+depend only on the previously fixed geometric constants. This completes
+the proof of Theorem~\ref{thm:geometry}.
+\end{proof}
+
+\subsection{The local-access contract}
+\label{geo:access}
+
+We next state the information needed to approximate this particular
+field. All probability statements below are for a fixed physical vector
+$X$, averaged over a uniform physical component $I\in[4m]$ and the
+public trial randomness. Write $J=1+((I-1)\bmod m)$ for its word
+coordinate. Fix a common threshold $\xi\in[2/5,3/5]$ and round the
+first two blocks at $\xi$. The two resulting binary words $y_1,y_2$
+are fixed, although the procedures only read requested coordinates.
+Any scalar-quantization offset may also be fixed throughout the following
+contract and accuracy analysis.
+
+\begin{definition}[Admissible local access]\label{geo:access-contract}
+An \emph{oracle access implementation} satisfies parts (a)--(c) below
+and the following operational condition. At its declared oracle level,
+each supplied bit or decision routine uses a bounded tree of completed
+Boolean answers, apart from public index computations. Every input,
+including an input outside a correctness promise, gives Boolean answers
+or public defaults. After its requested accuracies are fixed, the
+oracle tree and its symbolic query roles are bounded independently of
+$m$ and remain fixed under refinement of threshold-offset grids.
+These conditions impose no constant bound on the physical circuit
+realizing a completed oracle answer. Part (d) is an additional
+\emph{physical-read hypothesis}, required only when a conclusion about
+the fully expanded physical reads is explicitly invoked. All constants
+are uniform in $m$, $X$, and admissible fixed offsets. Geometric and
+validation radii and the reference rule below are fixed before
+reliability and numerical accuracy choices. Operational, sample,
+repetition, and read-density bounds may depend on the requested fixed
+accuracies as well as the encoding construction.
+\begin{enumerate}[label=\textup{(\alph*)},leftmargin=*]
+\item\label{geo:access-validation}
+Each seed has a validation decision whose failure probability can be
+made any prescribed fixed positive number. For fixed $y_s$ there is
+either no valid reference or a single fixed vertex $u_s$ such that, off
+the validation failure event, acceptance identifies $u_s$. This
+reference is independent of the actual random acceptance decision.
+It is also independent of requested confidence and numerical accuracy.
+If $\norm{X_s-E_u}\le5\tau$, off that failure event the seed is
+accepted with reference $u$. Intermediate seeds with a proper reference
+may be accepted or rejected. An accepted seed is used only through
+local routines, not through a globally decoded vertex name.
+\item\label{geo:access-bits}
+For any accepted reference, incident-edge flags can be supplied with
+arbitrarily small fixed error, and bits of its own and its actual
+immediate neighbors' encodings can be supplied with any prescribed
+fixed average error. The same bit guarantee holds for $E_{w_*}$ and
+the encoding of its actual successor, even if neither seed is accepted.
+These special words may be public or virtual.
+
+The bit guarantee is unconditioned for each fixed relevant symbolic
+role and branch: a request at a whole-word coordinate with density at
+most a fixed constant times $1/m$ has the asserted error under its
+given request law. It is not a guarantee conditional on selecting that
+branch. Requests to different word operands may use the same complete
+coordinate. The guarantee is required for the matching coordinate $J$
+and for all preallocated statistical coordinates below. A finite union
+of role or fixed block-type alternatives is allowed.
+\item\label{geo:access-statistics}
+For every finite family of bounded functions $f:[m]\to\mathbb R$
+fixed by $X$, the rounded seeds' possible actual references, and their
+actual candidate data, one can choose fixed sample counts so that each
+sample average estimates $m^{-1}\sum_jf(j)$ to any prescribed fixed
+additive tolerance with any prescribed fixed failure probability.
+The law is again unconditioned under $(I,\text{trial})$, for each fixed
+relevant role alternative. Every individual sampled word coordinate has
+uniform marginal, or the prescribed bounded-density marginal supported
+by the same bit-access guarantee. Corresponding word operands and
+physical blocks are evaluated at matching coordinates.
+
+Independent uniform samples satisfy this requirement. Other sample
+schemes are permitted if their concentration statement holds for the
+fixed underlying functions in this sense; samples and different batches
+need not be independent after averaging over $I$.
+\item\label{geo:access-finite}
+\emph{Additional physical-read hypothesis.} After a fixed accuracy has
+been specified, every supplied routine has a realization with a bounded
+number of primitive physical read answers and bounded Boolean
+processing, apart from public index computations. The bound is
+independent of $m$. On every fixed branch of its complete physical
+answer tree, including branches with wrong earlier answers, the
+unconditioned marginal of each raw address in a physical domain of
+size $M$ is at most $C/M$, with a fixed $C$. A branch that uses public
+data need not make a physical read. These bounds hold
+uniformly in admissible offsets, whose refinement changes comparison
+constants and not address primitives or numbers of hidden-answer slots.
+On each fixed symbolic answer branch, the address rule itself is
+independent of the numerical values of these offsets.
+\end{enumerate}
+\end{definition}
+
+Part~\ref{geo:access-bits} controls accuracy of a completed logical call
+on a fixed valid reference. The optional
+part~\ref{geo:access-finite} additionally controls physical reads on
+every wired branch, whether or not it is accurate or executed. The
+inner implementation of Section~\ref{sec:inner-gadget} satisfies this
+additional hypothesis. The outer construction invokes only the oracle
+contract: its ideal functional answers are subsequently replaced by
+gadgets of polynomial size in the port length. Its fixed direct-read
+count $H_{\rm out}$ and functional-invocation count $B_{\rm out}$,
+followed by Theorem~\ref{thm:port-gadget} and
+Lemma~\ref{out:routing}, account separately for that physical
+substitution. We do not assert a size-independent physical read count
+for the expanded outer construction. Neither application requires
+per-coordinate correctness.
+
+The physical completeness condition in part~\ref{geo:access-validation}
+is obtained from a binary completeness radius $\rho_{\rm comp}>0$ by
+the initial choice
+\begin{equation}\label{geo:rounding-radius}
+ \frac{(5\tau)^2}{(2/5)^2}<\rho_{\rm comp}.
+\end{equation}
+Indeed a rounded error relative to a binary coordinate costs squared
+physical error at least $(2/5)^2$ for every allowed $\xi$. An
+acceptance radius smaller than half the encoding's relative Hamming
+distance makes its reference unique. In the inner application this
+reference must lie on the path for the fixed true input tuple, exactly
+as established by the validation in Section~\ref{sec:inner-gadget}.
+
+\begin{remark}[Whole-word coordinates in the inner application]
+A matching target is a whole encoding coordinate. In a product block
+called $Z$, that coordinate includes both internal labels of the product.
+Using the same whole coordinate in $E_u$ and $E_v$ preserves the supplied
+joint request law; it does not replace the two internal labels of either
+word by a diagonal pair.
+
+For mixed public and variable products in the virtual source, the joint
+density bound gives the marginal bound
+\[
+ p(x,y)\le\frac{C}{|A||B|}
+ \quad\Longrightarrow\quad
+ \sum_{y\in B}p(x,y)\le\frac C{|A|}.
+\]
+The product becomes an additive form on the variable operand. That form
+may depend on the public operand and the discarded coordinate, but it
+does not use that coordinate to select a new variable address.
+Same-positive-tag products retain the whole pair, as required by
+Lemma~\ref{lem:inner-binary-interface}. Lemma~\ref{ig:virtual-source}
+proves this transport from the actual source and first successor to the
+fixed dummy ports. Thus their virtual supply has exactly the
+whole-coordinate access required by the contract.
+\end{remark}
+
+\begin{lemma}[Geometric procedures]\label{lem:geometric-procedures}
+Fix the geometric constants of Theorem~\ref{thm:geometry} and an
+oracle access implementation satisfying parts (a)--(c) and the
+operational condition of Definition~\ref{geo:access-contract}.
+For every fixed $a>0$ there is a finite local numerical schema with
+properties (i)--(iii) below. Property (iv) additionally assumes the
+physical-read hypothesis~\ref{geo:access-finite}.
+\begin{enumerate}[label=\textup{(\roman*)},leftmargin=*]
+\item It returns a signed value $\widehat v_I(X)$ on a fixed dyadic grid
+inside $[-B_1,B_1]$. It terminates and is bounded on every input and
+every trial. For every fixed $X$ with $\dist(X,\mathcal E)>10\tau$,
+and conditional on each fixed admissible binary threshold and
+scalar-quantization offset,
+\begin{equation}\label{geo:average-accuracy}
+ 4\E_{I,\,\mathrm{trial}}
+       |\widehat v_I(X)-v_I(X)|^2\le a^2.
+\end{equation}
+\item At fixed rounded seeds, correct decisions refer to at most eight
+true-word roles and 32 actual piece roles. Meshes, scalar tolerances,
+sample counts, and all statistical and final matching bit requests
+are fixed before choosing the smaller bit-error budgets, and all final
+matching requests are allocated before numerical minimization.
+\item Apart from the supplied routines and public index computations,
+the schema uses finitely many scalar comparisons and bounded Boolean
+lookup. Its only word-coordinate address primitives are the matching
+coordinate and the preallocated statistical coordinates. This remains
+true on every hypothetical wrong or default branch.
+If the oracle-level public spaces, probabilities, routine descriptions,
+and address computations are uniformly enumerable with polynomial
+support, time, and rational-probability bit complexity in a caller's
+size parameter, the composed oracle-level schema has those properties
+as well. This does not include physical expansion of an oracle call.
+\item If in addition the physical-read
+hypothesis~\ref{geo:access-finite} holds, then, once all accuracies and
+implementation counts have been fixed, the fully unfolded physical
+primitive tree has a fixed number $H$ of possible read slots. For each
+physical domain of size $M$, its unconditioned
+expected occurrence count at any fixed position, averaged over $I$
+and public trials, is at most $C/M$, for a fixed sum $C$ of slot
+density bounds. Both $H,C$ remain fixed under subsequent refinement
+of threshold-offset grids. Under the corresponding polynomial
+enumerability premise for these physical realizations, the fully
+expanded physical schema also has the enumerability properties in
+part (iii).
+\end{enumerate}
+Every choice concerns the unscaled field. In particular none depends
+on a subsequently chosen $0<\beta\le\beta_0$.
+\end{lemma}
+
+\subsection{A finite list covering every close piece}
+\label{geo:candidates}
+
+From a seed reference $u$, reserve the encoding roles for $u$, its
+predecessor, and its successor. Reserve its two possible incident edge
+roles. For each actual incident edge list the four retained straight
+segments and its three internal fillets. The trim convention uses the
+same $a_0$ at both outer ends regardless of whether those ends are
+shared, so these seven pieces need only their two endpoint words.
+When both graph edges at $u$ exist, also list the graph-junction fillet
+at $u$. The incoming direction there is the fixed downward third-block
+direction, so the junction needs only $E_u$, the outgoing neighbor,
+and the two flags. Apply this procedure to both seeds.
+
+Always add the special retained vertical and its bottom fillet using
+the distinguished source and its actual successor. These two pieces
+are present even on a trial with no accepted seed. The ordinary
+junction rule requires both graph edges and therefore never invents
+another junction at the source, which has no graph predecessor.
+There is no need to recognize the source by comparing hidden names.
+
+This is the following explicit reservation schedule; duplicate roles
+do no harm.
+\begin{center}
+\begin{tabular}{@{}lrr@{}}
+\toprule
+Reserved roles&Word roles&Piece roles\\
+\midrule
+Self, predecessor, successor for two seeds&6&0\\
+Four incident edges, seven pieces each&0&28\\
+Two ordinary junctions&0&2\\
+Distinguished source and actual successor&2&0\\
+Special vertical and bottom fillet&0&2\\
+\midrule
+Total upper bound&8&32\\
+\bottomrule
+\end{tabular}
+\end{center}
+On good validation and flag answers, the admitted roles are actual
+pieces. A gray acceptance merely chooses whether to include one fixed
+reference's list. For fixed $X,\xi$, all possible underlying actual
+words and pieces in the table are fixed independently of later accuracy
+choices and numerical decisions.
+
+\begin{lemma}[Two-seed coverage]\label{geo:coverage}
+Every non-special piece has either its first or its second block fixed
+to an encoding of a vertex whose seed generates it. Consequently,
+on the strong-completeness and flag good event, every piece at distance
+at most $4\tau$ from $X$ is included in the reserved candidate list.
+\end{lemma}
+\begin{proof}
+For $u\to v$, its first segment, first fillet, and upward toggle have
+first block $E_u$. The fillet from that toggle to the block-$1$
+segment, the block-$1$ segment, and its fillet to the downward toggle
+have second block $E_v$. The downward toggle has both blocks $E_v$.
+An ordinary graph-junction fillet at $u$ changes only blocks $3$ and
+$2$, so its first block is $E_u$. In each case the stated vertex
+generates the piece from its incident data. If the piece is within
+$4\tau$ of $X$, its fixed block is within $4\tau$ of the corresponding
+physical block. Part~\ref{geo:access-validation} of the contract
+therefore accepts that seed with its true reference. Correct flags
+include the piece. The two special pieces are included unconditionally.
+Gray acceptance cannot remove a piece required by this argument; it
+can only add or remove other actual pieces.
+\end{proof}
+
+\subsection{Numerical data and the order of their estimation}
+\label{geo:numerical}
+
+Put a finite parameter mesh, including endpoints, on each straight
+piece and each angle interval $[0,\pi/2]$ of a fillet. All original
+lengths are at most two, so a sufficiently fine fixed mesh gives
+maximum arclength spacing at most any prescribed fixed $D>0$.
+For the proof, a mesh entry means its exact point $Z'$ and its exact
+tangent on its underlying actual piece. The implementation computes
+approximations to these entries.
+
+At most five endpoint-word distance statistics are needed: the four
+ordinary edge roles and the special edge. Each estimates
+\begin{equation}\label{geo:length-statistic}
+ \ell_{uv}^2=m^{-1}\sum_{j=1}^m(E_u(j)-E_v(j))^2.
+\end{equation}
+Use the same coordinate in the two words. The true nonconstant
+lengths are in $[d_*,1]$; clamp their estimated square roots to
+$[d_*/2,2]$. The formulas for trim points, straight tangents, and
+\eqref{geo:fillet}--\eqref{geo:fillet-tangent} then have only bounded
+denominators separated from zero. Their coordinates are uniformly
+continuous functions of the endpoint bits at that coordinate, the
+lengths, and the finitely many fixed trigonometric mesh constants.
+Thus correct endpoint bits and sufficiently accurate length and public
+constant approximations give any desired component error, uniformly
+in the coordinate and in $m$. A component error at most $z$ everywhere
+gives full-vector norm error at most $2z$.
+
+Also estimate the fourth-block mean $\mu$. For each of the at most
+32 meshed roles, estimate each exact mesh point's squared distance by
+the sum of its four block means,
+\begin{equation}\label{geo:mesh-statistic}
+ \norm{X-Z'}^2
+   =\sum_{b=1}^4m^{-1}\sum_j(X_b(j)-Z'_b(j))^2.
+\end{equation}
+Finally reserve the corresponding outside-radius statistic for the
+exact point $Z_\mu$. Physical samples and endpoint-bit samples in a
+summand use the same $j$. One may use separate batches for every
+statistic and alternative role. Before a numerical winner is selected,
+reserve the final matching-coordinate requests for all eight word
+roles, as well as the matching physical coordinate. Inactive requests
+may later be omitted or use public defaults.
+
+These data define the numerical estimator. Use the estimated mean to
+choose the inside or outside prescription, with a fixed rule at equality.
+For the inside prescription, select an admitted mesh entry with minimum
+estimated squared distance, again with a fixed rule for ties. Clamp that
+estimate at zero and take its square root. Substitute this scalar radius,
+the computed tangent component, and the computed component of $Z'-X$
+into the component formula~\eqref{geo:Phi}; its last regime gives the
+default value. The radius is supplied by the statistic, not by a
+full-vector norm computation. For the outside prescription, clamp the
+outside squared-radius estimate at zero and take its square root. Use
+this radius, the estimated mean, and the special matching bits in
+\eqref{geo:outside}. Bounded clamping and final dyadic rounding
+will make this a bounded procedure on every trial. We next prove that
+the estimates are accurate and that the resulting component is stable
+under approximate minimization and side selection.
+
+Here is a precise reason that estimates based on earlier approximate
+lengths still satisfy the required concentration statement. Fix $X$, the
+rounded seeds, and one possible actual reference assignment. The exact
+functions in~\eqref{geo:length-statistic}, $\mu$,
+\eqref{geo:mesh-statistic}, and the outside-radius statistic are fixed
+functions of the sample index; they do not depend on an estimated
+length or a numerical winner. They have bounded ranges: a squared
+physical difference is at most nine, and the sum of four such terms
+is at most 36. Apply part~\ref{geo:access-statistics} to these exact
+functions. On the event of correct sampled bits, earlier good length
+estimates and good numerical constants perturb every implemented
+summand by a uniformly small amount. For example if $|z-z'|\le t$
+and $x,z,z'\in[-2,3]$, then
+\[
+ |(x-z)^2-(x-z')^2|
+   =|z-z'||2x-z-z'|\le10t.
+\]
+Clamps keep all approximations in one such fixed range. The analogous
+bound for replacing a physical read by its quantized representative
+is the same. For the outside point, a good mean estimate changes all
+fourth coordinates by at most its scalar error; the other coordinates
+use special-word bits. It follows that concentration of the exact
+statistics yields accuracy of all implemented statistics after these
+small deterministic perturbations. No assertion conditioned on the
+earlier estimates is being used.
+
+For independent uniform sampling, this concentration needs no external
+input. If $|f|\le M$, independence gives
+$\operatorname{Var}(q^{-1}\sum_{j=1}^q f(K_j))\le M^2/q$.
+Chebyshev's inequality gives failure probability at most
+$M^2/(qz^2)$ at tolerance $z$; a union bound handles the fixed finite
+list. In the outer application the batches are instead stationary
+under the output-coordinate average. The concentration calculation in
+Section~\ref{sec:outer-composition} supplies the same required small
+failure probabilities for these fixed functions. The geometry uses
+that conclusion, and does not condition on a gray acceptance pattern,
+a realized mesh winner, or a particular output coordinate.
+
+\begin{lemma}[Uniform component stability]\label{geo:stability}
+Fix a positive desired component error $e$. Mesh resolution,
+quantization width, scalar tolerances, and precision of public constants
+can be chosen so that, whenever all relevant decisions, exact scalar
+statistics, and requested bits are good, the local procedure's component
+error is at most $e$ at every $X$ outside the endpoint neighborhood.
+The bound holds for every output coordinate, even if the approximate
+winner depends on that coordinate. It also holds with arbitrary tie
+choices between numerically minimizing entries.
+\end{lemma}
+\begin{proof}
+For the inside prescription, consider the selected mesh entry.
+Let $z>0$ bound every squared-distance error, including the parameter,
+bit, quantization, and scalar-statistic errors just discussed. Suppose
+first that $r=\dist(X,\mathcal L)\le3\tau$. By
+Lemma~\ref{geo:coverage}, a minimizing piece is present. A mesh point
+at arclength at most $D$ from its projection has squared excess at most
+$2rD+D^2$. Every admitted exact entry is on the actual curve, so none
+has negative excess. The exact point $Z'$ underlying the approximate
+winner therefore has
+\begin{equation}\label{geo:winner-excess}
+ 0\le g=\norm{X-Z'}^2-r^2\le2rD+D^2+2z.
+\end{equation}
+Choose $D,z$ first small enough that the right side is at most
+$\tau^2$, and then still smaller as required. Equations
+\eqref{geo:point-error}--\eqref{geo:tangent-error} show that both its
+point and tangent approach those of the true projection in infinity
+norm, uniformly. Correct endpoint bits at the final matching coordinate
+also make its computed point and tangent close to their exact values.
+Clamping a squared-radius estimate at zero and taking square root is
+uniformly continuous on the fixed bounded interval. Consequently its
+estimated radius approaches $r$.
+
+Each component of~\eqref{geo:Phi} is a bounded continuous expression
+in that radius, the tangent component, and the components of $Z-X$.
+The formulas match at $\tau,2\tau,3\tau$, and the only reciprocal
+radius is protected by $r\ge\tau$ in its regime. Safe clamping can be
+used at approximate regime boundaries. Uniform continuity on these
+fixed compact ranges proves the desired component error. Crucially,
+the infinity-norm bounds hold for every good candidate satisfying
+\eqref{geo:winner-excess}; they therefore also hold when the chosen
+candidate is correlated with the requested coordinate.
+
+If the true inside radius is $r>3\tau$, the true field is default.
+Every actual candidate entry has distance at least $r$. Good squared
+distance estimates can yield a nondefault computed expression only
+arbitrarily near the $3\tau$ cutoff as $z$ tends to zero. Near that
+cutoff the coefficient of $T$ tends to zero and the coefficient of
+$\mathbf e$ tends to one, uniformly on the clamped coordinate ranges.
+Thus the computed value approaches default. If its radius estimate is
+at least $3\tau$, it is already default. Coverage far from the curve
+is not needed in either case.
+
+For the outside prescription, use the special bits, the estimated mean
+and outside squared radius in~\eqref{geo:outside}. Correct special
+matching bits and good statistics make all its scalar and coordinate
+arguments close to their exact values. The same protected denominators
+and compact-range continuity apply, and $h_\mu$ is continuous. This
+includes the special top, where the exact value is $-\mathbf e$.
+
+Finally consider the choice between the inside and outside computations.
+If it differs from the side of the true mean, a good
+estimate forces $|\mu-1/2|$ to be small. In the interface slab all
+endpoints remain far, and the only possible close piece is the always
+included vertical. The inside computation is therefore valid for the
+inside prescription extended to that slab. The outside computation is
+valid for its extension using the clamped $h_\mu$. By
+Lemma~\ref{geo:interface}, the two target values differ componentwise
+by at most $6|\mu-1/2|$. Adding that error proves stability even with
+the wrong side choice. These cases include exact mean, radius, and
+minimum ties, without requiring a reliable hidden-real comparison at
+zero gap.
+\end{proof}
+
+\subsection{Average accuracy and finite answer trees}
+\label{geo:finite-schema}
+
+\begin{proof}[Proof of Lemma~\ref{lem:geometric-procedures}]
+\paragraph{Oracle accuracy with fixed references.}
+Fix a desired $a>0$. Choose a component error $e>0$ and failure budget
+$p>0$ so that
+\begin{equation}\label{geo:accuracy-budget}
+ 4e^2+16B_1^2p<a^2.
+\end{equation}
+First choose the mesh, scalar tolerances, quantization width, and public
+numerical precision for Lemma~\ref{geo:stability}, with slack for final
+output rounding. Fix scalar sample counts that give its concentration
+events a total failure probability below a chosen part of $p$.
+The eight word roles, 32 piece roles, five possible length statistics,
+mean, mesh-distance statistics, and outside-radius statistic now
+determine a fixed number $N_{\rm call}$ of high-level completed calls. This count
+includes all sampled bit requests and all matching final bit requests
+over the possible relevant true references and flag alternatives.
+
+Make validation and edge-flag errors small enough for their total budget.
+For every remaining high-level bit call require average error smaller
+than the allocated part of $p/N_{\rm call}$. Analyze each call on its possible
+fixed reference before conditioning on acceptance or on the numerical
+winner. Parts~\ref{geo:access-validation}--\ref{geo:access-statistics}
+of the access contract and a union bound then charge the event of
+\emph{any} required bit, decision, or statistical error by at most
+$p$. Independence of these events is unnecessary. A fixed reference
+does not have to be selected on every good trial; the unconditioned
+analysis simply pays for its possible calls in advance. Validation
+failures, including any acceptance with no proper reference, are
+charged separately. The two special-word roles are charged in exactly
+the same way when their supply is virtual.
+
+On the complement of the charged failure event, component error is
+at most $e$ by Lemma~\ref{geo:stability} and the planned final rounding.
+On every outcome clamp the numerical answer to $[-B_1,B_1]$ and round
+to a sufficiently fine fixed dyadic grid there. Detected invalid
+numerical inputs may return any fixed bounded default. Undetected
+incorrect choices are already included in the failure event. The true
+field satisfies~\eqref{geo:bounded-field}, so the component error on
+an arbitrary outcome is at most $2B_1$. Hence
+\[
+ 4\E|\widehat v_I(X)-v_I(X)|^2
+    \le4e^2+16B_1^2p<a^2.
+\]
+This proves~\eqref{geo:average-accuracy} unconditionally over the
+output-coordinate and trial law. It does not assert correctness
+conditioned on a rare selected coordinate or branch. The proof began
+with arbitrary fixed admissible offsets, so it applies to each such
+pair, and therefore also after averaging any chosen laws on them.
+
+\paragraph{Finite oracle implementation.}
+Here are the finite implementation and raw-address details. Fix a small
+rational bin width $h>0$ and compare each physical scalar read to
+thresholds $a+kh+\zeta$ on a fixed interval larger than $[-1,2]$,
+where $\zeta$ is an offset modulo $h$. Clamp at the endpoints.
+Representatives can depend just on the bin number, on a fixed rational
+grid, and are within $O(h)$ of the input for every $\zeta$. This
+error was included in the earlier tolerance choice. Comparisons whose
+answers are forced by the allowed physical range can be public.
+Seed bits use the common threshold $\xi$. Each offset may eventually
+range over a fine finite rational grid. No numerical assertion above
+requires a continuous offset distribution.
+
+There are now only finitely many bits and bin indices in the geometric
+computation. Public rational approximations to the fixed mesh sine and
+cosine values, protected length normalizations, clamped square roots,
+and all other numerical expressions can be tabulated to the chosen
+precision. Bounded Boolean lookup implements the resulting answer
+function. In particular neither an exact curve minimum nor an
+unbounded hidden vertex name or transcript is processed by the local
+Boolean computation. Public index arithmetic and any caller-specific
+field or interpolation calculations are performed when constructing
+the corresponding fixed branch and its wiring.
+
+Wrong validation, flag, or type answers can invoke a reserved symbolic
+routine on an unsuitable seed or select another reserved role. Wrong
+length, radius, side, or minimum answers select a fixed mesh or lookup
+case. None of these answers supplies a new word-coordinate address:
+statistical coordinates remain the preallocated coordinates and final
+data remain at $J$. Inactive or nonsensical roles may use public-zero
+defaults, which require no query to a distinguished physical position.
+Consequently direct scalar reads have the matching or prescribed
+statistical marginals on every fixed hypothetical oracle branch. This
+proves the structural assertion of part (iii), without inspecting the
+physical realization of a supplied oracle answer. A constant number
+of choices from polynomially enumerable oracle-level public spaces
+has polynomial support; a constant product of rational probabilities
+of polynomial bit complexity has polynomial bit complexity. Public
+indices, oracle descriptions, and bounded answer tables retain the
+same polynomial construction bound. This proves the oracle-level
+enumerability assertion of part (iii).
+
+\paragraph{Physical read load under the additional hypothesis.}
+For part (iv), now assume the additional physical-read
+hypothesis~\ref{geo:access-finite}. The supplied routines then have
+its all-branch primitive bounds, even on unsuitable seeds.
+
+Only at this point, after all requested accuracies and the repetitions
+implementing them have been fixed, unfold the full lower-level answer
+tree. Give unused possible nodes their prescribed public random choices
+as well. It has a fixed finite number $H$ of potential primitive slots.
+For a domain of size $M$, let $C_\ell/M$ bound the probability that
+slot $\ell$ reads any specified position, for its fixed symbolic branch
+under the unconditioned public law. A slot not using this domain has
+$C_\ell=0$. Summing gives
+\begin{equation}\label{geo:raw-load}
+ \E_{I,\,\mathrm{trial}}
+       [\text{number of potential reads of position }j]
+ \le\frac{\sum_{\ell=1}^H C_\ell}{M}
+ =\frac C M.
+\end{equation}
+This counts all wired possibilities rather than conditioning on the
+executed path. Omitting an inactive occurrence only reduces its
+unconditioned mass. In particular this argument does not require
+accuracy on wrong branches. The high-level count $N_{\rm call}$ used for the
+correctness union bound preceded amplification; the larger count $H$
+is used for raw load and subsequent gate-failure accounting. It is
+not fed back into the choice of the accuracy that determines $H$.
+
+Refining offset grids changes the values of thresholds for a fixed
+finite set of comparisons. It changes neither the bin width and bin
+count, nor the mesh, nor the symbolic answer-tree slots, nor a
+fixed branch's address primitive. Thus $H$ and $C$ remain uniform
+under that refinement. Under the stated polynomial enumerability
+premise for the physical routines, a constant number of their public
+choices, also at unused tree nodes, again has polynomial support and
+rational-probability bit complexity. Their public addresses and bounded
+answer tables are constructed in polynomial time. This proves the
+physical enumerability assertion of part (iv) and completes the proof.
+\end{proof}
+
+\begin{remark}[Late margins under physical realization]\label{geo:late-margins}
+Assume the additional physical-read
+hypothesis~\ref{geo:access-finite}. Its grid uniformity explains the
+margin order for the fully expanded physical schema in part (iv).
+Once the full number of physical comparison slots is fixed, a
+uniform $K$-point threshold grid on an interval of length $L_0>0$
+satisfies, for every fixed scalar $x$,
+\[
+ \Pr(|x-\xi|\le d)\le2/K+4d/L_0.
+\]
+For an offset grid modulo $h$, the probability of lying within $d$
+of any bin boundary is at most $4/K+8d/h$. To see these bounds,
+intersect the relevant interval of length $2d$ with the equally spaced
+grid; each interval contains at most its length divided by the mesh
+spacing plus two points. Modulo $h$ there are at most two intervals
+after wraparound. These enlarged constants cover the endpoints.
+
+Fix an output coordinate and all nonthreshold public choices on every
+hypothetical branch. Each possible raw address is then fixed under
+the address contract, regardless of the eventual branch. A union bound
+over their fixed finite list, first choosing the offset grids fine
+enough and then a positive rational margin small enough, makes the
+probability of any small-gap comparison as small as prescribed,
+uniformly in $X$. No search over offsets or enlargement of the
+hidden-answer tree is involved. These margin losses are separate from
+\eqref{geo:average-accuracy}; the latter already holds for each fixed
+admissible offset. Occurrence-dependent perturbations within the margin
+are allowed because the full list was charged before following any
+branch. For an application using only the oracle contract, this
+argument does not account for physical comparisons inside substituted
+oracle gadgets. The outer construction uses its own direct-read and
+functional-boundary margin analysis in
+Section~\ref{sec:outer-composition}, together with the port theorem
+and its separate physical substitution count.
+\end{remark}
+
+The geometric procedures are now proved independently of both circuit
+constructions. Together with the port construction in
+Section~\ref{sec:inner-gadget}, they establish the two interfaces used in
+Section~\ref{sec:outer-composition}, completing the proof of
+Theorem~\ref{thm:main}.

preprints/The-PCP-for-PPAD-conjecture-a-quasilinear-reduction-September-25-2026/build/sections/11-weak-nash.tex

@@ -0,0 +1,785 @@
+\section{Consequences for games and markets}\label{sec:weak-nash}
+
+Theorem~\ref{thm:main} supplies the positive failed-gate premise in
+the cited reductions to equilibrium problems. We first put its
+circuits into the forms those reductions use. This preparation, together
+with the finite-witness checks below, makes the transfers precise for
+our gate conventions and complete binary encodings. The applications
+then concern weak and relative Nash equilibria, allocation and market
+equilibria, and stationary equilibria in discounted games. Their
+approximation requirements differ, so we specify each target relation
+before stating its consequence.
+
+\subsection{Circuit preparation for the applications}
+
+The Boolean tolerance in Definition~\ref{def:gcircuit} is nonmonotone.
+We therefore remove Boolean gates before reducing numerical tolerances
+or controlling fan-out. The two stages of the following lemma are used
+separately below.
+
+\begin{lemma}[Common circuit interfaces]\label{lem:application-circuits}
+There are fixed positive rational constants $\eta,\gamma,\tau,\gamma_2$
+and deterministic polynomial-time constructions which, from any
+End-of-Line instance $I$, produce circuits $G_{\rm nb}$ and $G_2$
+using only Constant, Scale, Copy, Add, Subtract, and Less. Their parameters
+lie in $[0,1]$, and every used node has a defining gate. The circuit
+$G_2$ has fan-out at most two.
+
+Every assignment of $G_{\rm nb}$ satisfying all but a $\gamma$
+fraction of its gates at tolerance $\eta$, and every assignment of
+$G_2$ satisfying all but a $\gamma_2$ fraction at tolerance $\tau$,
+can be converted to an assignment of $R(I)$ meeting the tolerances of
+Theorem~\ref{thm:main}. For rational assignments the conversion and the
+resulting End-of-Line decoder take time polynomial in the input length
+plus the total assignment encoding length.
+\end{lemma}
+
+\begin{proof}
+Let $\eps_c,\delta_c$ be the constants in Theorem~\ref{thm:main}, and
+let $G=(V,T)=R(I)$.
+
+\paragraph{Removing Boolean gates.}
+First give every used node a defining gate, if necessary. A node used as
+an input but not defined by a gate can be given the self-copy gate
+$\mathrm{Copy}(v;v)$, which every assignment satisfies. There are at
+most $2|T|$ such nodes, because the original gates have at most $2|T|$
+input positions. Thus the new gate count is at most $3|T|$; reducing
+the permitted failed-gate fraction by a factor three compensates.
+Unused nodes may be omitted and later assigned zero.
+
+On this gate list, use the strong-Boolean relation of Schuldenzucker
+and Seuken at the same tolerance $\eps_c$. Their Proposition~2 shows
+gatewise that a strong-Boolean $\eps_c$ solution satisfies the
+weak-Boolean relations in Definition~\ref{def:gcircuit}, since
+$\eps_c<1/10<1/4$. Their Lemma~1 replaces each Boolean gate by a
+constant-size gadget with no Boolean gates, such that an
+$\eps_c/2$ solution restricts to a strong-Boolean $\eps_c$ solution.
+Their Section~6 records the same reduction with a positive failed-gate
+fraction: the constant-size gadgets lose only a constant factor in
+that fraction~\cite[Proposition~2, Lemma~1, and Section~6]{SchuldenzuckerSeuken2019}.
+The comparison gate in that paper has the opposite orientation from
+$\mathrm{Less}$ here, so we swap its two inputs. The new parameters
+remain bounded rationals.
+
+Consequently there are fixed positive rationals
+$\eta=\eps_c/2$ and $\gamma$ and a polynomial-time constructed circuit
+$G_{\rm nb}$ using only Constant, Scale, Copy, Add, Subtract, and Less
+such that any assignment satisfying its gates at tolerance $\eta$ with
+at most a $\gamma$ fraction of failures yields an assignment of $G$
+satisfying the original $\eps_c,\delta_c$ bounds. All scale factors
+are at most one. This no-Boolean relation is monotone in its numerical
+tolerance. We use that monotonicity only after this conversion; at no
+point is a weak-Boolean gate reclassified at a larger tolerance.
+
+\paragraph{Fan-out reduction with fixed slack.}
+We spell out the precision and failure accounting in the fan-out
+steps of~\cite[proof of Theorem~2, pp.~4--5]{BPR2016}.
+Suppose $G_{\rm nb}$ has $m$ gates, put
+$\gamma_0=\min\{\gamma,1/4\}$, and set
+\[
+ D=\lceil 8/\gamma_0\rceil,\qquad
+ \eta_1=\eta/3,\qquad \gamma_1=\gamma_0/4.
+\]
+There are at most $2m$ outgoing wire occurrences in total, hence at
+most $\gamma_0m/4$ gates with fan-out at least $D$. Delete those gates.
+For each wire from a deleted output into a retained gate, supply an
+independent $\mathrm{Constant}(0)$ gate. The resulting circuit has
+at most $3m$ gates and fan-out less than $D$.
+
+Given an assignment to this circuit, restore each deleted output as
+zero. A failed new constant can spoil only its one receiving gate. A
+new constant satisfied at tolerance $\eta_1$ changes its receiving
+input by at most $\eta_1$ under restoration. For a retained arithmetic
+gate satisfied at tolerance $\eta_1$, the residual after changing its
+two inputs increases by at most $2\eta_1$; this uses the scale bound
+one. For Less, an original input gap greater than $3\eta_1$ remains
+greater than $\eta_1$ after those two changes, so its required output
+also remains valid at tolerance $3\eta_1=\eta$. Therefore, if $f_1$
+gates failed before restoration, at most
+$\gamma_0m/4+f_1$ original gates can fail at tolerance $\eta$.
+The bound $f_1\leq\gamma_1(3m)$ makes this at most $\gamma_0m$.
+This also shows why a smaller numerical tolerance is used when the
+replacement constants are made exact. The large-multiplier step in
+the cited construction is unnecessary for our bounded scale factors.
+
+Next route the $d>2$ outgoing wires of each remaining gate through
+a balanced binary tree of Copy gates. One may use at most $2d$ new
+gates and depth at most
+\[
+ h=\lceil\log_2 D\rceil+1.
+\]
+If the preceding circuit has $m_1$ gates, there are at most $2m_1$
+wire occurrences, so the new fan-out-two circuit has at most $5m_1$
+gates. Use the fixed parameters
+\[
+ \tau=\frac{\eta_1}{2(2h+1)},\qquad
+ \gamma_2=\frac{\gamma_1}{5D}.
+\]
+When every Copy gate on a route is $\tau$-satisfied, its leaf differs
+from the root by at most $h\tau$. The same arithmetic and comparison
+check shows that a retained gate satisfied at tolerance $\tau$
+becomes satisfied at tolerance at most $(2h+1)\tau<\eta_1$ when its
+inputs are restored to the roots. A failed Copy gate has fewer than
+$D$ descendant receiving gates. Charging these receivers, and charging
+each failed retained gate to itself, loses at most $D$ retained gates
+per failed gate. Thus at most a $\gamma_2$ fraction of failures in
+the new circuit produces at most a $\gamma_1$ fraction in the preceding
+one. All depths, blowups, and tolerance losses are fixed independently
+of $I$.
+
+Denote the final fan-out-two circuit by $G_2$. All instance maps have
+polynomial bit cost: they copy bounded rational parameters and introduce
+only fixed rational constants. Their solution maps restrict assignments,
+restore zero values, or copy existing coordinates, so they also have
+polynomial bit cost on rational inputs.
+
+The resulting assignment of $G$ need not obey the particular witness
+polynomial $p(M)$ in Definition~\ref{def:gcircuit}. This causes no
+restriction on the decoder. The argument after
+\eqref{out:endpoint-proximity} proves soundness for every real assignment
+meeting the gate-failure budget, and gives a deterministic decoder for
+rational assignments in time polynomial in $N$ plus their total encoding
+length. Applying it proves the stated interface.
+\end{proof}
+
+\subsection{Weak approximate Nash equilibria}
+
+Babichenko, Papadimitriou, and Rubinstein reduce positive-gap generalized
+circuits to weak approximate Nash search, where a fixed fraction of
+players may have deviation gain larger than a fixed positive
+error~\cite[Definition~4 and Theorem~2]{BPR2016}.
+Lemma~\ref{lem:application-circuits} supplies the required circuit
+interface. We retain the normalization and finite-witness details of the
+transfer.
+
+An $n$-player two-action polymatrix game is specified by an undirected
+interaction graph and, for each edge, a pair of rational $2$-by-$2$
+payoff matrices. Each player chooses action $0$ or $1$ and receives the
+sum of its payoffs on incident edges. We use matrices with entries in
+$[0,1/3]$ and graphs of maximum degree at most three, so every total
+payoff lies in $[0,1]$. The encoding explicitly lists all players,
+edges, and matrices, so its length is at least $n$. For a product
+mixed-strategy profile $\sigma$,
+write
+\[
+ \operatorname{reg}_i(\sigma)
+ =\max_{a\in\{0,1\}}u_i(a,\sigma_{-i})-u_i(\sigma)
+\]
+for player $i$'s possible gain from deviation. A rational profile is
+encoded by the binary rational probabilities $\sigma_i(1)\in[0,1]$.
+
+\begin{corollary}[Fixed-gap weak approximate Nash hardness]
+\label{cor:weak-nash}
+There are fixed positive rational constants $\eps_{\rm w},\delta_{\rm w}$
+and a fixed polynomial $p_{\rm w}$ for which the following search problem
+is PPAD-hard under deterministic polynomial-time many-one search
+reductions. Given an explicitly encoded rational two-action polymatrix
+game of maximum degree at most three with the normalization above and
+complete binary length $L$, find a rational product profile of total
+encoding length at most $p_{\rm w}(L)$ such that
+\begin{equation}\label{eq:weak-nash}
+ \bigl|\{i:\operatorname{reg}_i(\sigma)>\eps_{\rm w}\}\bigr|
+ \leq \lfloor\delta_{\rm w}n\rfloor .
+\end{equation}
+No further regret bound is imposed on the players counted on the left.
+\end{corollary}
+
+\begin{proof}
+Use the fan-out-two circuit of Lemma~\ref{lem:application-circuits},
+with numerical tolerance $\tau$ and failure allowance $\gamma_2$.
+
+\paragraph{The game reduction.}
+For a fan-out-two generalized circuit, the gate construction in
+Rubinstein~\cite[Section~7]{Rubinstein2018}, used in
+the forward reduction of~\cite[Theorem~2]{BPR2016}, has two actions per
+player and maximum degree at most three. If the players of a gate
+gadget use only actions within $\xi_0$ of a best response, their node
+probabilities satisfy that gate at tolerance $2\xi_0$. Choose a fixed
+positive rational $\xi_0<\tau/2$. These bounds imply our no-Boolean
+gate relations at tolerance $\tau$: an antecedent for Less at
+tolerance $\tau$ is also an antecedent at tolerance $2\xi_0$, and
+the arithmetic and output bounds are stronger. The scale gadget's
+output $\min\{\alpha x,1\}$ equals $\alpha x$ here because $\alpha\leq1$.
+Each gate uses a fixed number of players, and a player belongs to only
+a fixed number of gate gadgets.
+Hence there are fixed integers $C_1,C_2$ such that the game has at most
+$C_1$ times as many players as circuit gates and each player not
+meeting the well-supported condition can spoil at most $C_2$ gates.
+Choose a fixed positive rational
+\[
+ \delta_{\rm w}\leq\frac{\gamma_2}{C_1C_2}.
+\]
+The decoded circuit assignment then meets its failure budget whenever
+all but a $\delta_{\rm w}$ fraction of players meet that condition.
+
+The gadget matrices have rational entries bounded in absolute value
+by a fixed rational $B\geq1$: their variable scale parameters lie in
+$[0,1]$, and all other gadget constants are fixed. Replacing every
+payoff entry $a$ by $(a+B)/(6B)$ puts the entries in $[0,1/3]$.
+For each player this adds an action-independent constant and scales
+every deviation gain by $\lambda=1/(6B)$. Put $\xi=\lambda\xi_0$.
+Thus the needed well-supported tolerance in the normalized game is
+the fixed positive rational $\xi$.
+
+Choose a fixed rational $t>0$ such that
+$t(t+13)<\xi$, and put $\eps_{\rm w}=t^2$. We use the local pruning
+argument of Rubinstein~\cite[Lemma~5]{Rubinstein2018}; this is the lemma
+cited with an earlier number in the proof of BPR's Theorem~2. For every
+player whose regret is at most $t^2$, remove actions whose payoff is
+more than $(1+1/t)t^2$ below its best response and renormalize.
+Leave the other players unchanged. The regret bound ensures that the
+removed probability is less than one. The lemma's calculation, with
+incoming degree at most three and total payoffs in $[0,1]$, shows that
+every retained action of each formerly happy player is within
+\[
+ t(t+1+4\cdot3)=t(t+13)<\xi
+\]
+of a best response after all these changes. Unchanged unhappy
+neighbors contribute no change to that calculation. The fraction of
+players not meeting the well-supported condition therefore does not
+increase. This gives the required circuit assignment from any profile
+satisfying~\eqref{eq:weak-nash}.
+
+\paragraph{Rational witnesses and decoding.}
+All instance maps above have polynomial bit cost. The auxiliary
+constants are fixed rationals, and the game matrices copy bounded
+rational gate parameters through fixed-size gadgets. For a rational
+profile, its regrets require only a fixed number of rational operations
+per player. The pruning threshold is rational, so exact comparisons,
+deletion of actions, and renormalization have polynomial bit cost.
+The extracted node probabilities, restrictions, and restored zero
+values consequently form a rational assignment of polynomial total
+encoding length.
+
+Lemma~\ref{lem:application-circuits} decodes this assignment in its
+total encoding length, so the particular witness polynomial $p(M)$ of
+Definition~\ref{def:gcircuit} imposes no additional condition here.
+
+Finally, the finite target search relation has a polynomial witness
+bound. Round the probabilities of a mixed Nash equilibrium to a grid
+of denominator $q$, where the fixed integer $q$ is greater than
+$3/\eps_{\rm w}$. With entries in $[0,1/3]$ and degree at most three,
+each pure-action payoff changes by at most $1/q$ when the neighboring
+probabilities are rounded. Changing the player's own probability
+changes its payoff by at most another $1/q$. Its resulting regret is
+therefore at most $3/q<\eps_{\rm w}$. A fixed polynomial $p_{\rm w}$
+can encode these $n$ rational probabilities. Satisfaction of
+\eqref{eq:weak-nash} is checked by rational arithmetic in polynomial
+time. The composed instance map and solution decoder establish the
+stated search reduction.
+\end{proof}
+
+
+\subsection{Relative approximate bimatrix equilibria}
+
+For a rational bimatrix game $(A,B)$ with $A\geq0$ and $B\leq0$
+entrywise, a pair of mixed strategies $(x,y)$ is a relative
+$\eps_{\rm r}$-approximate Nash equilibrium if
+\begin{equation}\label{eq:relative-nash}
+ x^{\mathsf T}Ay\geq(1-\eps_{\rm r})\max_i(Ay)_i,
+ \qquad
+ x^{\mathsf T}By\geq(1+\eps_{\rm r})\max_j(x^{\mathsf T}B)_j.
+\end{equation}
+The plus sign in the second inequality reflects the column player's
+nonpositive payoffs: each player's loss is at most $\eps_{\rm r}$
+times the absolute value of its best-response payoff. These are
+conditions on expected payoffs, not on every action in the support.
+
+\begin{corollary}[Relative approximate Nash hardness]
+\label{cor:relative-nash}
+There are a fixed rational $0<\eps_{\rm r}<1$ and a fixed polynomial
+$p_{\rm r}$ such that the following search problem is PPAD-complete
+under deterministic polynomial-time many-one search reductions.
+Given an explicitly encoded rational bimatrix game $(A,B)$ with
+$A\geq0$, $B\leq0$, and complete binary length $L$, find a rational
+mixed-strategy profile of total encoding length at most $p_{\rm r}(L)$
+satisfying~\eqref{eq:relative-nash}.
+\end{corollary}
+
+\begin{proof}
+Apply the implication from Conjecture~2 to Theorem~3 in
+\cite[Section~4]{BPR2016}. This published chain passes through a
+bipartite degree-three family of weak approximate Nash instances.
+Theorem~\ref{thm:main} supplies its fixed-tolerance circuit premise on
+a bounded-rational subclass, and Lemma~\ref{lem:application-circuits}
+makes the gate-basis and fan-out preparation explicit. No change of
+weak-Boolean tolerance is needed. The remaining polynomial reduction
+gives hardness for a fixed positive relative error;
+choosing a smaller positive rational error only strengthens the
+requirements. Its marginal and conditional-probability decoder uses
+polynomial-time rational arithmetic. Composing with the circuit
+decoder after~\eqref{out:endpoint-proximity}, which is polynomial in
+the total rational assignment length, gives the required search
+reduction.
+
+For membership, an exact Nash equilibrium of the same game satisfies
+\eqref{eq:relative-nash}. Exact rational bimatrix Nash belongs to
+PPAD, and every rational bimatrix game has an exact equilibrium of
+polynomial encoding length~\cite[Theorem~1.1 and Section~2]{CDT2007Preprint}.
+Choose $p_{\rm r}$ to accommodate that bound. The displayed inequalities
+are checked by polynomial-time rational arithmetic, so the stated
+finite search problem is total and belongs to PPAD.
+\end{proof}
+
+
+\subsection{Indivisible course allocation}
+
+We also apply the course-allocation implication of Babichenko,
+Papadimitriou, and Rubinstein~\cite[Theorem~4 and Appendix~B]{BPR2016}.
+There are $M\geq1$ courses with nonnegative integer capacities $q_j$
+and $N\geq1$ students. Each student's acceptable bundles are explicitly
+listed in strict preference order, with the empty bundle as a final,
+always-available choice; let $k$ be the maximum bundle size.
+At prices $p_j\geq0$ and budgets $b_i>0$, each student receives its
+most-preferred bundle $x_i\in\{0,1\}^M$ satisfying $p\cdot x_i\leq b_i$.
+Budish's clearing convention and guaranteed threshold are
+\cite[Definition~1 and Theorem~1]{Budish2011}
+\[
+ z_j=\begin{cases}
+ \sum_i x_{ij}-q_j,&p_j>0,\\
+ \max\{\sum_i x_{ij}-q_j,0\},&p_j=0,
+ \end{cases}
+ \qquad
+ \alpha_* = \frac{\sqrt{M\min\{2k,M\}}}{2}.
+\]
+For positive budgets, their Gini coefficient is
+\[
+ \operatorname{Gini}(b)
+ =\frac{\sum_{i,h}|b_i-b_h|}{2N\sum_i b_i}.
+\]
+
+\begin{corollary}[Low-Gini course-allocation hardness]
+\label{cor:course-allocation}
+There are a fixed rational $0<\gamma_{\rm c}<1/2$ and a fixed
+polynomial $p_{\rm c}$ such that the following search problem is
+PPAD-complete under deterministic polynomial-time many-one search
+reductions. Given the explicit lists and capacities above, of total
+binary length $L$, find rational prices and positive rational budgets,
+together with the demanded bundles, of total encoding length at most
+$p_{\rm c}(L)$, satisfying
+\[
+ \norm{z}_2\leq\alpha_*,\qquad
+ \operatorname{Gini}(b)\leq\gamma_{\rm c}.
+\]
+\end{corollary}
+
+\begin{proof}
+Use the no-Boolean, fan-out-two circuits of
+Lemma~\ref{lem:application-circuits}, with their fixed positive numerical
+and failed-gate tolerances. The published reduction in
+\cite[Appendix~B.2--B.3]{BPR2016} applies to this bounded-rational
+subclass: after common normalization of prices and budgets, any
+invalid circuit decoding forces the Gini coefficient to be at least a fixed
+positive constant. Choose $\gamma_{\rm c}$ strictly below that
+bound. Median normalization and price extraction have polynomial bit
+cost on rational outputs. Composing with the circuit decoder after
+\eqref{out:endpoint-proximity} therefore gives the hardness reduction,
+with time polynomial in the input and total witness length.
+
+For membership, Othman, Papadimitriou, and
+Rubinstein~\cite[Appendix~A, Remarks~1--2]{OPR2014} prove that computing
+an allocation at this clearing threshold with budgets in
+$[1,1+\beta]$ belongs to PPAD in the explicit-list model, for every
+fixed rational $\beta>0$. Choose a rational $0<\beta<\gamma_{\rm c}$.
+Such budgets have Gini coefficient at most $\beta/2<\gamma_{\rm c}$;
+existence is guaranteed by~\cite[Theorem~1]{Budish2011}.
+The budget interval selects solutions for membership and is not an
+additional requirement of the target problem.
+
+Polynomial-size rational witnesses can also be seen directly.
+Fix such an allocation and its zero-price pattern, and cap prices at
+$2+\beta$ without changing demands. Keep the budget interval, these
+price bounds and affordability of the chosen bundles. Introduce a
+common slack $s\in[0,1]$ for the unaffordability of every strictly
+preferred listed bundle and for the positivity of every positive
+price. Maximizing $s$ over the resulting bounded rational polytope
+has a positive optimum and a rational vertex optimum of polynomial
+bit length. Demands and the price-sign pattern, hence clearing error,
+are preserved. All target conditions are checked by rational
+arithmetic and enumeration of the explicit lists; the clearing test
+is equivalently $4\sum_jz_j^2\leq M\min\{2k,M\}$.
+\end{proof}
+
+
+\subsection{Markets with approximately optimal bundles}
+
+The market reduction of Deligkas, Fearnley, Hollender, and
+Melissourgos~\cite{Fisher2026} uses a ternary circuit intermediary.
+We record it first; it will also supply the input to the single-controller
+game reduction below.
+
+A Pure-Circuit instance consists of explicit finite lists of nodes and
+gates. An assignment $z$ gives each node a value in $\{0,1,\bot\}$.
+A gate $\mathrm{NAND}(u,v;w)$ requires $z_w=0$ if $z_u=z_v=1$,
+and $z_w=1$ if either input is zero; otherwise it imposes no condition.
+A gate $\mathrm{PURIFY}(u;v,w)$ requires at least one of $z_v,z_w$
+to lie in $\{0,1\}$, and requires $z_v=z_w=z_u$ whenever
+$z_u\in\{0,1\}$.
+Each gate uses three distinct nodes, and each node is the output of
+exactly one gate. We use the version in which each node is also an input
+to exactly one gate~\cite[Section~2.2.1]{Fisher2026}.
+
+\begin{lemma}[The input-once Pure-Circuit interface]
+\label{lem:application-pure-circuit}
+There are a fixed positive rational $\delta_F$ and a deterministic
+polynomial-time reduction from End-of-Line to NAND/PURIFY circuits of
+the form just defined. Every ternary assignment satisfying all but a
+$\delta_F$ fraction of the gates yields an End-of-Line solution in
+polynomial time.
+\end{lemma}
+\begin{proof}
+Apply \cite[Theorem~2.2 and Lemma~A.2]{Fisher2026} to the
+no-Boolean, fan-out-two circuits of
+Lemma~\ref{lem:application-circuits}. The published bridge gives a fixed positive failure allowance and makes
+every node an input to exactly one gate. Its solution decoder returns
+rational unary-count values, to which the decoder in
+Lemma~\ref{lem:application-circuits} applies. All comparisons and
+arithmetic have polynomial bit cost. No change of tolerance for a
+weak-Boolean gate is required.
+\end{proof}
+
+We now define the simple market families used in the reduction.
+There is one unit of each good, and utilities have the separable form
+\[
+ u_i(x_i)=\sum_j\min\{a_{ij}x_{ij},b_{ij}\},
+ \qquad a_{ij}\geq0,\quad b_{ij}\in[0,\infty].
+\]
+All finite data are rational and explicitly encoded. Each buyer has
+nonzero utility for only a bounded number of goods, each good has
+nonzero utility for only a bounded number of buyers, and the ratio of
+nonzero slopes for each buyer is bounded. These bounds are constants
+for the family. Every buyer has at least one uncapped term of positive
+slope. Fisher buyers have budget one. In the exchange version, every
+buyer initially owns $1/|B|$ units of every good, where $B$ is the set
+of buyers; in particular the endowments are identical and the economy
+is strongly connected.
+
+For errors $\eps_{\rm m},\delta_{\rm m}>0$, an approximate equilibrium
+consists of prices $p_j\geq0$ and bundles $x_{ij}\geq0$. Each bundle
+must be affordable and give its buyer at least $1-\delta_{\rm m}$
+times the maximum utility attainable within that budget; each good
+satisfies $\abs{\sum_i x_{ij}-1}\leq\eps_{\rm m}$. In the exchange
+case the budget is the value of the buyer's endowment at $p$.
+Affordability is exact. Outputs are rational prices and bundles of
+polynomial encoding length; the rational-witness formulation is
+discussed in~\cite[Section~2.1]{Fisher2026}.
+
+\begin{corollary}[Approximate market equilibrium hardness]
+\label{cor:market-hardness}
+For every fixed $0<\eps_{\rm m}<1/9$, there is a fixed
+rational $0<\delta_{\rm m}<1$ such that finding an
+$(\eps_{\rm m},\delta_{\rm m})$-approximate equilibrium in the simple
+Fisher markets above is PPAD-hard under deterministic polynomial-time
+many-one search reductions. The same holds for their exchange-market
+versions with identical endowments and strongly connected economies.
+\end{corollary}
+
+\begin{proof}
+Use the circuits of Lemma~\ref{lem:application-pure-circuit}.
+For each fixed $\eps_{\rm m}<1/9$,
+\cite[Proposition~4.4]{Fisher2026} supplies a fixed positive $\delta_{\rm m}$ and a polynomial-time
+reduction to the stated Fisher markets. Composing its solution decoder
+with the circuit decoders proves the Fisher assertion of
+\cite[Theorem~4.1]{Fisher2026} without its conjectural premise.
+For rational $\eps_{\rm m}$, the other constants may be chosen rational
+within the reduction's strict margins. For a nonrational fixed
+$\eps_{\rm m}$, choose a rational clearing tolerance strictly between
+$\eps_{\rm m}$ and $1/9$ and use its reduction: every solution with
+the smaller clearing error also satisfies the larger one.
+The Pure-Circuit bridge returns rational unary-count values;
+for rational market witnesses, all comparisons and decoding have
+polynomial bit cost. The decoder after~\eqref{out:endpoint-proximity}
+applies in the total length of the resulting rational assignment,
+without requiring the particular bound $p(M)$ in
+Definition~\ref{def:gcircuit}.
+
+For the exchange markets, normalize prices so that
+$\sum_jp_j=|B|$. Every buyer's endowment then has value one, so the
+same prices and bundles give a Fisher equilibrium with precisely the
+same two errors. The positive-slope uncapped terms ensure that the
+price sum is nonzero at any approximate equilibrium. This is the
+transfer in~\cite[Appendix~B and Theorem~B.1]{Fisher2026}; normalization
+also preserves polynomial rational encoding length.
+\end{proof}
+
+\subsection{Stationary equilibria in discounted games}
+
+Daskalakis, Golowich, and Zhang~\cite{DGZ2023} obtain conditional
+hardness for stationary equilibria whose incentive constraints are
+averaged over an initial distribution. Their premise also follows
+from Theorem~\ref{thm:main}.
+
+A finite two-player discounted stochastic game has explicitly encoded
+rational transition probabilities, rewards, and initial distribution
+$\mu$. A stationary joint policy $\pi$ assigns a distribution on action
+pairs to each state, sampled afresh at every visit. Write $\pi_{-i}$
+for the other player's marginal policy and use normalized values
+\[
+ V_i^\pi(\mu)=(1-\gamma)\E_{\mu,\pi}
+                 \sum_{t\geq0}\gamma^t r_i(s_t,a_t).
+\]
+The policy is an $\eps$-stationary Markov coarse correlated equilibrium
+(CCE) when
+\[
+ \sup_{\sigma_i}
+ \bigl(V_i^{\sigma_i\times\pi_{-i}}(\mu)-V_i^\pi(\mu)\bigr)
+ \leq\eps\qquad(i=1,2),
+\]
+where $\sigma_i$ ranges over stationary policies chosen independently
+of any action recommendation. These constraints use the supplied
+$\mu$, rather than requiring the bound at every state. An
+$\eps$-stationary Nash equilibrium has the same supplied-$\mu$
+incentive bound and additionally requires $\pi$ to be a product policy.
+
+\begin{corollary}[Stationary stochastic-game hardness]
+\label{cor:stationary-game-hardness}
+For some fixed rational $\eps_{\rm s}>0$, finding an
+$\eps_{\rm s}$-stationary Markov CCE is PPAD-hard under deterministic
+polynomial-time many-one search reductions, even for two-player
+turn-based general-sum games with discount $\gamma=1/2$, rewards in
+$[-1,1]$, and at most two actions for the active player at each state.
+The inactive player's action affects neither rewards nor transitions.
+The same hardness holds for $\eps_{\rm s}$-stationary Nash equilibria
+with the same supplied-initial-distribution convention. Outputs are
+rational policy tables of polynomial encoding length.
+\end{corollary}
+
+\begin{proof}
+Use the no-Boolean circuit $G_{\rm nb}$ of
+Lemma~\ref{lem:application-circuits}, with its fixed positive rational
+tolerance $\eta$ and failure allowance $\delta_{\rm nb}$ (the constant
+$\gamma$ of that lemma). The target basis in
+\cite[Definition~8 and Appendix~D]{DGZ2023} consists of exact zero/one
+constants, clipped linear gates, and comparison gates. For Scale and
+Copy, repeat the input with coefficient pairs $(\alpha/2,\alpha/2)$
+and $(1/2,1/2)$, respectively; Add and Subtract use $(1,1)$ and
+$(1,-1)$. A Constant$(\alpha)$ uses its own exact-one node followed
+by the scaled linear gate. Retain Less: each strict source comparison
+antecedent implies the corresponding weak target antecedent, and the
+target output error is smaller. Choose a positive rational target
+tolerance below $\min\{\eta,1/12\}$. Each original gate uses at most
+two target gates, and each failed target gate can spoil only its own
+original gate. A target failure fraction at most $\delta_{\rm nb}/2$ therefore
+suffices. This proves the positive-gap premise in
+\cite[Conjecture~10]{DGZ2023}; no weak-Boolean tolerance is changed.
+
+Apply the second assertion of~\cite[Theorem~4]{DGZ2023}. Its turn-based
+construction has the stated discount, reward bounds and binary
+choices. Taking statewise marginals of a joint policy in such a game
+gives a product policy with the same values and unilateral deviation
+gains, so its approximate stationary CCE decodes to an approximate
+stationary Nash equilibrium with the same error. Choose a smaller
+positive rational error if needed.
+The fixed reduction parameters can be chosen rational with slack.
+For rational policy tables, policy evaluation solves rational linear
+systems, and the gate decoder and restrictions have polynomial bit
+cost. The decoder after~\eqref{out:endpoint-proximity} then applies in
+the total rational assignment length. Polynomial-size rational
+approximate policy tables exist by rounding stationary Nash policies
+at this fixed discount, as in~\cite[Section~2.5]{DGZ2023}.
+\end{proof}
+
+\subsection{Restricted stationary-game families}
+
+The preceding consequence also supplies the input for the triangular
+zero-sum construction of Park, Zhang, and Ozdaglar~\cite{PZO2025}.
+Here the interaction graph concerns the whole auxiliary function,
+not just immediate rewards: for every $V:S\to\mathbb R$, each
+\[
+ Q_i^V(s,a)=r_i(s,a)+\gamma\sum_{s'}P(s'\mid s,a)V(s')
+\]
+must be a sum of functions of $(s,a_i,a_j)$ over neighbors $j$ of $i$.
+This is the Q-function separability of~\cite[Definition~1]{PZO2025}.
+Use the same normalized values and supplied-initial-distribution CCE
+inequalities as above, now for all three players, with $\pi_{-i}$ the
+joint marginal of the other two players.
+
+\begin{corollary}[Triangular zero-sum stationary-game hardness]
+\label{cor:triangle-stationary-hardness}
+For some fixed rational $\eps_{\triangle}>0$, finding an
+$\eps_{\triangle}$-stationary Markov CCE is PPAD-hard under deterministic
+polynomial-time many-one search reductions even for rational three-player
+zero-sum games with Q-function-separable interactions on the fixed
+triangle $K_3$, discount $1/2$, and each player's stage reward in $[-1,1]$.
+The hard games are turn-based, with at most two actions for the active
+player and a third player with only one action. Outputs are rational
+joint-policy tables of polynomial encoding length; incentives are
+required only from the supplied rational initial distribution.
+\end{corollary}
+
+\begin{proof}
+Apply the triangle construction in~\cite[Appendix~C, Case~1]{PZO2025}
+to Corollary~\ref{cor:stationary-game-hardness}.
+Write $c(s)\in\{1,2\}$ for the state controller and suppress inactive
+actions. Add a one-action third player and retain states, transitions
+and initial distribution. Specify oriented edge rewards before scaling by
+\[
+ (b_{12},b_{13},b_{23})=
+ \begin{cases}
+ (-r_2,r_1+r_2,0),&c(s)=1,\\
+ (r_1,0,r_1+r_2),&c(s)=2,
+ \end{cases}
+ \qquad b_{ji}=-b_{ij}.
+\]
+Each term depends only on the actions at its edge. Divide every edge
+reward by two. The resulting total rewards are
+$(r_1/2,r_2/2,-(r_1+r_2)/2)$, lie in $[-1,1]$, and sum to zero.
+For every test function $V$, the continuation term in $Q_i^V$ depends
+only on $a_{c(s)}$. Allocate it to edge $\{i,c(s)\}$ when $i\ne c(s)$,
+and to any incident edge when $i=c(s)$. This proves the required
+separability on $K_3$ for every player and every $V$.
+
+The state process of any new-game policy agrees with that of its
+first-two-player restriction in the original game. Each original
+player's normalized value and deviation gain in the new game is half
+the corresponding quantity in the original game; the dummy has no
+deviation. Thus error
+$\eps_{\triangle}=\eps_{\rm s}/2$ suffices at the unchanged initial
+distribution. The primary source uses unnormalized values, which are
+twice our normalized values at discount $1/2$; the stated error
+calculation uses our convention throughout. All new data use only
+rational addition and halving, and restriction has polynomial bit cost.
+Polynomial-size rational witnesses follow by the same fixed-discount
+rounding argument as above.
+\end{proof}
+
+The single-controller construction of Farina, Kontogiannis, Panageas,
+and Pollatos~\cite{FKPP2026} gives a different restricted family.
+One player controls transitions at every state, while both players'
+actions may affect rewards. Joint policies need not be product policies.
+
+\begin{corollary}[Single-controller stationary-game hardness]
+\label{cor:single-controller-hardness}
+For some fixed rational $\eps_{\rm sc}>0$, finding an
+$\eps_{\rm sc}$-stationary Markov CCE is PPAD-hard under deterministic
+polynomial-time many-one search reductions for rational two-player
+single-controller games with discount $1/16$, rewards in $[0,1]$,
+and at most two actions per player per state. In the hard instances,
+the initial law is uniform on the circuit-output variable states of
+the cited construction and gives no mass to its absorbing sinks.
+Outputs are rational joint-policy tables of polynomial encoding length.
+The same assertion, with the same error, holds for every fixed rational
+$\gamma\in[1/16,1)$.
+\end{corollary}
+
+\begin{proof}
+Lemma~\ref{lem:application-pure-circuit} supplies input-once
+NAND/PURIFY circuits with a fixed positive failed-gate allowance
+$\delta_F$. Convert each NAND$(u,v;w)$ to
+\[
+ \mathrm{NOT}(u;a),\qquad \mathrm{NOT}(v;b),\qquad
+ \mathrm{OR}(a,b;w),
+\]
+using fresh private nodes $a,b$, and keep PURIFY unchanged.
+These are the ternary relations on $\{0,1,\bot\}$
+of~\cite[Definition~2.6]{FKPP2026}: if $u=v=1$, the NOT outputs are
+both zero and OR forces $w=0$; if either input is zero, its NOT output
+is one and OR forces $w=1$, even when the other input is $\bot$.
+In all other cases NAND imposes no condition. PURIFY has the same
+relation in both bases.
+Every node remains the input of exactly one gate, and outputs remain
+uniquely defined. An original failed gate charges a failure in its own
+disjoint replacement block. Since $m$ original gates give at most $3m$
+new gates, a failed-gate allowance $\delta_F/3$ suffices. No numerical
+tolerance of a weak-Boolean gate is changed.
+
+This establishes the premise of~\cite[Conjecture~1]{FKPP2026}.
+Apply its Theorem~5.5, choosing the fixed reduction parameters and a
+positive rational equilibrium error within the strict bounds in its
+equations~(23)--(24). Its states consist of circuit-variable states
+and constantly many absorbing sinks; the initial law is uniform
+precisely on the output-variable states. For a rational joint policy,
+equation~(8) of that paper decodes a node from the reward player's
+marginal $p$: output zero for $p\leq1/4$, one for $p\geq3/4$, and
+$\bot$ otherwise. Marginalization, comparison and restriction have
+polynomial bit cost. Follow these by the Fisher unary-count decoder
+and the decoder after~\eqref{out:endpoint-proximity}.
+At fixed discount, rounding stationary equilibrium policies with
+slack gives polynomial-size rational witnesses; policy evaluation and
+best-response optimization are rational finite-state discounted MDP
+computations. No collapse of correlated policies to Nash policies is
+used in this reduction.
+
+Finally, the extension in~\cite[Remark~5.7]{FKPP2026} preserves the
+normalized incentive inequalities exactly. Put $\beta=1/16$ and, for
+a fixed rational $\gamma\in[\beta,1)$, set
+\[
+ \tau=\frac{\beta(1-\gamma)}{\gamma(1-\beta)},\qquad
+ \widetilde P=(1-\tau)I+\tau P.
+\]
+For every stationary joint or unilaterally deviated policy $\pi$,
+\[
+ I-\gamma\widetilde P^\pi
+ =\frac{1-\gamma}{1-\beta}(I-\beta P^\pi).
+\]
+Thus $(1-\gamma)(I-\gamma\widetilde P^\pi)^{-1}r_i^\pi$
+equals $(1-\beta)(I-\beta P^\pi)^{-1}r_i^\pi$.
+Rewards, actions and initial law are unchanged, and the lazy transition
+remains rational and single-controller with polynomial encoding length.
+\end{proof}
+
+\subsection{Hylland--Zeckhauser equilibria}
+
+The reduction of Braverman, Liu, Xue, and Zhou~\cite{HZ2026}
+also yields constant-error hardness for one-sided allocation with
+unit demand. An instance has $n$ agents, $n$ unit-supply goods, and
+an explicitly encoded rational utility matrix $u\in[0,1]^{n\times n}$.
+In their approximate Hylland--Zeckhauser (HZ) equilibrium
+\cite[Definition~2.1]{HZ2026}, an allocation $x\geq0$ and prices
+$p\geq0$ satisfy
+\[
+ \sum_i x_{ij}=1,\qquad \sum_j x_{ij}=1,\qquad \min_jp_j=0,
+\]
+and, for each agent $i$,
+\[
+ p\cdot x_i\leq1+\eps,\qquad
+ u_i\cdot x_i\geq
+ \max\{u_i\cdot y:y\geq0,\ \sum_jy_j=1,\ p\cdot y\leq1\}-\eps.
+\]
+Thus clearing and unit demand are exact; the additive errors concern
+expenditure and utility, with comparison budget one.
+
+\begin{corollary}[Approximate HZ equilibrium hardness]
+\label{cor:hz-hardness}
+There are a fixed rational $\eps_{\rm h}>0$ and a fixed polynomial
+$p_{\rm h}$ such that finding an $\eps_{\rm h}$-HZ equilibrium with
+rational prices and allocation of total encoding length at most
+$p_{\rm h}(L)$ is PPAD-hard under deterministic polynomial-time
+many-one search reductions, where $L$ is the input encoding length.
+\end{corollary}
+
+\begin{proof}
+Use the fan-out-two circuit of Lemma~\ref{lem:application-circuits},
+with its fixed positive numerical tolerance $\tau$ and failure
+allowance $\gamma_2$. Its no-Boolean gate relations agree with
+\cite[Table~1]{HZ2026}, so no Boolean tolerance is changed.
+The reduction underlying~\cite[Theorem~3.2]{HZ2026} has two stages:
+Lemma~3.4 of that paper converts these circuits to bounded-degree
+threshold games, and Lemma~3.25 converts those games to HZ instances.
+The construction uses a fixed integer $m$ and rational market data,
+with HZ error $m^{-40}$. In that construction, each threshold-game
+vertex $v$ has an associated group $G_{v,1}$ of goods. The first
+decoder assigns $v$ the value $\min\{1,m^2p(G_{v,1})\}$, where
+$p(G_{v,1})$ is the minimum price in that group. This decoder, the
+threshold-game-to-circuit decoder, and the subsequent circuit decoders
+have polynomial bit cost on rational witnesses. Composing
+with the decoder after~\eqref{out:endpoint-proximity} therefore gives
+an End-of-Line solution in time polynomial in the input and total
+rational witness length.
+
+For completeness, polynomial-size rational HZ witnesses preserve the
+exact equalities above. Start from a normalized exact equilibrium
+$(x,p)$, as in~\cite[Lemma~B.2]{HZ2026}; summing budgets and column
+sums gives $\sum_jp_j\leq n$.
+Round prices upward on a rational grid of mesh at most
+$\eps_{\rm h}/4$, retaining zero prices. Express $x$ as a convex
+combination of at most $n^2$ permutation matrices and round its
+weights, preserving nonnegativity and total weight one, so each row
+changes in $\ell_1$ norm by at most $\eps_{\rm h}/[4(n+1)]$.
+A denominator of order $n^3/\eps_{\rm h}$ suffices. Row and column
+sums remain exact, expenditures increase by less than $\eps_{\rm h}$,
+and utilities decrease by less than $\eps_{\rm h}$. Since prices only
+increase, the comparison optimum cannot increase. These rational
+witnesses have a fixed polynomial encoding bound; their utility
+conditions are checked by rational linear programming.
+\end{proof}

preprints/The-PCP-for-PPAD-conjecture-a-quasilinear-reduction-September-25-2026/build/sections/12-extensions.tex

@@ -0,0 +1,451 @@
+\section{Further algebraic and binary representations}
+\label{sec:extensions}
+
+The constructions above use separate algebraic representations at the
+outer and inner scales, and retain only the binary products needed for
+functional evaluation. We record two extensions of these representations.
+The first applies the inner algebraic charts directly to the End-of-Line
+macrograph. The second allows unrestricted binary products and checks
+proposed moves. These extensions are independent of the proof of
+Theorem~\ref{thm:main} and the consequences in Section~\ref{sec:weak-nash}.
+
+\subsection{A polynomial chart representation of the End-of-Line macrograph}
+\label{ia:eol-variant}
+
+The polynomial-chart construction also realizes the full reciprocal
+End-of-Line macrograph, including its other paths and cycles, without
+the standardized-port requirement. We record this separate application
+of the same algebraic tools. Its size
+is polynomial; the quasilinear outer representation used in the main
+reduction remains the construction of Sections~\ref{sec:outer-codes}--
+\ref{sec:outer-representation}.
+All field and chart notation in this subsection is local to this separate
+instantiation. The fields, allowance, and one-port templates fixed in
+Section~\ref{sec:inner-algebra} retain their original meaning outside
+this subsection.
+
+\begin{proposition}[Full macrograph polynomial representation]
+\label{ia:eol-algebra-interface}
+For a normalized End-of-Line instance of encoding length \(N\), one
+can construct in deterministic polynomial time a cyclic macrograph and
+its progress-path extension with the following properties.
+\begin{enumerate}[label=(\roman*)]
+\item The graph has indegree and outdegree at most one, no isolated
+vertices or self edges, and a public exceptional source with a public
+successor. Every other endpoint yields an End-of-Line answer from
+its logical tape in polynomial time. Other paths and cycles are
+retained.
+\item Its valid extended labels have unique canonical polynomial
+chart tuples on one fixed-dimensional domain \(K^u\). There are a
+fixed number of chart families, native dimensions \(O(\ell)\),
+a bounded number of \(O(\ell)\) slot axes, and honest coordinate
+degree \(O(\ell h)<D\). All full evaluation, mask, coefficient,
+and indexing arrays have polynomial size. The construction does
+not enumerate the graph's vertices.
+\item The tuples have a perfectly complete bounded binary-form
+algebraic validator with a constant exact gap on genuine
+degree-at-most-\(D\) polynomials. Linear probes are smooth; each
+unknown product uses two independent full point marginals. Canonical
+recipes with the same degree and covariance properties are defined
+on every Boolean logical tape at every specified frame, mode, and
+progress. An invalid label so represented has the same exact gap.
+\item Mode, empty/full progress, and the critical bits for a proposed
+edit admit bounded smooth decisions at any fixed reliability. Once
+the true decision branch is fixed, either proposed canonical tuple has the
+bounded affine update formulas of~\eqref{ia:affine-update}, including
+its full proof data, whether or not the proposed target is valid.
+The coefficients are independent of point and row position, and
+the updates commute with common linear row encodings and ordered-line
+coefficient encodings.
+\end{enumerate}
+Here the fields and all fixed constants are chosen for this whole
+construction. No common one-port source or radius-before-arity
+conclusion is asserted. Target validity is tested after prediction,
+as in Proposition~\ref{ib:proposal-validation}; it is not presumed
+by an affine update formula.
+\end{proposition}
+
+\begin{proof}
+\paragraph{Cyclic schedule and Boolean validity.}
+Use the normalized transcript positions and the two orders of
+Section~\ref{macro:transcripts}, renaming their length \(T=O(N)\).
+For this variant keep two content tapes \(A,B\); their occupied
+prefix lengths are specified by the public stage, so no occupation
+coordinates are needed. With the notation \(Q_x,U_y,V_y\) from
+that section, use the following four stage conditions:
+\[
+\begin{array}{c|l}
+ 0\le s\le T
+   & A=Q_x,\quad B\text{ a forward prefix of length }s
+                           \text{ from }S(x),\\
+ T\le s\le2T
+   & B=Q_y,\quad A\text{ a reconstruction prefix of length }2T-s
+                           \text{ from }y,\\
+ 2T\le s\le3T
+   & B\text{ a valid full transcript},\quad
+                    A=\operatorname{pref}_{\pi_{\mathrm f},s-2T}(B),\\
+ 3T\le s\le4T
+   & A\text{ a valid full transcript},\quad
+                    B=\operatorname{pref}_{\pi_{\mathrm f},4T-s}(A).
+\end{array}
+\]
+The boundary descriptions agree. At \(s=T\), the first gives
+\(A=Q_x,B=Q_y,y=S(x),P(y)=x\); hence reconstruction from \(y\)
+gives exactly \(A\). Conversely, a complete valid reconstruction
+from \(y\) gives \(x=P(y)\), \(S(x)=y\), and the same pair.
+At \(2T\), \(3T\), and \(4T\) the descriptions respectively say
+that \(A\) is empty, that \(A=B\) is full, and that \(B\) is empty.
+The last is the stage-zero hub condition.
+
+Choose a fixed polynomial scale \(n_G=C_G(N+2)^2\), with \(C_G\)
+large enough that \(n_G\ge4T+3\) and all fixed lower cutoffs and
+degree inequalities below hold, and use~\eqref{ia:parameters}--
+\eqref{ia:radix-size} with \(n'\) replaced by \(n_G\). In
+particular \(h^\ell\ge n_G^3+3\), \(M=h^\ell-1\), and
+\(C=K_1^\times=\langle\gamma\rangle\). Pad stages
+\(4T,\ldots,M-1\) by the hub condition, and identify stage \(s\)
+with \(c=\gamma^s\) by enumerating the group. No discrete logarithm
+is needed. All hub-to-hub steps, including the wrap to stage zero,
+are allowed.
+
+The forward tape proposals are, in order: fill the next forward
+\(B\) position; erase the last reconstruction \(A\) position;
+copy the next \(B\) position into \(A\); erase the last copied
+\(B\) position; or do nothing at a padded hub. The reverse proposals
+are respectively erasure, local reconstruction, erasure, copying,
+and the identity. A fill uses its computed bit even if the new
+terminal comparison fails. An actual edge is exactly a proposed
+pair whose two stage conditions hold. Prefix uniqueness from
+Section~\ref{macro:transcripts} makes the operations reciprocal
+on such pairs: earlier operands or the unchanged full tape determine
+each erased bit. Thus both degrees are at most one.
+
+Add a legacy vector \(e\) on \(C\), one-hot at \(c^{-1}\), and
+write \(b=(A,B,e)\). Impose bitness, the one-hot equations
+\eqref{ia:one-hot}, and, for each stage \(c\), multiply every tape
+constraint at that stage by \(e_{c^{-1}}\). Gate constraints have
+degree at most two; blank, copy, and terminal-comparison constraints
+are linear. The resulting \(O(MT+M^2)\) rows have degree at most
+three and describe exactly the union of valid stage tapes. Their row and
+column index sets are polynomial in \(N\), hence require only
+\(O(\ell)\) digit coordinates in the linear verification tool.
+Padding other tape coordinates by zero to full digit grids preserves
+these bounds. A proposal changes at most one content coordinate and the
+two legacy coordinates \(c^{-1},(\gamma c)^{-1}\). Use these three
+unit vectors, or a zero content vector at hubs, as a fixed binary
+flip alphabet. The coefficient of the content flip is the old bit
+plus its required new value. Either direction needs only the old
+bit, at most two operands or one copy bit, and finitely many public
+phase and rule bits. The same rule defines proposals on every
+Boolean tape, without a validity call.
+
+\paragraph{Endpoints and progress paths.}
+A forward prefix can always be shortened, so a missing successor
+is its first failed terminal comparison. With \(A=Q_x\), put
+\(y=S(x)\). Then \(P(S(y))\ne y\) and \(P(y)=x\). If \(y=0\),
+the source promises would give \(x=P(0)=0\) and \(S(0)=0\), a
+contradiction. A reconstruction prefix can always be shortened in
+the forward direction; a missing predecessor gives
+\(y=\operatorname{in}_S(B)\) and \(S(P(y))\ne y\). Copy and hub
+stages have both neighbors. Thus there are no isolated vertices.
+The only zero exception is \(B=Q_0\) with its longest valid
+reconstruction prefix from zero. This transcript exists because
+\(P(S(0))=0\), and its maximal length \(k_*<T\) follows from
+\(S(P(0))=S(0)\ne0\). Its stage is \(2T-k_*\), and its successor
+shortens the prefix or starts copying if \(k_*=0\). Direct circuit
+evaluation computes this source and successor.
+
+For each macro tape at stage \(c\), introduce the path
+\[
+ (\gamma^{-1}c,b,1,q-1),\ldots,(\gamma^{-1}c,b,1,0),
+ (c,b,0,0),\ldots,(c,b,0,q-1).
+\]
+Attach each actual macro edge from the full pre end of one path
+to the full post start of the next. This is the four-operation
+extension of Definition~\ref{ia:extended-path}, with actual
+macroedge validity in place of that definition's disabled-wrap
+rule. The replacement paths are disjoint, so they preserve all
+components, endpoints, and degree bounds. Every edge changes a
+mode or progress coordinate. If the exceptional macro source is
+\((c_*,b_*)\), its extended source and successor are
+\((\gamma^{-1}c_*,b_*,1,q-1)\) and
+\((\gamma^{-1}c_*,b_*,1,q-2)\). At another endpoint, scanning the
+legacy vector and transcript gives the appropriate nonzero vector
+above in polynomial time.
+
+\paragraph{Canonical arrays without positive tags.}
+Use the ordinary order-one lifts for \(A,B\), the cyclic lift of
+Lemma~\ref{ia:cyclic-lift} for \(e\), and the constant chart. Include
+\emph{every} ordered monomial type list of length at most three
+in these groups, with independent argument groups even for repeated
+types. There are constantly many lists. Recursive factorization
+fixes the monomials from the order-one charts. Apply
+Lemma~\ref{ia:linear-tool} to their extension, zero-padding, bitness,
+and cubic validity rows. A coincident-grid product in a validity
+row is a linear condition on a tensor entry, so its verification
+uses that lemma rather than a diagonal unknown-product query.
+This fixes all its canonical certificates uniquely.
+
+Adjoin \(J_c\), the clock-times-monomial charts, and the clock/legacy
+identity~\eqref{ia:clock-legacy}. The clock-indexed construction
+of Lemma~\ref{ia:clock-formula} provides every critical logical bit
+and every public rule bit for both proposal directions. Their
+addresses are indexed by the absolute frame \(cz\); they do not
+require reading the unknown frame as a binary word. All these
+canonical formulas are defined for arbitrary Boolean \(b\), even
+when a final validity row fails.
+
+For every persistent chart of positive assigned Boolean degree,
+including every verification auxiliary, and each of the eight
+flip masks, apply the derivative construction~\eqref{ia:derivative}
+using the same frame in both terms. The top Boolean degree cancels,
+so the derivative and all certificates introduced to verify it
+have degree at most one less. Construct the coefficient maps from
+the parent's specified canonical recipe by
+\eqref{ia:boolean-inversion}, and repeat on every new positive-degree
+family. After at most three drops the process stops. It updates
+the certificates themselves and does not assume their validity.
+The number of families is fixed. Each persistent family has at
+most one \(O(\ell)\) slot axis, and finitely many applications of
+the degree estimate~\eqref{ia:tool-degree} leave coordinate degree
+\(O(\ell h)<D\). Give all inherited projection axes consistent
+full \(K\) labels and reserve a new label for each derivative
+axis; a fixed common \(K^u\) suffices. No positive-tag reuse or
+one-port identification is needed.
+
+\paragraph{Validation, affine updates, and size.}
+Use exactly the mode, radix marker, and projection stacks of
+Lemma~\ref{ia:stacks}. Their triangular recurrences uniquely fix
+all coefficients. At full progress, the singleton-stack sums
+\eqref{ia:critical-projection} supply the constant-polynomial
+critical values for the finite proposal rule. For every persistent
+chart and every stack subset, the summed derivative stack in
+\eqref{ia:edit-increment} gives its exact change at the fixed frame.
+These simultaneous identities hold on every Boolean tape. Increasing
+and decreasing progress use~\eqref{ia:increase-zero}--
+\eqref{ia:decrease}; frame advancement occurs only with empty stacks
+and uses the designated scaling by \(\gamma\). Thus the complete
+proposed target, including every proof chart, obeys the bounded
+affine formulas. Public coefficients and mask pulls are independent
+of target point and row; a common linear row extension and
+line-coefficient extraction commute with them.
+
+The exact-gap argument of Lemma~\ref{ia:exact-validator} now applies
+to this finite family: first enforce shape, independence, and
+linearity, then monomial and clock identities, then all unique
+linear certificates, and finally the marker and stack recurrences.
+If every identity holds, there is one valid tape and its canonical
+tuple. Otherwise a nonzero polynomial discrepancy, after the stated
+earlier linearity conditions, is detected with a fixed positive
+probability by uniform points, separable masks, and binary forms.
+Products of nonconstant monomial factors use their independent
+argument groups; clock products use a fresh independent group;
+marker products are sampled at independent points after constancy
+is enforced. Ignored coordinates are filled independently. Hence
+all unknown products have independent full point marginals, and
+all linear probes retain the smoothness of the linear tool.
+On an invalid Boolean tape the recipes, covariance, and degree
+bounds still hold, but some final identity fails with the same gap.
+
+Finally, \(M,|K|,q^{O(\ell)}\), the validity matrices, and all
+mask spaces have polynomial size in \(N\). More explicitly,
+\(T\le M\), the stage rows number \(O(M^2)\), and the logical
+monomials of degree at most three number \(O(M^3)\). All later
+row and column sets have size a fixed power of these quantities
+or of \(|K|\); their base-\(h\) digit dimensions are therefore
+\(O(\ell)\). The fixed-depth degree estimates remain
+\(O(\ell h)<D\) after the chosen cutoff. Boolean interpolation
+through degree three, every array operation, and every fixed-depth
+certificate construction consequently take polynomial time. No step enumerates Boolean assignments
+or graph vertices. This proves all the asserted costs and interfaces.
+\end{proof}
+
+\subsection{Unrestricted products and validation of proposed moves}
+\label{ib:unrestricted-variants}
+
+We return to the binary construction and notation of
+Section~\ref{sec:inner-binary}. The restriction to compatible tags is
+needed for the standardized one-port source in
+Section~\ref{sec:inner-gadget}. The binary methods
+also give the following variant when that source requirement is absent.
+Its constants are fixed for the entire chosen chart family.
+
+\begin{proposition}[All ordered products and full logical recovery]
+\label{ib:all-pairs-interface}
+Fix an inner algebraic template, and replace the compatible-pair list
+by the list of \emph{all} ordered pairs of point-track types in
+\eqref{ib:product-bit}. The two positions remain independent even
+when the types coincide; no line-product blocks are added. Rebalance all blocks by the repetition construction of
+Section~\ref{ib:binary-blocks}. This encoding has polynomial length and a
+uniform polynomial-time construction, and has the following properties.
+\begin{enumerate}[label=(\roman*)]
+\item Valid canonical words have constant distance and the perfectly
+complete smooth tester and proximity guarantee of
+Proposition~\ref{ib:tester}. A structurally canonical word whose
+label is invalid has rejection probability at least one fixed
+\(\theta_{\rm inv}>0\).
+\item One fixed radius permits arbitrary fixed average accuracy for
+bounded-density requests, with joint pair density for products,
+arbitrary public masks and forms, and all-hypothetical-branch
+smoothness as in Proposition~\ref{ib:average-correction} and
+Lemma~\ref{ib:all-branch-density}.
+\item From a sufficiently small fixed neighborhood of any structurally
+canonical word, its \emph{entire} logical tape
+\(b=(w,e,(s_i)_i)\) is recoverable exactly in deterministic polynomial
+time, whether or not that tape is valid.
+\end{enumerate}
+The constants and the radius may depend on the whole fixed template
+and its block list. No common one-port layout or radius fixed before
+arity is asserted for this variant.
+
+More generally, the same binary transfer applies to a supplied fixed
+finite chart template with the field, degree, slot, and polynomial-size
+bounds used in Section~\ref{sec:inner-binary}, subject to these explicit
+conditions: its canonical
+recipes are defined also on invalid logical tapes; its bounded
+binary-form algebraic validator has perfect completeness and an exact
+constant gap on genuine degree-in-bound polynomial tuples; its linear
+probes have bounded point marginals and its products have independent
+full point marginals; and its order-one grid charts and constant
+mode/marker tensors separate labels. Full logical recovery requires
+order-one grid charts for every logical coordinate. This transfer
+assumes these algebraic properties; it does not assert that the
+fixed-input path of Section~\ref{sec:inner-algebra} realizes an
+arbitrary macrograph.
+\end{proposition}
+
+\begin{proof}
+There are still only constantly many blocks, each of polynomial
+size, so the balancing product remains polynomial. The point and
+line construction, rectangular action, and common row extension do
+not use tags. In the proof of Proposition~\ref{ib:tester}, all
+replacements before the product step concern individual \(Y\)
+tracks. For each added product block, the additivity test in \(Q\)
+and the two independent uniform operand positions give exactly the
+same product replacement: a false binary matrix is detected by a
+uniform outer-product form with probability at least \(1/4\).
+Taking a positive fixed mixture over the enlarged finite block list
+therefore gives the same vanishing-rejection replacement argument.
+The algebraic validator is then applied to the common target tuple;
+every requested point product is now present. Its exact gap forces
+validity. On an invalid structurally canonical word, no replacement
+is necessary: the final validator component already rejects with a
+fixed positive probability. Its mixture weight and the fixed losses
+in binary-form simulation define \(\theta_{\rm inv}\). Distinct
+logical or mode/marker data retain the positive distance witnessed
+by their \(Y\) blocks, with the new fixed block weights.
+
+For correction, the two-axis support and inner-table arguments
+apply to any ordered pair of point tracks. They use independent
+row coordinates and the two row-code actions, not tag compatibility.
+Use the new block count in \eqref{ib:uncertain-average} and choose
+one radius satisfying \eqref{ib:fixed-radius} for this block list.
+The bad-column bound, support contraction, finite-cone consistency,
+and amplification then prove the same average guarantee. Each pair
+of affine shifts remains a permutation of the full pair domain,
+even for a repeated type. The proof of
+Lemma~\ref{ib:all-branch-density} consequently applies to every
+added block, with a new finite summed density bound. In particular,
+no product request is converted to a diagonal pair.
+
+For full recovery, apply the proof of
+Proposition~\ref{ib:global-recovery} to each order-one chart
+\(W_w,W_e,W_{s_i}\). The work and port charts use ordinary grid
+extensions, and the legacy chart uses its degree-in-bound cyclic
+lift. In each case nearest Hadamard symbols, equal-fiber plurality,
+and plurality over ignored coordinates produce a native polynomial
+estimate of sufficiently small constant symbol error. For each
+logical grid point, the complete enumeration of lines and the
+verified Berlekamp--Welch systems in that proof recover its exact
+value by a strict majority. The argument uses degree and table
+structure, not the final validity equations. There are polynomially
+many grid points and a fixed number of charts. Reducing the radius
+by the fixed block-weight factors makes all these decodings valid
+simultaneously. The same proof applies to the supplied general
+chart template under the stated separation and grid hypotheses.
+\end{proof}
+
+\begin{proposition}[Prediction before target validation]
+\label{ib:proposal-validation}
+Use the unrestricted representation of
+Proposition~\ref{ib:all-pairs-interface}. Suppose a valid current
+label has a prescribed proposal chosen by finitely many decisions
+which are recoverable, to any fixed reliability, by bounded smooth
+bit procedures. On a valid current label, once its true decision
+branch is fixed, suppose the full canonical proposed tuple is a
+definite structurally canonical tuple, possibly invalid, whose
+contractions are bounded sums of public terms and old contractions
+at invertible \(K\)-affine target pulls. For each branch these pulls
+must belong to a fixed bounded public list, independent of the
+requested outer indices and mask or form values. Masks may select
+terms from that list and determine their slot pulls and forms.
+Require bounded slot expansions, coefficients independent of target
+point and row, compatibility with the common row and line encodings,
+and publicly computable actual encoded values for public terms.
+On hypothetical wrong branches only these bounded schedules and
+smooth defaults are required; a canonical target identity is asserted
+only when the branch predicates hold.
+All branch lists, supports, and public computations are to have the
+uniform polynomial-size bounds of Section~\ref{sec:inner-binary}.
+
+From a sufficiently small fixed neighborhood of the current valid
+word, its proposed word can be predicted to arbitrary fixed average
+accuracy on bounded-density requests \emph{before} deciding target
+validity. That validity can then be decided with arbitrary fixed
+reliability. If an actual-edge predicate is exactly target validity
+together with specified public or similarly recoverable guards,
+this also decides that edge flag. All routines are bounded on every
+received word and retain the complete hypothetical-tree smoothness
+bound. No reciprocity of a general proposed graph is inferred from
+target validity alone.
+\end{proposition}
+
+\begin{proof}
+Fix the true decision branch of the nearby current label, charging
+the selector's error separately. For a point or line \(Y\) request,
+commute the proposed update through the common row encoder and,
+for a line, through restriction and coefficient contraction.
+This gives a bounded XOR of old \(Y\) requests with publicly
+pulled-back forms, plus public bits. Each map in the fixed public
+list induces a commuting column permutation on the retained point
+and ordered-line domains, preserving bounded outer-index density.
+If masks select among these maps, sum their unconditioned slot
+bounds before selection; the list is bounded independently of the
+request. Apply average correction with the error divided among
+these boundedly many slots.
+
+For a proposed \(Z\) bit, expand both updated symbols by bilinearity.
+Every two-variable term is present because all ordered point-track
+pairs were included. Each side undergoes its own affine column
+permutation, preserving the joint density of the pair. A term with
+one public actual \(J\)-symbol is an arbitrary linear form of the
+other symbol, and \eqref{ib:public-marginal} supplies its complete
+surviving row/point density. Two public operands give a public bit.
+Thus the proposal is predicted using only the current close word;
+its target need not be valid or supplied as a received word.
+
+Now simulate the tester on the proposed word by supplying each bit
+from this predictor with fresh internal randomness. First fix the
+selector error below \(\theta_{\rm inv}/16\) and its complete
+bounded tree. Unfold that tree and the bounded tester tree. Each
+fixed hypothetical tester request has bounded density, so next
+choose the per-prediction errors to sum to less than
+\(\theta_{\rm inv}/16\) over all these slots. The simulated
+rejection probability differs from the true one by less than
+\(\theta_{\rm inv}/8\). A valid proposal has
+true rejection zero; an invalid structural proposal has rejection
+at least \(\theta_{\rm inv}\). Independent repeated simulations
+and the threshold \(\theta_{\rm inv}/2\) therefore give any
+specified fixed reliability. Include the stated guards separately
+when interpreting the result as an edge flag.
+
+All error estimates use the unconditioned request law on each
+fixed branch, followed by a union bound over the complete bounded
+tree. They do not renormalize after a successful comparison or
+condition on a rare selected branch. Wrong decisions and failed
+comparisons still select only from the supplied finite lists;
+public-zero defaults and fresh splitting preserve the raw-read
+bounds of Lemma~\ref{ib:all-branch-density}. At fixed accuracy,
+the number of simulations and all their supports are bounded and
+polynomially constructible, as required.
+\end{proof}

preprints/The-PCP-for-PPAD-conjecture-a-quasilinear-reduction-September-25-2026/paper.pdf

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:a7ab4d86183890be9bfaa845f2c0eddf4ba89956911fc8318c47ac35f504e84b
+size 1117288

preprints/The-Partition-Principle-does-not-imply-Choice-September-24-2026/README.md

@@ -0,0 +1,18 @@
+# [The Partition Principle does not imply Choice](partition-principle-without-choice.pdf)
+
+**Author:** OpenAI
+
+**Date:** September 24, 2026
+
+## Citation
+
+```bibtex
+@misc{OAI:The-Partition-Principle-does-not-imply-Choice-September-24-2026,
+  author = {{OpenAI}},
+  title = {{The Partition Principle does not imply Choice}},
+  howpublished = {OpenAI Math Release preprint
+                  \href{https://github.com/openai/math/blob/main/preprints/The-Partition-Principle-does-not-imply-Choice-September-24-2026/partition-principle-without-choice.pdf}{OAI:The-Partition-Principle-does-not-imply-Choice-September-24-2026}},
+  year = {2026}
+}
+```
+

preprints/The-Partition-Principle-does-not-imply-Choice-September-24-2026/build/figures/moved-generic-transport.tex

@@ -0,0 +1,19 @@
+\begin{figure}[htbp]
+\centering
+\begin{tikzpicture}[font=\small,>=Stealth]
+  \node (a) at (0,1.65) {$L_C$};
+  \node (b) at (6,1.65) {$B_C(\pi^{-1}F)$};
+  \node (c) at (0,0) {$L_C$};
+  \node (d) at (6,0) {$B_C(F)$};
+  \draw[->] (a) -- node[above] {$f_C^0(\pi^{-1}F)$} (b);
+  \draw[->] (a) -- node[left] {$\pi\restriction L_C$} (c);
+  \draw[->] (b) -- node[right] {$T_\pi^{\pi^{-1}F}$} (d);
+  \draw[->] (c) -- node[below] {$f_C^{\pi K_C}(F)$} (d);
+\end{tikzpicture}
+\caption{Transporting a preliminary bijection, for $\pi\in G_C$.
+The square commutes: to assign the bottom label $j$, first compute
+at $\pi^{-1}j$ in the generic $\pi^{-1}F$, then transport the result
+to $F$. The top and bottom computations use $\pi^{-1}F$ and $F$,
+respectively.}
+\label{fig:moved-generic-transport}
+\end{figure}

preprints/The-Partition-Principle-does-not-imply-Choice-September-24-2026/build/figures/probe-divisibility.tex

@@ -0,0 +1,20 @@
+\begin{figure}[htbp]
+\centering
+\begin{tikzpicture}[font=\small,>=Stealth]
+  \node (pathpoint) at (0,0) {$n_i$};
+  \node (oldpoint) at (3,1) {$a$};
+  \node (probe) at (3,-1) {$m=p_0$};
+  \node (commonmultiple) at (6,0) {$ae$};
+  \draw[->] (pathpoint) -- node[above left] {$c_i$} (oldpoint);
+  \draw[->] (pathpoint) -- node[below left] {$p_i$} (probe);
+  \draw[->] (oldpoint) -- node[above right] {$e$} (commonmultiple);
+  \draw[->] (probe) -- node[below right] {$b$} (commonmultiple);
+\end{tikzpicture}
+\caption{The relations defining the probe give two factorizations of
+the prescribed common multiple $ae$.  Each arrow means right
+multiplication by its label, not a covering relation.  This is not a
+Hasse diagram: path points may coincide with $a$, and $a$ may divide
+$m$.  The exact-profile assertion is that a divisor of $a$ omitted
+from the path's downward closure has join $ae$ with $m$.}
+\label{fig:probe-divisibility}
+\end{figure}

preprints/The-Partition-Principle-does-not-imply-Choice-September-24-2026/build/figures/support-descent.tex

@@ -0,0 +1,24 @@
+\begin{figure}[htbp]
+\centering
+\begin{tikzpicture}[font=\small,
+    diagramrange/.style={draw,rounded corners=2pt,fill=white,
+                  inner xsep=9pt,inner ysep=5pt},
+    overlap/.style={fill=white,inner sep=3pt}]
+  \node[diagramrange] (t) at (0,1.8) {$D_t$};
+  \node[diagramrange] (v) at (-3,-.8) {$D_{v'}$};
+  \node[diagramrange] (w) at (3,-.8) {$D_{v''}$};
+  \draw (t) -- node[overlap,pos=.52] {$D_{\bar s}$} (v);
+  \draw (t) -- node[overlap,pos=.52] {$D_{\bar s'}$} (w);
+  \draw (v) -- node[overlap] {$D_u$} (w);
+  \node[align=center,font=\footnotesize] at (0,-.15)
+    {$D_p$\\triple intersection};
+\end{tikzpicture}
+\caption{The independent-copy case of support descent. Each edge is
+labelled by the exact intersection of its endpoint ranges. The two
+placed equality witnesses $v',v''$ use the same fresh copy $u$ of $s_1$;
+$t$ carries the proposed inequality between the copies on
+$\bar s$ and $\bar s'$, whose ranges meet exactly in $D_p$.
+These exact intersections allow the three diagrams to be amalgamated
+over their common root $p$.}
+\label{fig:support-descent}
+\end{figure}

preprints/The-Partition-Principle-does-not-imply-Choice-September-24-2026/build/main.tex

@@ -0,0 +1,67 @@
+\documentclass[11pt]{article}
+\pdfinfoomitdate=1
+\pdftrailerid{}
+\pdfsuppressptexinfo=15
+\usepackage[T1]{fontenc}
+\usepackage{lmodern}
+\usepackage[expansion=false]{microtype}
+\usepackage[margin=1.04in]{geometry}
+\usepackage{amsmath,amssymb,amsthm,mathtools}
+\usepackage{enumitem,booktabs,array,needspace}
+\usepackage{tikz}
+\usetikzlibrary{arrows.meta,positioning,calc,fit}
+\usepackage[hyphens]{url}
+\usepackage[hidelinks,hypertexnames=false]{hyperref}
+\usepackage[nameinlink,capitalize,noabbrev]{cleveref}
+\hypersetup{pdftitle={The Partition Principle does not imply Choice},pdfauthor={OpenAI}}
+\setlength{\emergencystretch}{2em}
+\allowdisplaybreaks[1]
+\numberwithin{equation}{section}
+\newtheorem{theorem}{Theorem}[section]
+\newtheorem{lemma}[theorem]{Lemma}
+\newtheorem{proposition}[theorem]{Proposition}
+\newtheorem{corollary}[theorem]{Corollary}
+\theoremstyle{definition}
+\newtheorem{definition}[theorem]{Definition}
+\theoremstyle{remark}
+\newtheorem{remark}[theorem]{Remark}
+\newcommand{\ZF}{\mathsf{ZF}}
+\newcommand{\ZFC}{\mathsf{ZFC}}
+\newcommand{\AC}{\mathsf{AC}}
+\newcommand{\PP}{\mathsf{PP}}
+\newcommand{\ACwo}{\mathsf{AC}_{\mathrm{WO}}}
+\newcommand{\Con}{\operatorname{Con}}
+\newcommand{\Ord}{\mathrm{Ord}}
+\newcommand{\HS}{\mathrm{HS}}
+\newcommand{\forces}{\Vdash}
+\DeclareMathOperator{\Fix}{Fix}
+\DeclareMathOperator{\Sym}{Sym}
+\DeclareMathOperator{\dom}{dom}
+\DeclareMathOperator{\ran}{ran}
+\DeclareMathOperator{\supp}{supp}
+\DeclareMathOperator{\id}{id}
+\title{The Partition Principle does not imply Choice}
+\author{OpenAI}
+\date{September 24, 2026}
+\begin{document}
+\maketitle
+\begin{abstract}
+We prove that the Partition Principle does not imply the Axiom of Choice:
+if ZF is consistent, then so is ZF with the Partition Principle, Choice for
+ordinal-indexed families, and the negation of the Axiom of Choice.
+Separately, over every countable transitive model of ZFC, we construct a
+transitive symmetric model of this theory with the same ordinals and no new
+countable sequences of ground elements.
+\end{abstract}
+\tableofcontents
+\clearpage
+\input{sections/introduction}
+\input{sections/monoid}
+\input{sections/forcing}
+\input{sections/quotients}
+\input{sections/conclusion}
+\input{sections/metatheory}
+{\small\raggedright
+\bibliographystyle{plain}
+\bibliography{references}}
+\end{document}

preprints/The-Partition-Principle-does-not-imply-Choice-September-24-2026/build/references.bib

@@ -0,0 +1,201 @@
+@article{BanaschewskiMoore1990,
+  author = {Banaschewski, Bernhard and Moore, Gregory H.},
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+  journal = {Notre Dame Journal of Formal Logic},
+  volume = {31},
+  number = {3},
+  year = {1990},
+  pages = {375--381},
+  doi = {10.1305/ndjfl/1093635502},
+  url = {https://doi.org/10.1305/ndjfl/1093635502}
+}
+
+@article{daSilva2021,
+  author = {da Silva, Samuel G.},
+  title = {The Axiom of Choice and the Partition Principle from {Dialectica} Categories},
+  journal = {Logic Journal of the IGPL},
+  volume = {29},
+  number = {5},
+  year = {2021},
+  pages = {783--797},
+  doi = {10.1093/jigpal/jzaa023},
+  url = {https://doi.org/10.1093/jigpal/jzaa023}
+}
+
+@article{Karagila2026,
+  author = {Karagila, Asaf},
+  title = {Approaching a {Bristol} model},
+  journal = {The Bulletin of Symbolic Logic},
+  volume = {32},
+  number = {1},
+  year = {2026},
+  pages = {1--46},
+  doi = {10.1017/bsl.2025.10126},
+  url = {https://doi.org/10.1017/bsl.2025.10126}
+}
+
+@article{RyanSmith2025,
+  author = {Ryan-Smith, Calliope},
+  title = {Local reflections of choice},
+  journal = {Acta Mathematica Hungarica},
+  volume = {176},
+  number = {1},
+  year = {2025},
+  pages = {244--257},
+  doi = {10.1007/s10474-025-01533-3},
+  url = {https://doi.org/10.1007/s10474-025-01533-3}
+}
+
+@misc{HolySchilhan2025,
+  author = {Holy, Peter and Schilhan, Jonathan},
+  title = {The Ordering Principle and Higher Dependent Choice},
+  year = {2025},
+  eprint = {2510.14821v2},
+  howpublished = {\href{https://arxiv.org/abs/2510.14821v2}{arXiv:2510.14821v2}},
+  archivePrefix = {arXiv},
+  primaryClass = {math.LO},
+  note = {Version 2, 17 October 2025},
+  url = {https://arxiv.org/abs/2510.14821v2}
+}
+
+@misc{BlassKulshreshtha2026,
+  author = {Blass, Andreas and Kulshreshtha, Dhruv},
+  title = {Cardinal Well-foundedness and Choice},
+  year = {2026},
+  eprint = {2310.09643v3},
+  howpublished = {\href{https://arxiv.org/abs/2310.09643v3}{arXiv:2310.09643v3}},
+  archivePrefix = {arXiv},
+  primaryClass = {math.LO},
+  note = {Version 3, 21 August 2026},
+  url = {https://arxiv.org/html/2310.09643v3}
+}
+
+@misc{Welch2020,
+  author = {Welch, P. D.},
+  title = {Axiomatic Set Theory},
+  year = {2020},
+  howpublished = {\url{https://people.maths.bris.ac.uk/~mapdw/current-axiomatic-set-theory.pdf}},
+  note = {Lecture notes, 16 August 2020, line-numbered version},
+  url = {https://people.maths.bris.ac.uk/~mapdw/current-axiomatic-set-theory.pdf}
+}
+
+@misc{Williams2019,
+  author = {Williams, Kameryn J.},
+  title = {{Math655} Lecture Notes: Part 2.1, An Introduction to Forcing},
+  year = {2019},
+  howpublished = {\url{https://juliakw.net/teaching/2019/math655/part2.1.pdf}},
+  note = {Lecture notes, 7 March 2019},
+  url = {https://juliakw.net/teaching/2019/math655/part2.1.pdf}
+}
+
+@misc{KaragilaNotes2026,
+  author = {Karagila, Asaf},
+  title = {Lecture Notes: Forcing \& Symmetric Extensions},
+  year = {2026},
+  howpublished = {\url{https://karagila.org/files/Forcing-2023.pdf}},
+  note = {Updated 24 August 2026},
+  url = {https://karagila.org/files/Forcing-2023.pdf}
+}
+
+@book{Jech2003,
+  author = {Jech, Thomas},
+  title = {Set Theory},
+  edition = {Third millennium, revised and expanded},
+  series = {Springer Monographs in Mathematics},
+  publisher = {Springer-Verlag},
+  address = {Berlin},
+  year = {2003}
+}
+
+@article{Blass1979,
+  author = {Blass, Andreas},
+  title = {Injectivity, projectivity, and the axiom of choice},
+  journal = {Transactions of the American Mathematical Society},
+  volume = {255},
+  year = {1979},
+  pages = {31--59},
+  doi = {10.1090/S0002-9947-1979-0542870-6},
+  url = {https://doi.org/10.1090/S0002-9947-1979-0542870-6}
+}
+
+@misc{Usuba2021,
+  author = {Usuba, Toshimichi},
+  title = {Geology of symmetric grounds},
+  year = {2021},
+  eprint = {1912.10246v3},
+  howpublished = {\href{https://arxiv.org/abs/1912.10246v3}{arXiv:1912.10246v3}},
+  archivePrefix = {arXiv},
+  primaryClass = {math.LO},
+  note = {Version 3, 15 February 2021},
+  url = {https://arxiv.org/abs/1912.10246v3}
+}
+
+@misc{GilsonIterationCorrection2026,
+  author = {Gilson, Frank},
+  title = {A countable-support symmetric iteration separating {PP} from {AC}},
+  year = {2026},
+  eprint = {2601.01855v8},
+  howpublished = {\href{https://arxiv.org/abs/2601.01855v8}{arXiv:2601.01855v8}},
+  archivePrefix = {arXiv},
+  primaryClass = {math.LO},
+  note = {Version 8, 10 September 2026; construction withdrawn in the correction},
+  url = {https://arxiv.org/abs/2601.01855v8}
+}
+
+@misc{GilsonExternalCorrection2026,
+  author = {Gilson, Frank},
+  title = {From Internal to External: Classical Models of {ZF + PP + $\neg$AC}},
+  year = {2026},
+  eprint = {2511.09764v5},
+  howpublished = {\href{https://arxiv.org/abs/2511.09764v5}{arXiv:2511.09764v5}},
+  archivePrefix = {arXiv},
+  primaryClass = {math.LO},
+  note = {Version 5, 10 September 2026; model constructions withdrawn},
+  url = {https://arxiv.org/abs/2511.09764v5}
+}
+
+@misc{GilsonSheafCorrection2026,
+  author = {Gilson, Frank},
+  title = {Partition Principle without Choice via Symmetric Iterations and Sheaf-Toposes},
+  year = {2026},
+  eprint = {2511.07675v6},
+  howpublished = {\href{https://arxiv.org/abs/2511.07675v6}{arXiv:2511.07675v6}},
+  archivePrefix = {arXiv},
+  primaryClass = {math.LO},
+  note = {Version 6, 10 September 2026; model and preservation claims withdrawn},
+  url = {https://arxiv.org/abs/2511.07675v6}
+}
+
+@article{Pincus1977,
+  author = {Pincus, David},
+  title = {Adding dependent choice},
+  journal = {Annals of Mathematical Logic},
+  volume = {11},
+  number = {1},
+  year = {1977},
+  pages = {105--145},
+  doi = {10.1016/0003-4843(77)90011-0},
+  url = {https://doi.org/10.1016/0003-4843(77)90011-0}
+}
+
+@article{Newman1942,
+  author = {Newman, M. H. A.},
+  title = {On Theories with a Combinatorial Definition of ``Equivalence''},
+  journal = {Annals of Mathematics},
+  volume = {43},
+  number = {2},
+  pages = {223--243},
+  year = {1942},
+  doi = {10.2307/1968867},
+  url = {https://doi.org/10.2307/1968867}
+}
+
+@incollection{DershowitzPlaisted2001,
+  author = {Dershowitz, Nachum and Plaisted, David A.},
+  title = {Rewriting},
+  booktitle = {Handbook of Automated Reasoning},
+  year = {2001},
+  publisher = {Elsevier},
+  url = {https://www.cs.tau.ac.il/~nachum/papers/hand-final.pdf},
+  note = {\url{https://www.cs.tau.ac.il/~nachum/papers/hand-final.pdf}}
+}

preprints/The-Partition-Principle-does-not-imply-Choice-September-24-2026/build/sections/conclusion.tex

@@ -0,0 +1,303 @@
+\section{From quotient classification to the Partition Principle}
+\label{sec:conclusion}
+
+We now pass from the classification of quotients of the label set to
+arbitrary surjections.  The reduction uses only three statements inside
+the symmetric model: every nonempty set is an image of the label set
+times an ordinal, choice holds for ordinal-indexed families, and every
+nonwellorderable image of the label set is bijective to it.  We first
+prove this reduction in $\ZF$, independently of the construction.
+
+The ordinal slicing and assembly of injections follow the local-to-global
+method in Ryan-Smith's analysis of $\PP$
+\cite[Propositions~3.17 and~3.20]{RyanSmith2025}. We use the stronger
+classification of nonwellorderable images stated below and give the
+complete reduction under these hypotheses.
+
+\begin{lemma}[The all-set reduction]\label{lemma:pp-reduction}
+Work in $\ZF$.  Suppose that a nonempty set $A$ has the following
+properties.
+\begin{enumerate}
+\item For every nonempty set $X$, there are an ordinal $\eta$ and
+a surjection $A\times\eta\longrightarrow X$.
+\item Every ordinal-indexed family of nonempty sets has a choice
+function.
+\item Every nonwellorderable surjective image of $A$ is in bijection
+with $A$.
+\end{enumerate}
+Then $\PP$ holds: for all sets $X,Y$, every surjection
+$f:X\longrightarrow Y$ has an injection $j:Y\longrightarrow X$.
+\end{lemma}
+
+\begin{proof}
+We will use two elementary observations about images of $A$.
+If $h:A\longrightarrow B$ is onto and $\varnothing\ne S\subseteq B$,
+then $S$ is also an image of $A$.  Indeed, fix one $s_0\in S$ and
+define
+\[
+ h_S(a)=
+ \begin{cases}
+ h(a),&h(a)\in S,\\
+ s_0,&h(a)\notin S.
+ \end{cases}
+\]
+This proves an existence assertion for one given nonempty subset;
+it does not choose a default element simultaneously for a family.
+Also, the image of a wellorderable set under any function is
+wellorderable: order its image by the least preimages in a fixed
+wellordering of its domain.  Both observations are theorems of $\ZF$.
+
+Fix a surjection $f:X\longrightarrow Y$.  If $Y=\varnothing$, the
+empty function is the desired injection.  Otherwise $X$ is nonempty.
+Choose a surjection $s:A\times\eta\longrightarrow X$ from the first
+assumption.  Define, for $i<\eta$,
+\begin{align}
+ B_i&=s[A\times\{i\}],\notag\\
+ Y_i&=f[B_i]\setminus\bigcup_{k<i}f[B_k],\notag\\
+ S_i&=B_i\cap f^{-1}[Y_i].\label{eq:final-disjoint-slices}
+\end{align}
+Separation and Replacement form these sequences.  Every $y\in Y$
+belongs to $f[B_i]$ for some $i$, and its least such index is the
+unique index for which $y\in Y_i$.  Thus the $Y_i$ partition $Y$.
+The $S_i$ are pairwise disjoint because their $f$-images lie in the
+pairwise disjoint $Y_i$.  Moreover,
+\[
+ f\mathbin{\upharpoonright}S_i:S_i\longrightarrow Y_i
+\]
+is onto for every $i$.  The sets $S_i$ need not cover $X$.
+
+Before choosing any injections, define
+\begin{equation}\label{eq:final-candidate-injections}
+ \begin{split}
+ \mathcal J_i=\{j\in\mathcal P(Y\times X):{}&
+    j\text{ is an injective function},\\
+    &\dom(j)=Y_i,\quad\ran(j)\subseteq S_i\}.
+ \end{split}
+\end{equation}
+Power Set and Separation give each candidate set, and Replacement
+gives the sequence $\langle\mathcal J_i:i<\eta\rangle$.
+We show that every candidate set is nonempty, fixing an arbitrary
+$i<\eta$.
+
+If $Y_i$ is empty, its empty injection belongs to $\mathcal J_i$.
+Suppose next that $Y_i$ is nonempty and wellorderable.  Choose a
+wellordering of this one set.  The nonempty fibers
+\[
+ S_i\cap f^{-1}[\{y\}]\qquad(y\in Y_i)
+\]
+can be indexed by an ordinal using that wellordering.  The second
+assumption gives a choice function on them.  Since different fibers
+are disjoint, this function is an injection $Y_i\longrightarrow S_i$.
+It therefore belongs to $\mathcal J_i$.
+
+Finally, suppose that $Y_i$ is nonwellorderable.  Then $S_i$ is
+nonempty and nonwellorderable, by the second elementary observation
+and the surjectivity of $f|S_i$.  The map $a\mapsto s(a,i)$
+surjects from $A$ onto $B_i$.  The first observation makes the
+nonempty subset $S_i\subseteq B_i$ an image of $A$, and composition
+with $f|S_i$ makes $Y_i$ an image of $A$ as well.  By the third
+assumption both $S_i$ and $Y_i$ are bijective to $A$.  Composing
+one of these bijections with the inverse of the other gives an
+injection $Y_i\longrightarrow S_i$, again in $\mathcal J_i$.
+
+We have proved $(\forall i<\eta)\ \mathcal J_i\ne\varnothing$.
+The pointwise proof selected neither a sequence of wellorderings nor
+a sequence of quotient bijections.  Apply the second assumption now
+to the already defined sequence of candidate sets.  It supplies
+$\langle j_i:i<\eta\rangle$ with $j_i\in\mathcal J_i$.
+Set
+\[
+ j=\bigcup_{i<\eta}j_i.
+\]
+The disjointness of the $Y_i$ makes this a function with domain $Y$.
+If $j(y)=j(y')$, that value lies in both corresponding source pieces
+$S_i$.  Their disjointness makes the indices equal, and injectivity
+of that $j_i$ gives $y=y'$.  Thus $j:Y\longrightarrow X$ is an
+injection.
+
+Every construction in this proof takes place in the same model of
+$\ZF$.  No rank bound on $X$ or $Y$, and no uniform choice of auxiliary
+presentations for the slices, has been used.
+\end{proof}
+
+\begin{remark}\label{rem:quotients-versus-images}
+The third hypothesis of Lemma~\ref{lemma:pp-reduction} can equivalently
+be stated for quotients $A/E$ by equivalence relations on $A$.
+For a surjection $h:A\longrightarrow B$, define
+\[
+ a\mathrel{E_h}a'\quad\Longleftrightarrow\quad h(a)=h(a').
+\]
+The map $[a]_{E_h}\mapsto h(a)$ is a bijection $A/E_h\longrightarrow B$.
+Its graph is defined by the unique common value on each class; it
+does not require choosing class representatives.
+
+The first hypothesis is the surjective assertion $\mathsf{SVC}(A)$.
+Its injective counterpart $\mathsf{SVC}^{+}(A)$ would assert an
+injection $X\longrightarrow A\times\eta$ for every $X$ and some
+ordinal $\eta$.  The proof does not assume that counterpart.
+It follows afterward by applying $\PP$ to each presentation surjection.
+The quotient assertion $\mathsf{PP}(A)$ says that each surjective
+image of $A$ injects into $A$.  The subset-restricted assertion
+$\PP\mathbin{\upharpoonright}A$ concerns surjections between two
+arbitrary subsets of $A$.  These are different local statements.
+The reduction above uses the stronger classification of every
+nonwellorderable image of $A$, together with ordinal-indexed choice;
+it does not replace that classification by quotient injections alone.
+\end{remark}
+
+The injection obtained in Lemma~\ref{lemma:pp-reduction} need not
+respect the fibers of the original surjection.  In its wellorderable
+target case the local injection is a section, but in its
+nonwellorderable case the two bijections with $A$ give no such
+requirement.  This is precisely the freedom allowed by $\PP$.
+
+\subsection{Further cardinal-comparison consequences}
+
+The same model separates Choice from two weaker cardinal-comparison
+principles. Their consequences follow from $\PP$ alone and do not
+require any further forcing construction.
+
+\begin{remark}\label{rem:dual-csb-wpp}
+For sets $X,Y$, write $X\leq^{*}Y$ if $X=\varnothing$ or there is a
+surjection $Y\twoheadrightarrow X$, and write $X<Y$ if there is an
+injection $X\hookrightarrow Y$ but no bijection $X\to Y$. The dual
+Cantor--Schr\"oder--Bernstein principle $\mathsf{CSB}^{*}$ asserts that
+$X\leq^{*}Y$ and $Y\leq^{*}X$ imply a bijection $X\to Y$; the Weak
+Partition Principle $\mathsf{WPP}$ asserts that $X\leq^{*}Y$ implies
+$\neg(Y<X)$. Both statements quantify over all sets. If either set in
+the dual-CSB hypothesis is empty, so is the other.
+
+In $\ZF$ one has
+\[
+ \PP\ \Longrightarrow\ \mathsf{CSB}^{*}\ \Longrightarrow\ \mathsf{WPP}.
+\]
+Indeed, the empty case of $X\leq^{*}Y\Rightarrow X\hookrightarrow Y$
+has the empty injection, so the first implication follows by applying
+$\PP$ twice and ordinary Cantor--Schr\"oder--Bernstein. For the second,
+if $X\leq^{*}Y$ and $Y<X$, the injection $Y\hookrightarrow X$ gives
+$Y\leq^{*}X$: for nonempty $Y$, invert it on its image and extend by
+one fixed element of $Y$; for empty $Y$, use the definition. Then
+$\mathsf{CSB}^{*}$ gives a bijection, contrary to $Y<X$. See
+\cite[Definitions~2.1--2.2, Proposition~2.3, and the proofs of
+Corollary~8.2 and Proposition~9.8]{BlassKulshreshtha2026}.
+
+These $\ZF$ implications hold internally in each model. Thus
+Theorem~\ref{thm:main-consistency} also gives
+\[
+ \Con(\ZF)\ \Longrightarrow\
+ \Con(\ZF+\PP+\ACwo+\mathsf{CSB}^{*}+\mathsf{WPP}+\neg\AC).
+\]
+In particular, assuming $\Con(\ZF)$, neither
+$\mathsf{CSB}^{*}\Rightarrow\AC$ nor $\mathsf{WPP}\Rightarrow\AC$ is
+provable in $\ZF$. In the separate transitive setting of
+Theorem~\ref{thm:main-transitive}, the same $W$ also satisfies
+$\mathsf{CSB}^{*}$ and $\mathsf{WPP}$. The assertion $\ACwo$ remains
+true there but is not needed for the implication chain.
+\end{remark}
+
+\subsection{An explicit family without a choice function}
+
+The nonwellorderability of the label set already witnesses failure of
+$\AC$.  The following elementary argument also identifies a family of
+nonempty sets with no choice function.
+
+\begin{lemma}\label{lemma:explicit-choice-failure}
+In $\ZF$, if a set $A$ is not wellorderable, then
+$\mathcal P(A)\setminus\{\varnothing\}$ has no choice function.
+\end{lemma}
+
+\begin{proof}
+Suppose $c$ were a choice function for this family, and let $\lambda$
+be the Hartogs ordinal of $A$, so there is no injection
+$\lambda\longrightarrow A$.  Foundation gives $A\notin A$.
+By transfinite recursion define $a:\lambda\longrightarrow A\cup\{A\}$
+as follows.  At stage $\alpha$, put
+$S_\alpha=A\setminus\ran(a|\alpha)$ and set
+\[
+ a(\alpha)=
+ \begin{cases}
+ c(S_\alpha),&S_\alpha\ne\varnothing,\\
+ A,&S_\alpha=\varnothing.
+ \end{cases}
+\]
+If every $S_\alpha$ were nonempty, the recursion would choose a
+different member of $A$ at every stage, giving an injection
+$\lambda\longrightarrow A$.  Hence there is a least $\alpha<\lambda$
+with $S_\alpha=\varnothing$.  Before that stage no value is the
+sentinel $A$, and $a|\alpha$ is a bijection onto $A$.  This
+wellorders $A$, a contradiction.  Hartogs' theorem and the recursion
+used here are available in $\ZF$.
+\end{proof}
+
+\subsection{The resulting symmetric model}
+
+We return to the construction.  The cardinal choices in the next
+statement are computed in the ground model.  They allow the same
+argument to be used over a given choice ground as well as over a
+constructible ground in the consistency deduction.
+
+Over a general ground, the statement is read in its internal forcing
+interpretation. Ground objects are identified with their canonical
+check-name images, and ordinals are those of this interpretation.
+In the sequence assertion, $\omega$ is the ground's internal
+natural-number object, and $z\in V$ means that $z$ is represented
+by a ground check name. When $V$ is externally transitive, these
+are the usual evaluated objects and literal inclusions. The two
+interpretations are distinguished in Theorem~\ref{thm:symmetric-model}
+and constructed separately in Section~\ref{sec:metatheory}.
+
+\begin{theorem}[Construction summary]\label{thm:construction-summary}
+Let $V$ be a ground model of $\ZFC$, put
+\[
+ \kappa=\omega_1^V,\qquad
+ \mu=\bigl((2^{\aleph_0})^+\bigr)^V,
+ \qquad |I|^V=\mu^+,
+\]
+and use the monoid of Theorem~\ref{thm:monoid} and the symmetric
+system constructed above.  In an ordinary generic extension for
+its product forcing, the resulting symmetric model $W$ satisfies
+\[
+ W\models\ZF+\PP+\ACwo+\neg\AC.
+\]
+It contains $V$ and has the same ordinals as $V$.  It adds no
+countable sequences of ground elements: if $b\in V$ and
+$z\in W$ is a function $\omega\longrightarrow b$, then $z\in V$.
+The set of labels $A$ is not wellorderable in $W$, and
+\[
+ \mathcal P^W(A)\setminus\{\varnothing\}
+\]
+is an explicit family in $W$ with no choice function in $W$.
+\end{theorem}
+
+\begin{proof}
+The symmetric system is for one set forcing, with the set group and
+normal filter specified above.  The set-forcing symmetric-model
+theorem, in its internal form, therefore gives full $\ZF$, the
+canonical ground embedding, and preservation of the interpreted
+ordinals.  For an externally transitive ground its evaluated
+hereditarily symmetric interpretation is transitive.  These are
+pure-set models, so Foundation and the full Separation and Replacement
+schemata are included.
+
+Proposition~\ref{prop:presentation} gives the first hypothesis of
+Lemma~\ref{lemma:pp-reduction} inside $W$, and
+Proposition~\ref{prop:ordinal-choice} gives the second, for all
+ordinal-indexed families in $W$.  Theorem~\ref{thm:quotient-classification},
+together with Remark~\ref{rem:quotients-versus-images}, gives the third.
+The lemma proves $\PP$ for every surjection in $W$, with its witnessing
+injection in $W$.
+
+Proposition~\ref{prop:not-choice} proves that $A$ is not wellorderable
+in $W$.  Applying Lemma~\ref{lemma:explicit-choice-failure} inside
+$W$ gives the displayed family and failure of $\AC$ there.
+Finally, Proposition~\ref{prop:no-countable-sequences} supplies the
+assertion about countable sequences.  This last assertion concerns
+the symmetric model itself; it does not require countable closure
+of the full product forcing.
+\end{proof}
+
+The statement of Theorem~\ref{thm:construction-summary} concerns the
+model produced by the forcing construction.  Extracting an ordinary
+relative-consistency implication, and obtaining an externally
+transitive model under the stronger model-existence hypothesis,
+are separate deductions given next.

preprints/The-Partition-Principle-does-not-imply-Choice-September-24-2026/build/sections/forcing.tex

@@ -0,0 +1,797 @@
+\section{Product forcing and support locality}\label{sec:forcing}
+
+The monoid supplies copies of diagrams with controlled intersections.
+We now turn those copies into two locality principles.  After the Cohen
+generic is fixed, a diagram decides statements whose name parameters it
+supports.  A value represented on two independent diagrams can then be
+named on their common part.  These principles will classify quotients
+and give the choice and coding properties needed to pass from quotients
+to arbitrary sets.
+
+\subsection{The choice ground and the two forcings}
+
+The product separates the generic labels from their ground-model diagram
+geometry. A related Cohen/tower product appears in
+\cite[Section~3]{HolySchilhan2025}; the monoid and the diagram order used
+here are specified below.
+
+Work in a ground model \(V\) of \(\ZFC\).  Set
+\[
+ \kappa=\aleph_1^V,\qquad
+ \mathfrak c=(2^{\aleph_0})^V,\qquad
+ \mu=(\mathfrak c^+)^V,
+ \qquad |I|^V=\mu^+.
+\]
+In the constructible ground used for the consistency result,
+\(\mathfrak c=\aleph_1\), so these parameters are exactly
+\(\kappa=\aleph_1\) and \(\mu=\kappa^+\).
+The slightly more general choice of \(\mu\) will allow us to retain an
+arbitrary given transitive choice ground.
+All objects and cardinal calculations in the construction, unless
+explicitly placed in an extension, belong to \(V\).
+
+For every infinite cardinal \(\lambda<\mu\),
+\[
+ \lambda^{\aleph_0}\leq
+ \mathfrak c^{\aleph_0}=\mathfrak c<\mu.
+\]
+Since \(\mu\) is regular and uncountable, every countable sequence in
+\(\mu\) is bounded there.  It follows that
+\begin{equation}\label{eq:ground-countable-arithmetic}
+ \mu^{\aleph_0}=\mu.
+\end{equation}
+Take the monoid \(T\), with distinguished \(e\neq1\), from
+Theorem~\ref{thm:monoid}, using this \(\mu\).
+Let
+\[
+ \mathbb R=\operatorname{Add}(\kappa,I),\qquad
+ \mathbb P=\{r:T\hookrightarrow I\},
+ \qquad \mathbb Q=\mathbb R\times\mathbb P .
+\]
+Here a condition of \(\mathbb R\) is a partial function
+\(I\times\kappa\to2\) with countable domain, ordered by reverse
+inclusion.  Its coordinate support is
+\[
+ \supp_I(d)=
+ \{j\in I:(\exists\alpha<\kappa)\ (j,\alpha)\in\dom(d)\}.
+\]
+For \(r\in\mathbb P\) and \(n\in T\), put
+\[
+ r_n(x)=r(nx),\qquad D_r=r[T],
+ \qquad r\leq_{\mathbb P}s
+       \ \Longleftrightarrow\
+       (\exists n\in T)\ s=r_n .
+\]
+Thus a stronger diagram contains a specified copy of a weaker one.
+The point \(r(1)\) is its center.  Whenever \(s=r_n\), the element
+\(n\) is determined by the centers, since \(s(1)=r(n)\).
+Left cancellation makes every \(r_n\) injective.  If
+\(r=r_{nm}\), injectivity and conicality give \(n=m=1\); this
+verifies antisymmetry of the order.  Transitivity follows from
+\((r_n)_m=r_{nm}\).
+
+The poset \(\mathbb P\) always denotes this \emph{ground} set,
+including when it is forced over the Cohen extension.
+No closure or cardinal-preservation assertion about \(\mathbb P\)
+or the full product is needed.
+
+\begin{lemma}\label{lem:cohen-properties}
+The forcing \(\mathbb R\) is countably closed and satisfies the
+\(\mu\)-chain condition.  It adds no countable sequences of ground
+elements, preserves \(\kappa=\aleph_1^V\), and preserves every
+ground cardinal at least \(\mu\).  In particular it preserves
+\(\mu\) and \(\mu^+\).
+If \(V\models\mathsf{CH}\), it preserves all cardinals.
+\end{lemma}
+
+\begin{proof}
+A descending countable sequence has its union as a lower bound.
+Deciding the entries of a name for a sequence in a ground set and
+then taking such a lower bound shows that no new countable sequence
+in that set is added.  This also preserves \(\omega_1^V\).
+
+We recall the relevant delta-system argument to identify the exact
+cardinal bound.  A family of \(\mu\) countable sets has a delta-system
+subfamily of size \(\mu\).  Code its union into \(\mu\), and write
+its members as \(S_\alpha\subseteq\mu\), for \(\alpha<\mu\).
+There is a club of \(\delta<\mu\) such that
+\(S_\alpha\subseteq\delta\) whenever \(\alpha<\delta\).
+On the stationary part of this club of cofinality \(\omega_1\),
+each \(S_\alpha\cap\alpha\) is bounded in \(\alpha\).
+The pressing-down lemma makes a bound constant, say \(\gamma<\mu\),
+on a stationary subset.  There are at most
+\(|\gamma|^{\aleph_0}+\aleph_0<\mu\) possible countable subsets
+of \(\gamma\), by
+Equation~\eqref{eq:ground-countable-arithmetic} and its preceding
+bound.  Regularity of \(\mu\) therefore makes
+\(S_\alpha\cap\alpha\) one fixed set on a stationary subset.
+For \(\alpha<\beta\) in that subset,
+\(S_\alpha\subseteq\beta\), so their intersection is this fixed root.
+
+Apply this to the domains of \(\mu\) Cohen conditions.
+There are at most \(2^{\aleph_0}=\mathfrak c<\mu\)
+assignments on the countable root.  Thinning once more gives two,
+indeed \(\mu\), conditions agreeing on the root.  They are compatible,
+which proves the chain condition.
+The usual antichain bound then preserves all cardinals at least
+\(\mu\): the possible values at any one coordinate of a name for
+an ordinal-valued function are covered by fewer than \(\mu\)
+ground ordinals, and regularity handles the case of \(\mu\) itself.
+Countable closure and the chain condition together preserve all
+cardinals when \(\mu=\aleph_2\).
+In a general ground we assert no preservation of cardinals strictly
+between \(\kappa\) and \(\mu\).
+
+Finally, any sequence of ground elements has its values in some ground
+rank segment: their ranks have an ordinal bound, and forcing preserves
+ordinals.  Thus the assertion about sequences in ground sets covers
+all countable sequences of ground elements.
+\end{proof}
+
+\begin{lemma}[Extending and amalgamating diagrams]
+\label{lem:diagram-amalgamation}
+\begin{enumerate}
+\item
+For every \(p\in\mathbb P\) and ground \(J\subseteq I\) of size at
+most \(\mu\), there is \(r\in\mathbb P\) with
+\(r_e=p\) and \(J\subseteq D_r\).
+\item
+A finite family \(s_i\in\mathbb P\) has a common extension whenever
+each pair has either disjoint ranges or an intersection that is
+exactly the range of a common subdiagram
+\[
+ s_{i,a_{ij}}=s_{j,a_{ji}}.
+\]
+\item
+A partial placement of finitely many diagrams that is injective and
+consistent on their
+specified common parts, extends to a ground permutation of \(I\).
+Unspecified coordinates of these diagrams and of finitely many Cohen
+conditions can be placed outside any prescribed ground set of size
+at most \(\mu\).
+\end{enumerate}
+\end{lemma}
+
+\begin{proof}
+The probes from Theorem~\ref{thm:monoid} for
+\(a=1,b=e,U=\{1\}\) give \(\mu\) distinct elements \(m\) with
+\(me=e\).  None belongs to \(eT\): if \(m=ex\), left cancellation
+would give \(xe=1\), contrary to conicality and \(e\neq1\).
+Hence \(|T\setminus eT|=\mu\).
+Define \(r(ex)=p(x)\), using left cancellation, and fill the
+complement injectively in \(I\setminus D_p\), including
+\(J\setminus D_p\).  Both the size of that complement in \(T\)
+and the strict inequality \(\mu<|I|\) give the first assertion.
+
+For the second assertion, identify the copies of \(T\) exactly where
+their labels in \(I\) coincide.  The hypothesis says that these
+identifications have the form required by the finite-merger property;
+their transitivity and absence of collapse within a copy follow from
+actual equality of labels.  Let \(l_i\in T\) realize this merger.
+On \(\bigcup_i l_iT\), assign \(l_ix\) the value \(s_i(x)\).
+The exact-overlap property makes this well defined and injective.
+Extend it injectively to the rest of \(T\).  The resulting \(r\)
+satisfies \(r_{l_i}=s_i\).
+
+For the last assertion, the prescribed domain and range each have
+size at most \(\mu\).  Their complements in \(I\) both have
+cardinality \(\mu^+\), so a bijection between the complements
+extends the partial bijection to a permutation.
+Fresh images can first be chosen off the excluded set, whose size
+is still at most \(\mu\), and then the same argument applies.
+All these choices occur in \(V\).
+\end{proof}
+
+\subsection{The symmetric system and transport}
+
+Let \(G=\Sym(I)^V\).  A permutation acts simultaneously on diagrams
+and on Cohen coordinates:
+\[
+ (\pi r)(x)=\pi(r(x)),\qquad
+ (\pi d)(\pi j,\alpha)=d(j,\alpha).
+\]
+These are automorphisms of \(\mathbb Q\).
+For ground \(J\subseteq I\), let \(\Fix(J)\) be its pointwise
+stabilizer in \(G\), and set
+\[
+ \mathcal F=
+ \{K\leq G:(\exists J\in V)\,
+       [J\subseteq I,\ |J|^V\leq\mu,\ \Fix(J)\subseteq K]\}.
+\]
+This is a normal filter of subgroups.
+Finite intersections are handled by \(J\cup J'\), and
+\(\pi\Fix(J)\pi^{-1}=\Fix(\pi J)\).
+
+The action on names is recursive:
+\[
+ \pi\tau=\{(\pi\sigma,\pi u):(\sigma,u)\in\tau\}.
+\]
+A name has support \(J\) when \(\Fix(J)\) fixes it literally.
+A hereditarily symmetric name, abbreviated \(\HS\), has a support
+and only hereditarily symmetric member names.
+There is no restriction on name rank.
+Ground check names are fixed by every permutation.
+Canonical ordered-pair and sequence names commute with the action.
+When the poset has no largest condition, their usual definitions use
+all conditions in place of that condition; equivalently one may
+adjoin a fixed largest condition.
+
+For the evaluated presentation, assume that \(V\) is externally
+transitive and that the indicated generic filters are available in the
+ambient universe.  Over a general choice ground, the same construction
+and the generic notation below express internal forcing assertions;
+they do not require external recursion on its membership relation.
+Section~\ref{sec:metatheory} realizes these assertions over an arbitrary
+countable ground by a quotient of internal names.
+
+Write \(F\) for an \(\mathbb R\)-generic filter over \(V\), and
+\(H\) for a generic on the ground \(\mathbb P\) over \(V[F]\).
+The product extension is \(V[F][H]\).  A product name \(\tau\),
+after evaluating its Cohen part, becomes a \(\mathbb P\)-name
+denoted by \(\tau[F]\).
+Let
+\[
+ W=\{\tau^{F\times H}:\tau\in\HS^V\}.
+\]
+
+\begin{theorem}[The symmetric-model theorem]\label{thm:symmetric-model}
+The symmetric forcing relation is definable formula by formula in
+\(V\), and it forces every axiom instance of full pure-set \(\ZF\).
+If \(V\) is externally transitive, the evaluated class \(W\) is
+a transitive model of that theory, with
+\[
+ V\subseteq W\subseteq V[F][H],
+ \qquad \Ord^W=\Ord^{V[F][H]}.
+\]
+The truth lemma holds for this evaluated model.  Over an arbitrary
+countable choice ground, it holds for the symmetric quotient of internal
+names described in Section~\ref{sec:metatheory}.
+\end{theorem}
+
+\begin{proof}
+Apply the set-forcing symmetric-extension theorem
+\cite[Section~2.1, p.~4]{Karagila2026} to \(\mathbb Q\), \(G\),
+and the normal subgroup filter verified above.
+Its internal formulation concerns all hereditarily symmetric names
+and yields the full Separation and Replacement schemata, as well as
+Foundation; this formulation is recalled in
+Lemma~\ref{lem:symmetric-forcing-schema}.  It requires no
+countable-completeness assumption on the filter.  External transitivity
+of \(V\) permits the usual recursive name evaluation and gives the
+displayed literal inclusions.  For an arbitrary countable ground,
+Lemma~\ref{lem:arbitrary-model-symmetric-truth} instead gives truth
+in the membership-closed symmetric quotient; it makes no assertion
+of external well-foundedness.
+\end{proof}
+
+The generic terminology below expresses arguments with the ordinary
+and symmetric forcing relations.  In particular, each assertion about
+all generics can instead be read as its forcing-theoretic assertion
+over the choice ground.  The formal consistency deduction is given
+after the construction.
+
+We will use symmetry in its precise product form:
+\begin{equation}\label{eq:product-symmetry}
+ (d,s)\forces_{\mathbb Q}\varphi(\vec\tau)
+ \quad\Longleftrightarrow\quad
+ (\pi d,\pi s)\forces_{\mathbb Q}\varphi(\pi\vec\tau).
+\end{equation}
+The corresponding evaluation identity is
+\[
+ (\pi\tau)^{\pi F\times\pi H}=\tau^{F\times H}.
+\]
+It does not define an automorphism of one fixed extension by sending
+\(\tau^{F\times H}\) to \((\pi\tau)^{F\times H}\).
+The product/iteration equivalence also gives the factorized version
+of Equation~\eqref{eq:product-symmetry}: a Cohen condition forcing
+\(s\forces_{\mathbb P}\varphi(\vec\tau[F])\) is transported to a
+Cohen condition forcing the corresponding assertion at \(\pi s\)
+with names \(\pi\vec\tau\).
+Here \(\pi s\) is explicitly reindexed ground data; it is not the
+image of \(\check s\) under the name action, which fixes check names.
+
+Let \(\dot A_j\) be the canonical name for the Cohen subset of
+\(\kappa\) at coordinate \(j\), and put
+\[
+ A_j=\dot A_j^{F\times H},\qquad A=\{A_j:j\in I\}.
+\]
+The name \(\dot A_j\) has support \(\{j\}\), and
+\(\pi\dot A_j=\dot A_{\pi j}\).
+The name for \(A\) is invariant.  Distinct coordinates give distinct
+labels: below any Cohen condition, assign different bits at one
+ordinal unused at the two coordinates.  This proves forced
+distinctness, which persists after the second forcing.
+
+\subsection{Decision and descent}
+
+Restriction to supporting coordinates and descent to a common support
+have precedents in \cite[Lemmas~10 and~15]{HolySchilhan2025}.
+The following proofs establish the versions needed for our diagrams,
+including arbitrary-rank product names and exact compatibility of the
+moved conditions.
+
+The two factors play different roles in the first lemma.  Once the
+Cohen generic is fixed, strengthening a supporting diagram cannot change
+the truth of a supported assertion.  The assertion can still depend on
+the Cohen generic; the second part says that a Cohen condition witnessing
+it may discard coordinates outside the supporting diagram.
+
+\begin{lemma}[Decision over a supporting diagram]
+\label{lem:local-decision}
+Fix an \(\mathbb R\)-generic \(F\).
+\begin{enumerate}
+\item
+If \(p\in\mathbb P\) and all product-name parameters
+\(\tau_1,\ldots,\tau_k\) are supported by \(D_p\), then \(p\)
+decides each ordinary \(\mathbb P\)-forcing formula
+\(\varphi(\tau_1[F],\ldots,\tau_k[F])\) in \(V[F]\).
+The parameters may have arbitrary rank.
+\item
+Suppose a Cohen condition \(d\) forces that
+\(s\forces_{\mathbb P}\varphi(\vec\tau[F])\), where all the
+product names are supported by \(D_s\).
+Then \(d\restriction(D_s\times\kappa)\) forces the same assertion.
+\end{enumerate}
+For brevity we write \(d\restriction D_s\) for this restriction.
+\end{lemma}
+
+\begin{proof}
+If the first assertion fails, there are \(s_0,s_1\leq p\)
+forcing contrary answers in \(V[F]\).
+Choose \(d_0,d_1\in F\) witnessing those answers in the factorized
+forcing relation.  Their restrictions to \(D_p\times\kappa\)
+are compatible.
+By Lemma~\ref{lem:diagram-amalgamation}, choose
+\(\pi\in\Fix(D_p)\) so that, outside \(D_p\), the coordinates
+of \(s_1,d_1\) move away from those of \(s_0,d_0\).
+The diagrams \(s_0,\pi s_1\) meet exactly in their common
+subdiagram \(p\), so have a common extension.
+The conditions \(d_0,\pi d_1\) agree on the root and have
+disjoint supports off it, so are compatible.
+All parameter names are fixed by \(\pi\).
+Equation~\eqref{eq:product-symmetry} now gives two contrary assertions
+forced by compatible product conditions, a contradiction.
+This argument does not assert that \(\pi d_1\) belongs to \(F\).
+
+For the second assertion, put \(\delta=d\restriction D_s\).
+If \(\delta\) does not force the indicated factorized assertion,
+there is \(c\leq\delta\) forcing its negation.
+Choose \(\pi\in\Fix(D_s)\) moving
+\(\supp_I(d)\setminus D_s\) off \(\supp_I(c)\setminus D_s\).
+The condition \(\pi d\) is compatible with \(c\).
+It forces the original factorized assertion, by product symmetry,
+because \(s\) and all its parameter names are fixed.
+This contradicts the assertion forced by \(c\).
+\end{proof}
+
+We next show that equality between independently supported values
+can be represented on the intersection.  The proof uses a third copy
+to compare the two values while keeping an arbitrary potential
+counterexample compatible.
+
+\begin{lemma}[Support descent]\label{lem:support-descent}
+Suppose \(s_0,s_1\leq p\) and
+\(D_{s_0}\cap D_{s_1}=D_p\).
+Let \(\sigma_i\) be product \(\HS\) names supported by \(D_{s_i}\).
+Fix an \(\mathbb R\)-generic \(F\).
+If some \(v\leq s_0,s_1\) forces
+\(\sigma_0[F]=\sigma_1[F]\) in ordinary \(\mathbb P\)-forcing,
+then there is a product \(\HS\) name \(\sigma\), supported by
+\(D_p\), such that
+\[
+ s_i\forces_{\mathbb P}\sigma[F]=\sigma_i[F]
+ \qquad(i=0,1)
+\]
+in \(V[F]\).
+The names may have arbitrary rank.
+If, in \(V[F]\), each \(s_i\) also forces
+\(\sigma_i[F]\subseteq A\), the descended name can be normalized
+to a subset name with only label names as possible members.
+\end{lemma}
+
+\begin{proof}
+Take \(d\in F\) witnessing the equality forced at \(v\).
+Both parameter supports lie in \(D_v\), so
+Lemma~\ref{lem:local-decision} allows us to replace \(d\) by
+\(d\restriction D_v\).  Set
+\(\delta=d\restriction D_{s_0}\).
+We will combine the translates of \(\sigma_0\) into one
+\(D_p\)-supported name.  A translate contributes its value only when
+the generic contains the corresponding translate of
+\((\delta,s_0)\); the value of the new name will be the union of these
+contributing values.  Thus the main task is to prove that any two
+contributing copies agree.  For this coherence argument, allow both
+factors of the product generic to vary.  We will return to the fixed
+\(F\) after constructing the name.
+
+For every \(\pi\in\Fix(D_p)\), call the copy
+\(\pi\sigma_0\) \emph{active} when its trigger
+\((\pi\delta,\pi s_0)\) belongs to the product generic.
+We first prove that any two active copies have equal values.
+
+Consider two copies whose diagrams \(\bar s,\bar s'\) have
+intersection exactly \(D_p\).  If they can be unequal while active,
+some product condition \((c,t)\) extends both triggers and forces
+the inequality.
+Place a copy \(u\) of \(s_1\) over \(p\), with the rest of its
+range fresh relative to \(D_t\cup\supp_I(c)\).
+Extend the respective placements of
+\((s_0,s_1)\) onto \((\bar s,u)\) and \((\bar s',u)\) to
+placements of \(v\), producing diagrams \(v',v''\).
+Use the same placement on all of \(D_{s_1}\) in both cases.
+Choose all remaining coordinates freshly, so that the ranges have the
+intersections shown in Figure~\ref{fig:support-descent}:
+\begin{equation}\label{eq:descent-three-overlaps}
+ \begin{aligned}
+ D_{v'}\cap D_t&=D_{\bar s},&
+ D_{v''}\cap D_t&=D_{\bar s'},&
+ D_{v'}\cap D_{v''}&=D_u .
+ \end{aligned}
+\end{equation}
+\input{figures/support-descent}
+Every intersection in this display is its indicated common
+subdiagram.  The required partial maps agree on the root, and
+their domain and range sizes are at most \(\mu\);
+Lemma~\ref{lem:diagram-amalgamation} therefore supplies the
+ground permutations realizing these placements.
+
+The two placed copies of \(d\) agree on \(D_u\).
+They agree with \(c\) on \(D_{\bar s}\) and \(D_{\bar s'}\),
+respectively, because \(c\) extends the two placed copies of
+\(\delta\).  On the remaining coordinates freshness gives
+compatibility.  Thus the Cohen conditions have a common extension.
+The three diagrams \(v',v'',t\) also have a common extension by
+Equation~\eqref{eq:descent-three-overlaps} and finite amalgamation.
+The two transported \(\sigma_1\) names are literally equal,
+since the placements agree on the entire \(D_{s_1}\).
+The equalities transported from \((d,v)\) therefore contradict
+the inequality forced by \((c,t)\).
+
+For two active copies with arbitrary overlap, start again with
+any proposed inequality-forcing common extension \((c,t)\).
+Place a third copy of \(s_0\) over \(p\), independent of \(t\)
+off the root, and place its Cohen trigger compatibly with \(c\).
+It has a common product extension with \((c,t)\).
+The independent case just proved applies between this third copy
+and each of the original ones, contradicting the inequality.
+This proves coherence of all active copies, below every possible
+inequality witness.
+
+Define a mixture using all joint strengthenings:
+\begin{equation}\label{eq:descent-mixture}
+ \sigma=
+ \left\{(\pi\eta,(c,t)):
+ \begin{array}{l}
+ \pi\in\Fix(D_p),\quad (\eta,(a,u))\in\sigma_0,\\
+ (c,t)\leq(\pi\delta,\pi s_0),\quad
+ (c,t)\leq(\pi a,\pi u)
+ \end{array}\right\}.
+\end{equation}
+This is a set name.
+Left multiplication in \(\Fix(D_p)\) permutes its entries, so it
+is supported by \(D_p\).
+Its member names are translates of member names of \(\sigma_0\)
+and hence are \(\HS\).
+In any product generic its value is the union of the values of
+the active copies.  Both inclusions in this description follow
+from directedness of the generic: all common strengthenings,
+rather than a selected one, occur in
+Equation~\eqref{eq:descent-mixture}.
+When the original trigger is active, coherence makes this union
+exactly the value of \(\sigma_0\).
+Return now to the fixed \(F\).  Since \(\delta\in F\), the
+original trigger is active in every product generic with this Cohen
+factor and with \(s_0\) in its diagram filter.  It follows that
+\(s_0\forces_{\mathbb P}\sigma[F]=\sigma_0[F]\).
+The equality with \(\sigma_1[F]\) holds below \(v\).
+Its parameters are supported by \(D_{s_1}\), so
+Lemma~\ref{lem:local-decision} makes it hold already below \(s_1\).
+
+No step imposed a rank bound.
+For the subset assertion, replace the mixture by its intersection
+with \(A\), using the normalization in the next paragraph.
+\end{proof}
+
+For \(r\in\mathbb P\), define the following ground set of names:
+\begin{equation}\label{eq:normalized-local-names}
+ \mathcal X(r)=
+ \left\{\tau\subseteq
+     \{\dot A_j:j\in I\}\times\mathbb Q:
+       (\forall\pi\in\Fix(D_r))\ \pi\tau=\tau\right\}.
+\end{equation}
+Every member is \(\HS\).
+If a product \(\HS\) name \(\tau\) is supported by \(J\), define
+\[
+ \tau\mathbin{\upharpoonright} A=
+ \{(\dot A_j,u):j\in I,\ u\in\mathbb Q,\
+                         u\forces_{\mathbb Q}\dot A_j\in\tau\}.
+\]
+The ordinary truth lemma gives value \(\tau^{F\times H}\cap A\).
+Symmetry preserves the support \(J\).
+In particular a name supported by \(D_r\) whose actual value is
+a subset of \(A\) has the same value as some member of
+\(\mathcal X(r)\).
+This is a set-sized normalization, valid for an original name of
+any rank.
+
+\subsection{Coding arbitrary sets by the labels}
+
+The next argument extracts a partial function from an orbit of a name.
+The diagram condition in the function's definition ensures that equal
+inputs have equal values.
+
+\begin{proposition}\label{prop:presentation}
+In \(W\), every nonempty set \(X\) is the image of a surjection
+\(A\times\eta\to X\) for some ordinal \(\eta\).
+No bound on the rank of \(X\) is imposed.
+\end{proposition}
+
+\begin{proof}
+Let \(\sigma\) be a ground \(\HS\) name, and let \(p\) have
+\(D_p\) as a support for \(\sigma\).
+Form the graph name
+\begin{equation}\label{eq:partial-orbit-map}
+ \dot h_{\sigma,p}=
+ \left\{\left(
+    \operatorname{op}(\dot A_{\pi(p(1))},\pi\sigma),
+                      (1_{\mathbb R},\pi p)\right):\pi\in G\right\},
+\end{equation}
+where \(\operatorname{op}\) denotes the canonical ordered-pair name.
+This name is invariant, and all its member names are \(\HS\).
+
+Its value is a partial function on \(A\).
+Indeed, suppose two contributing pairs have equal first coordinates.
+Label distinctness gives
+\(\pi(p(1))=\rho(p(1))\).
+There is a common extension \(r\in H\) of \(\pi p,\rho p\);
+write \(\pi p=r_n\) and \(\rho p=r_m\).
+The equality of centers and injectivity of \(r\) give \(n=m\).
+Consequently \(\pi p=\rho p\) pointwise, so
+\(\rho^{-1}\pi\in\Fix(D_p)\).
+It fixes \(\sigma\), and hence \(\pi\sigma=\rho\sigma\)
+literally.  The two values therefore agree.
+This argument concerns names under compatible diagram conditions,
+not an action on one fixed symmetric model.
+
+The orbit construction gives invariant partial functions.  We now
+form a set-indexed family of them whose ranges cover \(X\).
+Choose an \(\HS\) name \(\tau\) for \(X\).
+The set \(M\) of member names appearing in \(\tau\) is a ground
+set of \(\HS\) names.  In \(V\), enumerate by an ordinal \(\eta\)
+the set
+\[
+ B=\{(\sigma,p)\in M\times\mathbb P:
+                         \Fix(D_p)\text{ fixes }\sigma\}.
+\]
+Every member name has a support of size at most \(\mu\), and
+Lemma~\ref{lem:diagram-amalgamation} provides supporting diagrams.
+We include all such diagrams, because one chosen diagram need not
+belong to the generic filter.
+The corresponding sequence of graph names
+\(\dot h_{\sigma,p}\) is a ground sequence of invariant
+\(\HS\) names.  Its canonical sequence name is itself invariant
+and \(\HS\), so its interpretation
+\(\langle h_i:i<\eta\rangle\) belongs to \(W\).
+
+These partial functions cover \(X\).
+For \(x\in X\), an active member name \(\sigma\) of \(\tau\)
+has value \(x\).
+The set of diagrams whose ranges contain a fixed support of
+\(\sigma\) is ground dense, by
+Lemma~\ref{lem:diagram-amalgamation}.
+Thus some such \(p\) belongs to \(H\).
+The identity entry of \(\dot h_{\sigma,p}\) maps \(A_{p(1)}\)
+to \(x\).
+
+All remaining definitions take place in \(W\).
+For nonempty \(X\), fix one \(x_0\in X\), and define
+\[
+ h(a,i)=
+ \begin{cases}
+ h_i(a),&a\in\dom(h_i)\text{ and }h_i(a)\in X,\\
+ x_0,&\text{otherwise}.
+ \end{cases}
+\]
+Separation and Replacement form this function on \(A\times\eta\),
+and coverage makes it surjective.
+The one default element uses only nonemptiness of \(X\).
+The enumeration above is over \(M\times\mathbb P\), a set
+depending on the particular name \(\tau\); no restriction to
+names below a fixed rank was made.
+\end{proof}
+
+\subsection{Choice for ordinal-indexed families}
+
+The presentation just proved reduces arbitrary families to families of
+subsets of $A$. We first find one wellorderable part of $A$ meeting every
+member of a given ordinal-indexed family of nonempty subsets.
+
+\begin{proposition}\label{prop:ordinal-choice}
+The model \(W\) satisfies \(\ACwo\): every ordinal-indexed
+family of nonempty sets in \(W\) has a choice function in \(W\).
+\end{proposition}
+
+\begin{proof}
+We first treat a family
+\(\langle X_\xi:\xi<\eta\rangle\) of nonempty subsets of \(A\).
+Start below a product condition \((d_0,p)\) forcing this
+assertion for an \(\HS\) family name.
+By extending the diagram, arrange that \(D_p\) contains its
+support and \(\supp_I(d_0)\).
+The assertions that a given graph is a function and has nonempty
+subset values are absolute between \(W\) and the full extension.
+Consequently this use of ordinary forcing follows equally from
+a symmetric-forcing hypothesis.
+Names for the individual \(X_\xi\) can be obtained by the
+membership forcing relation from the family and \(\check\xi\);
+they retain support \(D_p\).
+Choose once and for all \(q\) with \(q_e=p\).
+This same \(q\) will work for every family index.  Witnesses already
+available on \(p\) need no move; for witnesses on proper extensions
+of \(p\), the probes place independent copies inside \(q\) and
+Cohen genericity activates one of them.
+
+Fix \(F\) containing \(d_0\), and fix \(\xi<\eta\).
+In \(V[F]\), some \(t\leq p\) and \(j\in I\) satisfy
+\[
+ t\forces_{\mathbb P} A_j\in X_\xi.
+\]
+This follows by deciding a witness in the ground index set \(I\)
+for membership in \(A\).
+Extend \(t\) to include \(j\) in its range, and write \(p=t_b\).
+Choose \(d\in F\) witnessing the displayed forcing assertion,
+and restrict it to \(D_t\) by
+Lemma~\ref{lem:local-decision}.
+If \(b=1\), then \(t=p\), so \(q\) already forces a witness
+whose index belongs to \(D_q\).
+
+Suppose \(b\neq1\).
+The probes of Theorem~\ref{thm:monoid}, with \(a=1,U=\{1\}\),
+give \(\mu\) elements \(m\) satisfying \(mb=e\) and
+\(mT\cap m'T=eT\) for distinct probes.
+Thus
+\[
+ (q_m)_b=p,\qquad
+ D_{q_m}\cap D_{q_{m'}}=D_p\quad(m\neq m').
+\]
+Each \(q_m\) is a copy of \(t\) over \(p\).
+Choose in \(V\) a permutation \(\pi_m\in\Fix(D_p)\) taking
+\(t\) to \(q_m\).
+The conditions \(\pi_m d\) all restrict on \(D_p\) to
+\(d_*=d\restriction D_p\), and their coordinate supports off
+that root are pairwise disjoint.
+
+For these fixed ground data, the set
+\begin{equation}\label{eq:probe-cohen-dense}
+ \{c:c\perp d_*\}\ \cup\
+ \{c:(\exists m)\ c\leq\pi_m d\}
+\end{equation}
+is dense in \(\mathbb R\).
+Given \(c\) compatible with \(d_*\), first extend it by \(d_*\).
+Its support is countable, so it meets the nonroot support of
+only countably many of the pairwise disjoint copies.
+Choose a remaining probe and extend \(c\) by its placed
+condition \(\pi_m d\).
+This proves density.
+Because \(d_*\in F\), the generic meets the second alternative
+of Equation~\eqref{eq:probe-cohen-dense}.
+For the resulting \(m\), product symmetry and
+\(\pi_m X_\xi=X_\xi\) give
+\[
+ q_m\forces_{\mathbb P} A_{\pi_mj}\in X_\xi
+ \quad\text{in }V[F].
+\]
+Since \(q\leq q_m\) and \(\pi_mj\in D_q\), this supplies the
+required witness in \(A[D_q]=\{A_j:j\in D_q\}\).
+
+The condition \(q\) was chosen before \(\xi\).
+Although \(t,d,j\) were chosen in the argument after \(F\),
+each is an individual ground object.
+The dense set in Equation~\eqref{eq:probe-cohen-dense} therefore
+belongs to \(V\), and \(F\) meets it.
+No sequence of witnesses for all \(\xi\) is being selected or
+imported into \(W\).
+For each \(\xi<\eta\) the argument shows that \(q\) forces
+\(X_\xi\cap A[D_q]\neq\varnothing\) in \(V[F]\).
+It forces the universal statement as well: a condition forcing
+a counterexample could be strengthened to decide its index in
+\(\check\eta\) to one ground \(\xi\), contradicting the
+assertion for that index.
+Since this holds for every \(F\) containing \(d_0\), the product
+condition \((d_0,q)\) forces the universal meeting property.
+
+A ground enumeration of \(D_q\) yields a sequence of its labels
+with an \(\HS\) name supported by \(D_q\).
+Distinctness of labels turns that enumeration into a wellordering
+of \(A[D_q]\) in \(W\).
+Taking the first element of each \(X_\xi\) in this wellordering
+defines the required choice function by Replacement in \(W\).
+The argument works below an extension of every starting
+condition forcing the subset-family hypothesis, and hence proves
+choice for all such families in \(W\).
+
+Finally let \(\langle Y_\xi:\xi<\eta\rangle\) be any family
+of nonempty sets in \(W\).
+For \(\eta=0\) use the empty function.
+Otherwise Proposition~\ref{prop:presentation}, applied once to
+the union of the family, gives a surjection
+\(h:A\times\theta\to\bigcup_{\xi<\eta}Y_\xi\) in \(W\).
+For each \(\xi\), take the least \(\alpha_\xi<\theta\)
+whose slice meets \(Y_\xi\), and form
+\[
+ B_\xi=\{a\in A:h(a,\alpha_\xi)\in Y_\xi\}.
+\]
+These are a family of nonempty subsets of \(A\) in \(W\).
+The subset case chooses \(a_\xi\in B_\xi\);
+then \(h(a_\xi,\alpha_\xi)\) is the desired choice from \(Y_\xi\).
+\end{proof}
+
+\subsection{Failure of Choice and countable-sequence preservation}
+
+\begin{proposition}\label{prop:not-choice}
+The set \(A\) is not wellorderable in \(W\).
+In particular \(W\models\neg\AC\).
+\end{proposition}
+
+\begin{proof}
+Suppose an \(\HS\) name is forced to be a function from an
+ordinal \(\eta\) onto \(A\).
+Choose a forcing condition \((d,p)\) whose diagram range contains
+the name's support and \(\supp_I(d)\).
+Fix \(j\notin D_p\), which exists since
+\(|D_p|^V=\mu<|I|^V\).
+Strengthen to a product condition \((c,t)\) deciding
+a ground \(\beta<\eta\) with value \(A_j\), and extend the
+diagram further to have \(j\in D_t\).
+
+Choose \(\pi\in\Fix(D_p)\) moving all coordinates of \(t,c\)
+outside \(D_p\) to fresh ones.
+Then \(t,\pi t\) are independent over \(p\) and have a common
+extension; \(c,\pi c\) are also compatible.
+The function name and \(\check\beta\) are fixed.
+Their common product extension would therefore force that
+one entry has both values \(A_j\) and \(A_{\pi j}\).
+They are distinct because \(j\notin D_p\) and \(\pi\) moved it
+freshly.  This is a contradiction.
+
+\end{proof}
+
+\begin{proposition}\label{prop:no-countable-sequences}
+For every ground set \(X\), every function \(f:\omega\to X\)
+belonging to \(W\) belongs to \(V\).
+Consequently \(W\) adds no countable sequences of ground elements.
+\end{proposition}
+
+\begin{proof}
+Choose a product \(\HS\) name \(\tau\) for \(f\).
+In \(V[F]\), the ordinary \(\mathbb P\)-truth lemma supplies
+a condition in \(H\) forcing that \(\tau[F]\) is a function
+\(\check\omega\to\check X\).
+Extend within \(H\) to a diagram \(p\) whose range covers a
+support of \(\tau\).
+For each \(n<\omega\), the ordinary forcing theorem supplies
+an extension of \(p\) and some \(x\in X\) forcing
+\(\tau[F](n)=\check x\).
+Both \(\tau\) and the check-name parameters are supported by
+\(D_p\), so Lemma~\ref{lem:local-decision} makes \(p\)
+itself force that equality.
+There is exactly one such \(x\), since \(p\) forces
+functionhood.
+
+The forcing relation is definable in \(V[F]\).
+Replacement there therefore forms the sequence of these unique
+values, a function \(g:\omega\to X\) in \(V[F]\).
+Then \(g=f\) in \(V[F][H]\), because \(p\in H\).
+By Lemma~\ref{lem:cohen-properties}, \(\mathbb R\) adds no
+countable sequences in the ground set \(X\), so \(g\in V\).
+Thus \(f\in V\).
+The final assertion follows by bounding the ranks of the entries
+of any countable sequence of ground elements, as in the proof of
+Lemma~\ref{lem:cohen-properties}.
+\end{proof}
+
+We have obtained the all-rank presentation and internal ordinal choice
+needed for the final reduction to \(\PP\), while the labels already
+witness failure of full Choice.  What remains is the quotient
+classification: nonwellorderable images of \(A\) must be put in
+bijection with \(A\) by one coherent, hereditarily symmetric graph.

preprints/The-Partition-Principle-does-not-imply-Choice-September-24-2026/build/sections/introduction.tex

@@ -0,0 +1,213 @@
+\section{Introduction}\label{sec:introduction}
+
+The \emph{Partition Principle} asserts that a surjection between two sets
+implies the existence of an injection in the reverse direction:
+\[
+ \PP:\qquad
+ \forall X\,\forall Y\,
+ \bigl[(\exists f:X\twoheadrightarrow Y)
+       \Longrightarrow(\exists j:Y\hookrightarrow X)\bigr].
+\]
+Equivalently, every partition of a set admits an injection into that set.
+The Axiom of Choice implies this principle by choosing one element of each
+fiber. The converse is less direct: the injection supplied by $\PP$ need
+not take a part to one of its members. In particular, it need not satisfy
+$f\circ j=\id_Y$. The problem is whether this cardinal comparison alone
+can recover choice for arbitrary families.
+
+Write $\ACwo$ for Choice for wellorderable families: for every ordinal
+$\eta$ and every family $(X_\xi)_{\xi<\eta}$ of nonempty sets, there is a
+function $c$ with $c(\xi)\in X_\xi$ for all $\xi<\eta$.
+Our principal result is the following ordinary relative-consistency statement.
+
+\begin{theorem}\label{thm:main-consistency}
+If $\ZF$ is consistent, then so is $\ZF+\PP+\ACwo+\neg\AC$.
+\end{theorem}
+
+The construction also has a separate transitive-model form.
+
+\begin{theorem}\label{thm:main-transitive}
+Let $V$ be a countable transitive model of $\ZFC$. There are a set forcing
+$\mathbb Q\in V$, a $V$-generic filter $G$, and a transitive symmetric
+submodel $W$ of $V[G]$ such that
+\[
+ V\subseteq W\subseteq V[G],\qquad
+ \Ord^W=\Ord^V,\qquad
+ W\models\ZF+\PP+\ACwo+\neg\AC.
+\]
+Every countable sequence of elements of $V$ that belongs to $W$ already
+belongs to $V$.
+\end{theorem}
+
+Here $\ZF$ is classical first-order set theory with pure sets, Foundation,
+and the full Separation and Replacement schemata.
+Theorem~\ref{thm:main-consistency} gives a negative answer to the
+Partition Principle versus Choice problem under ordinary syntactic
+consistency. It permits externally ill-founded models, whereas
+Theorem~\ref{thm:main-transitive} starts with a transitive ground and
+preserves transitivity. In either conclusion the reverse injections and
+the family without a choice function belong to the same model.
+Section~\ref{sec:metatheory} gives the two deductions separately.
+The consequences for the dual Cantor--Schr\"oder--Bernstein principle
+and the Weak Partition Principle are developed in
+Remark~\ref{rem:dual-csb-wpp}.
+
+\subsection{History and related methods}
+
+The distinction between comparing the sizes of a partition and its
+underlying set, and actually choosing representatives, is longstanding.
+Da Silva traces the Partition Principle
+to Beppo Levi in 1902 \cite[Section~2, p.~787]{daSilva2021}.
+Banaschewski and Moore studied its relation to the dual
+Cantor--Bernstein theorem and the surrounding cardinal-comparison
+principles \cite{BanaschewskiMoore1990}. These questions concern what
+remains of cardinal arithmetic when sets need not be wellorderable.
+For example, Blass and Kulshreshtha distinguish several forms of
+cardinal well-foundedness and show that they become equivalent under
+$\PP$; their August 2026 version still lists $\PP\Rightarrow\AC$
+as open \cite[Introduction and Section~9]{BlassKulshreshtha2026}.
+Gilson's September 2026 corrections withdraw the model-construction
+claims of three earlier approaches
+\cite{GilsonSheafCorrection2026,GilsonExternalCorrection2026,GilsonIterationCorrection2026}.
+
+There is already substantial choice in $\PP$. The implication
+$\PP\Rightarrow\ACwo$ is classical: Ryan-Smith records the
+argument attributed to Pincus by Pelc
+\cite[Proposition~3.17, proof and footnote~4]{RyanSmith2025}.
+Thus $\ACwo$ in our main statements is not an additional strength
+beyond $\PP$. Its separate appearance records the order of the
+construction: we establish ordinal-indexed choice first and then use
+it to assemble reverse injections for arbitrary sets.
+
+The difficulty of preserving a principle quantified over all sets
+can be reduced to a fixed set of parameters. Blass introduced small
+violations of choice \cite{Blass1979}. For a nonempty set $A$,
+$\mathrm{SVC}(A)$ says that every nonempty set is a surjective image
+of $A\times\eta$ for some ordinal $\eta$. The injective version,
+$\mathrm{SVC}^{+}(A)$, says that every set embeds into some
+$A\times\eta$. See also
+\cite[Definition~3.13 and Theorem~3.14]{Usuba2021}.
+These presentations let a construction on one set control objects
+of arbitrary rank, but their different directions cannot be
+interchanged without proof.
+
+Ryan-Smith makes the local-to-global step precise. Under
+$\mathrm{SVC}^{+}(A)$, full $\PP$ is equivalent to $\ACwo$
+together with reverse injections for surjections between subsets of
+$A$ \cite[Proposition~3.17]{RyanSmith2025}. Reverse injections for
+surjective images of $A$ are a different local assertion: even with
+$\ACwo$ and $\mathrm{SVC}^{+}(A)$, they need not imply full
+$\PP$ \cite[Proposition~3.23]{RyanSmith2025}. Our construction
+instead makes every nonwellorderable image of $A$ bijective to $A$.
+Lemma~\ref{lemma:pp-reduction} uses this stronger classification,
+the surjective presentation, and ordinal-indexed choice to carry out
+the required all-set reduction. Its slicing argument follows the
+method of \cite[Propositions~3.17 and~3.20]{RyanSmith2025}, rather
+than identifying the two local principles.
+
+The ambient method is a symmetric submodel of a set-forcing
+extension: hereditarily symmetric names for a normal filter of
+subgroups give a model of full $\ZF$
+\cite[Section~2.1]{Karagila2026}. A more specific antecedent is
+Holy and Schilhan's product of Cohen forcing with a forcing of
+ground-model hereditarily almost-disjoint towers
+\cite[Section~3]{HolySchilhan2025}. Their construction yields
+models in which every set can be linearly ordered and higher forms
+of Dependent Choice hold, although full Choice fails. They develop
+the hereditary almost-disjointness method from Pincus's work on
+adding Dependent Choice \cite{Pincus1977}. Their restriction and
+support-intersection arguments
+\cite[Lemmas~10 and~15]{HolySchilhan2025} are antecedents of the
+locality arguments here. Our diagrams have a different geometry:
+the monoid must support both exact quotient multiplicities and the
+coherent assignment of bijections, not just the descent of supports.
+
+\subsection{Proof strategy}
+
+We construct a set $A$ of distinct generic subsets of the ground-model
+$\omega_1$. Symmetry
+prevents $A$ from being wellordered. Orbit maps of names show that every
+nonempty set is a surjective image of $A\times\eta$ for some ordinal
+$\eta$, with no restriction on its rank. The forcing also allows one
+wellorderable part of $A$ to meet every member of any given ordinal-indexed
+family of nonempty subsets of $A$. Together with the preceding
+presentation of arbitrary sets, this yields $\ACwo$.
+
+The central additional property is
+\begin{equation}\label{eq:intro-quotient}
+ \text{every nonwellorderable surjective image of $A$ is in bijection with $A$.}
+\end{equation}
+To pass from this property to full $\PP$, partition the target of an
+arbitrary surjection into ordinal-indexed disjoint nonempty pieces and
+use disjoint source pieces mapping onto them. Each resulting piece is
+an image of $A$, though the source pieces need not cover the original
+domain. A wellorderable target piece admits a reverse injection by
+$\ACwo$. Otherwise both corresponding pieces are in bijection with $A$
+by \eqref{eq:intro-quotient}. A further application of $\ACwo$ chooses
+the piecewise injections, whose union is the required injection. The
+empty target uses the empty injection.
+
+The difficult part is to construct the bijections in
+\eqref{eq:intro-quotient} with enough coherence to belong to the symmetric
+model. Fix one equivalence relation on $A$ whose quotient is not
+wellorderable. Counting its classes on
+each support will produce many possible local bijections, but arbitrary
+choices of them need not agree on overlapping supports or respect
+coordinate changes. The proof solves these two compatibility problems
+in three stages.
+
+\begin{enumerate}[leftmargin=*,label=\arabic*.]
+\item \textbf{A geometry of diagrams.}
+We build a left-cancellative monoid $T$. A diagram is an injective map
+$r:T\to I$, where $I$ is the set of label indices. Its subdiagram at $n\in T$
+is the map $r_n(x)=r(nx)$. The diagram forcing makes larger diagrams
+stronger conditions. Finite amalgamations join independently placed
+diagrams. A second extension operation places many copies with prescribed
+patterns of intersection. Together these operations give the forcing its
+compatibility and support-descent properties. Height and cost functions
+reduce the availability patterns introduced next to countable cofinal
+paths. A rearrangement of the same monoid construction will later bound
+the symmetries that survive along such a path.
+
+\item \textbf{Exact availability fibers.}
+A diagram supports names whose definitions are fixed by all permutations
+of labels outside its range. For a quotient of $A$, we record the quotient
+elements that can be named at each diagram. An element's availability
+pattern records the subdiagrams at which it can still be named.
+The labels themselves have analogous patterns, determined by which
+subdiagrams contain their indices. The copying and support-descent lemmas
+show that, for each exact pattern, the available labels and quotient
+elements have the same cardinality. Restricting to a subdiagram in that
+pattern identifies the corresponding fibers canonically. These
+identifications group the fibers into components and are already
+compatible; the choices of bijections between label and quotient fibers
+are not.
+
+\item \textbf{Covariant choices on each component.}
+Each component has a countable cofinal chain of nested diagrams. Codes
+along this chain give a preliminary bijection compatible with every
+permutation fixing a sufficiently late diagram pointwise. Modulo these
+permutations, the remaining symmetries form a countable group. Countably
+many bounded initial segments of Cohen subsets distinguish its elements
+and select among the transported preliminary bijections. Finally,
+transport between components gives assignments that agree under
+restriction and respect all required permutations. Their union is one
+hereditarily symmetric graph for the fixed quotient.
+\end{enumerate}
+
+The countable description of each availability pattern, the reduction of
+its symmetries to a countable group, and the final equivariant selection
+are the main technical ingredients. The selection is performed between
+moved Cohen generics: it does not require permutations to act as
+automorphisms of a single fixed forcing extension.
+
+Section~\ref{sec:monoid} constructs $T$ and proves its local rearrangement
+property. Section~\ref{sec:forcing} develops the forcing, locality, and the
+basic properties of $A$. The quotient construction follows in
+three stages: Section~\ref{sec:quotient-fibers} computes exact availability
+fibers, Section~\ref{sec:quotient-components} organizes them into components
+and controls their symmetries, and Section~\ref{sec:quotient-equivariance}
+constructs the coherent bijections. Section~\ref{sec:conclusion} derives
+full $\PP$ and identifies a family without a choice function.
+Section~\ref{sec:metatheory} gives the ordinary consistency deduction and
+the separate transitive-ground result.

preprints/The-Partition-Principle-does-not-imply-Choice-September-24-2026/build/sections/metatheory.tex

@@ -0,0 +1,439 @@
+\section{The consistency deduction}\label{sec:metatheory}
+
+Theorem~\ref{thm:construction-summary} has two uses. For ordinary
+relative consistency, we must obtain a choice ground from
+$\Con(\ZF)$ and interpret the construction even when that ground
+is externally ill founded. For the separate transitive-model result,
+we retain the given transitive choice ground and evaluate names in
+the usual way. We begin with the precise forcing assertion that
+both deductions use.
+
+\subsection{The internal output of the construction}
+
+Write $\mathcal S=(\mathbb Q,\Gamma,\mathcal F)$ for a set
+symmetric system: $\mathbb Q$ is a nonempty set poset, $\Gamma$
+is a set group of automorphisms of $\mathbb Q$, and $\mathcal F$
+is a normal filter of subgroups of $\Gamma$. Let
+$\HS_{\mathcal S}$ be its class of hereditarily symmetric names.
+We use a largest condition $1$. If necessary, adjoin a fresh
+largest condition fixed by every automorphism; the original poset
+remains dense.
+
+The preceding construction is uniform over choice grounds. In
+particular, Theorem~\ref{thm:construction-summary} gives the
+following internal assertion in $\ZFC+V=L$: there is a nonempty
+set symmetric system $\mathcal S$ such that
+\begin{equation}\label{eq:construction-internal-forcing}
+ 1\forces_{\HS_{\mathcal S}}
+          \PP\ \wedge\ \neg\AC\ \wedge\ \ACwo.
+\end{equation}
+Here the universal quantifiers of the three sentences range over
+all internal HS names, with no name-rank cutoff. The standard
+symmetric-model theorem supplies full $\ZF$ for this interpretation;
+its formula-by-formula justification is given below.
+
+The internal character of this particular construction matters.
+The monoid is built by recursion along a set ordinal, and its
+countable-base rearrangement uses an elementary submodel of a set
+structure $H_\Theta$. Its normal forms, diagrams, permutations and
+supports are sets. In the quotient argument,
+Equation~\eqref{eq:normalized-local-names} bounds the local name
+collections by ground sets. The assignment predicate
+$\Phi$ in Equation~\eqref{eq:assignment-predicate} uses those
+sets, fixed ground choices, and the ordinary forcing relation for
+the ground poset $\mathbb P$ in an $\mathbb R$-extension.
+Equation~\eqref{eq:quotient-bijection-name} then forms an actual
+ground HS graph name. Thus the generic notation in that argument
+expresses fixed-formula forcing assertions; it is not a construction
+available only over externally transitive models.
+
+Propositions~\ref{prop:presentation} and~\ref{prop:ordinal-choice}
+supply the all-rank presentation and ordinal-indexed choice.
+Together with the quotient graph, the internal $\ZF$ reduction in
+Lemma~\ref{lemma:pp-reduction} gives $\PP$.
+Proposition~\ref{prop:not-choice} supplies $\neg\AC$.
+The quotient witnesses are produced below a strengthening of every
+condition asserting the relevant hypothesis. This is the local-witness
+form of existential forcing, made explicit in
+Equation~\eqref{eq:symmetric-existential-density} below; no single
+symmetric witness is required to work below every condition.
+
+It remains to transfer this forcing assertion to a model obtained
+from ordinary consistency. We first obtain a choice ground by the
+constructible interpretation, then define the symmetric extension
+without external recursion on an ill-founded membership relation.
+
+\subsection{The constructible choice ground}
+
+\begin{lemma}\label{lem:constructible-consistency}
+For every axiom $\alpha$ of $\ZFC+V=L$, its relativization
+$\alpha^L$ is provable in $\ZF$. Consequently
+\[
+ \Con(\ZF)\ \Longrightarrow\ \Con(\ZFC+V=L).
+\]
+Moreover, $\ZFC+V=L$ proves the generalized continuum hypothesis.
+\end{lemma}
+
+\begin{proof}
+The constructible-universe theorem proves, in $\ZF$, the
+relativization to $L$ of every $\ZF$ axiom, of Choice, and of
+$V=L$. These are assertions for each individual axiom instance,
+including the instances of Separation and Replacement. See
+\cite[Theorem~4.7, Lemma~4.10, Corollary~4.13, and
+Theorems~4.14 and~4.18]{Welch2020}.
+
+Relativization commutes with Boolean connectives and preserves
+first-order inference when quantifiers are restricted to the
+nonempty interpreted domain. A finite contradiction proof from
+$\ZFC+V=L$ would therefore yield, by relativizing each line and
+using the finitely many required interpreted axioms, a
+contradiction proof from $\ZF$. This proves the consistency
+implication. The same constructible-universe theorem proves GCH
+in $L$, hence proves GCH from $\ZFC+V=L$.
+\end{proof}
+
+In this choice ground the parameters of the construction are
+\[
+ \kappa=\aleph_1,\qquad \mu=\aleph_2,\qquad I=\aleph_3.
+\]
+These are a convenient specialization of the arbitrary-ground
+parameters in Section~\ref{sec:forcing}. GCH gives
+\[
+ \mu^{\aleph_0}=(2^\kappa)^{\aleph_0}=\mu.
+\]
+Since $\kappa=\aleph_1$, this is the bound
+$\mu^{<\kappa}=\mu$ used in the construction. CH also gives
+$\kappa^{<\kappa}=\kappa$ here, but that additional equality is
+not an assumption of the arbitrary-ground argument. The required
+general bounds were proved in
+Equation~\eqref{eq:ground-countable-arithmetic} and its preceding
+paragraph. The set constructions and ground wellorderings used
+earlier are available in $\ZFC+V=L$. In particular, $\Theta$ in
+the monoid rearrangement need only be a sufficiently large regular
+cardinal; no inaccessible cardinal or satisfaction predicate for
+the universe is required.
+
+\subsection{Symmetric forcing as a first-order interpretation}
+
+We now justify the interpretation machinery used with
+Equation~\eqref{eq:construction-internal-forcing}. The point is
+not only to obtain the elementary set axioms: every instance of
+Separation and Replacement must hold for arbitrary-rank names.
+
+\begin{lemma}[The symmetric forcing schema]
+\label{lem:symmetric-forcing-schema}
+For each fixed formula $\varphi$ in the language of set theory,
+the relation
+\[
+ p\forces_{\HS_{\mathcal S}}\varphi(\tau_1,\ldots,\tau_n)
+\]
+is definable in $\ZFC$, with $\mathcal S$ as a parameter and
+$\tau_1,\ldots,\tau_n\in\HS_{\mathcal S}$.
+It satisfies the forcing rules for first-order logic with equality
+and the symmetry lemma. For each axiom $\alpha$ of $\ZF$,
+including each Separation and Replacement instance, $\ZFC$
+proves
+\[
+ \mathcal S\text{ is a set symmetric system}
+ \quad\Longrightarrow\quad
+ 1\forces_{\HS_{\mathcal S}}\alpha.
+\]
+\end{lemma}
+
+\begin{proof}
+Atomic forcing is defined by the usual recursion on names.
+For a fixed formula, extend these atomic definitions through its
+finitely many connectives and quantifiers, restricting quantified
+names to $\HS_{\mathcal S}$. For example, define
+\[
+ D_{\varphi,\vec\tau}
+ =\{q\in\mathbb Q:(\exists\sigma\in\HS_{\mathcal S})\
+              q\forces_{\HS_{\mathcal S}}\varphi(\sigma,\vec\tau)\}.
+\]
+Existential forcing is characterized by
+\begin{equation}\label{eq:symmetric-existential-density}
+ p\forces_{\HS_{\mathcal S}}\exists x\,\varphi(x,\vec\tau)
+ \quad\Longleftrightarrow\quad
+ D_{\varphi,\vec\tau}\text{ is dense below }p.
+\end{equation}
+The set $D_{\varphi,\vec\tau}$ exists by Separation from $\mathbb Q$.
+The class of names and the hereditary symmetry predicate are
+definable by set-theoretic recursion. Thus all the displayed
+relations are ordinary first-order predicates. This construction
+is a schema indexed externally by formulas; it asserts no
+single satisfaction predicate for all formulas.
+
+The logical and symmetry rules follow by induction on a fixed
+formula, with the internal name recursion supplying the atomic
+case. The resulting symmetric-model theorem and its forcing
+relation are standard; see
+\cite[Equations~(15.39)--(15.41) and Lemma~15.51]{Jech2003}
+and \cite[Section~2.1, p.~4]{Karagila2026}.
+We recall the parts that ensure the full axiom schemata.
+
+Fix an arbitrary formula $\varphi(x,\vec a)$ and
+hereditarily symmetric names $\sigma,\vec\tau$ for its set and
+parameters. A Separation name uses the pairs
+$(\rho,q)\in\dom(\sigma)\times\mathbb Q$ for which
+\[
+ q\forces_{\HS_{\mathcal S}}
+       \rho\in\sigma\ \wedge\ \varphi(\rho,\vec\tau).
+\]
+The intersection of the stabilizers of $\sigma$ and the finitely
+many parameter names fixes this name. That intersection belongs
+to $\mathcal F$, and every member name is hereditarily symmetric.
+The atomic forcing rules and
+Equation~\eqref{eq:symmetric-existential-density} verify that it
+is the required subset. No restriction on $\varphi$ or on the
+rank of $\sigma$ has been imposed.
+
+Write $\operatorname{nrk}(\eta)$ for the name-rank
+\[
+ \operatorname{nrk}(\eta)=
+ \sup\{\operatorname{nrk}(\zeta)+1:(\zeta,r)\in\eta\}.
+\]
+For Replacement, fix a formula $\psi(x,y,\vec a)$, a domain
+name $\sigma$, and parameter names $\vec\tau$. For each pair
+$(\rho,q)\in\dom(\sigma)\times\mathbb Q$ for which some
+hereditarily symmetric name $\eta$ satisfies
+\[
+ q\forces_{\HS_{\mathcal S}}\psi(\rho,\eta,\vec\tau),
+\]
+take the least possible name-rank of such an $\eta$.
+This defines an ordinal-valued function on a ground set.
+Replacement in the ground bounds all these ranks strictly below
+one ordinal $\beta$. All $\mathbb Q$-names of name-rank less
+than $\beta$ form a set. Consequently
+\[
+ \dot B_\beta=
+ \{(\eta,1):\eta\in\HS_{\mathcal S},\
+                         \operatorname{nrk}(\eta)<\beta\}
+\]
+is a set name. Every automorphism preserves name-rank and
+hereditary symmetry, so $\dot B_\beta$ is fixed by the whole
+group and belongs to $\HS_{\mathcal S}$.
+
+Below a condition forcing totality of the relation on $\sigma$,
+local witnesses from
+Equation~\eqref{eq:symmetric-existential-density} can therefore
+be replaced, at the same witnessing conditions, by names occurring
+in $\dot B_\beta$. This gives a set bounding the required
+witnesses. When the relation is functional, Separation in that
+set gives its image. The bound $\beta$ depends on the particular
+domain name and formula; it is not a bound on the universe of
+names.
+
+For Power Set, normalize a name for a subset of $\sigma$ using
+member names from $\dom(\sigma)$ and the forcing tests for
+membership in that subset. The normalized names lie in the
+ground set $\mathcal P(\dom(\sigma)\times\mathbb Q)$.
+Use all hereditarily symmetric candidates, under the conditions
+forcing that they are subsets of $\sigma$. The stabilizer of
+$\sigma$ permutes this collection, giving a hereditarily
+symmetric power-set name. Every symmetric subset has such a
+normalization, so the name supplies the entire internal power set.
+Check names and the canonical pairing and union constructions
+supply the remaining elementary existence axioms.
+
+The usual internal name-rank argument verifies Foundation.
+Equivalently, the interpreted hereditary class is membership-closed
+inside the ordinary forcing extension: an element of an
+HS-valued set has an active member name of an HS name.
+Extensionality and Foundation then inherit their witnesses from
+the ordinary extension. This argument concerns first-order
+Foundation and does not assume that an arbitrary model is
+externally well founded.
+
+All these name constructions and forcing verifications are proofs
+in the ground theory, for the chosen formula. They prove the
+asserted schema. They use local existential witnesses rather
+than a maximum principle for symmetric forcing; that maximum
+principle need not hold
+\cite[Proposition~10.26]{KaragilaNotes2026}.
+\end{proof}
+
+\subsection{The extension of an arbitrary countable model}
+
+Let $M=(|M|,E^M)$ be a countable, possibly ill-founded model of
+$\ZFC$. Suppose $M$ contains a set symmetric system
+$\mathcal S$. Its internal poset determines the external set
+\[
+ Q^M=\{p\in|M|:M\models p\in\mathbb Q\},
+\]
+with the order interpreted by $M$. An $M$-generic filter means
+a filter on this poset meeting the interpretation of every
+$D\in|M|$ for which $M$ asserts that $D$ is dense in
+$\mathbb Q$.
+
+Such a filter exists through any prescribed condition. Enumerate
+these dense-set codes externally and choose a descending sequence
+meeting them successively. Internal density really supplies an
+external strengthening in $Q^M$, since its witnesses are elements
+of the domain of $M$. The upward closure of the chosen sequence
+is the required generic filter $G$. No closure property of
+$\mathbb Q$ and no well-foundedness of $E^M$ are needed here.
+
+Let $\mathcal N^M$ be the external set of elements of $M$ which
+$M$ regards as $\mathbb Q$-names. Define
+\begin{align}
+ \sigma\sim_G\tau
+ &\quad\Longleftrightarrow\quad
+   (\exists p\in G)\ M\models p\forces\sigma=\tau,
+ \label{eq:internal-name-equivalence}\\
+ [\sigma]_G\ E_G\ [\tau]_G
+ &\quad\Longleftrightarrow\quad
+   (\exists p\in G)\ M\models p\forces\sigma\in\tau.
+ \label{eq:internal-name-membership}
+\end{align}
+The equality and substitution laws proved internally in $M$,
+together with directedness of $G$, make $\sim_G$ an equivalence
+relation and make $E_G$ independent of representatives.
+The quotient of all internal names is the ordinary extension
+structure $M[G]$. Its substructure $W$ consists of the equivalence
+classes having a representative which $M$ regards as hereditarily
+symmetric. Both are external set structures, since $|M|$ is a set.
+The ordinary forcing quotient over nontransitive models is
+described in \cite[Section~4, p.~12]{Williams2019}.
+
+\begin{lemma}[Truth for the symmetric quotient]
+\label{lem:arbitrary-model-symmetric-truth}
+For each standard first-order formula $\varphi$ and internal
+hereditarily symmetric names $\vec\tau$,
+\[
+ W\models\varphi([\vec\tau]_G)
+ \quad\Longleftrightarrow\quad
+ (\exists p\in G)\ M\models
+      p\forces_{\HS_{\mathcal S}}\varphi(\vec\tau).
+\]
+In particular $W\models\ZF$.
+\end{lemma}
+
+\begin{proof}
+Induct externally on the finite syntax of the standard formula,
+using atomic formulas, conjunction, negation, and existential
+quantification. The atomic case is
+Equations~\eqref{eq:internal-name-equivalence} and
+\eqref{eq:internal-name-membership}; atomic symmetric forcing
+agrees with ordinary forcing. Conjunction uses a common
+strengthening of two conditions in $G$ and monotonicity.
+
+For negation, the set of conditions deciding the particular
+formula is an internally definable dense subset of $\mathbb Q$.
+It belongs to $M$ by the Separation instance for its defining
+forcing predicate. Genericity gives a deciding condition in $G$.
+The induction hypothesis excludes the wrong decision, and
+directedness excludes conditions in $G$ forcing opposite answers.
+
+Suppose $p\in G$ symmetrically forces an existential statement.
+The set of local witness conditions in
+Equation~\eqref{eq:symmetric-existential-density}, together with
+the conditions incompatible with $p$, is another internally
+definable dense set. Genericity meets its first part. The internal
+witness is an actual element of $|M|$ which $M$ regards as an HS
+name. Its equivalence class is a witness in $W$ by induction.
+Conversely, a witness in $W$ has an internal HS representative;
+apply induction to that representative and the elementary
+existential forcing rule.
+
+This proves the truth equivalence for every standard formula.
+Lemma~\ref{lem:symmetric-forcing-schema}, applied in $M$ to each
+standard $\ZF$ axiom instance, now gives $W\models\ZF$.
+The proof never evaluates a name by external recursion over
+$E^M$. Nor does it require a truth predicate in $M$ for all
+formulas, including its possible nonstandard formula codes.
+\end{proof}
+
+The membership-closed assertion used above can also be seen
+directly in this quotient. If $[\sigma]_G E_G[\tau]_G$ and
+$\tau$ is internally HS, the atomic forcing rule and genericity
+provide an active member name $\rho$ of $\tau$ with
+$[\rho]_G=[\sigma]_G$. Heredity makes $\rho$ internally HS.
+Thus $W$ and $M[G]$ have the same members of each $W$-object.
+This relational form of transitivity does not make $E_G$ an
+externally well-founded relation.
+
+\subsection{Applying the construction}
+
+The internal forcing assertion
+\eqref{eq:construction-internal-forcing} can now be used in a
+countable choice ground whether or not its membership relation is
+externally well founded.
+
+\begin{proof}[Proof of Theorem~\ref{thm:main-consistency}]
+Assume $\Con(\ZF)$. By
+Lemma~\ref{lem:constructible-consistency},
+$\ZFC+V=L$ is consistent. Completeness and downward
+L\"owenheim--Skolem give a countable model $M$ of this theory.
+No transitivity or external well-foundedness is asserted.
+
+Interpret the construction inside $M$ and choose an external
+$M$-generic filter on its internal poset as above. Form the
+symmetric quotient $W$ using
+Equations~\eqref{eq:internal-name-equivalence} and
+\eqref{eq:internal-name-membership}.
+Lemma~\ref{lem:arbitrary-model-symmetric-truth} gives full $\ZF$
+in $W$. The same lemma, applied to
+Equation~\eqref{eq:construction-internal-forcing}, gives
+\[
+ W\models\ZF+\PP+\neg\AC+\ACwo.
+\]
+Soundness therefore yields
+$\Con(\ZF+\PP+\ACwo+\neg\AC)$, as required.
+
+Equivalently, one can extract the consistency implication directly
+from finite proofs. A purported contradiction proof from the
+target theory uses only finitely many $\ZF$ instances.
+Translate its lines into symmetric forcing. The logical forcing
+rules preserve its inferences;
+Lemma~\ref{lem:symmetric-forcing-schema} supplies its $\ZF$
+axioms; and Equation~\eqref{eq:construction-internal-forcing}
+supplies $\PP$, $\ACwo$, and $\neg\AC$. This would make a nonempty
+poset force a contradiction, contrary to its atomic forcing laws.
+It therefore gives a contradiction proof in $\ZFC+V=L$, which
+the constructible interpretation translates into one in $\ZF$.
+This argument uses no stronger model-existence assumption.
+\end{proof}
+
+\subsection{The separate transitive-model conclusion}
+
+\begin{proof}[Proof of Theorem~\ref{thm:main-transitive}]
+Work in a $\ZFC$ metatheory, and let $V$ be the given
+countable transitive model of $\ZFC$. Carry out
+Theorem~\ref{thm:construction-summary} inside $V$, using its
+parameters
+\[
+ \kappa=\omega_1^V,\qquad
+ \mu=\bigl((2^{\aleph_0})^+\bigr)^V,\qquad
+ I=(\mu^+)^V.
+\]
+These are the arbitrary-choice-ground parameters of the
+construction.
+
+Enumerate externally the dense subsets of the constructed poset
+which belong to $V$, and recursively choose a descending
+sequence meeting them. Its upward-generated filter $G$ is
+$V$-generic. Here genuine well-founded recursion on ground names
+is available, since $V$ is transitive. Define
+\[
+ W=\{\tau^G:\tau\in\HS^V\}.
+\]
+The set symmetric-extension theorem gives transitive models
+$V\subseteq W\subseteq V[G]$ with the same ordinals, and gives
+all of $\ZF$ in $W$.
+Theorem~\ref{thm:construction-summary} supplies
+$\PP$, $\neg\AC$, and $\ACwo$ in this same $W$, together
+with its stated preservation of countable sequences from any fixed
+ground set. To obtain the last assertion of
+Theorem~\ref{thm:main-transitive}, let $f\in W$ be a countable
+sequence of elements of $V$. Its rank in $V[G]$ is bounded by a
+ground ordinal $\alpha$, since forcing preserves the ordinals.
+Ranks of ground sets are absolute between these transitive models,
+so the range of $f$ lies in the ground set $(V_\alpha)^V$.
+The preservation assertion of Theorem~\ref{thm:construction-summary}
+applied to this set gives $f\in V$. This proves the stated
+transitive-model conclusion.
+
+This is a separate model-existence deduction. The ordinary
+consistency proof permits an externally ill-founded ground and
+does not itself assert the existence of a transitive model.
+\end{proof}

preprints/The-Partition-Principle-does-not-imply-Choice-September-24-2026/build/sections/monoid.tex

@@ -0,0 +1,801 @@
+\section{A monoid of diagrams}\label{sec:monoid}
+
+The forcing construction will compare a diagram with its translated
+subdiagrams.  We first construct the monoid that indexes these
+translations.  Its two extension operations serve different purposes:
+one realizes finitely many prescribed overlaps, and the other inserts
+many new positions with a prescribed set of old divisors.  A further
+property of the construction will control symmetries along countable
+systems of subdiagrams.
+
+All constructions in this section take place in a universe satisfying
+$\ZFC$.  No inaccessible cardinal is used.  Fix an infinite cardinal
+$\mu\geq\aleph_1$ such that $\mu^{\aleph_0}=\mu$.
+
+\subsection{The required algebra and the construction theorem}
+
+Let $M$ be a monoid with identity $1$. Write $x\sqsubseteq y$ if $xz=y$ for
+some $z$.  We use only left cancellation.  A monoid is \emph{conical}
+if $xy=1$ implies $x=y=1$.  Left cancellation and conicality make
+$\sqsubseteq$ a partial order and make the right factor in a comparison
+unique.  Write $x\sqsubset y$ for $x\sqsubseteq y$ and $x\ne y$,
+and $\downarrow a=\{x:x\sqsubseteq a\}$.
+A \emph{conditional join} is an element $x\vee y$ satisfying
+\[
+ xM\cap yM=(x\vee y)M
+\]
+whenever the intersection is nonempty.  Thus a join is required exactly
+when the two elements have a common right multiple.
+An \emph{ideal} in $\downarrow a$ will mean a nonempty downward closed
+subset closed under finite nonempty joins.  These joins exist because
+$a$ is an upper bound.
+
+An inclusion $B\subseteq M$ is \emph{conservative} if it preserves and
+reflects comparisons, preserves their right factors, preserves existing
+joins, and preserves the absence of a common right multiple.  In
+particular, if $x,y\in B$ and $xz=y$ in $M$, then $z\in B$ and the
+same equation holds in $B$.  Mere injectivity is not this property.
+
+We shall also keep real-valued height and cost functions $\rho,\epsilon$
+on $M$, writing $\epsilon_x=\epsilon(x)$, and satisfying
+\begin{equation}\label{eq:mono-margin}
+ \begin{gathered}
+ 0<\rho(x)<1,\qquad 0<\epsilon_x\leq1,\\
+ \rho(y)-\rho(x)>
+ \sup\{\epsilon_d:d\sqsubseteq y,\ d\not\sqsubseteq x\}
+ \qquad(x\sqsubset y).
+ \end{gathered}
+\end{equation}
+An empty supremum of costs is understood to be zero.
+
+\begin{lemma}\label{lem:mono-countable-ideal}
+If a monoid has conditional joins and satisfies
+Equation~\eqref{eq:mono-margin}, every ideal in $\downarrow a$ has a
+nondecreasing cofinal sequence of length $\omega$, with repetitions
+allowed.  The same conclusion holds in any $\ZFC$ extension in which
+the monoid, order, heights and costs retain their meanings, for ideals
+belonging to that extension.
+\end{lemma}
+\begin{proof}
+For an ideal $U$, choose elements whose heights approach
+$\sup\rho[U]$, and replace their successive initial segments by their
+finite joins.  This gives a nondecreasing sequence $(n_i)$ in $U$
+with heights approaching the same supremum.  If $d\in U$ divides
+none of the $n_i$, then
+\[
+ \rho(n_i\vee d)-\rho(n_i)>\epsilon_d
+\]
+for every $i$, contradicting convergence to the supremum.  Prepending
+$1$ makes $n_0=1$.  The proof applies in the stated extension as well;
+its choices are made in that $\ZFC$ universe.
+\end{proof}
+
+Here is the finite overlap condition used below.  Take finitely many
+copies $\{i\}\times M$.  For each pair of distinct copies prescribe
+either no overlap or elements $a_{ij},a_{ji}\in M$ and the
+identifications
+\[
+ (i,a_{ij}x)\sim(j,a_{ji}x)\qquad(x\in M).
+\]
+We call the prescription \emph{valid} if these pair identifications,
+together with equality, already form an equivalence relation and do
+not identify distinct elements of any one copy.  In particular,
+transitive closure is not allowed to create extra overlaps.
+
+\begin{theorem}\label{thm:monoid}
+There is a monoid $T$ of cardinality $\mu$, with a distinguished
+$e\ne1$, having the following properties.
+\begin{enumerate}
+\item $T$ is left cancellative and conical, has conditional joins,
+and has heights and costs satisfying Equation~\eqref{eq:mono-margin}.
+\item Every valid finite overlap prescription on copies of $T$ is
+realized by left translations $x\mapsto l_i x$ in $T$, with exactly
+the prescribed identifications.
+\item For $a,b\in T$ with $b\ne1$, and every ideal
+$U\subseteq\downarrow a$, there are $\mu$ distinct elements $m$ with
+\begin{equation}\label{eq:mono-probes}
+ \begin{gathered}
+ mb=ae,\qquad m\not\sqsubseteq a,\qquad
+ \downarrow m\cap\downarrow a=U,\\
+ n\vee m=ae\quad(n\sqsubseteq a,\ n\notin U).
+ \end{gathered}
+\end{equation}
+The cones of distinct elements in this family intersect exactly in
+$aeT$.  Moreover $|T\setminus eT|=\mu$.
+\item For every countable subset of $T$ there is a countable
+submonoid $T_0$ containing it and a continuous conservative tower
+starting at $T_0$ and ending at $T$.  Each successor step has one of
+the following properties:
+\begin{enumerate}
+\item no new element divides an old element;
+\item every new element $w$ with an old right multiple has a least
+old right multiple $g(w)$, and
+\begin{equation}\label{eq:mono-gate-old}
+ d\text{ old},\quad g(w)\sqsubseteq wd
+       \quad\Longrightarrow\quad wd\text{ old}.
+\end{equation}
+\end{enumerate}
+Here ``old'' means belonging to the base of that particular successor
+step, and the comparisons may be tested in the final monoid.
+\end{enumerate}
+\end{theorem}
+
+For diagrams indexed by $T$, the finite-merger clause supplies a common
+larger diagram for finitely many copies with compatible overlaps.  The
+probe clause supplies $\mu$ copies with one prescribed intersection
+pattern: for $n\sqsubseteq a$, the cone $nT$ contains $mT$ exactly when
+$n\in U$, and otherwise meets it in $aeT$.  The margin makes every such
+ideal countably cofinal.  The countable-base clause serves a different
+purpose: it will allow suitably extendible positions below a countable
+path to be moved into a countable submonoid.  We formulate that descent
+precisely at the end of the section and use it to bound the residual
+symmetries of diagrams.
+
+We prove the theorem through two extension lemmas.  They preserve more
+than the order: keeping right factors and nonjoins fixed is what permits
+the operations to be rearranged over a countable submonoid.
+
+\subsection{The extension that inserts a probe}
+
+Suppose $M$ has all the algebraic and margin properties above.
+Fix $a,b,e\in M$, with $b,e\ne1$, and a nondecreasing path
+\begin{equation}\label{eq:mono-path}
+ 1=n_0\sqsubseteq n_1\sqsubseteq\cdots\sqsubseteq a,
+ \qquad n_id_i=n_{i+1},\qquad n_ic_i=a.
+\end{equation}
+Left cancellation gives $d_ic_{i+1}=c_i$.
+Form the monoid presentation over $M$ with new generators $p_i$ and
+relations
+\begin{equation}\label{eq:mono-probe-relations}
+ p_i=d_ip_{i+1},\qquad p_ib=c_ie\qquad(i<\omega).
+\end{equation}
+Put $m=p_0$.  Since $n_0=1$ and $n_id_i=n_{i+1}$, the first
+relations give $n_ip_i=m$ for every $i$.  Thus every point of the path
+divides the new probe.  The second relations make both factorizations
+in Figure~\ref{fig:probe-divisibility} end at $ae$:
+\[
+ n_ip_ib=n_ic_ie=ae,\qquad mb=ae.
+\]
+These identities explain the presentation, but do not yet establish
+its exact divisor profile.  We must prove that no old divisor of $a$
+outside the path's downward closure is inserted below $m$, that $m$
+does not divide $a$, and that every omitted divisor meets $m$ first
+at $ae$.  The next lemma proves these
+claims even for new divisors of $a$ introduced by the same extension.
+
+\input{figures/probe-divisibility}
+
+\begin{lemma}\label{lem:mono-probe-algebra}
+The presented extension $M'$ is left cancellative and conical, has
+conditional joins, and conservatively contains $M$.  Every new element
+with an old right multiple has a least old right multiple, and
+Equation~\eqref{eq:mono-gate-old} holds at this extension.
+For $m=p_0$,
+\begin{equation}\label{eq:mono-probe-profile}
+ \begin{gathered}
+ mb=ae,\qquad m\not\sqsubseteq a,\qquad
+ \downarrow m\cap\downarrow a=\bigcup_i\downarrow n_i,\\
+ x\vee m=ae\qquad
+ (x\sqsubseteq a,\ x\not\sqsubseteq m),
+ \end{gathered}
+\end{equation}
+where the downsets and joins in this display are computed in $M'$.
+\end{lemma}
+
+\begin{proof}
+\smallskip\noindent\textit{Normal forms.}
+Let $S$ be the direct limit of the sets $M$ under the maps
+$x\mapsto xd_i$.  Write $[x,i]$ for the image of $x$ at stage $i$.
+Equality in this direct limit means equality at a finite common stage.
+The transition maps need not be injective: no right cancellation has
+been assumed.  Nevertheless the left $M$-action on $S$ is injective
+for each multiplier.  Indeed, equality after multiplying two letters
+by an old $u$ becomes $uX=uY$ at a common stage, where left
+cancellation gives $X=Y$.
+
+There are well-defined equivariant maps
+\[
+ k([x,i])=xc_i,\qquad h(s)=k(s)e.
+\]
+The identity $d_ic_{i+1}=c_i$ proves well-definedness.  Before imposing
+the second relation in Equation~\eqref{eq:mono-probe-relations}, the
+normal forms are strings
+\[
+ s_1\cdots s_j y\qquad(s_i\in S,\ y\in M),
+\]
+including $j=0$.  The letter $[x,i]$ represents $xp_i$.
+Old factors between new generators are absorbed into the following
+letter, and the transition relations are precisely equality in $S$.
+Multiplication concatenates strings, using the left action to absorb
+the terminal old factor of the first string.
+
+The remaining reductions are
+\begin{equation}\label{eq:mono-reduction}
+ s(bt)\longmapsto h(s)t,
+\end{equation}
+where $t$ is a letter of $S$ or a terminal old element.  The quotient
+$t$ is unique because the left $b$-action is injective.  Each reduction
+removes one new letter.  The only overlapping reductions occur in
+$s\,(bt)\,(bu)$, with the final factor possibly terminal.  Reducing
+in either order gives the same result by
+\[
+ h(bt)=b h(t),\qquad h(h(s)t)=h(s)h(t).
+\]
+Disjoint reductions commute by associativity of the action.  By Newman's
+lemma \cite{Newman1942} (see \cite[Lemma~5.13]{DershowitzPlaisted2001}), these
+local diamonds and termination give unique irreducibles: besides the
+old elements, they are strings whose first letter is arbitrary and
+whose later letters and terminal factor are not left-divisible by $b$.
+
+Old left multipliers act injectively on these forms.  Left multiplication
+by a letter $s$ either prepends $s$, when the input is not a $b$-multiple,
+or sends $bZ$ to $h(s)Z$.  Both cases are injective.  Their ranges are
+disjoint, because
+\begin{equation}\label{eq:mono-head-separation}
+ s\notin h(s)S.
+\end{equation}
+To see this, $s\in xS$ implies $x\sqsubseteq k(s)$, whereas
+$k(s)e\not\sqsubseteq k(s)$ by conicality and left cancellation.
+Thus every left multiplication is injective.  A new word can become
+old only by consuming its first letter, at which point it is a multiple
+of $h(s)\ne1$.  It cannot equal $1$, proving conicality.
+
+\smallskip\noindent\textit{Old cones and the first-letter blocks.}
+For $s\in S$, let $B_s$ consist of all irreducibles beginning with $s$.
+For old $x$ the normal forms give
+\begin{equation}\label{eq:mono-old-cone}
+ xM'=xM\ \cup\!\bigcup_{s\in xS}B_s.
+\end{equation}
+At a common stage in the direct limit,
+$xS\cap yS$ is $(x\vee y)S$ if the old join exists, and is empty
+if the old cones are disjoint.  Thus old comparisons, their right
+factors, joins and nonjoins are unchanged.  Notice that this conclusion
+has not used a join theorem for arbitrary new words.
+
+\smallskip\noindent\textit{Exits and gates.}
+Fix an irreducible $w=sY$.  Right multiplication either leaves its
+first letter intact or eventually consumes that letter.  We prove the
+following precise alternative: the set $wM'\setminus B_s$ is empty,
+or it is one cone $g(w)M'$ whose root $g(w)$ is old.  Thus the gate
+will describe every exit from the fixed-head block, not only the old
+multiples of $w$.
+
+The first letter is consumed exactly when the suffix becomes a
+$b$-multiple, so first calculate $YM'\cap bM'$.  If $Y$ is old,
+Equation~\eqref{eq:mono-old-cone} gives the old join $Y\vee b$,
+or an empty intersection.  If $Y$ is new with head $t$,
+irreducibility gives $t\notin bS$.  No member of $B_t$ is then a
+$b$-multiple.  By induction on the number of letters in the suffix,
+all possible exits of $Y$ form the old-rooted cone $g(Y)M'$, if
+there are any.  Its intersection with $bM'$ is calculated by the
+old join $g(Y)\vee b$.
+
+In either case the suffix intersection is empty or is $zM'$ for
+an old $z=bu$ with old $u$.  The factor $u$ is old by conservation
+of old comparisons.  In the nonempty case, multiplication by $s$
+and the reduction $sbu=h(s)u$ give
+\begin{equation}\label{eq:mono-gate-formula}
+ wM'\setminus B_s=g(w)M',\qquad g(w)=h(s)u.
+\end{equation}
+This cone cannot return to $B_s$: every new head in it belongs to
+$h(s)S$, whereas $s\notin h(s)S$ by
+Equation~\eqref{eq:mono-head-separation}.  This proves the exit
+alternative.  Every old multiple of $w$ lies outside $B_s$, so
+$g(w)$ is in particular its least old multiple.
+
+We record also the right factor $c(w)$ with $wc(w)=g(w)$.
+If $Y$ is old, it is the old quotient of $z$ by $Y$.
+If $Y$ is new and $Yc(Y)=g(Y)$, it is
+\begin{equation}\label{eq:mono-gate-quotient}
+ c(w)=c(Y)v,
+ \qquad g(Y)v=z.
+\end{equation}
+All quotients in this recursion are old except the already constructed
+$c(Y)$.  This explicit calculation will ensure conservation of right
+factors when the same presentation is simulated over a larger base.
+
+Append an old $d$ to $w$.  If the product leaves $B_s$, its first
+letter has been consumed; that reduction can occur only after all
+intervening letters have been consumed.  The result is then entirely
+old.  Since the gate cone is disjoint from $B_s$, the hypothesis
+$g(w)\sqsubseteq wd$ implies just such an exit.  This proves
+Equation~\eqref{eq:mono-gate-old}.
+
+\smallskip\noindent\textit{All conditional joins.}
+For two forms with the same head, cancel that head and intersect their
+suffix cones, using induction on the total number of new letters.
+For old $x$ and new $w\in B_s$, either $s\in xS$, in which case
+$x\sqsubseteq w$, or every common multiple must leave $B_s$.
+The latter intersection is computed through $g(w)$ and an old join.
+For $w\in B_s$ and $v\in B_t$ with $s\ne t$, an intersection
+inside $B_s$ requires $g(v)$ to divide a member of $B_s$.  By
+Equation~\eqref{eq:mono-old-cone} it then divides all of $B_s$, so
+$v\sqsubseteq w$.  The symmetric case is the same.  Otherwise a
+common multiple must leave both blocks, and the intersection is
+computed by the old join of their gates.  Missing gates or missing
+old joins give an empty intersection.  This proves conditional joins
+in every case.
+
+\smallskip\noindent\textit{The prescribed divisor profile.}
+For any letter $s$, the old divisors of every element of $B_s$ form
+\begin{equation}\label{eq:mono-letter-ideal}
+ P_s=\{x\in M:s\in xS\}\subseteq\downarrow k(s).
+\end{equation}
+This is an ideal: common-stage representatives and old joins prove
+join closure.  For the head $[1,0]$ of $m=p_0$, it is precisely
+$\bigcup_i\downarrow_M n_i$, and the gate is $ae$.  The defining
+relation gives $mb=ae$.  Since $ae\not\sqsubseteq a$, no member
+of $B_{[1,0]}$ divides $a$; in particular $m\not\sqsubseteq a$.
+
+Now let $x$ be a new divisor of $a$.  It is outside that block and
+has an old gate $g(x)\sqsubseteq a$.  If $x\sqsubseteq m$, the
+comparison leaves its block, so $g(x)\sqsubseteq m$, and the old
+profile puts $g(x)$ below some $n_i$.  Conversely $x\sqsubseteq n_i$
+implies $x\sqsubseteq m$.  If $x\not\sqsubseteq m$, its gate also
+fails to divide $m$.  The possibility $m\sqsubseteq x$ is excluded
+by $x\sqsubseteq a$.  The gate calculation consequently gives
+$x\vee m=g(x)\vee m=ae$.  The corresponding assertions for old
+$x$ follow directly from its block membership.  This proves
+Equation~\eqref{eq:mono-probe-profile}, including new divisors of $a$.
+\end{proof}
+
+\subsection{Extending the real heights}
+
+The gate description also controls the costs of all newly introduced
+divisors.  This is the reason for keeping a cost as well as a height:
+pointwise strict inequalities alone would not control suprema at a
+limit stage.
+
+\begin{lemma}\label{lem:mono-probe-heights}
+The height and cost functions on $M$ extend to the probe extension
+$M'$ so that Equation~\eqref{eq:mono-margin} holds.  For any old
+comparison, its missed-cost supremum is unchanged.
+\end{lemma}
+\begin{proof}
+Use $P_s$ from Equation~\eqref{eq:mono-letter-ideal}, and put
+$A_s=\sup\rho[P_s]$.  We place the whole block $B_s$ in a short
+height interval above its old divisors and below $\rho(h(s))$.
+Here $h(s)$ is the gate of the bare letter $s$; the gate of a longer
+word in $B_s$, when present, is a multiple of $h(s)$ and need not
+equal it.  The interval must leave enough room for the costs of both
+the old divisors and the new words.  More precisely, there are
+positive $u_s,\delta_s$ with
+\begin{equation}\label{eq:mono-height-interval}
+ \begin{split}
+ u_s-A_s&>\delta_s,\\
+ \rho(h(s))-u_s-\delta_s
+  &>\sup\{\epsilon_d:d\sqsubseteq h(s),\ d\in M\setminus P_s\}.
+ \end{split}
+\end{equation}
+Indeed, put $p=\rho(k(s))-A_s\geq0$ and
+$q=\rho(h(s))-\rho(k(s))>0$.  For $d\sqsubseteq k(s)$ missing
+$P_s$, every $x\in P_s$ misses $d$, and the old margin yields
+$\epsilon_d<\rho(k(s))-\rho(x)$.  The supremum of these costs is
+at most $p$.  The supremum for $d\sqsubseteq h(s)$ not dividing
+$k(s)$ is strictly below $q$.  Thus the full supremum on the right
+of Equation~\eqref{eq:mono-height-interval} is strictly less than
+$p+q=\rho(h(s))-A_s$, also when $p=0$.  This leaves the required
+positive room for $u_s$ and $\delta_s$.
+
+Define recursively on irreducibles
+\begin{equation}\label{eq:mono-probe-height-definition}
+ \begin{gathered}
+ \rho(sY)=u_s+\delta_s\rho(Y),\qquad
+ 0<\epsilon_{sY}\leq\delta_s\epsilon_Y,\\
+ \epsilon_{sY}\leq\epsilon_{g(sY)}
+       \quad\text{whenever the gate exists}.
+ \end{gathered}
+\end{equation}
+These choices give heights in $(0,1)$ and costs in $(0,1]$.
+We verify all comparisons, including the suprema.
+
+For two elements in one block $B_s$, a divisor of the upper element
+from outside that block also divides the lower one.  An old divisor
+belongs to $P_s$; a new divisor with a different head must enter by
+its old gate, which then divides every member of $B_s$.
+Inside the block, cancellation removes $s$, and the margin follows
+by induction from the factor $\delta_s$ in both height differences
+and cost bounds.
+
+Suppose an old $x$ lies strictly below $sY$.  Then $x\in P_s$.
+A missed divisor outside $B_s$ is old in $P_s$, or has an old gate
+there of at least its cost.  For such an old divisor $d$ missing
+$x$, the join $x\vee d$ lies in $P_s$, and hence
+\[
+ \epsilon_d<\rho(x\vee d)-\rho(x)\leq A_s-\rho(x).
+\]
+Their supremum is at most $A_s-\rho(x)$.  Divisors inside $B_s$
+have cost at most $\delta_s$.  Equation~\eqref{eq:mono-height-interval}
+makes $\rho(sY)-\rho(x)$ strictly larger than both bounds.
+
+Suppose $sY$ lies strictly below an old $y$.  Then
+$h(s)\sqsubseteq y$.  Every divisor of $y$ missing $sY$ is old
+outside $P_s$, or has a gate there with at least its cost: a gate
+in $P_s$ would put the divisor below all of $B_s$.
+Among these old costs, those for divisors of $h(s)$ are bounded by
+Equation~\eqref{eq:mono-height-interval}.  If $y\ne h(s)$, the
+remaining costs have supremum strictly below
+$\rho(y)-\rho(h(s))$ by the old margin.  The total gap
+$\rho(y)-\rho(sY)$ is the sum of two positive gaps and exceeds both
+cost suprema.  This proves this comparison case.
+
+Distinct comparable new blocks have an intervening old gate, so the
+preceding cases apply transitively.  To justify this use, if
+$x<z<y$, any divisor of $y$ missing $x$ either divides $z$ and
+misses $x$, or misses $z$; the sum of the two positive height gaps
+strictly dominates both relevant suprema.
+
+Finally, for old $x<y$, any new divisor $d$ of $y$ missing $x$
+has an old gate $g(d)\sqsubseteq y$ missing $x$, with
+$\epsilon_d\leq\epsilon_{g(d)}$.  No new cost increases the old
+supremum, while every old divisor remains present.  The supremum is
+therefore exactly unchanged.  This also proves persistence of the
+margins at unions of a conservative tower of these extensions.
+\end{proof}
+
+\subsection{The extension that merges finitely many copies}
+
+\begin{lemma}\label{lem:mono-merger}
+Let $M$ satisfy the algebraic and margin hypotheses, and let a finite
+nonempty overlap prescription on copies of $M$ be valid.  There is a
+conservative extension realizing it by left translations.  The
+extension is left cancellative and conical, has conditional joins,
+and admits extending heights and costs satisfying
+Equation~\eqref{eq:mono-margin}.  No new element divides an old one.
+\end{lemma}
+\begin{proof}
+Let $D$ be the prescribed quotient of the copies, writing $t_i x$
+for the image of $x$ in copy $i$.  This quotient carries one layer
+of the required overlaps; strings of its elements will complete that
+layer to a monoid.  It is a right $M$-set and each
+map $x\mapsto t_i x$ is injective.  On $D$ use the order
+$z\sqsubseteq w$ when $zq=w$ for some $q\in M$; left cancellation
+and conicality inside a representing copy make this a partial order.
+Its right cones have conditional joins.  Within one copy this is the
+old join.  For elements represented in different copies, every common
+multiple must lie in their specified overlap, if there is one.
+In each copy join the element with the start of that overlap; transfer
+the two resulting elements into the overlap and join there.  A missing
+join at any step means that the original cones are disjoint.  This
+calculates exactly their intersection, because every common multiple
+must pass through both of those preliminary joins.
+
+Adjoin generators $l_i$ with the specified relations
+$l_i a_{ij}=l_j a_{ji}$.  The normal forms are
+\begin{equation}\label{eq:mono-merger-forms}
+ xz_1\cdots z_j\qquad(x\in M,\ z_i\in D),
+\end{equation}
+including $j=0$.  Each relation acts inside a new-generator/old-factor
+pair, exactly as in $D$; it changes neither the number of new letters
+nor the preceding old coefficient.  This proves uniqueness of the
+forms.  Multiplication acts by the leading old coefficient of the
+second word on the last $D$-letter of the first, and then appends the
+remaining letters.  For a fixed $z=t_i a$, the map $y\mapsto zy$
+is injective by left cancellation in copy $i$.  Together with old
+left cancellation, this proves cancellation for arbitrary products of
+forms.  Letter count proves conicality and that no new word divides
+an old element.
+
+Unlike the probe extension, this construction never consumes a new
+letter.  Its cones are therefore read from the final letter rather
+than from the first-letter blocks used above.  For a positive-length
+word, right multiplication fixes the prefix before its final letter,
+advances that letter within its $D$-cone, and possibly appends further
+letters.  More explicitly, compare
+$xz_1\cdots z_p$ and $yt_1\cdots t_q$ with $p,q>0$.
+A common multiple requires $x=y$ and equality of the letters before
+position $\min(p,q)$.  If $p=q$, their intersection is obtained by
+joining $z_p$ and $t_p$ in $D$, or is empty if that join is absent.
+If $p<q$, a common multiple exists exactly when $z_p$ divides $t_p$
+in the right $M$-set $D$; then the first word divides the second and
+the second is their join.  The case $q<p$ is symmetric.
+An old $x$ and a positive-length word with head $y$ have a common
+multiple exactly when $x\sqsubseteq y$, in which case $x$ divides
+that word.  Two old words use their old join, because the heads of
+common multiples are precisely their old common multiples.
+These descriptions prove conditional joins and preserve old comparisons,
+quotients and nonjoins.  The words $l_i x$ have exactly the
+identifications in $D$, as required.
+
+It remains to supply margins.  If the number of copies is $r$, set
+\[
+ \rho_D(z)=\frac1r\sum_{i<r}\rho_i(z),\qquad
+ \rho_i(z)=
+ \begin{cases}
+ \rho(x),&z=t_i x,\\
+ 0,&z\text{ has no representation in copy }i.
+ \end{cases}
+\]
+Each coordinate is nondecreasing along a comparison, and some
+coordinate increases strictly in a proper comparison.  Choose positive
+costs with
+\[
+ \epsilon^D_z\leq\frac1r
+ \min\bigl(\{\rho(1)/2\}\cup
+           \{\epsilon_x:z=t_i x\text{ for some }i\}\bigr).
+\]
+For $u<v$ in $D$, classify a divisor $d$ of $v$ missing $u$ by a
+copy representing $d$.  That copy also represents $v$.  If it
+represents $u$, its old margin bounds the supremum of these costs
+strictly below its coordinate increase divided by $r$.  Otherwise
+that increase is at least $\rho(1)$, while each such cost is at most
+$\rho(1)/(2r)$.  There are only finitely many copies, so their strict
+bounds still give the required strict supremum bound for
+$\rho_D(v)-\rho_D(u)$.
+
+Choose $\theta,\phi>0$ with $\theta+\phi<1$, and recursively set
+\begin{equation}\label{eq:mono-merger-heights}
+ \begin{split}
+ \rho(wz)&=\rho(w)+(1-\rho(w))(\theta+\phi\rho_D(z)),\\
+ 0<\epsilon_{wz}&\leq
+ (1-\rho(w))\min(\theta/2,\phi\epsilon^D_z).
+ \end{split}
+\end{equation}
+For comparable new words of equal length only their last letters
+vary; missed divisors of that length are handled by the scaled
+$D$-margin, and all shorter divisors divide both words.  For a word
+$w$ compared with $wz$, every missed divisor has the new length and
+prefix $w$, so its cost is at most $(1-\rho(w))\theta/2$, strictly
+less than the height increase.  An arbitrary comparison factors
+through these comparisons, and, when the lower word is old, an old
+comparison of leading coefficients.  Transitivity of the margin,
+proved in Lemma~\ref{lem:mono-probe-heights}, finishes the check.
+Old cost suprema are unchanged because there are no new old divisors.
+\end{proof}
+
+\subsection{Conservation under simulation}
+
+We next establish the compatibility needed to reorder the construction.
+This step also explains why overlap prescriptions remain valid when
+operations are scheduled before they are executed.
+
+\begin{lemma}\label{lem:mono-simulation}
+Suppose $B\subseteq M$ is conservative and both monoids have the
+algebraic properties above.  Either extension operation with parameters
+in $B$ is valid over $M$, and the extension of $B$ conservatively
+embeds into the corresponding extension of $M$ by the same generators.
+A finite overlap prescription with parameters in $B$ is valid in $B$
+if and only if it is valid in $M$.
+\end{lemma}
+\begin{proof}
+For the last assertion, the only nontrivial transitivity check passes
+through three copies.  In the middle copy, intersect the two prescribed
+overlap starts.  If their old cones are disjoint, conservativity keeps
+them disjoint.  Otherwise their old join and its right quotients are
+unchanged.  Every later common multiple is a multiple of that join.
+The two endpoint identifications at the join already agree in a valid
+prescription over $B$, so right multiplication gives agreement at all
+later common multiples.  This proves validity over $M$.  Conversely,
+restrict a valid prescription to $B$.  Comparisons between its old
+parameters have their quotients in $B$, so no identification or
+transitivity witness between elements of $B$ requires leaving $B$.
+The restriction has exactly the indicated pair relations and is valid.
+
+For probes, equality of two letters in the smaller direct limit is
+preserved and reflected because it is an equality at a common finite
+stage.  If an old $b\in B$ divides a smaller letter $[x,i]$ in the
+larger direct limit, there is a common stage at which
+\[
+ by=x d_i\cdots d_{j-1}.
+\]
+Both endpoints of this comparison belong to $B$, so its right factor
+$y$ belongs to $B$.  Thus divisibility of a smaller letter by an old
+smaller multiplier, including its quotient, is unchanged.  The same
+irreducible words therefore embed.  The gate recursion uses only the
+previous gates and old joins and quotients, so it has the same outcome,
+including the absence of a gate.  The join calculations in
+Lemma~\ref{lem:mono-probe-algebra} now preserve and reflect joins and
+nonjoins.
+
+For clarity, preservation of right quotients between new forms follows
+by a separate division argument.  If both forms are old, use the base
+hypothesis.  If the divisor is old and the dividend new, divide its
+first direct-limit letter by the preceding common-stage calculation,
+and retain its suffix.  If both forms have the same new head, cancel
+that head and recurse on their suffixes.  In the remaining cases the
+new divisor must leave its first-letter block.  Factor through its
+gate, using the quotient in Equation~\eqref{eq:mono-gate-quotient},
+and divide that old gate into the dividend by the first two cases.
+Every factor obtained lies in the smaller extension.  Left
+cancellation makes their product the unique required quotient.
+
+For mergers, representation in an overlap and intersections of cones
+in $D$ were calculated solely using old joins and their right
+quotients.  These calculations are consequently unchanged.  The
+normal forms in Equation~\eqref{eq:mono-merger-forms} then preserve
+comparisons, their right factors, joins and nonjoins.  This proves
+conservative simulation for both operations.
+\end{proof}
+
+\begin{lemma}\label{lem:mono-profile-persistence}
+Suppose a probe satisfies Equation~\eqref{eq:mono-probe-profile} in
+one stage of a tower of the two operations.  It continues to satisfy
+that equation at every later stage and at the union, with the full
+later downward closure of its original path.  Previously constructed
+joins between probes are also preserved.
+\end{lemma}
+\begin{proof}
+A merger introduces no new divisor of the old $a$.  At a probe
+extension, a new $x\sqsubseteq a$ has an old gate
+$g(x)\sqsubseteq a$.  If $x$ also divides the old probe, its gate
+lies below that probe and hence below some original path element;
+then $x$ does too.  If $x$ does not divide the old probe, its gate
+does not either.  The old probe cannot divide $x$, because it does
+not divide $a$.  The join calculation through the gate therefore
+gives $ae$ in this case.  This proves both alternatives at a successor
+step.  At a limit every tested element has already appeared, and all
+comparisons and joins have been preserved.  Conservativity directly
+preserves joins between existing probes.
+\end{proof}
+
+\subsection{The schedule and the countable-base rearrangement}
+
+We now assemble the operations.  The height bound ensures that the
+apparently arbitrary ideal requirements can be scheduled using only
+countable paths.  The rearrangement proved afterward concerns the
+same resulting monoid, not a separately constructed substitute.
+
+\begin{proof}[Proof of Theorem~\ref{thm:monoid}]
+\smallskip\noindent\textit{Saturating the geometric requirements.}
+Start with the free monoid on $e$.  For $n<\omega$, for example, put
+\[
+ \rho(e^n)=1-2^{-n-1},\qquad \epsilon_{e^n}=2^{-n-3}.
+\]
+These satisfy Equation~\eqref{eq:mono-margin}.  Perform $\omega_1$
+rounds.  At the start of a round schedule every available probe datum
+$(a,b,(n_i))$ with the fixed $e$, and every available finite valid
+merger.  For each probe datum schedule $\mu$ successive insertions.
+Execute the resulting wellordered list with the two lemmas, and take
+unions at limits.  Lemma~\ref{lem:mono-simulation} keeps every
+scheduled merger valid until it is executed.
+
+A monoid of size at most $\mu$ has at most $\mu^{\aleph_0}=\mu$
+countable paths and at most $\mu$ finite prescriptions.  A probe
+extension uses a quotient of $M\times\omega$ and finite strings;
+a merger uses finitely many copies and finite strings.  Thus every
+operation preserves size at most $\mu$, and each round has at most
+$\mu$ steps.  The union of $\omega_1$ such rounds still has size
+at most $\mu$.  The margin lemmas ensure that old values are never
+changed and that old missed-cost suprema never increase, so the final
+union $T$ has Equation~\eqref{eq:mono-margin} as well as all the
+algebraic properties.
+
+Given an ideal in $\downarrow_T a$, use
+Lemma~\ref{lem:mono-countable-ideal} to obtain a cofinal path starting
+at $1$.  Its elements, $a,b$, and all its transition and terminal
+quotients occur before one round: the set is countable and the rounds
+have cofinality $\omega_1$.  Alternatively, quotient conservativity
+already puts every quotient into any stage containing its endpoints.
+The corresponding requirement is scheduled.  Its final lower closure
+is the given ideal, and Lemma~\ref{lem:mono-profile-persistence}
+therefore gives Equation~\eqref{eq:mono-probes} for the $\mu$
+insertions.  For any earlier and later insertion with the same datum, the
+earlier probe is old but is not a divisor of $a$, so it is not an old
+divisor of the new probe.  Joining through the new probe's gate gives
+$ae$.  This join persists in the final union, proving independence.
+
+Every final finite overlap prescription has its parameters in one
+stage.  Its validity reflects to that stage by
+Lemma~\ref{lem:mono-simulation}, and the schedule realizes it later.
+The realization remains exact by conservation of comparisons and
+quotients: equality $l_i x=l_j y$ in a later stage must still factor
+through the preserved join of the two roots, with its prescribed
+quotients.  Thus later right factors add exactly the stipulated
+identifications and no others.  Finally use $a=1,b=e$ and $U=\{1\}$.
+Its probes have no old divisor $e$, so there are $\mu$ elements of
+$T\setminus eT$.  This proves both cardinality assertions.
+
+\smallskip\noindent\textit{Extracting a countable initial base.}
+It remains to prove the local construction property for this same $T$.
+We first select countably many operations containing any prescribed
+countable set of elements, and then perform all omitted operations over
+their union.  The resulting tower must retain the gate rule: this is what
+will let a later descent replace a position by one born at an earlier
+stage.  Regard the
+entire schedule as a continuous tower $(M_\alpha)_{\alpha\leq\lambda}$
+of individual operations.  Record each parameter by a finite word in
+previous generators.  Every element of the presented monoid has such
+a word, and each equality has a finite derivation from the relations.
+This does not assert that a whole countable path has finite support.
+
+Choose a sufficiently large regular $\Theta$ and a countable elementary
+submodel $N$ of $H_\Theta$ containing the tower, its presentations and
+an enumeration of the prescribed countable subset of $T$.
+Countable objects in $N$, in particular the parameter paths of
+operations indexed in $N$, have all their entries in $N$.
+First perform just the operations whose indices belong to $N$, in
+their original order.  Let $S_\alpha$ denote this subconstruction
+using selected indices below the cut $\alpha$.
+
+At cuts $\alpha\in N$ we have
+\begin{equation}\label{eq:mono-elementary-cut}
+ S_\alpha=M_\alpha\cap N,
+\end{equation}
+with a conservative inclusion in $M_\alpha$.  Here is the induction
+justifying the assertion.  Old comparisons, their quotients and joins
+in an elementary intersection are reflected by elementarity; a
+nonjoin remains a nonjoin in the larger monoid.  Before a selected
+operation, every parameter is already present: its finite term code
+belongs to $N$, and all entries of a countable path in $N$ belong to
+$N$.  Conservative simulation makes that operation legal in the
+subconstruction and embeds its result.  Conversely, every element of
+$M_\alpha\cap N$ has by elementarity a finite term code in $N$;
+all of its finitely many generator indices belong to $N$ and precede
+$\alpha$.  It is therefore produced by the selected subconstruction.
+This proves Equation~\eqref{eq:mono-elementary-cut} and the successor
+step of the induction.  At a cut $\alpha\notin N$, fix the finitely
+many words in a comparison or join test.  If they use selected operation
+indices, let $\gamma$ be the greatest such index and put
+$\beta=\gamma+1$; if there are no such indices, put $\beta=0$.
+Then $\beta\in N$, $\beta<\alpha$, and all the words belong to
+$S_\beta=M_\beta\cap N$.  The original inclusion
+$M_\beta\subseteq M_\alpha$ is conservative.  Any right quotient or
+conditional join required by the test therefore already lies in
+$M_\beta$ and, by elementarity, in $N$, hence in $S_\beta$.
+This also preserves nonjoins and proves conservation at $\alpha$.
+Unions introduce no additional finite relation.  This completes the
+induction.
+
+The resulting selected submonoid $T_0$ is countable: the initial
+monoid is countable, there are countably many selected steps, and each
+operation over a countable base adds only countably many elements.
+It contains the prescribed sequence by its finite term codes.
+The set of selected indices need not be an initial segment; the
+argument has used Equation~\eqref{eq:mono-elementary-cut} only at
+cuts belonging to $N$.
+
+\smallskip\noindent\textit{Reinserting the omitted operations.}
+Append now the omitted steps in their original order.  Before an
+omitted step $\alpha$, the current presentation is the union of the
+original relations below $\alpha$ and all selected relations.  It has
+an alternative realization: start with $M_\alpha$, and then perform
+the selected steps at indices at least $\alpha$ in their order.
+Indeed $S_\alpha\subseteq M_\alpha$ is conservative, and repeatedly
+simulating the selected subconstruction makes all these operations
+legal.  Each extends its whole base conservatively, so this alternative
+realization conservatively contains $M_\alpha$.  It has precisely
+the same generators and relations as the current reordered
+presentation.  Consequently the original operation at $\alpha$ is
+legal over the current base, by conservative simulation again.
+Induction validates every appended step.  At limits the presentations
+are their unions; at the end their union is exactly the original
+presentation of $T$.
+
+Each step of this reordered tower is one of the two operations.
+The merger has no new old divisors, and the probe extension has its
+least-old-multiple rule, including
+Equation~\eqref{eq:mono-gate-old}.  Conservation allows those
+comparisons to be tested in $T$.  This proves the last assertion and
+completes the theorem.
+\end{proof}
+
+We isolate the consequence of the local construction property that will
+be used for symmetries.  It makes explicit why a position below a
+countable path can be moved to the countable base after finitely many
+steps; it does not claim that every such position was in that base
+already.
+
+\begin{lemma}\label{lem:mono-descent}
+Let $(n_i)$ be a nondecreasing sequence in $T$, and choose a
+countable-base tower from Theorem~\ref{thm:monoid} whose initial
+submonoid contains all $n_i$ and all right quotients between them.
+Let $\mathcal A\subseteq\omega\times T$ be a set of pairs such that
+each $(l,m)\in\mathcal A$ has $m\sqsubseteq n_N$ for some $N$.
+Assume that for each $(l,m)\in\mathcal A$ and each $N$ there is
+$(l',md)\in\mathcal A$ with $l'\geq l$, $n_ld=n_{l'}$, and
+$n_N\sqsubseteq md$.  Then, starting from any pair in $\mathcal A$,
+a finite succession of these replacements reaches a pair whose second
+coordinate belongs to the initial countable submonoid.
+\end{lemma}
+\begin{proof}
+If $m$ is not in that submonoid, take its first birth step in the
+reordered tower.  It has the old multiple $n_N$, so this is a probe
+step and its gate satisfies $g(m)\sqsubseteq n_N$.  Choose an
+admissible replacement with $n_N\sqsubseteq md$.  The quotient
+$d$ belongs to the initial submonoid and is therefore old at this
+step.  Conservation and Equation~\eqref{eq:mono-gate-old} imply
+that $md$ belongs to the strictly earlier base.  Repeat if necessary.
+Each repetition strictly lowers the birth ordinal, so the process
+terminates after finitely many repetitions.  The resulting position
+lies in the countable initial submonoid.
+\end{proof}

preprints/The-Partition-Principle-does-not-imply-Choice-September-24-2026/build/sections/quotients.tex

@@ -0,0 +1,746 @@
+\section{Exact fibers of a symbolic quotient}\label{sec:quotient-fibers}\label{sec:quotients}
+
+Our next objective is a bijection between $A$ and each of its
+nonwellorderable quotients. A local count of names is insufficient:
+assignments made in two diagrams must agree whenever those diagrams
+occur in one generic filter. We first classify local elements by the
+subdiagrams on which they are available. We then construct bijections
+which respect every restriction and every change of coordinates.
+
+Throughout the quotient construction, the cardinal parameters are those
+of Section~\ref{sec:forcing}, with the monoid $T$ supplied there by
+Theorem~\ref{thm:monoid}. The forcing $\mathbb P$ remains the
+ground set of diagrams, including when it is used after forcing with
+$\mathbb R$. We use countable closure of $\mathbb R$, regularity of
+$\kappa=\omega_1$, preservation of the ground cardinals $\mu,\mu^+$,
+and the ground equality $\mu^{\aleph_0}=\mu$. Preservation of smaller
+cardinals is unnecessary. In particular, $\mu$ always denotes the
+chosen ground ordinal, not a cardinal recomputed from a new continuum.
+All ground selections below are choices over specified sets in $V$.
+
+Fix a hereditarily symmetric name $\dot E$ and a product condition
+$(d_0,p)$ symmetrically forcing that $\dot E$ is an equivalence relation
+on $A$ whose quotient is not wellorderable. Enlarge the diagram so that
+$D_p$ supports $\dot E$ and contains $\supp_I(d_0)$. Choose $q$ with
+$q_e=p$. Our bijection will have support $D_q$ and will be forced below
+$(d_0,q)$. Write $G_q=\Fix(D_q)$. This group fixes $q,p,d_0$ and
+$\dot E$ literally.
+
+We write $F$ for an $\mathbb R$-generic filter containing $d_0$, and
+work temporarily in $V[F]$. A product name $\tau$ determines an ordinary
+$\mathbb P$-name $\tau[F]$ there. Thus an assertion such as
+$r\forces\tau=\sigma$ in this section means ordinary $\mathbb P$-forcing
+in $V[F]$, using these sections of the product names. The generic
+notation abbreviates forcing arguments uniform below $d_0$; no claim
+that a ground permutation acts on one fixed evaluated universe is used.
+
+\subsection{Local names and their cardinal bound}
+
+Recall the local name collection from
+Equation~\eqref{eq:normalized-local-names}. A normalized subset name
+means a subset of the ground set
+\[
+ \{(\dot A_j,(d,s)):j\in I,
+                 \ (d,s)\in\mathbb R\times\mathbb P\}.
+\]
+For a diagram $r$, $\mathcal X(r)$ is the ground set of all such
+names fixed literally by $\Fix(D_r)$. Each is hereditarily symmetric,
+since its member names are the individual label names. Normalization by
+the forcing relation retains any given support. When $r\le q$, define
+in $V[F]$
+\[
+ \mathcal D_F(r)=
+ \{\tau\in\mathcal X(r):
+       r\forces\tau\text{ is an }\dot E\text{-class}\}/\mathord\sim_r,
+ \qquad
+ \tau\sim_r\sigma\ \Longleftrightarrow\ r\forces\tau=\sigma.
+\]
+These are quotients of ground sets, so they are sets. We sometimes omit
+$F$ from the notation. Their elements are symbolic classes of names,
+not yet the evaluated members of $A/E$ in a product extension.
+Lemma~\ref{lem:local-decision} applies to the
+class and equality tests. Consequently, for $r\le s\le q$, retaining
+the same name defines an injection
+\begin{equation}\label{eq:symbolic-inclusion}
+ \iota_{s,r}:\mathcal D_F(s)\longrightarrow\mathcal D_F(r).
+\end{equation}
+Indeed, if a stronger diagram forced two local names equal, the decision
+at $s$ would already be equality. These maps compose literally on
+representatives. Every $j\in D_r$ has a normalized name, supported by
+$D_r$, for its $E$-class: use the forcing relation to name the labels
+$E$-related to $\dot A_j$. Below the present hypothesis it is an
+eligible member of $\mathcal D_F(r)$.
+
+\begin{lemma}[Local quotient bound]\label{lem:local-quotient-bound}
+For every $F$ as above and every ground $r\le q$,
+$|\mathcal D_F(r)|\le\mu$ in $V[F]$.
+\end{lemma}
+\begin{proof}
+For an element choose a representative $\tau$ and a diagram $t\le r$
+forcing $\tau=[\dot A_j]_E$ for some $j\in D_t$. The ordinary forcing
+theorem permits deciding a representative label; extending the diagram
+includes its index. Write $r=t_b$. By Cohen restriction in
+Lemma~\ref{lem:local-decision}, there is $d\in F$, supported in $D_t$,
+witnessing this $\mathbb P$-forcing assertion. Record the triple
+\[
+ (b,x,t^{-1}d),\qquad t(x)=j,
+\]
+where $t^{-1}d$ is the partial binary function on $T\times\kappa$
+obtained by replacing each coordinate $t(y)$ with $y$. Its domain has
+size less than $\kappa$. The ground set of possible records has size
+at most $\mu$, since $|T|=\mu$, $\kappa\le\mu$, and
+$\mu^{<\kappa}=\mu$. Its ground enumeration remains an enumeration in
+$V[F]$.
+
+Suppose two chosen representatives $\tau_1,\tau_2$ have the same
+record, witnessed by $t_1,t_2,d_1,d_2,j_1,j_2$. The ground bijection
+$t_1(y)\mapsto t_2(y)$ agrees with the identity on $D_r$: in both
+diagrams $r=(t_i)_b$. It maps points outside $D_r$ to points outside
+$D_r$. Extend it to a ground permutation $\pi$ of $I$ fixing $D_r$.
+This is possible since the two ranges have size $\mu<|I|$ and their
+complements have the same cardinality $|I|$. Then
+\[
+ \pi t_1=t_2,\qquad \pi d_1=d_2,\qquad \pi j_1=j_2,
+ \qquad \pi\tau_1=\tau_1.
+\]
+Product symmetry makes $d_2$ witness at $t_2$ equality of both
+$\tau_1$ and $\tau_2$ with $[\dot A_{j_2}]_E$. Since $d_2\in F$,
+$t_2$ forces $\tau_1=\tau_2$ in $V[F]$. Decision at $r$ gives the
+same equality there. Thus distinct elements have distinct records.
+Choosing representatives and records uses Choice in the ordinary
+extension $V[F]$, not in the symmetric model.
+\end{proof}
+
+\subsection{Availability and exact multiplicities}
+
+For $r\le q$, write $q=r_a$. For $h\in\mathcal D_F(r)$ and
+$j\in D_r$, respectively, define
+\begin{align}
+ U_h&=\{n\sqsubseteq a:
+             h\in\ran(\iota_{r_n,r})\},\label{eq:class-availability}\\
+ U_j&=\{n\sqsubseteq a:j\in D_{r_n}\}.
+             \label{eq:label-availability}
+\end{align}
+An ideal here is nonempty, downward closed, and closed under finite
+joins. All joins needed within $\downarrow a$ exist.
+
+\begin{lemma}[Availability ideals]\label{lem:availability-ideals}
+Each set in Equations~\eqref{eq:class-availability} and
+\eqref{eq:label-availability} is an ideal of $\downarrow a$.
+Every ideal of $\downarrow a$ in $V[F]$ belongs to $V$.
+\end{lemma}
+\begin{proof}
+Both sets contain $1$ and are downward closed by inclusion of
+subdiagrams. Suppose $n,n'\in U_h$ and put $k=n\vee n'$. The ranges
+of $r_n,r_{n'}$ intersect exactly in $D_{r_k}$. Choose supported names
+for their representations of $h$. They are forced equal by $r$.
+Lemma~\ref{lem:support-descent}, with root $r_k$, supplies a
+$D_{r_k}$-supported name equal to them. Normalize it without changing
+support. Decision at $r_k$ shows it is an $E$-class there: its
+negation would persist to the extension $r$. Thus $k\in U_h$.
+For labels the identical argument is the literal equality of cone
+intersections, without any name descent.
+
+For the final assertion, apply Lemma~\ref{lem:mono-countable-ideal}
+in $V[F]$. The ground monoid, order, heights and costs retain their
+meanings there, so every nonempty ideal has a countable cofinal sequence.
+It consists of ground elements of $T$, so countable closure of
+$\mathbb R$ puts it in $V$. Its downward closure, computed with the
+unchanged ground order, is the original ideal. The argument applies
+to every ideal in $V[F]$, not only to ground ideals.
+\end{proof}
+
+For a nonempty ideal $U\subseteq\downarrow a$, put
+\[
+ B_{r,U}(F)=\{h\in\mathcal D_F(r):U_h=U\},\qquad
+ L_{r,U}=\{j\in D_r:U_j=U\}.
+\]
+The second set is ground data; the first can depend on $F$.
+These exact fibers, rather than the whole local sets $\mathcal D_F(r)$,
+are the pieces on which a bijection can respect restriction: a label
+and its assigned class must remain available at precisely the same
+subdiagrams.
+
+\begin{lemma}[Exact fibers]\label{lem:exact-fibers}
+For every $r\le q$ and every nonempty ideal
+$U\subseteq\downarrow a$, where $q=r_a$, both $B_{r,U}(F)$ and
+$L_{r,U}$ have cardinality $\mu$ in $V[F]$. The label fiber already
+has that cardinality in $V$.
+\end{lemma}
+\begin{proof}
+The common root $p$ will separate the copies used in this proof.
+We first find a class with no representation supported by $D_p$.
+Copying it into diagrams whose unwanted intersections are exactly
+$D_p$ will then rule out both coincidences between copies and
+availability at forbidden subdiagrams, by support descent.
+
+\smallskip\noindent\textit{A class unavailable at the root.}
+Enumerate
+$\mathcal X(p)$ in $V$ as $(\tau_\xi:\xi<\lambda)$. The canonical
+name for the sequence of its evaluations is hereditarily symmetric
+with support $D_p$. Explicitly, its entries are the canonical pairs
+$(\check\xi,\tau_\xi)$, each placed under every product condition.
+Every member of $\Fix(D_p)$ fixes each entry literally. The range
+of this sequence is wellorderable inside $W$: map each value to its
+least index of occurrence. If it contained every $E$-class, the same
+map restricted to $A/E$ would wellorder that quotient. The hypothesis
+therefore implies, ordinarily below $(d_0,p)$, that some $E$-class
+lies outside the evaluations of all names in $\mathcal X(p)$.
+
+Fix $F$ containing $d_0$. By the ordinary forcing theorem choose
+$t\le p$ and $j\in D_t$ such that $t$ forces the class of $A_j$
+outside that entire evaluation list. Choose a witnessing Cohen
+condition $d\in F$ supported in $D_t$, using
+Lemma~\ref{lem:local-decision}. Write $p=t_b$. Necessarily $b\ne1$,
+since a class represented by a label in $D_p$ has a name in
+$\mathcal X(p)$.
+
+\smallskip\noindent\textit{Independent copies with active Cohen conditions.}
+As $p=r_{ae}$, the probe property of Theorem~\ref{thm:monoid}
+supplies $\mu$ elements $m$ with $mb=ae$ and the prescribed ideal
+$U$. The diagrams $r_m$ are copies of $t$ over $p$, with pairwise
+intersection exactly $D_p$. Choose the ground permutations taking
+$t$ to these diagrams and fixing $D_p$, and transport $d$ and the
+class name along them. Their nonroot coordinate sets are disjoint.
+This is the role of the earlier extension $q_e=p$: the availability
+pattern is prescribed among divisors of $a$, while the common root
+lies at $ae$, strictly beyond $a$.
+
+There are $\mu$ copies whose transported Cohen conditions belong to
+$F$. To see this without any $\mu$-closure assertion, partition the
+ground family into $\mu$ ground subfamilies of size $\mu$. In each
+subfamily the conditions extending at least one transported copy of
+$d$ form a dense set below $d\restriction D_p$. Indeed, a Cohen
+condition uses countably many coordinate indices; among the pairwise
+disjoint nonroot ranges there is one it misses, and on the root the
+two conditions agree. Their union is a Cohen condition. Genericity
+meets each of these ground dense sets. A hit in each disjoint
+subfamily yields $\mu$ different hits.
+
+\smallskip\noindent\textit{Distinct classes with the prescribed exact ideal.}
+Each hit gives a $D_{r_m}$-supported class name forced at $r_m$
+to lie outside the evaluations of $\mathcal X(p)$. The diagram
+$r_m$ extends $p$ but need not extend $q$; we use its class name,
+not a set $\mathcal D_F(r_m)$. As $r$ extends $r_m$, this name
+represents a symbolic class in the defined set $\mathcal D_F(r)$.
+Two such
+classes cannot be equal there: support descent over their common
+root $p$ would supply a name in $\mathcal X(p)$ for their common
+value, contrary to the outside assertion. If $n\in U$, then
+$n\sqsubseteq m$, so $r_n$ extends $r_m$ and the class is available
+at $r_n$. If $n\sqsubseteq a$ and $n\notin U$, the exact probe
+intersection gives
+\[
+ nT\cap mT=aeT,
+ \qquad D_{r_n}\cap D_{r_m}=D_p.
+\]
+Availability at $r_n$ would again descend to a name in
+$\mathcal X(p)$, a contradiction. Its exact availability ideal is
+therefore $U$. This gives $\mu$ distinct elements of $B_{r,U}(F)$;
+Lemma~\ref{lem:local-quotient-bound} bounds that fiber from above.
+
+\smallskip\noindent\textit{The label fiber.}
+For labels use the probe centers $r(m)$. Their availability ideals
+are exactly $\downarrow m\cap\downarrow a=U$, and distinct probes
+have distinct centers. This supplies a ground $\mu$-subset of
+$L_{r,U}$; the upper bound follows from $|D_r|=\mu$.
+\end{proof}
+
+\section{Components of subdiagrams and their stabilizers}
+\label{sec:quotient-components}
+
+The exact fibers have equal cardinality, but choosing their bijections
+independently would not preserve assignments when a diagram is replaced
+by a subdiagram. We first show that restriction identifies entire
+exact fibers, not merely subsets of them. Following these canonical
+identifications groups the fibers into components. Only after forming
+the components will we choose bijections; their stabilizers determine
+which symmetries those choices must respect.
+
+\subsection{Contractions and canonical identifications}
+
+A vertex is a pair $v=(r,U)$, where $r\le q$, $q=r_a$, and $U$ is a
+nonempty ideal of $\downarrow a$. Vertices form a ground set by
+Lemma~\ref{lem:availability-ideals}. For $n\in U$ define a
+\emph{contraction}
+\begin{equation}\label{eq:quotient-contraction}
+ (r,U)\longrightarrow(r_n,U_n),\qquad
+ U_n=\{x\sqsubseteq c:nx\in U\},\quad nc=a.
+\end{equation}
+Left multiplication preserves conditional joins. In fact cancellation
+and the common-cone description of a join give
+$n(x\vee y)=nx\vee ny$ whenever these elements have a common right
+multiple. It follows that $U_n$ is an ideal. Contractions compose;
+there is at most one contraction between fixed endpoints, since the
+center of $r_n$ is $r(n)$ and $r$ is injective. Write $B_v(F)$ and
+$L_v$ for the two fibers at a vertex.
+
+\begin{lemma}[Contraction identifications]\label{lem:contraction-fibers}
+For every contraction $v\to w$, the label fibers satisfy $L_v=L_w$,
+and inclusion of symbolic elements gives a bijection
+$B_w(F)\longrightarrow B_v(F)$. These identifications commute with
+composition.
+\end{lemma}
+\begin{proof}
+Let $v=(r,U)$, $w=(r_n,U_n)$, and first take a symbolic class
+available at $r_n$. Its full availability ideal $J$ at $r$ contains
+$n$. For $z\sqsubseteq a$, ideal closure gives
+\[
+ z\in J\quad\Longleftrightarrow\quad z\vee n\in J.
+\]
+Write $z\vee n=nx$. Availability at $r_{z\vee n}$ is exactly the
+corresponding availability at the subdiagram $(r_n)_x$ of $r_n$.
+Thus the restricted ideal at $r_n$ determines the full ideal at $r$.
+If that restricted ideal is $U_n$, the last test is equivalent to
+$z\vee n\in U$, and hence to $z\in U$. Conversely, an element of
+$B_v(F)$ is available at $r_n$ because $n\in U$, and its restricted
+ideal is $U_n$. This proves surjectivity; injectivity follows from
+Equation~\eqref{eq:symbolic-inclusion}. For labels use the same
+argument with literal membership in ranges. Commutation follows
+because all the symbolic maps retain the same representative names.
+\end{proof}
+
+Let a component be a connected component of the vertex system when
+the direction of contractions is ignored. The successors of each
+vertex are directed: contract at the join of two contraction indices.
+Any two vertices in one component have a common successor. Indeed,
+this holds along a single edge; along a finite path combine a common
+successor of its initial part with the final edge, using directedness
+of successors at the shared vertex. The same induction gives a common
+successor for any finite set of vertices in a component.
+
+Consequently all finite chains of the identifications in
+Lemma~\ref{lem:contraction-fibers} are consistent. One can compare a
+finite chain at a common successor, where both composites are the
+same map. In a component $C$, glue the disjoint union of its $B_v(F)$
+by these identifications and call the resulting set $B_C(F)$. Each
+fiber maps bijectively onto $B_C(F)$. In particular, the gluing creates
+no new identifications within one fiber. All label fibers in $C$
+are the same ground subset of $I$; denote it by $L_C$. Both sets have
+cardinality $\mu$ in $V[F]$.
+
+Choose in $V$ a vertex $(r,U)$ of $C$ and a nondecreasing cofinal
+sequence $(n_i:i<\omega)$ in $U$, with $n_0=1$. Let $v_i$ be its
+contraction at $n_i$. This chain is cofinal in the entire component.
+To check this, compare any vertex with $v_0$ at a common successor,
+which is a contraction of $v_0$ at some $n\in U$. Choose $i$ with
+$n\sqsubseteq n_i$. Then $v_i$ is a successor of that common
+successor and hence of the given vertex.
+
+\subsection{Only countably many residual symmetries}
+
+The group $G_q$ acts on the ground vertex system. On symbolic fibers
+it acts by transport between generics:
+\begin{equation}\label{eq:symbolic-transport}
+ T_\pi^F:\mathcal D_F(r)\longrightarrow
+                   \mathcal D_{\pi F}(\pi r),
+ \qquad [\tau]_r\longmapsto[\pi\tau]_{\pi r}.
+\end{equation}
+The forcing symmetry lemma makes this well defined, and it commutes
+with all contraction identifications. We use the same notation for
+transport on the glued fibers, and suppress the superscript when its
+source generic is specified by the argument.
+
+Let $G_C$ be the setwise stabilizer of $C$ in $G_q$. Define
+\[
+ K_C=\bigcup_{i<\omega}\Fix(D_{r_{n_i}}).
+\]
+Each displayed fixer is a subgroup of $G_q$, because $D_q$ is contained
+in every chain range. It fixes a vertex and therefore lies in $G_C$.
+The ranges decrease, so the subgroups increase and their union is a
+subgroup. For $\sigma\in G_C$, the moved chain is also cofinal.
+For every $i$ there is $j$ with
+$D_{r_{n_j}}\subseteq D_{\sigma r_{n_i}}$. Hence
+\[
+ \sigma\Fix(D_{r_{n_i}})\sigma^{-1}
+ =\Fix(D_{\sigma r_{n_i}})\subseteq\Fix(D_{r_{n_j}}).
+\]
+Applying the same reasoning to $\sigma^{-1}$ proves that $K_C$ is
+normal in $G_C$.
+
+\begin{lemma}[Countable stabilizer quotient]\label{lem:countable-germs}
+In the ground model, $|G_C/K_C|\le\aleph_0$.
+\end{lemma}
+\begin{proof}
+Choose the countable-base tower of Theorem~\ref{thm:monoid} so that
+its initial submonoid $T_0$ contains all $n_i$ and all right
+quotients between them. For $\pi\in G_C$, consider its alignments
+with contractions of the chosen vertex:
+\[
+ \mathcal A_\pi=
+ \{(l,m)\in\omega\times U:\pi r_{n_l}=r_m\}.
+\]
+This set is nonempty: the moved chain is cofinal in $C$, so one of
+its terms is a successor of $v_0$. Every second coordinate $m$ of
+an alignment divides some $n_N$, by cofinality of $(n_i)$ in $U$.
+
+To apply Lemma~\ref{lem:mono-descent}, fix $(l,m)\in\mathcal A_\pi$
+and any $N$. Choose $l'\ge l$ so that $\pi v_{l'}$ is a successor
+of $v_N$. Its diagram is $r_k$ for some $k\in U$ with
+$n_N\sqsubseteq k$. Let $d\in T_0$ be the right quotient
+$n_ld=n_{l'}$. Then
+\[
+ r_k=\pi r_{n_{l'}}=(\pi r_{n_l})_d=r_{md}.
+\]
+Injectivity of $r$ gives $k=md$. Consequently $md\in U$,
+$n_N\sqsubseteq md$, and $(l',md)\in\mathcal A_\pi$.
+These are exactly the replacement hypotheses of
+Lemma~\ref{lem:mono-descent}. Its conclusion supplies an alignment
+$(l,m)$ with $m\in T_0$.
+
+There are countably many resulting alignment pairs
+$(l,m)\in\omega\times T_0$. Two permutations giving the same pair
+agree pointwise on $D_{r_{n_l}}$; if they are $\pi$ and $\rho$,
+then $\pi^{-1}\rho\in\Fix(D_{r_{n_l}})\subseteq K_C$.
+Enumerate the pairs, and assign to each coset the first
+pair realized by any of its members. Distinct cosets receive distinct
+pairs. This proves the bound.
+\end{proof}
+
+\section{Equivariant assignments and the quotient bijection}
+\label{sec:quotient-equivariance}
+
+For each component $C$ we seek a bijection
+$f_C(F):L_C\to B_C(F)$ satisfying
+\begin{equation}\label{eq:component-covariance}
+ f_C(\sigma F)(\sigma j)=T_\sigma^F(f_C(F)(j))
+              \qquad(\sigma\in G_C).
+\end{equation}
+This is an identity between computations at different generics, not
+an action on one fixed extension. We first construct a bijection
+covariant under $K_C$: its elements fix a sufficiently late subdiagram,
+so they fix the tails of suitable name codes. We then handle the
+remaining countable quotient $G_C/K_C$. Bounded Cohen-row patterns
+distinguish its cosets and select a transported bijection with the
+full covariance in Equation~\eqref{eq:component-covariance}.
+
+\subsection{An assignment invariant under a tail fixer}
+
+For the chosen chain of $C$, let $\mathcal Q_C$ be the ground set of
+sequences
+\[
+ c=(\tau_i:i<\omega),\qquad \tau_i\in\mathcal X(r_{n_i}).
+\]
+Say that $c$ represents $b\in B_C(F)$ if, from some index onward,
+$\tau_i$ represents an element of the exact fiber $B_{v_i}(F)$ and
+its image in $B_C(F)$ is $b$. Finite initial segments are ignored.
+Every $b$ has such a code: choose representatives at all chain
+vertices in $V[F]$, using their bijections with $B_C(F)$. This is a
+countable sequence of ground name codes, and hence belongs to $V$
+by countable closure of $\mathbb R$.
+
+Fix a ground wellordering of $\mathcal Q_C$, of order type $\lambda_C$.
+Assign to $b$ its least representing code. Distinct $b$ have different
+codes, so this defines a wellorder $<_F$ of $B_C(F)$.
+
+\begin{lemma}[Covariance of the least-code order]
+\label{lem:least-code-covariance}
+For every $k\in K_C$, transport $T_k^F$ is an order isomorphism
+from $(B_C(F),<_F)$ to $(B_C(kF),<_{kF})$. The group $K_C$ fixes
+$L_C$ pointwise.
+\end{lemma}
+\begin{proof}
+Choose $N$ such that $k$ fixes $D_{r_{n_N}}$ pointwise. For $i\ge N$
+it fixes $r_{n_i}$ and every name in $\mathcal X(r_{n_i})$ literally.
+Consequently each particular ground code $c$ represents $b$ at $F$
+if and only if the \emph{same code} represents $T_k^F(b)$ at $kF$.
+The forcing symmetry lemma proves this on the tail; the definition
+ignores the remaining finite prefix. Thus the sets of functioning
+codes are identical, and their minima under the fixed ground
+wellorder are identical. This proves the order assertion. Also
+$L_C\subseteq D_{r_{n_i}}$ for every $i$, so $k$ fixes its every
+member.
+\end{proof}
+
+Let $\theta_C(F)$ be the order type of this order. It is at most
+$\lambda_C$, since it is the order induced on a subset of the ordered
+code set. It has cardinality $\mu$ in $V[F]$. Preservation of $\mu$
+and $\mu^+$ implies
+\[
+ \mu\le\theta_C(F)<\mu^+,
+ \qquad |\theta_C(F)|^V=\mu.
+\]
+Here the order type is an ordinal, hence a ground ordinal. The two
+preserved cardinal endpoints justify the last equality even if smaller
+cardinals have collapsed. In $V$, choose bijections
+$e_\theta:\mu\longrightarrow\theta$ for the set of ordinals
+$\theta\le\lambda_C$ with ground cardinality $\mu$. Also fix a ground
+bijection $l_C:\mu\longrightarrow L_C$. If
+$b_F:\theta_C(F)\longrightarrow B_C(F)$ is the unique order
+isomorphism, define
+\[
+ f_C^0(F)=b_F\circ e_{\theta_C(F)}\circ l_C^{-1}.
+\]
+This is a bijection $L_C\to B_C(F)$. All choices have been restricted
+to sets; no simultaneous choice over a proper class of ordinals occurs.
+Lemma~\ref{lem:least-code-covariance} gives
+\begin{equation}\label{eq:preliminary-covariance}
+ f_C^0(kF)(j)=T_k^F\bigl(f_C^0(F)(j)\bigr)
+       \qquad(k\in K_C,\ j\in L_C).
+\end{equation}
+The ground wellorder of all codes need not be invariant under $k$.
+The proof uses eventual fixation of each code's entries instead.
+
+\subsection{Detecting and choosing a coset}
+
+Put $\Gamma_C=G_C/K_C$, writing cosets as $\pi K_C$. It is a
+countable ground
+group by Lemma~\ref{lem:countable-germs}. For each nonidentity coset
+$\delta$, choose a ground representative $\gamma_\delta$ and points
+\[
+ s_{\delta,i}\in D_{r_{n_i}},\qquad
+ \gamma_\delta(s_{\delta,i})\ne s_{\delta,i}
+       \quad(i<\omega).
+\]
+Such a point exists for every $i$, since otherwise
+$\gamma_\delta\in K_C$. Fix an enumeration of the set of nonidentity
+cosets. For $\alpha<\kappa$ and $\pi\in G_C$, define the pattern
+\begin{equation}\label{eq:detector-pattern}
+ P_F^\alpha(\pi K_C)=
+ \left(
+  \left[\left(A_{\pi(s_{\delta,i})}^F
+                       \restriction\alpha:i<\omega\right)\right]_{\rm ev}
+              :\delta\in\Gamma_C\setminus\{1\}\right),
+\end{equation}
+where $[\cdot]_{\rm ev}$ denotes equivalence modulo eventual equality
+of sequences. We regard each row as its binary characteristic
+function. The equivalence classes in this formula are taken in the
+ground set of sequences of subsets of $\alpha$.
+
+This ground interpretation is legitimate. Every $\alpha<\omega_1$
+is countable. The bounded row restrictions and all their countably
+many coordinates, for all detectors, form a countable sequence of
+ground bits and indices. Since $\mathbb R$ adds no such sequences,
+these data belong to $V$. Eventual equality between ground sequences
+has the same meaning in the extension. For all $\alpha<\kappa$
+together the possible patterns form a ground set, on which choose a
+ground wellorder, separately for each $\alpha$ if desired.
+
+\begin{lemma}[A covariant coset selector]\label{lem:coset-selector}
+Formula~\eqref{eq:detector-pattern} is independent of the
+representative of $\pi K_C$. For each $F$ there is an
+$\alpha<\kappa$ at which all distinct cosets have distinct patterns.
+There is a definable selection $c_C(F)\in\Gamma_C$ satisfying
+\begin{equation}\label{eq:coset-selector-covariance}
+ c_C(\sigma F)=\sigma c_C(F)\qquad(\sigma\in G_C).
+\end{equation}
+\end{lemma}
+\begin{proof}
+If $k\in K_C$, it fixes all $s_{\delta,i}$ once $i$ is sufficiently
+large. Replacing $\pi$ with $\pi k$ therefore changes each detector
+sequence in only finitely many places, proving representative
+independence.
+
+Suppose $\pi K_C\ne\pi'K_C$. Use the detector for
+$\delta=(\pi^{-1}\pi')K_C$ and write
+$\pi^{-1}\pi'=\gamma_\delta k$ with $k\in K_C$. Eventually
+$k(s_{\delta,i})=s_{\delta,i}$, and hence
+\[
+ \pi(s_{\delta,i})\ne\pi'(s_{\delta,i}).
+\]
+Distinct label indices give distinct Cohen rows. For each of these
+tail positions choose a bit where the two rows differ. The countably
+many bit ordinals are bounded below regular $\kappa$. Above that
+bound the two restricted sequences disagree at every position on a
+tail, so are unequal modulo eventual equality. There are countably
+many pairs of cosets; a further countable supremum supplies one bound
+that distinguishes all pairs. These choices take place in $V[F]$.
+They use regularity of $\kappa$, not closure of $\mathbb P$.
+
+At the least distinguishing $\alpha$, choose the coset whose pattern
+is least in the fixed ground wellorder. If $\Gamma_C$ is trivial,
+choose its sole element and use $\alpha=0$. For $\sigma\in G_C$,
+label transport gives, for every $s\in I$,
+$A_{\sigma s}^{\sigma F}=A_s^F$. Thus
+\begin{equation}\label{eq:pattern-covariance}
+ P_{\sigma F}^\alpha((\sigma\pi)K_C)
+                    =P_F^\alpha(\pi K_C).
+\end{equation}
+The actual ground pattern objects on the two sides are equal. Left
+multiplication permutes the coset set, so the least distinguishing
+bound is unchanged and the same pattern object is minimal. This
+proves Equation~\eqref{eq:coset-selector-covariance}.
+\end{proof}
+
+For a coset $\gamma=\pi K_C$, define a transported preliminary
+bijection by
+\begin{equation}\label{eq:coset-assignment}
+ f_C^\gamma(F)(j)=
+ T_\pi^{\pi^{-1}F}
+       \left(f_C^0(\pi^{-1}F)(\pi^{-1}j)\right),\qquad j\in L_C.
+\end{equation}
+Equivalently, the square in Figure~\ref{fig:moved-generic-transport}
+commutes.
+
+\input{figures/moved-generic-transport}
+
+This is independent of $\pi$. Replacing $\pi$ with $\pi k$ inserts
+$k^{-1}$ in the source generic, and
+Equation~\eqref{eq:preliminary-covariance} cancels its transport;
+$k$ fixes every label of $L_C$. Define
+$f_C(F)=f_C^{c_C(F)}(F)$. Equations~\eqref{eq:coset-selector-covariance}
+and~\eqref{eq:coset-assignment} now give the required
+Equation~\eqref{eq:component-covariance}.
+This calculation explains why the preliminary computation in
+Equation~\eqref{eq:coset-assignment} is made at $\pi^{-1}F$.
+An arbitrary transport of the computation at $F$ would not give the
+required identity.
+
+\subsection{Transport over all components}
+
+Choose in $V$ one representative from each $G_q$-orbit of components
+and perform the preceding set-sized choices for these representatives.
+For $C'=\sigma C$, transport $f_C(\sigma^{-1}F)$ by $\sigma$ to
+obtain $f_{C'}(F)$. Two possible choices of $\sigma$ differ by an
+element of $G_C$, so Equation~\eqref{eq:component-covariance} proves
+that this definition is independent of the choice. It yields
+bijections for every component, covariant under all $G_q$. Pulling
+back along the canonical maps $B_v(F)\to B_C(F)$ gives bijections
+\begin{equation}\label{eq:coherent-fiber-bijections}
+ f_{r,U}(F):L_{r,U}\longrightarrow B_{r,U}(F).
+\end{equation}
+They commute with every contraction, and their transports under every
+$\pi\in G_q$ are the corresponding bijections at $\pi F$.
+
+We record the definability needed to convert these computations into
+one symmetric name. The ground sets of vertices, components, name
+codes, code wellorders, patterns and auxiliary bijections are all
+sets. The sections of the product names can be constructed in an
+$\mathbb R$-extension from its generic filter; the ordinary forcing
+relation for the ground set poset $\mathbb P$ is definable there.
+Symbolic quotients and their gluing use equivalence relations on sets.
+All remaining steps use set operations, least elements of prescribed
+wellorders, and unique order isomorphisms. They therefore define,
+by an ordinary set-theoretic formula in an $\mathbb R$-extension, the
+predicate
+\begin{equation}\label{eq:assignment-predicate}
+ \Phi(F,r,j,\tau):
+ \quad \tau\in\mathcal X(r)\text{ represents }
+ f_{r,U_j}(F)(j)\text{ in }\mathcal D_F(r).
+\end{equation}
+Here $U_j$ is ground data from Equation~\eqref{eq:label-availability}.
+One formal implementation uses a ground set of all normalized name
+codes and its ordinary name-evaluation relation to construct their
+$\mathbb P$-sections. The variable for a ground code and the actual
+forcing name that it codes are kept distinct. In particular, we never
+use the false assertion that an automorphism moves a check-name for a
+ground code to the check-name for its transported code.
+
+The proved semantic covariance is the identity
+\[
+ \Phi(F,r,j,\tau)\quad\Longleftrightarrow\quad
+ \Phi(\pi F,\pi r,\pi j,\pi\tau)
+       \qquad(\pi\in G_q).
+\]
+These computations are ordinary forcing arguments for this fixed
+formula. More explicitly, let
+\[
+ \dot F^{\pi}=
+ \{(\check{\pi d},d):d\in\mathbb R\},
+\]
+whose value at $F$ is $\pi F$. The preceding covariance proof gives
+the equivalence, forced by $d_0$ in $\mathbb R$, between
+$\Phi(\dot F,r,j,\tau)$ and
+$\Phi(\dot F^{\pi},\pi r,\pi j,\pi\tau)$, with all auxiliary
+ground choices fixed. Under the name action,
+$\pi\dot F^{\pi}=\dot F$ literally, while check names for the
+displayed ground arguments remain fixed. Applying ordinary forcing
+symmetry therefore gives
+\begin{equation}\label{eq:assignment-forcing-covariance}
+ d\forces_{\mathbb R}\Phi(\dot F,r,j,\tau)
+ \quad\Longleftrightarrow\quad
+ \pi d\forces_{\mathbb R}
+                  \Phi(\dot F,\pi r,\pi j,\pi\tau)
+ \qquad(d\le d_0).
+\end{equation}
+All auxiliary ground choices are already fixed in the defining
+formula; covariance was proved for this one construction. The
+$\mathbb R$-forcing values in this display are ground sets of
+conditions. Equivalently, if $F'$ contains $\pi d$, put
+$F=\pi^{-1}F'$ and apply covariance to the computation at $F$;
+the converse uses $\pi^{-1}$. This generic description abbreviates
+the fixed-formula argument above and requires no transitive-ground
+assumption in the consistency deduction.
+
+\subsection{One hereditarily symmetric graph}
+
+\begin{theorem}[Quotient classification]\label{thm:quotient-classification}
+In $W$, every nonwellorderable surjective image of $A$ is in bijection
+with $A$.
+\end{theorem}
+\begin{proof}
+We first finish the construction for the equivalence relation fixed
+at the start of Section~\ref{sec:quotient-fibers}. Use canonical
+equivariant ordered-pair names and form the product name
+\begin{equation}\label{eq:quotient-bijection-name}
+ \dot J=
+ \left\{
+ \left(\langle\dot A_j,\tau\rangle^{\bullet},(d,r)\right):
+ \begin{array}{l}
+ r\le q,\ j\in D_r,\ \tau\in\mathcal X(r),\ d\le d_0,\\
+ d\forces_{\mathbb R}\Phi(\dot F,r,j,\tau)
+ \end{array}\right\}.
+\end{equation}
+The superscript denotes the name for the ordered pair. This is a set
+name: every index ranges over a ground set and the defining forcing
+relation is a ground relation. Equation~\eqref{eq:assignment-forcing-covariance}
+sends each entry under $\pi\in G_q$ to another entry. Applying
+$\pi^{-1}$ gives equality of the sets of entries, so $G_q$ fixes
+$\dot J$ literally. Each member name is a pair of HS names; thus
+$\dot J$ is HS with support $D_q$. The individual member names may
+have different supports. Hereditary symmetry does not require all of
+them to use $D_q$.
+
+Take a product generic $(F,H)$ containing $(d_0,q)$. For every
+$r\in H$ below $q$, the graph contains exactly the local assignments
+of Equation~\eqref{eq:coherent-fiber-bijections}. In detail, whenever
+$\Phi(F,r,j,\tau)$ holds, the $\mathbb R$ truth lemma gives a
+condition of $F$ forcing it; strengthening with $d_0$ retains a
+condition of $F$, and Equation~\eqref{eq:quotient-bijection-name}
+includes its entry. Conversely, every active entry forces its
+asserted assignment. All classes are represented by names in their
+symbolic equivalence classes, so the local assignments are complete.
+
+We check the comparison when the diagram is strengthened. Suppose
+$t\le r\le q$, $r=t_b$, and $j\in D_r$. The availability ideal
+of $j$ at $t$ contains $b$. Hence
+\[
+ (t,U_j\text{ computed at }t)
+        \longrightarrow(r,U_j\text{ computed at }r)
+\]
+is a contraction. Equation~\eqref{eq:coherent-fiber-bijections}
+therefore gives the same assigned symbolic element under inclusion.
+In particular, its product-name evaluations are equal in every
+$\mathbb P$-generic containing $t$.
+
+Any two active entries of $\dot J$ occur together below a common
+stronger diagram in $H$. At that diagram the union of the
+exact-fiber bijections is a bijection
+$D_t\to\mathcal D_F(t)$: availability ideals partition both sets.
+The preceding comparison proves single-valuedness and injectivity of
+$\dot J^{F,H}$. Distinct symbolic elements are forced unequal there
+by local decision, and distinct indices have distinct Cohen labels.
+Thus these are also single-valuedness and injectivity for the actual
+sets of the graph.
+
+Every label index belongs to a diagram in $H$ below $q$, by the
+density of extending diagrams to contain prescribed supports.
+Therefore the graph has domain $A$. To reach an arbitrary $E$-class,
+take any label belonging to that class and include its index in such
+a diagram $r$. Its canonical supported class name represents an
+element of $\mathcal D_F(r)$. The local fiber bijection is onto the
+fiber containing that element, so the graph reaches the class.
+This is a pointwise existence argument; no simultaneous selection
+of representatives of all classes has been made. The range is
+exactly $A/E$. The graph belongs to $W$ because its name is HS.
+
+The construction was made on an extension of an arbitrary symmetric
+condition asserting the nonwellorderable-quotient hypothesis. Applying
+the symmetric forcing theorem and truth lemma therefore proves in
+$W$ that every such quotient $A/E$ is in bijection with $A$.
+Finally, for a surjection $h:A\to Y$ in $W$, form its kernel
+relation $E$ in $W$. The map $[a]_E\mapsto h(a)$ is a bijection
+$A/E\to Y$, defined by its unique values without choosing class
+representatives. If $Y$ is nonwellorderable, so is $A/E$, and composing
+this bijection with the one just constructed proves the theorem.
+\end{proof}

preprints/The-Partition-Principle-does-not-imply-Choice-September-24-2026/partition-principle-without-choice.pdf

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:7101fb4828d8d2ba26e000ee813f467bc38fb974f00ed38d8da37926e4386005
+size 610408

preprints/The-Penrose-inequality-for-maximal-asymptotically-hyperbolic-initial-data-October-5-2026/README.md

@@ -0,0 +1,18 @@
+# [The Penrose inequality for maximal asymptotically hyperbolic initial data](paper.pdf)
+
+**Author:** OpenAI
+
+**Date:** October 5, 2026
+
+## Citation
+
+```bibtex
+@misc{OAI:The-Penrose-inequality-for-maximal-asymptotically-hyperbolic-initial-data-October-5-2026,
+  author = {{OpenAI}},
+  title = {{The Penrose inequality for maximal asymptotically hyperbolic initial data}},
+  howpublished = {OpenAI Math Release preprint
+                  \href{https://github.com/openai/math/blob/main/preprints/The-Penrose-inequality-for-maximal-asymptotically-hyperbolic-initial-data-October-5-2026/paper.pdf}{OAI:The-Penrose-inequality-for-maximal-asymptotically-hyperbolic-initial-data-October-5-2026}},
+  year = {2026}
+}
+```
+

preprints/The-Penrose-inequality-for-maximal-asymptotically-hyperbolic-initial-data-October-5-2026/build/paper.tex

@@ -0,0 +1,32 @@
+\newcommand{\DoNotLoadEpstopdf}{}
+\documentclass[11pt]{article}
+\input{preamble}
+\title{The Penrose inequality for maximal\\asymptotically hyperbolic initial data}
+\author{OpenAI}
+\date{October 5, 2026}
+\begin{document}
+\maketitle
+\begin{abstract}
+We prove the sharp Penrose inequality for three-dimensional maximal
+asymptotically hyperbolic initial data with spherical conformal infinity.
+Under the dominant energy condition, the stated decay and integrability
+assumptions, and a future-timelike mass covector, the invariant mass is bounded below
+by the Schwarzschild--anti-de Sitter mass associated with the minimum
+enclosing area of a weakly future outer-trapped boundary.
+The boundary may be disconnected, and no restriction is imposed on the
+compact topology.
+\end{abstract}
+\tableofcontents
+\input{sections/01-introduction}
+\input{sections/02-preliminaries}
+\input{sections/03-graph-calculus}
+\input{sections/04-area-transfer}
+\input{sections/05-maximal-reduction}
+\input{sections/06-stress-extension}
+\input{sections/07-compact-construction}
+\input{sections/08-transmission-analysis}
+\input{sections/09-scalar-error}
+\input{sections/10-completion}
+\bibliographystyle{plainnat}
+\bibliography{references/references}
+\end{document}

preprints/The-Penrose-inequality-for-maximal-asymptotically-hyperbolic-initial-data-October-5-2026/build/preamble.tex

@@ -0,0 +1,45 @@
+\usepackage[T1]{fontenc}
+\usepackage[utf8]{inputenc}
+\usepackage{lmodern,microtype}
+\usepackage[margin=1.05in]{geometry}
+\usepackage{amsmath,amssymb,amsthm,mathtools,mathrsfs}
+\usepackage{enumitem,booktabs,array,graphicx,xcolor,tikz}
+\usetikzlibrary{arrows.meta,positioning,calc}
+\usepackage[numbers,sort&compress]{natbib}
+\usepackage{xurl}
+\usepackage[colorlinks=true,linkcolor=teal!55!black,citecolor=teal!55!black,
+  urlcolor=teal!55!black,bookmarksnumbered=true]{hyperref}
+\hypersetup{pdftitle={The Penrose inequality for maximal asymptotically hyperbolic initial data},pdfauthor={OpenAI},
+  bookmarksdepth=2}
+\input{glyphtounicode}
+\input{glyphtounicode-cmex}
+\pdfgentounicode=1
+\numberwithin{equation}{section}
+\newtheorem{theorem}{Theorem}[section]
+\newtheorem{proposition}[theorem]{Proposition}
+\newtheorem{lemma}[theorem]{Lemma}
+\newtheorem{corollary}[theorem]{Corollary}
+\theoremstyle{definition}
+\newtheorem{definition}[theorem]{Definition}
+\theoremstyle{remark}
+\newtheorem{remark}[theorem]{Remark}
+\newcommand{\R}{\mathbb R}
+\newcommand{\N}{\mathbb N}
+\newcommand{\Sph}{\mathbb S}
+\newcommand{\eps}{\varepsilon}
+\newcommand{\dd}{\mathop{}\!\mathrm d}
+\newcommand{\AH}{\ensuremath{\mathrm{AH}}}
+\newcommand{\AF}{\ensuremath{\mathrm{AF}}}
+\DeclareMathOperator{\tr}{tr}
+\DeclareMathOperator{\Ric}{Ric}
+\DeclareMathOperator{\Scal}{Scal}
+\DeclareMathOperator{\divg}{div}
+\DeclareMathOperator{\Per}{Per}
+\DeclareMathOperator{\Area}{Area}
+\DeclareMathOperator{\Vol}{Vol}
+\DeclareMathOperator{\supp}{supp}
+\DeclareMathOperator{\dist}{dist}
+\DeclareMathOperator{\Id}{Id}
+\setlist{topsep=4pt,itemsep=2pt,parsep=0pt}
+\setcounter{tocdepth}{1}
+\emergencystretch=1.2em

preprints/The-Penrose-inequality-for-maximal-asymptotically-hyperbolic-initial-data-October-5-2026/build/references/references.bib

@@ -0,0 +1,257 @@
+
+@article{Penrose1973,
+  author = {Penrose, Roger},
+  title = {Naked singularities},
+  journal = {Annals of the New York Academy of Sciences},
+  volume = {224},
+  number = {1},
+  year = {1973},
+  pages = {125--134},
+  doi = {10.1111/j.1749-6632.1973.tb41447.x}
+}
+
+@article{HI2001,
+  author = {Huisken, Gerhard and Ilmanen, Tom},
+  title = {The inverse mean curvature flow and the {Riemannian Penrose} inequality},
+  journal = {Journal of Differential Geometry},
+  volume = {59},
+  number = {3},
+  year = {2001},
+  pages = {353--437},
+  doi = {10.4310/jdg/1090349447},
+  url = {https://math.jhu.edu/~js/Math646/huisken-ilmanen.pdf}
+}
+
+@article{Bray2001,
+  author = {Bray, Hubert L.},
+  title = {Proof of the {Riemannian Penrose} inequality using the positive mass theorem},
+  journal = {Journal of Differential Geometry},
+  volume = {59},
+  number = {2},
+  year = {2001},
+  pages = {177--267},
+  doi = {10.4310/jdg/1090349428},
+  url = {https://www.maths.tcd.ie/EMIS/journals/NYJM/jdg/p/2001/59-2-1.pdf}
+}
+
+@article{BrayLee2009,
+  author = {Bray, Hubert L. and Lee, Dan A.},
+  title = {On the {Riemannian Penrose} inequality in dimensions less than eight},
+  journal = {Duke Mathematical Journal},
+  volume = {148},
+  number = {1},
+  year = {2009},
+  pages = {81--106},
+  doi = {10.1215/00127094-2009-020},
+  url = {https://math.duke.edu/~bray/PE/euclid.dmj.1240432192.pdf}
+}
+
+@article{CH2003,
+  author = {Chru{\'s}ciel, Piotr T. and Herzlich, Marc},
+  title = {The mass of asymptotically hyperbolic {Riemannian} manifolds},
+  journal = {Pacific Journal of Mathematics},
+  volume = {212},
+  number = {2},
+  year = {2003},
+  pages = {231--264},
+  doi = {10.2140/pjm.2003.212.231},
+  url = {https://msp.org/pjm/2003/212-2/pjm-v212-n2-p02-p.pdf}
+}
+
+@article{Neves2010,
+  author = {Neves, Andr{\'e}},
+  title = {Insufficient convergence of inverse mean curvature flow on asymptotically hyperbolic manifolds},
+  journal = {Journal of Differential Geometry},
+  volume = {84},
+  number = {1},
+  year = {2010},
+  pages = {191--229},
+  doi = {10.4310/jdg/1271271798},
+  eprint = {0711.4335},
+  archivePrefix = {arXiv},
+  primaryClass = {math.DG},
+  url = {https://arxiv.org/abs/0711.4335}
+}
+
+@article{DGS2013,
+  author = {Dahl, Mattias and Gicquaud, Romain and Sakovich, Anna},
+  title = {{Penrose} type inequalities for asymptotically hyperbolic graphs},
+  journal = {Annales Henri Poincar{\'e}},
+  volume = {14},
+  number = {5},
+  year = {2013},
+  pages = {1135--1168},
+  doi = {10.1007/s00023-012-0218-4},
+  eprint = {1201.3321},
+  archivePrefix = {arXiv},
+  primaryClass = {math.DG},
+  url = {https://arxiv.org/abs/1201.3321}
+}
+
+@article{deLimaGirao2016,
+  author = {de Lima, Levi Lopes and Gir{\~a}o, Frederico},
+  title = {An {Alexandrov--Fenchel}-type inequality in hyperbolic space with an application to a {Penrose} inequality},
+  journal = {Annales Henri Poincar{\'e}},
+  volume = {17},
+  number = {4},
+  year = {2016},
+  pages = {979--1002},
+  doi = {10.1007/s00023-015-0414-0},
+  eprint = {1209.0438},
+  archivePrefix = {arXiv},
+  primaryClass = {math.DG},
+  url = {https://arxiv.org/abs/1209.0438}
+}
+
+@article{ShiTam2002,
+  author = {Shi, Yuguang and Tam, Luen-Fai},
+  title = {Positive mass theorem and the boundary behaviors of compact manifolds with nonnegative scalar curvature},
+  journal = {Journal of Differential Geometry},
+  volume = {62},
+  number = {1},
+  year = {2002},
+  pages = {79--125},
+  doi = {10.4310/jdg/1090425530},
+  url = {https://www.maths.tcd.ie/EMIS/journals/NYJM/jdg/p/2002/62-1-5.pdf}
+}
+
+@article{Nirenberg1953,
+  author = {Nirenberg, Louis},
+  title = {The {Weyl} and {Minkowski} problems in differential geometry in the large},
+  journal = {Communications on Pure and Applied Mathematics},
+  volume = {6},
+  number = {3},
+  year = {1953},
+  pages = {337--394},
+  doi = {10.1002/cpa.3160060303}
+}
+
+@book{GT2001,
+  author = {Gilbarg, David and Trudinger, Neil S.},
+  title = {Elliptic Partial Differential Equations of Second Order},
+  series = {Classics in Mathematics},
+  publisher = {Springer},
+  address = {Berlin},
+  year = {2001},
+  edition = {Second},
+  note = {Reprint of the 1998 edition},
+  isbn = {978-3-540-41160-4},
+  doi = {10.1007/978-3-642-61798-0}
+}
+
+@article{AraujoZhang2020,
+  author = {Ara{\'u}jo, Dami{\~a}o J. and Zhang, Lei},
+  title = {Optimal {$C^{1,\alpha}$} estimates for a class of elliptic quasilinear equations},
+  journal = {Communications in Contemporary Mathematics},
+  volume = {22},
+  number = {5},
+  year = {2020},
+  pages = {1950014},
+  doi = {10.1142/S0219199719500147},
+  eprint = {1507.06898},
+  archivePrefix = {arXiv},
+  primaryClass = {math.AP},
+  url = {https://arxiv.org/pdf/1507.06898v4}
+}
+
+@article{LSW1963,
+  author = {Littman, Walter and Stampacchia, Guido and Weinberger, Hans F.},
+  title = {Regular points for elliptic equations with discontinuous coefficients},
+  journal = {Annali della Scuola Normale Superiore di Pisa, Classe di Scienze},
+  series = {3},
+  volume = {17},
+  number = {1--2},
+  year = {1963},
+  pages = {43--77},
+  url = {https://www.numdam.org/item/ASNSP_1963_3_17_1-2_43_0.pdf}
+}
+
+@article{Serrin1964,
+  author = {Serrin, James},
+  title = {Local behavior of solutions of quasi-linear equations},
+  journal = {Acta Mathematica},
+  volume = {111},
+  year = {1964},
+  pages = {247--302},
+  doi = {10.1007/BF02391014}
+}
+
+@article{Trudinger1973,
+  author = {Trudinger, Neil S.},
+  title = {Linear elliptic operators with measurable coefficients},
+  journal = {Annali della Scuola Normale Superiore di Pisa, Classe di Scienze},
+  series = {3},
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+  number = {2},
+  year = {1973},
+  pages = {265--308},
+  url = {https://www.numdam.org/article/ASNSP_1973_3_27_2_265_0.pdf}
+}
+
+@article{CalderonZygmund1952,
+  author = {Calder{\'o}n, Alberto P. and Zygmund, Antoni},
+  title = {On the existence of certain singular integrals},
+  journal = {Acta Mathematica},
+  volume = {88},
+  year = {1952},
+  pages = {85--139},
+  doi = {10.1007/BF02392130}
+}
+
+@book{Stein1970,
+  author = {Stein, Elias M.},
+  title = {Singular Integrals and Differentiability Properties of Functions},
+  series = {Princeton Mathematical Series},
+  volume = {30},
+  publisher = {Princeton University Press},
+  address = {Princeton, NJ},
+  year = {1970},
+  url = {https://assets.press.princeton.edu/about_pup/PUP100/book/5gStein.pdf}
+}
+
+@book{Maggi2012,
+  author = {Maggi, Francesco},
+  title = {Sets of Finite Perimeter and Geometric Variational Problems: An Introduction to Geometric Measure Theory},
+  series = {Cambridge Studies in Advanced Mathematics},
+  volume = {135},
+  publisher = {Cambridge University Press},
+  address = {Cambridge},
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+  doi = {10.1017/CBO9781139108133},
+  isbn = {978-1-107-02103-7},
+  url = {https://cvgmt.sns.it/paper/1880/}
+}
+
+@article{ChoiKim2013,
+  author = {Choi, Jongkeun and Kim, Seick},
+  title = {Neumann functions for second order elliptic systems with measurable coefficients},
+  journal = {Transactions of the American Mathematical Society},
+  volume = {365},
+  number = {12},
+  year = {2013},
+  pages = {6283--6307},
+  doi = {10.1090/S0002-9947-2013-05886-2},
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+}
+
+@misc{Simon2014,
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+}
+
+@article{Miao2002,
+  author = {Miao, Pengzi},
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+  url = {https://arxiv.org/abs/math-ph/0212025}
+}

preprints/The-Penrose-inequality-for-maximal-asymptotically-hyperbolic-initial-data-October-5-2026/build/sections/01-introduction.tex

@@ -0,0 +1,239 @@
+\section{Introduction and statement of the result}
+\label{sec:hypotheses}
+
+The Penrose inequality relates the mass of an initial-data set to the
+area needed to enclose a trapped region. Penrose's original argument
+\citep{Penrose1973} connects such a bound with gravitational collapse
+and the expected behavior of black holes. In the asymptotically flat
+Riemannian case, Huisken and Ilmanen proved the connected-horizon
+inequality by weak inverse mean curvature flow \citep{HI2001}, and
+Bray proved the inequality for the total area of a possibly disconnected
+horizon \citep{Bray2001}. Bray and Lee extended the Riemannian result to
+dimensions below eight \citep{BrayLee2009}.
+
+For negative cosmological constant the natural comparison metric is
+Schwarzschild--anti-de Sitter. Its horizon of area $4\pi a^2$ has mass
+$(a+a^3)/2$ when the cosmological constant is $-3$.
+The additional cubic term is accompanied by a different asymptotic mass:
+the four fluxes at spherical conformal infinity form a Lorentz covector
+\citep{CH2003}. For asymptotically hyperbolic graphs,
+Dahl, Gicquaud, and Sakovich established mass identities and Penrose-type
+bounds \citep{DGS2013}; de Lima and Gir\~ao proved the sharp bound for
+a class of balanced graphs whose horizons are star-shaped and mean
+convex in the reference slice
+\citep{deLimaGirao2016}. Directly passing to infinity along
+inverse mean curvature flow has a distinct obstruction in this setting
+\citep{Neves2010}.
+
+Here we prove the global maximal asymptotically hyperbolic Penrose
+inequality in the class specified below. Thus the corresponding maximal
+formulation of the Penrose conjecture is resolved positively, with
+arbitrary compact topology and without an outermostness assumption on
+the original boundary. The area in the statement is the minimum
+\emph{enclosing} area, not necessarily the area of that boundary.
+The proof does not require a spacetime development or an additional
+null-expansion condition. We make no assertion about nonmaximal
+initial data or equality rigidity.
+
+\subsection{Geometry, decay, and mass}
+
+Let $(\Omega,g)$ be a smooth connected orientable three-manifold with
+nonempty compact smooth boundary $S$. The metric space obtained by
+including $S$ is complete. There is one end, with coordinates
+$(r,\omega)\in(r_0,\infty)\times\Sph^2$ outside a compact region.
+Write
+\begin{equation}\label{eq:background}
+ b=\frac{\dd r^2}{1+r^2}+r^2\sigma,
+ \qquad \eta=\operatorname{arsinh}r,
+ \qquad V_0=\sqrt{1+r^2},\qquad V_i=r\omega_i\quad(1\le i\le3),
+\end{equation}
+where $\sigma$ is the unit round metric and $\omega_i$ are the coordinate
+functions of the unit sphere. The Laplacian is $\Delta=\divg\nabla$;
+mean curvature is the sum of the principal curvatures.
+
+Fix $0<\alpha<1$ and $3/2<\tau<3$. On the end, a tensor belongs to
+$C^{k,\alpha}_\tau$ if its $b$-covariant derivatives through order $k$
+are $O(e^{-\tau\eta})$, and its order-$k$ local H\"older seminorms
+on uniform $b$-unit coordinate balls have the same weight. Equivalently,
+the $C^{k,\alpha}$ norms of its components in uniform background charts,
+multiplied by $e^{\tau\eta}$ at the chart centers, are bounded.
+This convention is a tensor-norm convention. It does not mean
+Euclidean coordinate-component decay. On compact subsets, including
+the finite boundary, all fields below are smooth.
+
+Let $K$ be a smooth symmetric covariant two-tensor on $\Omega$, smooth
+up to $S$, and assume
+\begin{equation}\label{eq:decay}
+ g-b\in C^{2,\alpha}_\tau,
+ \qquad K\in C^{1,\alpha}_\tau.
+\end{equation}
+For $e=g-b$, define the mass fluxes in the chosen chart by
+\begin{align}\label{eq:ah-mass}
+ p_\mu(g)&=\frac1{16\pi}\lim_{R\to\infty}
+       \int_{\{r=R\}}\mathbb U(V_\mu,e)(\nu_b)\dd A_b,
+       &&0\le\mu\le3,\\
+ \mathbb U(V,e)&=V(\divg_b e-\dd\tr_b e)
+              +(\tr_b e)\dd V-e(\nabla_b V,\cdot).
+              \label{eq:mass-integrand}
+\end{align}
+Here $(\divg_b e)_i=\nabla_b^j e_{ji}$, and $\nu_b$ points toward
+increasing $r$. Every operator and measure in this definition uses $b$.
+We require the four limits to exist and be finite. When the covector
+is future timelike, its invariant mass is
+\begin{equation}\label{eq:invariant-mass}
+ m_{\AH}(g)=\sqrt{p_0(g)^2-p_1(g)^2-p_2(g)^2-p_3(g)^2},
+ \qquad p_0(g)>\sqrt{p_1(g)^2+p_2(g)^2+p_3(g)^2}.
+\end{equation}
+The spatial entries in this covector are metric mass charges; they are
+not ADM momentum or angular momentum.
+
+\subsection{Constraints, boundary, and enclosing area}
+
+We use units $G=c=1$ and cosmological constant $\Lambda=-3$.
+Maximality and the noncosmological constraint densities are
+\begin{equation}\label{eq:constraints}
+ \tr_gK=0,
+ \qquad 16\pi\mu=R_g+6-|K|_g^2,
+ \qquad 8\pi J_i=\nabla_g^jK_{ij}.
+\end{equation}
+Assume the dominant energy condition and weighted integrability
+\begin{equation}\label{eq:dec-integrability}
+ \mu\ge|J|_g,
+ \qquad \int_{\mathrm{end}}V_0
+       \bigl(\mu+|J|_g+|K|_g^2\bigr)\dd V_g<\infty.
+\end{equation}
+In particular $R_g+6\ge0$ and its $V_0$-weighted integral is finite.
+
+On each component of $S$, choose the unit normal $\nu$ pointing into
+$\Omega$, toward the designated exterior. Set
+\begin{equation}\label{eq:future-trapping}
+ H=\divg_S\nu,
+ \qquad \tr_SK=(g^{ij}-\nu^i\nu^j)K_{ij},
+ \qquad \theta_+=H+\tr_SK.
+\end{equation}
+We assume only $\theta_+\le0$ on every component, with this same future
+orientation. For comparison with a spacetime convention, the sign is
+$K(X,Y)=\overline g(\overline\nabla_Xn_{\mathrm{future}},Y)$.
+There is no assumption on the inward null expansion.
+
+\begin{definition}[Minimum enclosing area]\label{def:minimum-area}
+An admissible end-side domain is a connected smooth codimension-zero
+submanifold-with-boundary $D\subset\Omega$, closed as a subset of
+$\Omega$, such that
+\[
+ \operatorname{int}D\subset\operatorname{int}\Omega
+ \quad\hbox{and}\quad
+ \{r>R\}\subset D\quad\hbox{for some }R.
+\]
+Its entire intrinsic manifold boundary $\partial D$ must be compact,
+smooth, embedded, and two-sided, with any portions coinciding with $S$
+included. Define
+\begin{equation}\label{eq:minimum-area}
+ A_{\min}(S;g)=\inf_D\Area_g(\partial D).
+\end{equation}
+In particular $D=\Omega$ is admissible and contributes $\partial D=S$.
+An empty relative frontier is not an admissible replacement for this
+intrinsic boundary.
+\end{definition}
+
+Neither the number nor the genus of the boundary components is
+restricted. No trapping condition is imposed on the competitors in
+Definition~\ref{def:minimum-area}.
+
+\begin{theorem}[Global maximal AH Penrose inequality]\label{thm:main}
+Under \eqref{eq:decay}--\eqref{eq:future-trapping}, the completeness,
+one-end, and smoothness assumptions above, and the future-timelike
+condition \eqref{eq:invariant-mass}, one has
+\begin{equation}\label{eq:main-inequality}
+ m_{\AH}(g)\ge\frac12\bigl(r_A+r_A^3\bigr),
+ \qquad r_A=\sqrt{\frac{A_{\min}(S;g)}{4\pi}}.
+\end{equation}
+The coefficient is sharp: equality holds for every positive-mass
+time-symmetric Schwarzschild--anti-de Sitter exterior bounded by its
+minimal horizon.
+\end{theorem}
+
+The proof establishes a time-component statement for Riemannian data
+which is useful independently and which does not require a causal
+condition on their mass covector.
+
+\begin{theorem}[Time-symmetric time-component inequality]\label{thm:ts}
+Let $(\Omega,g)$ have the smoothness, topology, completeness, and end
+structure above. Suppose $g-b\in C^{2,\alpha}_\tau$, $3/2<\tau<3$,
+the four fluxes \eqref{eq:ah-mass} are finite, and
+\[
+ R_g\ge-6,\qquad H_g(S)\le0,
+ \qquad \int_{\mathrm{end}}V_0(R_g+6)\dd V_g<\infty.
+\]
+In every such fixed end chart,
+\begin{equation}\label{eq:ts-inequality}
+ p_0(g)\ge\frac12(a+a^3),
+ \qquad a=\sqrt{A_{\min}(S;g)/(4\pi)}.
+\end{equation}
+\end{theorem}
+
+\subsection{Structure and new ingredients}
+
+The first deformation solves a finite-height graph problem together
+with a positive lapse and a conformal factor. It changes maximal data
+into Riemannian AH data with scalar curvature at least $-6$ and a free
+minimal boundary. Its mass cost tends to zero. The conformal factor
+can contract area, so pointwise comparison alone is insufficient.
+A volume penalty and a finite-time weak-flow argument recover the
+enclosing-area comparison. A continuous choice of inner flux, based
+on all possible perimeter-minimizer contact sets, removes contact with
+the original weakly trapped boundary.
+
+For Theorem~\ref{thm:ts}, a nonlinear electrostatic potential produces a
+divergence-free stress. Its algebraic constraint encodes the cubic
+term $a^3/2$ in the mass budget of an AF attachment. A second graph
+construction on a compact interior joined to that attachment yields
+an AF comparison exterior to which the ordinary Riemannian Penrose
+inequality applies. We prove the coupled transmission estimates and
+continuation needed for this construction. Where a trace cutoff can
+spoil the scalar-curvature sign, blowup gives uniformly controlled
+metric neighborhoods; additional obstacles then put these regions
+strictly inside a free minimizing enclosure.
+
+The area comparison uses weak inverse mean curvature flow only over
+a fixed interval of relative areas. Its curvature estimate depends on
+a scalar lower bound and fixed topology, and accommodates disconnected
+leaves. It therefore has a role different from taking an asymptotic
+limit of hyperbolic Hawking mass.
+
+The numerical argument can be seen before entering these constructions.
+In the positive-area time-symmetric case, fix a metric after the
+preliminary approximation and write
+$A_{\min}(S;g)=4\pi a^2$. Let $E_0$ denote the comparison enclosure of
+the auxiliary caps alone and $E$ the enclosure that also contains the
+regions where scalar curvature may be negative. Given small positive
+tolerances $\eps_m,\eps_A$, the attachment and comparison constructions,
+followed by area transfer, give
+\[
+ \begin{aligned}
+ m_o&\le p_0(g)-a^3/2+\eps_m,
+ &m_{\AF}(h')&\le m_o+\eps_m,\\
+ \Area_{h'}(\partial E)&\ge\Area_{h'}(\partial E_0)
+                    \ge(1-\eps_A)4\pi a^2.
+ \end{aligned}
+\]
+(Propositions~\ref{prop:af-attachment}, \ref{prop:af-comparison},
+and~\ref{prop:area-transfer}).
+The AF Penrose inequality applies to the exterior of $E$ and yields
+\[
+ p_0(g)\ge \frac{a^3}{2}
+       +\frac a2\sqrt{1-\eps_A}-2\eps_m.
+\]
+Thus the attachment supplies the cubic term and the AF inequality the
+linear term. The errors tend to zero through successive finite choices;
+no convergence of the comparison metrics is required.
+Theorem~\ref{thm:ah-reduction} then transfers the time-symmetric bound
+to maximal data. The original mass covector is put in a rest chart
+before any deformation, so its time component is the invariant mass.
+
+Sections~\ref{sec:graph-calculus} and~\ref{sec:area-transfer} develop
+the shared identities and area comparison. Section~\ref{sec:maximal-reduction}
+reduces Theorem~\ref{thm:main} to Theorem~\ref{thm:ts}.
+Sections~\ref{sec:stress-extension}--\ref{sec:scalar-error} construct
+the AF comparison for the latter. Section~\ref{sec:completion}
+assembles the inequalities and checks sharpness.

preprints/The-Penrose-inequality-for-maximal-asymptotically-hyperbolic-initial-data-October-5-2026/build/sections/02-preliminaries.tex

@@ -0,0 +1,260 @@
+\section{Preliminaries}
+\label{sec:preliminaries}
+
+\subsection{Compact cores and auxiliary caps}
+
+\begin{lemma}\label{lem:compact-core}
+The sufficiently large coordinate spheres bound connected compact
+truncations of $\Omega$. Each component of $S$ can be capped on its
+other side to produce a smooth complete connected one-ended manifold
+$X$ without boundary, with the orientation extending that of $\Omega$.
+Let $O$ denote the compact cap set, including its
+boundary. For competitors containing $O$, perimeter depends only on
+the metric on $\Omega$ and its boundary trace.
+\end{lemma}
+\begin{proof}
+The end metric is uniformly comparable to $b$ sufficiently far out.
+Radial infinity escapes compact subsets: otherwise a sequence tending
+to radial infinity would accumulate in a compact coordinate
+neighborhood, while pairwise disjoint background balls of a fixed small
+radius about a subsequence would have uniformly positive metric
+volumes and lie in a fixed compact neighborhood. This is impossible.
+A large coordinate sphere separates the product end from an inner
+side. If the latter were noncompact, a smooth compact exhaustion would
+yield an end separated from the given product end. One may choose this
+exhaustion with finitely many complement components at each stage,
+since its smooth compact frontier has finitely many components.
+Nested extraction then contradicts the one-end assumption. The inner
+side is therefore compact. Paths between its points can be pushed
+back across the product end, so the truncations can be taken connected.
+
+Every component of $S$ is a closed orientable surface and bounds a
+handlebody. Orient each handlebody and glue by a map reversing the
+induced boundary orientations.  The collars then extend the orientation
+of $\Omega$ across each seam. Extend the metric smoothly across $S$.
+The added region is compact; completeness
+at infinity is unchanged. For a set containing $O$ up to a null set,
+its characteristic function is constant almost everywhere in the
+cap interiors. Its perimeter therefore uses no metric values there.
+\end{proof}
+
+We use the ordinary perimeter of a finite-perimeter set in the capped
+manifold. In particular it is not a relative perimeter that discards
+the original boundary.
+
+\begin{lemma}\label{lem:cut-enclosure}
+Every admissible cut from Definition~\ref{def:minimum-area} determines
+a precompact finite-perimeter enclosure of $O$ whose perimeter is at
+most the area of that cut. Conversely, a precompact enclosure of $O$
+with smooth free boundary in $\operatorname{int}\Omega$ and connected
+exterior gives an admissible cut with the same area. If this enclosure
+is outward minimizing, its boundary realizes the minimum enclosing
+area of its own closed exterior.
+\end{lemma}
+\begin{proof}
+For an admissible $D$, complement its manifold interior in $X$.
+The result contains $O$ and is precompact because $D$ contains the
+whole sufficiently distant end. Local boundary coordinates show that
+its perimeter is bounded by the entire intrinsic area of $\partial D$,
+including any portions on $S$. This also follows by smoothing corners
+and passing to the perimeter lower limit. For the converse, take the
+closed exterior of the smooth enclosure. It is connected, has the
+required end, and has precisely the stated intrinsic boundary.
+Any admissible cut of this new exterior, complemented in $X$, encloses
+the original enclosure; outward minimization gives the final claim.
+\end{proof}
+
+The compactness, lower semicontinuity, coarea, slicing, reduced-boundary
+structure, and union/intersection inequalities for finite-perimeter
+sets will be used below. These are local facts and hold in smooth
+Riemannian coordinate charts; see \citet[Chapters 12--18]{Maggi2012}.
+They also give compactness and lower semicontinuity for a uniformly
+convergent family of uniformly positive continuous metrics on a fixed
+compact set, by comparison with smooth metrics. A free locally
+perimeter-minimizing boundary is smooth in ambient dimension three;
+one may use the smooth-ambient regularity theorem in
+\citet[Chapter 7, Section 5, Theorem 5.8]{Simon2014}, followed by
+elliptic bootstrapping. We invoke this regularity only after excluding
+contact with every obstacle and outer truncation.
+
+\subsection{Mass covariance and small metric changes}
+
+\begin{proposition}\label{prop:mass-covariance}
+Suppose $g-b\in C^{2,\alpha}_\tau$, $\tau>3/2$, and
+$\int V_0|R_g+6|\dd V_g<\infty$ on the end. The fluxes
+\eqref{eq:ah-mass} transform as a Lorentz covector under hyperbolic
+isometries of the asymptotic chart. Such chart changes preserve the
+weighted decay and integrability hypotheses used here. Consequently
+a future-timelike original covector admits a chart with
+$p_0=m_{\AH}$ and $p_1=p_2=p_3=0$.
+\end{proposition}
+\begin{proof}
+The potentials in \eqref{eq:background} are the ambient coordinate
+functions of the future unit hyperboloid. Hyperbolic isometries act on
+their span by Lorentz transformations. The flux integrand is tensorial
+in $b$, linear in $V$, and natural under these isometries.
+It remains to justify replacing the transformed integration surfaces
+by the original large coordinate spheres. The static identity
+$\nabla_b^2V=Vb$ gives
+\[
+ \divg_b\mathbb U(V,e)=V\,D R_b(e).
+\]
+The difference between $D R_b(e)$ and $R_g+6$ is bounded by a constant
+times
+\[
+ |e|_b^2+|\nabla_be|_b^2+|e|_b|\nabla_b^2e|_b.
+\]
+Its $V_0$-weighted integral over $r>D$ is
+$O(D^{3-2\tau})$, since $\dd V_b$ is comparable to
+$r\dd r\dd A_\sigma$. Also $|V_i|\le V_0$.
+Thus the divergence is absolutely integrable on the end for each
+potential. The divergence theorem on regions between the two
+families of spheres makes their flux difference tend to zero.
+This is the covariance mechanism of
+\citet[Proposition 2.2, Theorem 2.3, and Section 3]{CH2003} in the
+normalization \eqref{eq:ah-mass}.
+
+A fixed hyperbolic isometry changes distance from the chosen origin
+by a bounded amount. The exponential weights and $V_0$ are therefore
+comparable before and after the change, and uniform background balls
+are carried to uniform background balls. The rest-chart assertion is
+the elementary Lorentz normal form of a future-timelike covector.
+\end{proof}
+
+The boundary sign and the enclosing-area definition are intrinsic.
+We will apply Proposition~\ref{prop:mass-covariance} to the original
+data before constructing any comparison metric. No assertion that
+intermediate covectors remain timelike is required.
+
+If $(1-\eps)g\le\widetilde g\le(1+\eps)g$ as quadratic forms,
+then for every two-surface its area changes by factors between
+$1-\eps$ and $1+\eps$. Taking infima over the same admissible domains
+gives the same comparison for $A_{\min}$. This observation will be
+used when passing through the preliminary conformal approximation
+and the later corner smoothing.
+
+\subsection{The AF comparison theorem}
+
+For an AF metric $h$ in a Euclidean end chart $y$, our normalization is
+\begin{equation}\label{eq:adm-mass}
+ m_{\AF}(h)=\frac1{16\pi}\lim_{R\to\infty}
+ \int_{|y|=R}(\partial_jh_{ij}-\partial_ih_{jj})
+                  n^i\dd A_{\mathrm{Eucl}}.
+\end{equation}
+
+\begin{theorem}[AF Riemannian Penrose inequality]\label{thm:af-penrose}
+Let $(Y,h)$ be a complete connected one-ended three-dimensional AF
+manifold with smooth compact outer-minimizing minimal boundary.
+Assume $R_h\ge0$, $h-\delta=O_2(|y|^{-\beta})$ for some
+$\beta>1/2$, and $R_h=O(|y|^{-q})$ for some $q>3$.
+If the total boundary area is $A$, then
+\begin{equation}\label{eq:af-penrose}
+ m_{\AF}(h)\ge\sqrt{A/(16\pi)}.
+\end{equation}
+Disconnected boundary is allowed.
+\end{theorem}
+
+This is the dimension-three case of the published
+\citet[Theorem 1.4]{BrayLee2009}; its asymptotic definition is on
+page 82 and its boundary convention on page 83.
+Only the inequality is used. The smoothness, scalar sign, and
+outer-minimization required here will be established for the AF
+comparison exterior, rather than assumed for either original AH
+metric.
+
+\subsection{Local scalar elliptic estimates}
+
+We specify the standard local analytic inputs, separating them from
+the coupled transmission estimates proved in
+Section~\ref{sec:transmission-analysis}.
+
+\begin{proposition}[Scalar local estimates]\label{prop:elliptic-inputs}
+On fixed coordinate balls in dimension three the following estimates
+hold, with constants depending on the indicated ellipticity,
+coefficient, and geometry bounds.
+\begin{enumerate}[label=\textup{(\roman*)}]
+\item Bounded measurable uniformly elliptic divergence matrices admit
+local subsolution bounds, weak Harnack estimates for nonnegative
+supersolutions, and inhomogeneous Harnack and H\"older estimates for
+nonnegative solutions. Bounded divergence drifts multiplying the
+unknown, vector forcing in $L^q$, $q>3$, and scalar forcing in $L^p$,
+$p>3/2$, are permitted in the corresponding estimates.
+\item For Laplace equations with uniformly positive Lipschitz metric
+coefficients in interior or flattened smooth boundary charts,
+divergence forcing in $L^q$ gives local $W^{1,q}$ estimates.
+Scalar forcing in $L^2$ can be included for $3<q<6$.
+The same statements hold with local zero Dirichlet data or natural
+Neumann data, with the divergence forcing included in the conormal
+current. A boundary datum in $L^{p_1}$, $p_1>2$, permits a choice
+$3<q<3p_1/2$ sufficiently close to $3$.
+\item Smooth-coefficient scalar Dirichlet and conormal problems have
+the usual Schauder estimates, including divergence-form
+$C^{1,\gamma}$ estimates when the uniformly elliptic coefficients
+and vector forcing are $C^\gamma$, with boundary data of the
+corresponding regularity. These estimates are used only after their
+coefficient hypotheses have been obtained.
+\item For a bounded measurable symmetric uniformly elliptic matrix
+$a$ on a ball, its positive Dirichlet Green kernel is bounded above
+by $C|z-z'|^{-1}$, and below by $c|z-z'|^{-1}$ when the pole and
+the nearby evaluation point lie in a fixed smaller ball.
+\end{enumerate}
+\end{proposition}
+
+The local boundedness, Harnack, and H\"older estimates are the scalar
+divergence theory of \citet{Serrin1964} and
+\citet[Chapter 8]{GT2001}. The Laplace divergence estimates are the
+local Calder\'on--Zygmund estimates, with smooth-boundary
+localization; see \citet{Stein1970,GT2001}.
+For natural Neumann data, an explicit estimate is given by
+\citet[Appendix, proof of Lemma 6.1, (6.2), p.\ 19
+of arXiv v3]{ChoiKim2013} on bounded $C^1$ localizations.
+Our Lipschitz metric coefficients are VMO. A cutoff retains the
+full conormal current and the larger-patch $W^{1,2}$ norm.
+In dimension three the scalar and boundary forcing exponents are
+$3q/(q+3)$ and $2q/3$, respectively, so approximation permits
+$3<q<\min\{6,3p_1/2\}$ for the data in part~(ii).
+For the boundary exponent in part~(ii), the trace embedding of
+$W^{1,q'}$ into $L^{2q/(2q-3)}$ on a two-dimensional face allows
+pairing with $L^{p_1}$ whenever $q<3p_1/2$.
+Scalar $L^2$ forcing belongs locally to $W^{-1,q}$ when $q<6$.
+Part~(iv) is the local Green comparison of
+\citet[Theorem 7.1 and the discussion on pp.\ 66--67]{LSW1963}.
+
+Natural homogeneous flux permits even reflection in a flattened
+boundary chart, with the full current reflected. Smooth prescribed
+flux errors can be placed in a bounded smooth vector field.
+Fermi-coordinate metric matrices reflected at a smooth face are
+Lipschitz. Constant Dirichlet data for a scalar height permit odd
+reflection after subtraction of that constant. We spell out each
+reflection where it is needed; none of these scalar statements is a
+general nonlinear transmission theorem.
+
+For the electrostatic equation we also use an exact degenerate
+gradient regularity statement. Suppose
+\[
+ A(z,0)=0,\qquad
+ \lambda|\xi|I\le A_\xi(z,\xi)\le C|\xi|I,
+ \qquad
+ |A(z,\xi)-A(z',\xi)|\le C|z-z'|\,|\xi|^2,
+\]
+with symmetric $A_\xi$, continuous dependence, and smooth dependence
+on $\xi\ne0$. Then a bounded weak $W^{1,3}$ solution of
+$\partial_iA^i(z,\dd u)=0$ is locally $C^{1,\gamma}$ for some
+$\gamma>0$, with uniform estimates from the structural constants
+and a local energy bound. Indeed $A_\xi(z,0)=0$ extends continuously,
+and integration along a radial segment gives $|A(z,\xi)|\le C|\xi|^2$.
+The assumptions of \citet[Theorem 1.1, (1.5)--(1.8)]{AraujoZhang2020}
+therefore hold with $n=p=3$, zero forcing in $L^\infty$, and spatial
+H\"older exponent $\sigma=1/2$.  We use only an exponent
+$0<\gamma<\min\{\alpha_m,1/4\}$, where $\alpha_m>0$ is a
+constant-coefficient gradient exponent for the same structural constants.
+No optimal exponent or endpoint estimate is required.
+If these inequalities initially hold only on a known
+bounded gradient range, the constitutive law must first be extended
+with the same structure; that extension is constructed in
+Section~\ref{sec:stress-extension}.
+
+All global invertibility, weighted end estimates, seam regularity,
+continuation, and uniform large-parameter conclusions used later are
+proved below. The local estimates in this subsection supply only
+their explicitly stated scalar ingredients.

preprints/The-Penrose-inequality-for-maximal-asymptotically-hyperbolic-initial-data-October-5-2026/build/sections/03-graph-calculus.tex

@@ -0,0 +1,828 @@
+\section{Static graph identities and differential estimates}
+\label{sec:graph-calculus}
+
+This Section proves the differential identities used in both constructions.
+They are identities on a Riemannian base; no spacetime satisfying field
+equations is assumed.  In particular, the reference tensor in this Section
+may be either the given maximal tensor or the synthetic tensor constructed
+later.
+
+The curvature, current, and boundary identities give the scalar, mass,
+and boundary inequalities in both deformations. The maximum estimate
+controls finite slopes without differentiating the trace cutoff; the
+energy and Newton-current estimates supply the seam-continuity argument
+in Section~\ref{sec:transmission-analysis}.
+
+Let $(U,g)$ be a smooth three-dimensional Riemannian manifold.  Unless a
+metric is displayed, covariant derivatives, norms, contractions, volume
+forms, and divergences in this Section use $g$.  For a symmetric covariant
+tensor $C$, the notation $C p$ means the vector obtained by contracting with
+$p$ and raising the remaining index with $g$.  For a positive smooth
+function $N$ and a smooth function $f$, define $s>0$ and $p$ by
+\begin{equation}
+ p=s^2\nabla f,\qquad s^2+|\dd f|^2s^4=N^2,
+ \qquad v=\frac pN,\qquad W=\frac Ns,
+ \qquad \bar g=g+s^2\dd f\otimes\dd f.
+ \label{eq:calculus-constitutive}
+\end{equation}
+The positive root is unique, and it depends smoothly on $(N,\dd f)$,
+including at $\dd f=0$, because
+\begin{equation}
+ s^2=\frac{2N^2}{1+\sqrt{1+4N^2|\dd f|^2}}.
+ \label{eq:calculus-positive-root}
+\end{equation}
+We use $d=s/N=\sqrt{1-|v|^2}$ for a scalar in $(0,1]$; differentials are
+written $\dd$.  Direct rank-one inversion and the determinant formula give
+\begin{equation}
+ \bar g^{-1}=g^{-1}-v\otimes v,\qquad
+ \dd V_{\bar g}=W\,\dd V_g.
+ \label{eq:calculus-volume-inverse}
+\end{equation}
+For a further positive function $w$ put
+\begin{equation}
+ x=sw,\qquad h=x\bar g,\qquad A=Ns\bar g^{-1},
+ \qquad \pi=N^{-1}\operatorname{sym}\nabla p^\flat,
+ \label{eq:calculus-conformal-definitions}
+\end{equation}
+where $\operatorname{sym}$ is the half-sum.  The tensor $\pi$ throughout
+this Section is the graph second fundamental form, not the electrostatic
+stress introduced in Section~\ref{sec:stress-extension}.  The elementary
+identities
+\begin{equation}
+ p\cdot\dd f=W^2-1,\qquad
+ A(\dd f,\dd f)=W-W^{-1}\le p\cdot\dd f,
+ \qquad 0<A\le N^2g^{-1}
+ \label{eq:calculus-energy-algebra}
+\end{equation}
+will also be used in the penalty estimates.
+
+\subsection{Curvature and the conserved current}
+
+\begin{lemma}[Static graph curvature]\label{lem:graph-curvature}
+Suppose that $\divg_g p=TN$, where $T$ is a smooth scalar function.
+Then $\tr_g\pi=T$ and
+\begin{align}
+ \bar\nabla\log s&=\nabla\log N-\pi(v),
+ \label{eq:calculus-lapse-differential}\\
+ R_{\bar g}
+ &=R_g+T^2-|\pi|^2-2(\divg_g\pi-\dd T)\cdot v.
+ \label{eq:calculus-static-curvature}
+\end{align}
+More generally, let $\mathcal K$ be any smooth symmetric covariant tensor,
+and define
+\begin{equation}
+ \kappa=\tr_g\mathcal K,\qquad
+ \ell=T-\kappa,\qquad \lambda=\pi-\mathcal K,
+ \qquad C=\lambda-\ell g,
+ \label{eq:calculus-trace-definitions}
+\end{equation}
+and
+\begin{equation}
+ \mathscr D(v)=R_g+\kappa^2-|\mathcal K|^2
+       -2(\divg_g\mathcal K-\dd\kappa)\cdot v.
+ \label{eq:calculus-dominant-expression}
+\end{equation}
+Then
+\begin{equation}
+ R_{\bar g}=\mathscr D(v)+|\lambda|^2-\ell^2
+                   -\frac2N\divg_g(Cp).
+ \label{eq:calculus-completed-curvature}
+\end{equation}
+\end{lemma}
+
+\begin{proof}
+The trace assertion follows immediately from the definition of $\pi$.
+To prove the first identity without any geometric interpretation, set
+$a=\dd\log s$.  Since $p^\flat=s^2\dd f$,
+\[
+ \nabla_i p_j=N\pi_{ij}+a_i p_j-a_j p_i.
+\]
+Differentiating $s^2=N^2-|p|^2$ and substituting this formula gives
+\[
+ N^2a_i-(a\cdot p)p_i=N\nabla_iN-N\pi_{ij}p^j.
+\]
+Division by $N^2$ and Equation~\eqref{eq:calculus-volume-inverse} give
+Equation~\eqref{eq:calculus-lapse-differential}, as an equality of vectors.
+
+For the curvature identity consider, solely for this computation, the
+Lorentzian metric
+\[
+ \widetilde g=-s^2\dd t^2+\bar g
+\]
+on $\R\times U$.  The graph $t=f$ has induced metric $g$.  If
+$X^\mathrm{tan}=X+\dd f(X)\partial_t$, its future unit normal is
+$n=(\partial_t+p^\mathrm{tan})/N$.  Thus
+$\partial_t=Nn-p^\mathrm{tan}$.  The tangential part of the Killing
+equation for $\partial_t$ is
+$N\widetilde g(\widetilde\nabla_{X^\mathrm{tan}}n,Y^\mathrm{tan})
+=\operatorname{sym}\nabla p^\flat(X,Y)$.
+Its second fundamental form, with the sign convention specified after
+\eqref{eq:future-trapping},
+is therefore $\pi$.
+
+Write $\widetilde G$ for the Einstein tensor and
+$n_0=s^{-1}\partial_t$ for the normal to a static slice.  The contracted
+Gauss and Codazzi identities with timelike unit normal and second form
+$\widetilde g(\widetilde\nabla n,\cdot)$ read
+\[
+ 2\widetilde G(n,n)=R_g+T^2-|\pi|^2,
+ \qquad
+ \widetilde G(n,X^\mathrm{tan})
+       =(\divg_g\pi-\dd T)(X).
+\]
+The static slices have zero second form.  Applying these same contractions
+to a static slice gives
+$\widetilde G(n_0,n_0)=R_{\bar g}/2$ and
+$\widetilde G(n_0,X)=0$ for spatial $X$.  The static time component of
+$n$ is $N/s$, whereas that of $\partial_t/N$ is $s/N$.  Consequently
+\[
+ \frac12R_{\bar g}
+ =\widetilde G(n,\partial_t/N)
+ =\widetilde G(n,n)-\widetilde G(n,v^\mathrm{tan}),
+\]
+which proves Equation~\eqref{eq:calculus-static-curvature} with the
+claimed sign of the momentum term.
+
+For completeness, the completion of the square uses symmetry to give
+\[
+ \frac1N\divg_g(Cp)
+ =(\divg_g\lambda-\dd\ell)\cdot v
+                         +\lambda:\pi-\ell T.
+\]
+Also $2\ell T-\ell^2=T^2-\kappa^2$ and
+$|\lambda|^2-2\lambda:\pi=|\mathcal K|^2-|\pi|^2$.
+Substituting these two equalities in the right side of
+Equation~\eqref{eq:calculus-completed-curvature} recovers
+Equation~\eqref{eq:calculus-static-curvature} term by term.
+\end{proof}
+
+\begin{lemma}[Graph current and conformal curvature]\label{lem:graph-current}
+Under the hypotheses and notation of Lemma~\ref{lem:graph-curvature}, set
+\begin{equation}
+ B=\nabla N-(\mathcal K+\ell g)p,\qquad
+ \divg_g B=\Phi,\qquad -\divg_g(A\dd w)=\Psi,
+ \qquad Q=B+A\dd w.
+ \label{eq:calculus-general-system}
+\end{equation}
+Then
+\begin{align}
+ Q&=N\bar\nabla x+xCp-(x-1)B,\notag\\
+ \divg_g Q&=\Phi-\Psi,
+ \label{eq:calculus-general-current}\\
+ N\Delta_{\bar g}x+x\divg_g(Cp)&=x\Phi-\Psi,
+ \label{eq:calculus-general-x}\\
+ xR_h&=\mathscr D(v)+|\lambda|^2-\ell^2
+       -\frac{2\Phi}{N}+\frac{2\Psi}{Nx}
+       +\frac32|\dd\log x|_{\bar g}^2.
+ \label{eq:calculus-general-conformal}
+\end{align}
+In particular, the following two specializations hold.
+
+\smallskip\noindent
+\emph{The maximal AH system.}  If $\tr_gK=0$ and
+\begin{equation}
+ \divg_g p=0,\quad B=\nabla N-Kp,\quad
+ \divg_g B=3N+\Sigma,\quad
+ -\divg_g(A\dd w)=3N+\Sigma x+P,
+ \label{eq:calculus-ah-system}
+\end{equation}
+then, writing
+$D=R_g+6-|K|^2-2\divg_gK\cdot v$, one has
+\begin{align}
+ Q&=N\bar\nabla x+x(\pi-K)p-(x-1)B,\notag\\
+ \divg_g Q&=-(x-1)\Sigma-P,
+ \label{eq:calculus-ah-current}\\
+ \left(\Delta_{\bar g}-3+
+          N^{-1}\divg_g((\pi-K)p)\right)x&=-3-P/N,
+ \label{eq:calculus-ah-x}\\
+ x^2(R_h+6)
+ &=x\bigl(D+|\pi-K|^2\bigr)
+   +\frac{3}{2x}|\dd x|_{\bar g}^2
+   +6(x-1)^2+\frac{2P}{N}.
+ \label{eq:calculus-ah-curvature}
+\end{align}
+For the constraint densities in \eqref{eq:constraints},
+$D=16\pi(\mu-J\cdot v)\ge0$ under the dominant energy condition.
+
+\smallskip\noindent
+\emph{The compact trace system.}  Let $0\le u_*\le1$ be constant and
+take $\mathcal K=u_*K$, $\kappa=u_*\tr_gK$, $\ell=T-\kappa$.
+If
+\begin{equation}
+ \begin{aligned}
+ \divg_g p&=TN,&
+ B&=\nabla N-(u_*K+\ell g)p,\\
+ \divg_g B&=\Sigma_++\Sigma_-,&
+ -\divg_g(A\dd w)&=\Sigma_+x+P,
+ \end{aligned}
+ \label{eq:calculus-compact-system}
+\end{equation}
+then
+\begin{align}
+ Q&=N\bar\nabla x+x(\lambda-\ell g)p-(x-1)B,\notag\\
+ \divg_gQ&=-(x-1)\Sigma_+-P+\Sigma_-,
+ \label{eq:calculus-compact-current}\\
+ xR_h&=\mathscr D(v)+|\lambda|^2-\ell^2
+             -\frac{2\Sigma_-}{N}+\frac{2P}{Nx}
+             +\frac32|\dd\log x|_{\bar g}^2.
+ \label{eq:calculus-compact-curvature}
+\end{align}
+Here $\lambda=\pi-u_*K$.  At $u_*=1$ this is the scalar identity for
+the synthetic data.  It also holds on the outer region with
+$p=T=K=\ell=0$, $\Sigma_-=0$, and $\mathscr D=R_{g_o}$.
+\end{lemma}
+
+\begin{proof}
+Since $w=x/s$, Equation~\eqref{eq:calculus-lapse-differential} gives
+\[
+ A\dd w=N\bar\nabla x-Nx\bar\nabla\log s
+        =N\bar\nabla x-x\nabla N+x\pi p.
+\]
+Adding $B$ proves the first formula for $Q$.  Its divergence from the
+defining equations is $\Phi-\Psi$.  To compute the divergence of its
+other expression, Equation~\eqref{eq:calculus-volume-inverse} gives
+\begin{align*}
+ \divg_g(N\bar\nabla x)
+ &=N\Delta_{\bar g}x
+       +N\bar g^{-1}(\dd\log s,\dd x)\\
+ &=N\Delta_{\bar g}x+(\nabla N-\pi p)\cdot\dd x.
+\end{align*}
+The derivative-of-$x$ terms in $\divg_g(xCp+(1-x)B)$ are
+$(Cp-B)\cdot\dd x=(\pi p-\nabla N)\cdot\dd x$.
+They cancel exactly, leaving
+\[
+ \divg_gQ=N\Delta_{\bar g}x+x\divg_g(Cp)+(1-x)\Phi.
+\]
+This proves Equation~\eqref{eq:calculus-general-x}.  In dimension three
+the conformal change $h=x\bar g$ has scalar curvature
+\[
+ xR_h=R_{\bar g}-2x^{-1}\Delta_{\bar g}x
+                     +\frac32x^{-2}|\dd x|_{\bar g}^2.
+\]
+Substitution of Equations~\eqref{eq:calculus-completed-curvature} and
+\eqref{eq:calculus-general-x} cancels the two occurrences of
+$2N^{-1}\divg_g(Cp)$ and proves
+Equation~\eqref{eq:calculus-general-conformal}.
+The two stated systems now follow by substituting their values of
+$T,\mathcal K,\Phi,\Psi$.  In the AH case, the nondifferential
+cosmological terms after multiplication by $x$ are
+$6x^2-12x+6=6(x-1)^2$.
+Finally, $|v|<1$ and the constraint definitions give the asserted sign
+of $D$.
+\end{proof}
+
+For later use, the divergence theorem makes these current formulas into
+exact finite-domain flux budgets.  Let $S$ be the inner boundary, with
+normal pointing into the domain, let $S_R$ have the outward normal, and
+suppose $B_n=-b_S$ and $(A\dd w)_n=0$ on $S$.  The scalar $b_S$ is a
+boundary datum, unrelated to the hyperbolic reference metric $b$.
+In the compact construction assume also that the two currents have
+matching normal flux at their common face, measured toward the end.
+Then
+\begin{equation}
+ -\int_{S_R}Q_n\,\dd A=
+ \begin{cases}
+ \displaystyle\int_S b_S\,\dd A+
+    \int_{U_R}\bigl[(x-1)\Sigma+P\bigr]\,\dd V,
+       &\text{for \eqref{eq:calculus-ah-system}},\\[3pt]
+ \displaystyle\int_S b_S\,\dd A+
+    \int_{U_R}\bigl[(x-1)\Sigma_++P-\Sigma_-\bigr]\,\dd V,
+       &\text{for \eqref{eq:calculus-compact-system}}.
+ \end{cases}
+ \label{eq:calculus-flux-budget}
+\end{equation}
+Indeed, the outward normal at $S$ is $-n$, so its contribution to the
+outward flux is $\int_S b_S$.  The two face contributions cancel since
+the base induced area forms and the directed fluxes agree.  Conversion
+of the outer flux into the relevant mass difference requires the end
+estimates proved in the corresponding construction.
+
+\subsection{Boundary identities}
+
+\begin{lemma}[Constant-height boundary]\label{lem:graph-boundary}
+Let $\mathcal S$ be a smooth two-sided hypersurface on which $f$ is
+constant.  Choose a $g$-unit normal $n$, and write $H=\divg_{\mathcal S}n$,
+$\hat z=v\cdot n$, $y=\hat z^2$, and
+$k_T=\tr_{\mathcal S}\mathcal K$.  The same directed normal is used on
+each side if $\mathcal S$ is a gluing face.  Then
+\begin{align}
+ d^2\partial_n\log s&=B_n/N-k_T\hat z+yH,
+ \label{eq:calculus-boundary-lapse}\\
+ s\sqrt{x}\,H_h
+ &=N(H-k_T\hat z)+B_n+(A\dd w)_n/x.
+ \label{eq:calculus-boundary}
+\end{align}
+In particular, in the maximal AH system with $v=-z n$,
+$k=K(n,n)$, $B_n=-b_S$, and $(A\dd w)_n=0$,
+\begin{equation}
+ s\sqrt{x}\,H_h=N(H-zk)-b_S=NH+\partial_nN.
+ \label{eq:calculus-ah-boundary}
+\end{equation}
+In the compact system, if $H\le0$, $k_T<0$, $\hat z\le0$,
+$B_n=-b_*<0$, and $(A\dd w)_n=0$, then $H_h<0$.
+
+Suppose, at a compact gluing face, that $N_o=s_i$, that $w$ and the
+normal fluxes of $B,A\dd w$ agree, that $f$ is constant on the inner
+face, and that $p=0$ outside.  The induced $h$-metrics then agree.
+If the inner base mean curvature satisfies $H_i>|k_T|$ and the outer
+one is $H_o=\sqrt{H_i^2-k_T^2}$, then
+$H_{h,i}\ge H_{h,o}$, both measured toward the end.
+\end{lemma}
+
+\begin{proof}
+On $\mathcal S$, $p=N\hat z n$.  For tangent vectors $X,Y$,
+$\nabla^2f(X,Y)=f_n\operatorname{II}(X,Y)$; consequently
+$\pi(X,Y)=\hat z\operatorname{II}(X,Y)$ and
+$\pi(n,n)=T-\hat z H$.  Taking the normal component in
+Equation~\eqref{eq:calculus-lapse-differential} therefore yields
+\[
+ d^2\partial_n\log s
+ =\partial_n\log N-\hat z T+yH.
+\]
+Since $\partial_nN=B_n+N(\mathcal K(n,n)+\ell)\hat z$ and
+$\mathcal K(n,n)+\ell-T=-k_T$, this proves
+Equation~\eqref{eq:calculus-boundary-lapse}.
+
+In base normal coordinates, the tangential entries of $\bar g$ and
+their first normal derivatives agree with those of $g$ at the face:
+the extra products contain at least one tangential derivative of $f$.
+The mixed entries vanish along the entire face, and the normal length
+factor is $W$.  Hence $\bar n=dn$ and $H_{\bar g}=dH$.  The
+mean-curvature formula for $h=x\bar g$ gives
+\[
+ \sqrt{x}\,H_h=d(H+\partial_n\log x).
+\]
+Multiplication by $s=Nd$, followed by
+Equation~\eqref{eq:calculus-boundary-lapse}, gives
+\[
+ s\sqrt{x}\,H_h
+ =N(H-k_T\hat z)+B_n+Nd^2w_n/w.
+\]
+The final term equals $(A\dd w)_n/x$, proving
+Equation~\eqref{eq:calculus-boundary}.  In the maximal case
+$k_T=-k$, $\hat z=-z$, and $\partial_nN=-Nzk-b_S$;
+this proves Equation~\eqref{eq:calculus-ah-boundary}.
+The asserted strict sign in the compact case follows because
+$H-k_T\hat z\le0$ and $B_n<0$.
+
+At the gluing face the tangential graph metrics equal their base
+metrics, which agree, and $x_i=s_iw=N_ow=x_o$.
+The remaining inequality reduces by
+Equation~\eqref{eq:calculus-boundary} to
+\[
+ N_i(H_i-k_T\hat z)\ge s_i\sqrt{H_i^2-k_T^2}.
+\]
+Both sides are nonnegative, and after division by $N_i$ its squared
+difference is
+\[
+ (H_i-k_T\hat z)^2-(1-\hat z^2)(H_i^2-k_T^2)
+       =(k_T-H_i\hat z)^2\ge0.
+\]
+This proves the mean-curvature ordering.
+\end{proof}
+
+\subsection{A maximum estimate without differentiating the cutoff}
+
+\begin{lemma}[Gradient maximum estimate]\label{lem:gradient-maximum}
+Let $N>0,f,\mathcal K,T$ be smooth and satisfy
+$\divg_g p=TN$ and
+$\divg_g[\nabla N-(\mathcal K+\ell g)p]=\Phi$, with the definitions
+above.  Suppose on the region in question that
+\begin{equation}
+ |\Ric_g|+|\mathcal K|+
+       |\divg_g\mathcal K-\dd\tr_g\mathcal K|+|T|\le C_0,
+ \qquad \Phi\le C_0N+\Sigma,\qquad \Sigma\ge0.
+ \label{eq:calculus-maximum-hypotheses}
+\end{equation}
+At a point where $m=|\dd f|>0$, set
+\begin{equation}
+ \begin{aligned}
+ l&=\log m,\quad u=\log N,\quad e=\nabla f/m,\quad y=|v|^2,\\
+ q&=\frac{1-y}{1+y},\quad \beta=1+q,\quad
+ \mathcal P=g^{-1}+(q-1)e\otimes e,\quad G=-\log s.
+ \end{aligned}
+ \label{eq:calculus-gradient-notation}
+\end{equation}
+Let $j$ be twice continuously differentiable with $j''>0$, with either
+sign allowed for $j'$, and let $\eta$ be a twice continuously
+differentiable localization function.  At an interior local maximum of
+$G+j(f)+\eta$ with $y\ge1/2$,
+\begin{equation}
+ j''q m^2\le C\left[1+\frac\Sigma N
+       +(1-y)^2(j'm)^2+|\dd\eta|^2+|\nabla^2\eta|\right],
+ \label{eq:calculus-gradient-maximum}
+\end{equation}
+where $C$ depends only on $C_0$.  In particular, at that point,
+\begin{equation}
+ W^2\le\frac C{j''}\left[
+     N^2+N\Sigma+(j')^2
+       +N^2\bigl(|\dd\eta|^2+|\nabla^2\eta|\bigr)\right].
+ \label{eq:calculus-gradient-weighted}
+\end{equation}
+No positive lower bound for $N$, no derivative bound for $T$, and no
+second derivative bound for $N$ enter these two estimates.
+
+At a constant-height boundary, with the notation of
+Lemma~\ref{lem:graph-boundary}, the exact normal derivative is
+\begin{equation}
+ d^2\partial_n\bigl(G+j(f)+\eta\bigr)
+   =-B_n/N+k_T\hat z-yH+j'\hat z/N+d^2\partial_n\eta.
+ \label{eq:calculus-gradient-boundary}
+\end{equation}
+For the maximal AH inner boundary, if $H\le k$, $b_S\ge0$, and
+$v=-zn$, this implies
+\begin{equation}
+ G_n=-\frac{yH+u_n}{1-y}
+       =\frac{b_S/N+zk-z^2H}{1-z^2}
+       \ge\frac{zk}{1+z}.
+ \label{eq:calculus-ah-gradient-boundary}
+\end{equation}
+\end{lemma}
+
+\begin{proof}
+Use a $g$-orthonormal frame at the point with first vector $e$ and
+transverse indices $a,b\in\{2,3\}$.  Put
+$U_{ab}=\nabla^2_{ab}f/m$ and $C_T=T/z$, where $z=\sqrt y$.
+Differentiation of the constitutive law gives
+\begin{equation}
+ \begin{aligned}
+ Nm&=\frac z{1-y},\qquad
+ q'=\frac{\dd q}{\dd(l+u)}=-\frac{4yq}{(1+y)^2},\\
+ \mathcal P:\nabla^2f&=mC_T-\beta\nabla f\cdot\dd u,
+ \qquad \tr U=C_T-ql_1-\beta u_1.
+ \end{aligned}
+ \label{eq:calculus-height-nondivergence}
+\end{equation}
+Here and below $q'$ differentiates the scalar constitutive function,
+not a spatial coordinate.  The derivative of the height flux with
+respect to $\dd f$ is $s^2\mathcal P$; its derivative with respect to
+$N$ is $s^2(\beta/N)\nabla f$.  This also proves that the
+nondivergence equation is smooth and elliptic through zero gradient.
+
+Differentiating the height equation in direction $e$, commuting the
+third derivatives of a scalar, and differentiating its norm gives
+\begin{align}
+ \mathcal P:\nabla^2l={}&\Ric(e,e)-\beta u_{11}
+   -q'(l_1+u_1)^2-\beta\dd l\cdot\dd u
+   +|U|^2+(1-q)|\dd_\perp l|^2-ql_1^2 \notag\\
+ &+\frac{\partial_1T}{z}
+        +C_T\bigl[(1-q)l_1-qu_1\bigr].
+ \label{eq:calculus-log-gradient-equation}
+\end{align}
+Here is an explicit account of the norm and coefficient terms in this
+calculation.  In $m^{-1}\mathcal P:\nabla^2m$, the second derivatives
+of the norm contribute $q|\dd_\perp l|^2+|U|^2$.
+The contraction $m^{-1}(\nabla_e\mathcal P):\nabla^2f$ is
+$q'(l_1+u_1)l_1+2(q-1)|\dd_\perp l|^2$.
+Differentiating $-\beta m u_1$ contributes
+$-\beta u_{11}-\beta\dd l\cdot\dd u
+-q'(l_1+u_1)u_1$.
+Passing from $m$ to $l$ subtracts
+$\mathcal P(\dd l,\dd l)=ql_1^2+|\dd_\perp l|^2$.
+Finally,
+$m^{-1}\partial_1(mT/z)=T_1/z+C_T[(1-q)l_1-qu_1]$.
+These are precisely all terms of
+Equation~\eqref{eq:calculus-log-gradient-equation}.
+
+Set $I=(1+y)^{-1}$ and $J=y/(1+y)$.  Then
+$G_l=J$, $G_u=-I$, and all four second partial derivatives of $G$
+as a function of $(l,u)$ equal $-q'/2$.
+Since $-I(q-1)=J\beta$, the Hessian terms in $u$ in
+$\mathcal P:\nabla^2G$ reduce exactly to $-I\Delta u$.
+The lapse equation, expanded using the symmetry of
+$\mathcal K+\ell g$, is
+\begin{equation}
+ \Delta u=\Phi/N+
+     [\divg_g\mathcal K-\dd\kappa+\dd T]\cdot v
+     +(\mathcal K+\ell g):\pi-|\dd u|^2.
+ \label{eq:calculus-log-lapse}
+\end{equation}
+The coefficient of $T_1$ is $J/z-Iz=0$.  Thus no derivative of $T$
+remains.  More precisely, with $a_y=-q'/2$, the resulting identity is
+\begin{align}
+ \mathcal P:\nabla^2G={}&J\Ric(e,e)-I\Phi/N
+   -I(\divg_g\mathcal K-\dd\kappa)\cdot v
+   -I(\mathcal K+\ell g):\pi \notag\\
+ &+\mathcal Q+JC_T[(1-q)l_1-qu_1],
+ \label{eq:calculus-master-gradient}\\
+ \mathcal Q={}&J|U|^2+a_y|\dd l+\dd u|^2
+  -J\beta\dd l\cdot\dd u
+  +J(1-q)|\dd_\perp l|^2-Jql_1^2+I|\dd u|^2.
+ \label{eq:calculus-gradient-quadratic}
+\end{align}
+In obtaining the coefficient $a_y$ of the normal square, one uses
+$2J+q=1$: the two contributions are $-Jq'$ and $-qq'/2$.
+
+At the specified maximum put $D_0=j'm$.  The vanishing first derivative
+is equivalent to
+\begin{equation}
+ J(l_i+u_i)=u_i-D_0\delta_{i1}-\eta_i.
+ \label{eq:calculus-maximum-substitution}
+\end{equation}
+First take $\dd\eta=0$ and separate
+$|U|^2=|U_{\rm tf}|^2+(\tr U)^2/2$.
+The portion of
+$\mathcal Q+j'\mathcal P:\nabla^2f$ independent of $C_T$ is exactly
+\begin{align}
+ J|U_{\rm tf}|^2
+ &+\frac{(3-2y)u_1^2-6(1-y)D_0u_1
+                         +3(1-y)^2D_0^2}{2y(1+y)} \notag\\
+ &+\frac{2-y}{y(1+y)}|\dd_\perp u|^2.
+ \label{eq:calculus-maximum-positive-form}
+\end{align}
+This follows by inserting
+$l_1=(Iu_1-D_0)/J$, $l_a=Iu_a/J$, and
+$\tr U=C_T+[-u_1+(1-y)D_0]/y$ in
+Equation~\eqref{eq:calculus-gradient-quadratic}.
+The terms containing $C_T$, including the last term of
+Equation~\eqref{eq:calculus-master-gradient}, are
+\begin{align}
+ \frac J2 C_T^2+
+ C_T[J(1-2q)l_1-J(\beta+q)u_1+D_0]
+   =\frac J2C_T^2+qC_T(2D_0-u_1).
+ \label{eq:calculus-trace-cancellation}
+\end{align}
+This last simplification is the gain needed for a bounded trace target:
+$q|D_0|\le(1-y)|D_0|$, rather than an unweighted $|D_0|$.
+
+For $y\in[1/2,1)$ the positive coefficients in
+Equation~\eqref{eq:calculus-maximum-positive-form} are bounded below.
+Young's inequality therefore bounds that expression below by
+$c(|U_{\rm tf}|^2+|\dd u|^2)-C(1-y)^2D_0^2$.
+The trace terms in Equation~\eqref{eq:calculus-trace-cancellation}
+have the same lower bound after reducing $c$ and adding a constant,
+because $|C_T|\le\sqrt2 C_0$.
+For general $\dd\eta$, use $D_0+\eta_1$ in the normal substitutions.
+The unchanged term $-\beta D_0u_1$ adds $\beta\eta_1u_1$ relative to
+that substitution.  The transverse substitutions add terms bounded by
+$C(|\dd\eta|^2+|\dd\eta||\dd u|)$, and the additional trace term is
+$-(1-2q)C_T\eta_1$.  Another application of Young's inequality thus
+adds only $C|\dd\eta|^2$ to the lower bound.
+
+The components of the graph second form are
+\begin{equation}
+ \pi_{11}=T-z\tr U,\qquad \pi_{ab}=zU_{ab},\qquad
+ \pi_{1a}=\frac{z\beta}{2}(u_a+l_a).
+ \label{eq:calculus-second-form-components}
+\end{equation}
+Equation~\eqref{eq:calculus-maximum-substitution} also gives
+\[
+ |\tr U|\le C\bigl(1+|\dd u|+(1-y)|D_0|+|\dd\eta|\bigr).
+\]
+Thus the term $(\mathcal K+\ell g):\pi$ is absorbed in the same
+positive quadratic expression, with an error
+$C[1+(1-y)^2D_0^2+|\dd\eta|^2]$.
+The curvature and momentum terms are bounded, and
+$-I\Phi/N\ge-C-\Sigma/N$.
+At a local maximum the positive definite tensor $\mathcal P$ has
+nonpositive contraction with $\nabla^2(G+j(f)+\eta)$.
+Its remaining height term is $j''\mathcal P(\dd f,\dd f)=j''qm^2$,
+and $|\mathcal P:\nabla^2\eta|\le C|\nabla^2\eta|$.
+This proves Equation~\eqref{eq:calculus-gradient-maximum}.
+
+Multiply it by $N^2$ and use
+$N^2qm^2=J/(1-y)$, $J\ge1/3$, and
+$N^2(1-y)^2(j'm)^2=y(j')^2$.
+This proves Equation~\eqref{eq:calculus-gradient-weighted} without a
+lapse floor.  Lastly, negate
+Equation~\eqref{eq:calculus-boundary-lapse} and use
+$d^2f_n=\hat z/N$ to obtain
+Equation~\eqref{eq:calculus-gradient-boundary}.
+In the AH case, $\tr\pi=0$ also gives
+$d^2\partial_n\log s=u_n+yH$.
+Substitution of $u_n=-zk-b_S/N$ and $H\le k$ proves
+Equation~\eqref{eq:calculus-ah-gradient-boundary}.
+\end{proof}
+
+\subsection{Energy and Newton flux identities at finite slope}
+
+\begin{lemma}[Differential energy inequality]\label{lem:gradient-energy}
+Assume the two differential equations and the geometric coefficient
+bounds of Lemma~\ref{lem:gradient-maximum}.  On a region suppose, in
+addition, that $N,s$ have positive lower bounds and finite upper bounds,
+that $|\dd f|$ has a finite upper bound, and that $\Phi/N$ is bounded
+above.  Constants in this Lemma may depend on these finite bounds.
+With
+$E_i=|\nabla N|^2+|\nabla^2f|^2$ and
+$\rho=(1+y)/d$, one has
+\begin{equation}
+ \mathcal P:\nabla^2G\ge cE_i-C(1+|\dd G|^2).
+ \label{eq:calculus-energy-nondivergence}
+\end{equation}
+There is a finite $k_0>0$, depending only on the stated bounds, such
+that for $Y=s^{-k_0}$,
+\begin{equation}
+ \divg_g(\rho\mathcal P\dd Y)\ge cE_i-C.
+ \label{eq:calculus-energy-divergence}
+\end{equation}
+These inequalities hold also at $\dd f=0$, and their constants use no
+derivatives of $T$.
+
+On a region where $p=0$, $s=N$, and $\Delta N=\Phi$, the corresponding
+inequality is $\Delta Y\ge c|\nabla N|^2-C$.
+At a constant-height gluing face satisfying $N_o=s_i$ and matching
+flux of $B$, the directed inner flux $(\rho\mathcal P\dd Y)_n$ and
+outer flux $\partial_nY$ differ by a bounded amount.  The bound
+depends only on the finite field bounds and the fixed face geometry.
+\end{lemma}
+
+\begin{proof}
+At nonzero gradient write $\gamma=\dd G$.  The identity
+$\gamma=J\dd l-I\dd u$ gives
+$\dd u=y\dd l-(1+y)\gamma$.
+Inserting it into Equation~\eqref{eq:calculus-gradient-quadratic}
+gives the exact expression
+\begin{align}
+ \mathcal Q={}&J\left[|U|^2+(1-y)l_1^2
+                                  +(2-y)|\dd_\perp l|^2\right]\notag\\
+ &-\frac{2y(2-y)}{1+y}\dd l\cdot\gamma
+          +\frac{1+4y-y^2}{1+y}|\gamma|^2.
+ \label{eq:calculus-energy-positive-form}
+\end{align}
+On the bounded field range, $1-y$ has a positive lower bound and
+\[
+ \frac J{m^2}=\frac{N^2(1-y)^2}{1+y},\qquad
+ |\nabla^2f|^2=m^2(l_1^2+2|\dd_\perp l|^2+|U|^2).
+\]
+The first line of Equation~\eqref{eq:calculus-energy-positive-form}
+therefore controls $|\nabla^2f|^2$.  Its mixed term is at most
+$Cy|\dd l||\gamma|$ and is absorbed with an error $C|\gamma|^2$.
+Moreover,
+$|\dd u|^2\le C(y^2|\dd l|^2+|\gamma|^2)$, so after reducing the
+positive coefficient the same lower bound controls $E_i$.
+
+In Equation~\eqref{eq:calculus-master-gradient}, the trace term has
+$|JC_T|\le C z$.  Its absolute value is bounded by
+$C\sqrt{E_i}$, since $z$ is comparable to $m$ on this field range.
+Equation~\eqref{eq:calculus-second-form-components}, or direct
+differentiation of $p(N,\dd f)$, gives
+$|\pi|\le C\sqrt{E_i}$.
+The remaining terms in the master identity are bounded below by a
+constant minus $C\sqrt{E_i}$.  Young's inequality proves
+Equation~\eqref{eq:calculus-energy-nondivergence}.
+
+There is no singularity at zero slope.  The expansion
+$G=-\log N+\tfrac12N^2|\dd f|^2+O(|\dd f|^4)$ and the lapse equation
+give, directly at a point with $\dd f=0$,
+\[
+ \mathcal P:\nabla^2G
+ =N^2|\nabla^2f|^2+|\dd u|^2-\Phi/N
+                    -N(\mathcal K+\ell g):\nabla^2f.
+\]
+This proves the same inequality there.  The matrices
+$\mathcal P$ and $\rho\mathcal P$ are smooth functions of
+$(N,\dd f)$, despite their expressions in terms of $e$; for example
+$q-1=-2y/(1+y)$ cancels the denominator in $e\otimes e$.
+Their derivatives consequently satisfy
+$|\divg_g(\rho\mathcal P)|\le C\sqrt{E_i}$.
+
+For $Y=e^{k_0G}$ the product rule yields
+\begin{align*}
+ \divg_g(\rho\mathcal P\dd Y)
+ =k_0Y\bigl[\rho\mathcal P:\nabla^2G
+      +\divg_g(\rho\mathcal P)\cdot\dd G
+      +k_0\rho\mathcal P(\dd G,\dd G)\bigr].
+\end{align*}
+The mixed coefficient term is bounded by $C\sqrt{E_i}|\dd G|$.
+Absorb half the positive $E_i$ term with Young's inequality, and then
+choose $k_0$ so the final term absorbs the resulting multiple of
+$|\dd G|^2$.  The upper and lower bounds on $s$ bound $Y$ on this
+range and prove Equation~\eqref{eq:calculus-energy-divergence}.
+Outside, direct differentiation gives
+\[
+ \Delta N^{-k_0}=k_0N^{-k_0}
+          \left[(k_0+1)|\dd\log N|^2-\Phi/N\right],
+\]
+which proves the stated outer inequality.
+
+On the face, $\rho q=d$ and
+Equation~\eqref{eq:calculus-boundary-lapse} gives
+\begin{equation}
+ (\rho\mathcal P\dd Y)_n
+   =-\frac{k_0YB_n}{s}
+          +\frac{k_0Y}{d}(k_T\hat z-yH),
+ \qquad
+ (\partial_nY)_o=-\frac{k_0YB_n}{s}.
+ \label{eq:calculus-energy-seam-flux}
+\end{equation}
+The second term in the inner expression is bounded under the stated
+finite bounds; the first terms agree.  This proves the flux assertion.
+\end{proof}
+
+\begin{lemma}[Newton flux and freezing its coefficient]\label{lem:newton-flux}
+In the setting of Lemma~\ref{lem:gradient-energy}, let
+$\mathcal T_f=\nabla^2f-(\Delta f)g$.  The exact smooth vector identity
+is
+\begin{equation}
+ \rho\mathcal P\nabla G
+   =-\frac{\nabla N}{s}
+      +\mathcal T_f\left(\frac{s^3}{N}\nabla f\right)
+      +Ts\nabla f,
+ \qquad
+ \divg_g\mathcal T_f=\Ric_g(\nabla f,\cdot).
+ \label{eq:calculus-newton}
+\end{equation}
+The coefficient of $\mathcal T_f$ is smooth at $\dd f=0$.
+
+More specifically, in a fixed smooth face chart take compatible
+reference fields
+$\mathcal V_*=(n_*,a_*\dd t,s_*)$ for
+$\mathcal V=(N_i,\dd f,N_o)$, where $t$ is inner signed unit normal
+distance, $n_*>0$, and $s_*=s(n_*,|a_*|)$.
+Assume the reference ranges and the actual fields lie in a fixed
+bounded positive lapse and finite slope range.  Put
+$Y_*=s_*^{-k_0}$ and
+\[
+ c_*=(k_0Y_*/s_*) ,\qquad
+ \mathfrak b_*=k_0Y_*\frac{s_*^3}{n_*}a_*\nabla t.
+\]
+For the following assertion assume also that $\Phi$ is bounded
+on each side.  Modify the inner and outer energy fluxes by
+\begin{equation}
+ \mathcal J_i=\rho\mathcal P\dd Y+c_*B-\mathcal T_f\mathfrak b_* ,
+ \qquad \mathcal J_o=\nabla Y+c_*B.
+ \label{eq:calculus-frozen-flux}
+\end{equation}
+They have bounded directed flux difference across the face, and
+\begin{align}
+ |\mathcal J|&\le C\bigl(1+|\mathcal V-\mathcal V_*|\sqrt E\bigr),
+ \label{eq:calculus-frozen-flux-size}\\
+ \divg\mathcal J&\ge c'E-C-C\delta_{\mathcal C}
+ \label{eq:calculus-frozen-flux-divergence}
+\end{align}
+in the joined weak formulation, with $E=E_i$ inside and
+$E=|\nabla N_o|^2$ outside.  The second line is understood with the
+smooth metric densities in coordinates, and $\delta_{\mathcal C}$ is
+surface measure on the face.  The same conclusion holds on an interior
+chart with an arbitrary constant covector reference and no face term;
+its metric-dependent reference coefficients are taken as smooth
+functions of position.
+\end{lemma}
+
+\begin{proof}
+At nonzero gradient the vector multiplying $\mathcal T_f$ can be
+written as
+\[
+ \frac{y\nabla f}{d m^2}=
+       \frac{s^3}{N}\nabla f,
+ \qquad \frac{zT}{d}e=Ts\nabla f.
+\]
+For a transverse component, multiplication of the proposed identity
+by $d$ gives $-u_a+yl_a$ on both sides.
+For its first component, the two terms preceding $Ts\nabla f$ on the
+right side, multiplied by $d$, are
+\[
+ -u_1-y\tr U=q(yl_1-u_1)-zT,
+\]
+by Equation~\eqref{eq:calculus-height-nondivergence}.
+The last term adds $zT$, giving exactly the first component of the
+left side.  This proves the first identity, including at zero gradient
+by smoothness.  Commuting the covariant derivatives of $f$ gives
+$\nabla^j\nabla^2_{ij}f=\nabla_i\Delta f+\Ric_{ik}\nabla^kf$,
+which is the second identity.
+
+Multiply the first identity by $k_0Y$.  In
+Equation~\eqref{eq:calculus-frozen-flux}, the coefficients of
+$\nabla N$ and of $\mathcal T_f$ then vanish at the reference state:
+\begin{align*}
+ \mathcal J_i={}&
+   \left(c_*-\frac{k_0Y}{s}\right)\nabla N
+   +\mathcal T_f\left(k_0Y\frac{s^3}{N}\nabla f-\mathfrak b_*\right)\\
+ &+c_*(B-\nabla N)+k_0YTs\nabla f.
+\end{align*}
+The last line is bounded, since
+$B-\nabla N=-(\mathcal K+\ell g)p$.
+The two coefficient differences in the first line are Lipschitz in
+the bounded field variables, so they are bounded by
+$C|\mathcal V-\mathcal V_*|$.
+On the outer side,
+$\mathcal J_o=(c_*-k_0N_o^{-k_0-1})\nabla N_o$,
+whose coefficient also vanishes at $N_o=s_*$.
+This proves Equation~\eqref{eq:calculus-frozen-flux-size}.
+
+The reference vector $\mathfrak b_*$ and its covariant derivative are uniformly
+bounded in the fixed chart.  Equation~\eqref{eq:calculus-newton} gives
+\[
+ |\divg_g(\mathcal T_f\mathfrak b_*)|
+ \le C(1+|\nabla^2f|).
+\]
+Adding $c_*B$ changes the divergence by the bounded scalar $c_*\Phi$.
+Thus Equation~\eqref{eq:calculus-energy-divergence} and Young's
+inequality give $\divg\mathcal J\ge c'E-C$ on each open side.
+If reference coefficients vary smoothly with the metric in an interior
+chart, their derivatives add only terms bounded by $C(1+\sqrt E)$;
+these are absorbed in the same way.
+
+Across the face, the energy flux difference is bounded by
+Equation~\eqref{eq:calculus-energy-seam-flux}, and the added current has
+matching normal flux.  The vector $\mathfrak b_*$ is normal to the face.  Its
+Newton contraction there uses only
+\[
+ \mathcal T_f(n,n)=-\tr_{\mathcal C}\nabla^2f=-f_nH_i,
+\]
+which is bounded by the constant height trace and the slope bound.
+Consequently the modified normal flux difference is bounded.  Integration
+by parts on the two sides, whose face area densities agree, gives
+the surface error in
+Equation~\eqref{eq:calculus-frozen-flux-divergence}.
+\end{proof}

preprints/The-Penrose-inequality-for-maximal-asymptotically-hyperbolic-initial-data-October-5-2026/build/sections/04-area-transfer.tex

@@ -0,0 +1,605 @@
+\section{Finite-time area transfer}\label{sec:area-transfer}
+
+The two deformations used below can contract a metric on a set of small
+weighted volume.  The following argument shows that this contraction
+cannot substantially decrease the minimum enclosing area.  Its constants
+use a scalar-curvature lower bound and the topology of the original
+exterior.  No curvature bound for the changing metrics, and no limit of a
+hyperbolic Hawking mass, is needed.
+
+\subsection{Caps, intrinsic boundaries, and the initial surface}
+
+Attach a compact orientable handlebody to each component of the original
+boundary $S$.  Write $O$ for the union of these caps and $X$ for the
+resulting connected manifold without boundary.  This operation is
+topological: whenever a comparison metric is given, extend it smoothly
+across $S$ into the caps.  The end is unchanged, and the extension is
+complete.  The topology of $X$ and $O$ will remain fixed.  In particular,
+\begin{equation}\label{eq:flow-topology-constant}
+ b_2=\dim_{\mathbb F_2}H_2(X\setminus\operatorname{int}O;\mathbb F_2)<\infty.
+\end{equation}
+Finiteness follows by retracting the product end onto its inner sphere;
+what remains is a compact manifold with boundary.
+
+We will start from a largest perimeter-minimizing enclosure $E_0$ of
+$O$ alone.  In each application, the obstacle and end barriers establish
+that its boundary is smooth, free, minimal, and disjoint from $O$.
+Thus $O\subset E_0$ and $\partial O\subset\operatorname{int}E_0$.
+The open representative $E_0$ is precompact.  Each component meets $O$,
+since an extra component could be removed with a strict decrease of
+perimeter.  Bounded components of its complement can similarly be
+filled.  A largest minimizer is strictly outward minimizing: a larger
+set with equal perimeter would also be a minimizer, contrary to
+maximality.  This assertion concerns all precompact competitors, because
+the far-out barriers allow any such competitor to be trimmed into the
+fixed truncation used in constructing $E_0$.
+
+The cap construction respects the intrinsic-boundary convention in
+Section~\ref{sec:hypotheses}.  If $D$ is any admissible closed end-side
+domain in the original exterior, then
+$X\setminus\operatorname{int}_X D$ is a precompact finite-perimeter
+enclosure of $O$.  Its perimeter is bounded by the area of the
+\emph{entire} intrinsic boundary of $D$.  In particular, portions along
+$S$ are counted; for $D=\Omega$ the perimeter is $|S|$, not zero.
+Conversely, a smooth enclosure of $O$ whose boundary is disjoint from
+$O$ and whose exterior is connected gives an admissible cut by taking
+the closure of its exterior.  These observations also apply when the
+surfaces or the caps have several components.
+
+\begin{lemma}[Adjustment at the initial surface]\label{lem:flow-initial-adjustment}
+Let $\widehat h$ be a smooth complete metric on $X$, and suppose $E_0$
+has the properties just described.  If
+$\Scal_{\widehat h}\ge-C_R$ on $X\setminus E_0$, then an arbitrarily
+small, compactly supported smooth conformal change gives a metric
+$\widetilde h$ for which
+\begin{equation}\label{eq:flow-initial-adjustment}
+ |\partial E_0|_{\widetilde h}=|\partial E_0|_{\widehat h},\qquad
+ H_{\widetilde h}>0,\qquad
+ \int_{\partial E_0}H_{\widetilde h}^{2}\,\dd A_{\widetilde h}\le1,
+ \qquad \Scal_{\widetilde h}\ge-C_R-1
+ \quad\hbox{on }X\setminus E_0.
+\end{equation}
+The set $E_0$ remains strictly outward minimizing.  The size of the
+change may be chosen separately for every comparison metric.
+\end{lemma}
+
+\begin{proof}
+In a two-sided signed-distance collar of $\partial E_0$, let $q$ be
+positive on the exterior and choose
+$\chi=q\zeta(q)$, where $\zeta$ is a nonnegative smooth cutoff equal to
+one near zero.  Extend $\chi$ by zero beyond the collar.  Then
+$\chi=0$ and $\partial_\nu\chi=1$ at the initial surface, while
+$\chi\ge0$ outside $E_0$.  Set
+$\widetilde h=e^{2a\chi}\widehat h$, with $a>0$.  The conformal
+mean-curvature formula gives $H_{\widetilde h}=2a$ on $\partial E_0$.
+Consequently its area is unchanged and its squared-mean-curvature
+integral is $4a^2|\partial E_0|_{\widehat h}$.
+
+Every outward competitor has its boundary outside $E_0$, where the
+metric has only increased, so strict outward minimality is preserved.
+Finally,
+\begin{equation}\label{eq:flow-conformal-scalar}
+ \Scal_{\widetilde h}
+ =e^{-2a\chi}\bigl(\Scal_{\widehat h}
+       -4a\Delta_{\widehat h}\chi
+       -2a^2|\dd\chi|_{\widehat h}^2\bigr).
+\end{equation}
+Choose $a$ so that
+$4a\|\Delta\chi\|_\infty+2a^2\|\dd\chi\|_\infty^2\le1$ and
+$4a^2|\partial E_0|\le1$.  Further decreasing $a$ gives any prescribed
+$C^2$ or tensor-comparison tolerance.  There is no need for a uniform
+collar width or a uniform positive lower bound for the initial area.
+This modification will be used only for the flow calculation; mass
+inequalities are applied to the original comparison metric.
+\end{proof}
+
+\subsection{Weak flow and its topology}
+
+We specify the weak-flow input.  Theorem~3.1 of
+\citet{HI2001} applies on a smooth complete connected
+boundaryless manifold possessing a proper locally Lipschitz weak
+subsolution with precompact initial set.  It supplies a proper weak
+solution from every nonempty smooth precompact open set.  In dimension
+three, Regularity~1.3, Hull Property~1.4, and Growth Lemma~1.6 give
+compact $C^{1,\gamma}$ level boundaries for $0<\gamma\le1/2$,
+outward minimality, and
+exponential area growth from a minimizing hull.  Smooth Start
+Lemma~2.4 applies when that hull is strict and its smooth boundary has
+$H>0$.  We use the $H^2$ inequality from Step~1 of the proof of
+Growth Formula~5.7, Equation~(5.22): it holds for almost every pair of
+positive endpoints and precompact levels, without a connectedness or
+scalar-curvature-sign assumption.  This form suffices for a
+distributional inequality in time.  Our smooth initial adjustment
+and Smooth Start Lemma~2.4 supply its initial upper trace.  Thus no
+restart from an arbitrary nonsmooth positive-time level is needed.
+
+The end hypothesis in this existence theorem holds for all metrics
+used here.  On an $\AH$ end the coordinate-sphere mean curvature is
+$2+o(1)$ and $|\dd\log r|=1+o(1)$; on an $\AF$ end the corresponding
+quantities are $2/r+o(r^{-1})$ and $r^{-1}+o(r^{-1})$.
+Thus $u_\infty=c\log r$ with $0<c<2$ has
+\begin{equation}\label{eq:flow-end-subsolution}
+ \divg_{\widehat h}\left(
+    \frac{\nabla u_\infty}{|\nabla u_\infty|}\right)
+ \ge |\nabla u_\infty|
+\end{equation}
+beyond a sufficiently large sphere $r=R$.  Use
+$v=\max\{c\log(r/R),0\}-1$, extended by $-1$ throughout the inner
+region.  Its negative sublevel is a nonempty smooth precompact set,
+bounded by $r=R e^{1/c}$, and outside this set the displayed inequality
+holds smoothly.  Extending the radial unit normal field by zero
+inside $r=R$ gives a nonnegative distributional divergence
+contribution at that sphere.  The Gauss--Green inequality against
+outward variations proves the weak subsolution property as well.
+This subsolution is proper toward infinity.  The compact
+metric adjustment in Lemma~\ref{lem:flow-initial-adjustment} leaves it
+unchanged.
+
+For positive-time notation, extend the exterior level function $u$ by
+zero on the initial set and, for $t>0$, write
+\begin{equation}\label{eq:flow-levels}
+ E_t=E_0\cup\{u<t\},\qquad N_t=\partial E_t,\qquad
+ A(t)=|N_t|_{\widetilde h}=e^t A(0).
+\end{equation}
+Use the open perimeter representatives of these sets.  At almost every
+time their boundary is $C^{1,\gamma}$, its weak mean curvature is
+$H=|\nabla u|$, and its area measure agrees with the level measure in
+the coarea formula.  Indeed, differentiability and nonvanishing
+gradient hold at almost every point of almost every level in the
+coarea sense; those points belong to the reduced boundary of the
+sublevel set.  The equivalent statement follows directly from the BV
+coarea formula.  The variational property used below is local
+minimality of
+\begin{equation}\label{eq:flow-variational-functional}
+ F\longmapsto \Per_{\widetilde h}(F)
+       -\int_F|\nabla u|_{\widetilde h}\,\dd V_{\widetilde h}
+\end{equation}
+under compact changes away from the initial set; differences of this
+expression are taken in a compact region containing the change.
+
+\begin{lemma}[A component bound on fixed topology]\label{lem:hull-components}
+At almost every positive time, every component of $E_t$ meets $E_0$,
+the open exterior of $E_t$ is connected, and
+\begin{equation}\label{eq:flow-component-bound}
+ b_0(N_t)\le b_2,\qquad \chi(N_t)\le2b_2,
+\end{equation}
+where $b_2$ is the fixed number in
+\eqref{eq:flow-topology-constant}.
+\end{lemma}
+
+\begin{proof}
+Fix a time $t$ with the stated regularity and with
+$\Vol\{u=t\}=0$.  Excluding the latter exceptional times removes at
+most a countable set.  Suppose a component $B$ of $E_t$ did not meet
+$E_0$.  For almost every $0<s<t$, the set $E_s\cap B$ is compactly
+contained in $B$ and separated from $E_0$.  To check this, continuity
+gives $u\le s<t$ on its closure, and a neighborhood of
+$\overline E_0$ belongs to $E_t$.  A connected component of that
+neighborhood cannot meet $B$ without placing part of $E_0$ in $B$.
+The boundaries here are perimeter representatives; the same
+containment holds modulo null sets.
+
+Removing $E_s\cap B$ is consequently an allowed variation in
+\eqref{eq:flow-variational-functional}.  If
+$p(s)=\Per_{\widetilde h}(E_s;B)$, minimality and coarea give
+\begin{equation}\label{eq:flow-no-new-components}
+ p(s)\le\int_{E_s\cap B}|\nabla u|\,\dd V
+       =\int_0^s p(\ell)\,\dd\ell.
+\end{equation}
+There is no initial contribution because $B\cap E_0=\varnothing$.
+Gr\"onwall's inequality implies $p=0$ almost everywhere.  A
+finite-perimeter indicator with zero variation on the connected open
+set $B$ is constant there.  Compact containment rules out the value
+one, so $E_s\cap B$ is null for almost every $s<t$.  Letting such $s$
+increase to $t$ contradicts that $B$ is a nonempty open component of
+$E_t$.  This proves the first assertion.  It also shows that each such
+component contains an entire component of $E_0$, and hence meets $O$.
+
+Outward minimality excludes a bounded open component of the
+complement: filling it removes its nonempty boundary and strictly
+decreases perimeter.  Because $X$ has just one product end and $E_t$
+is precompact, only one component of the exterior can be unbounded.
+Its open exterior is therefore connected.
+
+Let $N_t^1,\ldots,N_t^m$ be the boundary components.  Choose a large
+compact truncation
+$Y\subset X\setminus\operatorname{int}O$ containing all of them.
+For each $j$, an arc can start on $\partial O$, travel in the adjacent
+component of $E_t$, cross $N_t^j$ once transversely, and then travel in
+the connected exterior to the outer boundary of $Y$, without meeting
+any other $N_t^k$.  To construct its inner part, join a point just
+inside $N_t^j$ to $O$ in the relevant connected open component and
+stop at the first cap encountered; thus the arc stays outside the cap
+interiors.  Small perturbations give the asserted transversality.
+
+These arcs define relative one-cycles in $(Y,\partial Y)$ whose
+mod-two intersection pairings with the closed surfaces $N_t^k$ are
+the separate coordinate vectors.  Hence the classes $[N_t^k]$ are
+linearly independent in $H_2(Y;\mathbb F_2)$.  Extending the
+truncation along its product end changes no homology, so $m\le b_2$.
+Each closed orientable surface has Euler characteristic at most two,
+which proves the second estimate.  The arcs and the intersection
+pairings also make explicit why disconnected caps and arbitrary
+compact interior topology cause no difficulty.
+\end{proof}
+
+\begin{lemma}[Curvature control up to a prescribed area]\label{lem:flow-curvature}
+Suppose the adjusted metric satisfies
+$\Scal_{\widetilde h}\ge-C_0$ outside $E_0$, and let $A_*>0$.
+At almost every time for which $A(t)\le A_*$,
+\begin{equation}\label{eq:flow-uniform-curvature}
+ J(t):=\int_{N_t}|\nabla u|^2\,\dd A_{\widetilde h}
+ \le C_H:=1+\frac23 C_0 A_*+16\pi b_2.
+\end{equation}
+In particular this bound is independent of $A(0)$, even if the
+initial areas tend to zero in a family.
+\end{lemma}
+
+\begin{proof}
+Let $\mathcal A_t$ denote the weak second fundamental form of $N_t$.
+For almost every pair $0<r<t$, the cited growth formula, after
+dropping its nonnegative tangential gradient term, says
+\begin{equation}\label{eq:flow-hi-growth}
+ J(r)\ge J(t)+\int_r^t\int_{N_s}
+ \bigl(2|\mathcal A_s|^2+2\Ric(\nu,\nu)-H^2\bigr)
+ \,\dd A\,\dd s.
+\end{equation}
+After our adjustment, Smooth Start Lemma~2.4 identifies the weak
+flow with the smooth flow on an initial interval.  Hence
+$J(r)\to J(0)\le1$ as $r\downarrow0$.  Letting admissible
+Lebesgue endpoints tend to zero supplies the initial upper trace in
+the integrated inequality below.  At positive times only the
+almost-everywhere inequality is used.
+
+For completeness, Gauss and Gauss--Bonnet apply to these levels with
+their weak curvature.  A local $C^{1,\gamma}$ graph with bounded weak
+mean curvature belongs to $W^{2,2}$ by difference quotients in its
+uniformly elliptic mean-curvature equation.  Smoothing graph
+representations gives smooth embeddings converging in
+$C^1\cap W^{2,2}$; the graph formula for the second fundamental form
+then gives convergence of the quadratic curvature terms in $L^1$.
+Thus the smooth Gauss identity and Gauss--Bonnet pass to the limit.
+This is also the content of Lemmas~5.4--5.5 of
+\citet{HI2001}.
+
+The Gauss equation and $|\mathcal A_s|^2\ge H^2/2$ give
+\begin{align}\label{eq:flow-gauss-bound}
+ \int_{N_s}\bigl(2|\mathcal A_s|^2+2\Ric(\nu,\nu)-H^2\bigr)
+ &=\int_{N_s}\bigl(|\mathcal A_s|^2+
+                   \Scal_{\widetilde h}-\Scal_{N_s}\bigr)\notag\\
+ &\ge\tfrac12J(s)-C_0A(s)-8\pi b_2.
+\end{align}
+Consequently, in distributions in time,
+$J'+J/2\le C_0A+8\pi b_2$, with initial upper value at most one.
+One can see this directly from \eqref{eq:flow-hi-growth} at Lebesgue
+endpoints: $J$ plus the integral of the lower bound in
+\eqref{eq:flow-gauss-bound} has a nonincreasing representative.  Any
+singular part of its derivative is nonpositive, so the integrating
+factor is valid.  It yields
+\begin{equation}\label{eq:flow-integrated-curvature}
+ J(t)\le e^{-t/2}
+       +\int_0^t e^{(s-t)/2}\bigl(C_0A(s)+8\pi b_2\bigr)\,\dd s.
+\end{equation}
+Using $A(s)=e^{s-t}A(t)$ bounds the two integrals by
+$\frac23C_0A(t)$ and $16\pi b_2$, respectively.
+\end{proof}
+
+\subsection{Projecting enclosing surfaces}
+
+In the next lemma, $A_*$ is the original minimum enclosing area,
+defined using complete intrinsic boundaries.  The comparison end may
+be the original $\AH$ end, or the $\AF$ extension of a fixed compact
+inner region.  In the latter case $g_{\mathrm{base}}$ denotes $g$ on
+the inner region and $g_o$ on the extension.  Statements involving
+this base metric are interpreted separately on the two sides of the
+joining sphere $\mathcal C$.
+
+\begin{lemma}[Enclosing-area projection]\label{lem:area-projection}
+Let $N$ be a compact embedded $C^1$ enclosure of $O$, disjoint from
+$S$, with connected exterior.
+\begin{enumerate}[label=\textup{(\roman*)}]
+\item On the original $\AH$ exterior, fix a large radius $r_p$ and put
+\begin{equation}\label{eq:flow-ah-weight}
+ \rho=\min\{1,r_p^2/r^2\},
+\end{equation}
+with $\rho=1$ on the compact part.  There is $q(r_p)\downarrow0$,
+depending only on the original end metric, such that
+\begin{equation}\label{eq:flow-ah-projection}
+ \int_N\rho\,\dd A_g\ge(1-q(r_p))A_*.
+\end{equation}
+\item Suppose the $\AF$ extension has the parallel-surface form
+$g_o=U^2\dd t^2+\gamma_t$, where $\gamma_t$ is induced on outward
+Euclidean parallels of a strictly convex embedding of
+$(\mathcal C,\gamma_0)$.  Then, with $\rho=1$,
+\begin{equation}\label{eq:flow-af-projection}
+ \int_N\dd A_{g_{\mathrm{base}}}\ge A_*.
+\end{equation}
+For a surface crossing the joining sphere, the assertion holds for
+the piecewise area; it is enough that its intersection with the
+joining sphere have zero two-dimensional measure.  In particular
+this holds for almost every weak-flow level.
+\end{enumerate}
+\end{lemma}
+
+\begin{proof}
+First take $N$ smooth and transverse to the spheres used in the
+construction.  Let $D$ be its closed end-side domain.  For part
+\textup{(i)}, choose a transverse sphere $r=r_1$ with
+$r_p/2\le r_1\le r_p$, and replace $D$ by
+\begin{equation}\label{eq:flow-projected-domain}
+ D'=D\cup\{r\ge r_1\}.
+\end{equation}
+This domain is connected because both sets contain the entire remote
+end.  Its boundary inside $r_1$ is part of $N$; the remaining boundary
+is a portion of the sphere $r=r_1$.  Every point of this newly exposed
+sphere portion is covered by radial projection of $N\cap\{r\ge r_1\}$:
+the ray starts outside $D$ and eventually lies in its end, so must
+cross $N$.
+
+In the hyperbolic reference metric the radial map from radius $r$ to
+$r_1$ has two-dimensional Jacobian at most $r_1^2/r^2$ on every
+two-plane.  The original metric has uniform relative error $o(1)$
+on $r\ge r_p/2$.  Its corresponding Jacobian is therefore at most
+\begin{equation}\label{eq:flow-projection-jacobian}
+ (1+o_{r_p}(1))\frac{r_1^2}{r^2}
+ \le(1+o_{r_p}(1))\rho.
+\end{equation}
+The area formula, allowing multiplicity, bounds the new sphere area
+by the weighted area of the discarded portion of $N$.
+
+The boundary of $D'$ has transverse corners along finitely many
+smooth curves.  In local coordinates given by defining functions for
+the two faces, round each quadrant edge in a shrinking tubular
+neighborhood, toward the side enlarging $D'$.  The added area tends
+to zero: the modification has vanishing width along a fixed compact
+edge, with bounded local slopes.  The rounded end-side domain stays
+connected and gives an admissible cut, disjoint from $S$.  Since the
+definition of $A_*$ uses its full intrinsic boundary, it follows that
+$A_*\le(1+o_{r_p}(1))\int_N\rho\,\dd A_g$.  This proves
+\eqref{eq:flow-ah-projection}.
+
+For part \textup{(ii)}, keep the part of $D$ in the original compact
+inner region and attach the whole original $\AH$ region outside
+$\mathcal C$.  The resulting end-side domain is connected.  Indeed,
+every point of its inner part can be joined inside the old $D$ to
+infinity, and the initial portion of such a path reaches
+$\mathcal C$.  Its newly exposed boundary in $\mathcal C$ is covered
+by projection of the old outer boundary along the parallel rays.
+This projection has two-dimensional Jacobian at most one:
+$\gamma_t\ge\gamma_0$ by strict convexity, its normal derivative is
+zero, and the positive coefficient $U^2$ creates no mixed terms.
+Rounding the resulting transverse corners as above gives admissible
+original cuts with area at most
+$\int_N\dd A_{g_{\mathrm{base}}}$ plus a vanishing error.
+
+Finally, a $C^1$ embedded enclosing surface can be approximated by
+smooth embeddings in $C^1$, within a collar disjoint from $S$, by an
+ambient isotopy.  This preserves the end side and its connectedness.
+Choose the approximations transverse to the cutting sphere.  Areas
+and weighted areas converge.  In the joined case, first approximate
+on the two smooth sides and make the approximations transverse to
+$\mathcal C$; round across a shrinking seam neighborhood.  The base
+metric is bounded and smooth on either side at this fixed
+construction, and the surface has zero area in the seam.  The area
+of its portion in that neighborhood tends to zero, so these local
+roundings contribute a vanishing area error.  This procedure also
+applies when the smooth collar coordinates chosen for the comparison
+metric differ from the original side coordinates: the
+identifications are smooth on each closed side and preserve the
+joining surface.  For almost every flow level the zero-area
+intersection follows by applying the level-measure decomposition to
+the fixed finite-area seam.  Passing to the limit proves both
+assertions.
+\end{proof}
+
+\subsection{The uniform transfer criterion}
+
+The following formulation isolates all requirements on the
+deformations.  A reference metric may be piecewise smooth across one
+fixed compact joining surface, as in
+Lemma~\ref{lem:area-projection}\textup{(ii)}.  Its area and volume
+measures below are then the sums of the two side measures.
+
+\begin{proposition}[Uniform area transfer]\label{prop:area-transfer}
+Fix the capped topology $(X,O)$, a number $A_*>0$, and a scalar lower
+bound $C_R\ge0$.  Suppose a comparison has the following properties.
+\begin{enumerate}[label=\textup{(\roman*)}]
+\item The smooth complete metric $\widehat h$ has one $\AH$ or $\AF$
+end with the mean-curvature and radial-gradient asymptotics used in
+\eqref{eq:flow-end-subsolution}.  The cap-only enclosure $E_0$ is a
+smooth precompact strictly outward-minimizing set, with minimal
+boundary disjoint from $O$, every component meeting $O$, and
+$\Scal_{\widehat h}\ge-C_R$ outside $E_0$.
+\item On $X\setminus O$ there are a reference metric
+$g_{\mathrm{base}}$, positive functions $x,W$, a metric $\bar g$, and
+a weight $0<\rho\le1$, such that
+\begin{equation}\label{eq:flow-metric-comparison}
+ \bar g\ge g_{\mathrm{base}},\qquad
+ \dd V_{\bar g}=W\,\dd V_{g_{\mathrm{base}}},\qquad
+ (1-\eta)x\bar g\le\widehat h\le(1+\eta)x\bar g.
+\end{equation}
+\item Every enclosing $C^1$ surface with connected exterior, as in
+Lemma~\ref{lem:area-projection}, satisfies
+\begin{equation}\label{eq:flow-abstract-projection}
+ \int_N\rho\,\dd A_{g_{\mathrm{base}}}\ge(1-q)A_*.
+\end{equation}
+In the piecewise case this is required for surfaces having zero area
+in the joining surface.
+\end{enumerate}
+For every $0<\eps<1$ there are positive tolerances
+$\eta_0,q_0,\delta_0$ and $v_0$, depending only on
+$\eps,A_*,C_R$ and the integer $b_2$ in
+\eqref{eq:flow-topology-constant}, with the following consequence.
+If $\eta\le\eta_0$, $q\le q_0$, $0<\delta\le\delta_0$, and
+\begin{equation}\label{eq:flow-bad-volume}
+ \int_{\{x<1-\delta\}}\rho^{3/2}W\,\dd V_{g_{\mathrm{base}}}<v_0,
+\end{equation}
+then
+\begin{equation}\label{eq:flow-area-conclusion}
+ |\partial E_0|_{\widehat h}\ge(1-\eps)A_*.
+\end{equation}
+The tolerances are uniform over all comparisons satisfying these
+conditions; the geometry and initial area of $E_0$ may vary.
+\end{proposition}
+
+\begin{proof}
+Make the change from Lemma~\ref{lem:flow-initial-adjustment}.  It
+preserves the initial area and all topological properties.  Take its
+size sufficiently small that the comparison
+\eqref{eq:flow-metric-comparison} for the resulting $\widetilde h$
+holds with an error $\bar\eta\le\eps/32$, by initially requiring,
+for example, $\eta\le\eta_0=\eps/64$.  Its scalar lower bound is
+$-C_0$, where $C_0=C_R+1$.  Also take
+$q_0=\delta_0=\eps/32$.
+
+Suppose, to the contrary, that $A(0)<(1-\eps)A_*$.  Exponential area
+growth gives an interval $I=[t_-,t_+]$ on which the endpoint areas
+are $(1-\eps)A_*$ and $(1-\eps/2)A_*$.  Its length is the fixed
+positive number
+\begin{equation}\label{eq:flow-time-window}
+ \Delta_\eps=\log\frac{1-\eps/2}{1-\eps}.
+\end{equation}
+Although $t_-$ may tend to infinity when $A(0)$ tends to zero,
+Lemma~\ref{lem:flow-curvature} bounds $J(t)$ throughout $I$ by the
+same constant $C_H$.
+
+Write $\mathcal B=\{x<1-\delta\}$.  On the good part of a level,
+\eqref{eq:flow-metric-comparison} and $\bar g\ge g_{\mathrm{base}}$
+give
+\begin{equation}\label{eq:flow-good-area}
+ \int_{N_t\setminus\mathcal B}\rho\,\dd A_{g_{\mathrm{base}}}
+ \le\frac{A(t)}{(1-\bar\eta)(1-\delta)}.
+\end{equation}
+Combine this with \eqref{eq:flow-abstract-projection}.  Since
+$(1-\bar\eta)(1-\delta)\ge1-\eps/16$, for almost every $t\in I$
+we obtain
+\begin{equation}\label{eq:flow-bad-area-lower}
+ \int_{N_t\cap\mathcal B}\rho\,\dd A_{g_{\mathrm{base}}}
+ \ge (1-q)A_*
+       -\frac{(1-\eps/2)A_*}{(1-\bar\eta)(1-\delta)}
+ \ge \frac{\eps A_*}{4}=:c_A.
+\end{equation}
+The same metric comparison on the bad part therefore implies
+\begin{equation}\label{eq:flow-weighted-area-lower}
+ \int_{N_t\cap\mathcal B}\rho x^{-1}\,\dd A_{\widetilde h}
+ \ge(1-\bar\eta)c_A\ge c_A/2.
+\end{equation}
+
+Here is the precise weighted coarea step.  Put $F=\rho x^{-1}$ and
+$H=|\nabla u|_{\widetilde h}$.  For almost every level $H>0$ at
+almost every point in its area measure.  H\"older with exponents
+$3/2$ and $3$ gives
+\begin{align}\label{eq:flow-holder-exponents}
+ \int_{N_t\cap\mathcal B}F\,\dd A
+ &=\int_{N_t\cap\mathcal B}
+       \left(\frac{F^{3/2}}H\right)^{2/3}(H^2)^{1/3}\,\dd A\notag\\
+ &\le
+ \left(\int_{N_t\cap\mathcal B}\frac{F^{3/2}}H\,\dd A\right)^{2/3}
+ J(t)^{1/3}.
+\end{align}
+Infinite integrals cause no difficulty; otherwise this identity can
+first be applied where $H$ is bounded away from zero and passed to
+the limit.  Equations~\eqref{eq:flow-uniform-curvature} and
+\eqref{eq:flow-weighted-area-lower} yield
+\begin{equation}\label{eq:flow-level-volume-lower}
+ \int_{N_t\cap\mathcal B}
+       \frac{\rho^{3/2}x^{-3/2}}{|\nabla u|}\,\dd A_{\widetilde h}
+ \ge \frac{(c_A/2)^{3/2}}{\sqrt{C_H}}=:c_V.
+\end{equation}
+Integrating over $I$ and applying Lipschitz coarea gives
+\begin{align}\label{eq:flow-coarea-contradiction}
+ \Delta_\eps c_V
+ &\le\int_{\mathcal B\cap\{t_-<u<t_+\}\cap\{|\nabla u|>0\}}
+                  \rho^{3/2}x^{-3/2}\,\dd V_{\widetilde h}\notag\\
+ &\le(1+\bar\eta)^{3/2}
+       \int_{\mathcal B}\rho^{3/2}W\,\dd V_{g_{\mathrm{base}}}.
+\end{align}
+The last inequality uses
+$\dd V_{\widetilde h}\le
+(1+\bar\eta)^{3/2}x^{3/2}W\,\dd V_{g_{\mathrm{base}}}$.
+Thus it is enough to take
+$v_0=\Delta_\eps c_V/2^{3/2}$.
+Notice that no contribution from a plateau of the level function
+has been assumed: coarea integrates only where its gradient is
+nonzero, and the full bad-set volume is an upper bound.  This proves
+the proposition.
+\end{proof}
+
+\subsection{The two applications and their parameter order}
+
+For the maximal $\AH$ reduction, the base is the original $g$ and
+\begin{equation}\label{eq:flow-graph-volume}
+ \bar g=g+s^2\dd f^2,\qquad h=x\bar g,\qquad
+ W^2=1+s^2|\dd f|_g^2,\qquad
+ \dd V_{\bar g}=W\,\dd V_g.
+\end{equation}
+The curvature bound is $\Scal_h\ge-6$, and the cap-only largest hull
+is the enclosure used in that reduction.  Fix the area tolerance
+$\eps$, then $r_p$ so that the error in
+\eqref{eq:flow-ah-projection} is at most $q_0$, and then
+$\delta\le\delta_0$.  The quantitative estimates of
+Section~\ref{sec:maximal-reduction} give, with
+$R_s=M^{3/10}$ and $L=M^2$,
+\begin{equation}\label{eq:flow-ah-bad-volume}
+ \int_{\{x<1-\delta\}}\rho^{3/2}W\,\dd V_g
+ \le C\left(L^{-1}+\sqrt{M/L}
+              +R_s^{-1}+\frac{\sqrt M}{R_s^2}\right)
+ \longrightarrow0.
+\end{equation}
+To explain the powers, the unweighted bad volume inside $R_s$ is
+$O(L^{-1})$, and $\int(W^2-1)\,\dd V_g\le CM$; Cauchy--Schwarz
+gives the first two terms.  On the tail the $\AH$ volume element is
+comparable to $r\,\dd r\,\dd A_\sigma$ and
+$\rho^{3/2}=r_p^3/r^3$.  Hence
+$\int_{r>R_s}\rho^{3/2}\,\dd V_g=O(R_s^{-1})$, whereas
+$\int_{r>R_s}\rho^3\,\dd V_g=O(R_s^{-4})$; another
+Cauchy--Schwarz inequality gives the last term.  The constants here
+may depend on the already fixed $r_p$ and original data.
+Proposition~\ref{prop:area-transfer} therefore yields the required
+lower bound for the comparison boundary area.
+
+For the compact construction with the $\AF$ attachment,
+$g_{\mathrm{base}}=g$ inside and $g_o$ outside,
+$\bar g=g+s^2\dd f^2$ inside and $\bar g=g_o$ outside.  Thus $W=1$
+outside, $\rho=1$, and Lemma~\ref{lem:area-projection} has zero
+projection error.  Section~\ref{sec:scalar-error} constructs a smooth
+metric $h'$ arbitrarily close to $h=x\bar g$, with a uniform scalar
+lower bound, together with two enclosures.  The enclosure $E$
+contains the caps and all unsafe balls and is used for the
+$\AF$ mass inequality.  The largest cap-only minimizer $E_0$ is
+used for the flow.  In particular
+\begin{equation}\label{eq:flow-af-two-enclosures}
+ |\partial E|_{h'}\ge|\partial E_0|_{h'}.
+\end{equation}
+No topological bound on the unsafe balls or on $E$ is required.
+
+The bad set is confined to a fixed outer radius by the capacitary
+lower bound for $x$.  On its designated penalized region,
+\begin{equation}\label{eq:flow-af-bad-volume}
+ \int_{\{x<1-\delta\}\cap\mathrm{designated}}
+                       W\,\dd V_{g_{\mathrm{base}}}\le C/L.
+\end{equation}
+The omitted fixed exterior collar has $W=1$, so its contribution is
+at most its base volume.  The omitted inner collar has base width
+$O(M^{-4})$.  Since the compact construction gives
+$\int_{\Omega_i}W^2\,\dd V_g\le C(L)M^2$, its contribution is at
+most
+\begin{equation}\label{eq:flow-inner-collar-cost}
+ O(M^{-2})\left(\int_{\Omega_i}W^2\,\dd V_g\right)^{1/2}
+ =O_L(M^{-1}).
+\end{equation}
+Choose the exterior collar sufficiently thin and then $L$
+sufficiently large, both before $M$.  Next choose $M$ large and the
+remaining finite penalty parameters as in
+Section~\ref{sec:compact-construction}.  Finally choose the individual
+corner-smoothing tolerance small enough for
+\eqref{eq:flow-metric-comparison}.  These choices make the bad volume
+smaller than $v_0$.  The flow-only initial adjustment does not change
+$|\partial E_0|_{h'}$.  Equations~\eqref{eq:flow-area-conclusion} and
+\eqref{eq:flow-af-two-enclosures} give
+\begin{equation}\label{eq:flow-af-area-conclusion}
+ |\partial E|_{h'}\ge|\partial E_0|_{h'}\ge(1-\eps)A_*.
+\end{equation}
+If $A_*=0$ there is no area-transfer assertion to prove; the
+nonnegative mass conclusion from the $\AF$ comparison supplies the
+needed zero-area bound.

preprints/The-Penrose-inequality-for-maximal-asymptotically-hyperbolic-initial-data-October-5-2026/build/sections/05-maximal-reduction.tex

@@ -0,0 +1,1305 @@
+\section{Reduction of maximal data to a minimal boundary}
+\label{sec:maximal-reduction}
+
+The result of this section is a reduction within the asymptotically
+hyperbolic class.  In particular, it requires no uniform statement about
+the metrics to which the time-symmetric inequality is applied.
+
+\begin{theorem}[Maximal reduction]\label{thm:ah-reduction}
+Let \(F:[0,\infty)\to[0,\infty)\) be continuous and nondecreasing.
+Suppose that every smooth, complete, connected, orientable, one-ended
+asymptotically hyperbolic three-dimensional exterior with nonempty
+compact minimal boundary, scalar curvature at least \(-6\),
+\(g-b\in C^{2,\gamma}_\tau\) for some \(0<\gamma<1\) and
+\(3/2<\tau<3\), the weighted scalar integrability of
+Section~\ref{sec:hypotheses}, and four finite mass fluxes satisfies
+\[
+ p_0\ge F(A_{\min}).
+\]
+The exponent \(\gamma\) may depend on the comparison metric.
+This hypothesis is to hold in every designated asymptotic chart,
+without a causal-character assumption on the mass covector.
+Then every maximal initial-data exterior of
+Section~\ref{sec:hypotheses} satisfies, in each designated chart,
+\[
+ p_0(g)\ge F\bigl(A_{\min}(S;g)\bigr).
+\]
+\end{theorem}
+
+Fix the original data and chart, and put \(A_*=A_{\min}(S;g)\).
+Extend the end radius \(r\) to a smooth positive function bounded away
+from zero on \(\Omega\).  Constants may depend on this extension and
+the fixed data.  Derivatives, volume measures, and fluxes in this
+section use \(g\), unless a different metric is indicated.
+The letter \(b_S\) below denotes a scalar inner flux datum; it is
+unrelated to the hyperbolic reference metric \(b\).
+
+\subsection{The equations and the order of choices}
+
+For \(N>0\) and \(f\), define
+\begin{equation}\label{eq:ah-fields}
+ \begin{gathered}
+ s^2+s^4|df|_g^2=N^2,\qquad p=s^2\nabla f,\qquad
+ v=p/N,\qquad W=N/s,\\
+ \bar g=g+s^2df^2,\qquad
+ \pi=N^{-1}\operatorname{sym}\nabla p^\flat,\qquad
+ x=sw,\qquad h=x\bar g,\qquad A=Ns\bar g^{-1}.
+ \end{gathered}
+\end{equation}
+The first relation uniquely defines \(s>0\), smoothly also at \(df=0\).
+The vector \(p\) in this display is not a component of the mass
+covector.  In particular
+\(\bar g^{-1}=g^{-1}-v\otimes v\) and
+\(\dd V_{\bar g}=W\,\dd V_g\).
+We solve
+\begin{equation}\label{eq:ah-system}
+ \divg_g p=0,\qquad B=\nabla N-Kp,\qquad
+ \divg_g B=3N+\Sigma,\qquad
+ -\divg_g(A\,dw)=3N+\Sigma x+P ,
+\end{equation}
+where \(\Sigma,P\ge0\).  At the inner boundary, whose normal
+\(\nu\) points into \(\Omega\), prescribe
+\begin{equation}\label{eq:ah-inner-data}
+ f=tM,\qquad B\cdot\nu=-t b_S,\qquad (A\,dw)\cdot\nu=0 .
+\end{equation}
+Here \(0\le t\le1\) is a continuation parameter.  On the outer
+sphere \(S_R=\{r=R\}\), with outward \(g\)-normal \(\nu_R\), prescribe
+\begin{equation}\label{eq:ah-outer-data}
+ f=0,\qquad B\cdot\nu_R=\partial_{\nu_R}V_0,\qquad
+ w=V_0^{-1}.
+\end{equation}
+We keep \(K\) fixed throughout the continuation; this preserves
+\(H_g(S)\le k\), where \(k=K(\nu,\nu)\).
+
+The identities of Lemmas~\ref{lem:graph-curvature},
+\ref{lem:graph-current}, and~\ref{lem:graph-boundary} apply with
+\(\tr_gK=\tr_g\pi=0\).  Writing \(Q=B+A\,dw\), they give
+\begin{align}
+ \divg_g Q&=-(x-1)\Sigma-P,\nonumber\\
+ Q&=N\bar\nabla x+x(\pi-K)p-(x-1)B,\label{eq:ah-current}\\
+ x^2(\Scal_h+6)
+ &=x\{D+|\pi-K|_g^2\}
+   +\frac{3}{2x}|dx|_{\bar g}^2+6(x-1)^2+\frac{2P}{N},
+ \label{eq:ah-scalar}\\
+ D&=16\pi(\mu-J\cdot v)\ge0,\nonumber\\
+ \left(\Delta_{\bar g}-3+
+  N^{-1}\divg_g((\pi-K)p)\right)x&=-3-P/N .
+ \label{eq:ah-x-equation}
+\end{align}
+The dominant energy condition and \(|v|<1\) give the displayed
+sign of \(D\).  At \(S\), the height maximum principle will give
+\(v=-z\nu\), \(0\le z<1\), and the boundary formula is
+\begin{equation}\label{eq:ah-boundary-curvature}
+ s\sqrt{x}\,H_h=N(H_g-zk)-t b_S .
+\end{equation}
+Consequently a suitable small inner flux will make the obstacle
+boundary strictly mean concave wherever an enclosing minimizer
+could touch it.
+
+Choose a fixed \(p_1>2\) and an area tolerance \(0<\delta<1/4\).
+For a large finite height \(M\), set
+\begin{equation}\label{eq:ah-parameter-scales}
+ R_s=M^{3/10},\qquad T=M^{4/5},\qquad L=M^2.
+\end{equation}
+The allowable smooth inner data form the convex class
+\begin{equation}\label{eq:ah-flux-class}
+ \mathcal B_M=
+ \left\{b_S\in C^\infty(S):
+ 0\le b_S\le B_*,\
+ \|b_S\|_{L^{p_1}(S,g)}\le M^{-1/(10p_1)}\right\}.
+\end{equation}
+The fixed constant \(B_*\) will be chosen using an upper lapse
+estimate which depends only on the norm constraint in
+\eqref{eq:ah-flux-class}, not on \(B_*\).
+
+Let \(\epsilon>0\) be chosen after \(M,L\).  The unscaled volume
+source is a product of smooth cutoffs, between \(0\) and \(L\),
+which vanishes when \(r\ge2R_s\), \(x\ge1-\delta/2\), or
+\(N\le\epsilon\), and equals \(L\) when
+\[
+ r\le R_s,\qquad x\le1-\delta,\qquad N\ge2\epsilon .
+\]
+The smooth cutoffs are defined also for negative arguments.
+Two further sources, each bounded by a finite strength \(d_*\),
+have the following support and full-strength regions:
+\begin{equation}\label{eq:ah-penalty-cutoffs}
+ \begin{array}{c|c|c}
+ &\text{vanishes if}&\text{equals }d_*\text{ if}\\ \hline
+ P_1& f\le M/2\ \text{or }w\ge4T&
+       f\ge3M/4,\quad w\le2T\\
+ P_2& N\ge4\epsilon\ \text{or }w\ge4T_2&
+       N\le3\epsilon,\quad w\le2T_2 .
+ \end{array}
+\end{equation}
+Here \(T_2=C_2(M,L)/\epsilon\), with \(C_2\) fixed below before
+\(\epsilon\) is chosen.  During the continuation, multiply both
+the volume source and \(P_1+P_2\) by \(t\); we continue to denote
+the scaled sources by \(\Sigma\), \(P_1\), \(P_2\),
+and \(P=P_1+P_2\).
+All strengths and heights remain finite when solving the equations.
+The limit \(R\to\infty\) is always taken with these internal
+parameters fixed.
+
+\subsection{Lapse estimates with only bounded drift}
+
+\begin{lemma}\label{lem:ah-lapse}
+For smooth solutions on sufficiently large truncations, uniformly
+over \(0\le t\le1\) and \(b_S\in\mathcal B_M\),
+\begin{equation}\label{eq:ah-lapse-bounds}
+ N\le Cr,\qquad N\ge c(M)\epsilon r,\qquad
+ \frac{|N-V_0|}{r}\le
+ C(R_s/r)^{1+\kappa}\quad(r\ge2R_s),
+\end{equation}
+where \(0<\kappa<\tau-1\) is fixed and small.  The upper constant
+uses only the \(L^{p_1}\) bound of \(b_S\).  On compact sets,
+\(N\) has uniform Hölder and \(W^{1,q}\) bounds for some \(q>3\),
+depending on \(M,L\), but independent of \(\epsilon,d_*\) and
+derivatives of \(b_S\).
+
+The positivity and lower bound remain true at a fixed point in
+which the drift \(Z=K(v)\) is computed using a smooth positive
+extension of the constitutive lapse argument, provided that the
+actual lapse equation is
+\(\divg_g(\nabla N-ZN)=3N+\Sigma\) and the cutoff
+\(N\le\epsilon\) uses the actual lapse.
+\end{lemma}
+
+\begin{proof}
+Only \(|Z|\le |K|\) is used.  First consider the density equation
+\begin{equation}\label{eq:ah-homogeneous-lapse}
+ \divg_g(\nabla H_1-ZH_1)=3H_1
+\end{equation}
+with the prescribed outer flux and inner outward flux \(t b_S\).
+Its adjoint is \(\Delta_g+Z\cdot\nabla-3\), with ordinary
+homogeneous Neumann boundary condition.  The maximum and boundary
+point principles give zero adjoint kernel.  Scalar elliptic
+Fredholm theory, deforming the lower terms to those of
+\(\Delta_g-3\), gives index zero and hence unique solvability of
+the primal problem.  Green's identity with the adjoint solution
+of a nonnegative forcing, which is nonpositive, proves
+\(H_1\ge0\): both prescribed outward fluxes are nonnegative, and
+the outer one is strictly positive.  Local Harnack then gives
+strict positivity.
+
+We give the weighted estimate used here, including its uniformity
+in the outer radius.  For \(-\Delta_g+3\), scalar and divergence
+forcing of size \(O(r^{-j})\), \(0<j<3\), with natural flux data
+in \(L^{p_1}(S)\) and of size \(O(R^{-j})\) on \(S_R\), yield
+an \(O(r^{-j})\) solution, with a constant independent of \(R\).
+The natural datum includes the flux of the divergence forcing.
+To see this, construct a positive strict supersolution comparable
+to \(r^{-j}\), constant on a fixed inner region.  Change its
+logarithmic slope slowly from \(0\) to \(-j\) in a long interval
+of \(\log r\) lying sufficiently far out.  The limiting radial
+indicial slopes are \(-3\) and \(1\), so this retains
+\((-\Delta_g+3)\varphi\ge c\varphi\).  Add a fixed sufficiently
+large multiple of
+\(R^{-j-a}r^a\), similarly continued as a constant inside, where
+\(0<a<1\), to make its outer normal derivative nonnegative.
+The resulting barrier stays comparable to \(r^{-j}\) on \(r\le R\).
+
+For divergence forcing the barrier argument is justified by
+normalization and compactness.  If the asserted bound fails,
+divide a violating solution by the maximum of its absolute
+value relative to the barrier, and normalize locally by the
+barrier's size at a maximizing point.  The normalized forcing
+and boundary data tend to zero.  Local Laplace \(W^{1,q}\)
+estimates are available with
+\(3<q<3p_1/2\): the boundary \(L^{p_1}\) datum belongs to the
+required negative trace space by the two-dimensional boundary
+Sobolev embedding.  These estimates give local uniform compactness.
+The outer sphere charts have uniformly controlled geometry.
+They may be formed by flowing
+\(\nabla\eta/|d\eta|^2\), \(\eta=\operatorname{arsinh}r\);
+the normal direction is orthogonal and the reflected principal
+metric is uniformly Lipschitz.  Thus no high derivatives of
+geodesic normal coordinates are needed.
+At points escaping to infinity, subsequences of the normalized
+geometries converge to hyperbolic geometry, the relative radius
+to an exponential horospherical coordinate, and a nearby outer
+boundary to a horosphere.  The strict supersolution and its
+Neumann sign persist in these limits.  A nonzero maximum of the
+limiting solution divided by that supersolution contradicts the
+maximum or boundary point principle.  The same argument in a
+fixed chart handles bounded maximizing points.  This proves the
+claimed estimate.
+
+Apply it to \(H_1-V_0\), retaining the drift as divergence forcing.
+The background errors are \(O(r^{1-\tau})\), and
+\(Z=O(r^{-\tau})\).  It follows that
+\begin{equation}\label{eq:ah-homogeneous-asymptotic}
+ |H_1-V_0|\le Cr^{-\kappa}.
+\end{equation}
+Here and below \(V_0\) is extended smoothly across the compact
+part.  There is one additional compactness point in this
+application: a failure with bounded maximizing points could
+produce a nonzero limit \(H_\circ\ge0\), with bounded measurable
+limit drift, zero natural flux, and
+\(H_\circ=O(r^{-\kappa})\), solving
+\eqref{eq:ah-homogeneous-lapse}.  The sign follows from positivity
+of \(H_1\) before subtracting \(V_0\) and dividing by an unbounded
+normalizing constant.  Regard \(ZH_\circ\) as divergence forcing
+for \(-\Delta_g+3\) and apply the preceding estimate repeatedly.
+This improves the decay to \(O(r^{-j'})\) for some \(2<j'<3\).
+To justify each improvement on the full exterior, solve the
+baseline problem on truncations with zero natural outer flux
+and pass to a limit; its difference from \(H_\circ\) decays in
+the previous weight and vanishes by the maximum principle.
+The improved decay is integrable.  Testing the density equation
+with radial cutoffs, equal to one near the finite boundary,
+and moving the Laplacian onto the cutoffs gives
+\(3\int_\Omega H_\circ=0\).  All shell terms vanish since
+\(j'>2\).  This is impossible.  On bounded limiting truncations
+the same contradiction follows by integrating the zero-flux
+equation directly.  This proves
+\eqref{eq:ah-homogeneous-asymptotic} with the asserted uniformity.
+For \(b_S=0\), Harnack, also by homogeneous-flux reflection at
+\(S\), and connectedness now give \(H_1\ge cr\).
+
+The dual maximum principle compares \(N\) to the homogeneous
+solution with the same drift and boundary data, giving
+\(N\le H_1\le Cr\).  For the lower bound divide \(N\) by the
+homogeneous solution with the same drift and \(b_S=0\).
+If this ratio is below \(c\epsilon/R_s\), with \(c\) sufficiently
+small, then \(N\le\epsilon\) wherever the spatial cutoff permits
+\(\Sigma\ne0\).  Hence \(\Sigma=0\) on this sublevel.  The ratio
+there solves an elliptic drift equation with no zeroth term.
+Its outward derivative is nonnegative at the inner face and
+strictly positive at the outer face when the ratio is below
+the threshold.  The minimum and boundary point principles
+therefore exclude such a minimum.  This proves the positive
+floor, and also excludes negative lapses when constitutive
+arguments have been extended.
+
+Finally, on \(2R_s\le r\le R\), subtract \(V_0\) and treat \(ZN\)
+as divergence forcing of size \(O(r^{1-\tau})\).
+The inner Dirichlet values have size \(O(R_s)\), and the natural
+outer error has the corresponding decaying size.  The weighted
+argument above, now with the inner Dirichlet face, gives
+\[
+ |N-V_0|\le C R_s^{1+\kappa}r^{-\kappa}.
+\]
+The homogeneous contribution of the inner data is bounded by
+the same barriers.  In the normalized argument the zero
+Dirichlet trace also excludes a limit maximum at that face.
+This proves the final estimate in \eqref{eq:ah-lapse-bounds}.
+On fixed patches write the lapse equation as a metric Laplace
+equation with bounded divergence forcing \(ZN\), bounded scalar
+forcing \(3N+\Sigma\), and the prescribed \(L^{p_1}\) boundary
+data.  The local \(W^{1,q}\) and Hölder estimates from
+Proposition~\ref{prop:elliptic-inputs} prove the last assertion.
+\end{proof}
+
+\subsection{Finite slopes and uniform estimates at infinity}
+
+\begin{lemma}\label{lem:ah-graph-estimates}
+The solutions satisfy \(0\le f\le tM\).  On every fixed compact
+subset, including \(S\),
+\begin{equation}\label{eq:ah-s-floor}
+ s\ge c_{\mathrm{loc}}(M,L)\epsilon ,
+\end{equation}
+and their height gradients have finite bounds for fixed
+\(M,L,\epsilon\), independent of \(d_*\), \(R\), and derivatives
+of \(b_S\).  Moreover,
+\begin{align}
+ \int_{\{f\in I\}}p\cdot df&\le C|I|,\qquad
+ \int_{\Omega_R}(W^2-1)\,\dd V_g\le CM,\label{eq:ah-height-energy}\\
+ |f|&\le C R_s^{-1}(R_s/r)^{4-\xi},\nonumber\\
+ |v|+|\pi|+|\nabla\pi|&\le C(R_s/r)^{3-\xi}
+       \quad(r\ge C_0R_s),\label{eq:ah-end-graph}
+\end{align}
+where \(I\) is any height interval, \(\xi>0\) can be fixed
+sufficiently small, and the constants in
+\eqref{eq:ah-height-energy}--\eqref{eq:ah-end-graph} are
+independent of \(M,L,\epsilon,d_*\) for large \(M\).
+The end estimates include the corresponding local Hölder
+control and derivatives of \(v\) through order two.
+\end{lemma}
+
+\begin{proof}
+The maximum principle for \(\divg_g p=0\) gives the height bound.
+We spell out how the differential estimate in
+Lemma~\ref{lem:gradient-maximum} supplies all the finite bounds
+used here.  At nonzero gradient write
+\[
+ z=|v|,\quad y=z^2,\quad m_f=|df|,\quad
+ q=\frac{1-y}{1+y},\quad
+ \beta=1+q,\quad
+ \mathcal P=g^{-1}+(q-1)e\otimes e,\quad e=\nabla f/m_f .
+\]
+The height equation is
+\(\mathcal P:\nabla^2 f+\beta\,d\log N\cdot df=0\).
+At an interior maximum of
+\(G+j(f)+d_0\), where \(G=-\log s\), \(j'<0\),
+\(j''>0\), and \(y\ge1/2\), that lemma gives
+\begin{equation}\label{eq:ah-gradient-maximum}
+ j''q m_f^2\le
+ C\left[1+\frac{\Sigma}{N}
+ +(1-y)^2(j'm_f)^2+|dd_0|^2+|\nabla^2d_0|\right].
+\end{equation}
+Its hypotheses hold because the lapse equation has precisely
+the drift \(K(v)\), the tensor \(K\) and its divergence are
+bounded on the relevant patches, and \(\Sigma\ge0\).
+The cancellation of the Hessian of \(\log N\) in that lemma
+requires no prior second-derivative estimate for \(N\).
+Multiplying \eqref{eq:ah-gradient-maximum} by \(N^2\) controls
+\((1-y)^{-1}\), using
+\(Nm_f=z/(1-y)\) and the upper lapse bound; a positive lower
+lapse is not needed in this step.
+
+At either constant-height face let \(n\) point towards increasing
+end direction.  When \(df\ne0\), \(e=-n\), and the boundary part
+of the same lemma gives
+\[
+ G_n=-\frac{yH_g+\partial_n\log N}{1-y}.
+\]
+At \(S\), \(\partial_\nu\log N=-zk-tb_S/N\), so
+\(G_\nu\ge zk/(1+z)\), a uniform lower bound.
+Since \(j'f_\nu=-j'm_f>0\), a maximum at this face bounds \(m_f\).
+At \(S_R\), \(H_g\) is bounded below by a positive constant,
+\(K=o(1)\), and
+\(N\partial_{\nu_R}\log N=K(p,\nu_R)+\partial_{\nu_R}V_0\).
+Taking \(|j'|\) sufficiently small makes the outward derivative
+of \(G+j(f)\) negative at large \(y\), bounded above by
+\(-c/(1-y)\).  Thus an outer maximum is excluded, or its slope
+is bounded by a localization derivative.  A sufficiently small
+decreasing exponential \(j\) has both required derivative signs.
+
+On a compact truncation these arguments give a finite global
+slope bound.  To make it independent of \(R\) locally, set
+\(d_0=k_0\log\chi\), \(k_0\ge2\), where \(\chi\) is a smooth
+cutoff on a slightly larger compact region, constant normally
+near \(S\) if that face is present.  Equation
+\eqref{eq:ah-gradient-maximum} bounds \(W\chi\) at an interior
+maximum; the preceding face argument does the same at \(S\).
+When \(y<1/2\) this bound is immediate.  Applying the lapse floor
+to the weighted maximum gives \eqref{eq:ah-s-floor}.
+
+Since \(|p|\le N\), the flux through \(S\) is bounded independently
+of all internal large parameters.  Test \(\divg_g p=0\) with
+a clipped height whose derivative is one on \(I\) and zero
+elsewhere.  Its boundary oscillation is at most \(|I|\), giving
+the first estimate in \eqref{eq:ah-height-energy}.
+The identity \(p\cdot df=W^2-1\), tested with the entire height,
+gives the second one.
+
+We next prove the initial uniform end bound \(f\le C/r\);
+it cannot be obtained from a compact-region bound depending
+arbitrarily on \(M\).  By Lemma~\ref{lem:ah-lapse}, \(N\asymp r\)
+when \(r\ge CR_s\), for a fixed large \(C\).
+Fix a radial segment of \(\eta\)-length at most one starting
+at radius \(r_*\) there, and set \(F=r_*f\).  In a bounded
+number of uniformly controlled metric charts covering its
+neighborhood, consider the subgraph of \(F\) in the product
+with a Euclidean line.  The product vector
+\[
+ \mathcal Z=(-p/r_*,1)
+\]
+is bounded and divergence free.  Its pairing with the upward
+graph normal is
+\[
+ \frac{1+p\cdot df}{\sqrt{1+r_*^2|df|^2}}
+ =\frac{W^2}{\sqrt{1+r_*^2|df|^2}}\ge c>0,
+\]
+using \(N/r_*\asymp1\) and the constitutive relation.
+The divergence theorem therefore makes the subgraph locally
+quasiminimizing for perimeter, with a uniform constant.
+Filling or emptying a product ball bounds the graph perimeter
+there by a constant times the trace area of either phase.
+The four-dimensional isoperimetric inequality then gives,
+for the positive volume \(V(h)\) of either phase in a ball,
+\(V'(h)\ge cV(h)^{3/4}\).
+Integration yields uniform two-sided volume density, and
+relative isoperimetry yields a fixed positive graph-area
+lower bound in each sufficiently small ball centered on
+the graph.
+
+If the height span of \(F\) on the radial segment is \(D\),
+choose graph points with separated heights and disjoint
+fixed-size product balls.  The preceding estimate bounds
+the graph area in the surrounding tube and relevant height
+band below by \(cD-C\).  Conversely, the constitutive relation
+gives
+\(\sqrt{1+r_*^2|df|^2}\le C(1+p\cdot df)\).
+The base tube has bounded volume and the relevant interval
+of \(f\) has length at most \(C(D+1)/r_*\).
+Equation~\eqref{eq:ah-height-energy} bounds the graph area
+above by \(C+C(D+1)/r_*\).  For large fixed starting radius
+this bounds \(D\) uniformly.  If the segment meets \(S_R\),
+oddly reflect \(f\), evenly reflect \(N\), and reflect the
+metric in the orthogonal charts described in the lapse proof.
+The zero trace of \(f\) makes the divergence equation valid
+weakly after reflection.  The reflected height band adds
+only an interval of comparable length to the energy estimate.
+Thus the same estimate holds there.  Summing the bounds
+\(C/r_*\) over consecutive outward unit segments, up to
+the zero height at \(S_R\), proves \(f\le C/r\).
+
+On far-out unit balls rescale \((f,N,p,s)\) by
+\((r_*,r_*^{-1},r_*^{-1},r_*^{-1})\), respectively.
+The equations there have \(\Sigma=0\) and retain their form.
+The localized gradient estimate now gives uniform bounds
+for the rescaled height gradient and uniform ellipticity.
+The outer-face argument applies with its rescaled positive
+flux datum.  The rescaled lapse has uniform \(W^{1,q}\) bounds.
+Differentiating the divergence height equation, or using
+difference quotients, gives scalar uniformly elliptic
+divergence equations for its first derivatives, with divergence
+forcing in \(L^q\), \(q>3\).  The inhomogeneous Hölder estimate
+therefore bounds \(df\) in a Hölder norm, also at a
+constant-height boundary by odd reflection.  The lapse
+divergence equation then gives \(C^{1,\gamma}\) control,
+followed by the nondivergence height equation and successive
+Schauder estimates.  These steps use only the stated
+\(C^{2,\alpha}\) metric and \(C^{1,\alpha}\) tensor bounds.
+They also prove compact-set gradient Hölder estimates away
+from infinity without differentiating \(b_S\).
+
+On each far-out ball, the energy estimate for a height interval
+of length \(C/r_*\), together with this Hölder control, implies
+\(z=o(1)\) uniformly as \(r_*\to\infty\).
+The derivative version of \eqref{eq:ah-lapse-bounds} implies
+that \(\nabla\log N\) tends to the hyperbolic radial direction
+as \(r/R_s\to\infty\).  The limiting height operator is
+\(\Delta_b+2\,d\eta\cdot d\); its decaying radial indicial
+root is \(-4\).  For any sufficiently small fixed \(\xi>0\),
+a positive multiple of \(r^{-4+\xi}\) is therefore a strict
+supersolution sufficiently far beyond \(R_s\).
+Comparison with the already proved \(C/r\) bound at a fixed
+multiple of \(R_s\) gives the first estimate in
+\eqref{eq:ah-end-graph}.  Treating the actual height equation
+as a linear equation on these smaller height ranges gives
+the corresponding derivative bounds by Schauder theory.
+Inserting them in \(p=s^2\nabla f\) and
+\(\pi=N^{-1}\operatorname{sym}\nabla p^\flat\) gives the
+remaining assertions.
+\end{proof}
+
+For a fixed sufficiently large \(C_0\), all sources vanish when
+\(r\ge C_0R_s\).  Indeed the height is then below \(M/2\),
+the lapse exceeds \(4\epsilon\), and the spatial cutoff
+for \(\Sigma\) has ended.  In this region
+\begin{equation}\label{eq:ah-end-x}
+ (\Delta_{\bar g}-3-U_0)x=-3,\qquad
+ U_0=-N^{-1}\divg_g((\pi-K)p),\qquad
+ |U_0|_{C^{0,\gamma}_{\mathrm{loc}}}
+       \le C(R_s/r)^\lambda
+\end{equation}
+for some \(\lambda>3\).  For example one can take any exponent
+below \(\min(6-2\xi,\tau+3-\xi)\), by
+\eqref{eq:ah-end-graph}.  The constants here retain the
+independence asserted in that lemma.
+
+\subsection{Costs and enforcement of the finite penalties}
+
+\begin{lemma}\label{lem:ah-penalties}
+The parameters can be chosen in the order already specified,
+with \(\epsilon\) sufficiently small after \(M,L\), and then
+\(d_*\) sufficiently large but finite, such that
+\begin{equation}\label{eq:ah-penalty-costs}
+ \int_{\Omega_R}P_1\,\dd V_g\le C T/M,\qquad
+ \int_{\Omega_R}P_2\,\dd V_g=o_{M,L}(1)
+       \quad(\epsilon\downarrow0),
+\end{equation}
+uniformly in \(R,b_S\), and in the strengths.
+At \(t=1\), the same choices enforce
+\begin{equation}\label{eq:ah-penalty-targets}
+ w|_S\ge T,\qquad
+ w\ge T_2\quad\hbox{on }\{N\le2\epsilon\},\qquad
+ sT_2\ge1\quad\hbox{on }\{N\le4\epsilon\}.
+\end{equation}
+Consequently \(x<1-\delta\) and \(r\le R_s\) imply
+\(\Sigma=L\).
+\end{lemma}
+
+\begin{proof}
+The lapse floor confines \(N<6\epsilon\) to a compact set
+depending on \(M\), independently of \(\epsilon\).
+Equation~\eqref{eq:ah-s-floor} on a slightly larger compact set
+therefore permits a choice of \(C_2(M,L)\) such that
+\(sT_2\ge1\) throughout the required low-lapse region.
+
+For every Lipschitz test \(\zeta\) vanishing at \(S_R\), positivity
+of \(w\), its equation, and its zero inner flux give
+\begin{equation}\label{eq:ah-penalty-test}
+ \int P\,\frac{\zeta^2}{w}\,\dd V_g
+       \le \int A(d\zeta,d\zeta)\,\dd V_g .
+\end{equation}
+Indeed test with \(\zeta^2/w\) and complete the square in
+\(2\zeta\,dw\cdot A\,d\zeta/w-\zeta^2 A(dw,dw)/w^2\).
+All the other terms on the right side of the \(w\)-equation
+are nonnegative.  On the support of \(P_1\),
+\(f/M\ge1/2\) and \(w\le4T\).  With \(\zeta=f/M\),
+the algebraic inequality
+\(A(df,df)=W-W^{-1}\le W^2-1=p\cdot df\) and
+\eqref{eq:ah-height-energy} prove
+\(\int P_1\le CT/M\).
+
+For \(P_2\) take a cutoff of \(N/\epsilon\), equal to one
+below \(4\) and zero above \(6\).  It vanishes near \(S_R\)
+for large \(R\).  Since \(A\le N^2g^{-1}\) and \(w\le4T_2\)
+on this source's support,
+\begin{equation}\label{eq:ah-low-lapse-cost}
+ \int P_2\le
+ C T_2\int_{\{N<6\epsilon\}}|\nabla N|^2\,\dd V_g.
+\end{equation}
+Test the lapse equation with \((6\epsilon-N)^+\).
+The outward current flux at \(S\) is \(t b_S\ge0\), so its
+boundary contribution has the favorable sign.  On the
+sublevel set \(|Kp|\le CN\le C\epsilon\).  Cauchy's inequality
+then gives
+\begin{equation}\label{eq:ah-low-lapse-energy}
+ \int_{\{N<6\epsilon\}}|\nabla N|^2
+ \le C(M)\epsilon^2
+       +C\epsilon\int_{\{N<6\epsilon\}}\Sigma .
+\end{equation}
+
+We must show that the final integral tends uniformly to zero;
+boundedness of \(\Sigma\) alone would not suffice.
+For any potentially violating sequence, use the uniform
+local lapse Hölder bounds to take \(N_j\to N_0\ge0\)
+uniformly on the compact region containing these sublevels.
+After a subsequence,
+\[
+ Kp_j\rightharpoonup Z_0N_0,\qquad
+ \Sigma_j\rightharpoonup^*\Sigma_0,\qquad
+ |Z_0|\le C,\quad 0\le\Sigma_0\le L .
+\]
+The weak limit equation is
+\(\Delta_gN_0=\divg_g(Z_0N_0)+3N_0+\Sigma_0\).
+We claim \(\Sigma_0=0\) almost everywhere on \(\{N_0=0\}\).
+Choose an interior density point of that zero set which is
+also a Lebesgue point of \(\Sigma_0\).  In balls of radius
+\(h\downarrow0\), rescale \(N_0\) by \(h^{-2}\).
+The rescaled fields are nonnegative and vanish at the center.
+Their divergence drift has size \(O(h)\) and their scalar
+forcing is bounded.  The inhomogeneous Harnack estimate
+therefore gives a uniform bound on smaller balls.
+Because the rescaled zero sets have asymptotically full
+measure, these functions tend to zero in local \(L^1\).
+Testing the rescaled equation against compactly supported
+smooth functions shows that the limiting constant scalar
+forcing is zero, proving the claim.  Boundary points need
+not be considered, since \(S\) has zero volume.
+
+For completeness, the changing sublevel sets cause no loss:
+for every \(a>0\), uniform convergence gives
+\(\{N_j<6\epsilon_j\}\subset\{N_0\le a\}\) eventually.
+Nonnegativity and weak convergence, using continuous cutoffs
+of \(N_0\) that majorize this fixed sublevel, imply
+\[
+ \limsup_j\int_{\{N_j<6\epsilon_j\}}\Sigma_j
+ \le \int_{\{N_0\le2a\}}\Sigma_0.
+\]
+Letting \(a\downarrow0\) proves the asserted uniform convergence.
+Equations~\eqref{eq:ah-low-lapse-cost} and
+\eqref{eq:ah-low-lapse-energy}, with \(T_2=C_2/\epsilon\),
+prove the second cost in \eqref{eq:ah-penalty-costs}.
+Choose \(\epsilon=\epsilon(M)\) sufficiently small that this
+cost, as well as \(T/M=M^{-1/5}\), tends to zero as \(M\to\infty\).
+
+Now fix \(M,L,\epsilon\) and increase the finite strengths.
+The previously proved continuity of \(f,N\) supplies uniform
+neighborhoods of the targets \(S\) and \(N\le2\epsilon\)
+inside, respectively, the weaker triggering regions
+\(f\ge3M/4\) and \(N\le3\epsilon\).
+The cost bounds imply that the volume in these neighborhoods
+where \(w\le2T\), or \(w\le2T_2\), tends to zero as
+\(d_*\to\infty\).
+For either threshold \(D=T,T_2\), the bounded nonnegative
+function \((2D-w)^+\) is a weak subsolution of the homogeneous
+divergence operator \(\divg_g(A\,d\cdot)\).
+Its coefficients are uniformly elliptic on the relevant
+compact region for these fixed parameters, independently of
+the strength.  The scalar local supremum estimate bounds its
+supremum on a smaller neighborhood by its \(L^2\) mean on the
+larger one.  At \(S\), even reflection using the zero natural
+flux gives the same estimate.  Choosing \(d_*\) sufficiently
+large makes that supremum smaller than \(D\).
+This proves \eqref{eq:ah-penalty-targets}, with room in the
+thresholds if desired.  Finally, if \(x<1-\delta\), the
+second target excludes \(N\le2\epsilon\), so the volume
+penalty is at full strength when also \(r\le R_s\).
+\end{proof}
+
+\subsection{Compactness, mass, and a uniform comparison sphere}
+
+Until the existence argument below, a \emph{limit solution} means
+any limit along expanding truncations of smooth solutions at
+the fixed internal parameters and fixed \(t\), allowing arbitrary
+\(b_S\in\mathcal B_M\).  Statements about this family are a priori
+statements and do not presuppose that it is nonempty.
+Pass also to a weak \(L^{p_1}(S)\) limit of the boundary data.
+Until the selector is imposed, we use \(b_S\) for this inherited
+datum in a limit solution: it need not be smooth, but its
+pointwise bounds, norm bound, and natural-flux identity pass
+to the limit.  Its integral is the limit of the integrals of
+the smooth data.
+
+\begin{lemma}\label{lem:ah-limit-mass}
+With internal parameters fixed, the fields \(N,f,df,w\)
+are uniformly precompact in local uniform topology, including
+at \(S\).  Limit solutions have \(N,w>0\) and are classical
+away from \(S\).  They obey
+\begin{equation}\label{eq:ah-conformal-decay}
+ x-1=O_{\mathrm{fixed},2,\gamma}(r^{-3+\xi}).
+\end{equation}
+Here \(\xi\) is taken sufficiently small, in particular
+\(0<\xi<\min(3-\tau,\tau-1)\).
+Their metrics \(h\) satisfy the required asymptotically
+hyperbolic decay, all four mass fluxes are finite, and
+\(V_0(\Scal_h+6)\) is integrable near infinity.  Their time
+components satisfy
+\begin{equation}\label{eq:ah-mass-budget}
+ 8\pi\{p_0(h)-p_0(g)\}
+   =t\int_S b_S\,\dd A_g
+      +\int_\Omega\{(x-1)\Sigma+P\}\,\dd V_g
+   \le o(1)\quad(M\to\infty).
+\end{equation}
+The \(o(1)\) is uniform over the allowed data, with the
+subsequent choices of \(\epsilon,d_*\) just described.
+\end{lemma}
+
+\begin{proof}
+Only the upper bound for \(w\) remains to be established.
+The right side of its equation is bounded on compact sets:
+\(\Sigma x\le L\), since a nonzero \(\Sigma\) requires \(x<1\),
+and \(P\) has fixed finite strength.
+By \eqref{eq:ah-lapse-bounds} and \eqref{eq:ah-end-graph},
+a multiple of \(r^{-1}\) is a supersolution for
+\(-\divg_g(A\,dw)=3N\) sufficiently far beyond \(R_s\).
+Comparison therefore bounds the end values of \(w\) by
+\[
+ C\left(1+\sup_{\{r\le C_0R_s\}}w\right)r^{-1}.
+\]
+If its compact supremum were unbounded, divide by that
+supremum.  Uniform ellipticity, local divergence estimates,
+and convergence of \(N,df\) then give a nonzero nonnegative
+limit satisfying the homogeneous \(w\)-equation, with zero
+inner flux and decaying to zero at infinity.  The maximum
+principle excludes such a function; Harnack and reflection
+at \(S\) handle a maximum at that face.  Thus the compact
+upper bounds hold.  Local Hölder estimates for \(w\) and
+the already proved estimates for \(N,df\) give the stated
+precompactness.  The lapse floor and the strictly positive
+forcing \(3N\) in the supersolution equation imply positivity
+of both limiting fields.  Scalar elliptic bootstrapping gives
+classical interior regularity.
+
+On the end, \eqref{eq:ah-end-x} and these upper bounds permit
+comparison for \(x-1\) with a multiple of \(r^{-3+\xi}\).
+For a finite truncation add \(C(r/R)^{a_0}\), \(0<a_0<1\),
+to dominate its outer boundary values.  The first term
+dominates the decaying forcing \(U_0\), while the second
+vanishes on fixed sets as \(R\to\infty\).
+The scalar zeroth coefficient is positive after increasing
+the fixed end radius.  The maximum principle proves the
+zeroth-order bound in \eqref{eq:ah-conformal-decay}, and
+interior Schauder estimates prove its derivative and Hölder
+versions.  These estimates are uniform over all limit
+solutions at the fixed parameters, without bounds on
+derivatives of \(b_S\).
+
+Since
+\(\bar g-g=O_{\mathrm{fixed},2,\gamma}(r^{-6+2\xi})\),
+the choice \(\xi<3-\tau\) ensures that \(h-b\) has the
+original admissible decay exponent, with a possibly smaller
+Hölder exponent.  Put \(\varphi=x-1\).
+The change in each defining mass flux is, up to an error
+tending to zero, the flux of the one-form
+\[
+ -2\{V_\mu\,d\varphi-\varphi\,dV_\mu\}.
+\]
+All nonlinear conformal errors and graph terms disappear
+by the displayed decay.  This flux has a finite limit:
+its divergence is
+\(-2V_\mu(\Delta_b-3)\varphi\), and
+\eqref{eq:ah-end-x} gives absolute weighted integrability.
+Indeed \(U_0=O(r^{-\lambda})\), \(\lambda>3\);
+the difference between \(\Delta_{\bar g}\) and \(\Delta_b\)
+on \(\varphi\) is
+\(O(r^{-\tau-3+\xi})+O(r^{-9+3\xi})\).
+Multiplication by \(V_0\) and integration against the
+hyperbolic volume, of radial size \(r\,\dd r\), is integrable.
+The scalar identity \eqref{eq:ah-scalar}, the original
+weighted matter and \(K^2\) integrability, and
+\eqref{eq:ah-end-graph}--\eqref{eq:ah-conformal-decay}
+similarly prove integrability of \(V_0(\Scal_h+6)\).
+
+On each limit end the lapse equation also gives
+\[
+ (\Delta_g-3)(N-V_0)
+  =O(r^{1-\tau})+\divg_g(Kp).
+\]
+Together with \eqref{eq:ah-lapse-bounds} and the end
+derivative estimates this yields
+\(|\nabla(N-V_0)|=O_{\mathrm{fixed}}(r^{-\kappa})\).
+Insert the second expression for \(Q\) in
+\eqref{eq:ah-current}.  Its sphere flux differs from
+that of \(V_0\,d\varphi-\varphi\,dV_0\) by a quantity
+tending to zero.  The \(N-V_0\) terms have integrated
+size \(O(r^{-1-\kappa+\xi})\); the
+\((\pi-K)p\) terms have integrated sizes bounded by
+\(O(r^{-3+2\xi})+O(r^{-\tau+\xi})\).
+Metric replacement errors vanish by the same decay.
+It follows that
+\[
+ \lim_{r\to\infty}\int_{\{r=\mathrm{const}\}}Q\cdot\nu_r
+       \,\dd A_g=-8\pi\{p_0(h)-p_0(g)\}.
+\]
+At the inner boundary the outward normal of the exterior
+is \(-\nu\), and \(Q\cdot(-\nu)=t b_S\).
+Integrating \(\divg_g Q=-(x-1)\Sigma-P\) proves
+\eqref{eq:ah-mass-budget}.  This integration remains valid
+for limits with only continuous fields at \(S\): use the
+first expression \(Q=A\,dw+B\), the weak equations and
+their prescribed natural fluxes on compact regions.
+The second expression is needed only at infinity, where
+the fields are classical.  Finally,
+\((x-1)\Sigma\le0\), the \(L^{p_1}\) bound makes
+\(\int_S b_S=o(1)\), and
+Lemma~\ref{lem:ah-penalties} gives \(\int P=o(1)\).
+\end{proof}
+
+The next estimate is needed before choosing \(b_S\); compact
+upper bounds for \(w\) at fixed parameters alone would not
+give it.
+
+\begin{lemma}\label{lem:ah-comparison-sphere}
+There is a fixed large \(C_1\) such that, at
+\(r_1=C_1R_s\), every limit solution at \(t=1\) satisfies
+\begin{equation}\label{eq:ah-comparison-area}
+ \Area_h(\{r=r_1\})\le C R_s^2,
+\end{equation}
+with \(C\) independent of the internal large parameters.
+\end{lemma}
+
+\begin{proof}
+On \(r>r_1\) solve
+\[
+ (\Delta_{\bar g}-3-U_0)Y=0,\qquad
+ Y|_{r=r_1}=0,\qquad Y/V_0\longrightarrow1 .
+\]
+Existence follows by solving with outer Dirichlet datum
+\(V_0\) on larger truncations, applying positive barriers,
+and passing to a limit.  Choose
+\(\xi<b_0<\min(1,\tau)\).
+Positive and negative barriers of size
+\(Cr(r_1/r)^{b_0}\) for \(Y-V_0\) work by the end estimates.
+They give \(0\le Y\le Cr\), and the corresponding first
+derivative control follows from elliptic estimates.
+On an annulus from \(r_1\) to a large fixed multiple of
+\(r_1\), compare from below with a small positive multiple
+of \(r_1((r/r_1)^2-1)\).  This is a subsolution with zero
+inner value, and its outer value can be made smaller than
+that of \(Y\).  The boundary point principle and local
+boundary estimates therefore give, in the end direction,
+\[
+ cr_1\le\partial_nY\le Cr_1
+      \quad\hbox{on }\{r=r_1\},
+\]
+where \(n\) is the \(\bar g\)-unit normal.
+
+For \(\varphi=x-1\), the operator just used satisfies
+\((\Delta_{\bar g}-3-U_0)\varphi=U_0\).
+Green's identity, the mass-flux calculation, and
+\(b_0>\xi\) give
+\begin{equation}\label{eq:ah-green-comparison}
+ \int_{\{r=r_1\}}(x-1)\partial_nY\,\dd A_{\bar g}
+ =\int_{\{r>r_1\}}YU_0\,\dd V_{\bar g}
+       +8\pi\{p_0(h)-p_0(g)\}
+ \le Cr_1^3 .
+\end{equation}
+The source integral is bounded by \(Cr_1^3\), since
+\(\lambda>3\), \(Y\le Cr\), and \(W\) is uniformly bounded
+on this end region.  The mass increment is bounded above
+by \eqref{eq:ah-mass-budget}.  Add
+\(\int\partial_nY\le Cr_1^3\) to both sides of
+\eqref{eq:ah-green-comparison}, and then use \(x>0\) and
+\(\partial_nY\ge cr_1\).  This yields
+\(\int_{\{r=r_1\}}x\,\dd A_{\bar g}\le Cr_1^2\).
+Two-dimensional conformal scaling makes this integral
+exactly the area in \eqref{eq:ah-comparison-area}.
+\end{proof}
+
+\subsection{A continuous choice of inner flux}
+
+\begin{lemma}\label{lem:ah-contact-selector}
+For all sufficiently large \(M\), at the finite parameters
+chosen above, there is a continuous rule \(\mathfrak b\) on
+the uniform topology of the fields \((N,f,df,w)\) on a fixed
+compact original subdomain, with values in a finite convex
+hull in \(\mathcal B_M\), such that every limit solution at
+\(t=1\) satisfies
+\begin{equation}\label{eq:ah-contact-majorant}
+ \mathfrak b(N,f,df,w)>N(H_g-zk)
+\end{equation}
+at every possible contact on \(S\) of a perimeter-minimizing
+enclosure of \(S\).
+\end{lemma}
+
+\begin{proof}
+Consider all limits at \(t=1\) arising from arbitrary smooth
+choices \(b_S\in\mathcal B_M\) on arbitrary expanding
+truncations.  Their restrictions to each fixed compact
+region form a compact family in the uniform field topology.
+Precompactness follows from Lemma~\ref{lem:ah-limit-mass};
+closedness follows by a diagonal choice of actual truncation
+solutions from the defining sequences of a convergent
+sequence of limits.  This definition uses no existence
+assertion, and the family may initially be empty.
+
+The fixed-parameter end estimates are uniform over this
+family.  Choose a large fixed radial sphere, beyond \(r_1\),
+so that all spheres from slightly before it to infinity are
+strictly outward mean convex in every limit metric.
+Adjoin smooth compact orientable handlebodies to all components of
+\(S\), possible because each is a closed orientable surface.  Choose
+the gluing maps to reverse the induced boundary orientations, so the
+orientation of \(\Omega\) extends over the caps.
+Let \(O\) denote their union and \(X\) the resulting
+one-ended smooth manifold without boundary; let \(K_c\)
+be the compact region cut off by the chosen large sphere.
+Extend each continuous positive metric through \(S\) to
+the cap side, continuously with respect to its uniform
+field values.  For example reflect in a fixed base collar
+and interpolate to a fixed cap metric.  The mixed metric
+entries vanish at \(S\), because \(f\) has constant trace.
+Perimeters of sets containing \(O\) depend only on the
+metric outside \(O\) and its boundary values.
+
+For each of these metrics minimize perimeter among
+finite-perimeter subsets of \(K_c\) containing \(O\)
+up to null sets.  Compactness and lower semicontinuity
+give a minimizer.  These facts hold for a continuous
+positive metric by uniform smooth approximation of the
+area integrands.  The sphere at \(r_1\) is an admissible
+comparison, so every minimizer has perimeter at most
+\(CR_s^2\).
+
+Define contact points on \(S\) as points in the support
+of a minimizer's perimeter measure.  The union of all
+such contacts equals the contact set of one minimizer.
+To prove this, minimize volume in an auxiliary fixed
+smooth metric among the perimeter minimizers.  Perimeter
+submodularity shows that the intersection and union of
+any two minimizers are still minimizers.  The
+volume-minimizing one, denoted \(E_-\), is therefore
+contained almost everywhere in every other minimizer.
+If a contact of a second minimizer were absent from the
+perimeter support of \(E_-\), a small ball there would
+have zero \(E_-\)-perimeter.  Since \(O\) occupies a
+positive part of that ball, the ball would be filled
+by \(E_-\), and hence by the second minimizer, a
+contradiction.  The contact union is consequently the
+closed contact set of \(E_-\).
+
+These contact unions are upper semicontinuous in the
+uniform field topology.  Indeed, along uniformly
+convergent metrics, any sequence of minimizers has an
+\(L^1\) subsequence converging to a minimizer for the
+limit metric.  At a contact point, the complement has
+uniform positive volume density: filling a chart ball
+is an allowed competitor, so its perimeter in the ball
+is bounded by a constant times its trace area on the
+sphere.  Isoperimetry, applied to the complement inside
+the ball, gives \(V^{2/3}\le CV'\), and thus
+\(V(r)\ge cr^3\).  Constants are uniform on the compact
+family of positive continuous metrics.  The cap supplies
+positive density for the set on the other side.
+These two density bounds persist when both the metric
+and the contact point converge, so the limit point is
+in the perimeter support of a limit minimizer.
+
+The same density bounds place all contacts in the
+measure-theoretic boundary.  Up to a set of
+two-dimensional Hausdorff measure zero they therefore
+belong to the reduced boundary, by the finite-perimeter
+structure theorem.  At almost every point of their
+intersection with the smooth surface \(S\), the tangents
+are those of \(S\).  Since \(df\) is tangentially zero,
+the area density there is \(x\,\dd A_g\).
+Writing \(C_h\) for the contact union, we obtain
+\begin{equation}\label{eq:ah-contact-area}
+ \int_{C_h}x\,\dd A_g\le CR_s^2.
+\end{equation}
+
+Set \(\Theta=[N(H_g-zk)]_+\) on \(S\).
+Since \(H_g\le k\), its positive part is bounded by \(Cs\).
+For example, when \(k\ge0\),
+\(N(H_g-zk)\le Nk(1-z)\le ks\); when \(k<0\)
+the left side is nonpositive.  The upper lapse bound
+also gives \(\Theta\le C\), with \(C\) independent of
+the pointwise cap \(B_*\).  Fix \(B_*\) strictly above
+this latter bound.  By \eqref{eq:ah-penalty-targets},
+\(s\le x/T\) at \(S\), and hence
+\begin{equation}\label{eq:ah-contact-flux-cost}
+ \int_{C_h}\Theta^{p_1}\,\dd A_g
+ \le C\int_{C_h}s\,\dd A_g
+ \le C R_s^2/T=CM^{-1/5}.
+\end{equation}
+The admissible \(p_1\)-power budget in
+\eqref{eq:ah-flux-class} is \(M^{-1/10}\).
+Thus for sufficiently large \(M\) there is strict slack.
+For each field in the compact family, approximate its
+continuous nonnegative threshold on the closed set \(C_h\)
+from above by a smooth function, with a small positive
+margin, and taper it to zero in a sufficiently small
+neighborhood of that set.  Regularity of surface measure
+and \eqref{eq:ah-contact-flux-cost} keep its \(L^{p_1}\)
+norm within the budget; the fixed gap below \(B_*\)
+keeps its pointwise values admissible.  This constructs
+a datum strictly majorizing the threshold on \(C_h\).
+When \(C_h\) is empty the condition is vacuous.
+
+Upper semicontinuity of contacts and continuity of the
+threshold show that the same datum works in an open
+neighborhood of the given field in the compact limit
+family.  Take a finite such cover, with associated
+smooth data \(b_1,\dots,b_m\), and enlarge the neighborhoods
+to open subsets of the ambient uniform field space so
+that their intersections with the limit family retain
+this property.  A continuous partition of unity,
+supplemented by the complement of the compact limit
+family carrying datum zero, yields \(\mathfrak b\).
+Explicitly, distance to the complements of these finitely
+many open neighborhoods, together with distance to the
+compact limit family, gives nonnegative continuous weights
+with nonzero sum after a harmless subordinate refinement.
+Normalize and take the corresponding convex combination.
+At a field in the limit family the zero datum has weight
+zero, and every active \(b_i\) is a strict majorant
+at all its contacts.  Convexity preserves both constraints
+in \eqref{eq:ah-flux-class}.  If the family is empty, use
+the constant zero rule.  Only the original compact
+subdomain's fields enter this rule; no trial fields on
+the caps are required.
+\end{proof}
+
+\subsection{Existence at fixed finite parameters}
+
+\begin{proposition}\label{prop:ah-finite-existence}
+With the choices above, the system
+\eqref{eq:ah-system}--\eqref{eq:ah-outer-data}, using
+\(b_S=\mathfrak b(N,f,df,w)\), has a smooth positive
+solution at \(t=1\) on every sufficiently large fixed
+truncation.  Expanding truncations have a subsequence
+converging to a solution smooth up to \(S\) on the
+whole exterior.  It satisfies all the preceding estimates
+and realizes the rule \(\mathfrak b\).
+\end{proposition}
+
+\begin{proof}
+We describe the compact map precisely, because a frozen
+source need not preserve positivity of a lapse output
+away from fixed points.  Fix a small Hölder exponent
+\(\gamma>0\), to be decreased below the exponents in
+the a priori estimates, and use triples of
+\(C^{1,\gamma}\) functions on the fixed truncation.
+Choose \(a>0\) so that \(2a\) is below the fixed-point
+floor in Lemma~\ref{lem:ah-lapse}, and a smooth function
+\(\mathcal E:\R\to[a,\infty)\) equal to the identity
+on \([2a,\infty)\).
+
+Given trial fields and \(t\), first solve the linear
+nondivergence height equation, freezing
+\(\mathcal P,\beta,d\log\mathcal E(N)\) from the
+trial arguments, with Dirichlet values \(tM\) and zero.
+At zero trial gradient the smooth continuations are
+\(\mathcal P=g^{-1}\) and \(\beta=2\).
+The first-order coefficient acts on the new height.
+Next compute \(v\) from this new height and the
+extended trial lapse, and set \(Z=K(v)\).
+Solve the linear density equation for the raw new lapse:
+\[
+ \divg_g(\nabla N_{\mathrm{new}}-ZN_{\mathrm{new}})
+    =3N_{\mathrm{new}}+\Sigma_{\mathrm{trial}} .
+\]
+Use the prescribed outer current flux and inner current
+flux \(-t\mathfrak b\) evaluated on the trial fields.
+The source here is frozen from the trial fields,
+including the raw trial lapse in its cutoff at
+\(N\le\epsilon\).  The scalar multiplying the drift
+and the zeroth term is the raw output lapse, not
+\(\mathcal E(N_{\mathrm{new}})\).
+The adjoint argument in Lemma~\ref{lem:ah-lapse}
+proves unique linear solvability.
+
+Finally solve a linear \(w\)-equation whose coefficient
+\(A\) is formed from the new height and
+\(\mathcal E(N_{\mathrm{new}})\).  Its right side is
+\[
+ 3\mathcal E(N_{\mathrm{new}})
+    +\Sigma_{\mathrm{trial}}x_{\mathrm{trial}}^+
+    +P_{\mathrm{trial}},
+ \qquad
+ x_{\mathrm{trial}}^+
+  =s(\mathcal E(N_{\mathrm{trial}}),df_{\mathrm{trial}})
+      (w_{\mathrm{trial}})^+ .
+\]
+It has the zero inner natural flux and the prescribed
+positive outer Dirichlet datum.  The three solves are
+sequential, and define a continuous compact map
+\(\mathcal T_t\).
+Both frozen sources include their prescribed factor \(t\).
+Indeed bounded sets of input fields give uniform
+ellipticity and Hölder coefficients.  The height output
+is bounded in \(C^{2,\gamma}\), so the drift in the
+lapse step is \(C^{1,\gamma}\); its inverse estimates
+are uniform on bounded input sequences by Schauder
+theory, coefficient compactness in a lower exponent,
+and the zero-kernel adjoint argument.  The lapse output
+and then the \(w\) output have second-order Hölder
+bounds.  The rule \(\mathfrak b\) has values in a
+fixed finite convex hull of smooth functions, hence
+uniform bounds of every spatial derivative.  The
+compact embedding of these \(C^{2,\gamma}\) bounds
+into \(C^{1,\gamma}\) proves compactness.  The same estimates and
+uniqueness prove continuity without differentiating
+the selection rule.
+
+At a fixed point, the cutoff in the actual lapse
+variable and the bounded drift give the positive floor
+in Lemma~\ref{lem:ah-lapse}.  Thus every extension
+\(\mathcal E\) becomes the identity.  The last equation
+and positive outer datum give \(w>0\), so the positive
+part also disappears.  Every fixed point therefore
+satisfies the original equations and boundary data.
+
+All such fixed points for \(0\le t\le1\) have bounded
+\(C^{1,\gamma}\) norms for the fixed truncation and
+fixed parameters.  Here is the regularity chain
+underlying this assertion.  The lapse bound, height
+gradient bound, and bounded sources give lapse
+\(W^{1,q}\) estimates, \(q>3\), and Hölder estimates
+for the height gradient, as in
+Lemma~\ref{lem:ah-graph-estimates}.  The lapse divergence
+equation then gives \(C^{1,\gamma'}\) estimates for
+some \(\gamma'>0\); its scalar-forcing portion can
+equivalently be estimated by the Laplace \(W^{2,q}\)
+theorem.  The selected boundary data have bounded
+smooth norms.  The nondivergence height equation
+gives \(C^{2,\gamma'}\) estimates, after decreasing
+\(\gamma'\) if necessary.  The \(w\)-equation has
+bounded forcing, since \(\Sigma x\le L\) and \(P\)
+has finite strength.  Mixed-boundary coercivity first
+gives an \(H^1\) bound, then scalar local boundedness
+and Hölder estimates give a uniform bound and
+continuity.  Successive Schauder estimates now give
+second-order Hölder bounds for all fields.  Take
+the Banach exponent \(\gamma<\gamma'\).
+The same bootstrap makes every fixed point smooth.
+
+At \(t=0\), the map is constant.  The first output
+is \(f_0=0\), so the drift in the second step is
+zero independently of the input.  Its output is
+the unique positive solution \(N_0\) of
+\(\Delta_gN_0=3N_0\), with zero inner flux and the
+prescribed outer flux.  The third output \(w_0\)
+uses \(\mathcal E(N_0)=N_0\), by the homogeneous lapse
+floor, and is the unique solution of
+\[
+ -\divg_g(N_0^2\nabla w_0)=3N_0,\qquad
+ \partial_\nu w_0|_S=0,\qquad w_0|_{S_R}=1/V_0 .
+\]
+In general \(w_0\ne1/N_0\), because the outer lapse
+condition is Neumann.  Choose a ball in the Banach
+space containing this constant output and all
+homotopy fixed points.  On its boundary
+\(\Id-\mathcal T_t\) never vanishes, and its
+Leray--Schauder degree is \(+1\) at \(t=0\).
+Homotopy invariance gives a fixed point at \(t=1\).
+
+Now let \(R\to\infty\).  The fixed-parameter compact
+estimates give a subsequential limit on every compact
+set, and a diagonal subsequence gives a global limit.
+It belongs to the family used to define \(\mathfrak b\).
+Uniform convergence of \((N,f,df,w)\) on its defining
+compact domain implies convergence of the selected
+datum, in every spatial norm because its range is
+a finite smooth convex hull.  Thus the limit realizes
+the rule, and the elliptic bootstrap is valid up to
+\(S\) as well as in the interior.
+\end{proof}
+
+\subsection{A free minimal boundary}
+
+Apply Proposition~\ref{prop:ah-finite-existence} at \(t=1\).
+For this smooth metric, extend through the cap side
+smoothly, and consider the perimeter minimization in
+Lemma~\ref{lem:ah-contact-selector}.  Equation
+\eqref{eq:ah-boundary-curvature} and the strict majorant
+give \(H_h(S)<0\) at every possible contact.
+This excludes contact without any assumed regularity
+of the minimizing surface at the obstacle.
+
+Here is the finite-perimeter argument.  Near the closed
+contact set, the unit normals to sufficiently small
+parallel outward collars of \(O\) have negative
+divergence.  Off a neighborhood of that set, a thin
+collar is already filled up to null sets: there is no
+perimeter support on \(S\), and the cap side is filled.
+Adjoin the entire collar of width \(a\) to a minimizer.
+For almost every sufficiently small \(a\), Gauss--Green
+and slicing bound the perimeter change above by the
+integral of that negative divergence on the added part.
+The comparison field has unit flux on the new outer
+leaf, and its flux across the removed old boundary is
+bounded by the area removed.  If contact existed,
+the complement density proved above would give
+positive added volume, so the perimeter would strictly
+decrease.  This is impossible.
+
+There is likewise no contact with the outer constraint
+\(\partial K_c\).  The outward normals to the far
+coordinate spheres have positive divergence, so
+cutting away the outside part of a competitor decreases
+perimeter whenever it removes positive volume.
+Gauss--Green and slicing make this statement valid
+for finite-perimeter competitors at almost every
+cutting radius.  It also shows that no larger precompact
+competitor, unconstrained by \(K_c\), can have smaller
+perimeter: first trim it by such a sphere.
+
+All minimizing boundaries are therefore free, locally
+perimeter minimizing in a smooth three-dimensional
+metric.  Interior codimension-one minimizing-boundary
+regularity gives smooth compact embedded minimal
+boundaries.  Choose a largest minimizer \(E\), by
+maximizing auxiliary volume among the minimizers in
+\(K_c\).  Its open representative contains \(O\)
+compactly in its interior and is strictly outward
+minimizing: equality for an enclosure differing by
+positive volume would
+contradict volume maximality after trimming.
+Every component of \(E\) meets \(O\), since an
+unattached component could be removed with a strict
+perimeter saving.  Its open exterior is connected,
+since filling any bounded exterior component would
+also save perimeter.
+
+Write
+\begin{equation}\label{eq:ah-hull-area}
+ A_h=\Area_h(\partial E)>0.
+\end{equation}
+The closed exterior \(X\setminus E\), with its boundary
+included, is smooth, connected, complete, orientable, and one-ended.
+Its orientation is the restriction of the orientation of \(X\);
+the graph and conformal changes alter the metric, not this orientation.
+Its compact truncations are compact, and its end is
+asymptotically hyperbolic.  It satisfies
+\(\Scal_h\ge-6\) by \eqref{eq:ah-scalar}, and its
+weighted scalar integrability and finite mass fluxes
+were verified in Lemma~\ref{lem:ah-limit-mass}.
+Its minimum enclosing area is exactly \(A_h\).
+Indeed any admissible end-side domain in this closed
+exterior, on taking the complement of its interior
+in \(X\), supplies a precompact finite-perimeter
+enclosure of \(E\).  Its perimeter is bounded by the
+entire intrinsic boundary area specified in the
+definition, including any part coinciding with
+\(\partial E\).  Outward minimization gives the
+lower bound \(A_h\), and the original boundary
+itself gives equality.
+
+The time-symmetric hypothesis of
+Theorem~\ref{thm:ah-reduction} is thus applicable to
+this fixed metric and gives
+\begin{equation}\label{eq:ah-ts-application}
+ p_0(h)\ge F(A_h)\ge0.
+\end{equation}
+No causal property of the intermediate mass covector
+has been used.
+
+\subsection{The weighted bad-volume estimate and conclusion}
+
+Suppose first that \(A_*>0\), fix \(0<\varepsilon_*<1\),
+and choose the preceding \(\delta\) sufficiently small
+for Proposition~\ref{prop:area-transfer}.
+Fix a sufficiently large projection radius \(r_p\)
+and put
+\[
+ \rho=\min(1,r_p^2/r^2),\qquad
+ \mathcal D=\{x<1-\delta\}.
+\]
+By \eqref{eq:ah-mass-budget} and
+\eqref{eq:ah-ts-application},
+\[
+ \int_\Omega(1-x)\Sigma
+ =\int_S b_S+\int_\Omega P
+       -8\pi\{p_0(h)-p_0(g)\}\le C.
+\]
+The constant is independent of \(M\).
+Lemma~\ref{lem:ah-penalties} makes
+\(\Sigma=L\) on \(\mathcal D\cap\{r\le R_s\}\),
+and \(1-x\ge\delta\) there.  Consequently
+\[
+ \Vol_g(\mathcal D\cap\{r\le R_s\})\le C/L.
+\]
+Since \(W\le1+\sqrt{W^2-1}\), Cauchy--Schwarz and
+\eqref{eq:ah-height-energy} give an inner weighted
+volume bound \(C(L^{-1}+\sqrt{M/L})\).
+For large \(M\), \(R_s>r_p\).  The asymptotically
+hyperbolic volume element gives
+\[
+ \int_{\{r>R_s\}}\rho^{3/2}\,\dd V_g\le C/R_s,\qquad
+ \int_{\{r>R_s\}}\rho^3\,\dd V_g\le C/R_s^4 ,
+\]
+where \(r_p\) is fixed and absorbed into \(C\).
+A second application of Cauchy--Schwarz yields
+\begin{equation}\label{eq:ah-bad-volume}
+ \int_{\mathcal D}\rho^{3/2}W\,\dd V_g
+ \le C\left(L^{-1}+\sqrt{M/L}
+                  +R_s^{-1}+\frac{\sqrt M}{R_s^2}\right)
+ \longrightarrow0 .
+\end{equation}
+For the choices \eqref{eq:ah-parameter-scales},
+all four powers tend to zero.
+
+We check the hypotheses of the area-transfer proposition
+explicitly.  The capped smooth manifold \(X\) and cap
+set \(O\) are fixed, so the relevant second Betti number
+of \(X\setminus\operatorname{int}O\) is finite and fixed.
+The chosen \(E\) is a cap-only largest minimizing
+enclosure, has a free smooth compact minimal boundary,
+contains \(O\) compactly, is strictly outward minimizing,
+and each component meets \(O\).  The comparison metric
+is the smooth \(h=x\bar g\); here there is no seam or
+smoothing error.  It obeys
+\(\bar g\ge g\), \(\dd V_{\bar g}=W\,\dd V_g\), and
+\(\Scal_h\ge-6\) outside \(E\).
+Its asymptotically hyperbolic end supplies the proper
+subsolution required in Proposition~\ref{prop:area-transfer}.
+Lemma~\ref{lem:area-projection} gives, on every relevant
+enclosing level with connected exterior,
+\[
+ \int\rho\,\dd A_g\ge(1-o_{r_p}(1))A_*.
+\]
+Choose \(r_p\) first to make this error sufficiently
+small for the fixed \(\varepsilon_*\), then choose
+\(\delta\), and finally take \(M\) large.
+The proposition's temporary conformal modification
+for starting the flow may be arbitrarily small at
+each fixed metric and leaves the initial area
+unchanged.  Its uniform scalar lower bound therefore
+has the required fixed constant, for example \(7\).
+Equation~\eqref{eq:ah-bad-volume} meets the remaining
+smallness hypothesis.  Proposition~\ref{prop:area-transfer}
+gives
+\begin{equation}\label{eq:ah-area-conclusion}
+ A_h\ge(1-\varepsilon_*)A_* .
+\end{equation}
+
+Combine \eqref{eq:ah-mass-budget},
+\eqref{eq:ah-ts-application}, and
+\eqref{eq:ah-area-conclusion}.  For every fixed
+\(\varepsilon_*>0\) and arbitrarily large \(M\),
+\[
+ p_0(g)+o(1)\ge F\bigl((1-\varepsilon_*)A_*\bigr).
+\]
+First let \(M\to\infty\), then
+\(\varepsilon_*\downarrow0\), using continuity of \(F\).
+If \(A_*=0\), no area transfer is needed:
+\(F(A_h)\ge F(0)\) and the mass budget give the
+same conclusion directly.  This proves
+Theorem~\ref{thm:ah-reduction}.

preprints/The-Penrose-inequality-for-maximal-asymptotically-hyperbolic-initial-data-October-5-2026/build/sections/06-stress-extension.tex

@@ -0,0 +1,1019 @@
+\section{Electrostatic stress and the asymptotically flat attachment}
+\label{sec:stress-extension}
+
+Throughout this section, $(\Omega,g)$ is a smooth connected exterior with
+the topology, completeness, spherical end, tensor decay, and weighted
+scalar integrability specified in Section~\ref{sec:hypotheses}.  We assume
+\begin{equation}
+ R_g\ge-6,\qquad H_g(S)\le0,
+ \label{eq:stress-base-assumptions}
+\end{equation}
+and make no causal assumption on its mass covector.  The normal on $S$
+points into $\Omega$.  Fix the end chart, and put
+\begin{equation}
+ A_* = A_{\min}(S;g),\qquad a=\sqrt{A_*/(4\pi)}.
+ \label{eq:stress-area-radius}
+\end{equation}
+The objective is to construct a compact inner region and a scalar-flat
+\AF\ exterior whose mass is at most $p_0(g)-a^3/2$, up to an arbitrarily
+small error.  The tensor introduced for this purpose is synthetic: it is
+not the second fundamental form of the original data.  We denote the
+electrostatic stress by $\Pi$, reserving $\pi$ for the graph second form in
+the subsequent construction.
+
+\subsection{A conformal approximation of the end}
+
+\begin{proposition}[End approximation]
+\label{prop:conformal-approximation}
+Under \eqref{eq:stress-base-assumptions}, there are smooth metrics $g_j$
+on the same exterior such that $R_{g_j}>-6$, $H_{g_j}(S)\le0$,
+$g_j\to g$ locally in $C^2$, and
+\begin{equation}
+ (1-\epsilon_j)g\le g_j\le(1+\epsilon_j)g,
+ \quad \epsilon_j\downarrow0,
+ \qquad p_0(g_j)\longrightarrow p_0(g).
+ \label{eq:stress-density-convergence}
+\end{equation}
+Each $g_j$ has the required weighted scalar integrability and finite mass
+fluxes.  Outside a compact set, depending on $j$, it is of the form
+\begin{equation}
+ g_j=\phi_j^4 b,\qquad
+ \phi_j=1+\alpha_j(\omega)r^{-3}+O(r^{-3-\xi}),
+ \label{eq:stress-conformal-end}
+\end{equation}
+where $\xi>0$ can be fixed and the remainder has the same decay after
+any fixed finite number of angular and radial-logarithmic derivatives.
+In particular $A_{\min}(S;g_j)\to A_{\min}(S;g)$.
+\end{proposition}
+
+\begin{proof}
+Choose $j_0$ with $3/2<j_0<\tau$, where $\tau$ is the original end
+decay exponent.  Extend $r$ to a smooth positive function on the compact
+part.  For large $B$, let $\chi_B$ be a smooth radial cutoff equal to one
+for $r\le B$ and zero for $r\ge2B$, with a fixed profile in $r/B$.
+Set
+\[
+ g_B=b+\chi_B(g-b)
+\]
+on the end and $g_B=g$ on the compact part.  For sufficiently large $B$
+this is positive definite, and its uniform metric charts and
+$C^{2,\alpha}$ bounds can be chosen independently of $B$.  Fix a smooth
+$k>0$ equal to $r^{-4}$ far out.  We seek a small $u$ with
+$\partial_\nu u=0$ on $S$ satisfying
+\begin{equation}
+ -8\Delta_{g_B}u+(R_{g_B}+6)(1+u)
+ +6\big((1+u)^5-(1+u)\big)
+ =\chi_B(R_g+6)+\delta k.
+ \label{eq:stress-density-equation}
+\end{equation}
+The constant-term discrepancy, excluding $\delta k$, is supported in
+$B<r<2B$ and is $O(r^{-\tau})$ in local $C^{0,\alpha}$ norms.
+Consequently it tends to zero in the scalar weighted norm of weight
+$r^{-j_0}$.  Also $R_{g_B}+6\ge-o(1)$ uniformly, because it equals the
+nonnegative $R_g+6$ on the inner part, vanishes beyond $2B$, and is
+$O(B^{-\tau})$ on the intervening annulus.
+
+For completeness, the linearized operator divided by eight is
+\[
+ \mathscr L_B=-\Delta_{g_B}+3+(R_{g_B}+6)/8.
+\]
+It has an inverse, with homogeneous Neumann condition, bounded uniformly
+from weighted $C^{0,\alpha}$ to weighted $C^{2,\alpha}$ at weight
+$j_0$ (a smaller H\"older exponent may be used).  To prove the required
+supremum bound, take a positive function $\beta$ constant on a large
+compact set, comparable to $r^{-j_0}$ at infinity, and let its logarithmic
+slope decrease slowly from zero to $-j_0$ as a function of $\log r$.
+For the limiting radial operator,
+\[
+ (-\partial_\eta^2-2\partial_\eta+3)e^{-j_0\eta}
+ =(3+2j_0-j_0^2)e^{-j_0\eta}>0.
+\]
+Every intermediate slope has the same strict sign.  Slow variation of
+the slope, placement of the transition sufficiently far out, and the
+uniformly small negative part of $(R_{g_B}+6)/8$ therefore give
+$\mathscr L_B\beta\ge c\beta$ with $c>0$ independent of large $B$.
+Here $\partial_\nu\beta=0$ on $S$.  Solve on compact truncations with
+zero outer Dirichlet data.  The maximum principle, applied to positive
+and negative multiples of $\beta$, bounds the solution by the weighted
+norm of its forcing.  Uniform local interior and Neumann Schauder
+estimates give the stated second-order bound.  Compactness gives a
+solution on $\Omega$, and the maximum principle gives uniqueness.
+This proves the inverse assertion without an assumption about solvability
+on the noncompact manifold.
+
+The nonlinear remainder in \eqref{eq:stress-density-equation} is
+$O(u^2)$ and is locally Lipschitz with Lipschitz constant tending to zero
+on small weighted balls.  The uniform inverse and contraction mapping
+therefore give $u=u_{B,\delta}$ tending to zero in weighted
+$C^{2,\alpha}_{j_0}$ as $B\to\infty$ and $\delta\downarrow0$.
+Local elliptic bootstrapping makes $u$ smooth.  For small parameters
+$1+u>0$, and the conformal scalar-curvature formula gives, for
+$\widetilde g=(1+u)^4g_B$,
+\begin{equation}
+ R_{\widetilde g}+6
+ =(1+u)^{-5}\big(\chi_B(R_g+6)+\delta k\big)>0.
+ \label{eq:stress-density-scalar}
+\end{equation}
+The boundary conformal formula and $\partial_\nu u=0$ give
+$H_{\widetilde g}=(1+u)^{-2}H_g\le0$.  The uniform metric convergence in
+\eqref{eq:stress-density-convergence} follows from the weighted bound and
+the cutoff construction.
+
+We next justify the end expansion for each fixed $B,\delta$.
+Beyond $2B$, $g_B=b$ and
+\begin{equation}
+ (-\Delta_b+3)u=F(u)+\delta r^{-4}/8,
+ \qquad F(u)=O(u^2),\quad F'(u)=O(u).
+ \label{eq:stress-density-end-ode}
+\end{equation}
+All fixed orders of angular derivatives of $u$ are $O(r^{-j_0})$.
+Indeed rotations commute with $\Delta_b$.  Their difference quotients
+solve a linear equation with small potential $O(u)$, with smooth data
+on a fixed inner sphere and decaying data at infinity.  The same
+$r^{-j_0}$ barriers and local weighted estimates bound these quotients.
+Induction, with the lower angular derivatives as forcing, proves the
+claim at every fixed order.  Radial derivatives at the initial weighted
+order follow from local elliptic estimates.  Since
+\[
+ \Delta_b=\partial_\eta^2+2\coth\eta\,\partial_\eta
+            +r^{-2}\Delta_\sigma,
+\]
+Equation~\eqref{eq:stress-density-end-ode} reduces, with all these angular
+derivatives, to
+\[
+ -u_{\eta\eta}-2u_\eta+3u=O(e^{-(3+\xi)\eta})
+\]
+for any fixed sufficiently small
+$0<\xi<\min\{2j_0-3,j_0-1,1\}$.  Variation of constants for the two
+solutions $e^\eta,e^{-3\eta}$ excludes the growing mode by the known
+decay and gives $u=c(\omega)e^{-3\eta}+O(e^{-(3+\xi)\eta})$.
+The same argument after angular differentiation makes $c$ smooth;
+the equation gives all stated radial derivatives.  Changing from
+$\eta$ to $r$ gives \eqref{eq:stress-conformal-end}.
+
+We finally check mass convergence, since convergence only in a weight
+less than three would not suffice.  Write $h=b+e$ for either the original
+metric or one of the approximants, and let $\mathbb U_{V_0}(e)$ be the
+one-form in the mass definition.  The linearized scalar curvature is
+\[
+ (DR)_b(e)=\divg_b\divg_b e-\Delta_b\tr_b e+2\tr_b e.
+\]
+Differentiation, using $\nabla_b^2V_0=V_0b$, yields the exact linear
+identity $\divg_b\mathbb U_{V_0}(e)=V_0(DR)_b(e)$.
+The difference between $R_h+6$ and $(DR)_b(e)$ is bounded by
+$C(|e|\,|\nabla_b^2e|+|\nabla_be|^2+|e|^2)$.  The uniform weighted
+second-order bound and $dV_b\asymp r\,dr\,dA_\sigma$ thus give
+\begin{equation}
+ 16\pi p_0(h)-\int_{r=D}\mathbb U_{V_0}(h-b)(\nu_b)\,\dd A_b
+ =\int_{r>D}V_0(R_h+6)\,\dd V_b+O(D^{3-2j_0}),
+ \label{eq:stress-density-mass-tail}
+\end{equation}
+uniformly in the approximants.  The scalar integrands in
+\eqref{eq:stress-density-scalar} are dominated by a fixed multiple of
+$V_0(R_g+6+k)$, which is integrable; they converge pointwise to
+$V_0(R_g+6)$.  The same divergence calculation with $V_i$, using
+$|V_i|\le V_0$ and the corresponding derivative bounds, proves existence
+of all fluxes for the approximants.  At a fixed $D$ their surface fluxes
+converge by local $C^2$ convergence.  Dominated convergence in
+\eqref{eq:stress-density-mass-tail}, followed by $D\to\infty$, proves
+the asserted time-component convergence.  This is also the scalar
+linearization underlying the mass definition in \citet{CH2003}.
+Finally, the two-sided metric comparison compares the areas of every
+admissible intrinsic boundary with the same multiplicative factors;
+taking infima proves convergence of the minimum enclosing areas.
+\end{proof}
+
+We henceforth fix one approximating metric, write it again as $g$, and
+write $\alpha$ for its smooth coefficient in
+\eqref{eq:stress-conformal-end}.  Thus $R_g+6>0$ everywhere.  All
+subsequent constants may depend on this fixed metric; no uniformity
+across the approximation is needed.
+
+\subsection{The nonlinear electrostatic problem}
+
+For $0\le e\le1$ define
+\begin{equation}
+ t(e)=\sqrt{1-e^{3/2}},\qquad
+ L(w)=\max_{0\le e\le1}\{t(e)+we\},\qquad w\ge0.
+ \label{eq:stress-legendre-law}
+\end{equation}
+Strict concavity of $t$, together with $t'(0)=0$ and
+$t'(e)\to-\infty$ as $e\uparrow1$, gives a unique maximizing $e=e(w)$,
+with $e(0)=0$, $0<e(w)<1$ for $w>0$, and
+\begin{equation}
+ w=-t'(e)=\frac{3\sqrt e}{4t(e)},\qquad
+ L'(w)=e(w),\qquad L=t+we.
+ \label{eq:stress-constitutive-law}
+\end{equation}
+In particular $L(|\xi|)$ is convex as a function of a covector $\xi$.
+
+\begin{proposition}[Electrostatic potential, stress, and charge]
+\label{prop:electrostatic-stress}
+For every $M_e>0$ there is a weak solution $u$ of
+\begin{equation}
+ \divg_g E=0,\qquad
+ E=e(w)\frac{\nabla u}{w},\quad w=|du|_g,
+ \qquad u|_S=-M_e,\quad u\longrightarrow0\text{ at infinity},
+ \label{eq:stress-electrostatics}
+\end{equation}
+where $E=0$ at $w=0$.  It satisfies $-M_e\le u\le0$,
+$|u|+|du|_g=O(r^{-1})$, and is $C^{1,\beta}$ locally up to $S$ and
+in compactified coordinates $(\rho,\omega)$ up to $\rho=0$, for some
+$\beta>0$, where $\rho=e^{-\eta}$.
+The continuous symmetric tensor
+\begin{equation}
+ \Pi=L(w)g-du\otimes E^\flat
+ \label{eq:stress-tensor}
+\end{equation}
+is distributionally divergence free, is positive definite, and obeys
+\begin{equation}
+ (\tr_g\Pi)^2-2|\Pi|_g^2=3.
+ \label{eq:stress-algebraic-identity}
+\end{equation}
+There is a continuous nonnegative function $e_0$ on $\Sph^2$ such that,
+uniformly in angle,
+\begin{equation}
+ r^2|E|_g\longrightarrow e_0,\qquad
+ r^3\big(\Pi(\nu,\nu)-1\big)
+       \longrightarrow-\tfrac12 e_0^{3/2}.
+ \label{eq:stress-end-asymptotics}
+\end{equation}
+Here $\nu$ points toward increasing $r$.  The charge
+\begin{equation}
+ F_{M_e}=\int_{r=R}g(E,\nu)\,\dd A_g
+ \label{eq:stress-charge-definition}
+\end{equation}
+is independent of large $R$, satisfies
+$F_{M_e}\le\int_{\Sph^2}e_0\,\dd A_\sigma$, and
+\begin{equation}
+ \liminf_{M_e\to\infty}F_{M_e}\ge A_*.
+ \label{eq:stress-charge-saturation}
+\end{equation}
+\end{proposition}
+
+\begin{proof}
+We give the construction and its regularity before establishing the
+stress and charge assertions.
+
+\emph{Constitutive estimates and regularization.}
+For $w>0$ put $Q(w)=we'(w)/e(w)$.  Logarithmic differentiation of
+\eqref{eq:stress-constitutive-law} gives
+\begin{equation}
+ Q(w)^{-1}=\frac12+\frac{3e(w)^{3/2}}{4(1-e(w)^{3/2})}.
+ \label{eq:stress-Q-formula}
+\end{equation}
+Thus $0<Q\le2$.  At zero, inversion in the variable $\sqrt e$ gives
+\[
+ e(w)=\frac{16}{9}w^2+O(w^5);
+\]
+in particular $e(w)/w^2$ extends smoothly and positively to $w=0$.
+At infinity, inversion in $w^{-2}$ gives
+\begin{equation}
+ e(w)=1-\frac{3}{8w^2}+O(w^{-4}),\qquad
+ Q(w)=\frac{3}{4w^2}+O(w^{-4}),\qquad
+ wQ'(w)=-\frac{3}{2w^2}+O(w^{-4}).
+ \label{eq:stress-large-gradient-law}
+\end{equation}
+Regularize only at zero by
+\begin{equation}
+ e_\varepsilon(w)=w\frac{e(z)}z,\quad
+ z=(w^2+\varepsilon^2)^{1/2},\quad
+ Q_\varepsilon(w)=1+\frac{w^2}{z^2}(Q(z)-1),\quad 0<\varepsilon<1.
+ \label{eq:stress-regularized-law}
+\end{equation}
+The flux $e_\varepsilon(|\xi|)\xi^\sharp/|\xi|$ is smooth and strictly
+elliptic on every finite gradient range, including zero.  Its radial
+and transverse derivative eigenvalues are $e_\varepsilon'$ and
+$e_\varepsilon/w$, respectively.  We have $0<Q_\varepsilon\le2$;
+uniformly for sufficiently large $w$,
+\begin{equation}
+ c\le w^2Q_\varepsilon(w)\le C,\qquad
+ -w^2\big(wQ_\varepsilon'(w)+Q_\varepsilon(w)\big)\ge c.
+ \label{eq:stress-uniform-Q}
+\end{equation}
+These follow by substituting \eqref{eq:stress-large-gradient-law} in
+\eqref{eq:stress-regularized-law}; the leading coefficient is
+$3/4+\varepsilon^2$, bounded above and below independently of
+$\varepsilon$.
+
+\emph{Boundary and interior gradient bounds.}
+On a connected smooth truncation $\Omega_R$ prescribe $u=-M_e$ on $S$
+and $u=0$ on $r=R$.  The maximum principle gives $-M_e\le u\le0$.
+At points with nonzero gradient the regularized equation is
+\begin{equation}
+ P:\nabla^2u=0,\qquad
+ P=g^{-1}+(Q_\varepsilon(w)-1)n\otimes n,
+ \quad n=\nabla u/w.
+ \label{eq:stress-regularized-nondivergence}
+\end{equation}
+Let $h$ be inward distance from $S$.  Since $H(S)\le0$, smoothness
+gives $\Delta h\le C h$ in a fixed collar.  A supersolution above the
+inner boundary datum is $-M_e+B(h)$, where
+\[
+ B(h)=b_0^{-1}\log(1+Dh),\qquad B''=-b_0(B')^2.
+\]
+For $p=B'$ sufficiently large, \eqref{eq:stress-uniform-Q} gives
+\[
+ P:\nabla^2B=Q_\varepsilon(p)B''+p\Delta h
+ \le-cb_0+C/b_0,
+\]
+because $hp\le1/b_0$.  Choose $b_0$ first to make this negative,
+then a collar width $h_0$ so small that
+$(2b_0h_0)^{-1}$ is above the required gradient threshold, and finally
+$D$ large so that $B(h_0)>M_e$ and $B'\ge(2b_0h_0)^{-1}$ there.
+Comparison with the constant lower barrier bounds the inward derivative
+on $S$, independently of $R$ and $\varepsilon$.
+At the outer sphere, inward distance may be replaced by
+$h=\eta_R-\eta$ in a uniform collar.  For gradients parallel to
+$d\eta$, asymptotic hyperbolicity and $Q_\varepsilon\le2$ give
+$P:\nabla^2h\le-1$ for sufficiently large $R$.  The linear lower
+barrier $-C h$, together with $u\le0$, bounds the outer derivative
+with $C$ depending on $M_e$ and the fixed collar width only.
+
+For the interior estimate use an orthonormal frame with first vector
+$n$, and write transverse indices as $a,b$.  Differentiating
+\eqref{eq:stress-regularized-nondivergence}, commuting one derivative,
+and differentiating the norm of $du$ gives
+\begin{align}
+ P:\nabla^2\log w
+ ={}&\Ric(n,n)
+ +(1-Q_\varepsilon)\frac{|d_\perp w|^2}{w^2}
+ +\frac{\sum_{a,b}u_{ab}^2}{w^2} \notag\\
+ &-\big(wQ_\varepsilon'+Q_\varepsilon\big)\frac{w_1^2}{w^2}.
+ \label{eq:stress-gradient-identity}
+\end{align}
+To specify the cancellation, the differentiated coefficient term is
+$Q_\varepsilon'w_1^2+2(Q_\varepsilon-1)|d_\perp w|^2/w$;
+the differentiated norm contributes
+$[Q_\varepsilon\sum_a u_{1a}^2+\sum_{a,b}u_{ab}^2]/w$ before
+division by $w$.  Subtracting $P(dw,dw)/w^2$ produces exactly
+\eqref{eq:stress-gradient-identity}.
+At a positive interior maximum of $w\exp(B_0u)$ we have
+$d_\perp w=0$ and $w_1/w=-B_0w$.  The left side of
+\eqref{eq:stress-gradient-identity} is nonpositive, since
+$P:\nabla^2u=0$.  The global lower Ricci bound and
+\eqref{eq:stress-uniform-Q} make the right side at least
+$-C+cB_0^2$ when $w$ is large.  A fixed sufficiently large $B_0$
+excludes such a maximum.  Combining the remaining interior maxima
+with the boundary estimates and the height bound gives
+\begin{equation}
+ \sup_{\Omega_R}|du|_g\le C(M_e),
+ \label{eq:stress-global-gradient-bound}
+\end{equation}
+uniformly in large $R$ and $0<\varepsilon<1$.
+
+\emph{Existence on truncations and passage to the limit.}
+For fixed $R,\varepsilon$, vary the inner Dirichlet value from zero
+to $-M_e$.  The derivative of the divergence equation is a strictly
+elliptic divergence operator without zeroth-order term and is
+invertible with zero Dirichlet data by the maximum principle and
+linear elliptic theory.  The bounds already proved hold on this
+entire path.  On its bounded gradient range, ordinary uniformly
+elliptic quasilinear gradient H\"older estimates apply.  For the
+boundary estimate subtract its constant value and reflect oddly in
+boundary normal coordinates, reflecting the metric as well.  The
+reflected metric is Lipschitz; its mixed normal--tangential entries
+vanish on the face.  The reflected equation is weakly valid by the
+zero trace.  Equivalently, difference quotients give uniformly
+elliptic equations for the first derivatives with bounded divergence
+forcing.  Scalar H\"older estimates followed by Schauder estimates
+on the original smooth side give closedness of the continuation.
+Here constants may depend on the fixed $R,\varepsilon$;
+\citet{GT2001} supplies the ordinary elliptic estimates being used.
+Thus smooth regularized solutions exist.
+
+A decay bound is uniform in both regularizations.  On the end put
+$v=C(e^{-\eta}-e^{-2\eta})$.  It is positive for large $\eta$, and
+\[
+ \frac{v''}{-v'}=\frac{1-4e^{-\eta}}{1-2e^{-\eta}}
+ \le1-e^{-\eta}.
+\]
+Also $|d\eta|_g^2=1+O(e^{-\tau\eta})$ and
+$P:\nabla^2\eta=2+O(e^{-\min\{\tau,2\}\eta})$ for the
+radial test direction, uniformly for $0<Q_\varepsilon\le2$.
+Since $\tau>1$, these inequalities imply
+$P:\nabla^2v\le0$ sufficiently far out, independently of $C$.
+Comparison of the nonnegative solution $-u$ with $v$, choosing $C$
+on a fixed inner sphere and using $u=0$ on the truncating edge,
+gives $|u|\le Ce^{-\eta}$.
+
+Take a subsequence with $R\to\infty$ and $\varepsilon\downarrow0$.
+The uniform Lipschitz bound gives local uniform convergence and weak
+gradient convergence to a locally Lipschitz function with the required
+traces.  To pass to the equation, let $\chi$ be a compactly supported
+cutoff and test the regularized equation with $\chi(u_j-u)$.
+The terms containing $d\chi$ tend to zero by uniform convergence.
+Subtracting the limiting-law flux evaluated at $du$ gives
+\[
+ \int\chi\,
+ \big(A(du_j)-A(du)\big)\cdot(du_j-du)\,\dd V_g\longrightarrow0,
+ \qquad A(\xi)=e(|\xi|)\xi^\sharp/|\xi|.
+\]
+Uniform convergence of the constitutive functions on bounded ranges
+justifies replacing the regularized flux here.  The law $A$ is
+strictly monotone.  On the fixed compact gradient range its
+monotonicity pairing has a positive minimum whenever the two
+arguments are separated by a prescribed positive distance.
+Thus $du_j\to du$ in measure and strongly in every finite local
+$L^p$.  This proves the limiting weak equation.
+
+\emph{Gradient regularity and compactification.}
+We apply the scalar $p=3$ gradient theorem recorded in
+Section~\ref{sec:preliminaries}, namely Theorem~1.1 and estimate~(1.8)
+of \citet{AraujoZhang2020}.  The coordinate flux must satisfy globally
+\begin{align}
+ \lambda|\xi|\,|\zeta|^2
+ &\le \mathcal A_\xi(x,\xi)[\zeta,\zeta],
+ &|\mathcal A|+|\mathcal A_\xi|\,|\xi|&\le C|\xi|^2,
+ \label{eq:stress-p3-structure}\\
+ |\mathcal A(x,\xi)-\mathcal A(y,\xi)|
+ &\le C|x-y|\,|\xi|^2.
+ \notag
+\end{align}
+It must also be $C^1$ in $\xi$ and the solution must have bounded
+local $W^{1,3}$ energy.  The Lipschitz spatial bound supplies a
+$C^{0,1/2}$ modulus, so $n=p=3$ and zero forcing meet all the
+parameters of that theorem.  Throughout this argument choose only
+$0<\beta<\min\{\alpha_m,1/4\}$ as in
+Section~\ref{sec:preliminaries}; the decay and boundary limits below
+require a positive exponent, not an optimal one.
+
+We verify global, rather than merely bounded-gradient, structure before
+applying this theorem.  On the gradient range in
+\eqref{eq:stress-global-gradient-bound}, $e(w)/w^2$ and $Q(w)$ have
+positive upper and lower bounds.  Thus $e'(w)$ is comparable to $w$.
+Choose a smooth cutoff $\chi=1$ below the known range and zero above
+twice a larger fixed bound, and replace $e'$ outside that range by
+\[
+ \widetilde e'(w)=\chi(w)e'(w)+(1-\chi(w))c_1w,
+ \qquad \widetilde e(0)=0,
+\]
+where $c_1>0$.  Then $\widetilde e=e$ on the range attained by the
+solution, $\widetilde e'\asymp w$, and $\widetilde e\asymp w^2$
+globally.  In coordinates with metric matrix $G$ the density flux is
+\begin{equation}
+ \mathcal A^i(x,\xi)=\sqrt{\det G}\,
+ \frac{\widetilde e(|\xi|_G)}{|\xi|_G}\,G^{ij}\xi_j,
+ \qquad |\xi|_G=(G^{ij}\xi_i\xi_j)^{1/2}.
+ \label{eq:stress-coordinate-flux}
+\end{equation}
+Uniform positive metric bounds give
+\eqref{eq:stress-p3-structure}; Lipschitz metric coefficients give
+the spatial bound.  The flux is $C^1$ at zero because it is
+$O(|\xi|^2)$ and its derivative is $O(|\xi|)$.  Its derivative is
+symmetric, as is also apparent from its convex radial primitive.
+The solution has finite local cubic energy by its Lipschitz bound.
+The quoted theorem therefore proves local $C^{1,\beta}$ regularity.
+
+The same argument works at $S$.  In orthogonal boundary coordinates
+with reflection matrix $J=\operatorname{diag}(-1,1,1)$, extend the
+metric matrix by $G(-h,x)=JG(h,x)J$, and extend $u+M_e$ oddly.
+The mixed metric entries vanish on the face, so this extension is
+Lipschitz.  The reflected flux satisfies
+$\mathcal A_-(\xi)=J\mathcal A_+(J\xi)$.  Evaluated on the odd
+solution, its normal component is even and its tangential components
+are odd.  A test on the reflected ball consequently reduces to a
+test on the original half-ball with zero trace.  Such tests are
+allowed: cutoff at distance $h=\delta$ creates an error tending to
+zero since the test is $O(h)$, the flux is bounded, and the strip
+volume tends to zero.  Thus the reflected weak equation satisfies
+all the same hypotheses, proving the asserted boundary regularity.
+
+To obtain gradient decay, take a uniform metric ball far out with
+central radius $r_*$, and put $s_*=C/r_*$, so that $|u|\le s_*$ on a
+slightly larger ball.  Apply the preceding fixed global extension
+to the equation for $u/s_*$; its flux is
+\[
+ \mathcal A_{s_*}(x,\xi)=s_*^{-2}\mathcal A(x,s_*\xi).
+\]
+The cubic structure constants and spatial modulus are unchanged.
+The required energy bound does not presume a bound on the normalized
+gradient.  Testing with $\chi^3(u/s_*)$, and using coercivity and
+Young's inequality, gives
+\begin{equation}
+ \int\chi^3|d(u/s_*)|^3
+ \le C\int |u/s_*|^3|d\chi|^3\le C.
+ \label{eq:stress-scaled-energy}
+\end{equation}
+The H\"older estimate for the gradient, together with this energy
+bound, gives its supremum on a smaller ball: some point has gradient
+bounded by the cubic mean, and the H\"older bound controls its
+difference from every other point.  Therefore $|du|_g\le C/r$.
+
+Now let $g_c=\rho^2g$ on the end.  It extends to a positive Lipschitz
+metric up to $\rho=0$, with vanishing mixed normal--tangential entries
+there.  In the conformal end at hand this follows directly from
+\[
+ g_c=\phi^4\left(d\rho^2+\tfrac14(1-\rho^2)^2\sigma\right)
+\]
+and \eqref{eq:stress-conformal-end}.  The estimate just obtained gives
+$|du|_{g_c}\le C$, and the height estimate gives the zero trace.
+Writing $\bar w=|du|_{g_c}$, the coordinate divergence equation in
+this metric has radial law
+\begin{equation}
+ q_\rho(\bar w)=\rho^{-2}e(\rho\bar w)
+              =\bar w^2 F(\rho\bar w),
+ \qquad F(t)=e(t)/t^2.
+ \label{eq:stress-compactified-law}
+\end{equation}
+On the known bounded range of $\bar w$, this satisfies uniform cubic
+structure and Lipschitz spatial dependence, including at $\rho=0$.
+For an explicit uniform extension, set
+$H(t)=2F(t)+tF'(t)>0$, choose a fixed cutoff equal to one below that
+range and zero above twice a larger bound, and define
+\[
+ \widetilde q_\rho'(w)
+ =w\big[\chi(w)H(\rho w)+(1-\chi(w))c_1\big],
+ \qquad \widetilde q_\rho(0)=0.
+\]
+It agrees with $q_\rho$ on the attained range, is comparable to
+$w^2$, has derivative comparable to $w$, and has the required uniform
+Lipschitz dependence in position.  Reflect $g_c$ as above, use
+$|\rho|$ in this coefficient, and reflect $u$ oddly.  The gradient
+bound ensures $W^{1,3}$ across the face.  The same zero-trace test
+argument proves the reflected weak equation.  The scalar theorem
+now gives $C^{1,\beta}$ in compactified coordinates.  No theorem
+about degenerate boundary equations is being assumed in addition
+to the stated interior result.
+
+\emph{Stress and asymptotic coefficients.}
+Convexity shows that the weak solution minimizes the integral of
+$L(|du|)$ against compactly supported Lipschitz variations: integrate
+the convex tangent inequality and use \eqref{eq:stress-electrostatics}.
+Apply this fact to compactly supported changes of independent variable.
+For the flow of a smooth vector field $X$, differentiating the local
+energy of the transported potential gives
+\[
+ \int\big(L(w)g-du\otimes E^\flat\big):\nabla X\,\dd V_g=0.
+\]
+The calculation is justified by bounded gradients on the support;
+no finite total energy of $L$ is needed.  This is
+$\divg_g\Pi=0$ in distributions.
+The stress eigenvalues in the gradient and transverse directions are
+$t$ and $l=L=t+we$, respectively, with $\Pi=g$ at zero gradient.
+They are positive at every finite gradient.  Moreover,
+\[
+ (t+2l)^2-2(t^2+2l^2)=4tl-t^2
+ =3t^2+4twe=3(1-e^{3/2})+3e^{3/2}=3,
+\]
+proving \eqref{eq:stress-algebraic-identity}.
+
+Since $u$ has constant trace at conformal infinity, all its tangential
+derivatives in compactified coordinates vanish there.  Its scaled
+gradient consequently has a continuous radial limit.  Together with
+$r\rho\to1/2$ and $e(w)=(16/9)w^2+O(w^5)$, this gives a continuous
+$e_0\ge0$ with $r^2e(w)\to e_0$ uniformly.  At a point where
+$e_0>0$ the gradient direction approaches the radial direction, so
+$\Pi(\nu,\nu)=t+o(r^{-3})$.  At a point where $e_0=0$, both
+$t-1$ and $l-t=we$ are $O(e^{3/2})=o(r^{-3})$, and the same conclusion
+holds.  Uniformity follows by splitting into the sets where $e_0$
+is larger or smaller than an arbitrary fixed positive threshold.
+Since $t=1-e^{3/2}/2+O(e^3)$, this proves
+\eqref{eq:stress-end-asymptotics}, as well as
+$E=O(r^{-2})$ and $\Pi-g=O(r^{-3})$ in metric norm.
+
+\emph{Flux saturation.}
+Continuous distributionally divergence-free fields have the usual
+flux identities on smooth compact domains: use smooth cutoffs
+approaching their boundaries and continuity in the shrinking collars.
+This proves independence in \eqref{eq:stress-charge-definition}.
+The asymptotics and $dA_g/r^2\to dA_\sigma$ give
+$F_{M_e}\le\int e_0\,dA_\sigma$.  Integration by parts in the
+electrostatic equation gives
+\begin{equation}
+ M_eF_{M_e}=\int_\Omega w e(w)\,\dd V_g.
+ \label{eq:stress-energy-charge}
+\end{equation}
+Indeed the infinity term tends to zero because
+$u=O(r^{-1})$, $E=O(r^{-2})$, and the sphere area is $O(r^2)$.
+The inner term has the indicated sign: the outward normal of the
+exterior at $S$ is $-\nu$, while $u=-M_e$ there.  The integral is
+finite since $we=O(r^{-3})$ and $dV_g\asymp r\,dr\,dA_\sigma$.
+
+Fix a large $R_0$, choose $4/3<s_0<2$, and set
+\begin{equation}
+ \zeta=1\quad(r\le R_0),\qquad
+ \zeta=(R_0/r)^{s_0}\quad(r>R_0),\qquad
+ C_\zeta=\int_\Omega(1-t(\zeta))\,\dd V_g<\infty.
+ \label{eq:stress-charge-weight}
+\end{equation}
+The last integrability follows from
+$1-t(\zeta)=O(\zeta^{3/2})$ and $s_0>4/3$.
+For every smooth function $v$ equal to $-M_e$ near $S$ and zero
+outside a compact set, each regular level $v=q$, $-M_e<q<0$, obeys
+\begin{equation}
+ \int_{v=q}\zeta\,\dd A_g
+ \ge(1-o_{R_0}(1))A_*.
+ \label{eq:stress-weighted-level-area}
+\end{equation}
+Here is the topological and geometric comparison for this assertion.
+Take the closure $D$ of the component of $\{v>q\}$ containing the
+end.  It is a connected closed end-side domain, with boundary a
+subset of the level; it misses a neighborhood of all of $S$.
+Choose a sphere $r=R_1$, with $R_1\in[R_0/2,R_0]$, transverse to its
+boundary, and adjoin to $D$ the whole region $r\ge R_1$.
+The union is still connected because the two domains have a common
+far-end region.  Each newly exposed sphere-cap point lies on a
+radial ray meeting the old boundary at some $r\ge R_1$: initially
+the ray is outside $D$, and eventually it lies in $D$.
+Radial projection onto $r=R_1$ has upper two-dimensional Jacobian
+\[
+ (1+o_{R_0}(1))(R_1/r)^2
+ \le(1+o_{R_0}(1))\zeta(r).
+\]
+The first estimate is uniform over tangent two-planes by the
+asymptotic tensor bounds; the second uses $s_0<2$ and $R_1\le R_0$.
+The area formula bounds the new caps by the weighted old boundary
+outside the sphere.  Inside, the weight is one.  The transverse
+corners can be rounded toward the side enlarging this union, in
+shrinking tubular neighborhoods of their intersection curves; the
+added area tends to zero.  The rounded exterior is connected and
+its full intrinsic boundary is an admissible cut in the definition
+of $A_*$.  This proves \eqref{eq:stress-weighted-level-area} without
+any connectedness assertion about the level itself.
+
+Coarea therefore gives
+$\int\zeta|dv|\ge M_e(1-o_{R_0}(1))A_*$.
+The same holds for $u$.  To justify approximation, first replace
+$u$ by its constant trace in a shrinking collar of $S$, interpolating
+in a collar of comparable width.  The error in its weighted gradient
+integral tends to zero by the bounded gradient and constant trace.
+Next multiply by a cutoff tending to zero in $R<r<2R$ and set it
+equal to zero beyond $2R$.  The height and gradient decay make both
+the lost tail and cutoff error $O(R^{1-s_0})$, tending to zero.
+Smooth the remaining compactly supported transition while retaining
+the constant boundary neighborhoods.  This gives convergence in
+the required weighted $W^{1,1}$ norm, and coarea applies to the
+smooth approximants.
+
+Finally, the definition of $L$ gives $L(w)\ge\zeta w+t(\zeta)$,
+whereas $L(w)-1\le we(w)$ since $t(e)\le1$.  Combining these facts
+with \eqref{eq:stress-energy-charge} yields
+\begin{equation}
+ F_{M_e}\ge(1-o_{R_0}(1))A_*-C_\zeta/M_e.
+ \label{eq:stress-charge-lower-bound}
+\end{equation}
+Let $M_e\to\infty$ at fixed $R_0$ and then $R_0\to\infty$.
+This proves \eqref{eq:stress-charge-saturation} and completes the
+proposition.
+\end{proof}
+
+\subsection{Smoothing the synthetic tensor and attaching an exterior}
+
+\begin{proposition}[Compact stress data and AF attachment]
+\label{prop:af-attachment}
+Let $g$ be the fixed approximating metric above.  For every
+$\varepsilon_0>0$ there are a sufficiently large coordinate sphere
+$\mathcal C$, its compact connected inner region $\Omega_i$, a
+smooth symmetric tensor $K$ on $\Omega_i$, and a smooth scalar-flat
+\AF\ exterior $(\Omega_o,g_o)$ attached along $\mathcal C$, with
+the following properties.  Put $\tau_K=\tr_g K$ and
+$k_T=\tr_{\mathcal C}K$.  Then
+\begin{align}
+ K&<0,\qquad
+ R_g+\tau_K^2-|K|_g^2
+   -2(\divg_gK-d\tau_K)\cdot v>0
+       &&(|v|_g\le1),
+ \label{eq:stress-synthetic-DEC}\\
+ H_i&>|k_T|,\qquad
+ H_o=\sqrt{H_i^2-k_T^2}>0,
+ \label{eq:stress-matching-curvatures}\\
+ m_o&\le p_0(g)-\tfrac12 a^3+\varepsilon_0.
+ \label{eq:stress-attachment-budget}
+\end{align}
+Here $H_i$ is the inner metric's mean curvature on $\mathcal C$,
+and $H_o$ is the outer metric's inner-boundary mean curvature, both
+in the direction toward the respective end.  The induced metrics
+agree.  The exterior is complete with its boundary included and
+has one end, with $g_o-\delta=O_k(|x'|^{-1})$ for every fixed $k$,
+where an ordinary derivative improves decay by one power.
+Projection from $\Omega_o$ to $\mathcal C$ along its parallel rays
+has upper area Jacobian at most one on every tangent two-plane.
+\end{proposition}
+
+\begin{proof}
+Choose an electrostatic height $M_e$ large enough that the preceding
+proposition gives
+\begin{equation}
+ \langle e_0\rangle\ge a^2-o(1),\qquad
+ \langle f\rangle=(4\pi)^{-1}\int_{\Sph^2}f\,\dd A_\sigma,
+ \label{eq:stress-charge-average}
+\end{equation}
+where the error can be arbitrarily small.  On the whole exterior
+first define a continuous tensor $K^{(0)}$ by
+\begin{equation}
+ K^{(0)}-(\tr_gK^{(0)})g=2\Pi,
+ \qquad K^{(0)}=2\Pi-(\tr_g\Pi)g.
+ \label{eq:stress-synthetic-tensor}
+\end{equation}
+Its eigenvalues are $t-2l=-t-2we<0$ in the gradient direction and
+$-t<0$ transversely.  Directly from
+\eqref{eq:stress-algebraic-identity},
+\begin{equation}
+ (\tr_gK^{(0)})^2-|K^{(0)}|_g^2=6,
+ \qquad
+ \divg_g\big(K^{(0)}-(\tr_gK^{(0)})g\big)=0
+ \label{eq:stress-synthetic-identities}
+\end{equation}
+in distributions.  For example, if $s_\Pi=\tr_g\Pi$, then
+$\tr_gK^{(0)}=-s_\Pi$ and
+$|K^{(0)}|^2=4|\Pi|^2-s_\Pi^2$, giving the first identity.
+
+On a coordinate sphere of radius $r$, the induced metric is
+$\phi^4r^2\sigma$, hence its Gauss curvature is positive for large
+$r$.  The conformal mean-curvature formula gives
+\begin{align}
+ H_i
+ &=\phi^{-2}\left(\frac{2\sqrt{1+r^2}}r
+       +4\sqrt{1+r^2}\,\partial_r\log\phi\right)\notag\\
+ &=2+r^{-2}-16\alpha r^{-3}+o(r^{-3}).
+ \label{eq:stress-inner-H-expansion}
+\end{align}
+The relation \eqref{eq:stress-synthetic-tensor} gives exactly
+$\tr_{\mathcal C}K^{(0)}=-2\Pi(\nu,\nu)$, so
+\eqref{eq:stress-end-asymptotics} yields
+\begin{equation}
+ k_T^{(0)}=-2+e_0^{3/2}r^{-3}+o(r^{-3}).
+ \label{eq:stress-tangential-K-expansion}
+\end{equation}
+All these remainders are uniform in angle.  Consequently
+$H_i>|k_T^{(0)}|$ on every sufficiently large sphere, and
+\begin{equation}
+ \sqrt{H_i^2-(k_T^{(0)})^2}
+ =\frac2r+\frac{-16\alpha+e_0^{3/2}}{r^2}+o(r^{-2}).
+ \label{eq:stress-outer-H-expansion}
+\end{equation}
+Indeed the expression under the root is
+$4r^{-2}+(-64\alpha+4e_0^{3/2})r^{-3}+o(r^{-3})$.
+
+We explain carefully the smoothing used after fixing such a sphere
+$\mathcal C$.  On the compact $\Omega_i$, continuity and strict
+positivity give positive lower bounds for the eigenvalues of $\Pi$,
+the absolute negative eigenvalues of $K^{(0)}$, and $R_g+6$.
+There are smooth tensors $\Pi_h$ on $\Omega_i$, up to its boundary,
+such that
+\begin{equation}
+ \|\Pi_h-\Pi\|_{C^0}\longrightarrow0,
+ \qquad \|\divg_g\Pi_h\|_{C^0}\longrightarrow0.
+ \label{eq:stress-divergence-smoothing}
+\end{equation}
+Here is the needed commutator proof, including the boundary.
+In an interior chart convolve the tensor components with a smooth
+kernel $\eta_h$.  A divergence term has form
+$a(x)\partial_jT+b(x)T$, with smooth coefficients.  For the principal
+part, integration by parts in the commutator produces
+$(a(y)-a(x))\partial\eta_h(x-y)$ and a term with
+$(\partial a)(y)\eta_h(x-y)$.  Together these cancel on constant
+$T$.  After subtracting $T(x)$ their integral is bounded by a fixed
+constant times the modulus of continuity of $T$ on a ball of radius
+$O(h)$, and hence tends uniformly to zero.  The lower-order terms
+commute up to the same type of error.
+
+In a boundary chart evaluate the convolution instead at
+$x+hc_0n_0$, with $n_0$ a constant inward coordinate direction and
+$c_0$ large enough that the kernel stays on the original side.
+The same proof applies: $|a(y)-a(x)|=O(h)$ throughout its support,
+and the combined commutator still vanishes on a constant tensor.
+It therefore extends the uniform estimate to the boundary without
+extending the original stress as a divergence-free tensor.  Take
+these local approximations on slightly larger charts and recombine
+with a smooth partition of unity.  The extra differentiated
+partition terms tend to zero, because their limiting sum is
+$(\sum d\chi)\Pi=0$.  This proves
+\eqref{eq:stress-divergence-smoothing}.
+
+Set $K_h=2\Pi_h-(\tr_g\Pi_h)g$.  Then
+$\divg K_h-d\tr K_h=2\divg\Pi_h\to0$ uniformly, and the algebraic
+part $R_g+(\tr K_h)^2-|K_h|^2$ tends uniformly to $R_g+6>0$.
+For sufficiently small $h$, \eqref{eq:stress-synthetic-DEC} follows
+uniformly for $|v|\le1$, as does $K_h<0$.  The strict gap
+$H_i>|k_T^{(0)}|$ at the now fixed sphere also persists.
+Write $K$ for this smooth tensor.  The error in any integral
+involving $\sqrt{H_i^2-k_T^2}$ can be made arbitrarily small at this
+fixed sphere, because the square root is smooth on its positive
+gap.  This order of choices is essential; we require no smoothing
+estimate uniform as the sphere tends to infinity.
+
+The positive-Gauss metric on $\mathcal C\simeq\Sph^2$ has a smooth
+strictly convex Euclidean isometric embedding by the smooth Weyl
+embedding theorem \citep{Nirenberg1953}.  This is precisely the
+spherical positive-curvature version also stated in the introduction
+of \citet{ShiTam2002}.  Denote its outward Euclidean mean curvature
+by $H_E$.  We will construct the Shi--Tam scalar-flat extension with
+the smooth positive prescribed boundary curvature
+$H_o=\sqrt{H_i^2-k_T^2}$.  The required estimate is
+\begin{equation}
+ m_o\le\frac1{8\pi}\int_{\mathcal C}(H_E-H_o)\,\dd A.
+ \label{eq:stress-shitam-mass-bound}
+\end{equation}
+Before proving this extension assertion, we evaluate its right side
+and record the exact coefficient of the charge term.
+
+Scale the induced metric on $\mathcal C$ by $r^{-2}$.
+Its area is $4\pi+O(r^{-3})$ and its Gauss curvature is
+$1+O(r^{-3})$, with smooth angular control.  For its convex Euclidean
+embedding choose an interior point as origin and use Gauss-map
+coordinates $\omega$.  Let $l(\omega)>0$ be the support function.
+Then the embedding is $l\omega+\nabla_\sigma l$, its inverse shape
+operator is $\nabla_\sigma^2l+l\sigma$, and
+\begin{equation}
+ \int H_E^{\mathrm{scaled}}\,\dd A
+   =\int_{\Sph^2}\tr_\sigma(\nabla_\sigma^2l+l\sigma)\,\dd A_\sigma
+   =2\int_{\Sph^2}l\,\dd A_\sigma.
+ \label{eq:stress-support-mean}
+\end{equation}
+The scaled enclosed volume is
+$\frac13\int l\,dA$, and $dA=(1+O(r^{-3}))dA_\sigma$ by the Gauss
+curvature estimate.  Euclidean isoperimetry bounds this volume by
+$4\pi/3+O(r^{-3})$.  Positivity of $l$ now implies
+$\int l\,dA_\sigma\le4\pi+O(r^{-3})$.  Scaling back in
+\eqref{eq:stress-support-mean} gives
+\begin{equation}
+ \int_{\mathcal C}H_E\,\dd A\le8\pi r+O(r^{-2}).
+ \label{eq:stress-Euclidean-H-bound}
+\end{equation}
+This uses no quantitative stability theorem for the Weyl embedding.
+
+Before smoothing the stress,
+\eqref{eq:stress-outer-H-expansion} and
+$dA=\phi^4r^2dA_\sigma$ give
+\[
+ \int_{\mathcal C}H_o\,\dd A
+ =8\pi r+\int_{\Sph^2}(-16\alpha+e_0^{3/2})\,\dd A_\sigma+o(1).
+\]
+The smoothing error in this equality can be made arbitrarily small
+after fixing $r$.  Moreover, direct substitution of
+$g-b=(4\alpha r^{-3}+O(r^{-3-\xi}))b$ in the mass flux gives
+\begin{equation}
+ p_0(g)=8\langle\alpha\rangle.
+ \label{eq:stress-conformal-mass}
+\end{equation}
+For clarity, if $e=fb$, the flux one-form is
+$-2(V_0\,df-f\,dV_0)$; for $f=4\alpha r^{-3}$ its radial sphere
+integral tends to $32\int\alpha\,dA_\sigma$.
+Consequently \eqref{eq:stress-Euclidean-H-bound} gives
+\begin{align}
+ \frac1{8\pi}\int_{\mathcal C}(H_E-H_o)\,\dd A
+ &\le p_0(g)-\tfrac12\langle e_0^{3/2}\rangle+o(1)\notag\\
+ &\le p_0(g)-\tfrac12\langle e_0\rangle^{3/2}+o(1)
+ \le p_0(g)-\tfrac12a^3+o(1).
+ \label{eq:stress-cubic-budget}
+\end{align}
+The second inequality is Jensen's inequality.  If $a=0$, its charge
+term is simply nonnegative.  In all cases first choosing $M_e$,
+then $r$, then the compact smoothing tolerance makes the final error
+smaller than the prescribed $\varepsilon_0$.
+
+\emph{The scalar-flat extension and its mass.}
+We include the extension argument to fix both the existence hypotheses
+and the normalization in \eqref{eq:stress-shitam-mass-bound}.  It is
+the quasi-spherical construction of \citet[Section~2, Theorem~2.1,
+and Lemma~4.2]{ShiTam2002}.  The input here is exactly a smooth
+strictly convex Euclidean sphere and a smooth positive prescribed
+mean curvature; no compact fill-in is used.
+
+Let $t\ge0$ be Euclidean distance along the outward normals to the
+convex embedding.  The parallels foliate its entire exterior.  Write
+the Euclidean metric as $dt^2+\gamma_t$, and denote the Euclidean
+mean curvature and surface scalar curvature by $H_E(t)$ and $R_t$.
+Both are positive.  If $\lambda_1,\lambda_2>0$ are the initial
+principal curvatures, then those at time $t$ are
+$\lambda_j/(1+t\lambda_j)$, whence
+\begin{equation}
+ \frac{R_t}{2H_E(t)}
+ =\frac1{2t+c(\omega)},\qquad
+ c=\lambda_1^{-1}+\lambda_2^{-1}>0.
+ \label{eq:stress-parallel-curvature}
+\end{equation}
+Solve the scalar parabolic equation
+\begin{equation}
+ 2H_E\partial_tU
+ =2U^2\Delta_{\gamma_t}U+(U-U^3)R_t,
+ \qquad U(0,\cdot)=H_E(0,\cdot)/H_o.
+ \label{eq:stress-quasispherical-equation}
+\end{equation}
+Smooth positive initial data give a local solution by strictly
+parabolic theory.  The reaction term is nonpositive above one and
+nonnegative below one.  The maximum principle therefore keeps $U$
+between fixed positive constants.  On every finite time interval
+the equation is uniformly parabolic with smooth background geometry.
+Scalar parabolic H\"older estimates for bounded solutions, followed
+by Schauder estimates and bootstrapping, give continuation for all
+finite times.
+
+For the asymptotic estimate, choose positive constants respectively
+above and below the entire initial range and one.  Solve, for each
+of these constants as initial value,
+\begin{equation}
+ \dot z=(z-z^3)/(2t+c_{\max}),\qquad
+ c_{\max}=\max c.
+ \label{eq:stress-parabolic-barriers}
+\end{equation}
+By \eqref{eq:stress-parallel-curvature} these are respectively upper
+and lower barriers: the sign of $z-z^3$ reverses the inequality in
+the upper case.  Since
+\[
+ z^{-2}-1=(z(0)^{-2}-1)\frac{c_{\max}}{2t+c_{\max}},
+\]
+comparison gives $U-1=O((1+t)^{-1})$.
+Put $s=\log(1+t)$ and $\widehat\gamma_t=(1+t)^{-2}\gamma_t$.
+In the Gauss-map parametrization the parallel embedding is
+$X(\omega)+t\omega$, so these scaled metrics and all their
+derivatives have bounded geometry.  Equation
+\eqref{eq:stress-quasispherical-equation} in $s$ is uniformly
+parabolic.  Its local H\"older and then Schauder estimates on fixed
+$s$-strips first bound all derivatives of $U$ uniformly.  Next apply
+the linear estimates to $U-1$ with the actual coefficients of this
+solution; its equation is homogeneous, with a bounded smooth
+zeroth-order coefficient.  The supremum bound just proved gives
+the same $O(e^{-s})$ decay at every fixed derivative order.  This
+proves the stated ordinary AF derivative bounds in the Euclidean
+coordinates $x'=X(\omega)+t\omega$.
+
+Set
+\begin{equation}
+ g_o=U^2dt^2+\gamma_t.
+ \label{eq:stress-af-extension-metric}
+\end{equation}
+It has the required induced metric and inner mean curvature
+$H_E/U=H_o$.  Its scalar curvature is, by the hypersurface Gauss
+and mean-variation formulas,
+\begin{equation}
+ R_{g_o}=R_t(1-U^{-2})+2H_EU^{-3}\partial_tU
+                    -2U^{-1}\Delta_{\gamma_t}U=0,
+ \label{eq:stress-af-scalar-zero}
+\end{equation}
+where the final equality is exactly
+\eqref{eq:stress-quasispherical-equation}.  The positive bounds on
+$U$ and the Euclidean completeness of the parallel exterior give
+completeness with the boundary included.  There is precisely one
+end, and $g_o-\delta=(U^2-1)dt^2$ has the stated decay.
+
+Define
+\[
+ \mathcal M(t)=\int_{\mathcal C_t}H_E(1-U^{-1})\,\dd A_{\gamma_t}.
+\]
+The Euclidean identities
+$\partial_tH_E=-|\mathrm{II}_E|^2$,
+$\partial_t dA=H_EdA$, and
+$H_E^2-|\mathrm{II}_E|^2=R_t$, together with
+\eqref{eq:stress-quasispherical-equation}, give
+\begin{align}
+ \mathcal M'(t)
+ &=\int_{\mathcal C_t}\left[
+       R_t(1-U^{-1})+H_EU^{-2}\partial_tU\right]\,\dd A\notag\\
+ &=-\frac12\int_{\mathcal C_t}R_t(U+U^{-1}-2)\,\dd A\le0.
+ \label{eq:stress-shitam-monotonicity}
+\end{align}
+Here the integral of $\Delta_{\gamma_t}U$ vanishes.  The asymptotic
+bounds show that $\mathcal M$ is bounded and that its derivative
+is integrable at infinity, so it has a finite limit.
+
+We identify the limit with the ADM mass in the normalization
+\eqref{eq:adm-mass}.  For the purely normal Euclidean metric error
+$e'=(U^2-1)dt^2$, the ADM flux through the parallel surface is exactly
+\[
+ \int_{\mathcal C_t}
+   (\divg_\delta e'-d\tr_\delta e')(\partial_t)\,\dd A
+ =\int_{\mathcal C_t}H_E(U^2-1)\,\dd A.
+\]
+The normal derivatives of $U^2-1$ cancel, and
+$\nabla_{\partial_t}\partial_t=0$ in the Euclidean metric, leaving
+only the tangential divergence $H_E$.  The scalar-curvature
+linearization and $R_{g_o}=0$ show that the divergence of this
+Euclidean flux is $O(|x'|^{-4})$.  Its integral between a large
+parallel and a comparable large round sphere tends to zero; thus
+it computes the ordinary ADM mass.  Finally
+\[
+ (U^2-1)-2(1-U^{-1})=O((U-1)^2)
+\]
+and $H_E\,\Area(\mathcal C_t)=O(t)$, so
+\begin{equation}
+ 16\pi m_o
+ =\lim_{t\to\infty}\int_{\mathcal C_t}H_E(U^2-1)\,\dd A
+ =2\lim_{t\to\infty}\mathcal M(t)
+ \le2\mathcal M(0).
+ \label{eq:stress-adm-identification}
+\end{equation}
+As $H_E/U(0)=H_o$, this proves
+\eqref{eq:stress-shitam-mass-bound}.  Combining it with
+\eqref{eq:stress-cubic-budget} gives
+\eqref{eq:stress-attachment-budget}.
+
+Lastly, convexity gives
+$\gamma_t(V,V)=\gamma_0((\Id+t\mathrm{II}_E^{\sharp})V,
+(\Id+t\mathrm{II}_E^{\sharp})V)\ge\gamma_0(V,V)$.
+The metric in \eqref{eq:stress-af-extension-metric} has no mixed
+normal--tangential terms.  Its projection along the parallels to
+$(\mathcal C,\gamma_0)$ therefore has operator norm at most one,
+and hence upper two-dimensional Jacobian at most one.  This is
+the area comparison required in the later transfer argument.
+\end{proof}
+
+All data in Proposition~\ref{prop:af-attachment} are now fixed before
+the compact graph parameters are chosen.  On $S$ and $\mathcal C$
+we use the base normals directed toward the end; derivatives and
+fluxes on the two sides of $\mathcal C$ use $g$ and $g_o$ respectively.
+Their boundary area forms agree.  In the next section the notation
+$\tau=\tr_gK$ refers to the synthetic trace, not to the original
+asymptotic decay exponent.

preprints/The-Penrose-inequality-for-maximal-asymptotically-hyperbolic-initial-data-October-5-2026/build/sections/07-compact-construction.tex

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+\section{The compact graph construction}\label{sec:compact-construction}
+
+Fix the compact data and the asymptotically flat attachment supplied by
+Proposition~\ref{prop:af-attachment}. Thus the connected compact inner
+manifold $(\Omega_i,g)$ has boundary $S\sqcup\mathcal C$, where
+$\mathcal C$ is a sphere, and carries a smooth negative definite tensor
+$K$. Put $\tau=\tr_g K$. The smooth exterior $(\Omega_o,g_o)$ is scalar
+flat and asymptotically flat. The induced metrics on $\mathcal C$ agree.
+All normals $n$ at $S$ and $\mathcal C$ point towards the designated end;
+in particular the outward normal of $\Omega_i$ at $S$ is $-n$. Write
+$k_T=\tr_{T\mathcal C}K$ on the seam and use the analogous notation on
+$S$. The fixed data satisfy
+\begin{equation}\label{eq:compact-base-data}
+ \begin{gathered}
+ H_g(S)\le0,\qquad H_i>|k_T|,\qquad
+ H_o=\sqrt{H_i^2-k_T^2},\\
+ D(v):=R_g+\tau^2-|K|^2-2(\divg_gK-\dd\tau)(v)>0
+ \quad (|v|_g\le1).
+ \end{gathered}
+\end{equation}
+No deformation parameter changes these base data. In this section every
+integral without a metric subscript uses the base metric on its own side.
+Piecewise integrals mean the sum of the two such integrals. Constants may
+depend on these fixed smooth data.
+
+On $\Omega_i$ define the fields, for $N_i>0$, by
+\begin{equation}\label{eq:compact-constitutive}
+ \begin{gathered}
+ p=s^2\nabla f,\qquad s^2+|\dd f|^2s^4=N_i^2,\qquad v=p/N_i,\\
+ z=|v|,\quad y=z^2,\quad d=\sqrt{1-y},\quad W=d^{-1}=N_i/s.
+ \end{gathered}
+\end{equation}
+The positive root specifies $s$ smoothly, including at $\dd f=0$.
+On a constant-height boundary we write $\widehat z=v\cdot n$.
+On $\Omega_o$ put $f=p=0$, $s=N_o$, $W=1$, and $\bar g=g_o$;
+inside put $\bar g=g+s^2\dd f^2$. On either side set
+$x=sw$, $h=x\bar g$, and $A=Ns\bar g^{-1}$, with $N=N_i$ or $N_o$.
+Here $d$ is a scalar, whereas $\dd$ denotes differentiation.
+
+For a finite height $M\ge1$ and a homotopy parameter $u_*\in[0,1]$ the
+equations are
+\begin{equation}\label{eq:compact-system}
+\begin{aligned}
+ \divg_g p&=TN_i &&\text{on }\Omega_i,\\
+ B&=\nabla N_i-(u_*K+\ell g)p,\qquad
+ \ell=T-u_*\tau &&\text{on }\Omega_i,\\
+ B&=\nabla N_o,\qquad \Sigma_-=0 &&\text{on }\Omega_o,\\
+ \divg B&=\Sigma_++\Sigma_-,\qquad
+ -\divg(A\dd w)=\Sigma_+x+P &&\text{on each side}.
+\end{aligned}
+\end{equation}
+We first truncate the exterior at a smooth distant coordinate sphere
+$\mathcal S_R$. The complete boundary conditions are
+\begin{equation}\label{eq:compact-boundary}
+\begin{array}{c|l}
+ S&f=u_*M,\quad B_n=-b_*,\quad (A\dd w)_n=0,\\
+ \mathcal C&f=0,\quad N_o=s_i,\quad (B_i)_n=(B_o)_n,
+       \quad w_i=w_o,\quad (A_i\dd w_i)_n=(A_o\dd w_o)_n,\\
+ \mathcal S_R&N_o=w=1.
+\end{array}
+\end{equation}
+Here $b_*>0$ is fixed throughout the homotopy. Eventually $R\to\infty$
+with every other parameter fixed, and the last row becomes $N_o,w\to1$.
+
+We specify all cutoffs used in these equations. Let $d_S$ be a fixed
+smooth function, positive away from $S$ and equal to inward distance on
+a collar of $S$. Set $F=u_*M-f$. The quotient $F/d_S$ at $S$ is its
+continuous extension. For example, in that collar it equals
+$\int_0^1\partial_nF(q,td_S)\,\dd t$; hence it has one fewer controlled
+derivative than $F$ in the relevant H\"older spaces. Choose smooth
+cutoffs whose product $\chi_T$ is supported where
+\begin{equation}\label{eq:compact-trace-cutoff}
+ F<M/10,\qquad z<1/2,\qquad FN_i\sqrt{x}<M^{-1},\qquad
+ N_iF/d_S<M^{-1},
+\end{equation}
+and equals one if all four upper bounds hold with an additional factor
+$1/2$. The slope cutoff is a smooth function of $z^2$. The other scalar
+cutoffs are extended as one for sufficiently negative arguments. Define
+\begin{equation}\label{eq:compact-sources}
+ T=u_*\tau(1-\chi_T),\qquad
+ \Sigma_-=-\tfrac12N_i\ell^2\zeta(N_i),\qquad
+ \Sigma_+=u_*LW\eta_M\chi_x(x)\chi_N(N/\eps).
+\end{equation}
+The fixed smooth function $\zeta$ lies in $[0,1]$, equals one for
+$N\le\delta_0$, and vanishes for $N\ge2\delta_0$.
+Here $0<\delta_0,b_*,\delta<1$, $L\ge1$, $0<\eps<1$;
+$\chi_x=1$ on $x\le1-\delta$ and is zero on $x\ge1-\delta/2$,
+whereas $\chi_N=0$ on $N\le\eps$ and is one on $N\ge2\eps$.
+In particular
+\begin{equation}\label{eq:compact-source-size}
+ u_*\tau\le T\le0,\quad |\ell|\le C,\quad
+ -C N_i\le\Sigma_-\le0,\quad
+ \|\Sigma_-\|_{L^1}\le C\delta_0,\quad 0\le\Sigma_+\le LW.
+\end{equation}
+
+The spatial cutoff $\eta_M$ is chosen as follows. Let $\psi_\infty$
+be the exterior capacitary potential, equal to one on $\mathcal C$ and
+zero at infinity. Fix a large radius $R_\delta$ beyond which
+$\psi_\infty<\delta$. Such a potential and radius follow from scalar
+Dirichlet theory and the barriers $C|x'|^{-\beta}$, $0<\beta<1$,
+on the fixed asymptotically flat end. Fix an exterior collar width
+$a_o>0$. The designated set $\mathcal D_M$ consists of all of
+$\Omega_i$ except an inner seam collar of width $2M^{-4}$, together
+with the exterior between collar distance $2a_o$ and radius
+$R_\delta$. Enlarge $R_\delta$ if necessary to contain that collar.
+Take $\eta_M=1$ on a neighborhood of $\mathcal D_M$, supported away
+from the seam, with inner separation at least $M^{-4}$ and exterior
+separation at least $a_o$, and with support in a common compact region.
+All spatial cutoffs below have slightly nested supports of this kind.
+Thus $N_o$ is harmonic on the fixed exterior seam collar. The choice of
+$R_\delta$ does not involve an unknown solution: the exterior equations
+give $N_o\Delta_{g_o}x=-P\le0$, so comparison gives
+\begin{equation}\label{eq:compact-capacity-comparison}
+ x\ge1-\psi_\infty \quad\text{on }\Omega_o.
+\end{equation}
+The same comparison holds on truncations since the right side is at
+most one on $\mathcal S_R$. Consequently every exterior point with
+$x<1-\delta$ lies inside the fixed radius $R_\delta$.
+
+There are two further sources, $P=u_*(P_1+P_2)$. Put
+$T_1=M^{9/10}$ and $T_2=C_2/\eps$. The source $P_1$ is an adjustable
+finite constant $D_1$ times smooth cutoffs, supported inside where
+$f>M/2$ and $w<4T_1$, and equal to $D_1$ where
+$f\ge4M/5$ and $w\le2T_1$. The source $P_2$ is $D_2$ times smooth
+cutoffs on the two spatial supports just described, vanishes if
+$N\ge4\eps$ or $w\ge4T_2$, and equals $D_2$ on a neighborhood of
+$\mathcal D_M$ where $N\le3\eps$ and $w\le2T_2$.
+The constants $C_2,D_1,D_2$ will be chosen below. Every equation is a
+smooth finite equation on the open domain $N_i,N_o,w>0$.
+
+\begin{proposition}[Finite compact construction]\label{prop:compact-solutions}
+Fix base data satisfying \eqref{eq:compact-base-data}. Fix
+$b_*,\delta_0,\delta,a_o>0$ and $L\ge1$. For every sufficiently large
+finite $M$, $C_2$ can be chosen depending on these parameters, and then
+$\eps>0$ can be chosen arbitrarily small and $D_1,D_2$ sufficiently
+large but finite, so that \eqref{eq:compact-system}--\eqref{eq:compact-boundary}
+at $u_*=1$ has a positive solution on the full attached exterior.
+It is smooth on each closed side of $\mathcal C$ and tends to the
+specified values at infinity. It satisfies
+\begin{equation}\label{eq:compact-bounds}
+ \begin{gathered}
+ 0\le f\le M,\qquad N_i,N_o\le C(L)M,\qquad
+ \int_{\Omega_i}W^2\le C(L)M^2,\\
+ N_i\ge c\eps,\qquad s_i,N_o\ge c(M,L)\eps.
+ \end{gathered}
+\end{equation}
+The constants in these bounds do not depend on $\eps,D_1,D_2$ or the
+outer truncation. Dependence on the fixed data and on the preceding
+parameters is suppressed. At fixed $M,L,\eps$, the lapse and height
+have uniform local continuity bounds off the seam, including at $S$,
+independent of $D_1,D_2$. Moreover
+\begin{equation}\label{eq:compact-penalty-targets}
+ w\ge M^{9/10}\quad\text{where }f\ge9M/10,
+ \qquad x\ge1\quad\text{on }\mathcal D_M\cap\{N\le2\eps\},
+\end{equation}
+and
+\begin{equation}\label{eq:compact-penalty-cost}
+ \int P_1\le C(L)M^{-1/10},\qquad
+ \int P_2\le\omega_M(\eps),\qquad \omega_M(\eps)\longrightarrow0
+ \quad(\eps\downarrow0).
+\end{equation}
+The function $\omega_M$ is uniform in the finite strengths and outer
+radius. In particular
+\begin{equation}\label{eq:compact-bad-volume}
+ \int_{\mathcal D_M\cap\{x<1-\delta\}}W\le C/L,
+\end{equation}
+where $C$ is independent of $L,M,\eps,D_1,D_2$. On the smooth sides the
+metric $h=x\bar g$ obeys the scalar and boundary identities
+\eqref{eq:compact-scalar} and \eqref{eq:compact-mean-curvature} below.
+Its asymptotically flat mass satisfies
+\begin{equation}\label{eq:compact-mass-budget}
+ m_h\le m_o+C(b_*+\delta_0)+C(L)M^{-1/10}+C\omega_M(\eps).
+\end{equation}
+Thus the parameter order is: fix the base data; choose $b_*,\delta_0$
+for the mass tolerance; choose $\delta,a_o$ and then $L$ for the area
+tolerance; choose $M$ large; choose $C_2$; choose $\eps$ small; choose
+the finite strengths; and finally let the outer radius tend to infinity.
+\end{proposition}
+
+We prove estimates for positive zeros in the H\"older domain used for
+continuation in Section~\ref{sec:transmission-analysis}, before invoking
+existence.  Thus $f\in C^{3,\alpha'}(\overline\Omega_i)$ and
+$N,w\in C^{2,\alpha'}$ on each closed side, for the fixed positive
+exponent $\alpha'$ selected there.  This is membership in the domain,
+not an a priori bound on its norm.  No a priori derivative bound on a
+cutoff is used in the following finite estimates.
+
+\begin{lemma}[Lapse and slope bounds]\label{lem:compact-lapse}
+Every such positive zero, for $0\le u_*\le1$ and sufficiently large
+outer radius, satisfies \eqref{eq:compact-bounds}, with $M$ replacing
+the upper height bound by $u_*M$. These estimates are independent of
+the strengths of $P_1,P_2$ and of all derivatives of $T$.
+\end{lemma}
+
+\begin{proof}
+The minimum principle for the height equation gives $f\ge0$ because
+$T\le0$. A maximum exceeding $u_*M$ has $F<0$ and $z=0$, so every
+trace cutoff is fully active in its neighborhood and $T=0$ there.
+The strong maximum principle, applied on the exceeding set, excludes
+such a maximum. Hence $0\le f\le u_*M$.
+
+Let $\psi_R$ be harmonic on the truncated exterior, one on
+$\mathcal C$ and zero on $\mathcal S_R$, and set
+$\gamma_R=-\partial_n\psi_R|_{\mathcal C}$. The functions $\gamma_R$
+have uniform positive upper and lower bounds. Also $\psi_R$ has a
+uniform positive lower bound on all exterior source supports. To check
+uniformity, compare $\psi_R$ from below with a fixed finite annular
+capacitary solution and from above with $\psi_\infty$, and apply
+boundary estimates and the Hopf lemma on a fixed collar. Green's
+formula on the exterior and the divergence theorem on the inside give
+\begin{equation}\label{eq:compact-capacity}
+ \int_{\Omega_i}\Sigma_+
+ +\int_{\Omega_o}\psi_R\Sigma_+
+ +\int_{\mathcal C}\gamma_Rs_i
+ =\int_{\mathcal C}\gamma_R+\int_S b_*
+      -\int_{\Omega_i}\Sigma_-\le C.
+\end{equation}
+For example the seam contribution in the exterior Green formula is
+$-B_n-\gamma_RN_o$, and its far contribution is
+$-\partial_n\psi_R$; the harmonic flux identity for $\psi_R$ gives
+the displayed signs. The constant uses only
+$\int|\Sigma_-|\le C\delta_0$. In particular $\min_{\mathcal C}s_i\le C$.
+
+Testing the height equation with $f$ and using its two constant traces
+gives, since $p\cdot\dd f=W^2-1$,
+\begin{equation}\label{eq:compact-height-energy}
+ \int_{\Omega_i}(W^2-1)
+ =-u_*M\int_S p_n-\int_{\Omega_i}fTN_i
+ \le CM\sup_{\Omega_i}N_i.
+\end{equation}
+
+For the lower lapse estimate freeze
+$Z=(u_*K+\ell g)v$, which satisfies $|Z|\le C$, and freeze the positive
+seam function $d=s_i/N_i\le1$. Solve the homogeneous density problem
+with current $\nabla\widehat N_i-Z\widehat N_i$ inside, ordinary
+gradient outside, divergence zero, zero current flux at $S$, matching
+current at the seam, $\widehat N_o=d\widehat N_i$, and outer value one.
+This is an independent linear problem. Its scalar Fredholm assertion
+is proved in Lemma~\ref{lem:transmission-fredholm}, using only frozen
+linear estimates, without the present nonlinear estimates. Its adjoint
+has operator $\Delta+Z\cdot\nabla$ inside, ordinary Laplacian outside,
+ordinary Neumann condition at $S$, continuous value at $\mathcal C$,
+and $\partial_n h_i=d\partial_n h_o$ there. The maximum and Hopf
+principles give zero adjoint kernel, hence invertibility. Dual testing
+with nonpositive scalar forcing gives $\widehat N\ge0$. Harnack and
+the Hopf lemma give strict positivity, also at the seam: a zero there
+would be a zero on both sides and the opposite Hopf derivative signs
+contradict current matching. The H\"older domain gives
+$d\in C^{2,\alpha'}$ and $Z\in C^{1,\alpha'}$ (in fact
+$Z\in C^{2,\alpha'}$), so the independent scalar lemma applies
+separately to each frozen problem. Its inverse may depend on these
+coefficient norms. The uniform estimate below uses only $|Z|\le C$
+and $0<d\le1$ and does not use those inverse norms.
+
+Put $H_*:=\sup\widehat N_i$. The homogeneous version of
+\eqref{eq:compact-capacity} implies $H_*\ge1$, and exterior comparison
+gives $\widehat N_o\le H_*$. We claim, uniformly in the frozen fields,
+\begin{equation}\label{eq:compact-homogeneous-harnack}
+ \inf_{\Omega_i}\widehat N_i\ge cH_*.
+\end{equation}
+Here it is essential that no lower bound on $d$ is required. In Fermi
+coordinates on the two sides reflect the normalized exterior field
+into the inside and add it to $\widehat N_i/H_*$. The two coefficient
+matrices, including volume densities, agree on the face and differ by
+$O(|t|)$ at distance $|t|$ from it. For the positive exterior harmonic
+field, interior gradient estimates and Harnack on balls of radius
+comparable to $|t|$ give
+$|t|\,|\nabla\widehat N_o|\le C\widehat N_o$.
+Consequently the sum $U$ satisfies a uniformly elliptic divergence
+equation with flux correction bounded by $CU$ and zero natural flux
+on the face. Expressing that correction as a bounded vector times $U$
+and reflecting again gives the ordinary divergence drift Harnack
+inequality. Harnack chains on the fixed inner region, together with
+this collar estimate and normalization by the inner supremum, give a
+uniform positive lower bound for $U$ on the seam. The inner trace is
+at least $U/2$, because $\widehat N_o=d\widehat N_i\le\widehat N_i$.
+
+To carry this trace bound uniformly into a collar, take a smooth small
+half-domain of radius $h$ in an inner chart and replace the normalized
+inner field by its harmonic function with the same boundary data.
+The replacement error is at most $Ch$: the divergence forcing
+$Z\widehat N_i/H_*$ is bounded, and scaled zero-Dirichlet
+$W^{1,p}$ estimates with $p>3$ give precisely that bound. The harmonic
+replacement has a fixed positive lower bound centrally up to its
+flat face, by the lower face data and nonnegative remaining data.
+First choose $h$ small enough and then use interior Harnack chains.
+This proves \eqref{eq:compact-homogeneous-harnack}.
+
+The ratio $r_N=N/\widehat N$ is continuous across the seam. On a
+sublevel $r_N<\eps/H_*$ we have $N<\eps$, so $\Sigma_+=0$ and
+$\Sigma_-\le0$. Dividing the density equation by $\widehat N$ gives
+a scalar supersolution inequality with no zeroth order term.
+At the seam its normal derivatives match with positive weights
+$\widehat N_i,\widehat N_o$; the flux $-b_*$ at $S$ has the favorable
+sign for a minimum. The minimum and boundary point principles exclude
+a minimum below $\eps/H_*$. Therefore
+$N_i\ge(\eps/H_*)\widehat N_i\ge c\eps$.
+
+Next suppose, with $L$ fixed, that the upper estimate fails. Along a
+sequence write $N_*:=\sup N_i$ with $N_*/M\to\infty$. The exterior
+maximum is bounded by $\max(N_*,1)$ since $\Delta N_o\ge0$.
+By \eqref{eq:compact-height-energy},
+\[
+ \|\Sigma_+/N_*\|_{L^2(\Omega_i)}
+ \le CL\bigl(N_*^{-2}+M/N_*\bigr)^{1/2}\longrightarrow0.
+\]
+The normalized negative source also tends to zero, and the exterior
+source tends uniformly to zero. Equation~\eqref{eq:compact-capacity}
+shows that $N_o/N_*$ has seam trace tending to zero in $L^1$.
+The normalized exterior field tends to zero on every open exterior
+compact set. Indeed it is bounded above by $N_*^{-1}$ plus the positive
+harmonic extension, decaying at infinity, of its nonnegative seam
+trace. That extension is bounded by $\psi_\infty$ and tends locally
+to zero off the seam by the $L^1$ trace convergence, using a fixed
+annular Poisson estimate and then the decay of $\psi_\infty$.
+
+The reflected sum of the actual normalized lapses is uniformly
+H\"older continuous up to the seam by the preceding reflected-sum
+argument, now with scalar forcing tending to zero in $L^2$.
+Interior Laplace divergence estimates give the same compactness away
+from the seam and at $S$. After passage to a subsequence the open
+inner limit $V$ extends continuously to the seam using the limit of
+the sum. It is nonnegative and nonzero: an inner supremum equal to one
+either lies away from the seam or gives a sum at least one in the
+seam collar. With $Z$ converging weakly star, this limit satisfies the
+homogeneous density equation with natural zero flux on both inner
+faces. Here is a way to justify the latter assertion without individual
+inner trace compactness. Use inner smooth test functions with ordinary
+zero normal derivative at both faces, extended across the seam with
+matching values and outer support in the harmonic collar. Integrate
+the lapse derivatives onto the tests. All outer volume and trace
+terms vanish by the convergence just proved, as do the normalized
+sources and $b_*$. Thus
+\[
+ \int_{\Omega_i}\bigl(V\Delta\varphi+VZ_\infty\cdot\nabla\varphi\bigr)=0
+ \quad\text{for every smooth ordinary Neumann test }\varphi.
+\]
+Solve the Neumann Laplace problem with divergence datum $VZ_\infty$.
+Testing the difference by Neumann Poisson solutions shows that its
+solution differs from $V$ by a constant. Hence $V\in H^1$ and the
+natural zero-flux equation holds in the usual weak sense. Reflection
+and Harnack show $V>0$ on the whole closed inner region. In particular
+the original normalized inner fields are bounded below away from the
+seam; near it the sum, the trace inequality, and the harmonic
+replacement argument above give the same lower bound. The replacement
+error now has the additional vanishing scaled $L^2$ forcing term.
+
+We reach a contradiction by the differential estimate of
+Lemma~\ref{lem:gradient-maximum}. Put $G=-\log s$ and maximize
+$G+j(f)$ on $\Omega_i$, where
+$j(f)=D(f^2-2Mf)$ and $D>0$ is fixed sufficiently large depending on
+$L$. Since $j\le0$ on the height range and $j(0)=0$,
+at the maximum $s\le\min_{\mathcal C}s\le C$. Since
+$N_i\ge cN_*$ there, $y\to1$. At an interior maximum that lemma gives
+\begin{equation}\label{eq:compact-gradient-test}
+ j''q|\dd f|^2\le C\{1+\Sigma_+/N_i+
+                    (1-y)^2(j'|\dd f|)^2\},\qquad
+ q=\frac{1-y}{1+y}.
+\end{equation}
+Use $q|\dd f|^2=J/s^2$, $J=y/(1+y)$,
+$\Sigma_+/N_i\le L/s$, and
+$(1-y)^2(j'|\dd f|)^2\le(j'/N_i)^2=o(1)$.
+Multiplication by $s^2$ contradicts \eqref{eq:compact-gradient-test}
+once $D$ exceeds a fixed multiple of $1+L$ and the fixed upper bounds
+for $s^2$ and $s$ at this maximum.
+
+At a constant-height boundary, write $\widehat z=v\cdot n\le0$.
+Lemma~\ref{lem:graph-boundary} gives
+\begin{equation}\label{eq:compact-boundary-gradient}
+ d^2\partial_n(G+j(f))
+ =-B_n/N_i+u_*k_T\widehat z-yH+j'\widehat z/N_i.
+\end{equation}
+At $S$ this is strictly positive because $j'\le0$, $H\le0$,
+$k_T<0$, and $B_n=-b_*$. That is impossible for an inward derivative
+at a maximum. At a seam maximum $s$ is its seam minimum. The positive
+harmonic exterior lapse on the fixed collar satisfies
+$B_n\ge-Cs$ at that point: compare with the seam minimum times a
+fixed harmonic function equal to one on the seam and zero on the
+other collar boundary. Equation~\eqref{eq:compact-boundary-gradient}
+is then at most $Cd+|k_T|z-yH_i+o(1)<0$, using
+$H_i>|k_T|$. That is impossible for an outward derivative at a maximum.
+This proves $N_i,N_o\le C(L)M$ and, by
+\eqref{eq:compact-height-energy}, the stated $L^2$ estimate for $W$.
+
+Finally keep $M,L$ fixed and suppose $\inf s_i/\eps\to0$.
+Choose a fixed smooth $j$ on $[0,M]$ with $j''>0$ and
+$0<j'<b_*/2$. At a maximum of $G+j(f)$, bounded oscillation of $j$
+implies $s_i/\eps\to0$. The lapse floor gives $y\to1$.
+Multiply \eqref{eq:compact-gradient-test} by $s^2$: its left side
+stays positive, while its right side is bounded by
+$C(s^2+Ls+(j')^2d^2)$ and tends to zero. At $S$, the first and last
+terms of \eqref{eq:compact-boundary-gradient} have sum at least
+$(b_*-j')/N_i>0$. At the seam its last term is nonpositive, and the
+same harmonic collar comparison makes the derivative strictly
+negative. These contradictions prove $s_i\ge c(M,L)\eps$.
+On the exterior, below the level $\eps$ the positive source vanishes
+and the lapse is harmonic. Its boundary values and the minimum
+principle therefore give $N_o\ge c(M,L)\eps$ too.
+\end{proof}
+
+\begin{lemma}[Finite penalty selection]\label{lem:compact-penalties}
+For fixed preceding parameters the sources can be selected with finite
+strengths to obtain \eqref{eq:compact-penalty-targets}--\eqref{eq:compact-bad-volume}
+in every actual positive solution, uniformly in the outer truncation.
+Their costs obey \eqref{eq:compact-penalty-cost}.
+\end{lemma}
+
+\begin{proof}
+All assertions in this proof concern $u_*=1$. First choose
+$C_2\ge c(M,L)^{-1}$ from Lemma~\ref{lem:compact-lapse}, so that
+$sT_2\ge1$. This precedes the choice of $\eps$. For a piecewise
+Lipschitz test $\varphi$ with common seam value and zero outer value,
+testing the positive $w$ equation with $\varphi^2/w$ gives
+\begin{equation}\label{eq:compact-penalty-test}
+ \int P\frac{\varphi^2}{w}\le\int A(\dd\varphi,\dd\varphi).
+\end{equation}
+Indeed the omitted term $\Sigma_+x\varphi^2/w$ is nonnegative,
+and completing the square in the two derivative terms gives the
+right side. Flux matching cancels the seam terms and the inner flux
+is zero.
+
+Take $\varphi=f/M$, extended by zero outside. Directly from
+\eqref{eq:compact-constitutive},
+$A(\dd f,\dd f)=W-W^{-1}$. On the support of $P_1$ we have
+$\varphi>1/2$ and $w<4T_1$. Thus
+\[
+ \int P_1\le\frac{16T_1}{M^2}\int_{\Omega_i}W
+ \le C(L)M^{-1/10},
+\]
+by Cauchy--Schwarz and \eqref{eq:compact-bounds}.
+
+For $P_2$ take spatial localizers equal to one on its supports and
+vanishing on slightly larger collars of the seam, multiplied by a
+scalar cutoff of $N/\eps$ which is one below four and zero above five.
+These tests may meet $S$ and have zero seam value on each side. Since
+$A\le N^2g_{\rm base}^{-1}$ and $w<4T_2$ on the source support,
+\begin{equation}\label{eq:compact-low-lapse-cost}
+ \int P_2\le C_M\eps+
+ C_M\eps^{-1}\int_{\mathcal U\cap\{N<5\eps\}}|\nabla N|^2.
+\end{equation}
+The fixed localization sets $\mathcal U$ stay away from the seam;
+constants here may depend on $M$ and every preceding parameter.
+Testing the lapse equation with a squared spatial cutoff times
+$(6\eps-N)^+$ gives
+\begin{equation}\label{eq:compact-low-lapse-energy}
+ \int_{\mathcal U\cap\{N<5\eps\}}|\nabla N|^2
+ \le C_M\eps^2+
+ C\eps\int_{\mathcal U'\cap\{N<6\eps\}}\Sigma_+.
+\end{equation}
+To check the signs, on this sublevel the current correction has size
+at most $CN\le C\eps$, so Young's inequality absorbs its gradient
+term and the cutoff terms cost $C_M\eps^2$. The negative source can
+be discarded. At $S$ the outward current equals $b_*>0$, so its
+contribution decreases the right side of the energy inequality.
+
+We next prove that the last integral tends uniformly to zero as
+$\eps\downarrow0$. This step does not use a positive lapse floor.
+If uniformity failed, take a violating sequence. Equations
+\eqref{eq:compact-height-energy} and \eqref{eq:compact-bounds} bound
+the positive sources in $L^2$, independently of $\eps$ and the
+strengths. The lapse equation with bounded divergence datum $ZN$
+therefore gives local $W^{1,p}$ bounds for every $3<p<6$, also at $S$
+with the fixed flux data. After passing to a subsequence, the lapses
+converge uniformly to a nonnegative function $V$ on the localization
+sets, the drifts converge weakly star, and the sources converge weakly
+in $L^2$. The limit equation has the form
+\[
+ \Delta V=\divg(Z_\infty V)+g_1.
+\]
+The negative-source limit vanishes almost everywhere on $\{V=0\}$
+because $|\Sigma_-|\le CN$.
+
+For completeness the scalar source $g_1$ also vanishes there. Let
+$q$ be a density point of $\{V=0\}$ and a Lebesgue point of $g_1$
+and $|g_1|^2$ in an interior chart. Rescale a ball of radius $r$ to a
+unit ball and divide $V$ by $r^2$. The resulting nonnegative functions
+vanish at the center. The inhomogeneous Harnack estimate gives a
+uniform bound on a smaller ball: the rescaled drift is $rZ_\infty$,
+and the rescaled scalar forcing has bounded $L^2$ norm at this
+Lebesgue point. Since the zero sets occupy a fraction tending to one,
+the rescaled functions tend to zero in local $L^1$. Testing their
+equations against a fixed smooth bump, with derivatives transferred
+to that bump, gives $g_1(q)=0$. Thus the nonnegative positive-source
+limit $\sigma_+$ vanishes almost everywhere on $\{V=0\}$ as well.
+Boundary points do not matter for these volume integrals. For each
+$t>0$, uniform convergence implies
+$\{N<6\eps\}\subset\{V\le t\}$ along the sequence eventually.
+Weak $L^2$ convergence, tested against the characteristic function of
+this fixed set, and then $t\downarrow0$, show
+\[
+ \limsup\int_{\mathcal U'\cap\{N<6\eps\}}\Sigma_+
+ \le\int_{\mathcal U'\cap\{V=0\}}\sigma_+=0.
+\]
+Combining this with \eqref{eq:compact-low-lapse-cost} and
+\eqref{eq:compact-low-lapse-energy} proves the second cost in
+\eqref{eq:compact-penalty-cost}, uniformly in both strengths.
+
+Now fix such an $\eps$. The lapse and slope bounds give uniform
+ellipticity, independent of strengths. They also give uniform
+continuity of $N$ and $f$ on the localization sets. For $N$, use the
+same Laplace divergence estimates, now with bounded scalar forcing;
+for $f$ the already established gradient bound suffices. Consequently
+each point with $f\ge9M/10$ has a neighborhood of a uniform positive
+radius on which $f\ge4M/5$. It is uniformly separated from the seam
+unless the neighborhood ends at $S$, where the constant trace permits
+reflection. Likewise every point of
+$\mathcal D_M\cap\{N\le2\eps\}$ has a uniform neighborhood within
+the full spatial cutoff range and with $N\le3\eps$.
+
+In either neighborhood let $T_j$ be its corresponding threshold.
+The volume of $\{w\le2T_j\}$ there is at most
+$D_j^{-1}\int P_j$, hence tends uniformly to zero as $D_j\to\infty$.
+The function $(2T_j-w)^+$ is a bounded weak subsolution of the
+homogeneous scalar divergence equation with coefficient $A$.
+Its local supremum is bounded by its $L^2$ mean, with a uniform
+constant on these fixed neighborhoods; at $S$ use the homogeneous
+natural-flux reflection. Since its support has the preceding small
+volume and its value is at most $2T_j$, this supremum is strictly
+less than $T_j$ once the strength is sufficiently large. This gives
+$w\ge T_1$ on the height target and $w\ge T_2$ on the low-lapse
+target. The latter implies $x\ge1$ since $sT_2\ge1$.
+The choices can be made with a strict margin, so the weak inequalities
+survive limits of outer truncations.
+
+If $x<1-\delta$ at a designated point, the second target now forces
+$N>2\eps$. All cutoffs in $\Sigma_+$ equal one, so
+$\Sigma_+=LW$ there. Apply \eqref{eq:compact-capacity}, using the
+positive lower bound of $\psi_R$ on the common exterior support.
+This proves \eqref{eq:compact-bad-volume} with a constant independent
+of $L,M,\eps$ and the strengths.
+\end{proof}
+
+\begin{proof}[Proof of Proposition~\ref{prop:compact-solutions}]
+Choose the parameters in the stated order, using
+Lemmas~\ref{lem:compact-lapse} and \ref{lem:compact-penalties}.
+Proposition~\ref{prop:transmission-existence}, proved independently in
+the next section, now supplies a positive zero for each sufficiently
+large compact truncation. Its hypotheses are exactly the finite
+smooth cutoffs above, the homotopy bounds of
+Lemma~\ref{lem:compact-lapse}, and the fixed exterior harmonic collar.
+Its continuation starts with the ordinary scalar transmission problem
+at $u_*=0$; it does not assume solvability of the coupled problem.
+That proposition also supplies an expanding-truncation subsequence
+converging smoothly on compact subsets of each closed side, with
+$N_o,w\to1$ at infinity. All the bounds and penalty conclusions pass
+to this limit. This is the only invocation of nonlinear existence in
+the present proof.
+
+We record the geometric conclusions precisely. Set
+$\pi=N_i^{-1}\operatorname{sym}\nabla p^\flat$,
+$\lambda=\pi-K$ inside, and set these fields to zero outside.
+Lemma~\ref{lem:graph-current}, applied at $u_*=1$, gives
+\begin{equation}\label{eq:compact-scalar}
+ xR_h=D(v)+|\lambda|^2-\ell^2-2\Sigma_-/N
+       +2P/(Nx)+\tfrac32|\dd\log x|_{\bar g}^2.
+\end{equation}
+On the exterior take $D=R_{g_o}=0$. The current
+$Q=B+A\dd w$ has matching normal flux and
+$\divg Q=(1-x)\Sigma_++\Sigma_- -P$. At a constant-height face,
+Lemma~\ref{lem:graph-boundary} gives
+\begin{equation}\label{eq:compact-mean-curvature}
+ s\sqrt{x}\,H_h
+ =N(H-k_T\widehat z)+B_n+(A\dd w)_n/x.
+\end{equation}
+At $S$ this is strictly negative. At $\mathcal C$ the induced metrics
+agree, because $s_i=N_o$ and $w$ is continuous. The seam mean
+curvatures satisfy $H_{h,i}\ge H_{h,o}$ in the specified direction:
+the flux terms agree and
+\[
+ H_i-k_T\widehat z\ge
+ \sqrt{1-\widehat z^2}\sqrt{H_i^2-k_T^2},
+\]
+as follows by squaring, since the difference of squares is
+$(H_i\widehat z-k_T)^2$ and both relevant sides are nonnegative.
+
+Beyond the fixed source supports, both $N_o$ and $x=N_ow$ are
+harmonic and tend to one. Radial barriers and scaled estimates give
+$N_o-1,x-1=O_2(|x'|^{-\beta})$ for every fixed $1/2<\beta<1$.
+Thus $h=xg_o$ is asymptotically flat, and the flux of $Q$ equals
+the limiting flux of $\nabla x$; the remaining terms have vanishing
+integral by $2\beta>1$. The linear conformal ADM variation is
+$m_h-m_o=-(8\pi)^{-1}\lim\int\partial_nx$. Integration of the
+current equation therefore gives the exact identity
+\begin{equation}\label{eq:compact-mass}
+ 8\pi(m_h-m_o)
+ =\int_S b_*+\int\bigl[(x-1)\Sigma_++P-\Sigma_-\bigr].
+\end{equation}
+Since the support of $\Sigma_+$ has $x<1$, discarding its contribution
+and using the source costs proves \eqref{eq:compact-mass-budget}.
+The scalar curvature decays as $O(|x'|^{-2-2\beta})$ on the end,
+by \eqref{eq:compact-scalar}; this has integrable decay since
+$2+2\beta>3$.
+\end{proof}
+
+The omitted collars have a controlled cost in the area comparison.
+Indeed $W=1$ outside, and the inner omitted collar has volume
+$O(M^{-4})$. Hence
+\begin{equation}\label{eq:compact-full-bad-volume}
+ \int_{\{x<1-\delta\}}W
+ \le C/L+C a_o+C(L)M^{-1}.
+\end{equation}
+This uses \eqref{eq:compact-capacity-comparison} outside $R_\delta$
+and Cauchy--Schwarz with \eqref{eq:compact-bounds} in the inner
+collar. It explains why $a_o$ and $L$ are chosen before $M$.
+Finally the possible negative scalar curvature is confined to the
+closed set defined by the weak inequalities in
+\eqref{eq:compact-trace-cutoff} and $N_i\ge\delta_0$.
+Outside that set either $\ell=0$ or $\zeta=1$, and
+\eqref{eq:compact-scalar} is nonnegative. On that set the height
+penalty and $z\le1/2$ give
+$x\ge(\sqrt3/2)\delta_0M^{9/10}$, so
+$R_h\ge-C\delta_0^{-1}M^{-9/10}$ everywhere on the smooth sides.
+The geometric treatment of this exceptional set is the content of
+Section~\ref{sec:scalar-error}.

preprints/The-Penrose-inequality-for-maximal-asymptotically-hyperbolic-initial-data-October-5-2026/build/sections/08-transmission-analysis.tex

@@ -0,0 +1,958 @@
+\section{Regularity and continuation at the transmission surface}
+\label{sec:transmission-analysis}
+
+We prove the existence assertion needed in Section~\ref{sec:compact-construction}.
+Throughout this section all parameters, including the penalty strengths, are
+finite.  Constants in estimates for the lapse and height state may depend on
+the fixed parameters preceding the strengths, but not on those strengths.
+Constants involving the conformal factor or higher derivatives may also
+depend on the strengths.  Neither assertion requires uniform ellipticity as
+the internal parameters tend to their final limits.
+
+Write $\Omega_R=\Omega_i\cup_{\mathcal C}\Omega_{o,R}$, with far boundary
+$S_R$.  On both sides of $\mathcal C$ the normal $n$ points toward the end.
+Thus equality of the two $n$-fluxes is the condition for cancellation in
+Green's formula.  The two base metrics have the same induced metric at
+$\mathcal C$.  Joined Fermi charts have coordinates $(z',t)$, with
+$t<0$ inside, $t>0$ outside, and $n=\partial_t$ at the face.  Coordinate
+divergence currents below include the relevant volume density.  We retain
+the notation $d=s/N$, $y=1-d^2$, $q=(1-y)/(1+y)$,
+$I=(1+y)^{-1}$, $J=y/(1+y)$, and $\beta=2/(1+y)$ from the graph calculus.
+
+\begin{proposition}[Finite-parameter transmission existence]
+\label{prop:transmission-existence}
+Fix the smooth compact base data and AF attachment of
+Section~\ref{sec:compact-construction}, with $H_i>|k_T|$ on
+$\mathcal C$, $H_g(S)\leq0$, and the smooth negative definite synthetic
+tensor $K$.  Fix positive finite parameters and smooth cutoffs as in
+\eqref{eq:compact-system}--\eqref{eq:compact-trace-cutoff}, including a
+source-free exterior collar of $\mathcal C$ and the constant $b_*>0$.
+Then on every sufficiently large smooth truncation there is a positive
+piecewise smooth solution of \eqref{eq:compact-system} and
+\eqref{eq:compact-boundary} at $u_*=1$.
+
+At these fixed parameters a subsequence of such solutions as $R\to\infty$
+converges smoothly on each closed side on compact subsets to a positive
+piecewise smooth solution on the full joined exterior.  Its end values are
+$N_o,w\to1$.  The bounds in \eqref{eq:compact-bounds} hold, and all
+compact-set derivative bounds are uniform in $R$.  The penalty conclusions
+in Section~\ref{sec:compact-construction}, when the strengths have been
+chosen there, therefore hold for these solutions.
+\end{proposition}
+
+The proof uses the a priori estimates of Lemma~\ref{lem:compact-lapse},
+which were proved for positive classical zeros without assuming existence.
+The scalar linear transmission fact used in that lemma is established
+independently below as part of Lemma~\ref{lem:transmission-fredholm}.
+In particular there is no appeal to
+Proposition~\ref{prop:compact-solutions} in proving this proposition.
+
+\subsection{Energy and oscillation of the lapse and height state}
+
+For the moment consider any positive homotopy zero in the continuation
+domain: $f\in C^{3,\alpha'}$ on the closed inner side and
+$N,w\in C^{2,\alpha'}$ on each closed side, with $\alpha'>0$.
+No bound for these norms is assumed.  The exponent $\gamma$ obtained
+below depends only on the finite state bounds, so the continuation
+domain can subsequently be fixed with $\alpha'<\gamma$.
+The finite bounds in \eqref{eq:compact-bounds} give positive constants
+$c_0,C_0$ such that
+\begin{equation}
+ c_0\leq N_i,s_i,N_o\leq C_0,
+ \qquad |\dd f|\leq C_0.
+ \label{eq:transmission-state-range}
+\end{equation}
+Here and below an estimate for $s$ on the exterior means the same estimate
+for $N_o$.  The scalar lapse sources and the drift
+$(u_*K+\ell g)p$ are bounded at this stage; their derivatives are not used.
+
+\begin{lemma}[Continuity at the nonlinear seam]
+\label{lem:seam-holder}
+For the state range \eqref{eq:transmission-state-range}, there are
+$\gamma>0$ and $C<\infty$ such that
+\begin{equation}
+ \|N_i\|_{C^\gamma(\overline\Omega_i)}
+ +\|\dd f\|_{C^\gamma(\overline\Omega_i)}
+ +\|N_o\|_{C^\gamma(K\cap\overline\Omega_o)}\leq C
+ \label{eq:transmission-holder}
+\end{equation}
+on every fixed compact set $K$.  The constants are uniform over homotopy
+zeros and sufficiently large outer truncations.  They depend on the finite
+state range, bounds for the undifferentiated height and lapse sources,
+and the fixed geometry, but not on derivatives of $T$ or the strength of
+$P$.
+\end{lemma}
+
+\begin{proof}
+Away from $\mathcal C$, the lapse equation is a smooth metric Laplace
+equation with bounded divergence forcing and bounded scalar forcing.
+Its local $W^{1,p}$ bounds hold for some $p>3$, also up to $S$ with its
+natural current flux.  The prescribed constant flux at $S$ can be put in
+the current by a smooth bounded extension before reflection.  Differentiating
+the divergence height equation gives scalar uniformly elliptic equations
+for the coordinate derivatives of $f$.  Their divergence forcing contains
+first derivatives of $N$, bounded in $L^p$, and the bounded scalar height
+forcing, put into the corresponding component of the differentiated
+current.  In particular differentiating that scalar forcing in the weak
+equation does not require a bound for $\dd T$.
+The inhomogeneous scalar divergence estimate gives H\"older bounds for
+$\dd f$.  At $S$ subtract the constant height and reflect it oddly, the
+lapse evenly, and the metric with its tensor parities in an orthogonal
+collar.  The reflected metric is Lipschitz.  The height flux satisfies the
+weak reflected equation, since its normal component has the required even
+parity.  Difference quotients justify differentiation there.  These
+arguments prove the assertion away from the seam.
+
+At the seam we first bound energy weighted by inverse distance.
+A modified Newton current then gives an estimate relative to compatible
+constant states. The weighted bound supplies a uniformly positive
+scale of small excess; a blowup to an explicit linear transmission
+problem gives excess decay and hence H\"older continuity.
+
+\smallskip\noindent\emph{Weighted energy.}
+Set
+\begin{equation}
+ E=\begin{cases}|\nabla N_i|^2+|\nabla^2 f|^2,&t<0,\\
+                 |\nabla N_o|^2,&t>0,
+   \end{cases}
+ \qquad G=-\log s,\qquad Y=s^{-k_0},
+ \label{eq:transmission-energy-definition}
+\end{equation}
+where $k_0$ is sufficiently large for
+Lemma~\ref{lem:gradient-energy}.  On the inside its divergence matrix is
+$\rho\mathcal P$, where $\rho=(1+y)/d$, and on the outside it is the
+base cometric.  After including the volume densities, these give a bounded
+symmetric uniformly elliptic matrix $a$ in the joined chart.
+$Y$ is continuous across the face because $N_o=s_i$ there.
+The energy identity and the boundary identity give
+\begin{equation}
+ \partial_j(a^{jk}\partial_kY)
+ \geq cE-C-C\delta_{\mathcal C}.
+ \label{eq:transmission-joined-energy}
+\end{equation}
+This is a distributional inequality.  Indeed $\rho q=d$ on a
+Dirichlet-height face, and the inner normal $Y$-flux is
+$-k_0YB_n/s+O(1)$, whereas the outer one is $-k_0YB_n/s$.
+The bounded discrepancy is precisely the surface term in
+\eqref{eq:transmission-joined-energy}.
+
+Fix concentric chart balls strictly contained in a larger joined chart.
+Let $\theta_\varepsilon$ be the zero Dirichlet Green potential, for
+$-\partial(a\partial)$ on the larger ball, of a nonnegative normalized
+bump at a point $z_0$ in the smaller ball.  If $Y_H$ is the homogeneous
+replacement of $Y$ with its boundary values, then
+\begin{equation}
+ \big\langle\partial(a\partial Y),\theta_\varepsilon\big\rangle
+  =\int(Y_H-Y)\,\chi_\varepsilon\leq C.
+ \label{eq:transmission-green-pairing}
+\end{equation}
+The maximum principle bounds $Y_H$, and $Y$ is bounded by
+\eqref{eq:transmission-state-range}.  For each classical solution the
+identity follows first with regular potentials and then by Sobolev
+approximation.  The surface term is well defined by the trace theorem.
+The divergence Green bounds for symmetric measurable uniformly elliptic
+matrices in dimension three give an upper bound $C|z-z_0|^{-1}$ and a
+comparable lower bound on a sufficiently small interior ball
+\citep[Theorem~7.1 and pp.~66--67]{LSW1963}.  The surface integral of
+$|z-z_0|^{-1}$ is uniformly finite.  Letting the bump shrink, using
+\eqref{eq:transmission-joined-energy} and Fatou's lemma, proves
+\begin{equation}
+ \int_{B_{h_0}(z_0)}\frac{E(z)}{|z-z_0|}\,\dd z\leq C.
+ \label{eq:transmission-green-energy}
+\end{equation}
+Both $h_0>0$ and $C$ are uniform on the finite collection of seam charts.
+
+\smallskip\noindent\emph{Comparison with a constant state.}
+We next prove an energy bound relative to a compatible constant state.
+Such a state is
+\begin{equation}
+ \mathcal V_*=(n_*,a_*\dd t,s_*),\qquad
+ s_*=s(n_*,|a_*|),\qquad
+ \mathcal V=(N_i,\dd f,N_o).
+ \label{eq:transmission-reference}
+\end{equation}
+The pairs $(n_*,a_*)$ range over a compact rectangle containing the
+state ranges with a margin, with $n_*>0$.  Since $|\dd t|_g=1$, $s_*$ is
+constant in this Fermi chart.  Norms of the state mean the sum of the
+inner pair's squared norm on the inner half and the outer lapse's
+squared norm on the outer half.
+
+To derive the bound, put
+$\mathcal T_f=\nabla^2f-(\Delta f)g$.  The Newton-current identity in
+Lemma~\ref{lem:newton-flux} says
+\begin{equation}
+ \rho\mathcal P\nabla G
+ =-\frac{\nabla N_i}{s}
+   +\mathcal T_f\left(\frac{y\nabla f}{d|\dd f|^2}\right)+O(1).
+ \label{eq:transmission-newton-current}
+\end{equation}
+The coefficient vector extends smoothly through $\dd f=0$.  In the
+current $a\dd Y$, add $k_0Y_*B/s_*$ on both sides, where
+$Y_*=s_*^{-k_0}$, and on the inside subtract the Newton term in
+\eqref{eq:transmission-newton-current} with its full coefficient
+$k_0Y,y\nabla f/(d|\dd f|^2)$ frozen at the reference.  Include metric
+densities in all three currents.  Call the resulting joined current
+$\mathcal J_*$.  The cancellation of the coefficients of $\nabla N$
+and $\nabla^2f$ at the reference gives
+\begin{equation}
+ |\mathcal J_*|\leq C\bigl(1+|\mathcal V-\mathcal V_*|\sqrt E\bigr).
+ \label{eq:transmission-modified-current-size}
+\end{equation}
+Moreover $\divg\mathcal T_f=\Ric(\nabla f,\cdot)$, so the divergence
+of the subtracted term is bounded by $C(1+\sqrt E)$; its frozen vector
+is extended smoothly using $\partial_t$.  The added $B$-current has
+bounded divergence and no seam jump.  The frozen Newton vector is normal
+at the seam.  Its normal contraction there involves
+$-\tr_{\mathcal C}\nabla^2f$, which is bounded since $f$ is constant
+along that face and $|\dd f|$ is bounded.  Young's inequality therefore
+gives, after lowering $c$,
+\begin{equation}
+ \divg\mathcal J_*\geq cE-C-C\delta_{\mathcal C}.
+ \label{eq:transmission-modified-current-divergence}
+\end{equation}
+Test by $\chi^2$, with $\chi=1$ on $B_h$, supported in $B_{2h}$ and
+$|\dd\chi|\leq C/h$.  Absorb half the energy using
+\eqref{eq:transmission-modified-current-size}.  The volume, constant
+current, and surface contributions are at most $Ch^3$, $Ch^2$, and
+$Ch^2$, respectively.  We obtain
+\begin{equation}
+ h^{-1}\int_{B_h}E\leq
+ C\left(\frac{1}{|B_{2h}|}\int_{B_{2h}}
+                    |\mathcal V-\mathcal V_*|^2+h\right).
+ \label{eq:transmission-caccioppoli}
+\end{equation}
+On a pure inner ball the identical construction uses any constant
+covector reference.  The coefficients frozen against this reference may
+vary smoothly with the background metric; their derivatives give only
+$C(1+\sqrt E)$ errors.  On a pure outer ball the analogous assertion
+is the usual lapse Caccioppoli inequality.
+
+\smallskip\noindent\emph{A scale of small excess.}
+Fix $0<b_0<1$ and define the seam excess
+\begin{equation}
+ \mathcal E(z_0,h)=
+ \inf_{\mathcal V_*}\frac{1}{|B_h|}\int_{B_h(z_0)}
+          |\mathcal V-\mathcal V_*|^2+h^{b_0}.
+ \label{eq:transmission-excess}
+\end{equation}
+There is a uniformly positive scale at which this excess is arbitrarily
+small.  To see the quantitative content, put $h_j=2^{-j}h_0$.  For each
+$z\ne z_0$ the sum of $h_j^{-1}$ over the indices with $|z-z_0|<h_j$
+is at most $C|z-z_0|^{-1}$.  Thus
+\eqref{eq:transmission-green-energy} bounds
+$\sum_j h_j^{-1}\int_{B_{h_j}}E$ uniformly.  Given $\eta>0$, first
+make the starting ceiling small enough that $h_0^{b_0}+h_0^2<\eta$.
+Among a fixed finite number of its dyadic descendants there is a scale
+with $h^{-1}\int_{B_h}E<\eta$.  The number depends only on the
+Green bound and $\eta$, so the chosen scale has a uniform positive
+lower bound.
+
+Here is why this is also a small excess scale.  Scaled Poincar\'e and
+the half-ball trace inequality bound both the volume deviations and the
+mean-square face deviations from each side mean by
+\begin{equation}
+ C h^{-1}\int_{B_h}E+Ch^2.
+ \label{eq:transmission-trace-poincare}
+\end{equation}
+The additional $Ch^2$ allows for Christoffel terms when replacing
+covariant Hessians by coordinate derivatives of $\dd f$.  The
+tangential trace of $\dd f$ is zero; hence the tangential components of
+its mean are small.  The trace relation $N_o=s(N_i,\partial_t f)$ and
+the uniform Lipschitz bound for $s$ on the state range show that the
+outer mean differs from the value obtained from the two inner means by
+at most the square root of \eqref{eq:transmission-trace-poincare}.
+This provides a compatible reference with excess $C\eta$.  All
+references obtained this way lie in a compact subrange with room for
+the subsequent small updates.
+
+\smallskip\noindent\emph{Excess decay.}
+We claim that, for fixed sufficiently small $\vartheta>0$, whenever
+$h$ and $\mathcal E(z_0,h)$ are sufficiently small,
+\begin{equation}
+ \mathcal E(z_0,\vartheta h)
+ \leq 2\bigl(C\vartheta^2+\vartheta^{b_0}\bigr)
+                    \mathcal E(z_0,h).
+ \label{eq:transmission-excess-decay}
+\end{equation}
+The constant $C$ is independent of $\vartheta$.  We give the
+compactness argument, including the value trace.
+
+If the claim fails, choose violating balls with
+$\sigma_j^2=\mathcal E(z_j,h_j)\to0$ and best references
+$(n_j,a_j\dd t,s_j)$.  Translate and dilate these balls to unit size
+using $z=z_j+h_j\zeta$, and rotate tangential coordinates to be
+orthonormal at their centers.  Define the lapse differences divided by
+$\sigma_j$, and the height difference by
+\begin{equation}
+ u_{i,j}=\frac{N_i-n_j}{\sigma_j},\qquad
+ u_{o,j}=\frac{N_o-s_j}{\sigma_j},\qquad
+ \phi_j(\zeta)=
+       \frac{f(z_j+h_j\zeta)-a_jh_j\zeta_3}{h_j\sigma_j}.
+ \label{eq:transmission-normalized-fields}
+\end{equation}
+The trace of $\phi_j$ is zero.  Their state has bounded $L^2$ norm on
+the unit ball.  Applying \eqref{eq:transmission-caccioppoli} on smaller
+balls gives locally bounded $H^1$ norms for $u_{i,j},u_{o,j}$ and
+$\nabla\phi_j$.  Indeed its error after normalization is
+$h_j/\sigma_j^2\leq h_j^{1-b_0}\to0$.
+The $L^2$ norm of $\phi_j$ is controlled by its gradient and its zero
+trace.  Rellich compactness and compactness of the $H^1$ trace into
+$L^2$ on smaller face patches give strong local $L^2$ convergence of
+the state in the bulk and on the face, and weak $H^1$ convergence.
+
+Pass to a convergent reference $(n_*,a_*,s_*)$.  The derivative of
+the height flux with respect to the gradient is $s_*^2\mathcal P_*$;
+its lapse derivative is $s_*^2(\beta/n_*)a_*\partial_t$.
+Taylor remainders divided by $\sigma_j$ have $L^1$ norm tending to
+zero, since the unnormalized state difference is $\sigma_j$ times
+an $L^2$-bounded function.  The same assertion holds on the face by
+the trace bounds.  Smooth metric errors are $O(h_j/\sigma_j)$,
+which tends to zero.  The height source and lapse drift enter weak
+tests with this same factor; the scalar lapse source has the smaller
+factor $O(h_j^2/\sigma_j)$.  Thus no derivative of a source is being
+passed to the limit.  Joined tests have a common value on the face,
+so lapse current matching passes to equality of the limiting normal
+derivatives.  We arrive at
+\begin{align}
+ \Delta u_i&=\Delta u_o=0,
+ & (q\partial_t^2+\Delta_{z'})\phi+k\partial_tu_i&=0,
+ & k&=\beta a_*/n_*,\label{eq:transmission-frozen-interior}\\
+ \phi&=0,
+ &\partial_tu_i&=\partial_tu_o,
+ &u_o&=r_0u_i+r_0c\partial_t\phi \quad(t=0),
+ \label{eq:transmission-frozen-boundary}
+\end{align}
+where
+\begin{equation}
+ r_0=dI>0,\qquad c=-\frac{n_*y}{a_*},\qquad
+ ck=q-1,\qquad r_0c=-\frac{s_*J}{a_*}.
+ \label{eq:transmission-frozen-relations}
+\end{equation}
+All expressions have their continuous interpretations at $a_*=0$;
+in particular $c=k=0$ there.  Differentiating
+$s^2+|\dd f|^2s^4=N_i^2$ gives
+$\dot s=dI\dot N_i-sJ\partial_t\dot f/a_*$, which verifies
+the value relation directly.
+
+The elementary harmonic estimate proved just below gives uniform
+smooth bounds for these limits on smaller closed half-balls in terms
+of their $L^2$ state norms.  Consequently their deviations from the
+center values have mean square at most $C\vartheta^2$.  The center
+values satisfy the linearized compatibility relation.  Replace the
+old reference by
+\begin{equation}
+ n_j'=n_j+\sigma_j u_i(0),\qquad
+ a_j'=a_j+\sigma_j\partial_t\phi(0),\qquad
+ s_j'=s(n_j',|a_j'|).
+ \label{eq:transmission-reference-update}
+\end{equation}
+The nonlinear compatibility error in the outer value is
+$O(\sigma_j^2)$, and the tangential height gradient is zero at the
+center.  Strong local $L^2$ convergence therefore proves
+\eqref{eq:transmission-excess-decay}, contradicting its violation.
+The added radius term contributes at most
+$\vartheta^{b_0}\sigma_j^2$, as required.
+
+For completeness, the harmonic estimate used here is as follows.
+Reflect $u_o$ into $t<0$ and put $U=u_i+u_o^{\rm refl}$.  It is
+harmonic with zero Neumann data, so it is smooth with controlled local
+norms by even reflection.  On a smaller inner cylinder choose a
+harmonic primitive $D$ with $D_t=u_i$.  An explicit choice is
+\begin{equation}
+ D(z',t)=\int_{t_0}^{t}u_i(z',v)\,\dd v+a(z'),
+ \qquad \Delta_{z'}a=-\partial_tu_i(z',t_0),
+ \label{eq:transmission-harmonic-primitive}
+\end{equation}
+where $t_0<0$ is a fixed interior slice.  The slice data are controlled
+by interior harmonic estimates.  This formula gives $D\in H^2$
+locally up to the face: the mixed and double normal derivatives come
+from $u_i\in H^1$, and tangential Laplace estimates applied to
+$\Delta_{z'}D=-\partial_tu_i$ give its remaining second derivatives.
+The function $V=D+c\phi$ satisfies
+$(q\partial_t^2+\Delta_{z'})V=0$, since $ck=q-1$, and $V=D$
+on the face.  Set $\widetilde V(z',t)=V(z',\sqrt q\,t)$.
+It is harmonic and, with $a_0=r_0/\sqrt q>0$,
+\begin{equation}
+ (\widetilde V-D)|_{t=0}=0,
+ \qquad \partial_t(D+a_0\widetilde V)|_{t=0}=U.
+ \label{eq:transmission-positive-harmonic-combinations}
+\end{equation}
+Scalar harmonic Dirichlet and Neumann estimates control both
+combinations.  Their coefficient matrix is invertible because
+$1+a_0>0$; in fact $a_0=1/\sqrt{1+y}$.  They control $D$, then
+$u_i=D_t$ and $u_o=U-u_i^{\rm refl}$.  Finally the scalar Dirichlet
+equation in \eqref{eq:transmission-frozen-interior} controls $\phi$.
+This last step avoids division by $c$, so all estimates are uniform
+through zero slope.  The normal rescaling only reduces the patch by
+a uniform factor on the state range.
+
+Choose $\vartheta$ so small, and then $\gamma>0$ so small, that
+$2(C\vartheta^2+\vartheta^{b_0})\leq\vartheta^{2\gamma}$.
+Begin iteration at the small-excess scales with uniform positive
+lower bound constructed above.  The references stay in the permitted
+range because their successive changes are bounded by a convergent
+geometric sum.  This yields a mean-square oscillation bound
+$Ch^{2\gamma}$ at every seam center.  On a pure ball near the seam,
+the first scale comparable to its distance from the seam is controlled
+by a seam ball.  The corresponding interior excess argument, with an
+arbitrary constant gradient reference, then gives the bound at all
+smaller scales.  Its limit equations are a harmonic lapse and a
+constant uniformly elliptic height equation with the known lapse
+coupling; scalar interior estimates give the same decay.  Campanato's
+characterization on each closed half-chart proves
+\eqref{eq:transmission-holder}.  Finitely many charts and the earlier
+ordinary boundary estimates complete the proof.
+\end{proof}
+
+\subsection{Estimates with the full derivative orders}
+
+We first record the linear estimate, including the unequal orders of
+the unknowns.  This is also a proof of the relevant boundary estimate;
+it does not invoke a general theorem for the nonlinear transmission
+condition.
+
+\begin{lemma}[Local transmission Schauder estimate]
+\label{lem:transmission-schauder}
+Give a height variation $\phi$ order $3$, lapse variations $u_i,u_o$
+order $2$, and conformal-factor variations $v_i,v_o$ order $2$.
+For frozen coefficients from a state in
+\eqref{eq:transmission-state-range}, the principal equations are
+\begin{equation}
+ \mathcal P:\nabla^2\phi
+       +\frac\beta N\nabla f\cdot\nabla u_i,\qquad
+ \Delta u_i,\quad\Delta u_o,\qquad
+ A_i:\nabla^2v_i,\quad A_o:\nabla^2v_o.
+ \label{eq:transmission-principal-rows}
+\end{equation}
+Their residual norms are respectively $C^{1,\alpha}$ and
+$C^{0,\alpha}$ for the other four rows.  The height trace has
+$C^{3,\alpha}$ norm.  The lapse and conformal-factor value residuals
+have $C^{2,\alpha}$ norm, and their current-flux residuals have
+$C^{1,\alpha}$ norm.  On a smaller fixed patch, the field norms of
+the indicated orders are bounded by these residual norms and the
+supremum norms of the fields on a larger patch.  The constants are
+uniform on compact positive state ranges.
+
+At the seam the height/lapse principal conditions are
+\eqref{eq:transmission-frozen-boundary}; the conformal-factor block
+has value continuity and positive conormal matching.  At $S$ they
+are Dirichlet for height and Neumann for the other fields; at $S_R$
+they are Dirichlet.  The same estimates hold for scalar transmission
+with a fixed positive $C^{2,\alpha}$ value ratio, and while its outer
+normal-derivative coefficient is decreased to zero, the inner
+coefficient being kept strictly positive.
+\end{lemma}
+
+\begin{proof}
+It suffices first to work on a flat seam patch.  Subtract scalar
+particular solutions of the two lapse Poisson equations, with
+$C^{2,\alpha}$ control, and a $C^{3,\alpha}$ particular solution
+of the height row, including its $C^{3,\alpha}$ boundary trace.
+The height forcing includes the first derivative of the particular
+inner lapse, hence belongs to $C^{1,\alpha}$ as required.  Such
+particular solutions can be made on smooth domains having flat face
+portions containing the smaller patch, after extending localized
+data; ordinary scalar Dirichlet estimates give their stated norms.
+We are left with harmonic lapses and the homogeneous height row,
+zero height trace, and residuals
+\begin{equation}
+ \partial_tu_i-\partial_tu_o=h_1,
+ \qquad u_o-r_0u_i-r_0c\partial_t\phi=h_2,
+ \label{eq:transmission-inhomogeneous-seam}
+\end{equation}
+where $h_1\in C^{1,\alpha}$ and $h_2\in C^{2,\alpha}$.
+The reflected sum $U=u_i+u_o^{\rm refl}$ is harmonic with Neumann
+data $h_1$, and has a controlled $C^{2,\alpha}$ norm on a smaller
+patch.  Construct $D$ by
+\eqref{eq:transmission-harmonic-primitive} and use its same local
+supremum bound.  The two harmonic combinations in
+\eqref{eq:transmission-positive-harmonic-combinations} now have
+Dirichlet datum zero and Neumann datum $U-h_2$, respectively.
+The latter belongs to $C^{2,\alpha}$, so both combinations, and
+therefore $D$, are controlled in $C^{3,\alpha}$.  This controls
+both lapses in $C^{2,\alpha}$.  Scalar Dirichlet estimates for the
+height row finish the $C^{3,\alpha}$ estimate for $\phi$.
+
+For the conformal-factor block, the frozen inner matrix in orthonormal
+face coordinates is a positive multiple of
+$\operatorname{diag}(1,1,d^2)$, and the outer matrix is isotropic.
+Subtract particular solutions, rescale the two normal coordinates,
+and reflect the outer field.  The resulting fields are harmonic;
+their difference has the prescribed value datum and a positive
+weighted sum has the prescribed Neumann datum.  This gives their
+$C^{2,\alpha}$ bounds.  More generally, if the value relation is
+$v_o=r v_i+h$ with $r>0$, and the flux relation after reflection is
+$a\partial_tv_i+b\partial_tv_o^{\rm refl}=k$, then the harmonic
+combinations $v_o^{\rm refl}-rv_i$ and $av_i+bv_o^{\rm refl}$
+have Dirichlet and Neumann data.  Their coefficient determinant is
+$a+br>0$.  It stays positive if $b$ decreases to zero with $a>0$.
+Variable fixed $C^{2,\alpha}$ ratios are treated by freezing, as in the
+next paragraph.  Localization differentiates the value row at most
+twice and the flux row at most once; the constants depend only on
+the corresponding coefficient norms.  The ordinary boundary and interior estimates follow
+from the scalar estimates, with the lapse estimated before the
+height row.
+
+For fixed regular variable coefficients, localization gives the same
+estimate with lower-order terms.  In a sufficiently small chart
+the leading coefficients differ little in supremum norm from their
+frozen values.  Apply the constant-coefficient estimate to cutoff
+fields on that chart.  In a product such as
+$(a-a_*)D^2v$, the coefficient supremum multiplies the top H\"older
+seminorm and can be absorbed.  Its fixed H\"older seminorm multiplies
+the second-derivative supremum, which is bounded by an arbitrarily
+small multiple of the full high norm plus a fixed multiple of the
+field supremum, by interpolation.  The same argument applies to
+the order-$1$ height coupling and to the differentiated value
+relation.  Cutoff commutators have strictly lower orders.  A finite
+cover and interpolation for separated-point H\"older quotients give
+the claimed local or global variable-coefficient estimate.  This
+argument requires only fixed coefficient bounds and is uniform
+along compact families of such coefficients.
+\end{proof}
+
+We give the nonlinear application of this estimate in detail.  Fix
+$0<\alpha'<\alpha<\gamma$, decreasing $\gamma$ in
+Lemma~\ref{lem:seam-holder} if necessary.  On a fixed truncation use
+the Banach domain
+\begin{equation}
+ X=C^{3,\alpha'}(\overline\Omega_i)_f
+   \times C^{2,\alpha'}(\overline\Omega_i)_{N_i,w_i}
+   \times C^{2,\alpha'}(\overline\Omega_{o,R})_{N_o,w_o},
+ \label{eq:transmission-domain-space}
+\end{equation}
+with the two height Dirichlet values imposed by an affine lift of
+$u_*$, and with the open condition $N_i,N_o,w_i,w_o>0$.
+There are no other conditions built into this domain.  The output
+space, denoted $\mathscr Y$, contains the height equation in
+$C^{1,\alpha'}$, the two lapse and two conformal-factor equations
+in $C^{0,\alpha'}$, the seam value residuals and the far Dirichlet
+residuals in $C^{2,\alpha'}$, and the flux residuals at $S$ and
+$\mathcal C$ in $C^{1,\alpha'}$.  Height boundary values are not
+additional outputs because they are prescribed in $X$.
+
+The quotient in the trace cutoff causes no additional loss.  In a
+distance collar $r_S\geq0$ of $S$, with $F=u_*M-f=0$ on $S$,
+\begin{equation}
+ \frac{F(z',r_S)}{r_S}
+   =\int_0^1\partial_{r_S}F(z',\theta r_S)\,\dd\theta.
+ \label{eq:transmission-quotient}
+\end{equation}
+It is consequently an affine bounded map from the height domain
+into $C^{2,\alpha'}$.  The same formula controls it in all the
+order-by-order estimates below.  Outside this collar its denominator
+is smooth and positive.  Products and the finite smooth cutoffs
+therefore define a smooth residual map
+\begin{equation}
+ \mathscr F:[0,1]\times X_+\longrightarrow\mathscr Y,
+ \label{eq:transmission-nonlinear-map}
+\end{equation}
+where $X_+$ denotes the positive open domain after the height lift.
+
+At an exact zero, $w\geq1$.  Indeed its equation has nonnegative
+right-hand side, its outer value is $1$, and its natural fluxes
+match with zero inner flux.  Testing with $(1-w)^+$ gives the claim.
+Since $\Sigma_+x$ is bounded on its support by the finite state
+range and the cutoff $x<1-\delta/2$, and $P$ has finite strength,
+the forcing in this scalar joined equation is bounded independently
+of the particular zero.  Testing with $w-1$ and using the
+Dirichlet Poincar\'e inequality gives an $H^1$ bound for $w-1$ on
+the fixed truncation.  Local scalar boundedness and H\"older estimates
+for its measurable uniformly elliptic joined divergence matrix give
+a $C^\gamma$ bound, after reducing $\gamma>0$.  Reflection handles
+the homogeneous inner flux.  Thus
+\begin{equation}
+ \mathscr U=(\dd f,N_i,N_o,w_i,w_o)
+ \quad\hbox{is bounded piecewise in }C^\gamma.
+ \label{eq:transmission-state-holder-all}
+\end{equation}
+
+We now prove an a priori $C^{2,\alpha}$ bound for $\mathscr U$.
+For a smooth zero, differentiate the nondivergence height row once.
+Its highest terms are the height and lapse terms in
+\eqref{eq:transmission-principal-rows}.  Expanding the lapse and
+$w$ equations shows that their only highest terms at these column
+orders are their own second derivatives.  Every other term involves
+at most two factors of first derivatives of $\mathscr U$, with
+coefficients that are smooth functions of its bounded state and of
+the quotient \eqref{eq:transmission-quotient}.  At the seam,
+differentiating the value relation twice tangentially gives
+\begin{equation}
+ \partial_a\partial_b(N_o-s_i)
+  =\partial_a\partial_b N_o
+    -D s_i\,\partial_a\partial_b(N_i,\dd f)
+    -D^2s_i\bigl(\partial_a(N_i,\dd f),
+                 \partial_b(N_i,\dd f)\bigr)
+    +\mathcal R_{ab},
+ \label{eq:transmission-value-differentiation}
+\end{equation}
+where $\mathcal R_{ab}$ contains only bounded geometry coefficients
+and lower-order derivatives.  The leading first differential is
+exactly the principal value relation already used.  One derivative
+of a current-flux relation has only the indicated first-order
+principal boundary operators; its other terms are lower at these
+column orders.  Formula~\eqref{eq:transmission-quotient} gives the
+same derivative count for every occurrence of $F/d_S$.
+
+Here are the interpolation exponents needed for absorption, to make
+the dependence on the prior continuity estimate explicit.  Put
+$H=1+\|\mathscr U\|_{C^{2,\alpha}}$ and suppose
+$\|\mathscr U\|_{C^\gamma}\leq C_0$.  On fixed slightly enlarged
+patches, H\"older interpolation gives
+\begin{align}
+ \|D\mathscr U\,D\mathscr U\|_{C^{0,\alpha}}
+   &\leq C H^{\theta_1},
+ &\theta_1&=\frac{2+\alpha-2\gamma}{2+\alpha-\gamma}<1,
+ \label{eq:transmission-product-interpolation}\\
+ \|D^2\mathscr U\|_{C^0}
+   &\leq C H^{\theta_2},
+ &\theta_2&=\frac{2-\gamma}{2+\alpha-\gamma}<1.
+ \label{eq:transmission-sup-interpolation}
+\end{align}
+Terms of still lower order have smaller exponents or admit the usual
+$\eta H+C_\eta$ bound.  The first estimate follows by assigning
+one first derivative its supremum norm and the other its
+$C^{0,\alpha}$ norm; their interpolation numerators are
+$1-\gamma$ and $1+\alpha-\gamma$, respectively.
+
+Freeze the coefficients in a chart of radius chosen using only
+\eqref{eq:transmission-state-holder-all}.  Their oscillations in
+supremum norm are then as small as desired, while their
+$C^\gamma$ norms are bounded.  A principal coefficient error
+times a top derivative has $C^{0,\alpha}$ norm bounded by a small
+constant times $H$ plus a constant times $H^{\theta_2}$.
+The same estimate after two tangential derivatives applies to the
+value row in \eqref{eq:transmission-value-differentiation}; its
+quadratic part is controlled by
+\eqref{eq:transmission-product-interpolation}.  Apply
+Lemma~\ref{lem:transmission-schauder} to cutoff fields with matched
+cutoff values at a seam.  A second, slightly larger cutoff extends
+coefficient differences without losing their small supremum norm.
+Cutoff commutators contain only lower derivatives and have fixed
+patch-dependent bounds.  Taking the maximum over a finite cover,
+and using supremum interpolation for H\"older quotients with
+separated points, gives
+\begin{equation}
+ H\leq \eta H+C\bigl(1+H^{\theta_1}+H^{\theta_2}
+                         +H^{\theta_3}\bigr),\qquad
+ 0\leq\theta_3<1.
+ \label{eq:transmission-global-absorption}
+\end{equation}
+Choose $\eta<1/2$.  This bounds $H$.  Constants for differentiated
+cutoffs and source functions may depend on the finite parameter
+choices, which is permitted here.
+
+We must also justify using smooth-zero estimates for zeros initially
+in \eqref{eq:transmission-domain-space}.  The following difference
+quotient argument supplies that justification.  Off the seam,
+ordinary scalar bootstrapping applies, estimating lapse and $w$
+first and height next; at $S$ the $w$ conormal condition is its
+ordinary homogeneous Neumann condition because the height gradient
+is normal.  Near a seam flatten the face and take a tangential
+difference quotient of the equations.  For the value relation use
+the averaged first differential of $s$ between the two states.
+For a fixed zero these coefficients are uniformly
+$C^{2,\alpha'}$ and, on a sufficiently small patch and for
+sufficiently small differences, are supremum-close to the same
+frozen principal coefficients.  The difference quotients have
+uniformly bounded one-lower norms from the starting regularity.
+Apply the fixed-coefficient estimate to their cutoffs.  Principal
+oscillations times top seminorms are absorbed; top supremum terms
+are interpolated against the known one-lower norm.  All
+commutators and differentiated explicit data involve only these
+known orders.  This controls one tangential derivative in the same
+column spaces.
+
+The normal derivatives are recovered in a definite order.  Solve
+the lapse and $w$ equations for their double normal derivatives;
+their strictly positive normal coefficients permit division.
+Then solve the height row for its double normal derivative, using
+the newly controlled lapse derivatives.  Other second derivative
+terms have at least one tangential index, which is already
+controlled.  Iterating proves smoothness.  More explicitly, if
+orders $m+3,\alpha'$ for height and $m+2,\alpha'$ for the other
+fields are known, apply the same estimate to a tangential
+difference quotient of $m$ tangential derivatives.  In the height
+row, including its one residual derivative, every distributed
+coefficient term other than the principal row on the increments
+uses at most $m+2,\alpha'$ derivatives of $\mathscr U$.
+The other interior rows start with at most one derivative of
+$\mathscr U$ outside their own principal terms.  In the value row
+the only new-order term is its first state differential on the
+increment.  Thus the same argument gives the next derivatives with
+the largest number of tangential indices.  Recover the others with
+successively fewer tangential indices by the lapse and $w$
+equations first, then the height equation.  These identities hold
+on the open side and extend continuously to the face.  The quotient
+near $S$ obeys \eqref{eq:transmission-quotient} at every step.
+
+Consequently every zero of \eqref{eq:transmission-nonlinear-map}
+is smooth, and \eqref{eq:transmission-global-absorption} bounds it
+at the stronger exponent $\alpha$.  On a fixed truncation the
+positive zero set over $0\leq u_*\leq1$ is compact in $X$:
+the stronger H\"older bounds give precompactness, the residual map
+is continuous, and the uniform positive floors for $N$ and $w$
+keep limits in $X_+$.
+
+\subsection{Fredholm index and continuation}
+
+\begin{lemma}[Index zero for the linearized transmission problem]
+\label{lem:transmission-fredholm}
+The field derivative $D_X\mathscr F$ at each zero of
+\eqref{eq:transmission-nonlinear-map} is Fredholm of index zero
+between \eqref{eq:transmission-domain-space} and its specified
+residual space.  Independently, the scalar density and adjoint
+problems with smooth base metrics, fixed $C^{1,\alpha'}$ drift,
+a fixed positive $C^{2,\alpha'}$ value ratio, natural flux matching,
+inner Neumann and outer Dirichlet data are
+Fredholm of index zero.  Their estimates may depend on derivatives
+of the fixed ratio.
+\end{lemma}
+
+\begin{proof}
+Discard compact lower-order terms in the linearization.  This is
+legitimate with the column orders used above.  For example, a
+variation of the trace target in the expanded lapse row contains
+at most two derivatives of the height variation or one of the
+other variations, and therefore maps compactly to its
+$C^{0,\alpha'}$ residual.  Variations of $F/d_S$ lose one height
+derivative by \eqref{eq:transmission-quotient}, which still leaves
+this compactness.  In contrast, the first differential of
+$N_o-s(N_i,\dd f)$ in the $C^{2,\alpha'}$ seam residual is
+principal and is retained.  The principal height coupling to a
+lapse derivative in its $C^{1,\alpha'}$ residual is also retained.
+
+In this principal operator scale the background $\dd f$ continuously
+to zero and recompute $q,\mathcal P,\beta,s,A$, the height coupling,
+and the value-differential coefficients from that scaled background.
+This scales coefficient fields, not the variations.  In particular
+$ck=q-1$ in \eqref{eq:transmission-frozen-relations} holds all
+along the path.  The states remain in a fixed positive ellipticity
+range.  The local estimates of
+Lemma~\ref{lem:transmission-schauder}, followed by localization,
+give along this path
+\begin{equation}
+ \|v\|_X\leq C\bigl(\|L_\lambda v\|_{\mathscr Y}
+                                  +\|v\|_0\bigr),
+ \label{eq:transmission-linear-apriori}
+\end{equation}
+where $\|v\|_0$ is a sum of field supremum norms and $C$ is
+uniform in the path parameter $\lambda$.
+
+At zero background gradient the height decouples, and the lapse
+and $w$ blocks have scalar value and positive conormal transmission.
+Decrease each outer derivative coefficient in its flux relation to
+zero, leaving the inner coefficient positive.  The determinant
+$a+br$ in the proof of
+Lemma~\ref{lem:transmission-schauder} stays positive, so
+\eqref{eq:transmission-linear-apriori} remains valid.  At the
+endpoint each scalar block is triangular: solve the inner Neumann
+problem, then the outer Dirichlet problem with the trace prescribed
+from the inner solution.  On each connected inner component the
+scalar Neumann problem has equal kernel and compatibility-cokernel
+dimensions, both one; the outer Dirichlet and height Dirichlet
+problems have index zero.  Dividing by positive scalar principal
+factors and discarding connection terms reduces this assertion to
+ordinary Poisson theory.  Thus the endpoint index is zero.
+
+We justify index propagation using only
+\eqref{eq:transmission-linear-apriori}.  A sufficiently fine fixed
+finite grid of field evaluations defines
+$\mathfrak p:X\to\R^l$ with
+$\|v\|_0\leq\eta\|v\|_X+C_\eta|\mathfrak p v|$.
+Choose $\eta$ small in \eqref{eq:transmission-linear-apriori}.
+The augmented operators
+\begin{equation}
+ \mathfrak A_\lambda=(L_\lambda,\mathfrak p):
+                   X\longrightarrow\mathscr Y\times\R^l
+ \quad\hbox{satisfy}\quad
+ \|v\|_X\leq C\|\mathfrak A_\lambda v\|.
+ \label{eq:transmission-augmented-bound}
+\end{equation}
+They are injective with closed range.  At an index-zero Fredholm
+parameter their range has codimension $l$.  If parameters
+$\lambda_j$ of this kind tend to $\lambda$, and the limit range
+had codimension at least $l+1$, choose an $(l+1)$-dimensional
+subspace $Z_0$ disjoint from that range.  At each $\lambda_j$
+there is a unit vector $z_j\in Z_0\cap\operatorname{ran}
+\mathfrak A_{\lambda_j}$.  Its preimages $v_j$ are bounded by
+\eqref{eq:transmission-augmented-bound}.  A subsequence of $z_j$
+converges in the finite-dimensional space to a unit vector $z$,
+and operator-norm continuity gives
+$\mathfrak A_\lambda v_j\to z$.  Closedness of the limit range
+is a contradiction.  Its codimension is therefore finite.
+
+The kernel of $L_\lambda$ is finite dimensional because
+$\mathfrak p$ is injective on it.  Its range is closed and has
+finite codimension: add the finite-dimensional factor
+$\{0\}\times\R^l$ to the closed finite-codimensional augmented
+range, and then quotient by that factor.  This sum is closed, and
+the resulting quotient is exactly $\operatorname{ran}L_\lambda$.
+Thus $L_\lambda$ is Fredholm also at the limit.  Openness and local
+constancy of the Fredholm index, together with this closedness
+argument starting from the endpoint, propagate index zero along
+the whole path.  Restoring compact lower-order terms does not
+change it.
+
+For the independent scalar density problem take the fixed positive
+$C^{2,\alpha'}$ ratio and fixed $C^{1,\alpha'}$ drift as coefficients.
+Its local
+principal boundary estimate is the scalar positive-ratio estimate
+already proved, and its lower-order drift is compact at the scalar
+column order two.  Localization uses only the stated coefficient
+norms, so no approximation in a H\"older norm is needed.
+The same outer-flux deformation and augmentation
+argument applies without reference to a nonlinear zero or to an
+a priori estimate for such zeros.  For its adjoint the value ratio
+and positive derivative weight exchange their roles; the same
+scalar combinations apply.  This proves both independent index
+assertions.
+\end{proof}
+
+We now complete existence on a fixed truncation.  At $u_*=0$ the
+height is identically zero.  Indeed its Dirichlet values are zero,
+its trace target is zero, and testing its divergence equation by
+$f$ gives $\int p\cdot\dd f=0$.  All scaled sources vanish and
+$s_i=N_i$, so the lapse solves the scalar harmonic transmission
+problem with ordinary value and normal-derivative matching, inner
+flux $-b_*$ in the $n$ direction, and far value $1$.  The weak
+form on the joined manifold is coercive after subtracting the
+outer boundary value.  Equivalently its index is zero by
+Lemma~\ref{lem:transmission-fredholm} and its homogeneous kernel
+is zero by energy testing.  It therefore has a unique solution.
+Comparison with $1$, using the outward inner flux $+b_*$, gives
+$N\geq1$.  Scalar transmission regularity makes it piecewise
+smooth.  The remaining equation is
+$\divg(N^2\dd w)=0$ with matching value and flux, inner zero
+flux, and far value $1$; hence $w=1$.
+
+This initial zero is nondegenerate.  Its homogeneous linearization
+is triangular: first
+$\divg(N^2\dd\dot f)=0$ with zero height traces, then the
+ordinary harmonic lapse transmission problem with homogeneous
+boundary data, then the homogeneous $w$ divergence problem with
+zero far value.  Energy testing makes each variation zero.
+Terms differentiating the source cutoffs carry $u_*$ or vanish
+because $\ell=0$; constitutive-coefficient variations in these
+rows multiply $\dd f=0$ or $\dd w=0$.  Index zero now implies
+invertibility of the full field derivative.
+
+We supply the finite-dimensional continuation argument so that no
+orientation or uniqueness assumption at the target parameter is
+implicit.  Let $\mathcal K$ be the compact set of positive zeros
+of \eqref{eq:transmission-nonlinear-map} over $[0,1]$.  By the
+Fredholm assertion, a finite-dimensional subspace
+$\mathfrak Z\subset\mathscr Y$ can be chosen so that
+\begin{equation}
+ D_X\mathscr F(X)+\mathfrak Z=\mathscr Y
+ \label{eq:transmission-cokernel-span}
+\end{equation}
+at every point of $\mathcal K$: choose local cokernel complements
+and use a finite cover.  Surjectivity persists on a neighborhood
+$\mathcal O$ of $\mathcal K$.  If $Q$ is the quotient map to
+$\mathscr Y/\mathfrak Z$, then
+\begin{equation}
+ \mathcal M=\{(u_*,v)\in\mathcal O:
+                               Q\mathscr F(u_*,v)=0\}
+ \label{eq:transmission-reduction-manifold}
+\end{equation}
+is a smooth finite-dimensional manifold of dimension
+$\dim\mathfrak Z+1$.  The field derivative of $Q\mathscr F$
+is onto, so the parameter projection on $\mathcal M$ is a
+submersion.  Its only endpoint faces are therefore $u_*=0$ and
+$u_*=1$.  Choose a relatively open neighborhood $\mathcal U$ of
+$\mathcal K$ in $\mathcal M$ with compact closure contained in
+$\mathcal O$ and with relative boundary disjoint from the zero
+set.  This is possible in the finite-dimensional locally compact
+manifold.  The norm of $\mathscr F$ has a positive minimum on
+that relative boundary.
+
+Apply Sard's theorem simultaneously to $\mathscr F|_{\mathcal M}$,
+which takes values in $\mathfrak Z$, and its two endpoint faces.
+For a sufficiently small common regular value
+$z\in\mathfrak Z$, its preimage in $\mathcal U$ is a compact
+one-manifold whose boundary lies only on the endpoint faces.
+The implicit function theorem at the unique nondegenerate initial
+zero gives exactly one point on the $u_*=0$ face.  Compactness of
+$\overline{\mathcal U}$ excludes other points on that face for
+small $z$: a sequence of such extra points would limit to an
+additional zero, or to the initial zero in its uniqueness
+neighborhood.  A compact one-manifold has an even number of
+boundary points.  Consequently the $u_*=1$ face is nonempty.
+Let a sequence of these regular values tend to zero and use
+compactness of $\overline{\mathcal U}$ once more.  Its limiting
+point is an exact positive zero at $u_*=1$.  This proves the
+finite-truncation assertion of
+Proposition~\ref{prop:transmission-existence}.
+
+\subsection{Removal of the outer truncation}
+
+Keep every internal parameter fixed and let $R\to\infty$ through
+the solutions just obtained.  The lapse, slope, and low-order seam
+bounds are locally uniform by
+\eqref{eq:compact-bounds} and Lemma~\ref{lem:seam-holder}.  We
+explain why the local upper bound for $w$ is also independent of
+$R$, since the fixed-truncation energy bound alone does not imply
+this assertion.
+
+Choose a fixed sphere $\mathcal S$ outside all source supports.
+On its exterior both $N_o$ and $x=N_ow$ are harmonic.  For $x$
+this follows directly from the two equations:
+\begin{equation}
+ N_o\Delta_{g_o}(N_ow)=-P,
+ \label{eq:transmission-exterior-x}
+\end{equation}
+so the conclusion holds wherever $P=0$, in particular beyond
+$\mathcal S$.  Both fields have value $1$ at the truncating
+boundary.  Positive radial superharmonic barriers on the AF end
+of weight $|x'|^{-\beta}$, with any fixed $0<\beta<1$, bound
+their deviations from $1$ by their bounded values at
+$\mathcal S$ times such a weight.  Their construction uses
+$\Delta_{g_o}|x'|^{-\beta}
+=\beta(\beta-1)|x'|^{-\beta-2}
++O(|x'|^{-\beta-3})<0$ at large radius; enlarge $\mathcal S$
+if necessary.  At a finite outer edge the zero deviation is
+consistent with the positive barrier.
+
+Suppose the supremum of $w$ on the fixed region inside
+$\mathcal S$ were unbounded, and divide by this supremum $H_j$.
+Exterior harmonic comparison for $x$, together with the global
+positive floor for $N_o$, bounds the normalized $w$ everywhere
+and gives a decay bound $C|x'|^{-\beta}$ for any subsequential
+limit.  On compact sets its scalar forcing divided by $H_j$
+tends to zero: $\Sigma_+x$ is bounded by its cutoff and the
+finite state bounds, and $P$ has fixed finite strength.  The
+coefficient matrices converge locally along a subsequence by the
+H\"older estimates for $N$ and $\dd f$.  Scalar local
+boundedness, H\"older compactness and weak energy estimates
+then give a nonnegative limit $w_\infty$, with a nonzero
+maximum on the fixed compact region, satisfying
+\begin{equation}
+ -\divg(A_\infty\dd w_\infty)=0,
+ \qquad [w_\infty]_{\mathcal C}=0,
+ \qquad [(A_\infty\dd w_\infty)_n]_{\mathcal C}=0,
+ \qquad (A_\infty\dd w_\infty)_n|_S=0.
+ \label{eq:transmission-normalized-w-limit}
+\end{equation}
+It tends to zero at infinity.  This is impossible: on arbitrarily
+large compact joined domains, the weak maximum principle with
+natural inner flux bounds it by the supremum on the distant
+Dirichlet sphere, and that supremum tends to zero.  This argument
+uses only the scalar joined divergence form and remains valid for
+the limiting piecewise coefficients.  The contradiction proves
+the required compact upper bound for $w$.
+
+Without normalization the same exterior barriers show that every
+limit has $N_o,x\to1$, and hence $w\to1$.  They also give
+derivative estimates of every fixed order on annuli after scaling,
+by ordinary scalar harmonic estimates for the fixed smooth AF
+metric.  On the compact region containing the sources, the
+H\"older estimates and the absorption argument above are uniform
+in $R$: take an auxiliary fixed surrounding annulus, whose
+derivatives are already bounded by these harmonic estimates, and
+use the finite cover of compact patches inside it.  A diagonal
+subsequence therefore converges in every fixed smooth norm on
+each closed side on compact sets.  Its positive floors and its
+boundary and transmission conditions persist.  This proves the
+outer-limit assertion and completes
+Proposition~\ref{prop:transmission-existence}.

preprints/The-Penrose-inequality-for-maximal-asymptotically-hyperbolic-initial-data-October-5-2026/build/sections/09-scalar-error.tex

@@ -0,0 +1,1015 @@
+\section{Scalar errors, smoothing, and free enclosures}
+\label{sec:scalar-error}
+
+Fix the compact data and the scalar-flat attachment of
+Proposition~\ref{prop:af-attachment}.  All solutions in this section are
+the actual solutions, with homotopy parameter equal to one, of
+Proposition~\ref{prop:compact-solutions}.  In particular, the notation
+\(N_i,N_o,f,s,w,x,W,T,\ell,\Sigma_\pm,P\) has the meaning given in
+Section~\ref{sec:compact-construction}.  Constants can depend on the fixed
+compact data, the positive numbers \(b_*,\delta_0\), the chosen exterior
+collar, and the fixed value of \(L\).  They are independent of \(M\), of
+the sufficiently small subsequent lapse cutoff, and of the sufficiently
+large finite penalty strengths, unless stated otherwise.  None of the
+arguments in this section restricts the topology of the compact part.
+
+We first state the output needed in the final comparison.  The capped
+manifold in this statement is auxiliary; no extension of the original
+initial data across its boundary is asserted.
+
+\begin{proposition}[Smooth AF comparison and two enclosures]
+\label{prop:af-comparison}
+Let \((\Omega_i,g,K)\) and \((\Omega_o,g_o)\) be the fixed inner data
+and attachment of Proposition~\ref{prop:af-attachment}, with attachment
+mass \(m_o\).  Given \(0<\delta<1\) and positive tolerances
+\(\eta_m,\eta_v,\eta_g\), the parameters of
+Proposition~\ref{prop:compact-solutions} can be chosen so that there is
+a smooth complete connected one-ended AF manifold \((X,h')\), a fixed
+topological cap set \(O\) with \(\partial O=S\), and precompact smooth
+open sets \(E_0,E\) containing \(O\) in their interiors, with the
+following properties.
+\begin{enumerate}[label=\textup{(\roman*)}]
+\item The boundaries of \(E_0\) and \(E\) are compact embedded minimal
+surfaces.  Both sets are outward minimizing, their open exteriors are
+connected, and
+\begin{equation}
+ \Per_{h'}(E_0)\le \Per_{h'}(E).
+ \label{eq:unsafe-two-perimeters}
+\end{equation}
+The set \(E_0\) is a largest perimeter minimizing enclosure of \(O\)
+alone, is strictly outward minimizing, and every component of \(E_0\)
+meets \(O\).
+\item On the original side of the caps,
+\begin{equation}
+ \Scal_{h'}\ge-2,\qquad
+ \Scal_{h'}\ge0\quad\hbox{on }X\setminus E,\qquad
+ m_{h'}\le m_o+\eta_m .
+ \label{eq:unsafe-comparison-output}
+\end{equation}
+In particular the closed exterior of \(E\) is smooth and complete,
+has exactly one AF end, and has outer minimizing minimal boundary.
+For any fixed \(1/2<\beta<1\), its end has metric decay
+\(O_2(|x'|^{-\beta})\) and scalar decay
+\(O(|x'|^{-2-2\beta})\).
+\item Identify the two original base pieces with their images in
+\(X\setminus\operatorname{int}O\), smoothly on each closed side of
+the seam, and let \(g_{\rm base}\) denote their respective metrics
+\(g,g_o\).  For the piecewise graph metric
+\(h=x\bar g\), where \(\bar g=g+s^2\dd f^2\) inside and
+\(\bar g=g_o\) outside, one has
+\begin{equation}
+ (1-\eta_g)h\le h'\le(1+\eta_g)h,\qquad
+ \bar g\ge g_{\rm base},\qquad
+ \dd V_{\bar g}=W\,\dd V_{\rm base},
+ \label{eq:unsafe-metric-output}
+\end{equation}
+after decreasing \(\eta_g\) below one if necessary, and
+\begin{equation}
+ \int_{\{x<1-\delta\}}W\,\dd V_{\rm base}<\eta_v.
+ \label{eq:unsafe-bad-volume-output}
+\end{equation}
+The scalar lower bound and the topology of \(X\setminus
+\operatorname{int}O\) are independent of the large parameters.
+\end{enumerate}
+The order of choice is the following: choose \(b_*,\delta_0\) for the
+mass tolerance, then \(\delta\), the thin exterior omitted collar and
+\(L\) for the volume tolerance, then \(M\), then the lapse cutoff and
+the finite penalty strengths.  The corner smoothing is chosen last,
+separately for each resulting fixed solution.  In the enclosure
+construction a large fixed scaled ball radius is chosen before \(M\).
+\end{proposition}
+
+We prove the proposition in several steps.  Throughout, a closed unsafe
+set is used, so excluding it from the new exterior also excludes limits
+of the regions in which the scalar sign is uncertain.
+
+\subsection{Uniform geometry at unsafe points}
+
+Define \(\mathcal U_M\subset\overline\Omega_i\) by the closed
+versions of the four trace-cutoff inequalities in
+\eqref{eq:compact-trace-cutoff}, together with \(N_i\ge\delta_0\).
+The quotient at \(S\) has the continuous interpretation specified
+there.  This is a compact set for each fixed solution.  The scalar
+identity \eqref{eq:compact-scalar} and the definition of \(\Sigma_-\)
+show that
+\begin{equation}
+ \Scal_h\ge0\quad\hbox{off }\mathcal U_M
+ \quad\hbox{on each smooth side of the seam}.
+ \label{eq:unsafe-scalar-support}
+\end{equation}
+Indeed the only possibly negative term is
+\(-\ell^2(1-\zeta(N_i))\); it vanishes when the trace cutoff is
+inactive or when \(N_i\le\delta_0\).
+
+At \(P\in\mathcal U_M\), set
+\begin{equation}
+ N_0=N_i(P),\qquad r_P=x(P)^{-1/2}.
+ \label{eq:unsafe-scale}
+\end{equation}
+Since \(f(P)\ge9M/10\), \(z(P)\le1/2\), and
+\(w(P)\ge M^{9/10}\),
+\begin{equation}
+ x(P)\ge\frac{\sqrt3}{2}\delta_0 M^{9/10},\qquad
+ r_P\le C M^{-9/20}.
+ \label{eq:unsafe-radius-bound}
+\end{equation}
+The negative term in the scalar identity is bounded independently of
+\(M\), so \eqref{eq:unsafe-scalar-support} also gives
+\(\Scal_h\ge-1\) everywhere on the two smooth pieces for all
+sufficiently large \(M\).
+
+\begin{lemma}[Geometry at the unsafe scale]
+\label{lem:unsafe-blowup}
+There is a constant \(C\ge1\) with the following property.  For every
+fixed \(R<\infty\), all sufficiently large \(M\), and every
+\(P\in\mathcal U_M\), the base ball of radius \(Rr_P\) about \(P\)
+does not meet the seam.  In interior charts, or in charts flattened at
+\(S\) when necessary, whose unit of base length is \(r_P\),
+\begin{equation}
+ C^{-1}\Id\le h\le C\Id
+ \label{eq:unsafe-uniform-metric}
+\end{equation}
+on the corresponding ball, restricted to the original side of \(S\).
+The same comparison holds after reflection across \(S\).  The constant
+\(C\) is independent of \(R\); the required lower bound for \(M\)
+may depend on \(R\).  Equivalently, the order of the quantifiers is
+\begin{equation}
+ \exists C\ \forall R<\infty\ \exists M_0(R)\
+ \forall M\ge M_0(R)\ \forall P\in\mathcal U_M.
+ \label{eq:unsafe-quantifiers}
+\end{equation}
+\end{lemma}
+
+\begin{proof}
+We first control the normalized lapse without a slope bound.
+Height compactness and a sublevel argument then exclude positive
+limiting heights and a seam at bounded scaled distance.
+Only after this do we use the gradient estimate to obtain the flat
+graph limit. Finally a truncated Harnack argument controls the
+conformal factor with a constant independent of the fixed scaled radius.
+
+It is enough to examine an arbitrary sequence \(M\to\infty\) and
+marked centers \(P\in\mathcal U_M\).  Write \(r=r_P\).  Use smooth
+base charts, with Fermi charts at a boundary, and rescale their lengths
+by \(r\); equivalently use the base metric \(g_r=r^{-2}g\).  On each
+fixed scaled compact set the base metrics converge smoothly to the
+Euclidean metric, or to the Euclidean metric on a half-space.  The two
+compact boundary faces \(S\) and \(\mathcal C\) have positive base
+separation, so at most one can remain at a bounded scaled distance.
+
+\smallskip\noindent\emph{Lapse bounds before slope bounds.}
+We first obtain lapse estimates without using height-gradient bounds.
+Testing the height equation on nested base balls with a cutoff and a
+height function bounded in absolute value by \(M\) gives, in scaled
+volume units,
+\begin{equation}
+ \int_{B_H}(W^2-1)\,\dd V_{g_r}
+ \le C_H\frac{M}{r}
+       \int_{B_{H'}}N_i\,\dd V_{g_r},\qquad H<H'.
+ \label{eq:unsafe-local-height-energy}
+\end{equation}
+For a ball reaching a Dirichlet face, subtract its constant boundary
+height before testing; the boundary term then vanishes.  To verify the
+estimate, use \(p\cdot\dd f=W^2-1\), \(|p|\le N_i\),
+\(|T|\le C\), and a cutoff derivative of size \(C_H/r\).
+These are also the only estimates required if the ball meets a face.
+
+Put \(U=N_i/N_0\), and let \(A_{H'}\) be a supremum of this
+normalized lapse on a larger fixed scaled patch.  The scalar forcing
+in its rescaled equation obeys
+\begin{align}
+ \left\|\frac{r^2}{N_0}(\Sigma_++\Sigma_-)\right\|_{L^2(B_H,g_r)}
+ &\le C_H\left(r^2+
+       M^{1/2}r^{3/2}N_0^{-1/2}A_{H'}^{1/2}\right) \\
+ &\le C_H\left(r^2+M^{-7/40}A_{H'}^{1/2}\right).
+ \label{eq:unsafe-scaled-source}
+\end{align}
+Here \(\Sigma_+\le LW\), \(|\Sigma_-|\le C\delta_0\),
+\(N_0\ge\delta_0\), and \eqref{eq:unsafe-local-height-energy} were
+used.  The divergence drift has size \(O(r)\).  The natural flux
+error at \(S\) has size \(O(rb_*/N_0)\); a bounded extension of the
+normal datum incorporates it in the divergence forcing.
+
+If no seam is present on the patch, the inhomogeneous Harnack estimate
+for \(U\), including natural-flux reflection at \(S\), bounds the
+smaller supremum by a constant times its infimum plus the right side
+of \eqref{eq:unsafe-scaled-source}.  Choose the smaller patch to
+contain the center, or use a boundary-foot ball containing it, so that
+the infimum is at most \(U(P)=1\).  Thus, on finitely many nested
+patches,
+\begin{equation}
+ A_H\le C_H\left(1+r^2+M^{-7/40}A_{H'}^{1/2}\right).
+ \label{eq:unsafe-nonseam-iteration}
+\end{equation}
+The crude bound \(N_i\le C(L)M\) gives \(A_{H'}=O(M)\).
+The exponent of \(M\) improves by
+\(a\mapsto\max(0,a/2-7/40)\).  Two applications give
+\(1\mapsto13/40\mapsto0\).  We have therefore obtained local
+boundedness by a finite iteration, before asserting that the forcing
+tends to zero.
+
+There is a slightly different normalization argument if the seam has
+bounded scaled distance.  Reflect the exterior lapse into the inner
+side in the two base Fermi collars, and set
+\(V=U+N_o^{\rm refl}/N_0\).  Use the inner density-cometric matrix
+for the summed equation.  The common induced metric implies that the
+two matrices differ by \(O(r|t|)\) at scaled distance \(|t|\) from
+the face.  Positivity and the interior gradient estimate for the
+harmonic exterior lapse on balls of radius comparable to \(|t|\)
+give
+\begin{equation}
+ |(a_i-a_o^{\rm refl})\nabla(N_o^{\rm refl}/N_0)|
+ \le Cr(N_o^{\rm refl}/N_0).
+ \label{eq:unsafe-seam-drift}
+\end{equation}
+The exterior is harmonic on the fixed omitted collar, which contains
+every such scaled patch for large \(M\).  The inner drift obeys the
+same bound with \(U\).  Consequently the total correction is a
+bounded divergence drift of size \(Cr\) times \(V\), with zero
+natural flux at the face, and the scalar forcing is still bounded by
+\eqref{eq:unsafe-scaled-source} with a larger supremum of \(V\).
+
+The infimum in Harnack for \(V\) must be anchored to the inner
+center value.  Here is that step explicitly.  Let \(m\) be the
+infimum of \(V\) on a fixed smaller half-patch containing the
+center and a central disk in the face.  On the face,
+\(N_o=dN_i\), \(0<d\le1\), whence \(U\ge m/2\).
+Choose a fixed smooth half-domain with that flat disk in its boundary
+and containing the center.  The harmonic measure of a smaller disk
+at the center is bounded below by a fixed positive number; this
+remains true as the center approaches that disk.  Write \(U=H+e\),
+where \(H\) is the harmonic replacement with the same boundary
+values.  The zero-Dirichlet estimate for \(e\), using divergence
+forcing in \(L^p\), \(3<p<6\), and scalar forcing in \(L^2\),
+gives
+\begin{equation}
+ \|e\|_\infty\le
+ C_H\left(rA_{H'}+r^2+M^{-7/40}A_{H'}^{1/2}\right).
+ \label{eq:unsafe-replacement-error}
+\end{equation}
+The remaining boundary values of \(H\) are nonnegative, and
+\(U(P)=1\).  Harmonic measure therefore bounds \(m\) by a
+constant times one plus this error.  Combining this with the Harnack
+estimate for the reflected sum yields
+\begin{equation}
+ A_H\le C_H\left(1+rA_{H'}+r^2+
+                 M^{-7/40}A_{H'}^{1/2}\right).
+ \label{eq:unsafe-seam-iteration}
+\end{equation}
+Starting again from \(O(M)\), the exponent improves by
+\(a\mapsto\max(0,a-9/20,a/2-7/40)\).  Three nested applications
+give \(1\mapsto11/20\mapsto1/10\mapsto0\).  Thus the sum is
+locally bounded.  This argument uses no derivative of the seam trace
+ratio and no separate trace compactness for either lapse.
+
+The forcing in \eqref{eq:unsafe-scaled-source} now tends to zero.
+Local Laplace estimates give \(W^{1,p}\) bounds, \(p>3\), and
+H\"older compactness, including reflected natural boundary estimates
+when appropriate.  Without a seam, every subsequential limit is a
+positive harmonic function on all of \(\R^3\), or on a half-space
+with zero Neumann data.  Positive harmonic Liouville, after reflection
+in the latter case, and the center normalization show that the limit
+is one.  At a seam the reflected sum has instead a positive constant
+limit.  Its constant is positive because it is at least the inner
+center value.  Its trace lower bound, the relation \(U\ge V/2\)
+on the face, and harmonic replacement with the error
+\eqref{eq:unsafe-replacement-error} tending to zero give a positive
+lower bound for the inner lapse up to the face.  On compact open
+inner sets it converges, in a smaller H\"older exponent, to a smooth
+positive harmonic function.  These conclusions hold on every fixed
+patch by diagonal extraction.  In particular both upper and positive
+lower lapse bounds are now available in every case.
+
+\smallskip\noindent\emph{Height compactness.}
+We turn to the height, setting
+\begin{equation}
+ \mathcal F=\frac{N_0(M-f)}{r},\qquad
+ \mathcal U=\frac{N_i}{N_0},\qquad \widetilde s=\frac{s}{N_0}.
+ \label{eq:unsafe-normalized-height}
+\end{equation}
+It is nonnegative and \(\mathcal F(P)\le M^{-1}\) by the closed
+cutoff inequality.  Its flux in the metric \(g_r\) is
+\begin{equation}
+ \mathfrak p=\widetilde s^2\nabla_{g_r}\mathcal F
+       =-\frac r{N_0}p,\qquad
+ \divg_{g_r}\mathfrak p=-rT\mathcal U=o(1).
+ \label{eq:unsafe-height-equation}
+\end{equation}
+The constitutive rule is unchanged:
+\(\widetilde s^2+|\mathfrak p|_{g_r}^2=\mathcal U^2\).
+If \(\mathcal U\) is frozen at its actual value, the flux is
+strictly monotone in its gradient argument \(\xi\), and the positive
+upper and lower lapse bounds give
+\begin{equation}
+ \langle\mathfrak p(\xi),\xi\rangle
+   \ge c\min(|\xi|^2,|\xi|),\qquad
+ |\mathfrak p(\xi)|^2
+   \le C\langle\mathfrak p(\xi),\xi\rangle.
+ \label{eq:unsafe-flux-coercivity}
+\end{equation}
+For example, in an orthonormal frame
+\(\mathfrak p(\xi)=a\xi\), where
+\(a+a^2|\xi|^2=\mathcal U^2\); these assertions follow directly
+by differentiation and by considering \(|\xi|\le1\) and
+\(|\xi|\ge1\).
+
+We require a compactness argument for possibly very steep graphs.
+In the product of a scaled base chart with a height line, the bounded
+vector field \((-\mathfrak p,1)\) has bounded divergence and its
+pairing with the upward unit graph normal is bounded below by a
+positive constant, by \eqref{eq:unsafe-flux-coercivity}.  Gauss--Green
+applied to the subgraph and supergraph in product balls gives, for
+either side's volume function \(V(a)\), an upper bound
+\(C(V'(a)+V(a))\) for the graph perimeter in that ball.  It also
+gives local perimeter bounds by using fixed surrounding balls.
+The four-dimensional isoperimetric inequality, at sufficiently small
+radii to absorb the volume term, yields
+\(V'(a)\ge cV(a)^{3/4}\).  Thus at every interior graph point both
+sides have volume at least \(ca^4\).  These constants are uniform
+on each of the fixed patches under consideration.
+
+At \(S\), \(\mathcal F=0\).  For product balls at heights bounded
+away above zero, extend the subgraph as the empty set across the
+face.  There is no new face perimeter, because its trace is empty
+there.  The same subgraph density estimate is therefore valid even
+when the ball reaches \(S\).  BV compactness on open inner product
+patches and vertical nesting now give a subsequence such that all
+bounded height truncations converge in local measure to those of an
+extended measurable function \(\mathcal F_\infty\ge0\), whose
+value is allowed initially to be \(+\infty\).
+
+We next prove that a nonzero limiting height forces a positive lower
+bound.  On a ball \(B_H\) contained in the inner side, test
+\eqref{eq:unsafe-height-equation} with
+\(\chi^2(k-\mathcal F)^+\), where \(\chi=1\) on \(B_h\).
+By \eqref{eq:unsafe-flux-coercivity} and Young's inequality,
+\begin{equation}
+ \int_{B_h}\min(|\dd(k-\mathcal F)^+|^2,
+                        |\dd(k-\mathcal F)^+|)
+ \le C\left(\frac{k^2}{(H-h)^2}+\varepsilon_M k\right)
+              |\{\mathcal F<k\}\cap B_H|,
+ \label{eq:unsafe-sublevel-energy}
+\end{equation}
+where \(\varepsilon_M\to0\) on the fixed patch.  The same test is
+allowed on a half-patch with extension by zero across a Dirichlet
+face whose height exceeds \(k\).  On the portion with derivative at
+most one, use Cauchy--Schwarz; on its complement use the linear part
+of the energy.  The \(W^{1,1}\) Sobolev inequality for an
+intermediate cutoff then gives, for \(0<l<k\),
+\begin{equation}
+ (k-l)|\{\mathcal F<l\}\cap B_h|^{2/3}
+ \le C\left(\frac{k}{H-h}+
+             \frac{k^2}{(H-h)^2}+o(1)\right)
+                |\{\mathcal F<k\}\cap B_H|.
+ \label{eq:unsafe-sublevel-sobolev}
+\end{equation}
+The error here includes \((\varepsilon_M k)^{1/2}\) and
+\(\varepsilon_M k\), which tend to zero for fixed levels.
+
+For completeness, take radii decreasing from \(H\) to \(H/2\)
+and levels decreasing from \(k\) to \(k/2\) geometrically.  First
+take \(M\) sufficiently large at the fixed \(H,k\) to absorb the
+forcing into the first cutoff term.  This also absorbs it at every
+later step, since the level remains at least \(k/2\) and the cutoff
+derivatives only increase.  After dividing volumes by \(H^3\),
+\eqref{eq:unsafe-sublevel-sobolev} has the form
+\begin{equation}
+ v_{j+1}^{2/3}\le C^{j+1}(1+k/H)v_j.
+ \label{eq:unsafe-sublevel-iteration}
+\end{equation}
+For \(0<k\le H\), a sufficiently small initial fraction, with a
+threshold independent of \(k/H\), therefore forces \(v_j\to0\).
+If \(\mathcal F_\infty\) is not zero almost everywhere, choose
+a density ball of one of its positive superlevel sets and then a
+small fixed \(k>0\).  Convergence of truncations supplies that
+small initial fraction.  Consequently \(\mathcal F\ge k/2\)
+on a smaller open inner ball for all sufficiently large indices.
+
+This lower bound propagates to compact subsets of the connected
+limiting inner domain.  To justify propagation for the capped flux,
+choose a fixed smooth connecting subdomain with a small nonnegative
+Dirichlet bump on a boundary portion inside the known positive ball
+and zero data elsewhere.  On this subdomain the actual lapse fields
+converge in a H\"older exponent to smooth positive coefficients.
+The flux is smooth and uniformly elliptic near zero gradient.  The
+Dirichlet inverse of its linearization at zero maps divergence
+\(C^\gamma\) data to \(C^{1,\gamma}\); contraction in a small
+\(C^{1,\gamma}\) ball therefore supplies small-gradient barriers,
+uniformly for the sequence, with divergence a positive constant
+tending to zero that dominates the right side of
+\eqref{eq:unsafe-height-equation}.  Their limiting solutions have
+positive interior values by the strict maximum principle.  The
+constructed boundary values are below those of \(\mathcal F\).
+Strict gradient-flux monotonicity, testing the positive part of the
+barrier minus \(\mathcal F\), proves comparison in the required
+direction.  A finite chain of such domains and a finite covering
+prove propagation on every compact inner subset.
+
+In the \(S\) alternative, the lapse convergence is also H\"older
+up to that face.  The connecting subdomains may then reach a flat
+boundary patch, with zero data on that portion.  The limiting
+barrier is smooth there and has strictly positive inward derivative
+by Hopf's lemma.  The \(C^1\) convergence of the barriers gives
+a uniform lower bound by a positive constant times inward scaled
+distance.  This contradicts the closed unsafe condition
+\begin{equation}
+ \frac{\mathcal F(P)}{d_S(P)/r}\le M^{-1},
+ \label{eq:unsafe-boundary-touchdown}
+\end{equation}
+with the normal derivative interpretation if \(P\in S\).
+An interior limiting center is already excluded by
+\(\mathcal F(P)\le M^{-1}\).  Thus a nonzero limiting height
+is impossible in these two alternatives.
+
+\smallskip\noindent\emph{Excluding seam limits.}
+If the center approaches the seam, the boundary height is instead
+\(N_0M/r\to\infty\).  A positive limiting height propagates
+through the open half-space as above.  On a fixed ball reaching the
+face, omit first a thin strip adjacent to the face.  The lower
+bound on the remaining compact set, followed by a small choice of
+\(k\), makes the starting sublevel fraction arbitrarily small.
+Apply \eqref{eq:unsafe-sublevel-iteration} with the negative part
+extended by zero across the seam.  It supplies a positive lower
+bound up to the central face patch and contradicts center
+touchdown.  This step does not require H\"older convergence of
+the individual lapse traces.
+
+The zero almost everywhere alternative is also impossible at a
+seam.  On a fixed half-patch choose a smooth nonnegative function
+\(b\) with amplitude \(k\), derivative \(O(k)\), positive by
+\(ck\) on a central face disk and zero near the other edges.
+The test \((b-\mathcal F)^+\) has zero boundary trace, since the
+seam height tends to infinity.  Monotonicity and
+\eqref{eq:unsafe-flux-coercivity} give
+\begin{equation}
+ \int_{\{\mathcal F<b\}}\min(|\dd\mathcal F|^2,
+                                      |\dd\mathcal F|)
+       \le Ck^2+o(1)k.
+ \label{eq:unsafe-seam-energy}
+\end{equation}
+The function \(v=\min(\mathcal F,b)\) has trace \(b\) on
+that disk and tends to zero in measure in the interior.  On almost
+every line normal to a smaller central disk, the fundamental
+theorem of calculus, followed by averaging over interior slices of
+a fixed thin collar \(D_\varepsilon\), gives
+\(\int_{D_\varepsilon}|\dd v|\ge ck-o(1)\).  Here convergence
+in measure suffices because \(0\le v\le k\).  Since
+\(\min(t^2,t)\ge t-1/4\) for \(t\ge0\), the left side of
+\eqref{eq:unsafe-seam-energy} is at least
+\begin{equation}
+ ck-Ck|D_\varepsilon|-\tfrac14|D_\varepsilon|-o(1).
+ \label{eq:unsafe-seam-trace-cost}
+\end{equation}
+Choose first \(k>0\) small enough that \(ck\) dominates
+\(Ck^2\), and then the collar thin enough.  This contradicts
+\eqref{eq:unsafe-seam-energy}.  All possible seam limits have now
+been excluded.  Consequently
+\begin{equation}
+ \dist_g(P,\mathcal C)/r_P\longrightarrow\infty,
+ \qquad \mathcal F_\infty=0
+ \quad\hbox{almost everywhere in the remaining limits}.
+ \label{eq:unsafe-seam-escape}
+\end{equation}
+
+\smallskip\noindent\emph{Uniform height and slope convergence.}
+The height convergence is in fact uniform on every scaled compact
+set, including up to \(S\).  Otherwise continuity and local
+convergence in measure produce graph points at some fixed positive
+height in a slightly larger patch.  A fixed small product ball at
+such a point has uniformly positive subgraph volume by the density
+bound proved above.  But that volume tends to zero because the
+limiting height is zero.  If the points approach \(S\), use the
+empty extension across the face; alternatively omit a thin boundary
+strip before taking the limit.  The strip has too little product
+volume to account for the density lower bound.  This excludes
+positive spikes as well as bounded positive plateaus.
+
+We can now apply the differential gradient estimate
+Lemma~\ref{lem:gradient-maximum}.  Under the rescaling above the
+synthetic endomorphism and trace target become \(-rK\) and
+\(-rT\), and the lapse current becomes \((r^2/N_0)B\), so the
+same identity applies.  Maximize
+\begin{equation}
+ -\log\widetilde s+j(\mathcal F)+k_0\log\chi,
+ \label{eq:unsafe-local-gradient-test}
+\end{equation}
+where \(k_0\ge2\), \(\chi\) is a fixed local cutoff, and
+\(j',j''\) are positive, bounded, and bounded away from zero on
+the now bounded height range.  If \(m=|\dd\mathcal F|_{g_r}\),
+the constitutive relations give
+\begin{equation}
+ qm^2=\frac{J}{\widetilde s^2}
+       =\frac{JW^2}{\mathcal U^2},\qquad
+ (1-y)^2(j'm)^2=\frac{(j')^2z^2}{\mathcal U^2}.
+ \label{eq:unsafe-scaled-gradient-algebra}
+\end{equation}
+At an interior maximum with \(y\ge1/2\), the positive leading
+term therefore controls \(W^2\).  The source term has size
+\(r^2\Sigma_+/N_i\le CW\), and localization costs
+\(C\chi^{-2}\).  Hence
+\begin{equation}
+ cW^2\le C(1+W+\chi^{-2}).
+ \label{eq:unsafe-slope-bound}
+\end{equation}
+At \(S\), \eqref{eq:calculus-boundary-lapse}, \(B_n=-b_*\),
+\(k_T\hat z\ge0\), and \(H_g\le0\) give
+\(\partial_n(-\log\widetilde s)\ge0\) for the inward normal.
+Also \(\partial_n\mathcal F=m\ge0\).  The derivative condition
+at a boundary maximum in \eqref{eq:unsafe-local-gradient-test}
+therefore gives \(j'm\le C/\chi\), which controls that case.
+When \(y<1/2\), \(W\le\sqrt2\) directly.  The maximum test
+and \(k_0\ge2\) thus bound \(W\) on a smaller patch.
+
+After this finite slope bound, differentiating the height divergence
+equation gives uniformly elliptic scalar divergence equations for
+its first derivatives.  Their divergence forcing is controlled in
+\(L^p\), \(p>3\), by the normalized lapse estimates.  Odd
+reflection of \(\mathcal F\), with even reflection of lapse and
+the Fermi metric, gives the same estimate at \(S\).  The resulting
+gradient H\"older estimates and the uniform height convergence
+imply
+\begin{equation}
+ \mathcal U\longrightarrow1,\qquad
+ \dd\mathcal F\longrightarrow0,\qquad
+ \widetilde s\longrightarrow1
+ \label{eq:unsafe-flat-graph-limit}
+\end{equation}
+on every fixed scaled compact set.
+
+\smallskip\noindent\emph{The conformal factor.}
+It remains to control the conformal factor uniformly in the chosen
+fixed radius.  Put \(w'=w/w(P)\).  On every fixed scaled patch,
+the height just proved gives \(f\ge9M/10\), hence \(w\ge T_1\)
+with \(T_1=M^{9/10}\).  The lapse is eventually at least
+\(\delta_0/2\); choose the subsequent cutoff
+\(\eps<\delta_0/8\).  Also
+\(x=sw\to\infty\) there.  The volume source and the low-lapse
+penalty are consequently absent.  The positive function \(w'\)
+is a supersolution for its uniformly elliptic divergence principal
+part, while
+\(v'=(w'-a)^+\), \(a=4T_1/w(P)\), is a subsolution: above
+that truncation the remaining height penalty vanishes as well.
+Weak Harnack for \(w'\), combined with the local supremum bound
+for \(v'\) with the same positive mean exponent, gives on balls
+with fixed nesting ratios
+\begin{equation}
+ \sup_{B_R}w'
+ \le a+C\left(\frac1{|B_{2R}|}\int_{B_{2R}}(w')^p\right)^{1/p}
+ \le a+C'\inf_{B_R}w'\le(4+C')\inf_{B_R}w'.
+ \label{eq:unsafe-w-harnack}
+\end{equation}
+The last step uses \(w'\ge T_1/w(P)=a/4\).  Natural zero flux
+permits reflection at \(S\).
+
+These constants can be chosen independently of the fixed radius.
+Indeed, on each arbitrarily large fixed patch the normalized
+principal matrices tend to the identity by
+\eqref{eq:unsafe-flat-graph-limit}.  If the boundary is far enough,
+use concentric balls about the center with universal radius ratios.
+Otherwise use comparable larger balls about its boundary foot in a
+single flattened chart, and reflect.  The smaller ball contains both
+the center and the requested region.  Uniform ellipticity with fixed
+constants and dilation invariance of the homogeneous estimates give
+one common constant in \eqref{eq:unsafe-w-harnack}.  The index from
+which that constant applies may depend on the radius.  Since
+\(w'(P)=1\), it follows that \(C^{-1}\le w'\le C\).
+
+In the rescaled coordinates the metric is
+\begin{equation}
+ h=\frac{x}{x(P)}
+       \left(g_r+\widetilde s^2\dd\mathcal F^2\right),
+ \qquad
+ \frac{x}{x(P)}=\frac{\widetilde s}{\widetilde s(P)}w'.
+ \label{eq:unsafe-rescaled-metric}
+\end{equation}
+Here, as usual, the expression denotes its pullback by the coordinate
+rescaling.  Equations~\eqref{eq:unsafe-flat-graph-limit} and
+\eqref{eq:unsafe-w-harnack} prove
+\eqref{eq:unsafe-uniform-metric}.  At \(S\) the mixed metric
+entries vanish, so reflection preserves this comparison.  If the
+uniform statement of the lemma failed, a sequence violating it would
+admit precisely the subsequences just analyzed, a contradiction.
+\end{proof}
+
+\subsection{Mass and a fixed enclosing comparator}
+
+On the AF end outside a common compact support, \(N_o\) and
+\(x=N_ow\) are harmonic and tend to one.  Radial barriers of
+weight \(|x'|^{-\beta}\), \(1/2<\beta<1\), and estimates on
+annuli rescaled to unit size give
+\begin{equation}
+ N_o=1+O_2(|x'|^{-\beta}),\qquad
+ x=1+O_2(|x'|^{-\beta}).
+ \label{eq:unsafe-af-decay}
+\end{equation}
+The constants in these end bounds can depend on the fixed solution.
+Thus \(h=xg_o\) is AF of this order.  Since \(g_o\) is scalar
+flat and \(x\) is harmonic there, its scalar curvature is
+\(O(|x'|^{-2-2\beta})\), an integrable decay rate.
+
+The ADM flux of the conformal metric error gives
+\begin{equation}
+ m_h-m_o=-\frac1{8\pi}
+           \lim_{R\to\infty}\int_{|x'|=R}\partial_nx\,\dd A_{g_o}.
+ \label{eq:unsafe-adm-flux}
+\end{equation}
+One may replace the conormal-area product by its Euclidean version:
+the error is quadratic in the decaying fields and tends to zero.
+On this end the current in \eqref{eq:calculus-compact-current} is
+\(Q=N_o\nabla x+(1-x)\nabla N_o\), whose flux has the same
+limit as \(\nabla x\).  Integration of its divergence over both
+pieces, cancellation of the matched seam fluxes with their common
+base area form, and the outward flux \(+b_*\) at \(S\) yield
+\begin{equation}
+ 8\pi(m_h-m_o)
+   =\int_S b_*\,\dd A_g
+      +\int\big((x-1)\Sigma_++P-\Sigma_-\big)\,\dd V_{\rm base}.
+ \label{eq:unsafe-mass-budget}
+\end{equation}
+This also proves existence of the ADM limit: the current divergence
+has compact support.  By Lemma~\ref{lem:compact-penalties},
+\begin{equation}
+ \limsup_{M\to\infty}(m_h-m_o)\le C(b_*+\delta_0).
+ \label{eq:unsafe-mass-limsup}
+\end{equation}
+The constant on the right uses only the fixed compact data.  The
+penalty cutoff makes \((x-1)\Sigma_+\le0\), and the positive
+penalty costs tend to zero after the subsequent choices specified
+in Section~\ref{sec:compact-construction}.
+
+We will need a competitor whose area remains bounded as \(M\)
+increases.  Fix a sphere \(\mathcal S_1\) in the attached end
+beyond all source supports.  Let \(\psi_1\) be its exterior
+capacitary potential: it is harmonic, is one on \(\mathcal S_1\),
+and tends to zero at infinity.  Existence follows by solving on
+compact annuli, using the preceding barriers, and taking a limit.
+The function \(\gamma_1=-\partial_n\psi_1\) is strictly positive
+on the sphere by Hopf's lemma.  Green's formula for \(x-1\) and
+\(\psi_1\) gives
+\begin{equation}
+ \int_{\mathcal S_1}\gamma_1(x-1)\,\dd A_{g_o}
+       =-\lim_{R\to\infty}\int_{|x'|=R}\partial_nx\,\dd A_{g_o}
+       =8\pi(m_h-m_o).
+ \label{eq:unsafe-capacity-comparator}
+\end{equation}
+The boundary term at infinity in the first Green identity vanishes
+by \(\beta>1/2\); alternatively apply the harmonic flux identity
+for \(x\) on the intervening annulus.  Positivity of \(x\), the
+positive minimum of \(\gamma_1\), and the upper mass bound imply
+\begin{equation}
+ \Area_h(\mathcal S_1)
+       =\int_{\mathcal S_1}x\,\dd A_{g_o}\le C_*.
+ \label{eq:unsafe-comparator-bound}
+\end{equation}
+For the fixed earlier choices, \(C_*\) is independent of large
+\(M\) and of the later parameters.  It is also independent of
+the scaled radius of the obstacles introduced below.
+
+\subsection{Removing the corner}
+
+The collar mollification and conformal correction below use Miao's
+corner-smoothing method \cite{Miao2002}. We give the local argument for
+the present unsafe-set and cap construction, rather than invoke his
+global positive mass theorem.
+
+\begin{lemma}[Corner smoothing with controlled mass]
+\label{lem:corner-smoothing}
+For each sufficiently large fixed solution, the joined metric \(h\)
+admits smooth AF replacements \(h'\) on the original side of the
+caps such that, with errors made arbitrarily small separately at
+that solution,
+\begin{equation}
+ h'=(1+o(1))h\ \hbox{as quadratic forms},\qquad
+ m_{h'}=m_h+o(1),\qquad H_{h'}(S)<0.
+ \label{eq:unsafe-smoothing-estimates}
+\end{equation}
+They satisfy \(\Scal_{h'}\ge0\) off \(\mathcal U_M\) and
+\(\Scal_{h'}\ge-2\) everywhere on the original side of the
+caps.  Their AF metric and scalar decay are those of
+\eqref{eq:unsafe-af-decay}.
+\end{lemma}
+
+\begin{proof}
+By Lemma~\ref{lem:unsafe-blowup} the closed unsafe set is disjoint
+from the seam for large \(M\).  Glue the two metrics using their
+own Gaussian normal collars.  In the resulting smooth manifold
+structure the joined metric is continuous and has the form
+\(\dd t^2+\lambda(t)\), piecewise smooth, with the inner side
+at \(t<0\).  The common induced metric and the mean-curvature
+inequality in \eqref{eq:compact-mean-curvature} give
+\begin{equation}
+ \frac12\tr_{\lambda(0)}
+              \big(\lambda'(0^+)-\lambda'(0^-)\big)\le0.
+ \label{eq:unsafe-corner-sign}
+\end{equation}
+This change of collar identification retains a smooth identification
+of each original closed piece with its image; comparisons with the
+base metric will always use those identifications.
+
+Mollify \(\lambda\) in \(t\) with a nonnegative smooth even
+kernel of radius \(a\), and interpolate back to the original
+smooth metrics in \(2a\le|t|\le3a\).  The resulting \(h_a\)
+converges uniformly to \(h\); its first normal derivatives and
+fixed tangential derivatives are bounded.  Its second normal
+derivative is the jump of \(\lambda'\) times the mollifier plus
+a uniformly bounded remainder.  The interpolation has the same
+bound, since on those smooth one-sided regions the mollification
+error is \(O(a^2)\) before taking the two cutoff derivatives.
+In the hypersurface scalar-curvature formula the only potentially
+unbounded term is \(-\tr_\lambda\lambda''\).  Replacing the
+inverse metric in its jump term by \(\lambda(0)^{-1}\) costs
+\(O(a)O(a^{-1})=O(1)\); the leading term has favorable sign by
+\eqref{eq:unsafe-corner-sign}.  Thus \(\Scal_{h_a}\ge-C(h)\)
+in the smoothing collar, uniformly as \(a\downarrow0\).
+Choose a smooth \(c_a\ge0\), bounded by a constant depending
+on the fixed metric, supported in \(|t|<4a\), and majorizing
+the negative scalar part created in that collar.
+
+Each orientable component of \(S\) bounds a compact handlebody.
+Attach such caps, extend the metric smoothly through \(S\), and
+denote their union by \(O\).  This gives a smooth complete
+one-ended AF manifold on which to solve the correction.  Its
+homogeneous Sobolev inequality
+\begin{equation}
+ \|v\|_{L^6}\le C\|\nabla v\|_{L^2},
+       \qquad v\in C_c^\infty(X),
+ \label{eq:unsafe-af-sobolev}
+\end{equation}
+has a constant uniform for small \(a\).  Indeed all these metrics
+are uniformly equivalent to one fixed smooth AF metric.  The
+Euclidean Sobolev inequality on the end and compact local Sobolev
+inequalities leave only a compact \(L^2\) term.  Integration
+along radial rays, followed by Cauchy--Schwarz with the integrable
+weight \(r^{-2}\), controls the trace at a fixed end sphere by
+the end gradient energy.  Poincar\'e's inequality with this trace
+controls the remaining compact term, proving
+\eqref{eq:unsafe-af-sobolev}.
+
+Solve in the completion for the gradient norm
+\begin{equation}
+ -8\Delta_{h_a}u_a=c_a(1+u_a).
+ \label{eq:unsafe-conformal-correction}
+\end{equation}
+Since \(\|c_a\|_{3/2}\to0\), the bilinear form
+\(8\int|\nabla v|^2-\int c_av^2\) is coercive by
+\eqref{eq:unsafe-af-sobolev}.  The right side is a bounded
+functional with norm tending to zero, because
+\(\|c_a\|_{6/5}\to0\).  Lax--Milgram gives a solution with
+\(\|u_a\|_6+\|\nabla u_a\|_2=o(1)\).  Testing with its
+negative part and using coercivity proves \(u_a\ge0\).
+Local boundedness for scalar divergence equations, treating the
+bounded \(c_a u_a\) term as a zeroth-order coefficient and the
+\(L^2\)-small \(c_a\) as forcing, gives
+\(\|u_a\|_\infty=o(1)\) on a fixed compact set.  On the
+unchanged end, \(u_a\) is harmonic and tends to zero: this
+follows first from its \(L^6\) membership and scaled local
+estimates.  Exterior comparison then gives the global supremum
+bound.  Elliptic estimates imply smooth convergence to zero on
+compact sets off the seam.  The same end barriers as before give
+\(u_a=O_2(|x'|^{-\beta})\).
+
+Set \(h'=(1+u_a)^4h_a\) and restrict to the original side of
+the caps.  The conformal scalar formula is
+\begin{equation}
+ \Scal_{h'}=(1+u_a)^{-5}
+       \big(-8\Delta_{h_a}(1+u_a)+\Scal_{h_a}(1+u_a)\big)
+       =(1+u_a)^{-4}(\Scal_{h_a}+c_a).
+ \label{eq:unsafe-corrected-scalar}
+\end{equation}
+It is nonnegative off the unsafe set.  On that set the metric
+was not mollified, \(c_a=0\), and \(\Scal_h\ge-1\);
+since \(u_a\ge0\), the lower bound remains at least \(-1\).
+In particular the claimed fixed lower bound \(-2\) holds.
+The end scalar decay is preserved by the conformal formula.
+
+The smoothing itself changes no asymptotic metric.  The conformal
+ADM flux, followed by integration of
+\eqref{eq:unsafe-conformal-correction} on the capped manifold,
+gives
+\begin{equation}
+ m_{h'}-m_h
+   =-\frac1{2\pi}\lim_{R\to\infty}
+                   \int_{|x'|=R}\partial_nu_a\,\dd A_{h_a}
+   =\frac1{16\pi}\int_X c_a(1+u_a)\,\dd V_{h_a}=o(1).
+ \label{eq:unsafe-smoothing-mass}
+\end{equation}
+The decay \(\beta>1/2\) justifies replacing conormal-area
+products in this flux as well.  Uniform metric convergence gives
+the quadratic-form comparison.  Finally \(H_h(S)<0\) by
+\eqref{eq:compact-mean-curvature}; the smooth convergence of the
+correction near this disjoint compact surface preserves that
+strict inequality.  Each smallness requirement here is imposed
+after the fixed solution has been selected.
+\end{proof}
+
+\subsection{Enclosures whose boundaries avoid the unsafe set}
+
+\begin{lemma}[Free enclosing boundaries]
+\label{lem:free-enclosure}
+Choose a sufficiently large fixed scaled radius \(R_*\), then
+\(M\) sufficiently large depending on that radius, and perform
+the smoothing of Lemma~\ref{lem:corner-smoothing} sufficiently
+closely.  There is a perimeter minimizing enclosure \(E\) of
+\(O\) and the marked balls about all points of \(\mathcal U_M\),
+whose boundary is smooth, free, and disjoint from those balls and
+from \(S\).  There is also a largest perimeter minimizing
+enclosure \(E_0\) of \(O\) alone.  These sets have all the
+boundary, exterior, and minimizing properties in
+Proposition~\ref{prop:af-comparison}.
+\end{lemma}
+
+\begin{proof}
+Extend the smooth inner base geometry across \(S\) in a fixed
+collar for the purpose of defining balls.  Mark every closed base
+ball
+\begin{equation}
+ \overline B_{g_{\rm ext}}(P,R_*r_P),\qquad P\in\mathcal U_M,
+ \label{eq:unsafe-marked-balls}
+\end{equation}
+and include their union together with \(O\) as an obstacle.
+For fixed parameters the union is compact: the center set is
+compact and its positive radius function is continuous.  By
+Lemma~\ref{lem:unsafe-blowup}, for each fixed \(R_*\) these
+balls and the fixed larger multiples used below lie off the seam
+once \(M\) is sufficiently large.  All lie inside the comparator
+\(\mathcal S_1\).
+
+Minimize \(h'\)-perimeter among precompact finite-perimeter
+sets containing this obstacle modulo null sets.  A minimizer
+exists by BV compactness and lower semicontinuity inside a
+sufficiently large smooth truncation.  This is also a global
+minimizer in the stated class.  To see that the outer truncation
+causes no restriction, all sufficiently far AF spheres have
+strictly positive outward mean curvature.  The unit normal field
+of their foliation has positive divergence.  For almost every
+cutting radius, Gauss--Green on the removed part of a competitor
+shows that cutting inward saves perimeter if the removed volume
+is positive.  The same argument excludes contact with the outer
+constraint.  The fixed comparator and the small metric change
+give a uniform bound, after enlarging \(C_*\) once,
+\begin{equation}
+ \Per_{h'}(E)\le C_*.
+ \label{eq:unsafe-enclosure-bound}
+\end{equation}
+
+We show that its perimeter support cannot meet any marked ball.
+Suppose it meets the ball marked at \(P\), at a point \(q\).
+Use scaled smooth base coordinates with length unit \(r_P\)
+on a fixed multiple of radius \(R_*\).  The base metric and
+the comparison metric are equivalent to Euclidean metrics with
+constants independent of \(R_*\), by
+Lemma~\ref{lem:unsafe-blowup} and the chosen small smoothing.
+If this region reaches \(S\), reflect in a base collar for these
+comparisons.  Mixed entries of the graph metric vanish on the
+face.  No comparison for the arbitrary cap metric is needed:
+perimeters of sets containing \(O\) are independent of its
+interior extension, and a reflected continuous metric can be
+used there in the comparison argument.
+
+The complement of \(E\) has uniform lower volume density at
+\(q\) in these scaled coordinates.  Indeed filling a coordinate
+ball preserves the obstacle condition.  Minimality and slicing
+therefore bound its interior perimeter by a constant times the
+complement's trace area on the ball.  If
+\(V(a)=|B_a(q)\setminus E|\), Euclidean isoperimetry gives
+\begin{equation}
+ V(a)^{2/3}\le C V'(a)
+ \quad\hbox{for almost every }a,\qquad
+ V(a)\ge ca^3.
+ \label{eq:unsafe-complement-density}
+\end{equation}
+The second inequality follows by integrating the first from zero;
+the volume is positive at every radius because \(q\) is in
+the perimeter support.  Apply this on a ball of radius comparable
+to \(R_*\).  On a slightly larger common coordinate ball,
+\(E\) itself has volume at least \(cR_*^3\), since it contains
+the whole marked base ball modulo null sets.  The uniform
+comparison of the scaled base metric with Euclidean metric
+justifies this statement even when the marked ball crosses
+\(S\).  Relative isoperimetry on that common ball now gives
+\begin{equation}
+ \Per_{h'}(E)\ge cR_*^2.
+ \label{eq:unsafe-obstacle-perimeter}
+\end{equation}
+Choose \(R_*\) so that this exceeds \(C_*\), and only then
+choose \(M\) large enough for all the scaled charts and their
+fixed multiples.  This contradicts
+\eqref{eq:unsafe-enclosure-bound}.  Compactness of the center set
+also covers contacts with the closure of the union of marked balls.
+This is where the radius-independent constant in
+\eqref{eq:unsafe-quantifiers} is essential.
+
+There is no contact with \(S\) either.  The parallel outward
+normal field in a sufficiently thin collar of \(O\) has strictly
+negative divergence, since \(H_{h'}(S)<0\).  Adjoin that collar
+to a purported minimizer.  Gauss--Green on the added portion
+bounds the newly exposed leaf area minus the removed perimeter
+by the integral of this negative divergence.  The added volume
+is positive for every sufficiently small width at a contact
+point; otherwise that point would not be in the perimeter
+support.  Slicing allows an almost every width for which all
+perimeter formulas hold.  The resulting strict decrease
+contradicts minimality.  Adjoining a collar is permitted whether
+or not marked balls are present.
+
+The perimeter support is thus separated from both obstacles and
+the outer constraint.  It is locally perimeter minimizing in a
+smooth three-manifold.  Interior codimension-one minimizing
+boundary regularity gives a smooth compact embedded minimal
+surface.  Take its open set representative for \(E\).  Filling
+any bounded open exterior component would strictly reduce
+perimeter, so its open exterior is connected.  Since every
+precompact enlargement is an admissible competitor, \(E\) is
+outward minimizing.  Its boundary is disjoint from every marked
+ball, so the unsafe set lies in its interior.  The metric on
+its closed exterior has nonnegative scalar curvature, and its
+completeness and unique AF end follow from smoothness of compact
+truncations and the AF estimates.
+
+Now minimize with obstacle \(O\) alone.  The same compactness,
+outer barriers, and strict inner mean-curvature barrier give
+a smooth free minimal boundary and a connected open exterior.
+Every component of this open minimizer meets \(O\), because
+an open component disjoint from \(O\) can be removed, strictly
+reducing perimeter.  Choose a minimizer of maximal volume within
+the fixed truncation.  It is a largest minimizer modulo null
+sets: the perimeter submodularity inequality implies that the
+union and intersection of any two minimizers are minimizers,
+and maximal volume forces the union to agree with this one.
+Denote its open representative by \(E_0\).  Any proper
+precompact enlargement of equal perimeter would also be a
+minimizer and would contradict maximal volume.  The outer
+barriers allow this reasoning without the truncation.  Thus
+\(E_0\) is strictly outward minimizing.  Its boundary avoids
+\(S\), so \(O\Subset E_0\).  Finally \(E\) is itself an
+admissible enclosure of \(O\), proving
+\eqref{eq:unsafe-two-perimeters}.
+\end{proof}
+
+No bound on the number or topology of components created by the
+marked balls has been used.  In particular the subsequent area
+transfer starts from \(E_0\), not from a topological assertion
+about \(E\).  The hypotheses for the AF Riemannian Penrose theorem
+are satisfied by the closed exterior of \(E\): it is smooth,
+complete and one-ended, its scalar curvature is nonnegative and
+decays as \(O(|x'|^{-2-2\beta})\), with \(2+2\beta>3\),
+its metric decay has exponent \(\beta>1/2\),
+and its entire possibly disconnected minimal boundary is outer
+minimizing.  The boundary version of \citet[Theorem~1.4]{BrayLee2009}
+therefore gives
+\begin{equation}
+ m_{h'}\ge\sqrt{\frac{\Per_{h'}(E)}{16\pi}}.
+ \label{eq:unsafe-af-penrose}
+\end{equation}
+
+\subsection{Bad volume and completion of the comparison}
+
+We finish the proof of Proposition~\ref{prop:af-comparison} by
+checking the uniform quantitative inputs for
+Proposition~\ref{prop:area-transfer}.  The exterior capacity
+comparison in Section~\ref{sec:compact-construction} gives
+\(x\ge1-\psi_\infty\), so for fixed \(\delta>0\) the set
+\(\{x<1-\delta\}\) does not reach beyond a fixed outer sphere.
+On the designated region, \eqref{eq:compact-bad-volume} gives
+\(\int_{\{x<1-\delta\}}W\le C/L\).  The omitted exterior
+collar has \(W=1\), so its contribution is bounded by its base
+volume.  The omitted inner collar has base volume \(O(M^{-4})\).
+Cauchy--Schwarz and \eqref{eq:compact-height-energy} together
+with \eqref{eq:compact-bounds} therefore give
+\begin{equation}
+ \int_{\{x<1-\delta\}}W\,\dd V_{\rm base}
+ \le \frac CL+\Vol_{g_o}(\text{omitted exterior collar})
+       +C(L)M^{-1}.
+ \label{eq:unsafe-total-bad-volume}
+\end{equation}
+The coefficient in \(C/L\) is independent of \(L\).  The
+other constants may depend on the earlier collar choice and
+on \(L\), which is harmless: first make the exterior collar
+thin and \(L\) large, then let \(M\) be large.
+
+Choose \(b_*,\delta_0\) small enough in
+\eqref{eq:unsafe-mass-limsup} for the prescribed mass tolerance.
+For the prescribed \(\delta\) and volume tolerance choose the
+collar and \(L\) as just described.  Fix the common comparator
+bound, choose \(R_*\) in Lemma~\ref{lem:free-enclosure}, and
+then choose \(M\) large enough for that lemma, the scalar
+lower bound, the mass costs, and
+\eqref{eq:unsafe-total-bad-volume}.  The subsequent small lapse
+cutoff and finite strengths are those of
+Proposition~\ref{prop:compact-solutions}; all uniform estimates
+above hold for those choices.  Apply
+Lemma~\ref{lem:corner-smoothing} with mass and metric errors
+small enough for the remaining tolerances.  These choices prove
+\eqref{eq:unsafe-comparison-output}--\eqref{eq:unsafe-bad-volume-output}.
+
+The caps may be fixed once as topological handlebodies, regardless
+of all metric parameters.  The topology entering the area-transfer
+argument is consequently the fixed finite topology of
+\(X\setminus\operatorname{int}O\).  Positive-time flow levels
+outside \(E_0\) are allowed to enter the unsafe region: the
+uniform lower scalar bound there is precisely the input supplied
+by \eqref{eq:unsafe-comparison-output}.  The mass theorem is
+applied to the unmodified smooth exterior of \(E\), whereas
+the small initial mean-curvature change used solely for the flow
+is covered by Proposition~\ref{prop:area-transfer}.  This
+completes the AF comparison construction.

preprints/The-Penrose-inequality-for-maximal-asymptotically-hyperbolic-initial-data-October-5-2026/build/sections/10-completion.tex

@@ -0,0 +1,164 @@
+\section{Completion of the proof}
+\label{sec:completion}
+
+\subsection{The time-symmetric inequality}
+
+\begin{proof}[Proof of Theorem~\ref{thm:ts}]
+The approximation of Proposition~\ref{prop:conformal-approximation}
+preserves $H(S)\le0$, converges uniformly in metric comparison, and
+preserves the limiting time component. The comparison of enclosing
+areas in Section~\ref{sec:preliminaries} therefore allows us first to
+assume that $R_g+6>0$ and that the end has the smooth conformal form
+used in Section~\ref{sec:stress-extension}. Fix this approximated
+metric for the constructions that follow, and put
+\[
+ A_*=A_{\min}(S;g),\qquad a=\sqrt{A_*/(4\pi)}.
+\]
+
+Given an arbitrarily small mass tolerance $\varepsilon_m>0$,
+Proposition~\ref{prop:af-attachment} produces a fixed compact inner
+piece, a smooth synthetic tensor on it, and a scalar-flat AF
+attachment with mass $m_o$ satisfying
+\begin{equation}\label{eq:completion-attachment-budget}
+ m_o\le p_0(g)-\frac{a^3}{2}+\varepsilon_m.
+\end{equation}
+The electrostatic height, attachment radius, and smoothing tolerances
+used to obtain this inequality have already been fixed. They are
+independent of the large graph height in the next construction.
+
+Apply Proposition~\ref{prop:compact-solutions} and the comparison
+construction of Proposition~\ref{prop:af-comparison}. Their parameters
+are chosen in the proved order: the small inner flux and scalar
+cutoff costs first, then a desired relative-area tolerance
+$0<\varepsilon_A<1$, a sufficiently small contraction threshold,
+a thin omitted exterior collar, and a sufficiently large fixed
+volume penalty; next the graph height is taken large, and finally
+the lapse clipping, finite penalty strengths, and corner-smoothing
+tolerances are chosen. The costs can be made small enough that the
+resulting smooth AF comparison metric satisfies
+\begin{equation}\label{eq:completion-comparison-budget}
+ m_{\AF}(h')\le m_o+\varepsilon_m.
+\end{equation}
+
+Let $E$ be its minimizing enclosure containing the caps and the
+unsafe-region obstacles. Its free boundary is smooth, compact,
+minimal, and outer minimizing. The exterior is complete and has
+$R_{h'}\ge0$, the AF decay required in Theorem~\ref{thm:af-penrose},
+and one end. Let $E_0$ instead be a largest perimeter-minimizing
+enclosure of the caps alone. Then
+\begin{equation}\label{eq:completion-enclosure-order}
+ \Area_{h'}(\partial E)\ge\Area_{h'}(\partial E_0).
+\end{equation}
+The exterior of $E_0$ has a scalar lower bound independent of the
+large graph height, even where it meets the unsafe region.
+The bad-volume estimates from Sections~\ref{sec:compact-construction}
+and~\ref{sec:scalar-error}, together with the projection along the
+AF attachment, satisfy Proposition~\ref{prop:area-transfer}.
+Consequently, when $A_*>0$, the parameters above can be chosen so that
+\begin{equation}\label{eq:completion-area}
+ \Area_{h'}(\partial E_0)\ge(1-\varepsilon_A)A_*.
+\end{equation}
+The small initial conformal change used only for the flow fixes this
+initial area. It is not made in applying the mass theorem.
+
+Theorem~\ref{thm:af-penrose} applies to the exterior of $E$, not the
+possibly unsafe exterior of $E_0$. Equations
+\eqref{eq:completion-attachment-budget}--\eqref{eq:completion-area}
+give
+\begin{align*}
+ p_0(g)
+ &\ge \frac{a^3}{2}+m_o-\varepsilon_m\\
+ &\ge \frac{a^3}{2}+m_{\AF}(h')-2\varepsilon_m\\
+ &\ge \frac{a^3}{2}+\frac a2\sqrt{1-\varepsilon_A}
+                         -2\varepsilon_m.
+\end{align*}
+Let the two tolerances tend to zero, making fresh choices of the
+finite parameters as necessary. If $A_*=0$, the same construction
+and the nonnegativity consequence of Theorem~\ref{thm:af-penrose}
+give $p_0(g)\ge-2\varepsilon_m$ without the area step.
+Thus \eqref{eq:ts-inequality} holds for every fixed approximated
+metric. Passing back through the preliminary approximation proves
+it for the original metric.
+\end{proof}
+
+\subsection{Maximal data and invariant mass}
+
+\begin{proof}[Proof of the inequality in Theorem~\ref{thm:main}]
+The maximal constraint and \eqref{eq:dec-integrability} imply
+\[
+ \int_{\mathrm{end}} V_0|R_g+6|\dd V_g<\infty.
+\]
+Use Proposition~\ref{prop:mass-covariance} to choose a rest chart
+for the original covector. All hypotheses remain valid in that
+chart and $p_0(g)=m_{\AH}(g)$.
+
+Define
+\[
+ F(A)=\frac12\left(\sqrt{\frac A{4\pi}}
+                       +\left(\frac A{4\pi}\right)^{3/2}\right).
+\]
+This function is continuous, nonnegative, and nondecreasing on
+$[0,\infty)$. Theorem~\ref{thm:ts} supplies the time-symmetric input
+of Theorem~\ref{thm:ah-reduction}, including comparison metrics whose
+covectors are not timelike. The reduction therefore gives
+\[
+ p_0(g)\ge F\bigl(A_{\min}(S;g)\bigr).
+\]
+Since the chart was fixed before any construction, this is precisely
+\eqref{eq:main-inequality}. The one-sided time-component budgets
+suffice; there is no further assertion about the spatial components
+of intermediate comparison metrics.
+\end{proof}
+
+\subsection{Sharpness}
+
+Let $m>0$, and let $a$ be the positive solution of
+$a+a^3=2m$. The Schwarzschild--anti-de Sitter metric is
+\begin{equation}\label{eq:sads-metric}
+ g_m=\frac{\dd r^2}{1+r^2-2m/r}+r^2\sigma,
+ \qquad r\ge a,\qquad K=0.
+\end{equation}
+To include its boundary smoothly, use proper distance $s$ from
+$r=a$. Then
+\[
+ g_m=\dd s^2+r(s)^2\sigma,
+ \quad r(0)=a,\quad r'(0)=0,
+ \quad (r')^2=1+r^2-2m/r.
+\]
+The simple positive radial derivative of the expression on the
+right at $a$ gives a smooth even solution $r(s)$ near zero, increasing
+for $s>0$. Hence the horizon is a smooth minimal boundary and the
+exterior is complete with that boundary included. Since
+$r''=r+m/r^2$, the warped-product formula gives
+\[
+ R_{g_m}=-\frac{4r''}{r}
+             +\frac{2(1-(r')^2)}{r^2}=-6.
+\]
+The constraints are vacuum, the metric error and its required tensor
+derivatives are $O(r^{-3})$, and the weighted matter integral vanishes.
+
+For completeness the mass normalization can be checked directly.
+In the background distance coordinate, write
+$g_m-b=\zeta\dd\eta^2$, where
+\[
+ \zeta=\frac{2m}{r(1+r^2-2m/r)}.
+\]
+The radial component of $\divg_b e-\dd\tr_be$ is
+$2\coth\eta\,\zeta$, and the other two terms of
+\eqref{eq:mass-integrand} cancel for $V_0$.
+The normalized sphere flux therefore tends to
+\[
+ m\frac{1+r^2}{1+r^2-2m/r}\longrightarrow m.
+\]
+Rotational symmetry makes the three spatial fluxes zero, so
+$m_{\AH}(g_m)=m$.
+
+Radial projection onto the horizon has two-area Jacobian at most one,
+since $r\ge a$. Every radial ray meets the entire intrinsic boundary
+of any admissible end-side domain: either it enters that domain
+across its boundary, or its initial point already lies on the counted
+boundary at $S$. Thus projection of the cut covers the horizon.
+The area formula gives $\Area_{g_m}(\partial D)\ge4\pi a^2$.
+The horizon itself is an admissible competitor, and hence
+$A_{\min}(S;g_m)=4\pi a^2$. Equation $2m=a+a^3$ proves equality in
+\eqref{eq:main-inequality} and completes Theorem~\ref{thm:main}.

preprints/The-Penrose-inequality-for-maximal-asymptotically-hyperbolic-initial-data-October-5-2026/paper.pdf

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:641d4aa3892adbca1c14c7bca0f165edc6b508e88f6b18880e3494a5495c39c2
+size 853015

preprints/The-Poisson-Dirichlet-Law-for-Prime-Predecessors-September-24-2026/README.md

@@ -0,0 +1,18 @@
+# [The Poisson-Dirichlet law for prime predecessors](paper.pdf)
+
+**Author:** OpenAI
+
+**Date:** September 24, 2026
+
+## Citation
+
+```bibtex
+@misc{OAI:The-Poisson-Dirichlet-Law-for-Prime-Predecessors-September-24-2026,
+  author = {{OpenAI}},
+  title = {{The Poisson--Dirichlet law for prime predecessors}},
+  howpublished = {OpenAI Math Release preprint
+                  \href{https://github.com/openai/math/blob/main/preprints/The-Poisson-Dirichlet-Law-for-Prime-Predecessors-September-24-2026/paper.pdf}{OAI:The-Poisson-Dirichlet-Law-for-Prime-Predecessors-September-24-2026}},
+  year = {2026}
+}
+```
+

preprints/The-Poisson-Dirichlet-Law-for-Prime-Predecessors-September-24-2026/build/figures/lifespan.tex

@@ -0,0 +1,40 @@
+\begin{figure}[tb]
+\centering
+\begin{tikzpicture}[x=1.64cm,y=1cm,
+  hit/.style={circle,draw=black,fill=white,inner sep=2.4pt},
+  active/.style={circle,draw=blue!65!black,fill=blue!25,inner sep=2.4pt},
+  ghost/.style={diamond,draw=black,fill=black!15,inner sep=2.4pt}]
+\draw[thick,black!55] (0,0)--(7,0);
+\draw[very thick,blue!65!black] (1,0)--(2,0);
+\foreach \j in {0,...,7}
+  \node[font=\small,above] at (\j,.23) {$\j$};
+\node[ghost] at (0,0) {};
+\node[active] at (1,0) {};
+\node[active] at (2,0) {};
+\node[hit] at (3,0) {};
+\node[circle,draw=black,dashed,fill=white,inner sep=2.4pt] at (4,0) {};
+\node[active] at (5,0) {};
+\node[hit] at (6,0) {};
+\node[ghost] at (7,0) {};
+\node[font=\scriptsize,align=center,below] at (0,-.22) {ghost\\birth};
+\node[font=\scriptsize,below] at (1,-.22) {active};
+\node[font=\scriptsize,below] at (2,-.22) {active};
+\node[font=\scriptsize,align=center,below] at (3,-.22) {pending\\hit};
+\node[font=\scriptsize,align=center,below] at (4,-.22) {pending\\miss};
+\node[font=\scriptsize,below] at (5,-.22) {active};
+\node[font=\scriptsize,align=center,below] at (6,-.22) {pending\\hit};
+\node[font=\scriptsize,align=center,below] at (7,-.22) {ghost\\death};
+\draw[decorate,decoration={brace,mirror,amplitude=4pt}]
+  (.93,-.88)--(2.07,-.88)
+  node[midway,below=5pt,font=\scriptsize] {first active run};
+\node[font=\scriptsize,below] at (5,-.95) {second active run};
+\end{tikzpicture}
+\caption{A schematic lifespan of a prime from group $i$.
+All active visits and both ghost endpoints have its common line;
+the pending visit at $4$ misses that line. The line contributes
+$1/(p+1)$ once. After extracting $b'_p$, the ghost signs,
+pending-hit damping, and active-run normalizations give the weight
+$\eta'_0\eta'_7q_3q_6 V_i^{-2}/((p+1)b'_p)$.
+The two powers of $V_i^{-1}$ count active runs, not active visits.}
+\label{fig:lifespan}
+\end{figure}

preprints/The-Poisson-Dirichlet-Law-for-Prime-Predecessors-September-24-2026/build/figures/proof-map.tex

@@ -0,0 +1,40 @@
+\begin{figure}[tb]
+\centering
+\begin{tikzpicture}[
+  box/.style={draw=black!65,rounded corners=2pt,align=center,
+    text width=5.55cm,minimum height=1.03cm,inner sep=5pt,font=\small},
+  arr/.style={-{Latex[length=2mm]},thick},
+  node distance=8mm and 7mm]
+\node[box,fill=blue!5] (det)
+  {Determinant graph and signed memory\\
+   Sections~\ref{sec:determinant}--\ref{mem:section}};
+\node[box,fill=blue!5,right=of det] (dil)
+  {Dilation-graph inputs and endpoint estimates\\
+   Section~\ref{sec:correlations}};
+\node[box,below=of det] (one)
+  {One-sided marked correlation\\Theorem~\ref{thm:type-II}};
+\node[box,below=of dil] (two)
+  {Two-sided marked correlation\\Theorem~\ref{thm:two-sided}};
+\node[box,below=11mm of $(one.south)!.5!(two.south)$,
+      text width=9cm] (interior)
+  {Presieve, replacement of prime slots, and averaging of marks\\
+   Interior factorial moments: Theorem~\ref{thm:interior}};
+\node[box,below=of interior,text width=9cm,fill=blue!5] (pd)
+  {Size-biased sampling, full mass, and decreasing rearrangement\\
+   Poisson--Dirichlet limit: Theorem~\ref{thm:main}};
+\draw[arr] (det) -- node[right,font=\scriptsize,align=left]
+  {major-term estimate\\Section~\ref{sec:major-arcs}} (one);
+\draw[arr] (dil) -- (two);
+\draw[arr] (one.south) -- (interior.north west);
+\draw[arr] (two.south) -- (interior.north east);
+\draw[arr] (interior) -- (pd);
+\end{tikzpicture}
+\caption{The proof has two analytic inputs. The one-sided estimate
+replaces prime slots in composite terms by rough slots. Successful
+marking brings the resulting centered statistic under the two-sided
+estimate. Averaging the fixed marks then gives interior moments;
+the probability argument recovers the full law.
+Appendix~\ref{lp:section} supplies the long-prime and
+logarithmic-phase estimates used in the analytic branches.}
+\label{fig:proof-map}
+\end{figure}

preprints/The-Poisson-Dirichlet-Law-for-Prime-Predecessors-September-24-2026/build/paper.tex

@@ -0,0 +1,30 @@
+\documentclass[11pt]{article}
+\input{preamble}
+\title{The Poisson--Dirichlet law for prime predecessors}
+\author{OpenAI}
+\date{September 24, 2026}
+\begin{document}
+\maketitle
+\begin{abstract}
+For a prime $p$ chosen uniformly from $3\le p\le x$, list the prime factors of $p-1$ in decreasing order, with multiplicity. As $x\to\infty$, their logarithms, divided by $\log(p-1)$, converge in every finite joint distribution to the Poisson--Dirichlet distribution with parameter one. This proves the conjecture of Ford, Konyagin and Luca.
+\end{abstract}
+\tableofcontents
+\input{sections/01-introduction}
+\input{sections/02-preliminaries}
+\input{sections/03-determinant}
+\input{sections/04-memory}
+\input{sections/05-major-arcs}
+\input{sections/06-correlations}
+\input{sections/07-extraction}
+\input{sections/08-poisson-dirichlet}
+\appendix
+\input{sections/09-long-prime-appendix}
+\clearpage
+\phantomsection
+\addcontentsline{toc}{section}{References}
+\begingroup
+\raggedright
+\bibliographystyle{plain}
+\bibliography{references}
+\endgroup
+\end{document}

preprints/The-Poisson-Dirichlet-Law-for-Prime-Predecessors-September-24-2026/build/preamble.tex

@@ -0,0 +1,43 @@
+\usepackage[T1]{fontenc}
+\usepackage{lmodern}
+\usepackage{amsmath,amssymb,amsthm,mathtools,mathrsfs}
+\usepackage[margin=1in]{geometry}
+\usepackage{enumitem,microtype,booktabs,array,longtable}
+\usepackage{xcolor,graphicx,tikz}
+\usetikzlibrary{arrows.meta,positioning,calc,fit,decorations.pathreplacing,shapes.geometric}
+\usepackage[hyphens]{url}
+\usepackage[colorlinks=true,linkcolor=black,citecolor=blue!55!black,urlcolor=blue!55!black]{hyperref}
+\usepackage[nameinlink,noabbrev,capitalize]{cleveref}
+\hypersetup{pdftitle={The Poisson--Dirichlet law for prime predecessors},pdfauthor={OpenAI}}
+\newcommand{\1}{\mathbf{1}}
+\newcommand{\e}{\mathrm{e}}
+\newcommand{\E}{\mathbb E}
+\newcommand{\Prob}{\mathbb P}
+\newcommand{\N}{\mathbb N}
+\newcommand{\Z}{\mathbb Z}
+\newcommand{\R}{\mathbb R}
+\newcommand{\C}{\mathbb C}
+\newcommand{\eps}{\varepsilon}
+\newcommand{\dd}{\,\mathrm d}
+\DeclarePairedDelimiter{\norm}{\lVert}{\rVert}
+\DeclarePairedDelimiter{\abs}{\lvert}{\rvert}
+\newcommand{\inner}[2]{\langle #1,#2\rangle}
+\DeclareMathOperator{\Li}{Li}
+\DeclareMathOperator{\supp}{supp}
+\DeclareMathOperator{\cond}{cond}
+\DeclareMathOperator{\lcm}{lcm}
+\DeclareMathOperator{\rad}{rad}
+\DeclareMathOperator{\PD}{PD}
+\newtheorem{theorem}{Theorem}[section]
+\newtheorem{lemma}[theorem]{Lemma}
+\newtheorem{proposition}[theorem]{Proposition}
+\newtheorem{corollary}[theorem]{Corollary}
+\theoremstyle{definition}
+\newtheorem{definition}[theorem]{Definition}
+\theoremstyle{remark}
+\newtheorem{remark}[theorem]{Remark}
+\numberwithin{equation}{section}
+\setlist[enumerate]{itemsep=3pt,topsep=5pt}
+\setlist[itemize]{itemsep=3pt,topsep=5pt}
+\allowdisplaybreaks[1]
+\emergencystretch=1em