diff --git a/.gitattributes b/.gitattributes index ba214a2ec45abfca09aa0118dac7341d084573b8..7e671d473e9adb2bb47e30c6a3df475db8cac8db 100644 --- a/.gitattributes +++ b/.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 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diff --git a/preprints/The-PCP-for-PPAD-conjecture-a-quasilinear-reduction-September-25-2026/README.md b/preprints/The-PCP-for-PPAD-conjecture-a-quasilinear-reduction-September-25-2026/README.md new file mode 100644 index 0000000000000000000000000000000000000000..a3a04fe4103235f0cf2afddc6f3bdb6943acc81e --- /dev/null +++ b/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} +} +``` + diff --git a/preprints/The-PCP-for-PPAD-conjecture-a-quasilinear-reduction-September-25-2026/build/main.tex b/preprints/The-PCP-for-PPAD-conjecture-a-quasilinear-reduction-September-25-2026/build/main.tex new file mode 100644 index 0000000000000000000000000000000000000000..e4313f4b15d74f3b0c44926b39c37952537e3446 --- /dev/null +++ b/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} diff --git a/preprints/The-PCP-for-PPAD-conjecture-a-quasilinear-reduction-September-25-2026/build/references.bib b/preprints/The-PCP-for-PPAD-conjecture-a-quasilinear-reduction-September-25-2026/build/references.bib new file mode 100644 index 0000000000000000000000000000000000000000..eee66f218f7058f2f883bc5151b91f6bbabc53dd --- /dev/null +++ b/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. 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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}} +} diff --git a/preprints/The-PCP-for-PPAD-conjecture-a-quasilinear-reduction-September-25-2026/build/sections/01-introduction.tex b/preprints/The-PCP-for-PPAD-conjecture-a-quasilinear-reduction-September-25-2026/build/sections/01-introduction.tex new file mode 100644 index 0000000000000000000000000000000000000000..a4ab38db211310e2f24c53472cfa8ebf241b17e0 --- /dev/null +++ b/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 +\[ + x_ax_b+\eps\ \Longrightarrow\ x_z\leq\eps. +\] +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}. diff --git a/preprints/The-PCP-for-PPAD-conjecture-a-quasilinear-reduction-September-25-2026/build/sections/02-macrograph.tex b/preprints/The-PCP-for-PPAD-conjecture-a-quasilinear-reduction-September-25-2026/build/sections/02-macrograph.tex new file mode 100644 index 0000000000000000000000000000000000000000..07a67bf3643a300452223eb616e2d86c4e7870e5 --- /dev/null +++ b/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