翻訳待ち：A Forced-Structure Reduction and Verifiable Bounds for Conway's 99-Graph
AI サービスが一時的に利用できないため、復旧後に翻訳を補完します。ソース概要：arXiv:2608.11211v1 Announce Type: new Abstract: Conway's 99-graph problem asks whether a strongly regular graph with parameters $\mathrm{srg}(99,14,1,2)$ exists. We report a systematic, fully reproducible attack by an autonomous AI research agent, scored under the track's partial-credit metric. Our verifiable contributions are: (1) an exhaustive proof that no circulant graph on $\mathbb{Z}/99$ satisfies more than $3366/4950=68.0\%$ of the constraints ($33$ of $49$ difference-classes), with the same ceiling for the other abelian group of order $99$; (2) a forced-structure reduction: $\lambda=1$ makes each neighbourhood a perfect matching and $\mu=2$ puts the outer vertices in bijection with non-matched neighbour-pairs, collapsing existence to a $12$-regular graph on $84$ vertices, encoded for CP-SAT and validated by recovering the unique $\mathrm{srg}(9,4,1,2)$; (3) a validated prescribed-automorphism orbit-existence framework (fixed-point-free and single-fixed-point actions, checked on $\mathrm{srg}(9,4,1,2)$ and the Paley graph $\mathrm{srg}(13,6,2,3)$), and (4) a best verified artifact at $69.43\%$, with evidence that this is a robust frontier (fourteen distinct methods, none exceeding it) entangled with the open question, since any provable bound below $4950$ is a non-existence proof.
AI サービスが一時的に利用できないため、復旧後に翻訳を補完します。
--> [Submitted on 13 Jul 2026] Title:A Forced-Structure Reduction and Verifiable Bounds for Conway's 99-Graph View a PDF of the paper titled A Forced-Structure Reduction and Verifiable Bounds for Conway's 99-Graph, by Aalok Thakkar View PDF HTML (experimental) Abstract:Conway's 99-graph problem asks whether a strongly regular graph with parameters $\mathrm{srg}(99,14,1,2)$ exists. We report a systematic, fully reproducible attack by an autonomous AI research agent, scored under the track's partial-credit metric. Our verifiable contributions are: (1) an exhaustive proof that no circulant graph on $\mathbb{Z}/99$ satisfies more than $3366/4950=68.0\%$ of the constraints ($33$ of $49$ difference-classes), with the same ceiling for the other abelian group of order $99$; (2) a forced-structure reduction: $\lambda=1$ makes each neighbourhood a perfect matching and $\mu=2$ puts the outer vertices in bijection with non-matched neighbour-pairs, collapsing existence to a $12$-regular graph on $84$ vertices, encoded for CP-SAT and validated by recovering the unique $\mathrm{srg}(9,4,1,2)$; (3) a validated prescribed-automorphism orbit-existence framework (fixed-point-free and single-fixed-point actions, checked on $\mathrm{srg}(9,4,1,2)$ and the Paley graph $\mathrm{srg}(13,6,2,3)$), and (4) a best verified artifact at $69.43\%$, with evidence that this is a robust frontier (fourteen distinct methods, none exceeding it) entangled with the open question, since any provable bound below $4950$ is a non-existence proof. Comments: This paper is accepted to the first Conference For AI Scientists (CAISc) Subjects: Artificial Intelligence (cs.AI); Symbolic Computation (cs.SC); Combinatorics (math.CO) Cite as: arXiv:2608.11211 [cs.AI] (or arXiv:2608.11211v1 [cs.AI] for this version) https://doi.org/10.48550/arXiv.2608.11211 arXiv-issued DOI via DataCite Submission history From: Aalok Thakkar [view email] [v1] Mon, 13 Jul 2026 18:42:21 UTC (18 KB) Full-text links: Access Paper: View a PDF of the paper titled A Forced-Structure Reduction and Verifiable Bounds for Conway's 99-Graph, by Aalok Thakkar View PDF HTML (experimental) TeX Source view license Current browse context: cs.AI new | recent | 2026-08 Change to browse by: cs cs.SC math math.CO References & Citations NASA ADS Google Scholar Semantic Scholar Loading... Data provided by: Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer (What is the Explorer?) Connected Papers Toggle Connected Papers (What is Connected Papers?) Litmaps Toggle Litmaps (What is Litmaps?) scite.ai Toggle scite Smart Citations (What are Smart Citations?) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv (What is alphaXiv?) Links to Code Toggle CatalyzeX Code Finder for Papers (What is CatalyzeX?) DagsHub Toggle DagsHub (What is DagsHub?) GotitPub Toggle Gotit.pub (What is GotitPub?) Huggingface Toggle Hugging Face (What is Huggingface?) ScienceCast Toggle ScienceCast (What is ScienceCast?) Demos Demos Replicate Toggle Replicate (What is Replicate?) Spaces Toggle Hugging Face Spaces (What is Spaces?) Spaces Toggle TXYZ.AI (What is TXYZ.AI?) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower (What are Influence Flowers?) Core recommender toggle CORE Recommender (What is CORE?) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs. Which authors of this paper are endorsers? | Disable MathJax (What is MathJax?)