A Forced-Structure Reduction and Verifiable Bounds for Conway's 99-Graph

Computer Science > Artificial Intelligence [Submitted on 13 Jul 2026] A Forced-Structure Reduction and Verifiable Bounds for Conway's 99-Graph Aalok Thakkar Conway's 99-graph problem asks whether a strongly regular graph with parameters 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 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: λ=1 makes each neighbourhood a perfect matching and μ=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 srg(9,4,1,2) ; (3) a validated prescribed-automorphism orbit-existence framework (fixed-point-free and single-fixed-point actions, checked on srg(9,4,1,2) and the Paley graph 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 Focus to learn more Submission history From: Aalok Thakkar [view email] [v1] Mon, 13 Jul 2026 18:42:21 UTC (18 KB) Access Paper: HTML (experimental) view license Current browse context: cs.AI < prev   |   next > new | recent | 2026-08 Change to browse by: cs cs.SC math math.CO References & Citations NASA ADS Google Scholar Semantic Scholar Export BibTeX Citation Bookmark 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 Demos Related Papers About arXivLabs Which authors of this paper are endorsers? | Disable MathJax (What is MathJax?)