{"ok":true,"article":{"slug":"a-forced-structure-reduction-and-verifiable-bounds-for-conway-s-99-graph-7d2338d6","title":"A Forced-Structure Reduction and Verifiable Bounds for Conway's 99-Graph","url":"https://arxiv.org/abs/2608.11211","canonical":"https://www.aimode.news/article/a-forced-structure-reduction-and-verifiable-bounds-for-conway-s-99-graph-7d2338d6","sourceName":"arXiv cs.AI","summary":"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.","category":"AI","image":null,"lang":"en","publishedAt":"2026-08-13T04:00:00+00:00","createdAt":"2026-08-13T04:00:14.554378+00:00"}}