{"slug": "a-forced-structure-reduction-and-verifiable-bounds-for-conway-s-99-graph", "title": "A Forced-Structure Reduction and Verifiable Bounds for Conway's 99-Graph", "summary": "An autonomous AI research agent has reported a systematic attack on Conway's 99-graph problem, proving that no circulant graph on Z/99 satisfies more than 68.0% of the constraints (33 of 49 difference-classes), and achieving a best verified artifact at 69.43% with evidence of a robust frontier across fourteen methods. The agent also introduced a forced-structure reduction that collapses the existence problem to a 12-regular graph on 84 vertices, validated by recovering the unique srg(9,4,1,2).", "body_md": "arXiv:2608.11211v1 Announce Type: new\nAbstract: 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.", "url": "https://wpnews.pro/news/a-forced-structure-reduction-and-verifiable-bounds-for-conway-s-99-graph", "canonical_source": "https://arxiv.org/abs/2608.11211", "published_at": "2026-08-13 04:00:00+00:00", "updated_at": "2026-08-13 04:18:48.323086+00:00", "lang": "en", "topics": ["artificial-intelligence", "ai-research"], "entities": ["Conway's 99-graph", "arXiv", "AI research agent", "CP-SAT", "Paley graph"], "alternates": {"html": "https://wpnews.pro/news/a-forced-structure-reduction-and-verifiable-bounds-for-conway-s-99-graph", "markdown": "https://wpnews.pro/news/a-forced-structure-reduction-and-verifiable-bounds-for-conway-s-99-graph.md", "text": "https://wpnews.pro/news/a-forced-structure-reduction-and-verifiable-bounds-for-conway-s-99-graph.txt", "jsonld": "https://wpnews.pro/news/a-forced-structure-reduction-and-verifiable-bounds-for-conway-s-99-graph.jsonld"}}