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

arXiv:2608.11211v1 Announce Type: new Abstract: 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: 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 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.

Source: arXiv cs.AI | 2026-08-13