AI research agent achieves verifiable bound in Conway’s 99-graph problem

A new arXiv preprint reports that an autonomous AI research agent systematically attacked Conway’s 99-graph problem, a long-standing open question in combinatorics that asks whether a strongly regular graph with parameters srg(99,14,1,2) exists.

The agent’s verifiable contributions include an exhaustive proof that no circulant graph on Z/99 satisfies more than 3366/4950 (68.0%) of the constraints, with the same ceiling applying to the other abelian group of order 99.

The work introduces a forced-structure reduction showing that λ=1 implies each neighborhood is a perfect matching and μ=2 places outer vertices in bijection with non-matched neighbor pairs, reducing the existence question to a 12-regular graph on 84 vertices encoded for CP-SAT and validated by recovering the unique srg(9,4,1,2).

The paper also presents a validated prescribed-automorphism orbit-existence framework checked on srg(9,4,1,2) and the Paley graph srg(13,6,2,3), alongside a best verified artifact at 69.43% achieved by fourteen distinct methods, none of which exceeded this bound.

The results are entangled with the open problem because any provable bound below 4950 constitutes a non-existence proof for the target graph.