An AI agent did not solve the Conway problem, but it narrowed the search space

The AI agent did not prove either the existence or the impossibility of Conway's 99-graph. But it showed what a useful result from an autonomous researcher can look like: not a loud "solution" but checkable intermediate bounds that narrow the search space.

The problem asks whether a strongly regular graph with parameters srg(99,14,1,2) exists. In the work the agent obtained its best confirmed artefact at the level of 69.43% of constraints. The number itself sounds impressive only until you remember that the problem is still open and that partially satisfying the constraints is no proof the graph exists.

The most substantive part is the structural reduction. With λ=1 each vertex neighbourhood becomes a perfect matching, and with μ=2 outer vertices can be mapped to pairs of neighbours not joined by that matching. As a result the original problem was reduced to finding a 12-regular graph on 84 vertices. This construction was encoded for CP-SAT and tested by recovering the unique srg(9,4,1,2).

There is a narrower but useful negative result too: among circulant graphs on Z/99 not one was found satisfying more than 68% of the constraints — that is 33 of 49 difference classes. The same ceiling was obtained for another abelian group of order 99. The approach with prescribed automorphisms was tested separately on srg(9,4,1,2) and the Paley graph srg(13,6,2,3).

The limitations here are fundamental. Fourteen different methods did not exceed the 69.43% mark, but that is only a stable bound within partial credit, not a solution to Conway's 99-graph. What is more, a provable bound below 4,950 constraints would already amount to a proof of non-existence, so the problem is bound up with these results far more tightly than a single percentage might suggest.

Are you ready to count such a reproducible but unfinished result a success for an AI researcher — or does success for you begin only with a complete proof? Source: cs.AI updates on arXiv.org