An AI Tried to Crack a Famous Math Puzzle. It Found a New Wall Instead.

An autonomous AI research agent has hit 69.43% on a partial-credit score for Conway's 99-graph problem, a long-standing open question in combinatorics about whether a particular 99-vertex network with a very strict local pattern can exist at all. That is not a solution. It is a sharper, fully auditable wall, and the wall itself is the result.

The paper, posted to arXiv on August 11 by a self-described autonomous AI research agent, walks through four distinct contributions (arXiv:2608.11211). None of them resolves the existence question. Together, they narrow it.

Conway's 99-graph problem asks whether a graph on 99 vertices can be drawn such that every vertex connects to exactly 14 others, every two adjacent vertices share exactly one common neighbor, and every two non-adjacent vertices share exactly two. That kind of object is called a strongly regular graph, one of the cleanest ways to encode very rigid local structure in a finite network. Mathematicians have spent decades trying to find one with these specific numbers. None has, and no one has proved that none can.

The AI's first contribution is a ceiling. The authors exhaustively checked every circulant graph on the cyclic group of order 99, a family defined by a fixed pattern of connections that wraps around the 99 vertices, and showed that none of them satisfies more than 3,366 of 4,950 constraints, or 68.0%. They repeated the test on the other abelian group of order 99 with the same result. This is a brute-force, fully reproducible upper bound on a specific family of candidates.

The second contribution is a structural reduction. The two local rules that define the problem, "adjacent vertices share one neighbor, non-adjacent vertices share two," force the candidate graph into a much smaller shape. The neighborhood of any vertex must be a perfect matching, 14 vertices paired into 7 disjoint edges with no vertex appearing twice. The non-neighbors of any vertex, 84 of them, can be placed in one-to-one correspondence with the unchosen pairs in that matching, which collapses the existence question into whether a 12-regular graph on 84 vertices exists under a clean combinatorial encoding that a constraint solver can read. The authors encode this as a CP-SAT problem, the standard input format for satisfiability solvers, and validate the encoding by recovering a known 9-vertex example, the unique strongly regular graph with parameters (9,4,1,2), from scratch.

The third contribution is a framework for symmetry-preserving constructions. The authors formalize a prescribed-automorphism orbit-existence approach, a way of asking whether a candidate graph can survive a given symmetry, and check it on the same 9-vertex example and on the Paley graph of order 13, a classical example of a strongly regular graph built from arithmetic modulo 13. These are not new objects. They are sanity checks, used to confirm that the framework does what it claims.

The fourth contribution is the 69.43% frontier itself. Across fourteen distinct methods, none produced a higher verified score. The paper's authors are explicit about what this means and what it does not: any provable bound strictly below 100% on this partial-credit metric would already constitute a non-existence proof. The 69.43% figure is therefore not a consolation score. It is a real, auditable ceiling on what these methods can reach, sitting below the threshold at which a non-existence proof would be automatic.

The secondary story here is what an autonomous AI research agent can, and cannot, do on hard open mathematics. The paper is credited to an AI rather than a human author. Its value is not in solving Conway's 99-graph problem. It is in producing a chain of reproducible artifacts, an exhaustive search, a structural reduction, a symmetry framework, and a fourteen-method frontier, that other researchers can re-run, re-check, and build on. A human co-author has not been confirmed in the abstract, and the paper itself flags that the frontier may or may not be sharp.

On a problem that has resisted proofs of existence and non-existence for sixty years, a clean, reproducible wall is itself a contribution, and a useful one for the next attempt to climb it.