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: $λ=1$ makes each neighbourhood a perfect matching and $μ=2$ 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.
Rohan Pandey, Michael Ruofan Zeng, Weikun K. Zhang +5cs.LG cs.AI
Finding minimal arithmetic circuits for polynomials over finite fields is a combinatorially hard problem central to algebraic complexity theory. We formulate it as a reinforcement learning problem in two directions, bottom-up and top-down. To address the challenge of a fast-growing combinatorial search space, we introduce FactorLibrary, which stores factorizable subexpressions that serve as reusable subgoals across training episodes. We trained a bottom-up agent with Gumbel-PPO-MCTS and two top-down agents with PPO+MCTS and SAC. The PPO+MCTS top-down agent exhibited the most stable performance, finding certified optimal circuits up to complexity $8$ with a success rate of $91.8\%$.