Dual-Agent AI Tightens Autocorrelation and Erdős Bounds

The interesting result buried in a new arxiv preprint from Sungyoon Kim and Mert Pilanci is not the AI framing but the fact that two named constants moved. Their system, described as AI-Assisted Discovery of Convex Relaxations via Dual Agents, reports a tighter lower bound for a first autocorrelation constant (from 1.28 to 1.2937) and for an Erdős minimum-overlap constant (from 0.379005 to 0.37912). Both prior bounds are attributed to work by Tao et al., and both new bounds are certified using explicit dual-feasible points validated through interval arithmetic.

The mechanism is a division of labor. A coding agent proposes constraints for the convex relaxation of a nonconvex optimization problem, and a theory agent verifies those proposals and identifies counterexamples when a proposal is wrong. That two-agent loop is what the authors call an autoresearch paradigm, pitched as a way to grind out incremental progress on problems where the search space of possible constraints is too large or too subtle for one model to navigate cleanly.

Why this matters if you do not work on bounds for classical constants: this is an early data point for AI systems producing certified mathematical results rather than plausible-looking ones. The bound improvements themselves are small in absolute terms, but they are underwritten by dual-feasible witnesses, which means a human referee can check the certificate rather than trusting the model on faith.

The honest caveat is scope. The paper reports two constants and describes the framework at a high level. What the preprint does not give you, at least in the summary form, is how much compute the loop consumed, how many failed proposals the theory agent had to reject, or how cleanly the approach transfers beyond the specific optimization templates in play. Take the numbers as reported rather than as evidence the method generalizes to arbitrary problems.

The piece worth watching is whether the same coding-and-theory split can be pointed at other long-standing bounds where the underlying convex-relaxation structure is well understood. If it can, the payoff is not headline theorems but a steady drip of tightened constants, each arriving with a machine-checkable certificate attached.