A New Upper Bound for the First Autocorrelation Inequality and 306 Circle-Packing Records via Active-Set Escape from Equioscillating and KKT Optima

An upper bound on an extremal constant such as the first autocorrelation constant C₁ needs one construction, which exact arithmetic checks. Most published values come from systems that add machinery around a language model, such as MILO, which evolves agent code. The Einstein Arena leaderboard requires a new record to beat the leader by a fixed margin, but small changes to the best published constructions gain less than that. We introduce active-set escape, a search whose moves change a construction’s active set (the constraints it meets with equality) and then re-solve the whole construction. In our starting C₁ entry, the bound is the maximum of a piecewise-linear function, and 24,397 of its breakpoints come within the margin of that maximum. Since a new record must lower that maximum by more than the margin, it must lower them all. We give Claude Opus 5.5 agents hours, machines and an exact evaluator, and they write these moves themselves while the model’s weights stay fixed. Three agents sharing an archive of verified attempts certify C₁ ≤ 1.50266484…. As of October 30, 2026, this bound beats the Arena’s leader by 7,865 times the margin. A single agent’s packings of unequal circles in a square hold the best known sums of radii at 306 sizes as of that date. When the agents and a preregistered classical optimizer start from the same Arena entries on C₁ and Erdős’s minimum overlap problem, the agents improve on them about three thousand times as much. Before any search, counting the breakpoints near a construction’s maximum shows whether local descent from it stalls.