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

Lindell S. Cumes QWNTL Labs

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From the same stationary entries, active-set escape goes about 3,000 times further than classical descent.
Figure 1. From the same stationary entries, active-set escape goes about 3,000 times further than classical descent. Each panel plots the distance below the Einstein Arena entry the search started from, on a log scale, against working time, the matched axis of a preregistered study on one machine type. (a) Erdős’s M from entry #2507. The shared archive’s 19-attempt lineage (three Claude Opus 5.5 agents) ends at the certified construction (filled circle, 10.82 h, 1.836×10−6 below the entry). The dotted line is the compute-only control from the same entry, whose best stayed 5.76×10−10 below it. The dashed line is MILO’s printed value. (b) C1 from entry #2603: the same run’s best so far. Filled circles mark the certificates at 3.97 h, 3.10×10−6 below the entry, and at 40.75 h (C1 ≤ 1.5026648…). At the first, the control (cross) stood 1.05×10−9 below the entry. The dashed line marks #2615, the Arena’s leader on October 30, 2026, 7.86×10−5 above the second certificate. Each arrow’s label is the ratio of the certified distance to the control’s. Curves are the protected evaluator’s conservative floating-point scores; the certificates are exact.

Abstract

As of October 30, 2026, the Einstein Arena’s leader on the first autocorrelation constant C1 stands at 1.5027434867, and MILO’s evolved agent harnesses report 1.5027435984. Under a two-hour, four-thread budget, Claude Opus 5.5 surpasses none of the Endless Exam’s 30 published frontiers. In the published optima of these minimax problems, the maximum is attained, to within the venue’s minimum improvement, at thousands of points, or the circles sit at a KKT point. Any improvement larger than that minimum must lower the value at every one of these points at once. Active-set escape, which we introduce, is a search in which each move changes the near-active set and a global re-melt and a second-order solve finish the new basin. We gave Claude Opus 5.5 agents hours of autonomous work and an exact evaluator, and they wrote these moves themselves. Three agents sharing an archive of verified attempts, with no weight update, certify C1 ≤ 1.50266484… after 40.75 working hours. The bound lies 7,865 times the Arena’s minimum improvement below its leader. A single agent with no outer loop also passes the leader, by 7,200 times the minimum improvement. The same moves lower the signed autocorrelation constant below MILO’s value and set the best sum of radii for unequal circles in a square at 306 Packomania sizes up to N = 10,000. A preregistered classical optimizer started from the same constructions stalls after gains about three thousand times smaller. Witnesses and exact verifiers ship with the paper.

Code and data

Ancillary files: witnesses, exact verifiers and hashes

Cite

@misc{cumes2026new,
  title         = {{A New Upper Bound for the First Autocorrelation Inequality and 306 Circle-Packing Records via Active-Set Escape from Equioscillating and KKT Optima}},
  author        = {Cumes, Lindell S.},
  year          = {2026},
  url           = {https://qwntl.com/paper-new-constructions}
}
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