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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 Durable Intelligence Labs

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Classical descent stalls at the stationary entries; active-set escape goes about 3,000 times further.
Figure 1. Classical descent stalls at the stationary entries; active-set escape goes about 3,000 times further. Distance below the Einstein Arena entry each search started from (log scale) against working hours, the matched axis of a preregistered study on one machine type. (a) Erdős’s M from entry #2507: the 19-attempt lineage of three Claude Opus 5.5 agents on a shared archive, ending at the certified construction (star, 10.82 h, 1.836×10−6 below the entry), against the compute-only control from the same entry, whose best stayed 5.76×10−10 below it; dashed, MILO’s printed value. (b) C1 from entry #2603: the same run’s best so far. Stars mark the certificates at 3.97 h (3.10×10−6 below the entry; the control, diamond, stood 1.05×10−9 below it at the same elapsed time) and at 40.75 h (C1 ≤ 1.5026648…, 7.86×10−5 below #2615, the Arena’s leader on October 30, 2026, dashed). Curves are the protected evaluator’s conservative floating-point scores; stars are exact.

Abstract

As of October 30, 2026, the Einstein Arena’s leader on the first autocorrelation constant C1 stands at 1.5027434867; MILO’s evolved agent harnesses report 1.5027435984. Under a two-hour, four-thread budget, the Endless Exam finds that Claude Opus 5.5 surpasses none of its 30 published frontiers. We introduce active-set escape. Published optima of these minimax problems are stationary: their maximum is attained, to within the venue’s minimum improvement, at thousands of points, or their circles sit at a KKT point, so descent must lower every near-active constraint at once and stalls. An escape move changes that set, and a global re-melt and a second-order solve finish the new basin. With no weight update, three Claude Opus 5.5 agents sharing an archive of verified attempts certify C1 ≤ 1.50266484… after 40.75 working hours, 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 about three thousand times short. 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://durableintel.com/paper-new-constructions}
}
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