T-242: Daniel's exact descents of public Kaggle packings at 268 and 304, below the reports before them

S2 V3 C3 Upper bound confirmed

2026-10-24 published · Daniel after Lavrentev, Tymchishin · $n=268,304$$n=268,304$

Two complete rational source certificates report finite upper-bound improvements at the exact sides they state: $s(268)\le 16.874361715869345824484822296466$$s(268)\le 16.874361715869345824484822296466$, 4.48e-12 below T-240 and 4.45e-3 below T-174, the case's house best packing when it was reported, and $s(304)\le 17.92689890035674158972360842897$$s(304)\le 17.92689890035674158972360842897$, 3.04e-9 below T-241 and 2.53e-3 below T-186, the case's house best packing when it was reported.

Both certificates and their duplicate-square and outside-container controls were decided here on 24 October 2026 by both maintained exact routes. Those routes share components with the producer: both read the certificates through the evand-cert adapter, whose reader's line filter and corner construction are transcribed from Daniel's own checkers, the ones the certificates name. The 24 October review of T-241 and T-242 decided both again on a clean-room checker of its own, by separating axes and by clipped areas with twelve controls each, by a Claude agent, a system family the producer also used, and found no blocking defect, so the bounds are confirmed, independently re-implemented.

They are search/packer/candidates/s268_lightning.cert and s304_pub.cert in Daniel's watched repository (https://github.com/evand/square-packing), pushed on 24 October 2026 at 04:53:53Z. No issue reports them, so they are a report from their publication (the owner, 14 October 2026); the source says it did not post them because issue 763 already held the counts. The same commit's certificate at 303, of Deotte's public Kaggle packing, is 2.60e-4 above T-240's 303 and is not recorded.

Each certificate's JSON report states it valid (cert_valid) at 80 digits, its least wall and pair clearances (5e-21 and about 1e-20), and S_exact, the side of a Newton-converged KKT point of the descent. Those statements are the author's; no source program ran here. From the two sides it states, the certificate's side lies above S_exact by about 1.7e-19 at 268 and 1.8e-19 at 304, a relative padding of 1e-20.

Credit at 268 Lavrentev's public Kaggle packing, as the source credits it: "Lavrentev's (Kaggle lightningv08) public packing, exact descent (shrink-only) to 16.8743617159, the side a Kaggle per-puzzle side-only claim lists; the first public packing at that side". The source's JSON gives the descent's input side as that of its input file, Lavrentev's placement, which the certificate is to 1.41e-4 (N-1). Measured here, it is another placement of the arrangement T-240 refines, not T-240's packing re-solved.

At 304 credit the packing T-241 certifies, which the source calls "the public Kaggle packing at 17.9268989004 (finder unknown; earliest public copy: kozykappa's notebook)" and Couzo's index credits to Tymchishin's Kaggle notebook "CPU Alternating Reclosure" v3; measured here, the source's input is that packing to 2.05e-9. No source states where Tymchishin's notebook started (M-1). Daniel descended each exactly to the side stated and wrote the certificate. The source's credits say its work was produced by Claude (Anthropic) under human direction, and the commit carries a Co-Authored-By trailer naming Claude.