T-258: Daniel's lower packing of 299 squares from his T-248 packing, 2.46e-3 below it

S3 V3 C3 Upper bound confirmed

2026-10-29 published · Daniel after Deotte · $n=299$$n=299$

One complete rational source certificate reports a finite upper-bound improvement at the exact side it states: $s(299)\le 17.801238549388584084680137655023$$s(299)\le 17.801238549388584084680137655023$. It is 2.46e-3 below T-248, the source's opening certificate at 299, and 2.16e-2 below the case's house best packing, the Kingbird catalogue's 33/2 + √7/2.

The certificate and its duplicate-square and outside-container controls were decided here on 31 October 2026 by both maintained exact routes. Those routes share components with the producer: both read the certificate through the evand-cert reader, whose line filter and corner construction are transcribed from Daniel's own checkers, the ones the certificates name. The 31 October review decided it again on a clean-room checker of its own, by separating axes over every pair and by exact clipped areas with ten sharp controls each, by a Claude agent, a system family the producer also used, and found no blocking defect, so the bound is confirmed, independently re-implemented.

The update says the source's two exact checkers, search/exact/verify_cert.py and verify_cert2.py, byte-identical to the copies the n183 packet retains, both report VALID. It calls the certificate an exact KKT point whose 80-digit point, s299_1028.exact.txt, is a local minimum of S strict modulo 119 exact flat motions, with no corner-corner MILP descent, 64 corner-corner contacts and 33 rattlers, by numerical evidence at 80 digits and not interval proofs. The source's commit of 29 October says the regularized view failed its checkers and is not committed, and ends with a Co-Authored-By trailer naming Claude. Those statements are the author's; no source program ran here.

Credit Daniel (https://github.com/evand/square-packing), who reports a new packing found from his T-248 packing at 299 by four generations of refill (removing a line or disc of squares, re-placing them and relaxing; 17.803695 to 17.801240) and an exact descent (a corner-corner MILP kick and re-solve, to 17.801238549), and asks that the credit of the opening post stand: Deotte for the method, his multi-gadget fault-line DP, and for the start, his public Kaggle $s(334)$$s(334)$ packing. A Kaggle per-puzzle side the update quotes, 17.793636991 for Cats and boxes, states no packing and is no report. The source's credits say its work was produced by Claude (Anthropic) under human direction.