T-135: A packing of 308 squares below the grid, from Fang

S3 V3 C3 Upper bound confirmed superseded by T-221

2026-10-10 published · Fang after Arslanov, Mustafin, Shangitbayev, Ellsworth, Stead · $n=308$$n=308$

One complete rational source certificate by Fang, reported on jlevy/squares#484, proves a finite upper bound at the exact side it states: $s(308)\le 496906684150483564257053639569689586211617178631487101147707692165192097/27606985387162255916575391130373464080218179016903156277167119960899584$.

The same release states $s(343)\le 789017423383549575129979982295456201/41538374868278621028243970633760768$ and $s(344)\le 262204097424407923560664994905454377606094204426897481135606818230728643/13803492693581127593573619627880932674597656005283617050019749892718592$, beyond the case corpus; the source register tracks those as beyond-horizon rows.

Each certificate gives a rational side and, for each unit square, a rational centre and the tangent of its half-angle, in Daniel's format, with the same packing in SQUISH's JSON layout. The author reports that Daniel's verify_cert.py, Ellsworth's check_packing.py on 40-digit exports and the author's own verify.py accept all three; their outputs are retained.

All nine retained jobs, the three positives with their duplicate-square and outside-container controls, were decided again here on 10 October 2026 by both maintained exact routes and equal their retained rows, and a third exact route, half-extent containment and exact intersection area, decided the three with six controls each and found each text certificate equal to its JSON copy. No route here is the source's code, so the replay is independently re-implemented. The 10 October review found no blocking defect.

The side at 308 was the first below the grid's 18, which the case held in both lanes. On 11 October 2026 the case record adopted the smaller side of issue 489 (T-141), Daniel's move of this construction, and this certificate stays verified and is not selected.

Credit Fang (https://github.com/TheSnakeFang/squarepack-certs). The packings extend the k^2 - k + 1 two-wedge family of Arslanov, Mustafin and Shangitbayev (Electronic Journal of Combinatorics 28(4), P4.22, 2021) one k up and one square past it, grown from Ellsworth's and Stead's packings of 273 and 307 squares on the Kingbird table; the certificate format is Daniel's and the JSON layout SQUISH's. The search, the verifier and the bundle were written with Claude under the author's direction.