T-189: Francisco Couzo's refinement of Mykhailo Polkovskyi's 237, and fifteen beyond-horizon bounds

S3 V3 C3 Upper bound confirmed superseded by T-190

2026-10-14 published · Couzo after Polkovskyi, Fang, Daniel, Ellsworth, Levy · $n=237$$n=237$

One complete rational source certificate reports a finite upper-bound improvement at the exact side it states: $s(237)\le 12163897896660350527924222391/764899771970500000000000000$.

The three retained jobs at 237, the certificate with its duplicate-square and outside-container controls, were decided again here on 16 October 2026 by both maintained exact routes and equal their retained rows, and the frozen comparison rebuilt equal. One maintained route is the verifier the source reports running; the independent rational corner checker is independently re-implemented, and the replay rests on it. The 16 October review found no blocking defect.

The same release states $s(339)\le 15760492451744053967692664819/833191141347000000000000000$, $s(342)\le 13505979727081420240143041327/712163911851000000000000000$, $s(382)\le 11022991764484891182563636300647365246824336756792134646965777453391/551490929281812357791494555514909292818123577914945500000000000000$, $s(409)\le 12913362409176042797577197977/621808448863500000000000000$, $s(419)\le 80695119733625752827535766822771925130844543419314113774658870691/3849958446685324913565194998050235494743230122939250000000000000$, $s(450)\le 86015497401862821831433322593/3951569106671500000000000000$, $s(461)\le 4815728038236625594119195870229537391060756184031012912605279819357/219163872303991689276056426799981765896492324486478500000000000000$, $s(542)\le 5532991125316051254254297371/232201273560500000000000000$, $s(552)\le 1336913104446320995776430361526045060886790944910372943/55779003704893601939282009645665379846500000000000000$, $s(599)\le 26704722559059887305228107379/1069556443814500000000000000$, $s(648)\le 1580917747964295160131077049/60879375637000000000000000$, $s(649)\le 22005763134880310331656432423/847125729886500000000000000$, $s(651)\le 241476276408156387115385105463274782867431850070691071/9293083814480905136206318825678729558937500000000000$, $s(757)\le 2017013645520556066618921865312330242011007231621137760082264853190861249025432441426574553/72059361334628479177359726217958887411830602793896518335182698782951858151500000000000000$ and $s(804)\le 21316338098656938426429163392528974468810470714920720667861834801531/737069301316017949763472070077201158201889297143765500000000000000$, beyond the case corpus; the source register tracks those as beyond-horizon rows.

The author reports that all 23 certificates pass this repository's sqpack verify.py, run independently with exact rationals and spatial-bucket pair enumeration, and Daniel's verify_cert.py, each returning VALID with exit status 0, the replay of the 16 campaign and precision certificates taking 22.4 seconds of wall time on two processes; that the 16 also passed the browser campaign's Rust rational checker; and that the seven composite receipts were reused after their digests matched the replayed bytes. Two composites were copied from Fang's shipped certificates and five regenerated with his tools/glue.py, all seven with the digests his table lists. Those statements are the author's; no source program ran here.

Credit Couzo (https://github.com/franciscouzo/square-packing) with the 16 campaign and precision certificates, with the packings each continues, as the author asks: at 237 Polkovskyi's issue-593 certificate (T-183), whose construction credit he asks to keep; at 339 the author's own L-border construction of issue 506, refined in issues 583 and 586; at 382 and 651 Fang's published arrangements; at 409 and 450 the campaign's L-border lineage, whose 409 start includes Daniel's 298 packing, with 450 one L-border layer on the new 409; at 419, 461 and 804 campaign refinements and L-border or elimination constructions from starts that include the project's refinements of Fang's 380 and 381; at 552, 599 and 648 the campaign's 552 with zero, one and two L-border layers; at 649 elimination from 651; and at 757 elimination from 758.

Credit Fang (TheSnakeFang) with the seven composites at 703, 812, 930, 931, 992, 993 and 994, which the author replayed and does not claim.

Daniel's optimizer and verifier, Ellsworth's refinement ideas and this repository's exact verifier were used. Issue 607 states: "I used OpenAI Codex for search and verification orchestration, source review, certificate packaging, and preparation of this issue. Earlier work on the search also used Claude Code. I directed the work."