T-169: A further exact refinement of Xu's packing of 261 squares, after Barnett's

S2 V3 C3 Upper bound confirmed superseded by T-174

2026-10-12 published · PingYou Ltd after Xu, Barnett, Daniel, Levy · $n=261$$n=261$

One complete rational source certificate reports a finite upper-bound improvement at the exact side it states: $s(261)\le 60221701896183155925623822067552396531186006689382586124901485272245038570679102391768853738640845762968885190194646113914622817225143520151264873514153447594856784993916447668795594139686908064452561271291314933291180992874044736240078222111748672780052136222253269408041817329591669640100566654264077488796569417423990746316944258046820136508658141713282417630107002587683964892225712832132809051198459587754731837534654563549531425572967566922458827712106584551155379606891894276892745499879038466854033251780879739643554702103280446489851484546104056691503474152741213554587705922412237998514260452502621555989906039903465080299767718489292438151410266412241443553912996153501809246611773602786410917605511496451072010703261644140260354724734077108181090405074464644068238476104240402870056122000746777426989961280021283237882422984517186879213359668962009071281220892663202830388800939868731631999073946444449/3610677900079007969706765943565336768719447884288352204035356284560922367257758833639504829783888314773470186559856651703840826986312510315164755650194437807425149383349412670053592606596086384850274071672011088362441130659512400319497229936794767986753014863986710537052753629434552337631447921268887522107208292732259544414433641519431486273344988404799773016119041232200157601295450248940554312915607588240758183597134737218881654441888759199840243417937275746123252418387728815666095108047599852739142587192176714235571601282650194803742120186930924408146376687967975698205674166080147009184200896016800571473933793928015380986107693008261921638483793126847396768411768614291571832645828780798475334365381997818684448262987968974484285109167302223926052090540071153696848666844652463042625831311185308825340533519645238451116030713985979718105294854931486199059291092660241441782826348588921333492299220000000$.

All three retained jobs, the positive with its duplicate-square and outside-container controls, were decided again here on 14 October 2026 by both maintained exact routes and equal their retained rows. A third exact route, half-extent containment and exact intersection area, decided the certificate with six controls each and held the derived fact to the pinned file it is derived from, fetched at the pin. No route here is the source's code; the source's runs of this repository's two maintained routes on its corners witnesses are its own and decide nothing here, so the replay is independently re-implemented. The 14 October review found no blocking defect.

The author reports that its exact separating-axis checker and a separate rational polygon-intersection checker accept all 261 squares, every wall and all 33,930 pairs, that every one of the 34,974 declared support inequalities holds, with duplicate and outside controls refused and the conversion to its corners Witness/v2 lossless; and that this repository's sqpack exact_verify and check_rational_witness_independent, run unchanged, accept that witness. Those statements are the author's; no source program ran here.

Credit PingYou Ltd (https://github.com/BuddenPing/square-packing-certificates) with this further refinement, as the source asks, of Barnett's refinement (T-150, issue 505) of Xu's packing of 261 (ry-xu, issue 432); the source credits Daniel with the contact-solver source. The pinned source makes no statement about AI assistance.

Measured pose by pose by that review, it is Barnett's packing (T-150) re-solved, no square moved beyond 2e-06; squares touch the box and the closest pair touches, which the closed box allows. At 261 PR 574's certificate (T-174) is smaller, by 1.99e-21, and the house rule of 16 October 2026, which lifted the hold of 15 October, made T-174 the house there, and the adoption of 17 October 2026 (think-3mry) moved the case to it. This entry held the case until then, the smallest certificate there that the hold left in.