$n=155$$n=155$

s(155) — Square Packing Case

Hosono’s closed form, 14 October 2026. Hosono’s closed form (wand125, 13 October 2026, T-180) states $s(N)\ge a(N)$$s(N)\ge a(N)$ for every non-square $N\ge 49$$N\ge 49$, from a measure of line segments on 34 intervals of the fractional part. At $n=155$$n=155$, $k=12$$k=12$ and interval 27 give $a(155)=12.540954306468783969574031785279\dots$$a(155)=12.540954306468783969574031785279\dots$, so the reported lower bound here is now $s(155)\ge 12.540954306468$$s(155)\ge 12.540954306468$, rounded down, above Nagamochi’s closed form, $12.48912529307$$12.48912529307$. The source replayed its certificate with its own checker; no record of it has been replayed here, so the verified lower bound is unchanged. The source states that its search, certificates, checkers and text were produced with AI assistance (Claude and Codex), under the direction of and reviewed by Hosono.

House adoption, 19 October 2026. Daniel’s packing from a search started from Ostashev’s packing (T-197) is now this case’s house: its exact certificate proves $s(155)\le 12.9522609446505641$$s(155)\le 12.9522609446505641$, the reported and the verified upper bound, rounded outward from its exact side. The exact certificate side improves the previous verified exact side by about 1.64×10−4$1.64\times {10}^{-4}$. The earlier ceiling, displayed outward as $12.95242540695845614$$12.95242540695845614$ from Ostashev’s packing, improved by PingYou Ltd (T-174), applied until this adoption and stays retained as history (the certificate).

Reported certificate, 2026-10-17. Daniel reported a 155-square certificate at container side $12.952260944651$$12.952260944651$, rounded upward from the retained exact certificate. The exact rational side is registered in the result register. Its V0/C0 registration left both case upper lanes, the house witness and its drawing unchanged until the house adoption of 19 October 2026, and until then the rigidity and arrangement notes below described that retained house.

The source calls this a different numerical local minimum found from Ostashev’s T-149 packing as a search start. Its original and regularized display certificates have the same side and yield one result. The retained regularized JSON carries "by": ["Francisco Couzo"], as the views at 236 and 271 do, and the review of 19 October 2026 asked the author what it means (RV-7). His comment on issue 669 of 20 October answers that it was a bug in his tooling, which took the field from the record holder at that count in his 5 October batch instead of from the packing drawn, and asks that it be read as no credit. This case follows the issue and the prior-record file, which credit Ostashev’s packing as the start.

House adoption, 17 October 2026. PingYou Ltd’s refinement of Ostashev’s packing (T-174) was this case’s house until 19 October 2026: its exact certificate proves $s(155)\le 12.95242540695845614$$s(155)\le 12.95242540695845614$, the reported and the verified upper bound, rounded outward from its exact side. The exact certificate side improves the previous verified exact side by about 1.30×10−19$1.30\times {10}^{-19}$. The earlier ceiling, displayed outward as $12.9524254069584562$$12.9524254069584562$ from Ostashev’s packing (T-149), applied until this adoption and stays retained as history (the certificate).

House adoption, 11 October 2026. Ostashev’s packing (T-149) was this case’s house until 17 October 2026: its exact certificate proves $s(155)\le 12.9524254069584562$$s(155)\le 12.9524254069584562$, the reported and the verified upper bound, 7.78×10−5$7.78\times {10}^{-5}$ below the ceiling the case held until then, $12.9525032026045129$$12.9525032026045129$ from Chaoweeraprasit’s packing (T-115), which stays retained as history (the certificate).

The SQUISH Update

Chaoweeraprasit, using SQUISH, reported on 7 October 2026 a bound tighter than the case then held $s(155)\le 12.9525032026045129$$s(155)\le 12.9525032026045129$, with exact side $7291611074595605/562949953421312$$7291611074595605/562949953421312$. This update is confirmed at V3/C3 by the complete retained dual exact replay and separately prompted scoped AI reviews. It establishes a feasible upper bound; optimality and human oversight have not been established. The author reports the seed as Couzo’s s(156), squares removed. The revision-specific packet retains the source print $12.9525032026045128$$12.9525032026045128$ separately from the safe decimal ceiling. The atlas drawing below uses the update’s exact rational geometry, checked for the drawing independently of the frontier assurance. Earlier certified geometries, evidence and reviews remain retained as historical provenance, separate from this update’s confirmation.

Separately, this repository’s numerical translation screen finds a nontrivial feasible motion in the current update geometry drawn by the atlas. That numerical assessment concerns this update pose alone; it does not settle the optimum or describe the earlier certified pose.

The Earlier Verified Upper Bound

Chaoweeraprasit (itsnaka), using SQUISH, reports s(155)≤Sn​$s(155)\le S_n$, with Sn​=1290933978599768306573377536001672234187663680786096364585719​$S_n=\frac{1672234187663680786096364585719}{129093397859976830657337753600}$ in SQUISH’s ten-packing release. The certificate gives rational centres and half-angle parameters for 155 unit squares. The source’s original finite decimal display is $12.9536770693532741$$12.9536770693532741$; the rational fraction states the claimed bound exactly.

The earlier SQUISH packing improves the earlier reported side $12.95561959213445243801096476729$$12.95561959213445243801096476729$ from [evand exact optima 2026-10-05]. This repository confirmed the upper bound by independently re-implemented exact-rational checks: native exact_verify and devtools.check_rational_witness_independent accept every pair and wall of the earlier packing and reject both geometric controls. The verified display is $12.9536770693532733$$12.9536770693532733$, the exact source side rounded upward at sixteen decimals. Both accepted reviews cover the shared half-angle conversion and the complete roster. The submitted certificates establish feasible upper bounds and make no optimality or rigidity claim.

The author credits Ellsworth’s records and tools and states that SQUISH’s solver, verification and issue were written with Claude Opus 5.5 under his direction, management, review and steering. The source packet retains derived packing facts and source metadata.

Earlier Packing

Couzo found the earlier packing; Daniel improved it. Its reported side was $12.95561959213445243801096476729$$12.95561959213445243801096476729$. The earlier certified ceiling $12.95561959213445243801096476729$$12.95561959213445243801096476729$ was the verified upper bound before the SQUISH certificate (T-098), supported by Daniel’s certificate packet. This repository replayed Daniel’s two source checkers and independently decided the rational certificate with its own exact-rational routes; the packet records the shared premises and receipts. Any optimality or rigidity observations about that packing concern its own pose.

The lower bound

The strongest lower bound independently verified in this record is Karakuş’s general bound, which applies to every nonsquare $N\ge 8$$N\ge 8$:

Correction, 2 October 2026. Until that date the verified lower bound here was Nagamochi’s general closed form, s(N)≥min(⌈N​⌉,N−2⌊N​⌋+1​+1)$s(N)\ge min(\lceil \sqrt{N}\rceil ,\sqrt{N-2\lfloor \sqrt{N}\rfloor +1}+1)$, which is stronger at this $n$$n$ and is now recorded as a reported bound. Its published proof rests on Nagamochi’s Lemma 1, which Karakuş showed false; nothing is disproved, and no packing beating it is known (review of 2 October 2026). This register had recorded that proof as verified, its own error, logged as defect D-516.

Source-reported bounds are recorded separately from this independently verified theorem.

Verification Code

The programs behind this case’s verified bounds, by their evidence. The code column says how the code that ran stands to the code its producer used. VERIFIERS.md says what each program is and whose it is.

Daniil Ostashev’s certificate (T-149)

Ostashev’s packing, reported on jlevy/squares#501, at an exact rational side, displayed upward as $12.9524254069584562$$12.9524254069584562$ (T-149). The certificate gives a rational side and, for each of the 155 unit squares, a rational centre and the tangent of its half-angle; the retained packet holds it with the complete native jobs that decided it.

Both maintained exact routes decided it here, sqpack’s exact separating-axis verifier and the independent rational corner checker, with its duplicate-square and outside-container controls, and a third exact route, half-extent containment and exact intersection area, decided it again. No route here is the source’s code, so the replay is independently re-implemented. The review of 11 October 2026 accepted the replay for finite feasibility. That proves $s(155)\le 12.9524254069584562$$s(155)\le 12.9524254069584562$, the verified upper bound until 17 October 2026: confirmed, independently re-implemented. It says nothing about optimality, rigidity or a local minimum.

The source states nothing about AI assistance.

The house it replaces, $12.9525032026045129$$12.9525032026045129$ from Chaoweeraprasit’s packing (T-115), stays retained with its own replay as history. The other certificates confirmed here at this count, T-128 at $12.9524989440140074$$12.9524989440140074$ and T-113 at $12.9536770693532733$$12.9536770693532733$, stay verified and are not selected. T-174 at $12.9524254069584562$$12.9524254069584562$, 1.30×10−19$1.30\times {10}^{-19}$ below this house, is confirmed here and accepted by the review of 14 October 2026. From 15 October 2026 until the owner’s house rule of 16 October 2026 lifted the hold, PR #574’s houses were held for the owner, since their witnesses would add about 18.5 MB to a checkout and their custody inputs about 5.8 MB to every worker’s snapshot.

PingYou Ltd’s certificate (T-174)

PingYou Ltd’s refinement of Ostashev’s packing, reported in pull request #574, at an exact rational side, displayed upward as $12.95242540695845614$$12.95242540695845614$ (T-174). The certificate gives a rational side and, for each of the 155 unit squares, a rational centre and the tangent of its half-angle; the retained packet holds it with the complete native jobs that decided it.

Both maintained exact routes decided it here, sqpack’s exact separating-axis verifier and the independent rational corner checker, with its duplicate-square and outside-container controls, and a third exact route, half-extent containment and exact intersection area, decided it again. The source ran both maintained routes itself on its corners witnesses; those runs are its own, and the replay here is independently re-implemented. The review of 14 October 2026 accepted the replay for finite feasibility. That proves $s(155)\le 12.95242540695845614$$s(155)\le 12.95242540695845614$, the verified upper bound until 19 October 2026: confirmed, independently re-implemented. It says nothing about optimality, rigidity or a local minimum.

The certificate is PingYou Ltd’s tightening of the packing T-149 records, that packing by the source’s statement, not decided here (T-174’s regularizations), so the packing keeps its finder’s credit and PingYou Ltd is credited with the certificate.

The source makes no statement about AI assistance in this pull request; for its #503 and #504 certificates the author said on 12 October 2026 that they were AI assisted by ‘PY’, an unreleased agentic AI offering by PingYou Ltd.

The house it replaces, $12.9524254069584562$$12.9524254069584562$ from Ostashev’s packing (T-149), stays retained with its own replay as history. The other certificates confirmed here at this count, T-128 at $12.9524989440140074$$12.9524989440140074$, T-115 at $12.9525032026045129$$12.9525032026045129$ and T-113 at $12.9536770693532733$$12.9536770693532733$, stay verified and are not selected.

Evan Daniel’s certificate (T-197)

Daniel’s packing from a search started from Ostashev’s packing, reported on jlevy/squares#669, at exact side $2590452188930112807053435932173/200000000000000000000000000000$$2590452188930112807053435932173/200000000000000000000000000000$, displayed upward as $12.9522609446505641$$12.9522609446505641$ (T-197). The certificate gives a rational side and, for each of the 155 unit squares, a rational centre and the tangent of its half-angle; the retained packet holds it with the complete native jobs that decided it.

Both maintained exact routes decided it here, sqpack’s exact separating-axis verifier and the independent rational corner checker, with its duplicate-square and outside-container controls, and a third exact route, half-extent containment and exact intersection area, decided it again. No route here is the source’s code, so the replay is independently re-implemented. The review of 19 October 2026 accepted the replay for finite feasibility. That proves $s(155)\le 12.9522609446505641$$s(155)\le 12.9522609446505641$, the verified upper bound: confirmed, independently re-implemented. It says nothing about optimality, rigidity or a local minimum.

The issue says the work was done with Claude (Claude Code), directed and approved by Daniel.

The author’s basin explorer started from Ostashev’s packing, the register’s witness for 155 when the search ran, and reached this one after a chain of 18 moves; the issue credits that packing as the start. Measured here square by square, no symmetry of the box pairs the two one to one, and the best assignment moves 29 squares beyond a tenth, sharing no tilt: a new arrangement found from Ostashev’s packing (T-197).

The house it replaces, $12.95242540695845614$$12.95242540695845614$ from Ostashev’s packing, improved by PingYou Ltd (T-174), stays retained with its own replay as history. The other certificates confirmed here at this count, T-149 at $12.9524254069584562$$12.9524254069584562$, T-128 at $12.9524989440140074$$12.9524989440140074$, T-115 at $12.9525032026045129$$12.9525032026045129$ and T-113 at $12.9536770693532733$$12.9536770693532733$, stay verified and are not selected.