$n=154$$n=154$

s(154) — 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=154$$n=154$, $k=12$$k=12$ and interval 24 give $a(154)=12.486669250021968643133242987761\dots$$a(154)=12.486669250021968643133242987761\dots$, so the reported lower bound here is now $s(154)\ge 12.486669250021$$s(154)\ge 12.486669250021$, rounded down, above Nagamochi’s closed form, $12.44552314225$$12.44552314225$. 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, 20 October 2026. Daniel’s continuation of Couzo’s further arrangement of Daniel’s n = 154 packing in a search the source started from Ostashev’s n = 156 packing (T-222) is now this case’s house: its exact certificate proves $s(154)\le 12.9211009407616806$$s(154)\le 12.9211009407616806$, the reported and the verified upper bound, rounded outward from its exact side. The exact certificate side improves the previous verified exact side by about 1.06×10−4$1.06\times {10}^{-4}$. The earlier ceiling, displayed outward as $12.92120657713263105$$12.92120657713263105$ from Daniel’s packing, improved by Couzo and Mishapolk (T-186), applied until this adoption and stays retained as history (the certificate).

House adoption, 16 October 2026. Polkovskyi’s refinement of Couzo’s further arrangement of Daniel’s n = 154 packing (T-186) was this case’s house until 20 October 2026: its exact certificate proves $s(154)\le 12.92120657713263105$$s(154)\le 12.92120657713263105$, the reported and the verified upper bound, rounded outward from its exact side. The exact certificate side improves the previous verified exact side by about 1.29×10−19$1.29\times {10}^{-19}$. The earlier ceiling, displayed outward as $12.9212065771326311$$12.9212065771326311$ from Daniel’s packing, improved by Couzo (T-182), applied until this adoption and stays retained as history (the certificate).

House adoption, 15 October 2026. Couzo’s further arrangement of Daniel’s n = 154 packing (T-182) was this case’s house until 16 October 2026: its exact certificate proves $s(154)\le 12.9212065771326311$$s(154)\le 12.9212065771326311$, the reported and the verified upper bound, 1.36×10−3$1.36\times {10}^{-3}$ below the ceiling the case held until then, $12.9225664019340596$$12.9225664019340596$ from Daniel’s packing (T-160), which stays retained as history (the certificate).

House adoption, 12 October 2026. Daniel’s packing from his size-chaining explorer (T-160) was this case’s house until 15 October 2026: its exact certificate proves $s(154)\le 12.9225664019340596$$s(154)\le 12.9225664019340596$, the reported and the verified upper bound, 5.04×10−4$5.04\times {10}^{-4}$ below the ceiling the case held until then, $12.9230702023011408$$12.9230702023011408$ from Chaoweeraprasit’s packing (T-136), which stays retained as history (the certificate).

House adoption, 11 October 2026. Chaoweeraprasit’s SQUISH packing (T-136) was this case’s house until 12 October 2026: its exact certificate proves $s(154)\le 12.9230702023011408$$s(154)\le 12.9230702023011408$, the reported and the verified upper bound, 3.49×10−3$3.49\times {10}^{-3}$ below the ceiling the case held until then, $12.9265622458523470$$12.9265622458523470$ from Chaoweeraprasit’s packing, improved by Gupta (T-127), which stays retained as history (the certificate).

The SQUISH Update

Previously, Chaoweeraprasit, using SQUISH, reports a tighter bound $s(154)\le 12.9265622458535390$$s(154)\le 12.9265622458535390$, with exact side $1819251903550235/140737488355328$$1819251903550235/140737488355328$. 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 SQUISH’s own s(155), squares removed. The revision-specific packet retains the source print $12.9265622458535390$$12.9265622458535390$ 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

Previously, Chaoweeraprasit (itsnaka), using SQUISH, reports s(154)≤Sn​$s(154)\le S_n$, with Sn​=5629499534213127277982312406737​$S_n=\frac{7277982312406737}{562949953421312}$ in SQUISH’s ten-packing release. The certificate gives rational centres and half-angle parameters for 154 unit squares. The source’s original finite decimal display is $12.9282936576777576$$12.9282936576777576$; the rational fraction states the claimed bound exactly.

The earlier SQUISH packing improves the earlier reported side $12.931663721166847019324363825319$$12.931663721166847019324363825319$ from [evand exact optima 2026-10-05]. This repository confirmed the upper bound by independently re-implemented exact-rational checks: native exact_verify and devtools.check_rational_witness_independent accept every pair and wall of the earlier packing and reject both geometric controls. The verified display is $12.9282936576777577$$12.9282936576777577$, 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 $12.931663721166847019324363825319$$12.931663721166847019324363825319$. The earlier certified ceiling $12.931663721166847019324363825319$$12.931663721166847019324363825319$ 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.

The Gupta precision refinement

Gupta refines the earlier SQUISH construction at exact rational side $12926562245852346932527003441777/1000000000000000000000000000000$$12926562245852346932527003441777/1000000000000000000000000000000$, displayed upward as $12.9265622458523470$$12.9265622458523470$ (T-127). The complete factual packet retains every original certificate, complete comparator and actual deciding outcome for all seventeen source cases, including three withdrawals.

Complete dual-route native replay, full-roster controls and separately prompted source and production review confirm finite feasibility at V3/C3 using independently re-implemented deciding code. The repository routes share parsing, half-angle conversion, Fraction arithmetic and SAT methodology; the recorded code relation is to the source producer’s verification, not to each other. The full source poses are converted without dilation or rounding. The source credits Chaoweeraprasit for SQUISH and Daniel for the optimizer. No author program is executed here. Feasibility establishes no optimizer, local-minimum, rigidity, novelty, priority, global-optimality, formal proof or human-oversight assurance.

Nate Chaoweeraprasit’s certificate (T-136)

Chaoweeraprasit’s SQUISH packing, reported on jlevy/squares#481, at an exact rational side, displayed upward as $12.9230702023011408$$12.9230702023011408$ (T-136). The certificate gives a rational side and, for each of the 154 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 10 October 2026 and the final review of 11 October 2026 accepted the replay for finite feasibility. That proves $s(154)\le 12.9230702023011408$$s(154)\le 12.9230702023011408$, the verified upper bound until 12 October 2026: confirmed, independently re-implemented. It says nothing about optimality, rigidity or a local minimum.

The source states that SQUISH’s solver, its verification and the issue were written with Claude Opus 5.5 under the author’s direction.

The house it replaces, $12.9265622458523470$$12.9265622458523470$ from Chaoweeraprasit’s packing, improved by Gupta (T-127), stays retained with its own replay as history. The other certificates confirmed here at this count, T-115 at $12.9265622458535390$$12.9265622458535390$ and T-113 at $12.9282936576777577$$12.9282936576777577$, stay verified and are not selected.

Evan Daniel’s certificate (T-160)

Daniel’s packing from his size-chaining explorer, reported on jlevy/squares#552, at exact side $12922566401934059503377267965457/1000000000000000000000000000000$$12922566401934059503377267965457/1000000000000000000000000000000$, displayed upward as $12.9225664019340596$$12.9225664019340596$ (T-160). The certificate gives a rational side and, for each of the 154 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 12 October 2026 accepted the replay for finite feasibility. That proves $s(154)\le 12.9225664019340596$$s(154)\le 12.9225664019340596$, the verified upper bound until 15 October 2026: confirmed, independently re-implemented. It says nothing about optimality, rigidity or a local minimum.

The issue says the search code, the run and the request were done with Claude (Claude Code), which Daniel directed and reviewed.

The author states that the packing came from a time-travel test of his size-chaining explorer, restarted from this register as it stood on 5 October, and that it descends from Couzo’s 5 October packing of 155 squares with one square removed; he asks that the start be credited to that packing, and it is. Measured here, the packing is another arrangement than T-136’s: no symmetry of the box pairs the two square for square. The measurements are consistent with the author’s account of the start and do not establish it: 43 squares of Couzo’s 155 reappear to the digit, but so do 45 of his 154, and the round-by-round trace is not published.

The house it replaces, $12.9230702023011408$$12.9230702023011408$ from Chaoweeraprasit’s packing (T-136), stays retained with its own replay as history. The other certificates confirmed here at this count, T-127 at $12.9265622458523470$$12.9265622458523470$, T-115 at $12.9265622458535390$$12.9265622458535390$ and T-113 at $12.9282936576777577$$12.9282936576777577$, stay verified and are not selected.

Francisco Couzo’s certificate (T-182)

Couzo’s further arrangement of Daniel’s n = 154 packing, reported on jlevy/squares#586, at exact side $12921206577132631050071241337979/1000000000000000000000000000000$$12921206577132631050071241337979/1000000000000000000000000000000$, displayed upward as $12.9212065771326311$$12.9212065771326311$ (T-182). The certificate gives a rational side and, for each of the 154 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. One maintained route is the verifier the source ran; the independent rational corner checker and the third route are independently re-implemented. The review of 14 October 2026 accepted the replay for finite feasibility. That proves $s(154)\le 12.9212065771326311$$s(154)\le 12.9212065771326311$, the verified upper bound until 16 October 2026: confirmed, independently re-implemented. It says nothing about optimality, rigidity or a local minimum.

The author says OpenAI Codex assisted the search implementation, the verification orchestration and the requests, that earlier search work used Claude Code, and that he directed the work.

The house it replaces, $12.9225664019340596$$12.9225664019340596$ from Daniel’s packing (T-160), stays retained with its own replay as history. The other certificates confirmed here at this count, T-172 at $12.9224998146423975$$12.9224998146423975$, T-136 at $12.9230702023011408$$12.9230702023011408$, T-127 at $12.9265622458523470$$12.9265622458523470$, T-115 at $12.9265622458535390$$12.9265622458535390$ and T-113 at $12.9282936576777577$$12.9282936576777577$, stay verified and are not selected.

Mykhailo Polkovskyi’s certificate (T-186)

Polkovskyi’s refinement of Couzo’s further arrangement of Daniel’s n = 154 packing, reported in his repository on 14 October 2026, with no issue, at an exact rational side, displayed upward as $12.92120657713263105$$12.92120657713263105$ (T-186). The certificate gives a rational side and, for each of the 154 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 16 October 2026 accepted the replay for finite feasibility. That proves $s(154)\le 12.92120657713263105$$s(154)\le 12.92120657713263105$, the verified upper bound until 20 October 2026: confirmed, independently re-implemented. It says nothing about optimality, rigidity or a local minimum.

The certificate is Polkovskyi’s tightening of the packing T-182 records, measured here pose by pose as that packing (T-186’s regularizations), so the packing keeps its finder’s credit and Polkovskyi is credited with the certificate.

The commit makes no statement about AI assistance; the author’s issue 593 discloses Gemini assistance with solver scripting, precision checks and formatting.

The house it replaces, $12.9212065771326311$$12.9212065771326311$ from Daniel’s packing, improved by Couzo (T-182), stays retained with its own replay as history. The other certificates confirmed here at this count, T-172 at $12.9224998146423975$$12.9224998146423975$, T-160 at $12.9225664019340596$$12.9225664019340596$, T-136 at $12.9230702023011408$$12.9230702023011408$, T-127 at $12.9265622458523470$$12.9265622458523470$, T-115 at $12.9265622458535390$$12.9265622458535390$ and T-113 at $12.9282936576777577$$12.9282936576777577$, stay verified and are not selected.

Evan Daniel’s certificate (T-222)

Daniel’s continuation of Couzo’s further arrangement of Daniel’s n = 154 packing in a search the source started from Ostashev’s n = 156 packing, reported on jlevy/squares#694, at exact side $1615137617595210068732072760947/125000000000000000000000000000$$1615137617595210068732072760947/125000000000000000000000000000$, displayed upward as $12.9211009407616806$$12.9211009407616806$ (T-222). The certificate gives a rational side and, for each of the 154 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 20 October 2026 accepted the replay for finite feasibility. That proves $s(154)\le 12.9211009407616806$$s(154)\le 12.9211009407616806$, the verified upper bound: confirmed, independently re-implemented. It says nothing about optimality, rigidity or a local minimum.

The source’s credits say its work was done with Claude (Anthropic) under human direction, and the commit that publishes the certificate names Claude Opus 5.5 as co-author and says Daniel approved it.

The author’s chained explorer removed two squares from Ostashev’s n = 156 packing, the register’s witness for 156, and crossovers against close relatives of Couzo’s #586 packing and a later restart of his basin explorer reached this packing; the issue asks that #586’s arrangement, itself continued from Daniel’s #552, and Ostashev’s packing be credited as its starts, and calls it a different KKT point in #586’s family. Measured by the 20 October review, square by square, it pairs one to one with #586’s packing under a quarter turn, three squares beyond a tenth, at most 0.204: Couzo’s arrangement continued, not a refinement in this record’s sense (T-222), as Daniel’s packing at 271 (T-219) continues Couzo’s #561 arrangement. Ostashev’s packing lies within a tenth of it at only 94 of its 156 squares, so his credit as the start rests on the source’s account of the search, and Couzo’s on the measured arrangement.

The house it replaces, $12.92120657713263105$$12.92120657713263105$ from Daniel’s packing, improved by Couzo and Mishapolk (T-186), stays retained with its own replay as history. The other certificates confirmed here at this count, T-182 at $12.9212065771326311$$12.9212065771326311$, T-172 at $12.9224998146423975$$12.9224998146423975$, T-160 at $12.9225664019340596$$12.9225664019340596$, T-136 at $12.9230702023011408$$12.9230702023011408$, T-127 at $12.9265622458523470$$12.9265622458523470$, T-115 at $12.9265622458535390$$12.9265622458535390$ and T-113 at $12.9282936576777577$$12.9282936576777577$, stay verified and are not selected.