Case record

The best packing known for 237 squares, side

15.90260363 …

, Chaoweeraprasit 2026. Regularized by the Squares Project with its own axis tool (regularize_axis_components/2)

As published

The packing as published, Chaoweeraprasit 2026 [Couzo n237 2026-10-15]

$n = 237$

= corrects Nagamochi 2005open

$15.40805151 … \leq s (237) \leq 15.90260363 …$

15.4015.903

151617

15.39516.395

\sqrt{n}

\sqrt{n} + 1

Proven

$15.40805$ $\leq s (237) \leq$ $15.902604$

exact

Citation record n-237

lowerKarakuş 2026, arXiv corrects Nagamochi 2005 (confirmed T-083)

upperChaoweeraprasit et al., GitHub (confirmed T-190)

Open

optimality

Bounds

Best known packing

15.90260363…

15.9026036382595969

Found by: Chaoweeraprasit 2026
Improved by: Couzo, Polkovskyi
Construction: —
Source: [Couzo n237 2026-10-15]
Evidence: E-couzo-4e4c8ce-n237-certificate-report

Verified upper bound

15.90260363…

15.9026036382595969

The reported value, verified here.

Evidence: E-couzo-4e4c8ce-n237-exact-feasibility

Reported lower bound

15.45709095…

15.457090953095

Proved by: Hosono 2026
Kind: counting
Scope: Unrestricted unit-square packing with independent rotations and disjoint interiors.
Note: Hosono's closed form (wand125, 13 October 2026, T-180): $s (N)$ >= a(N) for every non-square N >= 49, from a measure of line segments on 34 intervals of the fractional part. At $n = 237$ , k = 15 and interval 18 give a(237) = 15.457090953095929454106610296136..., here rounded down at 12 and 30 places. The source replayed its certificate with its own checker; no replay has run here.
Source: [wand125 closed form 2026-10-14]
Evidence: E-wand125-closed-form-report

Verified lower bound

15.40805151…

15.4080515158

Corrects: Nagamochi 2005 (T-007)
Evidence: E-karakus-strip-lower, E-karakus-strip-measure-interval

Gap

0.49455212…

Verified upper minus verified lower.

Results in the register

T-007 V0 C1 Nagamochi · 2026-08-31 · 321 cases
$s (n) \geq min (⌈ \sqrt{n} ⌉, \sqrt{n - 2 ⌊ \sqrt{n} ⌋ + 1} + 1)$ for $4 \leq n \leq 324$
T-056 V3 C3 Couzo · 2026-09-29 · 49 cases
Smaller packings for 49 counts from $n = 68$ to $307$ , each certified two independent ways
T-083 V3 C3 Karakuş · 2026-10-02 · 301 cases
$s (n) \geq 1 / 2 + \sqrt{n - ⌊ \sqrt{n} ⌋ + 1 / 4}$ for every nonsquare $8 \leq n \leq 324$
T-085 V3 C3 Karakuş; Vlasenko · 2026-10-02 · 315 cases
Nagamochi 2005, Lemma 1 is false for every container with $a > 3$ and $b > 2$
T-098 V3 C3 Daniel after Couzo, de Winter, Ellsworth, Levy · 2026-10-05 · 48 cases
Exact optima of 48 known-best packings: s(n) ≤ S', 3.5e-13 to 5.0e-11 below each printed side
T-115 V3 C3 Chaoweeraprasit after Couzo and Ellsworth · 2026-10-07 · 12 cases
$s (n) \leq S_{n}$ at twelve counts, from the SQUISH update’s new and smaller rational packings
T-127 V3 C3 Gupta after Chaoweeraprasit, Daniel · 2026-10-08 · 17 cases
Fourteen rational refinements, seventeen complete source cases
T-134 V3 C3 Couzo after Polkovskyi, Chaoweeraprasit, Gupta, Xu, Daniel, Ellsworth, Levy · 2026-10-10 · 6 cases
Six exact rational certificates from Couzo
T-136 V3 C3 Chaoweeraprasit after Couzo, Xu, Göbel, Schadt, Ellsworth, Hajba, Levy · 2026-10-10 · 17 cases
Seventeen exact rational packings from SQUISH's third request
T-140 V3 C3 Couzo after Chaoweeraprasit, Polkovskyi, Xu, Daniel, Ellsworth, Levy · 2026-10-10 · 6 cases
Six more exact rational certificates from Couzo
T-161 V3 C3 Couzo after Ostashev, Chaoweeraprasit, Fang, Daniel, Ellsworth, Levy · 2026-10-14 · n = 156, 176, 237, 271
Exact rational certificates at 156, 176, 237 and 271 from Couzo
Claim and records
Claim
Four complete rational source certificates report finite upper-bound improvements at the exact sides they state: $s (156) \leq 12.979165853834276453965882038847$ , $s (176) \leq 42253500388043689581217825957 / 3060154296728500000000000000$ , $s (237) \leq 15.902947773658185869705573326239$ and $s (271) \leq 16.948992356451205603226618436627$ .

The same release states $s (338) \leq 18.906332809852093379991386331018$ , $s (344) \leq 83284325592732658265440964873 / 4384829811311500000000000000$ , $s (375) \leq 581837142778534347520704447657618356033117642159 / 29235669412085804585614997608579500000000000000$ , $s (376) \leq 19.904949736498672043413871274032$ and $s (378) \leq 28547984384116369335179568707 / 1431512228978500000000000000$ , beyond the case corpus; the source register tracks those as beyond-horizon rows.

All 27 retained jobs, the release's nine certificates with their 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 four certificates this entry cites with six controls each and held each retained file to its upstream bytes, fetched at the pin. No route here is the source's code; the author's runs of this repository's sqpack verifier are his and decide nothing here, so the replay is independently re-implemented. The 14 October review found no blocking defect.

The author reports that all nine requested certificates, fetched at the pin and checked against the digests and row counts the issue lists, pass this repository's sqpack verify.py and Daniel's verify_cert.py, at the revisions the issue names, in exact rational arithmetic, in 4.52 and 5.56 seconds; that 156, 237, 271, 338 and 376 are exact optima from Daniel's exact contact solver; and that 176, 344, 375 and 378 are rational witnesses of their decimal packings with centres dilated by 1 + 1e-11, 1 + 1e-12, 1 + 1e-10 and 1 + 1e-12. Those statements, and the author's replay script and receipt, are the author's; no source program ran here.

Credit Couzo (https://github.com/franciscouzo/square-packing), with the packings each started from, as the author asks: Ostashev's arrangements at 156 and 176, continued with the author's basin hopping and refinement and, at 176, through his web app; Chaoweeraprasit's SQUISH arrangements at 237 and 271, continued with the author's basin hopping and Daniel's fq refinement (237 improves the author's T-140 certificate).

Beyond the case corpus, 344 continues Fang's 344 (issue 484) from the packing reported in issue 506, and 338, 375, 376 and 378 the author's L-border constructions of issue 506 (338 and 375 from his 303, itself from SQUISH's issue 401; 376 from his earlier 304; 378 from his 306), refined with Daniel's fq at 338 and 376 and basin hopping at 378; the latest step at 375 came through the author's browser web app with Ellsworth's search. Daniel's optimizer, exact contact solver and verifier, Ellsworth's refinement and search ideas, and this project's sqpack verifier were used.

The author states that OpenAI Codex assisted the continuing search, the application work, the source audit, the verification orchestration and the request, that earlier search code and submissions were prepared with Claude Code, and that he directed the work.

Measured pose by pose by that review, the 156 continues Ostashev's packing (T-149), the 237 continues the author's T-140 design and the 271 refines Chaoweeraprasit's SQUISH packing (T-151), each paired square for square; the 176 moves Ostashev's issue-562 arrangement and pairs with no placement of it.

At 237 Polkovskyi's issue 593 (T-183) is smaller and held the case from 15 October 2026, and Couzo's T-190 has held it since 16 October 2026. At 156, 176 and 271 PR 574's certificates (T-174) are smaller, by 1.30e-19, 1.18e-02 and 1.69e-19, and were held for the owner from 15 October until the house rule of 16 October 2026 lifted the hold. Meanwhile this entry held the cases at 156 and 271, the smallest certificates there that the hold left in, and T-165, the author's re-solve of the 176, held 176. The adoption of 16 October 2026 (PR #657) made Polkovskyi's T-186, smaller than T-174, the house at all three.
Significance
Four continuations and refinements of earlier packings, 1.83e-06 to 5.22e-03 below their case ceilings, as T-128, T-130 and T-140 are; no lower bound or optimum.
Novelty
previously-published Present in an identified source
Records
n = 156 n = 176 n = 237 n = 271 register evidence 1 evidence 2
n = 156 n = 176 n = 237 n = 271 register evidence 1 evidence 2
T-174 V3 C3 PingYou Ltd after Friedman, Ellsworth, Daniel, Xu, Barnett, Couzo, Ostashev, Chaoweeraprasit, Gupta, Fang, Polkovskyi, Levy · 2026-10-14 · 68 cases
Tighter certificates at 68 counts from PingYou Ltd, of others' packings and its own
T-180 V0 C1 Hosono · 2026-10-14 · 264 cases
Closed-form lower bound $s (N) \geq a (N)$ for every non-square $N \geq 49$ , reported
T-183 V3 C3 Polkovskyi after Friedman, Daniel, Xu, Chaoweeraprasit, Barnett, Couzo, Ostashev, Ellsworth, Fang, Levy · 2026-10-14 · n = 237
Polkovskyi's certificate for 237 squares, the smallest known side there when registered
Claim and records
Claim
One complete rational source certificate reports a finite upper-bound improvement at the exact side it states: $s (237) \leq 15.902705057866366799436476453482$ .

All nine retained jobs, three certificates with their 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 this entry cites 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 author's runs of this repository's sqpack verifier, from copies outside the pinned tree, are his and decide nothing here, so the replay is independently re-implemented. The 14 October review found no blocking defect.

The author reports that Daniel's verify_cert.py and verify_cert2.py and this repository's sqpack/verify.py, run from copies outside the pinned tree, accept all 73 certificates with zero tolerance in 219 runs taking 43.9 seconds. Those statements are the author's; no source program ran here.

Credit Polkovskyi (Mishapolk), as he asks on the issue, with the certificate at 237. The issue's lineage column, headed 'Lineage / Baseline Holder', names Couzo's issue 561 (itself a continuation of Chaoweeraprasit's SQUISH packing) as the baseline at 237; it does not call the 237 a refinement, as it does at 146 and 261. Measured pose by pose by the 14 October review, the 237 is Couzo's issue-561 design moved, every square paired one to one, 63 moved beyond 1e-03 and seven beyond a tenth: another arrangement of that design.

The issue claims all 73 as beating or tying every standing and pending record; measured here, 13 of the files are byte for byte certificates by Daniel (70, 102, 123, 269, 292, 295) and Couzo (105, 108, 127, 209, 267, 305, 306), 40 have the case's exact side, and at 129, 176, 204, 303 and 304 the side is within 1e-30 of Ostashev's earlier issues 562 and 575, at 211 and 241 it equals Couzo's issue-586 certificates exactly, and at 156 and 271 it equals Couzo's issue 561 exactly.

The same review compared the poses: at 211 and 241, where the issue asks to be recorded jointly with issue 586, its certificates are issue 586's pose for pose, at 241 all but one square, which lies 1.9e-04 from issue 586's, and both are exact re-solves of Couzo's issue-583 arrangement, first committed at 13:53:57Z on 13 October, two hours before the issue's first commit of its 211 and 241 at 15:51:54Z; issue 586's commit followed at 17:30:53Z. Identity of poses is what was measured: two exact solves of one contact graph from one start give the same poses, so it establishes no copying, and the owner decides the joint credit. The source discloses Gemini assistance with solver scripting, precision checks and formatting.

On 15 October 2026 the case record at 237 adopted this entry as its house, its reported and verified upper bounds moving together and Couzo's certificate (T-140) kept as history, with T-161 verified and not selected; private-worker custody was closed in the same change (packing/devtools/house-custody-2026-10-11.json).
Significance
Another arrangement of Couzo's design at $n = 237$ , 2.45e-04 below the case ceiling and the smallest known side there, within T-128's range; no lower bound or optimum.
Novelty
previously-published Present in an identified source
Records
n = 237 register evidence 1 evidence 2
n = 237 register evidence 1 evidence 2
T-189 V3 C3 Couzo after Polkovskyi, Fang, Daniel, Ellsworth, Levy · 2026-10-15 · n = 237
Couzo's refinement of Polkovskyi's 237, and fifteen beyond-horizon bounds
Claim and records
Claim
One complete rational source certificate reports a finite upper-bound improvement at the exact side it states: $s (237) \leq 12163897896660350527924222391 / 764899771970500000000000000$ .

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 $s (339) \leq 15760492451744053967692664819 / 833191141347000000000000000$ , $s (342) \leq 13505979727081420240143041327 / 712163911851000000000000000$ , $s (382) \leq 11022991764484891182563636300647365246824336756792134646965777453391 / 551490929281812357791494555514909292818123577914945500000000000000$ , $s (409) \leq 12913362409176042797577197977 / 621808448863500000000000000$ , $s (419) \leq 80695119733625752827535766822771925130844543419314113774658870691 / 3849958446685324913565194998050235494743230122939250000000000000$ , $s (450) \leq 86015497401862821831433322593 / 3951569106671500000000000000$ , $s (461) \leq 4815728038236625594119195870229537391060756184031012912605279819357 / 219163872303991689276056426799981765896492324486478500000000000000$ , $s (542) \leq 5532991125316051254254297371 / 232201273560500000000000000$ , $s (552) \leq 1336913104446320995776430361526045060886790944910372943 / 55779003704893601939282009645665379846500000000000000$ , $s (599) \leq 26704722559059887305228107379 / 1069556443814500000000000000$ , $s (648) \leq 1580917747964295160131077049 / 60879375637000000000000000$ , $s (649) \leq 22005763134880310331656432423 / 847125729886500000000000000$ , $s (651) \leq 241476276408156387115385105463274782867431850070691071 / 9293083814480905136206318825678729558937500000000000$ , $s (757) \leq 2017013645520556066618921865312330242011007231621137760082264853190861249025432441426574553 / 72059361334628479177359726217958887411830602793896518335182698782951858151500000000000000$ and $s (804) \leq 21316338098656938426429163392528974468810470714920720667861834801531 / 737069301316017949763472070077201158201889297143765500000000000000$ , 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."
Significance
A refinement of issue 593's packing at $n = 237$ , 1.01e-04 below the case ceiling and the smallest known side there when published, as T-172 and T-183 are, with fifteen finite construction bounds beyond the case corpus; no lower bound or optimum.
Novelty
previously-published Present in an identified source
Records
n = 237 register evidence 1 evidence 2
n = 237 register evidence 1 evidence 2
T-190 V3 C3 Couzo after Polkovskyi, Chaoweeraprasit, Daniel, Levy · 2026-10-15 · n = 237
A tighter exact certificate for 237 squares from Couzo, below his issue-607 certificate
Claim and records
Claim
One complete rational source certificate reports a finite upper-bound improvement at the exact side it states: $s (237) \leq 15.902603638259596880179144254274$ .

All three retained jobs, 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 author published it in square-packing on 15 October 2026, "Update all verified bounds for n < 1024", with no issue, and it is unchanged at the pinned head.

The author's index (certificates/best/manifest.json at the pin) reports that his standard-library checker verify_bound_certificate.py, which uses Daniel's exact rational separating-axis geometry, accepts the certificate under both its closed-box and strict contracts (3,214 candidate pairs), and that this repository's sqpack verify.py is the first route of the index. The certificate's metadata calls it an exact optimum seeded from Polkovskyi's issue-593 certificate and found by the author's fq search. Those statements are the author's; no source program ran here.

Credit Couzo (https://github.com/franciscouzo/square-packing) with this refinement at 237 of Polkovskyi's issue-593 certificate (T-183), from which the certificate's metadata says the author's fq search was seeded, and whose credit he asks to keep; the index names Chaoweeraprasit (itsnaka) as the construction's origin. Daniel's solver and verifier and this repository's exact verifier were used. The pinned files state nothing about AI assistance; issue 607, filed the day before for the author's earlier certificate at 237 (T-189), states that he used OpenAI Codex for search and verification orchestration, that earlier search work used Claude Code, and that he directed the work.
Significance
A refinement of issue 607's packing at $n = 237$ to a new smallest side, 2.40e-11 below it and 1.01e-04 below the case ceiling, as T-165 is; not padding; no lower bound or optimum.
Novelty
previously-published Present in an identified source
Records
n = 237 register evidence 1 evidence 2
n = 237 register evidence 1 evidence 2

Verification

upper: replayed here; lower: external proof (read here), audited here

formal lower differs from report

Rigidity

not rigid, numerically checked, numerical multiprecision

Evidence: E-translation-escape-not-rigid

Scope

Square 28 of the retained witness (witness id 29) translates 0.000335 along (0, -1) with the packing still valid, so the configuration admits a non-trivial feasible motion; 21 of its 237 squares do. Every constraint is exactly affine in the slide parameter, so the arithmetic carries no linearization error, but the coordinates are the witness's own finite-precision transcription: this settles the retained configuration, not the true optimum. Rigidity and optimality are independent, and this bears only on the former.

Regularized drawing

Regularized by the Squares Project with its own axis tool (regularize_axis_components/2). A drawing of T-190's packing at the witness's own side: a pure regularization, which proves no new bound, so not a result. The house is itself its source's regularization of T-183's packing (Couzo after Polkovskyi, Chaoweeraprasit, Daniel, Levy), registered as T-190 for the upper bound its exact side proves.

Open questions

Priority: s(237) <= 15.91292783720833270, Casson's packing, larger than the reported side (Casson)
Priority: s(237) <= 15.911196348876377, the first of Couzo's packings for this count (Couzo)
Priority: Prior reported s(237) <= 15.911191683001783031028912780176 from [evand exact optima 2026-10-05]; its evidence and finder credit remain in the retained earlier packet and case history. (Couzo)
Priority: Earlier source ceiling 15.9036762351906287 from [SQUISH update 2026-10-07]; the complete original frontier and house remain retained in the packet. (Chaoweeraprasit)
Priority: Earlier house 15.9036762351891381 from [Gupta rational refinements 2026-10-08], the case’s ceiling until 11 October 2026; its certificate and replay remain retained. (Chaoweeraprasit (itsnaka), Gupta)
Priority: Earlier house 15.9029496076778317 from [Couzo certificates 2026-10-10], the case’s ceiling from 11 October 2026 until 15 October 2026; its certificate and replay remain retained. (Chaoweeraprasit (itsnaka), Couzo)
Priority: Earlier house 15.9027050578663668 from [Mishapolk records 2026-10-14], the case’s ceiling from 15 October 2026 until 16 October 2026; its certificate and replay remain retained. (Chaoweeraprasit (itsnaka), Couzo, Mishapolk)

Evidence and sources

34 evidence entries

[Gupta rational refinements 2026-10-08] upper bound report
[SQUISH update 2026-10-07] upper bound report
[evand exact optima 2026-10-05] upper bound report
[franciscouzo square-packing 2026-09-27] upper bound report
[griffcass square-packing 2026-09-23] upper bound report
[Kingbird] record catalogue
[Nagamochi 2005] lower bound proof
[Friedman DS7] survey
[Karakuş 2026] lower bound proof
[Couzo certificates 2026-10-10] upper bound report
[wand125 closed form 2026-10-14] lower bound proof
[Mishapolk records 2026-10-14] upper bound report
[Couzo n237 2026-10-15] upper bound report

In the frontier survey On GitHub

$s (237)$ — open

Hosono’s closed form, 14 October 2026. Hosono’s closed form (wand125, 13 October 2026, T-180) states $s (N) \geq a (N)$ for every non-square $N \geq 49$ , from a measure of line segments on 34 intervals of the fractional part. At $n = 237$ , $k = 15$ and interval 18 give $a (237) = 15.457090953095929454106610296136 \dots$ , so the reported lower bound here is now $s (237) \geq 15.457090953095$ , rounded down, above Nagamochi’s closed form, $15.42220510185$ . 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, 16 October 2026. Couzo’s refinement of Polkovskyi’s arrangement of Couzo’s design, after Chaoweeraprasit’s SQUISH packing (T-190) is now this case’s house: its exact certificate proves $s (237) \leq 15.9026036382595969$ , 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 $1.01 \times 10^{- 4}$ . The earlier ceiling, displayed outward as $15.9027050578663668$ from Chaoweeraprasit’s packing, improved by Couzo and Mishapolk (T-183), applied until this adoption and stays retained as history (the certificate).

House adoption, 15 October 2026. Polkovskyi’s arrangement of Couzo’s design, after Chaoweeraprasit’s SQUISH packing (T-183) was this case’s house until 16 October 2026: its exact certificate proves $s (237) \leq 15.9027050578663668$ , the reported and the verified upper bound, $2.45 \times 10^{- 4}$ below the ceiling the case held until then, $15.9029496076778317$ from Chaoweeraprasit’s packing, improved by Couzo (T-140), which stays retained as history (the certificate).

House adoption, 11 October 2026. Couzo’s refinement of Chaoweeraprasit’s SQUISH packing (T-140) was this case’s house until 15 October 2026: its exact certificate proves $s (237) \leq 15.9029496076778317$ , the reported and the verified upper bound, $7.27 \times 10^{- 4}$ below the ceiling the case held until then, $15.9036762351891381$ 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 $s (237) \leq 15.9036762351906287$ , with exact side $8952973795828191 / 562949953421312$ . 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(238), squares removed. The revision-specific packet retains the source print $15.9036762351906287$ 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.

Open. Previously, the earlier verified packing, Couzo’s at Daniel’s exact optimum (T-098), gives $s (237) \leq 15.9111916830017830311 \dots$ , and the strongest lower bound independently verified here is $15.408051$ from Karakuş’s general theorem, leaving a gap of $0.5031$ . General bound: $s (N) \geq 1 / 2 + \sqrt{N - ⌊ \sqrt{N} ⌋ + 1 / 4}$ .

The exact optimum

Daniel’s square-packing published on 5 October 2026 an exact rational certificate of this packing at its exact optimum (T-098): the same 237 squares in a square of side $15.9111916830017830311 \dots$ , $6.7 \times 10^{- 12}$ below the side Couzo prints. Square for square, its pose lies within $6.3 \times 10^{- 4}$ of the binary64 pose the atlas previously pictured for this count. 112 squares move by more than $1 \times 10^{- 8}$ , 13 of them squares the source lists as carrying no force and the other 99 by at most $6.5 \times 10^{- 6}$ ; every other square moves by at most $1.1 \times 10^{- 11}$ . The previous atlas witness was that binary64 pose, posed at its finder’s larger side; the side above is witnessed by the certificate itself. Daniel’s solver moves the binary64 pose to a nearby exact KKT point of the problem of minimizing the side under non-overlap, computed at 80 digits, and rounds it outward to rationals, each square a rational centre and a rational $t = \tan (θ / 2)$ .

This repository decides the certificate exactly. Converted without rounding, every pair and every wall is decided over $ℚ$ twice, by sqpack’s exact separating-axis test and by an independent checker that shares no code with it, and the source’s own two checkers, run here as retained, accept it as well (receipts). That previously proved $s (237) \leq 15.9111916830017830311 \dots$ , the verified upper bound; it says nothing about optimality. The source reports the exact point as a KKT local minimum: multipliers that keep it in equilibrium, a reduced Hessian positive definite once its exact flat motions are set aside (its per-count report), and no first-order descent across corner-to-corner contacts, each computed numerically at that point; none of that is verified here, and it bears on this packing alone, not on $s (237)$ . On jlevy/squares#375 its author wrote that the solver, its checkers and the batch “were written with Claude (Anthropic) as a coding and research agent, directed and reviewed by me.”

The packing

Couzo’s square-packing reports a packing of side $15.911191683008479$ for this count, dated 26 September 2026 and unchanged when this record retained the repository on 27 September 2026 (T-056). Its first packing for this count, of side $15.911196348876377$ , is dated 24 September 2026. The repository names no method and no tolerance, and itself states no AI assistance; its author said on issue #227 that he found the 102 and 103 packings “with the help of Claude”.

This repository certifies it exactly. The retained decimal pose rounds to an exact rational packing at centre dilation 1, of side $15.9111916830084787629 \dots$ , no larger than the printed side, and every pair and every wall is decided over $ℚ$ twice, by the promotion’s exact separating-axis test and by an independent checker that shares no code with it (receipt). That proves $s (237) \leq 15.911191683008479$ , which was the verified upper bound until Daniel’s exact optimum above replaced it; it says nothing about optimality. Interval arithmetic on the printed pose itself, with each angle’s true cosine and sine and no rational rounding, decides every pair and wall again and gives the same bound (interval route).

Casson’s square-packing, dated 23 September 2026 and made with the help of Claude as its README says, reports a packing of side $15.91292783720833270$ for this count, larger than Couzo’s by $0.00173615419985370$ . Casson’s is the earlier of the two by the timestamps this record’s priority notes keep: Couzo’s first packing for this count, of side $15.911196348876377$ , is dated 24 September 2026. The record states both dates and infers nothing about whether either packing derives from the other.

The previous best known packing

Before this intake the best known packing was the Kingbird catalogue’s, of side $15.91421356237309$ .

Found by Ellsworth in 2024, via an unrecorded method.

The lower bound

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

s (N) \geq \frac{1}{2} + \sqrt{N - ⌊ \sqrt{N} ⌋ + \frac{1}{4}}

Correction, 2 October 2026. Until that date the verified lower bound here was Nagamochi’s general closed form, $s (N) \geq min (⌈ \sqrt{N} ⌉, \sqrt{N - 2 ⌊ \sqrt{N} ⌋ + 1} + 1)$ , which is stronger at this $n$ 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.

bound	evidence	run	code	programs
verified lower	`E-karakus-strip-lower`	a published proof	no code	no verification code
verified lower	`E-karakus-strip-measure-interval`	audited here	independent	`V-check-karakus-strip-measure` (first-party)
verified upper	`E-couzo-4e4c8ce-n237-exact-feasibility`	replayed here	independent	`V-sqpack-verify`, `V-check-rational-witness-independent` (first-party); `V-evand-arrangement-receipts`, `V-upper-bound-reports`, `V-evand-exact-certificates` (first-party, premises)

Gupta refines the earlier SQUISH construction at exact rational side $3180735247037827607980371558401 / 200000000000000000000000000000$ , displayed upward as $15.9036762351891381$ (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.

Francisco Couzo’s certificate (T-140)

Couzo’s refinement of Chaoweeraprasit’s SQUISH packing, reported on jlevy/squares#488, at exact side $99393435047986448060704927533 / 6250000000000000000000000000$ , displayed upward as $15.9029496076778317$ (T-140). The certificate gives a rational side and, for each of the 237 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 10 October 2026 and the final review of 11 October 2026 accepted the replay for finite feasibility. That proves $s (237) \leq 15.9029496076778317$ , the verified upper bound until 15 October 2026: confirmed, independently re-implemented. It says nothing about optimality, rigidity or a local minimum.

The author says the search code and the request were written with Claude under his direction.

The house it replaces, $15.9036762351891381$ 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-136 at $15.9029892208749656$ , T-134 at $15.9036709055806231$ and T-115 at $15.9036762351906287$ , stay verified and are not selected.

Mykhailo Polkovskyi’s certificate (T-183)

Polkovskyi’s arrangement of Couzo’s design, after Chaoweeraprasit’s SQUISH packing, reported on jlevy/squares#593, at exact side $7951352528933183399718238226741 / 500000000000000000000000000000$ , displayed upward as $15.9027050578663668$ (T-183). The certificate gives a rational side and, for each of the 237 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 $s (237) \leq 15.9027050578663668$ , the verified upper bound until 16 October 2026: confirmed, independently re-implemented. It says nothing about optimality, rigidity or a local minimum.

The source discloses Gemini assistance with solver scripting, precision checks and formatting.

The house it replaces, $15.9029496076778317$ from Chaoweeraprasit’s packing, improved by Couzo (T-140), stays retained with its own replay as history. The other certificates confirmed here at this count, T-174 at $15.9029477736581859$ , T-161 at $15.9029477736581859$ , T-136 at $15.9029892208749656$ , T-134 at $15.9036709055806231$ , T-127 at $15.9036762351891381$ and T-115 at $15.9036762351906287$ , stay verified and are not selected.

Francisco Couzo’s certificate (T-190)

Couzo’s refinement of Polkovskyi’s arrangement of Couzo’s design, after Chaoweeraprasit’s SQUISH packing, reported in his repository on 15 October 2026, with no issue, at exact side $7951301819129798440089572127137 / 500000000000000000000000000000$ , displayed upward as $15.9026036382595969$ (T-190). The certificate gives a rational side and, for each of the 237 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 16 October 2026 accepted the replay for finite feasibility. That proves $s (237) \leq 15.9026036382595969$ , the verified upper bound: confirmed, independently re-implemented. It says nothing about optimality, rigidity or a local minimum.

The certificate is Couzo’s tightening of the packing T-183 records, measured here pose by pose as that packing (T-190’s regularizations), so the packing keeps its finder’s credit and Couzo is credited with the certificate.

The commit makes no statement about AI assistance; the author’s issue 607, of the day before, says OpenAI Codex assisted the search and verification orchestration, that earlier search work used Claude Code, and that he directed the work.

The house it replaces, $15.9027050578663668$ from Chaoweeraprasit’s packing, improved by Couzo and Mishapolk (T-183), stays retained with its own replay as history. The other certificates confirmed here at this count, T-189 at $15.9026036382835755$ , T-174 at $15.9029477736581859$ , T-161 at $15.9029477736581859$ , T-140 at $15.9029496076778317$ , T-136 at $15.9029892208749656$ , T-134 at $15.9036709055806231$ , T-127 at $15.9036762351891381$ and T-115 at $15.9036762351906287$ , stay verified and are not selected.