T-182: Exact rational certificates at 154, 211 and 241 from Couzo

V3 C3 Upper bound confirmed

2026-10-13 published · Couzo after Daniel, Ostashev, BuddenPing, Fang, Ellsworth, Levy · $n = 154, 211, 241$

Three complete rational source certificates report finite upper-bound improvements at the exact sides they state: $s (154) \leq 12.921206577132631050071241337979$ , $s (211) \leq 14.993558913974006171333205657646$ and $s (241) \leq 15.987117095150238758321758129189$ .

The same release states $s (339) \leq 7843211074020413644369172117 / 414429522933500000000000000$ , $s (379) \leq 3465647601101642309045474807 / 173685927478500000000000000$ , $s (380) \leq 10836602188256857315809318661 / 542872123174250000000000000$ and $s (382) \leq 16910059023533815728370360359 / 846022698554500000000000000$ , beyond the case corpus; the source register tracks those as beyond-horizon rows.

All 42 retained jobs of the packet, its fourteen certificates with their duplicate-square and outside-container controls, were decided again here on 14 and 15 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 three 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 every certificate of issues 583 and 586 passes this repository's sqpack verify.py and Daniel's verify_cert.py, at the revisions the issue names, in exact rational arithmetic; that 154, 211, 241, 302 and 381 are exact optima from Daniel's exact contact solver; and that 339, 379, 380 and 382 are rational witnesses of their decimal packings with centres dilated by 1 + 1e-11 at 339 and 1 + 1e-12 at the others. Issue 586 updates issue 583 at 154, 339, 379, 380 and 382 and tightens the certificates of the 211 and 241 arrangements, which the record lists as regularizations of issue 583's (T-181). Those statements 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: Daniel's 154 arrangement of issue 552, itself descended from the author's earlier 155 with one square removed, continued through the author's browser refinement, native basin hopping and exact finishing; Ostashev's 211 and 241 arrangements of issue 575, continued with native search and refinement.

Beyond the case corpus, 339 and 379 continue the author's L-border constructions of issue 506 (339 last through the browser's fq search, 379 through native SLP refinement), and 380 and 382 continue Fang's published arrangements (TheSnakeFang/squarepack-certs) through browser and native searches. Daniel's optimizer, exact contact solver and verifier, Ellsworth's search and refinement ideas, and this project's sqpack verifier were used.

The author states that 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.

Measured pose by pose by that review, the 211 and 241 are issue 583's packings (T-181), no square moved beyond 1e-05, and the 154 moves issue 583's design well beyond a refinement, 19 squares beyond a tenth. On 15 October 2026 the case records adopted this entry as the house at 154, 211 and 241, each case's reported and verified upper bounds moving together and the earlier houses kept as history, Daniel's at 154 (T-160), Ostashev's at 211 (T-149) and Daniel's exact optimum of Couzo's packing at 241 (T-098), with T-172, T-175 and T-181 verified and not selected; private-worker custody was closed in the same change (packing/devtools/house-custody-2026-10-11.json).

Polkovskyi's issue 593 states the same poses at 211 and 241 and asks to be recorded jointly; the owner decides that credit (RQ-1).

Significance, composition and next rung

Significance: Continuations at $n = 211$ and 241 and a further arrangement at 154, 1.02e-03 to 2.78e-03 below the case ceilings, as T-158 is; no lower bound or optimum.
Next rung: V4 and C4 need a second accepted adversarial review by a distinct reviewer and a retained human oversight record; one AI review, of 14 October, is retained.
Regularizations: n = 211: a regularization of T-181’s packing by Couzo (a tighter exact certificate of the same arrangement from Daniel's exact contact solver; solved to its side; identity: measured here; nearest centres one to one; 0 squares moved, at most 5.374e-10; turns up to 4.947e-08°).

n = 241: a regularization of T-181’s packing by Couzo (a tighter exact certificate of the same arrangement from Daniel's exact contact solver; solved to its side; identity: measured here; nearest centres one to one; 0 squares moved, at most 6.25e-06; turns up to 7.463e-11°).
Novelty: previously-published Present in an identified source

The cases

Case record

Regularized by the Squares Project with Daniel's regularizer (evand/square-packing, MIT)

As published

The packing as published, Daniel 2026 [Daniel n154 sweep 2026-10-20]

12.4212.921

121314

12.41013.410

\sqrt{n}

\sqrt{n} + 1

Proven

$12.42686$ $\leq s (154) \leq$ $12.921101$

exact

Citation record n-154

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

upperDaniel & Couzo, GitHub (confirmed T-222)

Open

optimality

The case record

LowerUpper

Verified12.42686044…12.92110094…

Reported12.48666925…12.92110094…

Gap0.49424049…

Case record

Regularized by the Squares Project with Daniel's regularizer (evand/square-packing, MIT)

As published

The packing as published, Ostashev 2026 [Couzo verified bounds 2026-10-14]

14.5414.994

141516

14.52615.526

\sqrt{n}

\sqrt{n} + 1

Proven

$14.54457$ $\leq s (211) \leq$ $14.993550$

exact

Citation record n-211

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

upperOstashev & Couzo, GitHub (confirmed T-188)

Open

optimality

The case record

LowerUpper

Verified14.54457190…14.99354962…

Reported14.60509269…14.99354962…

Gap0.44897772…

Case record

Regularized by the Squares Project with Daniel's regularizer (evand/square-packing, MIT)

As published

The packing as published, Ostashev 2026 [Couzo certificates 2026-10-13]

15.5415.987

151617

15.52416.524

\sqrt{n}

\sqrt{n} + 1

Proven

$15.54160$ $\leq s (241) \leq$ $15.987118$

exact

Citation record n-241

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

upperOstashev & Couzo, GitHub (confirmed T-182)

Open

optimality

The case record

LowerUpper

Verified15.54160895…15.98711709…

Reported15.60000000…15.98711709…

Gap0.44550813…

Results on these cases

22 results in the register on $n = 154, 211, 241$ , oldest first, each with what it established and how it stands now.

2005 published T-007 · $n = 154, 211, 241$
$s (n) \geq min (⌈ \sqrt{n} ⌉, \sqrt{n - 2 ⌊ \sqrt{n} ⌋ + 1} + 1)$ for $4 \leq n \leq 324$
V0 C1 Lower bound incomplete on these cases, superseded by T-083 and T-180 (reported)
Nagamochi · Nagamochi 2005 · source · register
2026-09-04 published T-085 · $n = 154, 211, 241$
Nagamochi 2005, Lemma 1 is false for every container with $a > 3$ and $b > 2$
V3 C3 Correction confirmed
Karakuş; Vlasenko · Karakuş 2026 · chelokot Nagamochi counterexample 2026 · packet · register
2026-09-16 published T-057 · $n = 211$
$s (211) \leq 14.99796070496771500150 < 15$ , the first packing of 211 squares below the grid on record
V3 C3 Upper bound confirmed superseded by T-188
de Winter · de Winter n211 2026-09-16 · packet · register
2026-09-27 published T-056 · $n = 154, 241$
Smaller packings for 49 counts from $n = 68$ to $307$ , each certified two independent ways
V3 C3 Upper bound confirmed superseded by T-182 and T-222
Couzo · franciscouzo square-packing 2026-09-27 · packet · register
2026-09-29 published T-083 · $n = 154, 211, 241$
$s (n) \geq 1 / 2 + \sqrt{n - ⌊ \sqrt{n} ⌋ + 1 / 4}$ for every nonsquare $8 \leq n \leq 324$
V3 C3 Lower bound confirmed
Karakuş · Karakuş 2026 · source · register
2026-10-05 published T-098 · $n = 154, 211, 241$
Exact optima of 48 known-best packings: s(n) ≤ S', 3.5e-13 to 5.0e-11 below each printed side
V3 C3 Upper bound confirmed on these cases, superseded by T-182, T-188 and T-222
Daniel after Couzo, de Winter, Ellsworth, Levy · evand exact optima 2026-10-05 · packet · register
2026-10-07 published T-113 · $n = 154$
$s (n) \leq S_{n}$ at ten counts, $n = 108$ to 303, from new SQUISH rational packings
V3 C3 Upper bound confirmed superseded by T-222
Chaoweeraprasit after Ellsworth · SQUISH ten packings 2026-10-07 · packet · register
2026-10-07 published T-115 · $n = 154$
$s (n) \leq S_{n}$ at twelve counts, from the SQUISH update’s new and smaller rational packings
V3 C3 Upper bound confirmed superseded by T-222
Chaoweeraprasit after Couzo and Ellsworth · SQUISH update 2026-10-07 · packet · register
2026-10-08 published T-127 · $n = 154$
Fourteen rational refinements, seventeen complete source cases
V3 C3 Upper bound confirmed on these cases, superseded by T-222
Gupta after Chaoweeraprasit, Daniel · Gupta rational refinements 2026-10-08 · packet · register
2026-10-09 published T-136 · $n = 154$
Seventeen exact rational packings from SQUISH's third request
V3 C3 Upper bound confirmed on these cases, superseded by T-222
Chaoweeraprasit after Couzo, Xu, Göbel, Schadt, Ellsworth, Hajba, Levy · SQUISH third request 2026-10-09 · packet · register
2026-10-11 published T-149 · $n = 211$
New packings of 155, 156 and 211 squares, each with an exact rational certificate
V3 C3 Upper bound confirmed superseded by T-188
Ostashev after Nakajima, Levy, Xu · Ostashev records 2026-10-11 · packet · register
2026-10-12 published T-160 · $n = 154$
A smaller packing of 154 squares from Daniel's size-chaining explorer, certified exactly
V3 C3 Upper bound confirmed superseded by T-222
Daniel after Couzo, Levy · Daniel n154 2026-10-12 · packet · register
2026-10-12 published T-163 · $n = 211, 241$
s(n² − n + 1) < n for every n ≥ 15, and n² − n + F(n) squares below side n, from Fang
V2 C1 Upper bound reviewed superseded by T-182 and T-188
Fang after Arslanov, Mustafin, Shangitbayev, Ellsworth, Stead, de Winter, Daniel, Couzo · Fang excess theorem 2026-10-13 · packet · register
2026-10-12 published T-172 · $n = 154$
A packing of 154 squares from Couzo, continuing Daniel's arrangement
V3 C3 Upper bound confirmed superseded by T-222
Couzo after Daniel, Ostashev, Fang, Ellsworth, Levy · Couzo issue-583 certificates 2026-10-13 · packet · register
2026-10-13 published T-174 · $n = 211, 241$
Tighter certificates at 68 counts from PingYou Ltd, of others' packings and its own
V3 C3 Upper bound confirmed on these cases, superseded by T-182 and T-188
PingYou Ltd after Friedman, Ellsworth, Daniel, Xu, Barnett, Couzo, Ostashev, Chaoweeraprasit, Gupta, Fang, Polkovskyi, Levy · PingYou PR 574 certificates 2026-10-13 · packet · register
2026-10-13 published T-175 · $n = 211, 241$
New packings of 211, 241 and 304 squares, two with merged tilted bands, each certified exactly
V3 C3 Upper bound confirmed superseded by T-182 and T-188
Ostashev after Nakajima, Daniel, de Winter, Couzo, Levy, Xu · Ostashev merged bands 2026-10-13 · packet · register
2026-10-13 published T-180 · $n = 154, 211, 241$
Closed-form lower bound $s (N) \geq a (N)$ for every non-square $N \geq 49$ , reported
V0 C1 Lower bound reviewed
Hosono · wand125 closed form 2026-10-14 · packet · packet · register
2026-10-13 published T-181 · $n = 211, 241$
Exact rational certificates at 211, 241 and 302 from Couzo
V3 C3 Upper bound confirmed superseded by T-182 and T-188
Couzo after Daniel, Ostashev, Fang, Ellsworth, Levy; Couzo after Daniel, Ostashev, BuddenPing, Fang, Ellsworth, Levy · Couzo issue-583 certificates 2026-10-13 · Couzo certificates 2026-10-13 · packet · packet · register
2026-10-13 published T-182 this result · $n = 154, 211, 241$
Exact rational certificates at 154, 211 and 241 from Couzo
V3 C3 Upper bound confirmed
Couzo after Daniel, Ostashev, BuddenPing, Fang, Ellsworth, Levy · Couzo certificates 2026-10-13 · packet · register
2026-10-14 published T-186 · $n = 154$
Exact certificates at twelve counts from Polkovskyi, tightening the smallest reported there
V3 C3 Upper bound confirmed on these cases, superseded by T-222
Polkovskyi after Ostashev, Couzo, PingYou Ltd, Levy · Mishapolk KKT refinements 2026-10-14 · packet · register
2026-10-14 published T-188 · $n = 211$
A tighter exact certificate for 211 squares from Couzo, of Ostashev's arrangement
V3 C3 Upper bound confirmed
Couzo after Ostashev, Daniel, Levy · Couzo verified bounds 2026-10-14 · packet · register
2026-10-20 published T-222 · $n = 154$
Daniel's smaller n154 packing, continuing Couzo's #586 arrangement, certified exactly
V3 C3 Upper bound confirmed
Daniel after Couzo, Ostashev, Levy · Daniel n154 sweep 2026-10-20 · packet · packet · register

Links

On this site: Case record, $n = 154$ · Frontier row, $n = 154$ · Case record, $n = 211$ · Frontier row, $n = 211$ · Case record, $n = 241$ · Frontier row, $n = 241$ · T-182 in the results table

On GitHub, at main

Register: T-182 in results.yaml, line 24715
Evidence: E-couzo-583-586-certificate-report · E-couzo-586-exact-feasibility
Proofs and certificates: certificate certificates
Sources: Couzo certificates 2026-10-13 (its own site, retained copy)
Source packet: resources/web/couzo-certificates-2026-10-13/README.md
Artifacts: 10 artifacts and controls
resources/web/couzo-certificates-2026-10-13/README.md resources/web/couzo-certificates-2026-10-13/acquisition/claims.json resources/web/couzo-certificates-2026-10-13/source/certificates/n154.cert resources/web/couzo-certificates-2026-10-13/source/certificates/n211.cert resources/web/couzo-certificates-2026-10-13/source/certificates/n241.cert devtools/upper_bound_reports.py devtools/check_half_angle_area.py docs/project/reviews/review-2026-10-14-upper-bound-imports-561-593.md tests/test_upper_bound_reports.py tests/test_check_half_angle_area.py
Case file: frontier/n-154.md (verified lower, verified upper, reported lower, reported upper) · frontier/n-211.md (verified lower, verified upper, reported lower, reported upper) · frontier/n-241.md (verified lower, verified upper, reported lower, reported upper)