T-181: Exact rational certificates at 211, 241 and 302 from Couzo

V3 C3 Upper bound confirmed superseded by T-182, T-186 and T-188

2026-10-13 published · Couzo after Daniel, Ostashev, Fang, Ellsworth, Levy; Couzo after Daniel, Ostashev, BuddenPing, Fang, Ellsworth, Levy · $n = 211, 241, 302$

Three complete rational source certificates report finite upper-bound improvements at the exact sides they state: $s (211) \leq 5735936523425659529936799651 / 382560041700500000000000000$ , $s (241) \leq 56654293892568867634658195801 / 3543746727775500000000000000$ and $s (302) \leq 17.870013850404464344638853539285$ .

The same release states $s (339) \leq 4247372425158980137376328899 / 224403568199500000000000000$ , $s (379) \leq 19.953531531602552149382778262085$ , $s (380) \leq 71405320581108789876116351057 / 3574993313028500000000000000$ , $s (381) \leq 19.974977397656713743580918961689$ and $s (382) \leq 94198950843216391147752843823 / 4712633609411500000000000000$ , beyond the case corpus; the source register tracks those as beyond-horizon rows.

All 21 retained jobs of the issue-583 packet and all 42 of the issue-586 packet, each certificate with its 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 certificates at 211, 241, 339, 379, 380 and 382 are those at the revision of franciscouzo/square-packing the issue pins, which issue 586's revision replaced; the couzo-issue583-2026-10-13 packet retains them. The 302 and 381 are unchanged at issue 586's revision, where the couzo-certificates-2026-10-13 packet retains them.

The author reports that every certificate of issue 583 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. Issue 570's 154 is this revision's n154.cert, byte for byte. 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, 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, and 380 and 382 continue Fang's published arrangements (TheSnakeFang/squarepack-certs). 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 continue Ostashev's packings of issue 575 (T-175) and the 302 continues PingYou Ltd's arrangement (T-164), each paired square for square. On 15 October 2026 the case record at 302 adopted the 302, as the issue-586 packet retains it, as its house, its reported and verified upper bounds moving together and Daniel's regularization of Chaoweeraprasit's packing (T-146) kept as history, with T-164 verified and not selected; private-worker custody was closed in the same change (packing/devtools/house-custody-2026-10-11.json). At 211 and 241 issue 586's certificates (T-182) are smaller and hold the cases.

Significance, composition and next rung

Significance: Continuations of earlier packings at $n = 211$ , 241 and 302, 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.
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, 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…

Case record

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

As published

The packing as published, PingYou Ltd 2026 [Mishapolk KKT refinements 2026-10-14]

17.3817.870

171819

17.37818.378

\sqrt{n}

\sqrt{n} + 1

Proven

$17.38934$ $\leq s (302) \leq$ $17.870014$

exact

Citation record n-302

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

upperPingYou Ltd et al., GitHub (confirmed T-186)

Open

optimality

The case record

LowerUpper

Verified17.38934575…17.87001385…

Reported17.43342608…17.87001385…

Gap0.48066809…

Results on these cases

21 results in the register on $n = 211, 241, 302$ , oldest first, each with what it established and how it stands now.

2005 published T-007 · $n = 211, 241, 302$
$s (n) \geq min (⌈ \sqrt{n} ⌉, \sqrt{n - 2 ⌊ \sqrt{n} ⌋ + 1} + 1)$ 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 · $n = 211, 241, 302$
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 · $n = 241, 302$
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-186
Couzo · franciscouzo square-packing 2026-09-27 · packet · register
2026-09-29 published T-083 · $n = 211, 241, 302$
$s (n) \geq 1 / 2 + \sqrt{n - ⌊ \sqrt{n} ⌋ + 1 / 4}$ for every nonsquare $8 \leq n \leq 324$
V3 C3 Lower bound confirmed
Karakuş · Karakuş 2026 · source · register
2026-10-05 published T-098 · $n = 211, 241, 302$
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-186 and T-188
Daniel after Couzo, de Winter, Ellsworth, Levy · evand exact optima 2026-10-05 · packet · register
2026-10-07 published T-116 · $n = 302$
$s (n) \leq S_{n}$ at nine counts, from the second SQUISH update’s new rational packings
V3 C3 Upper bound confirmed superseded by T-186
Chaoweeraprasit after Ellsworth, Couzo and SQUISH · SQUISH second update 2026-10-07 · packet · register
2026-10-09 published T-136 · $n = 302$
Seventeen exact rational packings from SQUISH's third request
V3 C3 Upper bound confirmed on these cases, superseded by T-186
Chaoweeraprasit after Couzo, Xu, Göbel, Schadt, Ellsworth, Hajba, Levy · SQUISH third request 2026-10-09 · packet · register
2026-10-09 published T-138 · $n = 302$
Exact witnesses at Mishapolk's printed ceilings, the smallest known at 103 and 258 when registered
V3 C3 Upper bound confirmed superseded by T-186
Polkovskyi after Stenlund, Friedman, Ellsworth, Chaoweeraprasit, Couzo, Xu, Levy · Mishapolk decimal poses 2026-10-09 · packet · register
2026-10-10 published T-146 · $n = 302$
Exact rational certificates at fifteen counts from Daniel's regularized record lists
V3 C3 Upper bound confirmed on these cases, superseded by T-186
Daniel after Xu, Chaoweeraprasit, Polkovskyi, Levy · Daniel regularized lists 2026-10-10 · 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-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-164 · $n = 302$
A packing of 302 squares from four boundary patches of Xu's packing of 70 and two upright banks
V3 C3 Upper bound confirmed superseded by T-186
PingYou Ltd after Xu, Daniel, Levy · PingYou n302 certificate 2026-10-12 · packet · register
2026-10-13 published T-174 · $n = 211, 241, 302$
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, T-186 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 · $n = 211, 241, 302$
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 this result · $n = 211, 241, 302$
Exact rational certificates at 211, 241 and 302 from Couzo
V3 C3 Upper bound confirmed superseded by T-182, T-186 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 · $n = 211, 241$
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 · $n = 302$
Exact certificates at twelve counts from Polkovskyi, tightening the smallest reported there
V3 C3 Upper bound confirmed
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
2026-10-16 published T-193 · $n = 302$
Exact algebraic sides at sixteen counts from Daniel, of packings the record credits
V3 C3 Upper bound confirmed
Daniel after Xu, Chaoweeraprasit, Ostashev, Couzo, PingYou Ltd, Gupta, Polkovskyi, Levy · Daniel exact current records 2026-10-16 · packet · register

Links

On this site: Case record, $n = 211$ · Frontier row, $n = 211$ · Case record, $n = 241$ · Frontier row, $n = 241$ · Case record, $n = 302$ · Frontier row, $n = 302$ · T-181 in the results table

On GitHub, at main

Register: T-181 in results.yaml, line 24514
Evidence: E-couzo-583-certificate-report · E-couzo-586-n302-report · E-couzo-583-exact-feasibility · E-couzo-586-n302-exact-feasibility
Proofs and certificates: certificate certificates · certificate certificates
Sources: Couzo issue-583 certificates 2026-10-13 · Couzo certificates 2026-10-13 (its own site, retained copy)
Source packet: resources/web/couzo-issue583-2026-10-13/README.md · resources/web/couzo-certificates-2026-10-13/README.md
Artifacts: 10 artifacts and controls
resources/web/couzo-issue583-2026-10-13/README.md resources/web/couzo-issue583-2026-10-13/acquisition/claims.json resources/web/couzo-issue583-2026-10-13/source/certificates/n211.cert resources/web/couzo-issue583-2026-10-13/source/certificates/n241.cert resources/web/couzo-certificates-2026-10-13/source/certificates/n302.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-211.md (verified lower, verified upper, reported lower, reported upper) · frontier/n-241.md (verified lower, verified upper, reported lower, reported upper) · frontier/n-302.md (verified lower, verified upper, reported lower, reported upper)