T-231: Polkovskyi's lower re-solve at 204 of Couzo's packing, 1.48e-11 below T-229

V3 C3 Upper bound confirmed superseded by T-257

2026-10-22 published · Polkovskyi after Couzo, Ostashev, PingYou Ltd, Daniel, Levy · $n = 204$

One complete rational source certificate of issue 719 reports a finite upper-bound improvement at the exact side it states: $s (204) \leq 14.808642830513087197802415708703$ , below T-229, the case's house best packing when it was registered here, by 1.48e-11, and below T-225, the house when the author claimed it, by 1.01e-8. Since 24 October 2026 the case holds it as its house best packing.

The certificate and its duplicate-square and outside-container controls were decided here on 23 October 2026 by both maintained exact routes, with sharp controls at exact contact and 1e-30 past it. The author says he imported this repository's sqpack/verify.py as a module, and the maintained exact_verify route decides through that file, so that replay shares the producer's named components. The 23 October stage 4 review decided the certificate again on a clean-room checker of its own that shares no code with either route or the source, and found no blocking defect: the bound is confirmed, independently re-implemented.

It is the second 204 file of issue 719's release, published on 22 October 2026 in the same commit as T-229's certificate. The issue named the other file, which T-229 registers; the author claimed this one in his comment of 22 October on the issue, after T-229 was registered. Measured here, the two files are one solved packing at two paddings: T-229's certificate is this one dilated by about 1 + 1e-12, so no square moves beyond 2.02e-11 between them.

Measured here, it is the exact solved side of T-229's packing. Its side and the 119 rows that Daniel's exact solver (exactsolve.py, MIT) determines equal that solver's output for T-229's witness, row for row, at the solver's padding of 1e-20; the other 85 rows are the solver's free squares and frozen flat variables. The source does not name the solver, so the record states it as a measured method, not as an antecedent of the packing, and the author is asked which program wrote the file. The project's regularization run of 22 October on T-229's witness reached the same side.

The author's comment of 23 October says he checked his certificates with sqpack/verify.py imported in a batch harness and with Daniel's verify_cert.py and verify_cert2.py as archived in Couzo's repository. Those statements are the author's; no source program ran here.

Credit Couzo's packing, first reported in #670 and refined in #675 and #678 (T-226, T-228, T-223), re-solved to a smaller side by Polkovskyi (https://github.com/Mishapolk/square-packing-records), whose continuous SQP optimization started from Couzo's #678 certificate in Couzo's frame, as his comment of 23 October on issue 719 says; Couzo's answer of 23 October on #670 traces the seed of his search to Polkovskyi's own T-186 packing, his refinement of Ostashev's packing as PingYou Ltd refined it. The issue discloses Gemini assistance.

Significance, composition and next rung

Significance: The solved side of T-229's packing at 204, at Daniel's exact solver's padding of 1e-20, 1.48e-11 below T-229, which certifies the same point padded by 1e-12 (T-229 is this certificate dilated by 1 + 1.000055e-12 to within 2.9e-15). That is T-225's case, the solved exact side of a packing already reported 1.76e-11 to 1.39e-9 below its certificate, S2. The step is padding alone, but eight orders above the padding the record scores S1 (T-170, T-174, T-186, at most 1.8e-19), and it moves the case's 12-place print (14.808642830528 to 14.808642830514), as at T-229's 105 (S2). A citable detail that moves no theorem; no lower bound, optimum or local minimum.
Next rung: None for the bound. V4 and C4 need a second accepted adversarial review by a distinct reviewer and a retained human oversight record; one AI review, of 23 October, is retained.
Regularizations: n = 204: a regularization of T-229’s packing by Polkovskyi (the exact solved side of T-229's packing, whose determined rows equal Daniel's exact solver's (exactsolve.py) row for row at its padding of 1e-20, as measured here; the author's comment of 23 October states a continuous SQP solve from Couzo's #678 certificate; solved to its side; identity: measured here; nearest centres one to one; 0 squares moved, at most 2.02e-11; no square turned).
Novelty: previously-published Present in an identified source

The case

Case record

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

As published

The packing as published, in its source’s pose, Daniel 2026 [Daniel issue 845 2026-10-29]

14.2914.808

141516

14.28315.283

\sqrt{n}

\sqrt{n} + 1

Proven

$14.29311$ $\leq s (204) \leq$ $14.807628$

exact

Citation record n-204

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

upperDaniel 2026, GitHub (confirmed T-257)

Open

optimality

The case record

LowerUpper

Verified14.29311422…14.80762747…

Reported14.36174789…14.80762747…

Gap0.51451324…

Results on the case

17 results in the register on $n = 204$ , oldest first, each with what it established and how it stands now.

2005 published T-007
$s (n) \geq min (⌈ \sqrt{n} ⌉, \sqrt{n - 2 ⌊ \sqrt{n} ⌋ + 1} + 1)$ for $4 \leq n \leq 324$
V0 C1 Lower bound incomplete on this case, superseded by T-083 and T-180 (reported)
Nagamochi · Nagamochi 2005 · source · register
2026-09-04 published T-085
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
2026-09-29 published T-083
$s (n) \geq 1 / 2 + \sqrt{n - ⌊ \sqrt{n} ⌋ + 1 / 4}$ for every nonsquare $8 \leq n \leq 324$
V3 C3 Lower bound confirmed
Karakuş · Karakuş 2026 · source · register
2026-10-05 published T-101
Exact certificates of 77 catalogue packings: s(n) ≤ S', 1.2e-16 to 9.8e-15 above each side
V3 C3 Upper bound confirmed on this case, superseded by T-257
Daniel after Couzo, de Winter, Ellsworth, Levy · evand exact optima 2026-10-05 · packet · register
2026-10-12 established T-155
Exact regularized sides at 59 counts, 5.4e-20 to 9.9e-15 below the ceilings they replace
V3 C3 Upper bound confirmed on this case, superseded by T-257
Levy after Daniel · register
2026-10-12 published T-162
New packings of 129, 204 and 303 squares, a cut-out and two compositions, each certified exactly
V3 C3 Upper bound confirmed superseded by T-257
Ostashev after Nakajima, Chaoweeraprasit, Daniel, Xu, Levy · Ostashev compositions 2026-10-12 · packet · register
2026-10-13 published T-174
Tighter certificates at 68 counts from PingYou Ltd, of others' packings and its own
V3 C3 Upper bound confirmed on this case, superseded by T-257
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-180
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
2026-10-14 published T-186
Exact certificates at twelve counts from Polkovskyi, tightening the smallest reported there
V3 C3 Upper bound confirmed on this case, superseded by T-257
Polkovskyi after Ostashev, Couzo, PingYou Ltd, Levy · Mishapolk KKT refinements 2026-10-14 · packet · register
2026-10-16 published T-193
Exact algebraic sides at sixteen counts from Daniel, of packings the record credits
V3 C3 Upper bound confirmed on this case, superseded by T-257
Daniel after Xu, Chaoweeraprasit, Ostashev, Couzo, PingYou Ltd, Gupta, Polkovskyi, Levy · Daniel exact current records 2026-10-16 · packet · register
2026-10-17 published T-226
Couzo's 204 of #670, below Polkovskyi's T-186, since superseded by his own #675 and #678
V3 C3 Upper bound confirmed superseded by T-257
Couzo after Polkovskyi, Ostashev, PingYou Ltd, Daniel, Levy · Couzo issue 670 2026-10-17 · packet · packet · register
2026-10-18 published T-223
Couzo's smaller packings at 204 and 260, the two in the corpus of issue 678's 85 bounds
V3 C3 Upper bound confirmed superseded by T-257
Couzo after de Winter, Polkovskyi, Ostashev, Daniel, Fang, PingYou Ltd, Levy · Couzo issue 678 2026-10-18 · packet · register
2026-10-18 published T-228
Couzo's 204 of #675, below his #670, since superseded by his #678
V3 C3 Upper bound confirmed superseded by T-257
Couzo after Polkovskyi, Ostashev, PingYou Ltd, Daniel, Levy · Couzo issue 675 2026-10-18 · packet · packet · register
2026-10-20 published T-225
Daniel's solved, regularized views at 204, 227, 229 and 260, each the smallest side reported there
V3 C3 Upper bound confirmed on this case, superseded by T-257
Daniel after Couzo, de Winter, Polkovskyi, Ostashev, PingYou Ltd, Levy · Daniel regularized lists 2026-10-20 · packet · register
2026-10-22 published T-229
Polkovskyi's re-solves of the house packings at 105 and 204, below T-174 and T-225
V3 C3 Upper bound confirmed on this case, superseded by T-257
Polkovskyi after Couzo, Muglich, Daniel, Xu, Ostashev, PingYou Ltd, Levy · Mishapolk issue 719 2026-10-22 · packet · register
2026-10-22 published T-231 this result
Polkovskyi's lower re-solve at 204 of Couzo's packing, 1.48e-11 below T-229
V3 C3 Upper bound confirmed superseded by T-257
Polkovskyi after Couzo, Ostashev, PingYou Ltd, Daniel, Levy · Mishapolk n204 2026-10-22 · packet · register
2026-10-29 published T-257
Daniel's new packings of 172, 204, 260 and 293 squares, 1.64e-4 to 1.18e-2 below each house
V3 C3 Upper bound confirmed
Daniel after Chaoweeraprasit, Gupta, Ostashev, kirilltym, Friedman · Daniel issue 845 2026-10-29 · packet · register

Links

On this site: Case record, $n = 204$ · Frontier row, $n = 204$ · T-231 in the results table
Issues: Issue #719

On GitHub, at main

Register: T-231 in results.yaml, line 32776
Evidence: E-mishapolk-n204-certificate-report · E-mishapolk-n204-exact-feasibility · E-mishapolk-n204-review-exact-feasibility
Proofs and certificates: certificate certificates · certificate n204.cert
Sources: Mishapolk n204 2026-10-22 (its own site, retained copy)
Source packet: resources/web/mishapolk-n204-2026-10-22/README.md
Artifacts: 16 artifacts and controls
resources/web/mishapolk-n204-2026-10-22/README.md resources/web/mishapolk-n204-2026-10-22/acquisition/claims.json resources/web/mishapolk-n204-2026-10-22/source/certificates/n204.cert resources/web/mishapolk-n204-2026-10-22/receipts/exact-certification.json.xz resources/web/mishapolk-n204-2026-10-22/receipts/sharp-controls.json resources/web/mishapolk-n204-2026-10-22/measure/identity.json campaign/issue-intake/upper-bounds-issue719-20261022/README.md devtools/upper_bound_reports.py docs/project/reviews/review-2026-10-23-mishapolk-n204.md resources/web/mishapolk-n204-2026-10-22/review/review-2026-10-23/README.md resources/web/mishapolk-n204-2026-10-22/review/review-2026-10-23/outputs/lineage.json tests/test_upper_bound_reports.py tests/test_upper_bound_reports_n204.py resources/web/mishapolk-n204-2026-10-22/receipts/exact-certification.json.xz resources/web/mishapolk-n204-2026-10-22/receipts/sharp-controls.json resources/web/mishapolk-n204-2026-10-22/review/review-2026-10-23/outputs/sharp-controls.json
Case file: frontier/n-204.md (verified lower, verified upper, reported lower, reported upper)