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V3 C3 Upper bound confirmed superseded by T-186 and T-225

2026-10-12 published · Ostashev after Nakajima, Chaoweeraprasit, Daniel, Xu, Levy · $n = 129, 204, 303$

Three complete rational source certificates report finite upper-bound improvements at the exact sides they state: $s (129) \leq 59350069252022321722894267696423193212052484284509904682889059350069252022321722894267696423193212052484284509904682889 / 5000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000$ , $s (204) \leq 74043316171934782832354950404729823976428779059647971914573074043316171934782832354950404729823976428779059647971914573 / 5000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000$ and $s (303) \leq 179099582910699079815957410813987360800486270805270137352241179099582910699079815957410813987360800486270805270137352241 / 10000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000$ .

All nine retained jobs, the three positives 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 three certificates 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, and the source ran none of them, so the replay is independently re-implemented. The 14 October review found no blocking defect.

The author reports that each packing passes its own C++ checker (sqverify --precise --certify, MPFR then GMP rationals) at delta = 1e-60, that sqverify --check-cert certifies the shipped certificates in GMP rationals with no dilation and no tolerance, and that a separately written 80-digit mpmath cross-check finds a least separation of 5.0e-21; the issue states that all three checkers are the author's, written with AI assistance, the first two sharing the C++ parser and geometry. It also states that each side is the exact first-order optimum of its contact topology, with nonnegative KKT multipliers at a 60-digit solution and the load-bearing squares scaled by 1 + 1e-20. Those statements are the author's; no source program ran here.

Credit Ostashev (https://github.com/dostashev/square-packing) with the three packings and their certificates. The issue says 204 and 303 are new arrangements: two copies of a new rectangle packing in opposite corners with aligned squares filling the rest, 44 unit squares in an (h + 3) x h rectangle at 204 and 62 in an (h + 6) x h rectangle at 303, both found by a GPU soft-to-rigid search after Nakajima's idea, started from corner staircases.

129 is a cut-out: a 163-square (h + 3) x h rectangle cut from the $s (302)$ house, Chaoweeraprasit's packing in Daniel's regularization, re-optimised as a rectangle, less its three rightmost unit columns; its aligned blocks are laid out like Xu's $s (129)$ and its tilted squares sit in a different band, so the issue calls it a different local optimum and not a refinement of Xu's. Each was solved to the exact optimum of its contact topology. The author asks to be credited and leaves the pieces' acknowledgement to the record.

The issue states AI assistance: an AI-assisted research loop the author drives with his own ideas, the rectangle library and composition being his; AI agents wrote and ran the search, composition and verification code, and an agent's check of the library against the register found 129.

Measured pose by pose by that review, no symmetry of the box pairs the 129 square for square with Xu's packing or T-146's, though 46 of Xu's squares lie in place under a quarter turn, and the 204 and 303 are new arrangements; that the 129 is cut from the $s (302)$ house is the author's statement, consistent with the measurements and established by none of them. At 129, 204 and 303 PR 574's certificates (T-174) are smaller, by 1.18e-19, 1.46e-19 and 1.79e-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 three cases, the smallest certificates there that the hold left in. The adoption of 16 October 2026 (PR #657) made Polkovskyi's T-186, smaller than T-174, the house at all three.

Significance, composition and next rung

Significance: New arrangements at 204 and 303 and a cut-out at 129, 2.02e-03 to 1.42e-02 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 [Mishapolk KKT refinements 2026-10-14]

11.3711.870

111213

11.35812.358

\sqrt{n}

\sqrt{n} + 1

Proven

$11.37428$ $\leq s (129) \leq$ $11.870014$

exact

Citation record n-129

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

upperOstashev et al., GitHub (confirmed T-186)

Open

optimality

The case record

LowerUpper

Verified11.37428158…11.87001385…

Reported11.44220661…11.87001385…

Gap0.49573226…

Case record

14.2914.809

141516

14.28315.283

\sqrt{n}

\sqrt{n} + 1

Proven

$14.29311$ $\leq s (204) \leq$ $14.808643$

exact

Citation record n-204

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

upperCouzo & Daniel, GitHub (confirmed T-225)

Open

optimality

The case record

LowerUpper

Verified14.29311422…14.80864284…

Reported14.36174789…14.80864284…

Gap0.51552861…

Case record

Regularized by the Squares Project with its own axis tool (regularize_axis_components/2)

As published

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

17.4117.910

171819

17.40718.407

\sqrt{n}

\sqrt{n} + 1

Proven

$17.41892$ $\leq s (303) \leq$ $17.909959$

exact

Citation record n-303

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

upperOstashev et al., GitHub (confirmed T-186)

Open

optimality

The case record

LowerUpper

Verified17.41892431…17.90995829…

Reported17.46198618…17.90995829…

Gap0.49103397…

Results on these cases

26 results in the register on $n = 129, 204, 303$ , oldest first, each with what it established and how it stands now.

2005 published T-007 · $n = 129, 204, 303$
$s (n) \geq min (⌈ \sqrt{n} ⌉, \sqrt{n - 2 ⌊ \sqrt{n} ⌋ + 1} + 1)$ 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 · $n = 129, 204, 303$
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
2026-09-27 published T-056 · $n = 303$
Smaller packings for 49 counts from $n = 68$ to $307$ , each certified two independent ways
V3 C3 Upper bound confirmed superseded by T-186
Couzo · franciscouzo square-packing 2026-09-27 · packet · register
2026-09-29 published T-083 · $n = 129, 204, 303$
$s (n) \geq 1 / 2 + \sqrt{n - ⌊ \sqrt{n} ⌋ + 1 / 4}$ for every nonsquare $8 \leq n \leq 324$
V3 C3 Lower bound confirmed
Karakuş · Karakuş 2026 · source · register
2026-10-03 published T-092 · $n = 303$
Smaller packings again at seven counts from $n = 208$ to $306$ , each certified two independent ways
V3 C3 Upper bound confirmed superseded by T-186
Couzo · franciscouzo square-packing 2026-10-03 · packet · register
2026-10-05 published T-098 · $n = 303$
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-186
Daniel after Couzo, de Winter, Ellsworth, Levy · evand exact optima 2026-10-05 · packet · register
2026-10-05 published T-101 · $n = 129, 204$
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 these cases, superseded by T-186 and T-225
Daniel after Couzo, de Winter, Ellsworth, Levy · evand exact optima 2026-10-05 · packet · register
2026-10-07 published T-113 · $n = 129, 303$
$s (n) \leq S_{n}$ at ten counts, $n = 108$ to 303, from new SQUISH rational packings
V3 C3 Upper bound confirmed superseded by T-186
Chaoweeraprasit after Ellsworth · SQUISH ten packings 2026-10-07 · packet · register
2026-10-07 published T-115 · $n = 129$
$s (n) \leq S_{n}$ at twelve counts, from the SQUISH update’s new and smaller rational packings
V3 C3 Upper bound confirmed superseded by T-186
Chaoweeraprasit after Couzo and Ellsworth · SQUISH update 2026-10-07 · packet · register
2026-10-08 published T-125 · $n = 129$
Complete rational construction reports at 25 counts
V3 C3 Upper bound confirmed superseded by T-186
Xu · ry-xu square packing 2026 · packet · register
2026-10-08 published T-127 · $n = 129$
Fourteen rational refinements, seventeen complete source cases
V3 C3 Upper bound confirmed on these cases, superseded by T-186
Gupta after Chaoweeraprasit, Daniel · Gupta rational refinements 2026-10-08 · packet · register
2026-10-09 published T-134 · $n = 303$
Six exact rational certificates from Couzo
V3 C3 Upper bound confirmed on these cases, superseded by T-186
Couzo after Polkovskyi, Chaoweeraprasit, Gupta, Xu, Daniel, Ellsworth, Levy · Couzo exact certificates 2026-10-09 · packet · register
2026-10-09 published T-136 · $n = 303$
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 · $n = 303$
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 · $n = 129, 303$
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
2026-10-12 established T-155 · $n = 204$
Exact regularized sides at 59 counts, 5.4e-20 to 9.9e-15 below the ceilings they replace
V3 C3 Upper bound confirmed on these cases, superseded by T-225
Levy after Daniel · register
2026-10-12 published T-162 this result · $n = 129, 204, 303$
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-186 and T-225
Ostashev after Nakajima, Chaoweeraprasit, Daniel, Xu, Levy · Ostashev compositions 2026-10-12 · packet · register
2026-10-13 published T-174 · $n = 129, 204, 303$
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-186 and T-225
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 · $n = 129, 204, 303$
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 · $n = 129, 204, 303$
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
2026-10-16 published T-193 · $n = 204$
Exact algebraic sides at sixteen counts from Daniel, of packings the record credits
V3 C3 Upper bound confirmed on these cases, superseded by T-225
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 · $n = 204$
Couzo's 204 of #670, below Polkovskyi's T-186, since superseded by his own #675 and #678
V0 C0 Upper bound recorded superseded by T-225
Couzo after Polkovskyi, Daniel, Levy · Couzo issue 670 2026-10-17 · packet · packet · register
2026-10-18 published T-223 · $n = 204$
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-225
Couzo after de Winter, Polkovskyi, Daniel, Fang, PingYou Ltd, Levy · Couzo issue 678 2026-10-18 · packet · register
2026-10-18 published T-228 · $n = 204$
Couzo's 204 of #675, below his #670, since superseded by his #678
V0 C0 Upper bound recorded superseded by T-225
Couzo after Polkovskyi, Daniel, Levy · Couzo issue 675 2026-10-18 · packet · packet · register
2026-10-20 published T-225 · $n = 204$
Daniel's solved, regularized views at 204, 227, 229 and 260, each the smallest side reported there
V3 C3 Upper bound confirmed
Daniel after Couzo, de Winter, Polkovskyi, Levy · Daniel regularized lists 2026-10-20 · packet · register
2026-10-22 published T-229 · $n = 204$
Polkovskyi's re-solves of the house packings at 105 and 204, below T-174 and T-225
V0 C0 Upper bound recorded
Polkovskyi after Couzo, Muglich, Daniel, Xu, PingYou Ltd, Levy · Mishapolk issue 719 2026-10-22 · packet · register

Links

On this site: Case record, $n = 129$ · Frontier row, $n = 129$ · Case record, $n = 204$ · Frontier row, $n = 204$ · Case record, $n = 303$ · Frontier row, $n = 303$ · T-162 in the results table

On GitHub, at main

Register: T-162 in results.yaml, line 20352
Evidence: E-ostashev-562-certificate-report · E-ostashev-562-exact-feasibility
Proofs and certificates: certificate records
Sources: Ostashev compositions 2026-10-12 (its own site, retained copy)
Source packet: resources/web/dostashev-compositions-2026-10-12/README.md
Artifacts: 10 artifacts and controls
resources/web/dostashev-compositions-2026-10-12/README.md resources/web/dostashev-compositions-2026-10-12/acquisition/claims.json resources/web/dostashev-compositions-2026-10-12/source/records/n0129.sqcert resources/web/dostashev-compositions-2026-10-12/source/records/n0204.sqcert resources/web/dostashev-compositions-2026-10-12/source/records/n0303.sqcert 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-129.md (verified lower, verified upper, reported lower, reported upper) · frontier/n-204.md (verified lower, verified upper, reported lower, reported upper) · frontier/n-303.md (verified lower, verified upper, reported lower, reported upper)