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

2026-10-29 published · Daniel after Chaoweeraprasit, Gupta, Ostashev, kirilltym, Friedman · $n = 172, 204, 260, 293$

Four complete rational source certificates report finite upper-bound improvements at the exact sides they state: $s (172) \leq 13.617935698381415404117316038995$ , $s (204) \leq 14.807627471881579627471866771703$ , $s (260) \leq 16.633982502121883982502105249666$ and $s (293) \leq 17.633982502122781982502105147646$ . They are 1.64e-4 to 1.18e-2 below each house: 1.05e-3 below T-174 at 172, 1.02e-3 below T-231 at 204, 1.18e-2 below T-225 at 260 and 1.64e-4 below the Kingbird catalogue's 723/41 at 293, which T-101 verifies; at 260 they are below issue 844's certificate too.

The four certificates and their duplicate-square and outside-container controls were decided here on 31 October 2026 by both maintained exact routes. Those routes share components with the producer: both read the certificates through the evand-cert reader, whose line filter and corner construction are transcribed from Daniel's own checkers, the ones the certificates name. The 31 October review decided them again on a clean-room checker of its own, by separating axes over every pair and by exact clipped areas with ten sharp controls each, by a Claude agent, a system family the producer also used, and found no blocking defect, so the bounds are confirmed, independently re-implemented.

The issue says the source's two exact checkers, search/exact/verify_cert.py (separating axes) and verify_cert2.py (polygon intersection), byte-identical to the copies the n183 packet retains, both report VALID on all five certificate files, and that they share an author and the certificate format. At 172 the certificate dilates the source's 80-digit KKT point by 1 + 1e-20, with numerical evidence at 80 digits, not interval proofs, that it is a local minimum of S strict modulo 69 exact flat motions; s172_1028_regularized.cert is a regularized view at the same side. At 204, 260 and 293 the exact solver did not converge and the certificates dilate the binary64 packings by 1.5e-11 to 1.8e-11; local optimality is not established there.

The commit ends with a Co-Authored-By trailer naming Claude. Those statements are the author's; no source program ran here.

Credit Daniel (https://github.com/evand/square-packing), whose population searches over his packing store found all four, after the starts the issue asks to be credited: at 172 the $s (199)$ record of Chaoweeraprasit (itsnaka), improved by Gupta, from which an explorer search and the removal of one L-shaped strip gave 172 at side - 1; at 204 Ostashev's $s (176)$ packing (#575), used as a seed; at 260 Tymchishin's (kirilltym) $s (260)$ from his public Kaggle notebook 'Square Packing Lab | CPU/CUDA/MPS' (side 16.638543986, 26 October), whose tilted island a fault-line composition and refill at 405 improved before a composition run put it in a 260 frame; and at 293 the 260 packing plus one L-shaped strip (Friedman's L step), with the same credit.

The issue calls all four new packings, none a refinement of the register's. Kaggle per-puzzle sides it quotes at 172, 204 and 293 state no packing and are no report. The source's credits say its work was produced by Claude (Anthropic) under human direction.

Significance, composition and next rung

Significance: New arrangements below each house: 1.05e-03 at 172 and 1.02e-03 at 204, 1.18e-02 at 260, and 1.64e-04 at 293 below the catalogue construction 723/41 that the case records as its conjectured optimum, none one to one with the house's packing under any symmetry, as T-243's 3.17e-03 at 182 (S3) is. No lower bound or optimum.
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 31 October, is retained. It is the smallest validated side at 172, 204, 260 and 293, and the case adopted it there on 31 October 2026 under the house rule, with both strong generators' regularization of each new house; the conjectured optimum 723/41 at 293 is withdrawn (N-9). The review's questions for the author (Tymchishin's identity, the 172's parent, the 204 exact point and an enclosure of the 172 KKT point) hold no rung.
Regularized drawings: n = 172: a regularization of T-257’s packing by Levy (regularize_grid_slack/2: Daniel's coincidence placement (evand/square-packing search/regularize reg.coincidences and reg.place, MIT), ported to start from the witness's exact frame, over its number field where it has one, and to keep its contacts, then the same-side exact re-solve, through devtools.regularize_grid_slack; the project’s modifications: port of his coincidence placement (regularize_grid_slack/2) rather than his code, same-side exact re-solve (devtools.regularize_same_side); at its own side; identity: measured here; nearest centres one to one; 49 squares moved, at most 0.0986551; turns up to 1.06934°).

n = 204: a regularization of T-257’s packing by Levy (regularize_grid_slack/2: Daniel's coincidence placement (evand/square-packing search/regularize reg.coincidences and reg.place, MIT), ported to start from the witness's exact frame, over its number field where it has one, and to keep its contacts, then the same-side exact re-solve, through devtools.regularize_grid_slack; the project’s modifications: port of his coincidence placement (regularize_grid_slack/2) rather than his code, same-side exact re-solve (devtools.regularize_same_side); at its own side; identity: measured here; nearest centres one to one; 36 squares moved, at most 0.329819; turns up to 1.70663°).

n = 260: a regularization of T-257’s packing by Levy (regularize_grid_slack/2: Daniel's coincidence placement (evand/square-packing search/regularize reg.coincidences and reg.place, MIT), ported to start from the witness's exact frame, over its number field where it has one, and to keep its contacts, then the same-side exact re-solve, through devtools.regularize_grid_slack; the project’s modifications: port of his coincidence placement (regularize_grid_slack/2) rather than his code, same-side exact re-solve (devtools.regularize_same_side); at its own side; identity: measured here; nearest centres one to one; 49 squares moved, at most 0.439552; turns up to 0.104723°).

n = 293: a regularization of T-257’s packing by Levy (regularize_grid_slack/2: Daniel's coincidence placement (evand/square-packing search/regularize reg.coincidences and reg.place, MIT), ported to start from the witness's exact frame, over its number field where it has one, and to keep its contacts, then the same-side exact re-solve, through devtools.regularize_grid_slack; the project’s modifications: port of his coincidence placement (regularize_grid_slack/2) rather than his code, same-side exact re-solve (devtools.regularize_same_side); at its own side; identity: measured here; nearest centres one to one; 52 squares moved, at most 0.439552; turns up to 0.104723°).
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, in its source’s pose, Daniel 2026 [Daniel issue 845 2026-10-29]

13.1113.618

131415

13.11514.115

\sqrt{n}

\sqrt{n} + 1

Proven

$13.11942$ $\leq s (172) \leq$ $13.617936$

exact

Citation record n-172

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

upperDaniel 2026, GitHub (confirmed T-257)

Open

optimality

The case record

LowerUpper

Verified13.11942946…13.61793569…

Reported

\frac{6 \sqrt{26}}{85} + 2 \sqrt{2} + \frac{851}{85}

13.61793569…

Gap0.49850623…

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…

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]

16.1216.634

161718

16.12517.125

\sqrt{n}

\sqrt{n} + 1

Proven

$16.12849$ $\leq s (260) \leq$ $16.633983$

exact

Citation record n-260

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

upperDaniel 2026, GitHub (confirmed T-257)

Open

optimality

The case record

LowerUpper

Verified16.12849960…16.63398250…

Reported

\frac{574 \sqrt{2}}{257} + \frac{3343}{257}

16.63398250…

Gap0.50548289…

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]

17.1217.634

171819

17.11718.117

\sqrt{n}

\sqrt{n} + 1

Proven

$17.12077$ $\leq s (293) \leq$ $17.633983$

exact

Citation record n-293

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

upperDaniel 2026, GitHub (confirmed T-257)

Open

optimality

The case record

LowerUpper

Verified17.12077013…17.63398250…

Reported

\frac{8 \sqrt{34}}{145} + 2 \sqrt{2} + \frac{2031}{145}

17.63398250…

Gap0.51321236…

Results on these cases

24 results in the register on $n = 172, 204, 260, 293$ , oldest first, each with what it established and how it stands now.

2005 published T-007 · $n = 172, 204, 260, 293$
$s (n) \geq min (⌈ \sqrt{n} ⌉, \sqrt{n - 2 ⌊ \sqrt{n} ⌋ + 1} + 1)$ 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 · $n = 172, 204, 260, 293$
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 · $n = 172$
Smaller packings for 49 counts from $n = 68$ to $307$ , each certified two independent ways
V3 C3 Upper bound confirmed superseded by T-257
Couzo · franciscouzo square-packing 2026-09-27 · packet · register
2026-09-29 published T-083 · $n = 172, 204, 260, 293$
$s (n) \geq 1 / 2 + \sqrt{n - ⌊ \sqrt{n} ⌋ + 1 / 4}$ for every nonsquare $8 \leq n \leq 324$
V3 C3 Lower bound confirmed
Karakuş · Karakuş 2026 · source · register
2026-10-05 published T-098 · $n = 172$
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-257
Daniel after Couzo, de Winter, Ellsworth, Levy · evand exact optima 2026-10-05 · packet · register
2026-10-05 published T-101 · $n = 204, 260, 293$
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-257
Daniel after Couzo, de Winter, Ellsworth, Levy · evand exact optima 2026-10-05 · packet · register
2026-10-12 established T-155 · $n = 204, 260, 293$
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-257
Levy after Daniel · register
2026-10-12 published T-162 · $n = 204$
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
2026-10-12 published T-261 · $n = 260, 293$
Reported grade-C local minima for 233 source configurations of field degree at most 4
V3 C1 Restricted optimality incomplete
Daniel after Schadt, Ellsworth, Levy · Daniel exact and local reports 2026-10-12 · packet · packet · register
2026-10-13 published T-174 · $n = 172, 204$
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-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 · $n = 172, 204, 260, 293$
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 · $n = 204$
Exact certificates at twelve counts from Polkovskyi, tightening the smallest reported there
V3 C3 Upper bound confirmed on these cases, 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 · $n = 204$
Exact algebraic sides at sixteen counts from Daniel, of packings the record credits
V3 C3 Upper bound confirmed on these cases, superseded by T-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-196 · $n = 260$
de Winter reports three smaller rational upper bounds at 227, 229 and 260
V3 C3 Upper bound confirmed superseded by T-257
de Winter after Levy · de Winter upper bounds 2026-10-17 · 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
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 · $n = 204, 260$
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 · $n = 204$
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 · $n = 204, 260$
Daniel's solved, regularized views at 204, 227, 229 and 260, each the smallest side reported there
V3 C3 Upper bound confirmed on these cases, 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 · $n = 204$
Polkovskyi's re-solves of the house packings at 105 and 204, below T-174 and T-225
V3 C3 Upper bound confirmed on these cases, 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 · $n = 204$
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
2026-10-25 published T-244 · $n = 172$
Kim's exact algebraic sides at 172 and 263, as reported, of packings the record credits
V3 C3 Upper bound confirmed on these cases, superseded by T-257
Kim after Couzo, Chaoweeraprasit, Daniel, PingYou Ltd, Levy · Kim exact forms 2026-10-25 · packet · packet · register
2026-10-28 published T-255 · $n = 260$
Couzo's certificates of ntumlnoob's Kaggle packings at 260 and 307, below T-225 and T-174
V3 C3 Upper bound confirmed superseded by T-257
Couzo after ntumlnoob, Daniel, Levy · Couzo a5a5d76 2026-10-28 · packet · packet · register
2026-10-28 published T-256 · $n = 260$
Couzo's packings from Kaggle notebook packings at 232, 260 and 307, below T-186, T-225 and T-174
V3 C3 Upper bound confirmed on these cases, superseded by T-257
Couzo after Chilton, ntumlnoob, Lavrentev, Ellsworth, Daniel, Levy · Couzo issue 844 2026-10-28 · packet · register
2026-10-29 published T-257 this result · $n = 172, 204, 260, 293$
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, $n = 172$ · Frontier row, $n = 172$ · Case record, $n = 204$ · Frontier row, $n = 204$ · Case record, $n = 260$ · Frontier row, $n = 260$ · Case record, $n = 293$ · Frontier row, $n = 293$ · T-257 in the results table
Issues: Issue #845

On GitHub, at main

Register: T-257 in results.yaml, line 37064
Evidence: E-evand-845-certificate-report · E-evand-845-exact-feasibility · E-evand-845-review-exact-feasibility
Proofs and certificates: certificate candidates
Sources: Daniel issue 845 2026-10-29 (its own site, retained copy)
Source packet: resources/web/evand-issue845-2026-10-29/README.md
Artifacts: 30 artifacts and controls in 11 directories. Complete artifact list in the result entry. docs/project/reviews packing/campaign/issue-intake/upper-bounds-843-846-20261031 packing/devtools packing/resources/web/evand-issue845-2026-10-29 packing/resources/web/evand-issue845-2026-10-29/acquisition packing/resources/web/evand-issue845-2026-10-29/measure packing/resources/web/evand-issue845-2026-10-29/receipts packing/resources/web/evand-issue845-2026-10-29/review/review-2026-10-31 packing/resources/web/evand-issue845-2026-10-29/review/review-2026-10-31/out packing/resources/web/evand-issue845-2026-10-29/source/search/packer/candidates packing/tests
Case file: frontier/n-172.md (verified lower, verified upper, reported lower, reported upper) · frontier/n-204.md (verified lower, verified upper, reported lower, reported upper) · frontier/n-260.md (verified lower, verified upper, reported lower, reported upper) · frontier/n-293.md (verified lower, verified upper, reported lower, reported upper)