T-146: Exact rational certificates at fifteen counts from Evan Daniel's regularized record lists

V3 C3 upper bound confirmed

2026-10-10 published · Daniel after Xu, Chaoweeraprasit, Mishapolk, Levy · 15 cases, $n = 70$ to 303

Fifteen complete rational certificates from Evan Daniel's display-regularized record lists, which no issue reports, prove finite upper bounds at their exact sides: $s (70) \leq 8.880960371555579594785501178639$ , $s (102) \leq 10.605828696431578360490734944053$ , $s (103) \leq 10.679232047361306442680936279997$ , $s (123) \leq 11.591378145497698159550863183468$ , $s (129) \leq 11.872029849081179973752429831968$ , $s (146) \leq 12.583782277415076881948309755914$ , $s (153) \leq 12.872029849081179973762429831968$ , $s (236) \leq 15.863955747159267422617702706295$ , $s (258) \leq 16.563448002133789379765816191273$ , $s (263) \leq 16.733166007899378970679613144039$ , $s (269) \leq 16.901513582189132198548012552701$ , $s (292) \leq 17.591378145497698159610863183468$ , $s (295) \leq 17.704232790736030358056092972112$ , $s (302) \leq 17.872029849081179973812429831968$ and $s (303) \leq 17.913065462738532305839753791206$ .

Each certificate gives a rational side and, for each unit square, a rational centre and the tangent of its half-angle, in Evan Daniel's format, retained as the original gzip the source publishes beside a JSON that prints the same side as S_cert. Each is below every value the record held or had pending at its count when read, by 2.21e-17 at 302 to 7.82e-10 at 295.

All 45 retained jobs, the fifteen positives with their duplicate-square and outside-container controls, were decided again here on 11 October 2026 by both maintained exact routes and equal their retained rows, and a third exact route, half-extent containment and exact intersection area, decided the fifteen with six controls each, decompressing each retained gzip itself, and held each retained file to its upstream bytes at the pin. No route here is the source's code, so the replay is independently re-implemented. The 11 October review found no blocking defect.

Each side is below the ceiling its case holds; the case records are unchanged until a house is adopted, so each is pending adoption.

The source presents each certificate as a display view and not a new bound. Its method log says every refreshed side is the exact optimum of the store's best packing at its count, below the posted side only by rounding, and classes each of the fifteen views as the same packing in a different arrangement, traversable without expansion, by a path test that samples angle paths in 0.01 degree steps and that it says is not a continuous proof; it reports all 324 certificates of the lists VALID under its two exact checkers and verified at 80 digits. Those statements are the author's: no source program ran here, and the solved point each view was made from is not published.

Measured pose by pose, every view differs from the certificate the record credits; at 123, 129 and 292 nearest centres match under no symmetry and only an optimal assignment pairs the squares, slid by up to 0.59, 0.83 and 0.59.

Credit the packings to their finders, as the source's provenance names them: Ryan Xu's (ry-xu, https://github.com/ry-xu/square_packing, issue 432) at 70, 102, 103, 123, 129, 146 and 295, at 103 as Mishapolk refined it (issue 470, T-138); Nate Chaoweeraprasit's SQUISH packing of issue 401 at 258, as Mishapolk refined it; and his SQUISH packings of issue 481 (T-136) at 153, 236, 263, 269, 292, 302 and 303, composed from pieces of Francisco Couzo's, Ryan Xu's and earlier packings. At 70 Eric Deleeuw's refinement of Ryan Xu's packing in place (issue 483, T-137) was the smallest side the record held.

Evan Daniel (https://github.com/evand/square-packing) refreshed each packing from his store, solved it to the side he reports as its exact optimum, regularized it for display at that side, at 70, 129, 153, 236, 263, 269 and 292 also merging determined rotation groups by up to 2.96 degrees, and wrote the certificate.

His lists name the finders of the arrangements the packings descend from: Joe DeVincentis at 70, Francisco Couzo at 102, 103, 123, 236, 263, 269, 292, 302 and 303, David Ellsworth at 129 and 153 and David W. Cantrell at 146, and no one at 258 and 295. The source claims no record, and its credits say the repository's work was produced by Claude (Anthropic) under human direction.

Significance, composition and next rung

Significance: Smaller finite construction sides at fifteen counts from 70 to 303, 2.21e-17 to 7.82e-10 below the least side the record held or had pending at each and 5.36e-12 to 9.28e-03 below the case ceilings, where the larger gaps are improvements SQUISH's issue 481 (T-136) already made: display-regularized arrangements at the sides the source reports as the exact optima of packings the record already credits, a citable detail of each case that moves no theorem, S2 as T-098 is; no lower bound, optimum or priority.
Next rung: V4 and C4 need a retained human oversight record: two accepted adversarial reviews by distinct AI reviewers are retained, the 11 October review and the 11 October final review (docs/project/reviews/review-2026-10-11-final-upper-bounds.md). Separately, once jlevy/squares#403 lands and private-worker custody is on file (RG-7), adopt this entry as the house at all fifteen counts, each earlier house kept as history and each packing credited as the review decided (RG-2).
Regularizations: $n = 70$ : a regularization of T-125’s packing by Daniel (fewest; solved to its side; rotation groups merged by up to 0.93°, 3 free angles moved; identity: the source’s claim; nearest centres one to one; 10 squares moved, at most 0.222; turns up to 22.5°).

$n = 102$ : a regularization of T-125’s packing by Daniel (conservative; solved to its side; no angle changed; identity: the source’s claim; nearest centres one to one; 18 squares moved, at most 0.169; turns up to 0.433°).

$n = 103$ : a regularization of T-138’s packing by Daniel (conservative; solved to its side; no angle changed; identity: the source’s claim; nearest centres one to one; 15 squares moved, at most 0.074; turns up to 0.411°).

$n = 123$ : a regularization of T-125’s packing by Daniel (conservative; solved to its side; no angle changed; identity: the source’s claim; nearest centres not one to one; 45 squares moved, at most 0.591; no square turned).

$n = 129$ : a regularization of T-125’s packing by Daniel (fewest; solved to its side; rotation groups merged by up to 2.96°, 16 free squares straightened, 5 free angles moved; identity: the source’s claim; nearest centres not one to one; 43 squares moved, at most 0.834; turns up to 39.4°).

$n = 146$ : a regularization of T-125’s packing by Daniel (conservative; solved to its side; no angle changed; identity: the source’s claim; nearest centres one to one; 25 squares moved, at most 0.124; no square turned).

$n = 153$ : a regularization of T-136’s packing by Daniel (fewest; solved to its side; rotation groups merged by up to 1.81°, 8 free angles moved; identity: the source’s claim; nearest centres one to one; 80 squares moved, at most 0.065; turns up to 2.43°).

$n = 236$ : a regularization of T-136’s packing by Daniel (fewest; solved to its side; rotation groups merged by up to 0.67°, 5 free angles moved; identity: the source’s claim; nearest centres one to one; 53 squares moved, at most 0.068; turns up to 0.705°).

$n = 258$ : a regularization of T-138’s packing by Daniel (conservative; solved to its side; no angle changed; identity: the source’s claim; nearest centres one to one; 48 squares moved, at most 0.496; turns up to 1.13°).

$n = 263$ : a regularization of T-136’s packing by Daniel (fewest; solved to its side; rotation groups merged by up to 1.06°, 14 free squares straightened, 18 free angles moved; identity: the source’s claim; nearest centres one to one; 132 squares moved, at most 0.049; turns up to 1.22°).

$n = 269$ : a regularization of T-136’s packing by Daniel (fewest; solved to its side; rotation groups merged by up to 0.12°, 4 free squares straightened, 11 free angles moved; identity: the source’s claim; nearest centres one to one; 54 squares moved, at most 0.088; turns up to 1.16°).

$n = 292$ : a regularization of T-136’s packing by Daniel (fewest; solved to its side; rotation groups merged by up to 0.03°, 9 free angles moved; identity: the source’s claim; nearest centres not one to one; 239 squares moved, at most 0.589; turns up to 4.6°).

$n = 295$ : a regularization of T-125’s packing by Daniel (conservative; solved to its side; no angle changed; identity: the source’s claim; nearest centres one to one; 122 squares moved, at most 0.138; no square turned).

$n = 302$ : a regularization of T-136’s packing by Daniel (conservative; solved to its side; 56 free squares straightened, 26 free angles moved; identity: the source’s claim; nearest centres one to one; 198 squares moved, at most 0.102; turns up to 2.62°).

$n = 303$ : a regularization of T-136’s packing by Daniel (conservative; solved to its side; 1 free square straightened; identity: the source’s claim; nearest centres one to one; 19 squares moved, at most 0.135; turns up to 1.21°).
Novelty: previously-published Present in an identified source

The cases

This result concerns 15 cases, too many to draw one by one. Each is listed with the film’s bounds, the proved lower bound and the best known side, and links to its case record, where its packing and number line are drawn.

n	Proved lower	Best known	Gap	Status	Records
70	8.657500	8.880961	0.22346037…	open=	frontier n-070.md
102	10.280000	10.605829	0.32582869…		frontier n-102.md
103	10.280000	10.679233	0.39923204…		frontier n-103.md
123	11.094810	11.591379	0.49656809…		frontier n-123.md
129	11.374281	11.872030	0.49774826…		frontier n-129.md
146	12.086630	12.583783	0.49715205…		frontier n-146.md
153	12.384864	12.879680	0.49481504…		frontier n-153.md
236	15.374474	15.867801	0.49332605…		frontier n-236.md
258	16.064382	16.563449	0.49906558…		frontier n-258.md
263	16.224185	16.740420	0.51623448…		frontier n-263.md
269	16.413830	16.905968	0.49213659…	open≈	frontier n-269.md
292	17.090660	17.597250	0.50658936…	open=	frontier n-292.md
295	17.180827	17.704233	0.52340547…		frontier n-295.md
302	17.389345	17.881307	0.49196046…		frontier n-302.md
303	17.418924	17.920313	0.50138805…		frontier n-303.md

Results on these cases

33 results in the register on these cases, oldest first

2005 published T-007 · 15 of these cases
$s (n) \geq min (⌈ \sqrt{n} ⌉, \sqrt{n - 2 ⌊ \sqrt{n} ⌋ + 1} + 1)$ for $4 \leq n \leq 324$
lower bound incomplete
Nagamochi · Nagamochi 2005 · source · register
2026-09-04 published T-085 · 15 of these cases
Nagamochi 2005, Lemma 1 is false for every container with $a > 3$ and $b > 2$
correction confirmed
Karakuş; chelokot · Karakuş 2026 · chelokot Nagamochi counterexample 2026 · packet · register
2026-09-22 published T-044 · case 70
Weighted point lower bounds for ten counts in $n = 26 \dots 72$ , plus seven from the same files
lower bound confirmed superseded by T-091
wand125 after Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 point bounds 2026 · packet · source · review · register
2026-09-27 published T-046 · case 70
Rectangle-density lower bounds reported for 48 counts in $n = 18 \dots 95$
lower bound recorded superseded by T-091
wand125 after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 rectangle bounds 2026 · wand125 rectangle bounds 2026-09-28 · packet · packet · register
2026-09-27 published T-056 · 9 of these cases
Smaller packings for 49 counts from $n = 68$ to $307$ , each certified two independent ways
upper bound confirmed superseded by T-098, T-113, T-116, T-117, T-125 and T-127
Couzo · franciscouzo square-packing 2026-09-27 · packet · register
2026-09-28 published T-070 · case 70
Rectangle-density lower bounds replayed at 25 counts in $n = 29 \dots 95$
lower bound confirmed superseded by T-091
wand125 after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 rectangle bounds 2026-09-28 · packet · packet · source · review · register
2026-09-29 published T-058 · case 70
Rectangle-certificate ceiling α·UB(n) proved for $n = 1$ ..100; B·UB(n) on 64 grid rows
method limit confirmed
wand125 after Tokoharu, Daniel · wand125 tools 2026 · packet · register
2026-09-29 published T-083 · 15 of these cases
$s (n) \geq 1 / 2 + \sqrt{n - ⌊ \sqrt{n} ⌋ + 1 / 4}$ for every nonsquare $8 \leq n \leq 324$
lower bound confirmed
Karakuş · Karakuş 2026 · source · register
2026-10-01 published T-068 · case 70
Rectangle-density lower bounds verified at 34 counts in $n = 19 \dots 95$
lower bound confirmed on these cases, superseded by T-091
wand125 after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 rectangle bounds 2026-10-01 · packet · source · review · register
2026-10-01 published T-074 · case 70
Rectangle-density lower bounds replayed at 31 counts in $n = 19 \dots 95$
lower bound confirmed on these cases, superseded by T-091
wand125 after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 rectangle bounds 2026-10-01 · packet · packet · source · review · register
2026-10-02 published T-073 · cases 102, 103
Linear-measure lower bounds replayed at $n = 83$ and $n = 101 \dots 105$
lower bound confirmed
wand125 after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 linear certificates 2026-10-02 · packet · source · review · register
2026-10-02 published T-077 · case 70
Rectangle-density lower bounds replayed at $n = 20$ , 42 and 70
lower bound confirmed superseded by T-091
wand125 after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 rectangle bounds 2026-10-02 · packet · packet · source 1 · source 2 · review 1 · review 2 · register
2026-10-02 published T-080 · cases 102, 103
Linear-measure lower bound replayed at $n = 101 \dots 105$
lower bound confirmed
wand125 after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 linear certificates 2026-10-02 · packet · source · review · register
2026-10-03 published T-082 · case 70
Mixed rectangle-measure lower bounds verified at 22 counts in $n = 51 \dots 96$
lower bound confirmed on these cases, superseded by T-091
wand125 after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 mixed bounds 2026-10-03 · packet · packet · source · review · register
2026-10-03 published T-092 · cases 263, 303
Smaller packings again at seven counts from $n = 208$ to $306$ , each certified two independent ways
upper bound confirmed superseded by T-113 and T-116
Couzo · franciscouzo square-packing 2026-10-03 · packet · register
2026-10-04 published T-091 · case 70
Mixed rectangle-measure lower bounds verified at 17 counts in $n = 53 \dots 95$
lower bound confirmed
wand125 after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 mixed bounds evening 2026-10-04 · packet · source · review · register
2026-10-05 published T-098 · 8 of these cases
Exact optima of 48 known-best packings: s(n) ≤ S', 3.5e-13 to 5.0e-11 below each printed side
upper bound confirmed
Daniel after Couzo, de Winter, Ellsworth, Levy · evand exact optima 2026-10-05 · packet · register
2026-10-05 published T-101 · 6 of these cases
Exact certificates of 77 catalogue packings: s(n) ≤ S', 1.2e-16 to 9.8e-15 above each side
upper bound confirmed on these cases, superseded by T-115, T-125 and T-127
Daniel after Couzo, de Winter, Ellsworth, Levy · evand exact optima 2026-10-05 · packet · register
2026-10-07 published T-113 · cases 129, 303
$s (n) \leq S_{n}$ at ten counts, $n = 108$ to 303, from new SQUISH rational packings
upper bound confirmed
Chaoweeraprasit after Ellsworth · SQUISH ten packings 2026-10-07 · packet · register
2026-10-07 published T-114 · case 153
$s (153) \leq 7250614903299225 / 562949953421312$ , from a SQUISH rational packing
upper bound confirmed superseded by T-127
Chaoweeraprasit after Ellsworth · SQUISH n153 2026-10-07 · packet · packet · register
2026-10-07 published T-115 · cases 123, 129, 258, 263
$s (n) \leq S_{n}$ at twelve counts, from the SQUISH update’s new and smaller rational packings
upper bound confirmed
Chaoweeraprasit after Couzo and Ellsworth · SQUISH update 2026-10-07 · packet · register
2026-10-07 published T-116 · cases 236, 263, 302
$s (n) \leq S_{n}$ at nine counts, from the second SQUISH update’s new rational packings
upper bound confirmed
Chaoweeraprasit after Ellsworth, Couzo and SQUISH · SQUISH second update 2026-10-07 · packet · register
2026-10-07 published T-117 · case 292
Exact rational ceiling refinements at $n = 105$ , 292
upper bound confirmed
Rehwaldt after Couzo and earlier contributors · Rehwaldt Couzo refinements 2026-10-07 · packet · register
2026-10-07 published T-120 · case 102
Reported s(102) ≤ alpha_102 with a degree-8 exact side form
upper bound recorded superseded by T-125
Daniel after Couzo · Daniel exact and local reports 2026 · packet · register
2026-10-07 published T-129 · case 292
Three dated Daniel certificate reports with historical source custody
upper bound recorded superseded by T-117
Daniel after Couzo, Levy · Daniel dated certificates 105 and 130 2026-10-07 · Daniel dated certificate 292 2026-10-07 · packet · packet · packet · register
2026-10-08 published T-125 · 11 of these cases
Complete rational construction reports at 25 counts
upper bound confirmed
ry-xu · ry-xu square packing 2026 · packet · register
2026-10-08 published T-127 · cases 123, 129, 153, 236
Fourteen rational refinements, seventeen complete source cases
upper bound confirmed
Gupta after Chaoweeraprasit, Daniel · Gupta rational refinements 2026-10-08 · packet · register
2026-10-09 published T-134 · cases 263, 303
Six exact rational certificates from Francisco Couzo
upper bound confirmed
Couzo after Mishapolk, Chaoweeraprasit, Gupta, Xu, Daniel, Ellsworth, Levy · Couzo exact certificates 2026-10-09 · packet · register
2026-10-09 published T-136 · 7 of these cases
Seventeen exact rational packings from SQUISH's third request
upper bound confirmed
Chaoweeraprasit after Couzo, Xu, Goebel, Schadt, Ellsworth, Hajba, Levy · SQUISH third request 2026-10-09 · packet · register
2026-10-09 published T-138 · cases 103, 258, 302, 303
Exact witnesses at Mishapolk's printed ceilings, the smallest known at 103 and 258 when registered
upper bound confirmed
Mishapolk after Stenlund, Friedman, Ellsworth, Chaoweeraprasit, Couzo, Xu, Levy · Mishapolk decimal poses 2026-10-09 · packet · register
2026-10-10 published T-137 · case 70
An exact rational refinement of Ryan Xu's packing of 70 squares
upper bound confirmed
Deleeuw after Xu, Levy · Deleeuw n70 refinement 2026-10-10 · packet · register
2026-10-10 published T-144 · case 123
Linear-measure lower bound $s (n) \geq 563 / 50 = 11.26$ for $n = 122 \dots 126$ , reported
lower bound recorded
wand125 after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 linear n122 2026-10-10 · packet · register
2026-10-10 published T-146 this result · 15 of these cases
Exact rational certificates at fifteen counts from Evan Daniel's regularized record lists
upper bound confirmed
Daniel after Xu, Chaoweeraprasit, Mishapolk, Levy · Daniel regularized lists 2026-10-10 · packet · register

Links

On this site: The frontier survey · T-146 in the results table

On GitHub, at main

Register: T-146 in results.yaml, line 15630
Evidence: E-evand-regularized-certificate-report · E-evand-regularized-exact-feasibility
Proofs and certificates: certificate certs
Sources: Daniel regularized lists 2026-10-10
Source packet: resources/web/evand-regularized-2026-10-10/README.md
Artifacts: 24 artifacts and controls in 7 directories. Complete artifact list in the result entry. docs/project/reviews packing/devtools packing/resources/web/evand-regularized-2026-10-10 packing/resources/web/evand-regularized-2026-10-10/acquisition packing/resources/web/evand-regularized-2026-10-10/receipts packing/resources/web/evand-regularized-2026-10-10/source/search/regularize/lists/certs packing/tests
Case file: each case’s file is linked from its row above