T-113: $s (n) \leq S_{n}$ at ten counts, $n = 108$ to 303, from new SQUISH rational packings

V3 C3 upper bound confirmed

2026-10-07 published · Chaoweeraprasit after Ellsworth · 10 cases, $n = 108$ to 303

Nate Chaoweeraprasit (itsnaka), using SQUISH, establishes $s (n)$ <= S_n for $n = 108$ , 126, 129,

130, 154, 155, 180, 209, 238 and 303 in the ten-packing release submitted on

jlevy/squares#401 on 7 October 2026. Each S_n is the exact rational side in the retained

packing facts, from about 0.0004126 to 0.0067047 below the earlier reported upper bound

here. Each certificate supplies rational centres and rational half-angle parameters for

closed unit squares in the rational container.

The claim establishes only upper bounds, with no optimality or rigidity conclusion. The

author credits David Ellsworth’s records and tools and states human-directed Claude Opus 5.5

assistance.

This repository confirmed the upper bounds by independently re-implemented exact-rational

checks: sqpack.witness.exact_verify and devtools.check_rational_witness_independent accept

every pair and container wall of each certificate, with both geometric controls rejected.

Significance, composition and next rung

Significance: New packings reduce the recorded upper bounds at ten counts by about 0.0004 to 0.0067, substantive case-bound improvements. They settle no exact optimum and introduce no proved general packing theorem.
Composition: Each source rational triple is converted without rounding to exact unit-square corners, at the source’s unchanged rational side. Both native exact_verify and the independently written rational-corner checker decide every pair and wall; the source-to-witness conversion and complete scope are accepted by the retained mathematical review. The wrapper binds the receipt to the decided witness, checks the full roster and upward decimal display, and replays both geometric controls. This is an independently re-implemented confirmation, sharing no verification code with the producer; the two first-party routes share the converted corners, YAML and Python rational arithmetic.
Next rung: V4 and C4 need a retained human oversight record for the claim, certificate, assumptions, two deciding implementations, controls and accepted adversarial AI reviews. The two separately prompted project AI reviews are retained; human oversight is not. Rung 5 needs a proof-assistant formalization with the open replay and two human expert reviews. No local or global optimality claim is made by these upper-bound certificates.
Novelty: previously-published Present in an identified source

The cases

This result concerns 10 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
108	10.412113	10.904825	0.49271098…	open=	frontier n-108.md
126	11.235455	11.742641	0.50718541…		frontier n-126.md
129	11.374281	11.872030	0.49774826…		frontier n-129.md
130	11.420164	11.904484	0.48431819…		frontier n-130.md
154	12.426860	12.926563	0.49970180…		frontier n-154.md
155	12.468709	12.952504	0.48379399…		frontier n-155.md
180	13.432517	13.917654	0.48513626…		frontier n-180.md
209	14.473188	14.949618	0.47642933…		frontier n-209.md
238	15.441552	15.926147	0.48459405…		frontier n-238.md
303	17.418924	17.920313	0.50138805…		frontier n-303.md

Results on these cases

21 results in the register on these cases, oldest first

2005 published T-007 · 10 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 · 10 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
2026-09-27 published T-056 · 7 of these cases
Smaller packings for 49 counts from $n = 68$ to $307$ , each certified two independent ways
upper bound confirmed superseded by T-113, T-115 and T-127
Couzo · franciscouzo square-packing 2026-09-27 · packet · register
2026-09-29 published T-083 · 10 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
2026-10-03 published T-092 · cases 209, 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-127
Couzo · franciscouzo square-packing 2026-10-03 · packet · register
2026-10-05 published T-098 · 7 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 on these cases, superseded by T-113, T-115, T-125 and T-127
Daniel after Couzo, de Winter, Ellsworth, Levy · evand exact optima 2026-10-05 · packet · register
2026-10-05 published T-101 · cases 108, 129
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-125
Daniel after Couzo, de Winter, Ellsworth, Levy · evand exact optima 2026-10-05 · packet · register
2026-10-07 published T-113 this result · 10 of these cases
$s (n) \leq S_{n}$ 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-115 · 5 of these cases
$s (n) \leq S_{n}$ 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 108, 180
$s (n) \leq S_{n}$ at nine counts, from the second SQUISH update’s new rational packings
upper bound confirmed on these cases, superseded by T-125 and T-127
Chaoweeraprasit after Ellsworth, Couzo and SQUISH · SQUISH second update 2026-10-07 · packet · register
2026-10-07 published T-129 · case 130
Three dated Daniel certificate reports with historical source custody
upper bound recorded superseded by T-127
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 · cases 108, 126, 129, 130
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 · 7 of these cases
Fourteen rational refinements, seventeen complete source cases
upper bound confirmed
Gupta after Chaoweeraprasit, Daniel · Gupta rational refinements 2026-10-08 · packet · register
2026-10-08 published T-128 · cases 108, 155, 180
Eight complete rational refinements from Francisco Couzo
upper bound confirmed
Couzo after Xu, Chaoweeraprasit, Gupta, Ellsworth, Daniel, Levy · Couzo exact refinements 2026-10-08 · packet · register
2026-10-09 published T-134 · case 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 · cases 154, 209, 303
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 108, 180, 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
2026-10-09 published T-139 · case 126
A more precise certificate of Ryan Xu's 126-square arrangement, from Evan Daniel
upper bound confirmed
Daniel after Xu, Chaoweeraprasit · Daniel trio126 2026-10-09 · packet · register
2026-10-10 published T-140 · case 209
Six more exact rational certificates from Francisco Couzo
upper bound confirmed
Couzo after Chaoweeraprasit, Mishapolk, Xu, Daniel, Ellsworth, Levy · Couzo certificates 2026-10-10 · packet · register
2026-10-10 published T-144 · case 126
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 · cases 129, 303
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-113 in the results table

On GitHub, at main

Register: T-113 in results.yaml, line 11784
Evidence: E-squish-ten-packings-2026-10-07-report · E-squish-ten-packings-2026-10-07-exact-replay
Proofs and certificates: certificate facts · certificate squish-401-2026
Sources: SQUISH ten packings 2026-10-07 (its own site, retained copy)
Source packet: resources/web/squish-401-2026-10-07/README.md
Artifacts: 32 artifacts and controls in 8 directories. Complete artifact list in the result entry. docs/project/reviews packing/devtools packing/resources/web/squish-401-2026-10-07 packing/resources/web/squish-401-2026-10-07/acquisition packing/resources/web/squish-401-2026-10-07/facts packing/resources/web/squish-401-2026-10-07/receipts packing/tests packing/witnesses/squish-401-2026
Case file: each case’s file is linked from its row above

T-113: s(n)≤Sn​s(n)≤Sn at ten counts, n=108n=108 to 303, from new SQUISH rational packings

The cases

Results on these cases

Links

T-113: $s (n) \leq S_{n}$ at ten counts, $n = 108$ to 303, from new SQUISH rational packings