T-114: $s (153) \leq 7250614903299225 / 562949953421312$ , from a SQUISH rational packing

V3 C3 upper bound confirmed superseded by T-127

2026-10-07 published · Chaoweeraprasit after Ellsworth · $n = 153$

Nate Chaoweeraprasit (itsnaka), using SQUISH, establishes $s (153)$ <=

7250614903299225/562949953421312, about 12.8796793733329640, in the supplement to

jlevy/squares#401 on 7 October 2026, about 0.0019874 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 $n = 153$ by about 0.00199, 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 case

Case record

12.3812.880

121314

12.36913.369

\sqrt{n}

\sqrt{n} + 1

Proven

$12.38486$ $\leq s (153) \leq$ $12.879680$

exact

Citation record n-153

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

upperChaoweeraprasit & Gupta, GitHub (confirmed T-127)

Open

optimality

The case record

LowerUpper

Verified12.38486432…12.87967937…

Reported12.40175425…12.87967937…

Gap0.49481504…

Results on the case

9 results in the register on $n = 153$ , 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
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; chelokot · 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-127
Daniel after Couzo, de Winter, Ellsworth, Levy · evand exact optima 2026-10-05 · packet · register
2026-10-07 published T-114 this result
$s (153) \leq 7250614903299225 / 562949953421312$ , from a SQUISH rational packing
V3 C3 upper bound confirmed superseded by T-127
Chaoweeraprasit after Ellsworth · SQUISH n153 2026-10-07 · packet · packet · register
2026-10-08 published T-125
Complete rational construction reports at 25 counts
V3 C3 upper bound confirmed on this case, superseded by T-127
ry-xu · ry-xu square packing 2026 · packet · register
2026-10-08 published T-127
Fourteen rational refinements, seventeen complete source cases
V3 C3 upper bound confirmed
Gupta after Chaoweeraprasit, Daniel · Gupta rational refinements 2026-10-08 · packet · register
2026-10-09 published T-136
Seventeen exact rational packings from SQUISH's third request
V3 C3 upper bound confirmed
Chaoweeraprasit after Couzo, Xu, Goebel, Schadt, Ellsworth, Hajba, Levy · SQUISH third request 2026-10-09 · packet · register
2026-10-10 published T-146
Exact rational certificates at fifteen counts from Evan Daniel's regularized record lists
V3 C3 upper bound confirmed
Daniel after Xu, Chaoweeraprasit, Mishapolk, Levy · Daniel regularized lists 2026-10-10 · packet · register

Links

On this site: Case record, $n = 153$ · Frontier row, $n = 153$ · T-114 in the results table

On GitHub, at main

Register: T-114 in results.yaml, line 11916
Evidence: E-squish-n153-2026-10-07-report · E-squish-n153-2026-10-07-exact-replay
Proofs and certificates: certificate facts · certificate squish-401-2026
Sources: SQUISH n153 2026-10-07 (its own site, retained copy)
Source packet: resources/web/squish-401-2026-10-07/README.md · resources/web/squish-401-update-2026-10-07/README.md
Artifacts: 16 artifacts and controls
resources/web/squish-401-2026-10-07/README.md resources/web/squish-401-2026-10-07/facts/n-153.json.gz resources/web/squish-401-2026-10-07/acquisition/sources.json resources/web/squish-401-update-2026-10-07/facts/n-153.json.gz resources/web/squish-401-update-2026-10-07/acquisition/n153-comparison.json resources/web/squish-401-2026-10-07/acquisition/frontier-comparison.json witnesses/squish-401-2026/n-153-rational.yaml.gz resources/web/squish-401-2026-10-07/receipts/certification.json.gz resources/web/squish-401-2026-10-07/receipts/negative-controls.json devtools/squish_upper_bound_packets.py docs/project/reviews/review-2026-10-06-squish-upper-bound-packings.md docs/project/reviews/review-2026-10-06-squish-import-correctness.md resources/web/squish-401-2026-10-07/receipts/negative-controls.json tests/test_squish_upper_bound_packets.py tests/test_squish_upper_bound_receipts.py tests/test_import_half_angle_witness.py
Case file: frontier/n-153.md (verified lower, verified upper, reported lower, reported upper)

T-114: s(153)≤7250614903299225/562949953421312s(153)≤7250614903299225/562949953421312, from a SQUISH rational packing

The case

Results on the case

Links

T-114: $s (153) \leq 7250614903299225 / 562949953421312$ , from a SQUISH rational packing