T-131: s(132)≤11.987099332245063… from Evan Daniel's record hunt

V3 C3 upper bound confirmed

2026-10-09 published · Daniel after Couzo, Chaoweeraprasit, Levy · n=132

One complete rational source certificate by Evan Daniel, reported on jlevy/squares#465, proves a finite upper bound at the exact side it states: s(132)≤11.987099332245063227179742877435.

The certificate gives a rational side and, for each unit square, a rational centre and the tangent of its half-angle. The source checks it with its own verify_cert.py and verify_cert2.py, and reports a different local minimum from the case's, reached from Francisco Couzo's packing by an uphill excursion.

All six retained jobs of the release, its n132 and n155 positives with their duplicate-square and outside-container controls, were decided again here on 10 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 both certificates with six controls each. No route here is the source's code, so the replay is independently re-implemented. The 10 October review found no blocking defect.

The side is below the case's ceiling, Evan Daniel's exact optimum of Francisco Couzo's packing (T-098); the case record is unchanged, so it is pending adoption.

Credit Evan Daniel (https://github.com/evand/square-packing), from Francisco Couzo's packing.

Significance, composition and next rung
Significance
A smaller finite construction side, 4.23e-3 below the case's ceiling; no lower bound or optimum.
Next rung
V4 and C4 need a retained human oversight record: two accepted adversarial reviews by distinct AI reviewers are retained, the 10 October review and the 11 October final review (docs/project/reviews/review-2026-10-11-final-upper-bounds.md). No house move is due: at 132 issue 489's smaller side (T-141) is the one the later review of issues 488 and 489 recommends adopting.
Novelty
previously-published Present in an identified source

The case

Case record

n=132

11.5111.991
111213
11.48912.489
nn+1

Proven

11.51135≤s(132)≤11.991328

  • numerical

Citation record n-132

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

upperCouzo & Daniel, GitHub (confirmed T-098)

Open

  • optimality
  • exact value

The case record

LowerUpper
Gap0.47997011…

Results on the case

10 results in the register on n=132, oldest first, each with what it established and how it stands now.

  1. 2005 published T-007

    s(n)≥min(⌈n⌉,n−2⌊n⌋+1+1) for 4≤n≤324

    V0 C1 lower bound incomplete

    Nagamochi · Nagamochi 2005 · source · register

  2. 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

  3. 2026-09-27 published T-056

    Smaller packings for 49 counts from n=68 to 307, each certified two independent ways

    V3 C3 upper bound confirmed superseded by T-098

    Couzo · franciscouzo square-packing 2026-09-27 · packet · register

  4. 2026-09-29 published T-083

    s(n)≥1/2+n−⌊n⌋+1/4 for every nonsquare 8≤n≤324

    V3 C3 lower bound confirmed

    Karakuş · Karakuş 2026 · source · register

  5. 2026-10-05 published T-098

    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

    Daniel after Couzo, de Winter, Ellsworth, Levy · evand exact optima 2026-10-05 · packet · register

  6. 2026-10-09 published T-131 this result

    s(132)≤11.987099332245063… from Evan Daniel's record hunt

    V3 C3 upper bound confirmed

    Daniel after Couzo, Chaoweeraprasit, Levy · Daniel record hunt 2026-10-09 · packet · register

  7. 2026-10-09 published T-134

    Six exact rational certificates from Francisco Couzo

    V3 C3 upper bound confirmed

    Couzo after Mishapolk, Chaoweeraprasit, Gupta, Xu, Daniel, Ellsworth, Levy · Couzo exact certificates 2026-10-09 · packet · register

  8. 2026-10-09 published T-138

    Exact witnesses at Mishapolk's printed ceilings, the smallest known at 103 and 258 when registered

    V3 C3 upper bound confirmed

    Mishapolk after Stenlund, Friedman, Ellsworth, Chaoweeraprasit, Couzo, Xu, Levy · Mishapolk decimal poses 2026-10-09 · packet · register

  9. 2026-10-10 published T-140

    Six more exact rational certificates from Francisco Couzo

    V3 C3 upper bound confirmed

    Couzo after Chaoweeraprasit, Mishapolk, Xu, Daniel, Ellsworth, Levy · Couzo certificates 2026-10-10 · packet · register

  10. 2026-10-10 published T-141

    Exact rational certificates at 132 and 308 from Evan Daniel's third record hunt

    V3 C3 upper bound confirmed

    Daniel after Fang, Arslanov, Mustafin, Shangitbayev, Couzo, Levy · Daniel record hunt 3 2026-10-10 · packet · register

Links

On this site
Case record, n=132 · Frontier row, n=132 · T-131 in the results table

On GitHub, at main

Register
T-131 in results.yaml, line 13406
Evidence
E-evand-465-record-hunt-report · E-evand-465-record-hunt-exact-feasibility
Proofs and certificates
certificate candidates
Sources
Daniel record hunt 2026-10-09
Source packet
resources/web/evand-record-hunt-2026-10-09/README.md
Artifacts
9 artifacts and controlsresources/web/evand-record-hunt-2026-10-09/README.md resources/web/evand-record-hunt-2026-10-09/acquisition/claims.json resources/web/evand-record-hunt-2026-10-09/source/search/packer/candidates/hunt1_n132.cert resources/web/evand-record-hunt-2026-10-09/receipts/exact-certification.json.xz devtools/evand_hunt_reports.py devtools/check_half_angle_area.py docs/project/reviews/review-2026-10-10-couzo-daniel-refinement-replays.md tests/test_evand_hunt_reports.py tests/test_check_half_angle_area.py
Case file
frontier/n-132.md (verified lower, verified upper, reported lower, reported upper)