T-130: Five follow-up rational refinements from Francisco Couzo

V3 C3 upper bound confirmed

2026-10-08 published · Couzo after Xu, Daniel, Ellsworth, Levy · 5 cases, $n = 84$ to 270

Five complete rational source certificates by Francisco Couzo, reported in a comment on pull request jlevy/squares#460, prove finite upper bounds at the exact sides they state: $s (84) \leq 9.697934799014921307163820128651$ , $s (86) \leq 9.820535407496742209971278280039$ , $s (105) \leq 10.789303783748158831034729697921$ , $s (175) \leq 13.767155163542549110085664417048$ and $s (270) \leq 16.9367230228761834072968722597$ .

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. The source reports passes from Evan Daniel's verify_cert.py and a copy of this repository's sqpack verifier.

All 15 retained jobs, the five 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 the five certificates with six controls each. One maintained route is the verifier the source ran; the independent rational corner checker and the third route are independently re-implemented. The 10 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.

Credit Francisco Couzo (https://github.com/franciscouzo/square-packing), refining Ryan Xu's packings at 84, 86, 105 and 175 and Evan Daniel's at 270 by his basin hopping and, at 175, David Ellsworth's refine_packing and Evan Daniel's fq; Daniel's exact contact solver wrote the certificates.

Significance, composition and next rung

Significance: Five smaller finite construction sides at $n = 84$ , 86, 105, 175 and 270, 3.03e-05 to 1.74e-03 below the case ceilings, each an earlier packing refined to a new optimum; 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). Separately, once jlevy/squares#403 lands and this packet's private-worker custody is on file (RD-5), adopt T-130 as the house at 84, 86 and 105 with each earlier house kept as history.
Novelty: previously-published Present in an identified source

The cases

This result concerns 5 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
84	9.411000	9.698053	0.28705206…	open=	frontier n-084.md
86	9.503000	9.820566	0.31756573…		frontier n-086.md
105	10.280000	10.790677	0.51067657…		frontier n-105.md
175	13.237739	13.768900	0.53116007…		frontier n-175.md
270	16.445218	16.937808	0.49258850…		frontier n-270.md

Results on these cases

27 results in the register on these cases, oldest first

2005 published T-007 · 5 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 · 5 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-046 · case 86
Rectangle-density lower bounds reported for 48 counts in $n = 18 \dots 95$
lower bound recorded superseded by T-090
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 · cases 105, 270
Smaller packings for 49 counts from $n = 68$ to $307$ , each certified two independent ways
upper bound confirmed superseded by T-119 and T-125
Couzo · franciscouzo square-packing 2026-09-27 · packet · register
2026-09-28 published T-070 · case 86
Rectangle-density lower bounds replayed at 25 counts in $n = 29 \dots 95$
lower bound confirmed superseded by T-090
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 · cases 84, 86
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 · 5 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 86
Rectangle-density lower bounds verified at 34 counts in $n = 19 \dots 95$
lower bound confirmed on these cases, superseded by T-090
wand125 after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 rectangle bounds 2026-10-01 · packet · source · review · register
2026-10-02 published T-071 · cases 84, 86
Mixed rectangle-measure lower bounds replayed at $n = 84 \dots 87$
lower bound confirmed superseded by T-090 and T-094
wand125 after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 mixed bounds 2026-10-02 · packet · source · review · register
2026-10-02 published T-073 · case 105
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-075 · case 86
Mixed rectangle-measure lower bounds replayed at nine counts in $n = 83 \dots 96$
lower bound confirmed on these cases, superseded by T-090
wand125 after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 mixed bounds afternoon 2026-10-02 · packet · source 1 · source 2 · review · register
2026-10-02 published T-080 · case 105
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 86
Mixed rectangle-measure lower bounds verified at 22 counts in $n = 51 \dots 96$
lower bound confirmed on these cases, superseded by T-090
wand125 after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 mixed bounds 2026-10-03 · packet · packet · source · review · register
2026-10-04 published T-090 · cases 84, 86
Mixed rectangle-measure lower bounds verified at 17 counts in $n = 42 \dots 96$
lower bound confirmed
wand125 after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 mixed bounds 2026-10-04 · packet · source · review · register
2026-10-05 published T-094 · case 84
Mixed rectangle-measure lower bounds verified at $n = 67$ and $84$
lower bound confirmed
wand125 after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 mixed bounds 2026-10-05 · packet · source · review · register
2026-10-05 published T-098 · case 270
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-119
Daniel after Couzo, de Winter, Ellsworth, Levy · evand exact optima 2026-10-05 · packet · register
2026-10-05 published T-101 · case 175
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-117 · case 105
Exact rational ceiling refinements at $n = 105$ , 292
upper bound confirmed on these cases, superseded by T-125
Rehwaldt after Couzo and earlier contributors · Rehwaldt Couzo refinements 2026-10-07 · packet · register
2026-10-07 published T-119 · case 270
Three exact feasible upper bounds at n=266,270,272 from new source arrangements
upper bound confirmed
Daniel after Levy, Ellsworth, Couzo, Stead · Daniel new arrangements 2026-10-07 · packet · register
2026-10-07 published T-129 · case 105
Three dated Daniel certificate reports with historical source custody
upper bound recorded superseded by T-125
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 84, 86, 105, 175
Complete rational construction reports at 25 counts
upper bound confirmed
ry-xu · ry-xu square packing 2026 · packet · register
2026-10-08 published T-128 · case 105
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
2026-10-08 published T-130 this result · 5 of these cases
Five follow-up rational refinements from Francisco Couzo
upper bound confirmed
Couzo after Xu, Daniel, Ellsworth, Levy · Couzo follow-up refinements 2026-10-08 · packet · register
2026-10-09 published T-134 · case 270
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 · case 270
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 · 5 of these cases
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-140 · cases 175, 270
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

Links

On this site: The frontier survey · T-130 in the results table

On GitHub, at main

Register: T-130 in results.yaml, line 13262
Evidence: E-couzo-460-followup-rational-report · E-couzo-460-followup-exact-feasibility
Proofs and certificates: certificate facts
Sources: Couzo follow-up refinements 2026-10-08
Source packet: resources/web/couzo-followup-refinements-2026-10-08/README.md
Artifacts: 13 artifacts and controls
resources/web/couzo-followup-refinements-2026-10-08/README.md resources/web/couzo-followup-refinements-2026-10-08/acquisition/sources.json resources/web/couzo-followup-refinements-2026-10-08/facts/n-084.yaml resources/web/couzo-followup-refinements-2026-10-08/facts/n-086.yaml resources/web/couzo-followup-refinements-2026-10-08/facts/n-105.yaml resources/web/couzo-followup-refinements-2026-10-08/facts/n-175.yaml resources/web/couzo-followup-refinements-2026-10-08/facts/n-270.yaml resources/web/couzo-followup-refinements-2026-10-08/receipts/exact-certification.json.xz devtools/couzo_followup_reports.py devtools/check_half_angle_area.py docs/project/reviews/review-2026-10-10-couzo-daniel-refinement-replays.md tests/test_couzo_followup_reports.py tests/test_check_half_angle_area.py
Case file: each case’s file is linked from its row above