T-128: Eight complete rational refinements from Francisco Couzo

V3 C3 upper bound confirmed

2026-10-08 published · Couzo after Xu, Chaoweeraprasit, Gupta, Ellsworth, Daniel, Levy · 8 cases, $n = 105$ to 306

Eight complete rational source certificates by Francisco Couzo, reported on jlevy/squares#451, prove finite upper bounds at the exact sides they state: $s (105) \leq 10.790618268107144505815379335866$ , $s (108) \leq 10.904821012320356429055628704222$ , $s (127) \leq 11.810878787589179839642367108002$ , $s (131) \leq 11.951105389414677460694240669403$ , $s (155) \leq 12.95249894401400738196585056694$ , $s (180) \leq 13.916993522477832483558124720393$ , $s (228) \leq 15.604601475729674284720634102188$ and $s (306) \leq 17.963433717491425739593840075522$ .

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 a copy of this repository's sqpack verifier in exact rational arithmetic.

All 24 retained jobs, the eight 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 eight 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 105, 108, 127 and 131, Nate Chaoweeraprasit's SQUISH packings at 155 and 180 (180 through Siddharth Gupta) and his own at 228 and 306, with David Ellsworth's refine_packing, his basin hopping and Evan Daniel's fq; Daniel's exact contact solver wrote every certificate.

Significance, composition and next rung

Significance: Eight smaller finite construction sides at $n = 105$ , 108, 127, 131, 155, 180, 228 and 306, 9.79e-07 to 6.60e-04 below the case ceilings, each an earlier packing refined to a new optimum by basin hopping and refine_packing; 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, adopt T-128 as the house at 108, 127, 155, 180, 228 and 306 with each earlier house kept as history.
Novelty: previously-published Present in an identified source

The cases

This result concerns 8 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
105	10.280000	10.790677	0.51067657…	open=	frontier n-105.md
108	10.412113	10.904825	0.49271098…		frontier n-108.md
127	11.281929	11.810937	0.52900732…		frontier n-127.md
131	11.465856	11.951151	0.48529394…		frontier n-131.md
155	12.468709	12.952504	0.48379399…		frontier n-155.md
180	13.432517	13.917654	0.48513626…		frontier n-180.md
228	15.103081	15.604603	0.50152058…	open≈	frontier n-228.md
306	17.507351	17.963439	0.45608678…	open≈	frontier n-306.md

Results on these cases

20 results in the register on these cases, oldest first

2005 published T-007 · 8 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 · 8 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 · 6 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-115, T-125 and T-127
Couzo · franciscouzo square-packing 2026-09-27 · packet · register
2026-09-29 published T-083 · 8 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
2026-10-02 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-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-092 · cases 228, 306
Smaller packings again at seven counts from $n = 208$ to $306$ , each certified two independent ways
upper bound confirmed superseded by T-098
Couzo · franciscouzo square-packing 2026-10-03 · packet · register
2026-10-05 published T-098 · 5 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 · cases 108, 127
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 · cases 108, 155, 180
$s (n) \leq S_{n}$ at ten counts, $n = 108$ to 303, from new SQUISH rational packings
upper bound confirmed on these cases, superseded by T-115, T-125 and T-127
Chaoweeraprasit after Ellsworth · SQUISH ten packings 2026-10-07 · packet · register
2026-10-07 published T-115 · case 155
$s (n) \leq S_{n}$ 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-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-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 105, 108, 127, 131
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 108, 180
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 this result · 8 of these cases
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 · case 105
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-136 · case 131
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 · 6 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

Links

On this site: The frontier survey · T-128 in the results table

On GitHub, at main

Register: T-128 in results.yaml, line 13062
Evidence: E-couzo-451-rational-report · E-couzo-451-exact-feasibility · E-evand-465-record-hunt-report · E-evand-465-record-hunt-exact-feasibility
Proofs and certificates: certificate complete-certificates-and-decimal-poses.json.xz · certificate candidates
Sources: Couzo exact refinements 2026-10-08 · Daniel record hunt 2026-10-09
Source packet: resources/web/couzo-exact-refinements-2026-10-08/README.md
Artifacts: 14 artifacts and controls
resources/web/couzo-exact-refinements-2026-10-08/README.md resources/web/couzo-exact-refinements-2026-10-08/acquisition/case-inputs.json resources/web/couzo-exact-refinements-2026-10-08/facts/complete-certificates-and-decimal-poses.json.xz resources/web/couzo-exact-refinements-2026-10-08/receipts/exact-certification.json.xz resources/web/couzo-exact-refinements-2026-10-08/receipts/exact-replay-incomplete-62268899-v1.json.xz resources/web/couzo-exact-refinements-2026-10-08/receipts/execution-attempts.json resources/web/couzo-exact-refinements-2026-10-08/receipts/private-custody-668cc1a-v1.json devtools/couzo_refinement_reports.py devtools/check_half_angle_area.py docs/project/reviews/review-2026-10-10-couzo-daniel-refinement-replays.md tests/test_couzo_refinement_reports.py tests/test_couzo_custody.py tests/test_negative_controls.py tests/test_check_half_angle_area.py
Case file: each case’s file is linked from its row above