T-134: Six exact rational certificates from Couzo

V3 C3 Upper bound confirmed

2026-10-09 published · Couzo after Polkovskyi, Chaoweeraprasit, Gupta, Xu, Daniel, Ellsworth, Levy · 6 cases, $n = 132$ to 303

Six complete rational source certificates by Couzo, reported on jlevy/squares#476, prove finite upper bounds at the exact sides they state: $s (132) \leq 11.986954193640392741450366579623$ , $s (237) \leq 15.903670905580623014470917594172$ , $s (263) \leq 7488939206954142477014939239 / 447356905819500000000000000$ , $s (267) \leq 16.838815269948262260434481342204$ , $s (270) \leq 16.936720031121015799532991452837$ and $s (303) \leq 17.917443925494235891916989336536$ .

Each certificate gives a rational side and, for each unit square, a rational centre and the tangent of its half-angle, in Daniel's format. The author reports that all six pass Daniel's verify_cert.py and a copy of this repository's sqpack verifier in exact rational arithmetic. Five are exact optima from Daniel's exact contact solver; the one at 263 is a rational witness, the decimal packing with its centres scaled apart by 1 + 1e-12.

All 51 retained jobs of the release's 17-count replay, the six positives with their duplicate-square and outside-container controls among them, 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 six 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.

On 11 October 2026 the case record at 267 adopted its certificate there as the house. At 132, 237, 263, 270 and 303 smaller certificates hold the cases (T-141, T-140, T-146 and, at 270, T-151), and these five stay verified and are not selected.

Credit Couzo (https://github.com/franciscouzo/square-packing), with the packings each started from, as the author asks: Mishapolk's at 132, Chaoweeraprasit's SQUISH packings at 237, 263 and 303 (237 through Gupta's refinement), Xu's at 267 and Daniel's at 270. The search used Ellsworth's refine_packing and Daniel's fq, and was written with Claude under the author's direction.

Significance, composition and next rung

Significance: Smaller finite construction sides at $n = 132$ , 237, 263, 267, 270 and 303, 5.33e-06 to 4.37e-03 below the case ceilings; 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).
Regularized drawings: n = 267: a regularization of T-134’s packing by Levy (regularize_grid_slack/1: Daniel's coincidence placement (evand/square-packing search/regularize reg.coincidences and reg.place, MIT), ported to start from the witness's exact frame and to keep its contacts, then the same-side exact re-solve, through devtools.regularize_grid_slack; the project’s modifications: port of his coincidence placement (regularize_grid_slack/1) rather than his code, same-side exact re-solve (devtools.regularize_same_side); at its own side; identity: measured here; nearest centres one to one; 208 squares moved, at most 0.130329; turns up to 0.584252°).
Novelty: previously-published Present in an identified source

The cases

This result concerns 6 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
132	11.511357	11.985681	0.47432242…	open=	frontier n-132.md
237	15.408051	15.902604	0.49455212…		frontier n-237.md
263	16.224185	16.733167	0.50898081…		frontier n-263.md
267	16.350867	16.838816	0.48794778…		frontier n-267.md
270	16.445218	16.928163	0.48294353…		frontier n-270.md
303	17.418924	17.909959	0.49103397…		frontier n-303.md

Results on these cases

32 results in the register on these cases, oldest first

2005 published T-007 · 6 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 on these cases, superseded by T-083 and T-180 (reported)
Nagamochi · Nagamochi 2005 · source · register
2026-09-04 published T-085 · 6 of these cases
Nagamochi 2005, Lemma 1 is false for every container with $a > 3$ and $b > 2$
Correction confirmed
Karakuş; Vlasenko · Karakuş 2026 · chelokot Nagamochi counterexample 2026 · packet · register
2026-09-27 published T-056 · 5 of these cases
Smaller packings for 49 counts from $n = 68$ to $307$ , each certified two independent ways
Upper bound confirmed superseded by T-174, T-186, T-190 and T-224
Couzo · franciscouzo square-packing 2026-09-27 · packet · register
2026-09-29 published T-083 · 6 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 263, 303
Smaller packings again at seven counts from $n = 208$ to $306$ , each certified two independent ways
Upper bound confirmed superseded by T-174 and T-186
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 on these cases, superseded by T-174, T-186, T-190 and T-224
Daniel after Couzo, de Winter, Ellsworth, Levy · evand exact optima 2026-10-05 · packet · register
2026-10-05 published T-101 · case 267
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-134
Daniel after Couzo, de Winter, Ellsworth, Levy · evand exact optima 2026-10-05 · packet · register
2026-10-07 published T-113 · case 303
$s (n) \leq S_{n}$ at ten counts, $n = 108$ to 303, from new SQUISH rational packings
Upper bound confirmed superseded by T-186
Chaoweeraprasit after Ellsworth · SQUISH ten packings 2026-10-07 · packet · register
2026-10-07 published T-115 · cases 237, 263
$s (n) \leq S_{n}$ at twelve counts, from the SQUISH update’s new and smaller rational packings
Upper bound confirmed superseded by T-174 and T-190
Chaoweeraprasit after Couzo and Ellsworth · SQUISH update 2026-10-07 · packet · register
2026-10-07 published T-116 · case 263
$s (n) \leq S_{n}$ at nine counts, from the second SQUISH update’s new rational packings
Upper bound confirmed superseded by T-174
Chaoweeraprasit after Ellsworth, Couzo and SQUISH · SQUISH second update 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 on these cases, superseded by T-224
Daniel after Levy, Ellsworth, Couzo, Stead · Daniel new arrangements 2026-10-07 · packet · register
2026-10-08 published T-125 · cases 263, 267
Complete rational construction reports at 25 counts
Upper bound confirmed superseded by T-134 and T-174
Xu · ry-xu square packing 2026 · packet · register
2026-10-08 published T-127 · case 237
Fourteen rational refinements, seventeen complete source cases
Upper bound confirmed on these cases, superseded by T-190
Gupta after Chaoweeraprasit, Daniel · Gupta rational refinements 2026-10-08 · packet · register
2026-10-08 published T-130 · case 270
Five follow-up rational refinements from Couzo
Upper bound confirmed superseded by T-224
Couzo after Xu, Daniel, Ellsworth, Levy · Couzo follow-up refinements 2026-10-08 · packet · register
2026-10-09 published T-131 · case 132
$s (132) \leq 11.987099332245063 \dots$ from Daniel's record hunt
Upper bound confirmed superseded by T-174
Daniel after Couzo, Chaoweeraprasit, Levy · Daniel record hunt 2026-10-09 · packet · register
2026-10-09 published T-134 this result · 6 of these cases
Six exact rational certificates from Couzo
Upper bound confirmed
Couzo after Polkovskyi, Chaoweeraprasit, Gupta, Xu, Daniel, Ellsworth, Levy · Couzo exact certificates 2026-10-09 · packet · register
2026-10-09 published T-136 · cases 237, 263, 270, 303
Seventeen exact rational packings from SQUISH's third request
Upper bound confirmed on these cases, superseded by T-174, T-186, T-190 and T-224
Chaoweeraprasit after Couzo, Xu, Göbel, Schadt, Ellsworth, Hajba, Levy · SQUISH third request 2026-10-09 · packet · register
2026-10-09 published T-138 · cases 132, 267, 270, 303
Exact witnesses at Mishapolk's printed ceilings, the smallest known at 103 and 258 when registered
Upper bound confirmed superseded by T-134, T-174, T-186 and T-224
Polkovskyi after Stenlund, Friedman, Ellsworth, Chaoweeraprasit, Couzo, Xu, Levy · Mishapolk decimal poses 2026-10-09 · packet · register
2026-10-10 published T-140 · cases 132, 237, 270
Six more exact rational certificates from Couzo
Upper bound confirmed superseded by T-174, T-190 and T-224
Couzo after Chaoweeraprasit, Polkovskyi, Xu, Daniel, Ellsworth, Levy · Couzo certificates 2026-10-10 · packet · register
2026-10-10 published T-141 · case 132
Exact rational certificates at 132 and 308 from Daniel's third record hunt
Upper bound confirmed superseded by T-174
Daniel after Fang, Arslanov, Mustafin, Shangitbayev, Couzo, Levy · Daniel record hunt 3 2026-10-10 · packet · register
2026-10-10 published T-146 · cases 263, 303
Exact rational certificates at fifteen counts from Daniel's regularized record lists
Upper bound confirmed on these cases, superseded by T-174 and T-186
Daniel after Xu, Chaoweeraprasit, Polkovskyi, Levy · Daniel regularized lists 2026-10-10 · packet · register
2026-10-11 published T-151 · case 270
Nine exact rational packings from SQUISH's fourth request
Upper bound confirmed superseded by T-224
Chaoweeraprasit after Couzo, Gupta, Ellsworth, Friedman, Levy · SQUISH fourth request 2026-10-11 · packet · register
2026-10-12 published T-161 · case 237
Exact rational certificates at 156, 176, 237 and 271 from Couzo
Upper bound confirmed superseded by T-190
Couzo after Ostashev, Chaoweeraprasit, Fang, Daniel, Ellsworth, Levy · Couzo certificates 2026-10-12 · packet · register
2026-10-12 published T-162 · case 303
New packings of 129, 204 and 303 squares, a cut-out and two compositions, each certified exactly
Upper bound confirmed superseded by T-186
Ostashev after Nakajima, Chaoweeraprasit, Daniel, Xu, Levy · Ostashev compositions 2026-10-12 · packet · register
2026-10-13 published T-174 · 6 of these cases
Tighter certificates at 68 counts from PingYou Ltd, of others' packings and its own
Upper bound confirmed
PingYou Ltd after Friedman, Ellsworth, Daniel, Xu, Barnett, Couzo, Ostashev, Chaoweeraprasit, Gupta, Fang, Polkovskyi, Levy · PingYou PR 574 certificates 2026-10-13 · packet · register
2026-10-13 published T-180 · 6 of these cases
Closed-form lower bound $s (N) \geq a (N)$ for every non-square $N \geq 49$ , reported
Lower bound reviewed
Hosono · wand125 closed form 2026-10-14 · packet · packet · register
2026-10-13 published T-183 · case 237
Polkovskyi's certificate for 237 squares, the smallest known side there when registered
Upper bound confirmed superseded by T-190
Polkovskyi after Friedman, Daniel, Xu, Chaoweeraprasit, Barnett, Couzo, Ostashev, Ellsworth, Fang, Levy · Mishapolk records 2026-10-14 · packet · register
2026-10-14 published T-186 · case 303
Exact certificates at twelve counts from Polkovskyi, tightening the smallest reported there
Upper bound confirmed
Polkovskyi after Ostashev, Couzo, PingYou Ltd, Levy · Mishapolk KKT refinements 2026-10-14 · packet · register
2026-10-14 published T-189 · case 237
Couzo's refinement of Polkovskyi's 237, and fifteen beyond-horizon bounds
Upper bound confirmed superseded by T-190
Couzo after Polkovskyi, Fang, Daniel, Ellsworth, Levy · Couzo certificates 2026-10-14 · packet · register
2026-10-15 published T-190 · case 237
A tighter exact certificate for 237 squares from Couzo, below his issue-607 certificate
Upper bound confirmed
Couzo after Polkovskyi, Chaoweeraprasit, Daniel, Levy · Couzo n237 2026-10-15 · packet · register
2026-10-16 published T-193 · case 267
Exact algebraic sides at sixteen counts from Daniel, of packings the record credits
Upper bound confirmed
Daniel after Xu, Chaoweeraprasit, Ostashev, Couzo, PingYou Ltd, Gupta, Polkovskyi, Levy · Daniel exact current records 2026-10-16 · packet · register
2026-10-20 published T-224 · case 270
Daniel's smaller n270 packing, another arrangement found from Chaoweeraprasit's, certified exactly
Upper bound confirmed
Daniel after Chaoweeraprasit, Levy · Daniel n270 sweep 2026-10-20 · packet · register

Links

On this site: The frontier survey · T-134 in the results table

On GitHub, at main

Register: T-134 in results.yaml, line 14199
Evidence: E-couzo-476-certificate-report · E-couzo-476-exact-feasibility
Proofs and certificates: certificate certificates
Sources: Couzo exact certificates 2026-10-09 (its own site, retained copy)
Source packet: resources/web/couzo-exact-certificates-2026-10-09/README.md
Artifacts: 14 artifacts and controls
resources/web/couzo-exact-certificates-2026-10-09/README.md resources/web/couzo-exact-certificates-2026-10-09/acquisition/claims.json resources/web/couzo-exact-certificates-2026-10-09/source/certificates/n132.cert resources/web/couzo-exact-certificates-2026-10-09/source/certificates/n237.cert resources/web/couzo-exact-certificates-2026-10-09/source/certificates/n263.cert resources/web/couzo-exact-certificates-2026-10-09/source/certificates/n267.cert resources/web/couzo-exact-certificates-2026-10-09/source/certificates/n270.cert resources/web/couzo-exact-certificates-2026-10-09/source/certificates/n303.cert resources/web/couzo-exact-certificates-2026-10-09/receipts/exact-certification.json.xz devtools/upper_bound_reports.py devtools/check_half_angle_area.py docs/project/reviews/review-2026-10-10-upper-bound-imports-476-481-483-484.md tests/test_upper_bound_reports.py tests/test_check_half_angle_area.py
Case file: each case’s file is linked from its row above