T-140: Six more exact rational certificates from Couzo

V3 C3 Upper bound confirmed superseded by T-174, T-186, T-190 and T-224

2026-10-10 published · Couzo after Chaoweeraprasit, Polkovskyi, Xu, Daniel, Ellsworth, Levy · 6 cases, $n = 132$ to 305

Six complete rational source certificates by Couzo, reported on jlevy/squares#488, prove finite upper bounds at the exact sides they state: $s (132) \leq 11.986953587254420950497823747297$ , $s (175) \leq 13.767120724692313217301436951952$ , $s (209) \leq 14.946223226446451055256712327571$ , $s (237) \leq 15.90294960767783168971278840528$ , $s (270) \leq 16.929775353541359582336028215624$ and $s (305) \leq 17.951139772206997545942513172626$ .

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, and that all six are exact optima from Daniel's exact contact solver at 80 digits, each within about 1e-19 of its KKT point, with no first-order descent found by the corner-corner MILP in any branch; those statements are the author's.

All 27 retained jobs of the release's nine-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 records adopted five of them as the house, at 175, 209, 237, 270 and 305. On 12 October the smaller certificates of issue 500 at 270 (T-151), of issue 506 at 305 (T-152), the latter refining issue 500's packing there, and of issue 528 at 175 (T-158), Ostashev's composition, replaced them at those three counts; there, and at 132, where the smaller certificate of issue 489 (T-141) holds the case, this entry stays verified and is not selected.

Credit Couzo (https://github.com/franciscouzo/square-packing), with the packings each started from, as the author asks: Chaoweeraprasit's SQUISH packings from issue 481 at 209, 237, 270 and 305, Mishapolk's at 132 (issue 470) and Xu's at 175 (issue 432). They were improved by Daniel's fq (132, 175, 209, 237 and 305), the author's basin hopping (175 and 270) and Ellsworth's refine_packing (132, 175 and 270), and Daniel's exact contact solver wrote all six. The search code and the request were written with Claude under the author's direction.

Significance, composition and next rung

Significance: Smaller finite construction sides at $n = 132$ , 175, 209, 237, 270 and 305, 7.27e-04 to 8.03e-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).
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
175	13.237739	13.764175	0.52643511…		frontier n-175.md
209	14.473188	14.946224	0.47303461…		frontier n-209.md
237	15.408051	15.902604	0.49455212…		frontier n-237.md
270	16.445218	16.928163	0.48294353…		frontier n-270.md
305	17.477926	17.950467	0.47253963…		frontier n-305.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-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 · case 209
Smaller packings again at seven counts from $n = 208$ to $306$ , each certified two independent ways
Upper bound confirmed superseded by T-174
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-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 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-186
Daniel after Couzo, de Winter, Ellsworth, Levy · evand exact optima 2026-10-05 · packet · register
2026-10-07 published T-113 · case 209
$s (n) \leq S_{n}$ at ten counts, $n = 108$ to 303, from new SQUISH rational packings
Upper bound confirmed superseded by T-174
Chaoweeraprasit after Ellsworth · SQUISH ten packings 2026-10-07 · packet · register
2026-10-07 published T-115 · case 237
$s (n) \leq S_{n}$ at twelve counts, from the SQUISH update’s new and smaller rational packings
Upper bound confirmed superseded by T-190
Chaoweeraprasit after Couzo and Ellsworth · SQUISH 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 · case 175
Complete rational construction reports at 25 counts
Upper bound confirmed superseded by T-186
Xu · ry-xu square packing 2026 · packet · register
2026-10-08 published T-127 · cases 209, 237
Fourteen rational refinements, seventeen complete source cases
Upper bound confirmed on these cases, superseded by T-174 and T-190
Gupta after Chaoweeraprasit, Daniel · Gupta rational refinements 2026-10-08 · packet · register
2026-10-08 published T-130 · cases 175, 270
Five follow-up rational refinements from Couzo
Upper bound confirmed superseded by T-186 and 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 · cases 132, 237, 270
Six exact rational certificates from Couzo
Upper bound confirmed on these cases, superseded by T-174, T-190 and T-224
Couzo after Polkovskyi, Chaoweeraprasit, Gupta, Xu, Daniel, Ellsworth, Levy · Couzo exact certificates 2026-10-09 · packet · register
2026-10-09 published T-136 · cases 209, 237, 270, 305
Seventeen exact rational packings from SQUISH's third request
Upper bound confirmed on these cases, superseded by T-174, 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, 175, 270
Exact witnesses at Mishapolk's printed ceilings, the smallest known at 103 and 258 when registered
Upper bound confirmed superseded by 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 this result · 6 of these cases
Six more exact rational certificates from Couzo
Upper bound confirmed superseded by T-174, T-186, 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
2026-10-11 published T-151 · cases 270, 305
Nine exact rational packings from SQUISH's fourth request
Upper bound confirmed superseded by T-174 and T-224
Chaoweeraprasit after Couzo, Gupta, Ellsworth, Friedman, Levy · SQUISH fourth request 2026-10-11 · packet · register
2026-10-11 published T-152 · case 305
Exact rational refinements of SQUISH's Mondrian packings of 304 and 305 squares
Upper bound confirmed superseded by T-174
Couzo after Chaoweeraprasit, Fang, Daniel, Ellsworth, Levy · Couzo certificates 2026-10-11 · packet · register
2026-10-11 published T-185 · case 175
A tighter packing of 175 squares from Couzo, superseded by Ostashev's composition
Upper bound confirmed superseded by T-186
Couzo after Xu, Daniel, Ellsworth, Levy · Couzo n175 2026-10-11 · packet · register
2026-10-12 published T-158 · case 175
New packings of 175, 232 and 264 squares built on a new 38-square rectangle, each certified exactly
Upper bound confirmed superseded by T-186
Ostashev after Nakajima, Ellsworth, Schadt, Friedman, de Winter, Daniel, Levy, Xu · Ostashev records 2026-10-12 · 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
2026-10-13 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 175
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 175
Exact algebraic sides at sixteen counts from Daniel, of packings the record credits
Upper bound confirmed on these cases, superseded by T-186
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-140 in the results table

On GitHub, at main

Register: T-140 in results.yaml, line 15250
Evidence: E-couzo-488-certificate-report · E-couzo-488-exact-feasibility
Proofs and certificates: certificate certificates
Sources: Couzo certificates 2026-10-10 (its own site, retained copy)
Source packet: resources/web/couzo-certificates-2026-10-10/README.md
Artifacts: 15 artifacts and controls
resources/web/couzo-certificates-2026-10-10/README.md resources/web/couzo-certificates-2026-10-10/acquisition/claims.json resources/web/couzo-certificates-2026-10-10/source/certificates/n132.cert resources/web/couzo-certificates-2026-10-10/source/certificates/n175.cert resources/web/couzo-certificates-2026-10-10/source/certificates/n209.cert resources/web/couzo-certificates-2026-10-10/source/certificates/n237.cert resources/web/couzo-certificates-2026-10-10/source/certificates/n270.cert resources/web/couzo-certificates-2026-10-10/source/certificates/n305.cert resources/web/couzo-certificates-2026-10-10/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-488-489.md tests/test_upper_bound_reports.py tests/test_upper_bound_reports_488_489.py tests/test_check_half_angle_area.py
Case file: each case’s file is linked from its row above