T-091: Mixed rectangle-measure lower bounds verified at 17 counts in $n = 53 \dots 95$

V3 C3 lower bound confirmed

2026-10-04 published · wand125 after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · 17 cases, $n = 53$ to 95

Twelve rectangle densities of wand125/square-packing-bounds, checked at coverage one and published on 4 October 2026, after those of T-090, prove $s (53) \geq 3051 / 400$ = 7.6275, $s (54) \geq 1537 / 200$ = 7.685, $s (58) \geq 1587 / 200$ = 7.935, $s (70) \geq 3463 / 400$ = 8.6575, $s (71) \geq 8721 / 1000$ = 8.721, $s (73) \geq 8813 / 1000$ = 8.813, $s (76) \geq 1793 / 200$ = 8.965, $s (87) \geq 479 / 50$ = 9.58, $s (88) \geq 481 / 50$ = 9.62, $s (90) \geq 973 / 100$ = 9.73, $s (91) \geq 781 / 80$ = 9.7625 and $s (94) \geq 199 / 20$ = 9.95. A packing of n + 1 squares contains one of n, so the $n = 71$ value also bounds $n = 72$ , the $n = 88$ value $n = 89$ , the $n = 91$ value $n = 92$ and 93, and the $n = 94$ value $n = 95$ , where each is above the verified bound the record held: 17 counts in all.

Each is a density of 312 to 853 uniform rectangles in D4 orbits and no point mass, of total mass n - 1/100000, at core side 9977/10000 on 201 net half-angles of step 83/40000, of the kind and checker of T-082 and T-090. The source reports each accepted at all 200 oblique net angles by code/mixed_rotated_verify.cpp, its research copy of Tokoharu's verify.cpp at coverage threshold one, and at the axis by exact integer tables. Ten supersede the source's certificates at the same count: eight of T-082's, and at $n = 88$ and 94 two of T-090's of earlier that day. At $n = 53$ and 54 they are the source's first mixed certificates.

All 12 values were above what the record reported at their counts, by 0.004 to 0.03: over T-082 at eight counts, T-090 at $n = 88$ and 94, and the source's rectangle certificates (T-068, T-074) at $n = 53$ and 54. Over the verified lower bounds they replace, at the twelve counts and the five they carry to, the rise is 0.0125 to 0.1525.

Each certificate was decided here on 5 and 6 October 2026 by sqverify-fast, this repository's clean-room measure verifier, at all 201 net directions of the retained candidate, and two mutants of each scaled below coverage one were refused: confirmed, independently re-implemented. The verifier shares no code with the source's checker and runs the same net-and-shrink method, so it is a second implementation and not a second method. The source's own checker was not run here.

wand125 after Tokoharu and Levy, square-packing-bounds. Registration was requested in 12 comments of 4 October 2026 on jlevy/squares#282. The source says parts of the work were produced with AI assistance under human direction.

Significance, composition and next rung

Significance: The strongest verified lower bounds on record at 17 counts from $n = 53$ to 95 when decided, and at 14 since T-090's certificates were decided at $n = 72$ , 93 and 95 on 6 October; the strongest reported at all 12 of its own. Further sizes from the generator and checker of T-069, T-071, T-072, T-075, T-082 and T-090, with no new technique, at S3 as those are; the 5 October review of these certificates confirmed S3.
Composition: 12 primary certificates, each on its own reported entry and its own replay entry. $n = 72$ , 89, 92, 93 and 95 take the $n = 71$ , 88, 91 and 94 certificates directly, their masses being below those counts, so no step is added. Each replay is sqverify-fast at all 201 net directions of the retained candidate: one interval-certified method, replayed here by an independent implementation, C3. The source's checker and axis tables were not run here, so no second method and no reproduction with the producer's code stands beside it.
Next rung: V4 and C4 need a human oversight record and two adversarial AI reviews of the claim by distinct reviewers: the 5 October review of these certificates is one, and the review of the census route of the same day decided the route, not these certificates' mathematics. Replays of the source's own checker (about 131 CPU-hours planned, devtools.audit_wand125_point_and_mixed mixed-shard wand125-mixed-bounds-evening-2026-10-04 --runners 6) would add a reproduction with the producer's code beside the rung, and only they bear on the five bundles that record their proof run on macOS arm64 (OF-2); a second machine method would be a method-distinct decision of rotated coverage.
Novelty: previously-published Present in an identified source

The cases

This result concerns 17 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
53	7.627500	7.822876	0.19537565…	open=	frontier n-053.md
54	7.685000	7.846668	0.16166719…		frontier n-054.md
58	7.935000	8	0.065		frontier n-058.md
70	8.657500	8.880961	0.22346037…		frontier n-070.md
71	8.721000	8.944072	0.22307155…	open≈	frontier n-071.md
72	8.760000	9	0.24	open=	frontier n-072.md
73	8.813000		0.187		frontier n-073.md
76	8.965000		0.035		frontier n-076.md
87	9.580000	9.838816	0.25881526…		frontier n-087.md
88	9.620000	9.882452	0.26245103…		frontier n-088.md
89	9.650000	9.949748	0.29974746…		frontier n-089.md
90	9.730000	10	0.27		frontier n-090.md
91	9.762500		0.2375		frontier n-091.md
92	9.770000		0.23		frontier n-092.md
93	9.880000		0.12		frontier n-093.md
94	9.950000		0.05		frontier n-094.md
95	9.965000		0.035		frontier n-095.md

Results on these cases

26 results in the register on these cases, oldest first

2005 published T-007 · 17 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-082, T-090 and T-091
Nagamochi · Nagamochi 2005 · source · register
2026-09-04 published T-085 · 17 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
2026-09-22 published T-044 · 6 of these cases
Weighted point lower bounds for ten counts in $n = 26 \dots 72$ , plus seven from the same files
lower bound confirmed superseded by T-090 and T-091
wand125 after Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 point bounds 2026 · packet · source · review · register
2026-09-24 published T-089 · case 87
$s (83) \leq 9.63475764863195$ and $s (87) \leq 9.83881526994915$ , Chang's packings, certified exactly
upper bound confirmed superseded by T-101
Chang, Ellsworth after Cantrell, Hajba, Stenlund, Bidwell; Chang after Ellsworth, Cantrell, Schadt, Hajba, DeVincentis · Chang n83 2026-09-24 · Chang n87 2026-09-24 · packet · packet · register
2026-09-27 published T-045 · case 76
Rectangle-density lower bounds replayed at 15 counts in $n = 18 \dots 78$
lower bound confirmed superseded by T-091
wand125 after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 rectangle bounds 2026 · packet · packet · source · review · register
2026-09-27 published T-046 · 14 of these cases
Rectangle-density lower bounds reported for 48 counts in $n = 18 \dots 95$
lower bound recorded superseded by T-082, T-090 and T-091
wand125 after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 rectangle bounds 2026 · wand125 rectangle bounds 2026-09-28 · packet · packet · register
2026-09-28 published T-070 · 7 of these cases
Rectangle-density lower bounds replayed at 25 counts in $n = 29 \dots 95$
lower bound confirmed superseded by T-090 and T-091
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 · 17 of these cases
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 · 17 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 on these cases, superseded by T-082, T-090 and T-091
Karakuş · Karakuş 2026 · source · register
2026-10-01 published T-068 · 11 of these cases
Rectangle-density lower bounds verified at 34 counts in $n = 19 \dots 95$
lower bound confirmed on these cases, superseded by T-082, T-090 and T-091
wand125 after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 rectangle bounds 2026-10-01 · packet · source · review · register
2026-10-01 published T-069 · cases 90, 92
Mixed rectangle-measure lower bounds replayed at $n = 37, 65, 66, 90, 92$
lower bound confirmed on these cases, superseded by T-082 and T-091
wand125 after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 mixed bounds 2026-10-01 · packet · source · review · register
2026-10-01 published T-074 · 9 of these cases
Rectangle-density lower bounds replayed at 31 counts in $n = 19 \dots 95$
lower bound confirmed on these cases, superseded by T-082, T-090 and T-091
wand125 after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 rectangle bounds 2026-10-01 · packet · packet · source · review · register
2026-10-02 published T-071 · case 87
Mixed rectangle-measure lower bounds replayed at $n = 84 \dots 87$
lower bound confirmed superseded by T-091
wand125 after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 mixed bounds 2026-10-02 · packet · source · review · register
2026-10-02 published T-072 · case 76
Mixed rectangle-measure lower bound replayed at $n = 76$
lower bound confirmed superseded by T-091
wand125 after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 mixed bounds n76 2026-10-02 · packet · packet · source · review · register
2026-10-02 published T-075 · 5 of these cases
Mixed rectangle-measure lower bounds replayed at nine counts in $n = 83 \dots 96$
lower bound confirmed on these cases, superseded by T-082, T-090 and T-091
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-077 · case 70
Rectangle-density lower bounds replayed at $n = 20$ , 42 and 70
lower bound confirmed superseded by T-091
wand125 after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 rectangle bounds 2026-10-02 · packet · packet · source 1 · source 2 · review 1 · review 2 · register
2026-10-03 published T-082 · 14 of these cases
Mixed rectangle-measure lower bounds verified at 22 counts in $n = 51 \dots 96$
lower bound confirmed
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 · 5 of these cases
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
2026-10-04 published T-091 this result · 17 of these cases
Mixed rectangle-measure lower bounds verified at 17 counts in $n = 53 \dots 95$
lower bound confirmed
wand125 after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 mixed bounds evening 2026-10-04 · packet · source · review · register
2026-10-05 published T-101 · 6 of these cases
Exact certificates of 77 catalogue packings: s(n) ≤ S', 1.2e-16 to 9.8e-15 above each side
upper bound confirmed
Daniel after Couzo, de Winter, Ellsworth, Levy · evand exact optima 2026-10-05 · packet · register
2026-10-07 published T-116 · case 88
$s (n) \leq S_{n}$ at nine counts, from the second SQUISH update’s new rational packings
upper bound confirmed on these cases, superseded by T-127
Chaoweeraprasit after Ellsworth, Couzo and SQUISH · SQUISH second update 2026-10-07 · packet · register
2026-10-07 published T-124 · 10 of these cases
Reported non-strict local minima for 178 source configurations
restricted optimality recorded
Daniel after Couzo · Daniel exact and local reports 2026 · packet · register
2026-10-08 published T-125 · cases 70, 88
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 · case 88
Fourteen rational refinements, seventeen complete source cases
upper bound confirmed
Gupta after Chaoweeraprasit, Daniel · Gupta rational refinements 2026-10-08 · packet · register
2026-10-10 published T-137 · case 70
An exact rational refinement of Ryan Xu's packing of 70 squares
upper bound confirmed
Deleeuw after Xu, Levy · Deleeuw n70 refinement 2026-10-10 · packet · register
2026-10-10 published T-146 · case 70
Exact rational certificates at fifteen counts from Evan Daniel's regularized record lists
upper bound confirmed
Daniel after Xu, Chaoweeraprasit, Mishapolk, Levy · Daniel regularized lists 2026-10-10 · packet · register

Links

On this site: The frontier survey · T-091 in the results table

On GitHub, at main

Register: T-091 in results.yaml, line 8919
Evidence: E-n053-wand125-mixed-76275-report · E-n053-wand125-mixed-76275-sqverify-fast-replay · E-n054-wand125-mixed-7685-report · E-n054-wand125-mixed-7685-sqverify-fast-replay · E-n058-wand125-mixed-7935-report · E-n058-wand125-mixed-7935-sqverify-fast-replay · E-n070-wand125-mixed-86575-report · E-n070-wand125-mixed-86575-sqverify-fast-replay · E-n071-wand125-mixed-8721-report · E-n071-wand125-mixed-8721-sqverify-fast-replay · E-n073-wand125-mixed-8813-report · E-n073-wand125-mixed-8813-sqverify-fast-replay · E-n076-wand125-mixed-8965-report · E-n076-wand125-mixed-8965-sqverify-fast-replay · E-n087-wand125-mixed-958-report · E-n087-wand125-mixed-958-sqverify-fast-replay · E-n088-wand125-mixed-962-report · E-n088-wand125-mixed-962-sqverify-fast-replay · E-n090-wand125-mixed-973-report · E-n090-wand125-mixed-973-sqverify-fast-replay · E-n091-wand125-mixed-97625-report · E-n091-wand125-mixed-97625-sqverify-fast-replay · E-n094-wand125-mixed-995-report · E-n094-wand125-mixed-995-sqverify-fast-replay
Proofs and certificates: certificate candidate.json.gz · proof SOUNDNESS.md · audit review-2026-10-05-wand125-october-4-certificates.md · certificate candidate.json.gz · certificate candidate.json.gz · certificate candidate.json.gz · certificate candidate.json.gz · certificate candidate.json.gz · certificate candidate.json.gz · certificate candidate.json.gz · certificate candidate.json.gz · certificate candidate.json.gz · certificate candidate.json.gz · certificate candidate.json.gz
Sources: wand125 mixed bounds evening 2026-10-04 (its own site, retained copy)
Source packet: resources/web/wand125-mixed-bounds-evening-2026-10-04/README.md
Artifacts: 32 artifacts and controls in 7 directories. Complete artifact list in the result entry. docs/project/reviews packing/benchmarks/measure-verifier/census-mixed packing/benchmarks/measure-verifier/census-mixed/wand125-mixed-bounds-evening-2026-10-04 packing/devtools packing/resources/web/wand125-mixed-bounds-evening-2026-10-04 packing/resources/web/wand125-mixed-bounds-evening-2026-10-04/receipts packing/tests
Case file: each case’s file is linked from its row above

T-091: Mixed rectangle-measure lower bounds verified at 17 counts in n=53…95n=53…95

The cases

Results on these cases

Links

T-091: Mixed rectangle-measure lower bounds verified at 17 counts in $n = 53 \dots 95$