T-082: Mixed rectangle-measure lower bounds verified at 22 counts in $n = 51 \dots 96$

V3 C3 lower bound confirmed

2026-10-03 published · wand125 after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · 22 cases, $n = 51$ to 96

Twenty-two rectangle densities of wand125/square-packing-bounds, checked at coverage one and published on 3 October 2026, prove $s (51) \geq 373 / 50$ = 7.46, $s (52) \geq 151 / 20$ = 7.55, $s (55) \geq 966 / 125$ = 7.728, $s (58) \geq 1581 / 200$ = 7.905, $s (69) \geq 2153 / 250$ = 8.612, $s (70) \geq 3459 / 400$ = 8.6475, $s (71) \geq 1741 / 200$ = 8.705, $s (73) \geq 8809 / 1000$ = 8.809, $s (74) \geq 3547 / 400$ = 8.8675, $s (75) \geq 223 / 25$ = 8.92, $s (76) \geq 224 / 25$ = 8.96, $s (86) \geq 19 / 2$ = 9.5, $s (87) \geq 191 / 20$ = 9.55, $s (88) \geq 48 / 5$ = 9.6, $s (89) \geq 193 / 20$ = 9.65, $s (90) \geq 389 / 40$ = 9.725, $s (91) \geq 39 / 4$ = 9.75, $s (92) \geq 977 / 100$ = 9.77, $s (93) \geq 493 / 50$ = 9.86, $s (94) \geq 248 / 25$ = 9.92, $s (95) \geq 249 / 25$ = 9.96 and $s (96) \geq 997 / 100$ = 9.97.

Each is a density of 303 to 571 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-069, T-071, T-072 and T-075. 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. Six supersede certificates of the same source this record holds verified at the same count: at $n = 76$ (T-072), 87, 91, 92 and 96 (T-075), and 90 (T-069).

21 of the values were above what the record reported at their counts, by 0.015 to 0.125: over the source's rectangle certificates (T-046, T-070, T-074, T-077) at 13 counts, and over its earlier mixed certificates directly at five and by monotonicity at $n = 86$ , 88 and 93. At $n = 96$ , 997/100 is below the side 10 that Evan Daniel's s(k^2 - 4) = k reports there (T-081).

The source's later certificates are higher at 16 of the counts, T-090's of 3 and 4 October at $n = 51$ , 69, 75, 86, 93 and 95 and T-091's of 4 October at $n = 58$ , 70, 71, 73, 76, 87, 88, 90, 91 and 94, each verified here, so the verified lower bound there is theirs. At the other six, $n = 52$ , 55, 74, 89, 92 and 96, these are the verified lower bounds, above those they replace by 0.005 to 0.03. A packing of n + 1 squares contains one of n, but at each next count the record already holds more, so none carries further.

Each certificate was decided here on 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. At $n = 96$ the first control failed closed, its mutant scaled to 99/100 verifying at the one direction it ran, which showed the certificate's slack there and not a fault (think-0uia); the repaired control, which runs that mutant at every net direction, refused it at 187 of the 201, each at an exact capture below one.

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 22 comments of 3 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 six counts, $n = 52$ , 55, 74, 89, 92 and 96, and the strongest reported at five of them (at $n = 96$ T-081 reports 10); at the other 16 the source's later certificates (T-090, T-091) are higher. Further sizes from the generator and checker of T-069, T-071, T-072 and T-075, with no new technique, at S3 as those are; the 3 October review of these certificates proposed S3, which the replays leave standing.
Composition: 22 primary certificates, each on its own reported entry and its own replay entry. 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 3 October review of these certificates is one, and the review of the census route of 5 October decided the route, not these certificates' mathematics. Replays of the source's own checker (about 162 CPU-hours planned, devtools.audit_wand125_point_and_mixed mixed-shard wand125-mixed-bounds-2026-10-03 --runners 8) would add a reproduction with the producer's code beside the rung; 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 22 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
51	7.470000	7.690356	0.22035593…	open=	frontier n-051.md
52	7.550000	7.707107	0.15710678…	open=	frontier n-052.md
55	7.728000	7.945772	0.21777100…	open≈	frontier n-055.md
58	7.935000	8	0.065	open=	frontier n-058.md
69	8.620000	8.827195	0.20719465…		frontier n-069.md
70	8.657500	8.880961	0.22346037…		frontier n-070.md
71	8.721000	8.944072	0.22307155…	open≈	frontier n-071.md
73	8.813000	9	0.187	open=	frontier n-073.md
74	8.867500		0.1325		frontier n-074.md
75	8.940000		0.06		frontier n-075.md
76	8.965000		0.035		frontier n-076.md
86	9.503000	9.820566	0.31756573…		frontier n-086.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
96	9.970000		0.03		frontier n-096.md

Results on these cases

32 results in the register on these cases, oldest first

2005 published T-007 · 22 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-081 (reported), T-082, T-090 and T-091
Nagamochi · Nagamochi 2005 · source · register
2026-09-04 published T-085 · 22 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-082, 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-088 · case 69
$s (69) \leq 8.82719465572975$ , Ellsworth's degree-38 packing, certified exactly from its picture
upper bound confirmed superseded by T-101
Ellsworth after hmbelvedere, Cantrell, Schadt, Morandi · Ellsworth n69 2026-09-24 · packet · packet · 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 · cases 75, 76
Rectangle-density lower bounds replayed at 15 counts in $n = 18 \dots 78$
lower bound confirmed superseded by T-090 and 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 · 18 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-048 · case 51
$s (50) \geq 37 / 5 = 7.4$
lower bound confirmed on these cases, superseded by T-090
wand125 after Daniel, Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 point and mixed bounds 2026-09-28 · packet · source · review · register
2026-09-28 published T-070 · 10 of these cases
Rectangle-density lower bounds replayed at 25 counts in $n = 29 \dots 95$
lower bound confirmed superseded by T-082, 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 · 22 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 · 22 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-081 (reported), T-082, T-090 and T-091
Karakuş · Karakuş 2026 · source · register
2026-10-01 published T-068 · 14 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 · 11 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 · cases 86, 87
Mixed rectangle-measure lower bounds replayed at $n = 84 \dots 87$
lower bound confirmed superseded by T-090 and 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 · 7 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-081 (reported), 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-081 · case 96
$s (k^{2} - 4) = k$ for every integer $k$ from 5 up; $k = 5 \dots 18$ are the cases held here
optimality reviewed
Daniel after Burns, Massaccesi · evand square-packing 2026-10-03 · packet · packet · source · review · register
2026-10-03 published T-082 this result · 22 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 · 9 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 · 14 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 · 7 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 · 12 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 51, 70, 86, 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-126 · case 51
Undilated n51 construction over Q(sqrt2)
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
2026-10-08 published T-130 · case 86
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-138 · case 86
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-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-082 in the results table

On GitHub, at main

Register: T-082 in results.yaml, line 7882
Evidence: E-n051-wand125-mixed-746-report · E-n051-wand125-mixed-746-sqverify-fast-replay · E-n052-wand125-mixed-755-report · E-n052-wand125-mixed-755-sqverify-fast-replay · E-n055-wand125-mixed-7728-report · E-n055-wand125-mixed-7728-sqverify-fast-replay · E-n058-wand125-mixed-7905-report · E-n058-wand125-mixed-7905-sqverify-fast-replay · E-n069-wand125-mixed-8612-report · E-n069-wand125-mixed-8612-sqverify-fast-replay · E-n070-wand125-mixed-86475-report · E-n070-wand125-mixed-86475-sqverify-fast-replay · E-n071-wand125-mixed-8705-report · E-n071-wand125-mixed-8705-sqverify-fast-replay · E-n073-wand125-mixed-8809-report · E-n073-wand125-mixed-8809-sqverify-fast-replay · E-n074-wand125-mixed-88675-report · E-n074-wand125-mixed-88675-sqverify-fast-replay · E-n075-wand125-mixed-892-report · E-n075-wand125-mixed-892-sqverify-fast-replay · E-n076-wand125-mixed-896-report · E-n076-wand125-mixed-896-sqverify-fast-replay · E-n086-wand125-mixed-950-report · E-n086-wand125-mixed-950-sqverify-fast-replay · E-n087-wand125-mixed-955-report · E-n087-wand125-mixed-955-sqverify-fast-replay · E-n088-wand125-mixed-960-report · E-n088-wand125-mixed-960-sqverify-fast-replay · E-n089-wand125-mixed-965-report · E-n089-wand125-mixed-965-sqverify-fast-replay · E-n090-wand125-mixed-9725-report · E-n090-wand125-mixed-9725-sqverify-fast-replay · E-n091-wand125-mixed-975-report · E-n091-wand125-mixed-975-sqverify-fast-replay · E-n092-wand125-mixed-977-report · E-n092-wand125-mixed-977-sqverify-fast-replay · E-n093-wand125-mixed-986-report · E-n093-wand125-mixed-986-sqverify-fast-replay · E-n094-wand125-mixed-992-report · E-n094-wand125-mixed-992-sqverify-fast-replay · E-n095-wand125-mixed-996-report · E-n095-wand125-mixed-996-sqverify-fast-replay · E-n096-wand125-mixed-997-report · E-n096-wand125-mixed-997-sqverify-fast-replay
Proofs and certificates: certificate candidate.json.gz · proof SOUNDNESS.md · audit review-2026-10-03-wand125-october-3-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 · 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 2026-10-03 (its own site, retained copy)
Source packet: resources/web/wand125-mixed-bounds-2026-10-03/README.md · resources/web/wand125-mixed-bounds-2026-10-04/README.md
Artifacts: 58 artifacts and controls in 8 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-2026-10-03 packing/devtools packing/resources/web/wand125-mixed-bounds-2026-10-03 packing/resources/web/wand125-mixed-bounds-2026-10-03/receipts packing/resources/web/wand125-mixed-bounds-2026-10-04 packing/tests
Case file: each case’s file is linked from its row above

T-082: Mixed rectangle-measure lower bounds verified at 22 counts in n=51…96n=51…96

The cases

Results on these cases

Links

T-082: Mixed rectangle-measure lower bounds verified at 22 counts in $n = 51 \dots 96$