T-074: Rectangle-density lower bounds replayed at 31 counts in $n = 19 \dots 95$

V3 C3 lower bound confirmed

2026-10-01 published · wand125 after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · 31 cases, $n = 19$ to 95

Twenty-nine rectangle-density certificates in wand125/square-packing-bounds, raised or added between 29 September and 1 October 2026, prove $s (19) \geq 1927 / 400$ , $s (20) \geq 1959 / 400$ , $s (26) \geq 2213 / 400$ , $s (27) \geq 1127 / 200$ , $s (28) \geq 2289 / 400$ , $s (29) \geq 2319 / 400$ , $s (30) \geq 47 / 8$ , $s (31) \geq 2381 / 400$ , $s (38) \geq 1309 / 200$ , $s (39) \geq 1327 / 200$ , $s (40) \geq 67 / 10$ , $s (41) \geq 169 / 25$ , $s (42) \geq 1363 / 200$ , $s (43) \geq 551 / 80$ , $s (44) \geq 2777 / 400$ , $s (53) \geq 3043 / 400$ , $s (54) \geq 3069 / 400$ , $s (55) \geq 617 / 80$ , $s (56) \geq 3113 / 400$ , $s (68) \geq 851 / 100$ , $s (69) \geq 1717 / 200$ , $s (70) \geq 69 / 8$ , $s (74) \geq 3539 / 400$ , $s (75) \geq 89 / 10$ , $s (88) \geq 3791 / 400$ , $s (89) \geq 1913 / 200$ , $s (93) \geq 3889 / 400$ , $s (94) \geq 1961 / 200$ and $s (95) \geq 49259 / 5000$ .

Each is an exact nonnegative rational density on rectangles, D4-expanded, of mass n - 1/100, such that the shrunken core of every rotated unit square at each of 201 net directions captures mass at least one. The $n = 56$ certificate's mass is below 57 and 58 too, which gives $s (57), s (58) \geq 3113 / 400$ : 31 counts in all.

Each passed a complete 201-direction replay here of Tokoharu's unchanged verify.cpp, in sixteen cloud batches on 2 October 2026 whose receipts were merged into the packet the same day, and each replay reproduced the upstream accepting run's node count, leaf count and lower bound at every direction. The inputs were regenerated by this repository's exact preflight, which checks that each is the input the upstream accepting run recorded and checks the exact mass, net, smoothing and axis-event premises. A thirtieth certificate replayed in the same batches, rect_n76_L8925, is below the mixed certificate that holds $n = 76$ (T-072) and is not registered here.

wand125, square-packing-bounds, built with and checked by Tokoharu's solver and interval verifier. The source keeps its credit to this project for the method lineage and says parts of the work were produced with AI assistance under human direction.

Significance, composition and next rung

Significance: Raised the verified lower bound at 31 counts from $n = 19$ to 95, over the replayed certificates of T-045 and T-070, wand125's point certificate at $n = 56$ (T-044) and Nagamochi's closed form at $n = 57$ and 58, on Tokoharu's method with larger certificates. Substantive case results, S3, as T-045 and T-070; the 2 October review of T-068 confirmed S3 for these certificates. The closed form has been a reported bound since Karakuş's finding (T-085, merged 2026-10-03); the floors it left at $n = 57$ and 58, T-044's 381/50 and Karakuş's general bound, are lower still.
Composition: Twenty-nine primary certificates. $n = 57$ and 58 take the $n = 56$ certificate directly, its mass being below those counts, so no step is added. One interval-certified method, replayed here: C3. The replay decides coverage with the source's own checker, so it confirms the source's run, reproduced with the producer's code, rather than adding a second method.
Next rung: V4 and C4 need two adversarial AI reviews by distinct reviewers and a human oversight record; one adversarial review is retained, read before the replays. A second machine method would be a method-distinct decision of rotated coverage; global coverage is decided by the source's C++ checker here and, under T-068, by sqverify-fast, one method in two implementations, and the exact preflight does not decide it.
Novelty: previously-published Present in an identified source

The cases

This result concerns 31 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
19	4.825000	4.885619	0.06061808…	open=	frontier n-019.md
20	4.905000	5	0.095		frontier n-020.md
26	5.545000	5.621321	0.07632034…		frontier n-026.md
27	5.643500	5.707107	0.06360678…		frontier n-027.md
28	5.735000	5.824445	0.08944461…	open=R	frontier n-028.md
29	5.810000	5.933834	0.12383346…	open≈	frontier n-029.md
30	5.883500	6	0.1165	open=	frontier n-030.md
31	5.952500	6	0.0475		frontier n-031.md
38	6.545000	6.707107	0.16210678…		frontier n-038.md
39	6.650000	6.810723	0.16072208…		frontier n-039.md
40	6.715815	6.828428	0.11261137…	open=R	frontier n-040.md
41	6.775000	6.926694	0.15169309…	open=	frontier n-041.md
42	6.847500	7	0.1525		frontier n-042.md
43	6.907500		0.0925		frontier n-043.md
44	6.972500		0.0275		frontier n-044.md
53	7.627500	7.822876	0.19537565…		frontier n-053.md
54	7.685000	7.846668	0.16166719…		frontier n-054.md
55	7.728000	7.945772	0.21777100…	open≈	frontier n-055.md
56	7.802500	8	0.1975	open=	frontier n-056.md
57	7.872500		0.1275		frontier n-057.md
58	7.935000		0.065		frontier n-058.md
68	8.510000	8.798796	0.28879523…		frontier n-068.md
69	8.620000	8.827195	0.20719465…		frontier n-069.md
70	8.657500	8.880961	0.22346037…		frontier n-070.md
74	8.867500	9	0.1325		frontier n-074.md
75	8.940000	9	0.06		frontier n-075.md
88	9.620000	9.882452	0.26245103…		frontier n-088.md
89	9.650000	9.949748	0.29974746…		frontier n-089.md
93	9.880000	10	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

48 results in the register on these cases, oldest first

2005 published T-007 · 31 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-068, T-074, T-082, T-090, T-091, T-103, T-104, T-105, T-106, T-107, T-108, T-109, T-110, T-111 and T-145
Nagamochi · Nagamochi 2005 · source · register
2026-08-21 published T-016 · case 19
$s (n) \geq 22529 / 5000$ for $n = 18, 19$ , by monotonicity from T-015
lower bound confirmed superseded by T-103
Massaccesi after Burns · Burns–Massaccesi n17 · packet · register
2026-08-29 established T-009 · case 29
$s (29) \leq 5.933833 \dots$ , by a Krawczyk interval certificate
upper bound confirmed
Levy · register
2026-08-30 established T-013 · case 40
Goebel's $n = 40$ packing: seven verified first-order flexes, each refused at second order
rigidity confirmed
Levy · register
2026-09-04 established T-019 · case 19
$s (n) \geq 459 / 100 = 4.59$ for $n = 17, 18, 19$
lower bound confirmed superseded by T-103
Levy after Burns, Massaccesi · register
2026-09-04 established T-020 · cases 19, 20
$s (n) \geq 24 / 5 = 4.80$ for $n = 19, 20, 21$
lower bound confirmed superseded by T-103 and T-104
Levy after Burns, Massaccesi · register
2026-09-04 published T-085 · 31 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
2026-09-05 established T-021 · case 20
$s (n) \geq 97 / 20 = 4.85$ for $n = 20, 21$
lower bound confirmed superseded by T-104
Levy after Burns, Massaccesi · register
2026-09-22 published T-044 · 13 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-068, T-074, T-082, T-090, T-091, T-105, T-108, T-110, T-111 and T-145
wand125 after Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 point bounds 2026 · packet · source · review · register
2026-09-22 published T-047 · 6 of these cases
$s (11) \geq 381 / 100$ ; $s (n) \geq 1377 / 250$ for $n = 26 \dots 28$ ; $s (n) \geq 571 / 100$ for $n = 29 \dots 31$
lower bound confirmed superseded by T-068, T-074, T-105, T-106, T-107, T-108 and T-109
Tokoharu after Levy, wand125, Stromquist, Nagamochi, Burns, Massaccesi · Tokoharu density 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
2026-09-27 published T-045 · 9 of these cases
Rectangle-density lower bounds replayed at 15 counts in $n = 18 \dots 78$
lower bound confirmed superseded by T-068, T-074, T-090, T-103, T-104, T-105, T-106, T-107, T-109 and T-145
wand125 after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 rectangle bounds 2026 · packet · packet · source · review · register
2026-09-27 published T-046 · 30 of these cases
Rectangle-density lower bounds reported for 48 counts in $n = 18 \dots 95$
lower bound recorded superseded by T-068, T-074, T-082, T-090, T-091, T-103, T-104, T-105, T-106, T-107, T-108, T-109, T-110, T-111 and T-145
wand125 after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 rectangle bounds 2026 · wand125 rectangle bounds 2026-09-28 · packet · packet · register
2026-09-27 published T-056 · case 68
Smaller packings for 49 counts from $n = 68$ to $307$ , each certified two independent ways
upper bound confirmed superseded by T-118
Couzo · franciscouzo square-packing 2026-09-27 · packet · register
2026-09-28 published T-070 · 16 of these cases
Rectangle-density lower bounds replayed at 25 counts in $n = 29 \dots 95$
lower bound confirmed superseded by T-068, T-074, T-082, T-090, T-091, T-108, T-110 and T-111
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 · 31 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 · 31 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-068, T-074, T-082, T-090, T-091, T-103, T-104, T-105, T-106, T-107, T-108, T-109, T-110, T-111 and T-145
Karakuş · Karakuş 2026 · source · register
2026-10-01 published T-068 · 29 of these cases
Rectangle-density lower bounds verified at 34 counts in $n = 19 \dots 95$
lower bound confirmed
wand125 after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 rectangle bounds 2026-10-01 · packet · source · review · register
2026-10-01 published T-074 this result · 31 of these cases
Rectangle-density lower bounds replayed at 31 counts in $n = 19 \dots 95$
lower bound confirmed
wand125 after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 rectangle bounds 2026-10-01 · packet · packet · source · review · register
2026-10-02 published T-075 · cases 88, 93
Mixed rectangle-measure lower bounds replayed at nine counts in $n = 83 \dots 96$
lower bound confirmed on these cases, superseded by 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 · cases 20, 42, 70
Rectangle-density lower bounds replayed at $n = 20$ , 42 and 70
lower bound confirmed superseded by T-090, T-091 and T-104
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 · 11 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 · 11 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 · 9 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-098 · case 68
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-118
Daniel after Couzo, de Winter, Ellsworth, Levy · evand exact optima 2026-10-05 · packet · register
2026-10-05 published T-101 · 9 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
2026-10-06 published T-100 · case 19
Mixed rectangle-measure lower bound verified at $n = 19$ , past the best 18-square packing
lower bound confirmed superseded by T-103
wand125 after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 mixed bounds finer net 2026-10-06 · packet · source · review · register
2026-10-06 published T-103 · case 19
Mixed rectangle-measure lower bound verified at $n = 19$ , on the finest declared net yet
lower bound confirmed
wand125 after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 mixed bounds check2 2026-10-06 · packet · source · review · register
2026-10-06 published T-104 · case 20
Mixed rectangle-measure lower bound verified at $n = 20$ , on a declared net
lower bound confirmed
wand125 after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 mixed bounds check2 2026-10-06 · packet · source · review · register
2026-10-06 published T-105 · case 26
Mixed rectangle-measure lower bound verified at $n = 26$ , on a declared net
lower bound confirmed
wand125 after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 mixed bounds check2 2026-10-06 · packet · source · review · register
2026-10-06 published T-106 · case 27
Mixed rectangle-measure lower bound verified at $n = 27$ , on a declared net
lower bound confirmed
wand125 after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 mixed bounds check2 2026-10-06 · packet · source · review · register
2026-10-06 published T-107 · case 28
Mixed rectangle-measure lower bound verified at $n = 28$ , on a declared net
lower bound confirmed
wand125 after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 mixed bounds check2 2026-10-06 · packet · source · review · register
2026-10-06 published T-108 · case 29
Mixed rectangle-measure lower bound verified at $n = 29$ , on a declared net
lower bound confirmed
wand125 after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 mixed bounds check2 2026-10-06 · packet · source · review · register
2026-10-06 published T-109 · case 30
Mixed rectangle-measure lower bound verified at $n = 30$ , on a declared net
lower bound confirmed
wand125 after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 mixed bounds check2 2026-10-06 · packet · source · review · register
2026-10-06 published T-110 · case 39
Mixed rectangle-measure lower bound verified at $n = 39$ , on a declared net
lower bound confirmed
wand125 after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 mixed bounds check2 2026-10-06 · packet · source · review · register
2026-10-06 published T-111 · case 41
Mixed rectangle-measure lower bound verified at $n = 41$ , on a declared net
lower bound confirmed
wand125 after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 mixed bounds check2 2026-10-06 · packet · source · review · 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-118 · case 68
Exact rational ceiling refinements at $n = 68$
upper bound confirmed
Rehwaldt after Couzo and earlier contributors · Rehwaldt n68 refinement 2026-10-07 · packet · register
2026-10-07 published T-124 · 15 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-132 · case 29
Mixed rectangle-measure lower bound $s (29) \geq 291 / 50$ reported, on a 2,073-direction declared net
lower bound recorded
wand125 after Tokoharu and Levy · wand125 fine-net n29 2026-10-10 · packet · register
2026-10-10 published T-133 · case 40
$s (40) > 335427 / 50000 = 6.70854$ , by a clipped-corner transfer on a 401-direction net
lower bound confirmed superseded by T-145
Guzhou0806 after wand125, Tokoharu, Levy, Stromquist, Burns, Massaccesi · Guzhou0806 n40 clipped corner 2026-10-10 · packet · packet · source 1 · source 2 · review · 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-142 · case 28
Mixed rectangle-measure lower bound $s (28) \geq 2297 / 400$ reported, on a 2,073-direction declared net
lower bound recorded
wand125 after Tokoharu and Levy · wand125 fine-net n28 n30 2026-10-10 · packet · register
2026-10-10 published T-143 · case 30
Mixed rectangle-measure lower bound $s (30) \geq 2357 / 400$ reported, on a 2,073-direction declared net
lower bound recorded
wand125 after Tokoharu and Levy · wand125 fine-net n28 n30 2026-10-10 · packet · register
2026-10-10 published T-145 · case 40
$s (40) > 1340000 / 199529 = 6.7158157 \dots$ , by a continuous-pose interval kernel
lower bound confirmed
Guzhou0806 after wand125, Tokoharu, Levy, Stromquist, Burns, Massaccesi · Guzhou0806 n40 continuous pose 2026-10-10 · packet · packet · source 1 · source 2 · review · 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-074 in the results table

On GitHub, at main

Register: T-074 in results.yaml, line 6882
Evidence: E-wand125-rectangle-2026-10-01-source-replay
Proofs and certificates: certificate certificates · proof continuous-density-certificate.ja.md · audit review-2026-09-22-tokoharu-density-mathematics.md
Sources: wand125 rectangle bounds 2026-10-01 (its own site, retained copy)
Source packet: resources/web/wand125-rectangle-certificates-2026-10-01/README.md · resources/web/external-square-certificates-2026-09-22/README.md
Artifacts: 12 artifacts and controls
resources/web/wand125-rectangle-certificates-2026-10-01/README.md resources/web/wand125-rectangle-certificates-2026-10-01/receipts/replay/audit.json.gz resources/web/wand125-rectangle-certificates-2026-10-01/receipts/preflight/audit.json.gz resources/web/external-square-certificates-2026-09-22/tokoharu-density/src/verify.cpp devtools/audit_wand125_rectangles.py devtools/apply_wand125_rectangles.py docs/project/reviews/review-2026-09-22-tokoharu-density-mathematics.md docs/project/reviews/review-2026-10-02-wand125-rectangle-bounds-t068.md tests/test_wand125_checker_controls.py tests/test_wand125_rectangle_audit.py tests/test_wand125_rectangle_merge.py resources/web/wand125-rectangle-certificates-2026-10-01/receipts/controls/rect_n41_L676.json
Case file: each case’s file is linked from its row above

T-074: Rectangle-density lower bounds replayed at 31 counts in n=19…95n=19…95

The cases

Results on these cases

Links

T-074: Rectangle-density lower bounds replayed at 31 counts in $n = 19 \dots 95$