T-068: Rectangle-density lower bounds verified at 34 counts in $n = 19 \dots 95$

V3 C3 Lower bound confirmed

2026-10-01 published · Hosono after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · 34 cases, $n = 19$ to 95

Thirty-four rectangle-density certificates of wand125/square-packing-bounds, one at each of 34 counts from $n = 19$ to $n = 95$ , published between 29 September and 1 October 2026, prove the values below. They raise 32 of the values T-046 reports and add $n = 87$ and $n = 93$ .

Up to $n = 44$ the values are $s (19) \geq 1927 / 400$ , $s (20) \geq 1959 / 400$ , $s (26) \geq 2213 / 400$ , $s (27) \geq 1127 / 200$ , $s (28) \geq 2289 / 400$ , $s (29) \geq 2319 / 400$ , $s (30) \geq 47 / 8$ , $s (31) \geq 2381 / 400$ , $s (38) \geq 1309 / 200$ , $s (39) \geq 1327 / 200$ , $s (40) \geq 67 / 10$ , $s (41) \geq 169 / 25$ , $s (42) \geq 1363 / 200$ , $s (43) \geq 551 / 80$ and $s (44) \geq 2777 / 400$ .

From $n = 53$ they are $s (53) \geq 3043 / 400$ , $s (54) \geq 3069 / 400$ , $s (55) \geq 617 / 80$ , $s (56) \geq 3113 / 400$ , $s (66) \geq 1677 / 200$ , $s (68) \geq 851 / 100$ , $s (69) \geq 1717 / 200$ , $s (70) \geq 69 / 8$ , $s (74) \geq 3539 / 400$ , $s (75) \geq 89 / 10$ , $s (76) \geq 357 / 40$ , $s (86) \geq 1873 / 200$ , $s (87) \geq 941 / 100$ , $s (88) \geq 3791 / 400$ , $s (89) \geq 1913 / 200$ , $s (90) \geq 3831 / 400$ , $s (93) \geq 3889 / 400$ , $s (94) \geq 1961 / 200$ and $s (95) \geq 49259 / 5000$ .

Each is an exact nonnegative rational density on rectangles, D4-expanded, of mass n - 1/100, at core side 9977/10000 on 201 net half-angles of step 83/40000, built with Tokoharu's solver, its weights scaled to that mass. The source reports each accepted at all 201 net directions by Tokoharu's unchanged interval checker, verify.cpp, and replayed in full from its published files on a second machine before publication.

Each certificate was decided here by sqverify-fast, this repository's clean-room measure verifier, at all 201 net directions of the retained candidate at Tokoharu's threshold 10001/10000, in the census of 3 October 2026, and on 6 October two mutants of each scaled below that threshold 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.

Thirty of the certificates, all but $n = 66$ , 86, 87 and 90, were also replayed here in full with the source's own checker on 2 October, each reproducing the upstream accepting run at every direction; 29 of them are registered again in T-074, and $n = 76$ 's in none. The case records decide which certificate is current at each count.

Hosono after Tokoharu and Levy, square-packing-bounds. Registration was requested in jlevy/squares#281. The source says parts of the work were produced with AI assistance under human direction.

Significance, composition and next rung

Significance: The strongest rectangle-density lower bounds on record at 34 counts when published, each above the certificate T-046 reports there; through T-074's replays, 29 of them raised the verified lower bound. Further sizes and raised rungs from one generator at S3. The 2 October review of these certificates confirmed S3. Confirmed on 11 October beside T-045, T-070, T-074 and T-077, further certificates of T-047's kind at S3.
Composition: One claim per count, each on its own certificate at one source revision, the packet's 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, on the census route for format T that the review of 6 October set and these certificates meet. The source's checker was also replayed here on 30 of them, a reproduction with the producer's code beside the rung, recorded in the packet's receipts and, for 29, under T-074; no second method stands beside it.
Next rung: V4 and C4 need two adversarial AI reviews of this result by distinct reviewers and a human oversight record: the 2 October review of these certificates is one, and the 6 October review of the format T census route decided the route, not these certificates' mathematics. A complete replay of the source's own checker on the four certificates it has not run on, rect_n66_L8385, rect_n86_L9365, rect_n87_L941 and rect_n90_L95775 (10.7 CPU-hours by the upstream per-angle times), would add a reproduction with the producer's code beside the rung for those counts; 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 34 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		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	6.718000	6.828428	0.11042712…	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
66	8.430000	8.656855	0.22685424…		frontier n-066.md
68	8.530000	8.798796	0.26879523…		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		0.06		frontier n-075.md
76	8.965000		0.035		frontier n-076.md
86	9.503000	9.820536	0.31753540…		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
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

76 results in the register on these cases, oldest first

2005 published T-007 · 34 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-097, T-103, T-104, T-105, T-106, T-107, T-108, T-109, T-110, T-111, T-157 and T-205
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
$s (29) \leq 5.93383346267692918974 \dots$ , by a Krawczyk interval certificate
Upper bound confirmed superseded by T-195
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 · 34 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
2026-09-05 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 · 12 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, T-091, T-105, T-108, T-110, T-111, T-157 and T-205
Hosono 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, Hosono, 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 and T-155
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 and T-155
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 · 10 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-091, T-103, T-104, T-105, T-106, T-107, T-109 and T-157
Hosono after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 rectangle bounds 2026 · packet · packet · source · review · register
2026-09-27 published T-046 · 32 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-097, T-103, T-104, T-105, T-106, T-107, T-108, T-109, T-110, T-111, T-157 and T-205
Hosono 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-174
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, T-111 and T-205
Hosono after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 rectangle bounds 2026-09-28 · packet · packet · source · review · register
2026-09-29 published T-058 · 34 of these cases
Rectangle-certificate ceiling α·UB(n) proved for $n = 1$ ..100; B·UB(n) on 64 grid rows
Method limit confirmed
Hosono after Tokoharu, Daniel · wand125 tools 2026 · packet · register
2026-09-29 published T-083 · 34 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-097, T-103, T-104, T-105, T-106, T-107, T-108, T-109, T-110, T-111, T-157 and T-205
Karakuş · Karakuş 2026 · source · register
2026-10-01 published T-068 this result · 34 of these cases
Rectangle-density lower bounds verified at 34 counts in $n = 19 \dots 95$
Lower bound confirmed
Hosono after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 rectangle bounds 2026-10-01 · packet · source · review · register
2026-10-01 published T-069 · cases 66, 90
Mixed rectangle-measure lower bounds replayed at $n = 37, 65, 66, 90, 92$
Lower bound confirmed on these cases, superseded by T-091 and T-097
Hosono after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 mixed bounds 2026-10-01 · packet · source · review · register
2026-10-01 published T-074 · 29 of these cases
Rectangle-density lower bounds replayed at 31 counts in $n = 19 \dots 95$
Lower bound confirmed
Hosono 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
Hosono 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
Hosono 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 · cases 86, 87, 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
Hosono 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
Hosono 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
Hosono 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
Hosono after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 mixed bounds 2026-10-04 · packet · source · review · register
2026-10-04 published T-091 · 11 of these cases
Mixed rectangle-measure lower bounds verified at 17 counts in $n = 53 \dots 95$
Lower bound confirmed
Hosono after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 mixed bounds evening 2026-10-04 · packet · source · review · register
2026-10-05 published T-097 · case 66
Mixed rectangle-measure lower bound verified at $n = 66$
Lower bound confirmed
Hosono after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 mixed bounds finer net 2026-10-05 · 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-174
Daniel after Couzo, de Winter, Ellsworth, Levy · evand exact optima 2026-10-05 · packet · register
2026-10-05 published T-101 · 10 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
Hosono 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
Hosono 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
Hosono 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
Hosono 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
Hosono 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
Hosono 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
Hosono 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
Hosono 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
Hosono 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
Hosono 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 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 superseded by T-174
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, 86, 88
Complete rational construction reports at 25 counts
Upper bound confirmed superseded by T-127, T-146 and T-174
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 Couzo
Upper bound confirmed superseded by T-174
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 superseded by T-174
Polkovskyi after Stenlund, Friedman, Ellsworth, Chaoweeraprasit, Couzo, Xu, Levy · Mishapolk decimal poses 2026-10-09 · 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
Hosono 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-157
Guzhou0806 after Hosono, 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 Xu's packing of 70 squares
Upper bound confirmed superseded by T-146
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
Hosono 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
Hosono 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 superseded by T-157
Guzhou0806 after Hosono, 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 Daniel's regularized record lists
Upper bound confirmed
Daniel after Xu, Chaoweeraprasit, Polkovskyi, Levy · Daniel regularized lists 2026-10-10 · packet · register
2026-10-11 published T-157 · case 40
$s (40) > 3359 / 500 = 6.718$ , by T-145's continuous-pose kernel on a density with 163 new supports
Lower bound confirmed
Guzhou0806 after Hosono, Tokoharu, Levy, Stromquist, Burns, Massaccesi · Guzhou0806 n40 new supports 2026-10-11 · packet · source 1 · source 2 · review · register
2026-10-12 established T-155 · 9 of these cases
Exact regularized sides at 59 counts, 5.4e-20 to 9.9e-15 below the ceilings they replace
Upper bound confirmed
Levy after Daniel · register
2026-10-13 published T-174 · cases 68, 70, 86, 88
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-176 · case 39
Mixed rectangle-measure lower bound $s (39) \geq 533 / 80$ reported, on a 2,073-direction declared net
Lower bound reviewed
Hosono after Tokoharu and Levy · wand125 fine-net n37 n39 n44 2026-10-13 · packet · register
2026-10-13 published T-177 · case 44
Mixed rectangle-measure lower bound $s (44) \geq 2799 / 400$ reported, on a 2,073-direction declared net
Lower bound reviewed
Hosono after Tokoharu and Levy · wand125 fine-net n37 n39 n44 2026-10-13 · packet · register
2026-10-13 published T-180 · 19 of these cases
Closed-form lower bound $s (N) \geq a (N)$ for every non-square $N \geq 49$ , reported
Lower bound reviewed on these cases, superseded by T-082, T-090, T-091, T-097 and T-205
Hosono · wand125 closed form 2026-10-14 · packet · packet · register
2026-10-14 published T-184 · case 76
$s (76) = 9$ , by a mixed cover of points and grid-line segments
Optimality reviewed
Greff after Daniel, Hosono · bengreff s76 certificate 2026-10-14 · packet · source · review · register
2026-10-15 published T-191 · case 95
$s (95) = 10$ , by a mixed cover of points and grid-line segments
Optimality reviewed
Greff after Daniel, Hosono · bengreff s95 certificate 2026-10-15 · packet · source · review · register
2026-10-15 published T-192 · case 28
Continuous-pose lower bound $s (28) > 1149 / 200$ reported, by two of the source's programs
Lower bound recorded
Hosono after Guzhou0806, Tokoharu, Levy · wand125 continuous n28 2026-10-15 · packet · register
2026-10-15 published T-195 · case 29
A tighter rational certificate of Schadt and Ellsworth's 29-square packing, 1.57e-36 below T-155
Upper bound confirmed
PingYou Ltd after Schadt, Ellsworth, Daniel, Levy · PingYou PR 622 certificate 2026-10-15 · packet · register
2026-10-16 published T-193 · case 88
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
2026-10-16 published T-194 · case 19
Fitzharding reports a Lean proof that Wainwright's packing is optimal, $s (19) = 3 + 4 \sqrt{2} / 3$
Optimality reviewed
Fitzharding after Levy, Ahmed · Fitzharding n19 Lean 2026 · packet · source · review · register
2026-10-16 published T-200 · case 38
Mixed rectangle-measure lower bound $s (38) \geq 2631 / 400$ reported, on a 2,073-direction declared net
Lower bound recorded
Hosono after Tokoharu and Levy · wand125 fine-net lower bounds 2026-10-16 · packet · register
2026-10-16 published T-201 · case 40
Mixed rectangle-measure lower bound $s (40) \geq 1347 / 200$ reported, on a 2,073-direction declared net
Lower bound recorded
Hosono after Tokoharu and Levy · wand125 fine-net lower bounds 2026-10-16 · packet · register
2026-10-16 published T-202 · case 41
Mixed rectangle-measure lower bound $s (41) \geq 2721 / 400$ reported, on a 2,073-direction declared net
Lower bound recorded
Hosono after Tokoharu and Levy · wand125 fine-net lower bounds 2026-10-16 · packet · register
2026-10-16 published T-203 · case 43
Mixed rectangle-measure lower bound $s (43) \geq 2777 / 400$ reported, on a 2,073-direction declared net
Lower bound recorded
Hosono after Tokoharu and Levy · wand125 fine-net lower bounds 2026-10-16 · packet · register
2026-10-16 published T-205 · case 68
Mixed rectangle-measure lower bound $s (68) \geq 853 / 100$ verified, on the 201-direction net
Lower bound confirmed
Hosono after Tokoharu and Levy · wand125 fine-net lower bounds 2026-10-16 · packet · source · review · register
2026-10-16 published T-208 · case 86
Mixed rectangle-measure lower bound $s (86) \geq 3811 / 400$ reported, on the 201-direction net
Lower bound recorded
Hosono after Tokoharu and Levy · wand125 fine-net lower bounds 2026-10-16 · packet · register
2026-10-16 published T-209 · case 88
Mixed rectangle-measure lower bound $s (88) \geq 3853 / 400$ reported, on the 201-direction net
Lower bound recorded
Hosono after Tokoharu and Levy · wand125 fine-net lower bounds 2026-10-16 · packet · register
2026-10-16 published T-210 · case 89
Mixed rectangle-measure lower bound $s (89) \geq 1937 / 200$ reported, on the 201-direction net
Lower bound recorded
Hosono after Tokoharu and Levy · wand125 fine-net lower bounds 2026-10-16 · packet · register
2026-10-16 published T-212 · case 90
Mixed rectangle-measure lower bound $s (90) \geq 489 / 50$ reported, on the 201-direction net
Lower bound recorded
Hosono after Tokoharu and Levy · wand125 fine-net lower bounds 2026-10-16 · packet · register

Links

On this site: The frontier survey · T-068 in the results table

On GitHub, at main

Register: T-068 in results.yaml, line 6346
Evidence: E-wand125-rectangle-2026-10-01-report · E-n019-wand125-rect-48175-sqverify-fast-replay · E-n020-wand125-rect-48975-sqverify-fast-replay · E-n026-wand125-rect-55325-sqverify-fast-replay · E-n027-wand125-rect-5635-sqverify-fast-replay · E-n028-wand125-rect-57225-sqverify-fast-replay · E-n029-wand125-rect-57975-sqverify-fast-replay · E-n030-wand125-rect-5875-sqverify-fast-replay · E-n031-wand125-rect-59525-sqverify-fast-replay · E-n038-wand125-rect-6545-sqverify-fast-replay · E-n039-wand125-rect-6635-sqverify-fast-replay · E-n040-wand125-rect-67-sqverify-fast-replay · E-n041-wand125-rect-676-sqverify-fast-replay · E-n042-wand125-rect-6815-sqverify-fast-replay · E-n043-wand125-rect-68875-sqverify-fast-replay · E-n044-wand125-rect-69425-sqverify-fast-replay · E-n053-wand125-rect-76075-sqverify-fast-replay · E-n054-wand125-rect-76725-sqverify-fast-replay · E-n055-wand125-rect-77125-sqverify-fast-replay · E-n056-wand125-rect-77825-sqverify-fast-replay · E-n066-wand125-rect-8385-sqverify-fast-replay · E-n068-wand125-rect-851-sqverify-fast-replay · E-n069-wand125-rect-8585-sqverify-fast-replay · E-n070-wand125-rect-8625-sqverify-fast-replay · E-n074-wand125-rect-88475-sqverify-fast-replay · E-n075-wand125-rect-89-sqverify-fast-replay · E-n076-wand125-rect-8925-sqverify-fast-replay · E-n086-wand125-rect-9365-sqverify-fast-replay · E-n087-wand125-rect-941-sqverify-fast-replay · E-n088-wand125-rect-94775-sqverify-fast-replay · E-n089-wand125-rect-9565-sqverify-fast-replay · E-n090-wand125-rect-95775-sqverify-fast-replay · E-n093-wand125-rect-97225-sqverify-fast-replay · E-n094-wand125-rect-9805-sqverify-fast-replay · E-n095-wand125-rect-98518-sqverify-fast-replay
Proofs and certificates: certificate certificates · certificate certified_candidate.json.gz · proof SOUNDNESS.md · audit review-2026-10-02-wand125-rectangle-bounds-t068.md · certificate certified_candidate.json.gz · certificate certified_candidate.json.gz · certificate certified_candidate.json.gz · certificate certified_candidate.json.gz · certificate certified_candidate.json.gz · certificate certified_candidate.json.gz · certificate certified_candidate.json.gz · certificate certified_candidate.json.gz · certificate certified_candidate.json.gz · certificate certified_candidate.json.gz · certificate certified_candidate.json.gz · certificate certified_candidate.json.gz · certificate certified_candidate.json.gz · certificate certified_candidate.json.gz · certificate certified_candidate.json.gz · certificate certified_candidate.json.gz · certificate certified_candidate.json.gz · certificate certified_candidate.json.gz · certificate certified_candidate.json.gz · certificate certified_candidate.json.gz · certificate certified_candidate.json.gz · certificate certified_candidate.json.gz · certificate certified_candidate.json.gz · certificate certified_candidate.json.gz · certificate certified_candidate.json.gz · certificate certified_candidate.json.gz · certificate certified_candidate.json.gz · certificate certified_candidate.json.gz · certificate certified_candidate.json.gz · certificate certified_candidate.json.gz · certificate certified_candidate.json.gz · certificate certified_candidate.json.gz · certificate certified_candidate.json.gz
Sources: wand125 rectangle bounds 2026-10-01 (its own site, retained copy)
Source packet: resources/web/wand125-rectangle-certificates-2026-10-01/README.md
Artifacts: 43 artifacts and controls in 7 directories. Complete artifact list in the result entry. docs/project/reviews packing/benchmarks/measure-verifier packing/benchmarks/measure-verifier/census packing/benchmarks/measure-verifier/census/2026-10-01 packing/devtools packing/resources/web/wand125-rectangle-certificates-2026-10-01 packing/tests
Case file: each case’s file is linked from its row above

T-068: Rectangle-density lower bounds verified at 34 counts in n=19…95n=19…95

The cases

Results on these cases

Links

T-068: Rectangle-density lower bounds verified at 34 counts in $n = 19 \dots 95$