T-046: Rectangle-density lower bounds reported for 48 counts in $n = 18 \dots 95$

V0 C0 lower bound recorded superseded by T-062, T-063, T-064, T-066, T-067, T-068, T-069, T-074, T-082, T-090, T-091, T-094, T-097, T-102, T-103, T-104, T-105, T-106, T-107, T-108, T-109, T-110, T-111 and T-145

2026-09-27 published · wand125 after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · 48 cases, $n = 18$ to 95

wand125/square-packing-bounds reports one standing rectangle-density certificate for each of 48 counts from $n = 18$ to $n = 95$ , added or raised between 26 and 28 September 2026.

Up to $n = 44$ the standing values are $s (18) \geq 939 / 200$ , $s (19) \geq 963 / 200$ , $s (20) \geq 979 / 200$ , $s (26) \geq 553 / 100$ , $s (27) \geq 28 / 5$ , $s (28) \geq 143 / 25$ , $s (29) \geq 579 / 100$ , $s (30) \geq 1173 / 200$ , $s (31) \geq 1187 / 200$ , $s (37) \geq 257 / 40$ , $s (38) \geq 327 / 50$ , $s (39) \geq 663 / 100$ , $s (40) \geq 1339 / 200$ , $s (41) \geq 1351 / 200$ , $s (42) \geq 679 / 100$ , $s (43) \geq 1373 / 200$ and $s (44) \geq 1387 / 200$ .

From $n = 51$ they are $s (51) \geq 2977 / 400$ , $s (52) \geq 1507 / 200$ , $s (53) \geq 1519 / 200$ , $s (54) \geq 3067 / 400$ , $s (55) \geq 771 / 100$ , $s (56) \geq 777 / 100$ , $s (57) \geq 1567 / 200$ , $s (58) \geq 789 / 100$ , $s (59) \geq 198 / 25$ , $s (60) \geq 397 / 50$ , $s (61) \geq 199 / 25$ , $s (66) \geq 67 / 8$ , $s (67) \geq 1691 / 200$ , $s (68) \geq 1699 / 200$ , $s (69) \geq 343 / 40$ , $s (70) \geq 431 / 50$ , $s (71) \geq 1737 / 200$ , $s (72) \geq 437 / 50$ , $s (73) \geq 439 / 50$ , $s (74) \geq 221 / 25$ , $s (75) \geq 889 / 100$ , $s (76) \geq 223 / 25$ , $s (77) \geq 223 / 25$ , $s (78) \geq 1791 / 200$ , $s (86) \geq 1871 / 200$ , $s (88) \geq 189 / 20$ , $s (89) \geq 191 / 20$ , $s (90) \geq 191 / 20$ , $s (91) \geq 1929 / 200$ , $s (94) \geq 1959 / 200$ and $s (95) \geq 49209 / 5000$ . Here $n = 77$ and $n = 90$ take the $n = 76$ and $n = 89$ certificates, whose masses lie below 77 and 90.

The certificates were built with Tokoharu's solver, their weights scaled by one exact rational factor to mass n - 1/100 (n - 1/1000 at $n = 27$ ) before the recorded run, and each was accepted there at every net direction by Tokoharu's unchanged interval checker, verify.cpp.

Here this repository's exact preflight checked that every checker input is the one the upstream run recorded, and checked the mass, net and partition premises. The values stand here as reported; those whose certificates have since been replayed in full are registered again as replayed results, in T-045 and T-070.

wand125 after Tokoharu and Levy, square-packing-bounds as of 27 September and 28 September 2026. The source says parts of the work were produced with AI assistance under human direction.

Significance, composition and next rung

Significance: Scored as a claim: if the replays pass, these are the strongest lower bounds on record at 48 counts, above Green's DS7 reports and every certificate the register holds at those sizes. The score is of what the reports would establish; V0/C0 says how far they stand, and the score gates nothing.
Composition: One claim per count, from two source revisions: the counts whose case cites E-wand125-rectangle-report keep the 27 September certificate, and those citing the 28 September entries the raised or new one. $n = 77$ and $n = 90$ are reported-lane transfers by the mass condition, which add no premise to the source claim they rest on. Every part stands at V0/C0.
Next rung: The complete 201-direction replay of the 18 standing certificates no replay here has carried. Twenty-one passed and were entered on 2026-10-02, and each count they carry is registered again as replayed while staying in this scope, as $n = 27$ did: $n = 18$ , 19, 20, 26, 30, 40, 61, 75 and 78, with 76 and 77 by mass, in T-045, and $n = 29$ , 38, 39, 41, 52, 53, 59, 67, 69, 71, 86 and 95, with 42 to 44, 54, 55, 60, 68, 70 and 72 to 74 by mass, in T-070. Six more passed on 3 October and were entered on 6 October: $n = 51$ , 57, 72 and 73 in T-070, and $n = 58$ and 91, which are below the mixed certificates that hold those counts.

Of the 18 left, the 1 October revision raised 16 (T-068), whose own certificates carry those counts, and $n = 37$ and 60 are held above these certificates by replayed bounds (T-069's $s (37) \geq 161 / 25$ and $s (60) = 8$ , T-062), so no replay of the source's checker is planned for them. sqverify-fast's census of 3 October verified all 48 standing certificates at all 201 directions; with control receipts and a review of the census route that reads them, it would carry every count of this entry. C1 before that, by recording the 2026-09-27 scaling review as an external_review on the report entries.
Unfinished confirmations: C1: the 2026-09-27 checker-scaling review recorded as an external_review on the report entries this result cites. Not priced.

C3: sqverify-fast's census of 3 October, which verified all 48 standing certificates at all 201 directions, recorded with its control receipts and a review of the census route that reads them. Not priced.
Novelty: previously-published Present in an identified source

The cases

This result concerns 48 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
18	4.705000	4.822876	0.11787565…	open=	frontier n-018.md
19	4.825000	4.885619	0.06061808…		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
37	6.440000	6.598620	0.15861960…		frontier n-037.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
51	7.470000	7.690356	0.22035593…		frontier n-051.md
52	7.550000	7.707107	0.15710678…		frontier n-052.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
59	8		0	provedO=	frontier n-059.md
60					frontier n-060.md
61					frontier n-061.md
66	8.430000	8.656855	0.22685424…	open=	frontier n-066.md
67	8.480000	8.707107	0.22710678…		frontier n-067.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
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
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
77	9		0	provedO=	frontier n-077.md
78	9		0	provedO=	frontier n-078.md
86	9.503000	9.820566	0.31756573…	open=	frontier n-086.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
94	9.950000		0.05		frontier n-094.md
95	9.965000		0.035		frontier n-095.md

Results on these cases

73 results in the register on these cases, oldest first

2005 published T-007 · 48 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-062, T-063, T-064, T-066, T-067, T-068, T-069, T-074, T-082, T-090, T-091, T-094, T-097, T-102, 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
2018 published T-087 · cases 37, 61
$s (37) \geq 5 \sqrt{3} / 2 + 2 \sqrt{2} - 1$ and $s (61) \geq 7 \sqrt{3} / 2 + 2 \sqrt{2} - 1$ , by optimal piercing
lower bound reviewed superseded by T-063 and T-069
Bašić, Slivková · Basic-Slivkova 2018 · source · register
2026-08-21 published T-016 · cases 18, 19
$s (n) \geq 22529 / 5000$ for $n = 18, 19$ , by monotonicity from T-015
lower bound confirmed superseded by T-102 and 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
2026-08-31 established T-002 · case 18
$s (18) \geq 4426213 / 1000000$ , by monotonicity from T-001
lower bound confirmed superseded by T-102
Levy after Bentz · register
2026-08-31 established T-003 · case 18
The sixteen-point set's unavoidability ceiling lies in $[4426213 / 1000000, 4427 / 1000)$
method limit confirmed
Levy after Bentz · register
2026-09-04 established T-019 · cases 18, 19
$s (n) \geq 459 / 100 = 4.59$ for $n = 17, 18, 19$
lower bound confirmed superseded by T-102 and 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 · 48 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
2026-09-18 established T-027 · case 18
$s (18) \geq 467 / 100 = 4.67$
lower bound confirmed superseded by T-102
Levy after Burns, Massaccesi · register
2026-09-19 established T-028 · case 18
$s (18) \geq 187 / 40 = 4.675$
lower bound confirmed superseded by T-102
Levy after Burns, Massaccesi · register
2026-09-19 established T-029 · case 18
$s (18) \geq 1871 / 400 = 4.6775$
lower bound confirmed superseded by T-102
Levy after Burns, Massaccesi · register
2026-09-19 established T-030 · case 18
$s (18) \geq 4679 / 1000 = 4.679$
lower bound confirmed superseded by T-102
Levy after Burns, Massaccesi · register
2026-09-22 published T-044 · 17 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 · 14 of these cases
Rectangle-density lower bounds replayed at 15 counts in $n = 18 \dots 78$
lower bound confirmed superseded by T-063, T-064, T-067, T-068, T-074, T-090, T-091, T-102, 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 this result · 48 of these cases
Rectangle-density lower bounds reported for 48 counts in $n = 18 \dots 95$
lower bound recorded superseded by T-062, T-063, T-064, T-066, T-067, T-068, T-069, T-074, T-082, T-090, T-091, T-094, T-097, T-102, 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-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-062 · case 60
$s (60) = 8$ , by a mixed cover of points and grid-line segments
optimality confirmed
Daniel after Burns, Massaccesi · evand square-packing 2026-10-01 · packet · source · review · register
2026-09-28 published T-063 · case 61
$s (61) = 8$ , by monotonicity from T-062
optimality confirmed
Daniel after Burns, Massaccesi · evand square-packing 2026-10-01 · packet · packet · packet · source 1 · source 2 · review 1 · review 2 · register
2026-09-28 published T-070 · 25 of these cases
Rectangle-density lower bounds replayed at 25 counts in $n = 29 \dots 95$
lower bound confirmed superseded by T-062, T-066, T-068, T-074, T-082, T-090, T-091, T-094, 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 · 48 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 · 48 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-062, T-063, T-064, T-066, T-067, T-068, T-069, T-074, T-082, T-090, T-091, T-094, T-097, T-102, 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
2026-09-30 published T-064 · cases 61, 78
$s (k^{2} - 3) = k$ for every integer $k$ from 6 up; $k = 6 \dots 18$ are the cases held here
optimality confirmed on these cases, current best, reported
Daniel after Burns, Massaccesi · evand square-packing 2026-10-01 · packet · packet · packet · packet · source 1 · source 2 · review 1 · review 2 · register
2026-10-01 published T-066 · case 59
$s (59) = 8$ , by a mixed cover of points and grid-line segments
optimality confirmed
wand125 after Daniel, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 exact covers 2026-10-01 · packet · source · review · register
2026-10-01 published T-067 · cases 77, 78
$s (77) = 9$ , by a mixed cover extended from Daniel's $s (60)$ cover
optimality confirmed
wand125 after Daniel, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 exact covers 2026-10-01 · packet · source · review · register
2026-10-01 published T-068 · 32 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-069 · cases 37, 66, 90
Mixed rectangle-measure lower bounds replayed at $n = 37, 65, 66, 90, 92$
lower bound confirmed
wand125 after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 mixed bounds 2026-10-01 · packet · source · review · register
2026-10-01 published T-074 · 30 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-071 · case 86
Mixed rectangle-measure lower bounds replayed at $n = 84 \dots 87$
lower bound confirmed superseded by T-090
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 · cases 86, 88, 91
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-081 · cases 60, 77
$s (k^{2} - 4) = k$ for every integer $k$ from 5 up; $k = 5 \dots 18$ are the cases held here
optimality reviewed on these cases, second certificate, reported
Daniel after Burns, Massaccesi · evand square-packing 2026-10-03 · packet · packet · source · review · register
2026-10-03 published T-082 · 18 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 · 14 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-094 · case 67
Mixed rectangle-measure lower bounds verified at $n = 67$ and $84$
lower bound confirmed
wand125 after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 mixed bounds 2026-10-05 · packet · source · review · register
2026-10-05 published T-096 · case 18
Mixed rectangle-measure lower bound replayed at $n = 18$ , on a net the certificate declares
lower bound confirmed superseded by T-102
wand125 after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 mixed bounds finer net 2026-10-05 · packet · source · review · register
2026-10-05 published T-097 · case 66
Mixed rectangle-measure lower bound verified at $n = 66$
lower bound confirmed
wand125 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-118
Daniel after Couzo, de Winter, Ellsworth, Levy · evand exact optima 2026-10-05 · packet · register
2026-10-05 published T-101 · 12 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-099 · case 18
Mixed rectangle-measure lower bound verified at $n = 18$ , on a finer declared net
lower bound confirmed superseded by T-102
wand125 after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 mixed bounds finer net 2026-10-06 · packet · source 1 · source 2 · review · 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-102 · case 18
Mixed rectangle-measure lower bound verified at $n = 18$ , 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-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 · 24 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-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-046 in the results table

On GitHub, at main

Register: T-046 in results.yaml, line 3973
Evidence: E-wand125-rectangle-report · E-wand125-rectangle-2026-09-28-report · E-wand125-rectangle-2026-09-28-monotone-report
Proofs and certificates: certificate certificates · certificate certificates
Sources: wand125 rectangle bounds 2026 (its own site, retained copy) · wand125 rectangle bounds 2026-09-28 (its own site, retained copy)
Source packet: resources/web/wand125-rectangle-certificates-2026-09-27/README.md · resources/web/wand125-rectangle-certificates-2026-09-28/README.md
Artifacts: 5 artifacts and controls
resources/web/wand125-rectangle-certificates-2026-09-27/README.md resources/web/wand125-rectangle-certificates-2026-09-28/README.md resources/web/wand125-rectangle-certificates-2026-09-28/receipts/preflight/audit.json.gz devtools/audit_wand125_rectangles.py devtools/apply_wand125_rectangles.py
Case file: each case’s file is linked from its row above

T-046: Rectangle-density lower bounds reported for 48 counts in n=18…95n=18…95

The cases

Results on these cases

Links

T-046: Rectangle-density lower bounds reported for 48 counts in $n = 18 \dots 95$