T-045: Rectangle-density lower bounds replayed at 15 counts in $n = 18 \dots 78$

V3 C3 lower bound confirmed superseded by T-051, T-063, T-064, T-067, T-068, T-074, T-090, T-091, T-102, T-103, T-104, T-105, T-106, T-107, T-109 and T-145

2026-09-27 published · wand125 after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · 15 cases, $n = 18$ to 78

Twelve rectangle-density certificates in wand125/square-packing-bounds, added on 26 and 27 September 2026, prove $s (18) \geq 939 / 200$ , $s (19) \geq 963 / 200$ , $s (20) \geq 979 / 200$ , $s (26) \geq 553 / 100$ , $s (27) \geq 28 / 5$ , $s (30) \geq 1173 / 200$ , $s (31) \geq 148 / 25$ , $s (32) \geq 119 / 20$ , $s (40) \geq 1339 / 200$ , $s (61) \geq 199 / 25$ , $s (75) \geq 889 / 100$ and $s (78) \geq 1791 / 200$ . They are rect_n18_L4695, rect_n19_L4815, rect_n20_L4895, rect_n26_L553, rect_n27_L56, rect_n30_L5865, rect_n31_L592, rect_n32_L595, rect_n40_L6695, rect_n61_L796, rect_n75_L889 and rect_n78_L8955.

Each is an exact nonnegative rational density on rectangles, D4-expanded, of mass n - 1/100 (n - 1/1000 at $n = 27$ ), such that the shrunken core of every rotated unit square at each of 201 net directions captures mass at least one. The $n = 27$ certificate's mass 26.999 is also below 28, and the $n = 75$ certificate's 74.99 below 76 and 77, so they give $s (28) \geq 28 / 5$ and $s (76), s (77) \geq 889 / 100$ too: 15 counts in all.

Each passed a complete 201-direction replay here of Tokoharu's unchanged verify.cpp: $n = 27$ , 31 and 32 on 27 September 2026, and the other nine in cloud batches on 29 September, whose receipts were merged into the packet on 2 October. 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.

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 $n = 27$ and 28 by 0.092 and at $n = 31$ by 0.21 over Tokoharu's values (T-047), on Tokoharu's method with larger certificates. Substantive case results, S3.
Composition: Twelve primary certificates. $n = 28$ takes the $n = 27$ certificate and $n = 76$ and 77 the $n = 75$ one directly, their masses being below those counts, so no step is added. The $n = 32$ part rests on the complete replay receipt retained beside the others (receipts/replay/rect_n32_L595), which E-wand125-rectangle-source-replay's limitations name; that entry's scope lists only the counts whose verified lower bound it carries, and $n = 32$ never held one. One interval-certified method, replayed here: C3.
Next rung: V4 and C4 need two adversarial AI reviews by distinct reviewers and a human oversight record, and none is retained. A second machine method would be a method-distinct decision of rotated coverage; global coverage is still decided by the source's C++ checker alone, and the exact preflight does not decide it.

Every certificate of this revision that T-046 reports is now replayed and registered here, and only $n = 18$ still holds this entry's value. At $n = 32$ superseded on intake, 2026-09-27, by Evan Daniel's $s (32) = 6$ (T-051), before it held the verified field; at $n = 61$ on 2026-10-02 by $s (61) = 8$ (T-063), when the $s (60)$ cover's replay landed; on the same day at $n = 76$ by $s (76) \geq 447 / 50$ (T-072) and at $n = 77$ and 78 by $s (77) = 9$ and $s (78) = 9$ (T-067), when their replays landed; and at $n = 19$ , 20, 26 to 28, 30, 31, 40 and 75 by wand125's raised certificates of 1 October (T-074), when their replays were merged. Stays true as stated.
Novelty: previously-published Present in an identified source

The cases

This result concerns 15 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
30	5.883500	6	0.1165	open=	frontier n-030.md
31	5.952500		0.0475	open=	frontier n-031.md
32	6		0	provedO=	frontier n-032.md
40	6.715815	6.828428	0.11261137…	open=R	frontier n-040.md
61	8	8	0	provedO=	frontier n-061.md
75	8.940000	9	0.06	open=	frontier n-075.md
76	8.965000		0.035	open=	frontier n-076.md
77	9		0	provedO=	frontier n-077.md
78	9		0	provedO=	frontier n-078.md

Results on these cases

48 results in the register on these cases, oldest first

2005 published T-007 · 15 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-051, 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
Nagamochi · Nagamochi 2005 · source · register
2018 published T-087 · case 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
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
2026-08-30 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 · 15 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 · cases 26, 40
Weighted point lower bounds for ten counts in $n = 26 \dots 72$ , plus seven from the same files
lower bound confirmed superseded by T-105 and T-145
wand125 after Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 point bounds 2026 · packet · source · review · register
2026-09-22 published T-047 · 5 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 and T-109
Tokoharu after Levy, wand125, Stromquist, Nagamochi, Burns, Massaccesi · Tokoharu density 2026 · packet · source · review · register
2026-09-26 published T-051 · case 32
$s (32) = 6$
optimality confirmed
Daniel after Burns, Massaccesi · evand square-packing 2026 · packet · packet · packet · packet · source 1 · source 2 · review 1 · review 2 · review 3 · register
2026-09-27 published T-045 this result · 15 of these cases
Rectangle-density lower bounds replayed at 15 counts in $n = 18 \dots 78$
lower bound confirmed superseded by T-051, 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 · 14 of these cases
Rectangle-density lower bounds reported for 48 counts in $n = 18 \dots 95$
lower bound recorded 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 · wand125 rectangle bounds 2026-09-28 · packet · packet · 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
2026-09-29 published T-058 · 15 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 · 15 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-051, 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
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-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 · 10 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 · 9 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-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-077 · case 20
Rectangle-density lower bounds replayed at $n = 20$ , 42 and 70
lower bound confirmed superseded by 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 32, 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 · cases 75, 76
Mixed rectangle-measure lower bounds verified at 22 counts in $n = 51 \dots 96$
lower bound confirmed on these cases, superseded by T-090 and T-091
wand125 after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 mixed bounds 2026-10-03 · packet · packet · source · review · register
2026-10-04 published T-090 · case 75
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 · case 76
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-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-101 · case 28
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-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
2026-10-07 published T-124 · 10 of these cases
Reported non-strict local minima for 178 source configurations
restricted optimality recorded
Daniel after Couzo · Daniel exact and local reports 2026 · packet · register
2026-10-10 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-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

Links

On this site: The frontier survey · T-045 in the results table

On GitHub, at main

Register: T-045 in results.yaml, line 3874
Evidence: E-wand125-rectangle-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 (its own site, retained copy)
Source packet: resources/web/wand125-rectangle-certificates-2026-09-27/README.md · resources/web/external-square-certificates-2026-09-22/README.md
Artifacts: 9 artifacts and controls
resources/web/wand125-rectangle-certificates-2026-09-27/README.md resources/web/wand125-rectangle-certificates-2026-09-27/receipts/replay/audit.json resources/web/external-square-certificates-2026-09-22/tokoharu-density/src/verify.cpp devtools/audit_wand125_rectangles.py docs/project/reviews/review-2026-09-27-wand125-rectangle-scaling.md docs/project/reviews/review-2026-09-22-tokoharu-density-mathematics.md tests/test_wand125_rectangle_audit.py tests/test_wand125_rectangle_merge.py tests/test_tokoharu_density_audit.py
Case file: each case’s file is linked from its row above

T-045: Rectangle-density lower bounds replayed at 15 counts in n=18…78n=18…78

The cases

Results on these cases

Links

T-045: Rectangle-density lower bounds replayed at 15 counts in $n = 18 \dots 78$