T-070: Rectangle-density lower bounds replayed at 25 counts in $n = 29 \dots 95$

V3 C3 Lower bound confirmed superseded by T-062, T-066, T-068, T-074, T-082, T-090, T-091, T-094, T-108, T-110, T-111 and T-205

2026-09-28 published · Hosono after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · 25 cases, $n = 29$ to 95

Sixteen rectangle-density certificates in wand125/square-packing-bounds, added or raised on 27 and 28 September 2026, prove $s (29) \geq 579 / 100$ , $s (38) \geq 327 / 50$ , $s (39) \geq 663 / 100$ , $s (41) \geq 1351 / 200$ , $s (51) \geq 2977 / 400$ , $s (52) \geq 1507 / 200$ , $s (53) \geq 1519 / 200$ , $s (57) \geq 1567 / 200$ , $s (59) \geq 198 / 25$ , $s (67) \geq 1691 / 200$ , $s (69) \geq 343 / 40$ , $s (71) \geq 1737 / 200$ , $s (72) \geq 437 / 50$ , $s (73) \geq 439 / 50$ , $s (86) \geq 1871 / 200$ and $s (95) \geq 49209 / 5000$ . They are rect_n29_L579, rect_n38_L654, rect_n39_L663, rect_n41_L6755, rect_n51_L74425, rect_n52_L7535, rect_n53_L7595, rect_n57_L7835, rect_n59_L792, rect_n67_L8455, rect_n69_L8575, rect_n71_L8685, rect_n72_L874, rect_n73_L878, rect_n86_L9355 and rect_n95_L98418.

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. A certificate's mass is below every larger count too, which gives nine more, 25 counts in all: $s (42), s (43), s (44) \geq 1351 / 200$ from $n = 41$ , $s (54), s (55) \geq 1519 / 200$ from $n = 53$ , $s (60) \geq 198 / 25$ from $n = 59$ , $s (68) \geq 1691 / 200$ from $n = 67$ , $s (70) \geq 343 / 40$ from $n = 69$ , and $s (74) \geq 439 / 50$ from $n = 73$ .

Each passed a complete 201-direction replay here of Tokoharu's unchanged verify.cpp, twelve in cloud batches on 29 September 2026 whose receipts were merged into the packet on 2 October, and rect_n51_L74425, rect_n57_L7835, rect_n72_L874 and rect_n73_L878 in batches on 3 October merged on 6 October; 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.

Two more certificates replayed in the 3 October batches, rect_n58_L789 and rect_n91_L9645, are below the mixed certificates that hold $n = 58$ and 91 (T-091) and are not registered here.

Hosono, 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 23 counts from $n = 29$ to 95, by 0.004 to 0.34, over Hosono's weighted point certificates (T-044), Tokoharu's $n = 29$ certificate (T-047) and Nagamochi's closed form, on Tokoharu's method with larger certificates, and on 6 October at $n = 51$ , 57 and 72, by 0.0425, 0.0525 and 0.019 over the mixed certificates of T-048 and T-091 and T-074's $n = 56$ certificate. Substantive case results, S3, as T-045. The closed form has been a reported bound since Karakuş's finding (T-085, merged 2026-10-03). Confirmed on 11 October beside T-045, T-068 and T-074, further certificates of T-047's kind at S3.
Composition: Sixteen primary certificates. The nine other counts take a smaller count's certificate directly, its mass being below them, 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, 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.

At $n = 59$ and 60 superseded on 2026-10-02 by the exact values $s (59) = 8$ (T-066) and $s (60) = 8$ (T-062), when their covers' replays landed, at $n = 86$ the same day by $s (86) \geq 471 / 50$ from the replayed $n = 85$ mixed certificate (T-071), and at $n = 29$ , 38, 39, 41 to 44, 53 to 55, 68 to 70, 74 and 95 by Hosono's raised certificates of 1 October (T-074), when their replays were merged; stays true as stated. The counts it took on the morning of 6 October, $n = 51$ , 57 and 72, went later that day to T-090's mixed certificates when sqverify-fast decided them, and $n = 52$ , the last count it held, to T-082's the same day.
Novelty: previously-published Present in an identified source

The cases

This result concerns 25 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
29	5.810000	5.933834	0.12383346…	open=	frontier n-029.md
38	6.545000	6.707107	0.16210678…		frontier n-038.md
39	6.650000	6.810723	0.16072208…		frontier n-039.md
41	6.775000	6.926694	0.15169309…		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
57	7.872500	8	0.1275	open=	frontier n-057.md
59	8		0	provedO=	frontier n-059.md
60	8		0	provedO=	frontier n-060.md
67	8.480000	8.707107	0.22710678…	open=	frontier n-067.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
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
86	9.503000	9.820536	0.31753540…		frontier n-086.md
95	9.965000	10	0.035		frontier n-095.md

Results on these cases

51 results in the register on these cases, oldest first

2005 published T-007 · 25 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-066, T-068, T-074, T-082, T-090, T-091, T-094, T-108, T-110, T-111 and T-205
Nagamochi · Nagamochi 2005 · source · 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
2026-09-04 published T-085 · 25 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
2026-09-22 published T-044 · 14 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-108, T-110, T-111 and T-205
Hosono after Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 point bounds 2026 · packet · source · review · register
2026-09-22 published T-047 · case 29
$s (11) \geq 381 / 100$ ; $s (n) \geq 1377 / 250$ for $n = 26 \dots 28$ ; $s (n) \geq 571 / 100$ for $n = 29 \dots 31$
Lower bound confirmed superseded by T-108
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
2026-09-27 published T-046 · 25 of these cases
Rectangle-density lower bounds reported for 48 counts in $n = 18 \dots 95$
Lower bound recorded superseded by T-062, T-066, T-068, T-074, T-082, T-090, T-091, T-094, T-108, T-110, T-111 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-048 · case 51
$s (50) \geq 37 / 5 = 7.4$
Lower bound confirmed on these cases, superseded by T-090
Hosono 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-070 this result · 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, 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 · 25 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 · 25 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-066, T-068, T-074, T-082, T-090, T-091, T-094, T-108, T-110, T-111 and T-205
Karakuş · Karakuş 2026 · source · register
2026-10-01 published T-066 · case 59
$s (59) = 8$ , by a mixed cover of points and grid-line segments
Optimality confirmed
Hosono after Daniel, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 exact covers 2026-10-01 · packet · source · review · register
2026-10-01 published T-068 · 16 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-074 · 16 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 · case 86
Mixed rectangle-measure lower bounds replayed at $n = 84 \dots 87$
Lower bound confirmed superseded by T-090
Hosono after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 mixed bounds 2026-10-02 · packet · source · review · register
2026-10-02 published T-075 · case 86
Mixed rectangle-measure lower bounds replayed at nine counts in $n = 83 \dots 96$
Lower bound confirmed on these cases, superseded by T-090
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 42, 70
Rectangle-density lower bounds replayed at $n = 20$ , 42 and 70
Lower bound confirmed superseded by T-090 and T-091
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-081 · case 60
$s (k^{2} - 4) = k$ 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 · 10 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 · 10 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 · 7 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-094 · case 67
Mixed rectangle-measure lower bounds verified at $n = 67$ and $84$
Lower bound confirmed
Hosono after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 mixed bounds 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 · 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-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-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-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 · 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
2026-10-08 published T-125 · cases 51, 70, 86
Complete rational construction reports at 25 counts
Upper bound confirmed superseded by T-126, T-146 and T-174
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
Xu · ry-xu square packing 2026 · 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-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-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
2026-10-12 established T-155 · 8 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
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 · 18 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-062, T-066, T-082, T-090, T-091, T-094 and T-205
Hosono · wand125 closed form 2026-10-14 · packet · packet · 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-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-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-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-204 · case 51
Mixed rectangle-measure lower bound $s (51) \geq 3011 / 400$ 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

Links

On this site: The frontier survey · T-070 in the results table

On GitHub, at main

Register: T-070 in results.yaml, line 6653
Evidence: E-wand125-rectangle-2026-09-28-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 2026-09-28 (its own site, retained copy)
Source packet: resources/web/wand125-rectangle-certificates-2026-09-28/README.md · resources/web/external-square-certificates-2026-09-22/README.md
Artifacts: 12 artifacts and controls
resources/web/wand125-rectangle-certificates-2026-09-28/README.md resources/web/wand125-rectangle-certificates-2026-09-28/receipts/replay/audit.json resources/web/wand125-rectangle-certificates-2026-09-28/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-2026-09-27-wand125-rectangle-scaling.md docs/project/reviews/review-2026-10-02-wand125-rectangle-bounds-t068.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-070: Rectangle-density lower bounds replayed at 25 counts in n=29…95n=29…95

The cases

Results on these cases

Links

T-070: Rectangle-density lower bounds replayed at 25 counts in $n = 29 \dots 95$