T-044: Weighted point lower bounds for ten counts in $n = 26 \dots 72$ , plus seven from the same files

V3 C3 Lower bound confirmed superseded by T-082, T-090, T-091, T-105, T-108, T-110, T-111, T-157 and T-205

2026-09-22 published · Hosono after Levy, Stromquist, Nagamochi, Burns, Massaccesi · 17 cases, $n = 26$ to 73

Ten exact weighted point certificates in wand125/square-packing-bounds, of 22 September 2026, prove $s (26) \geq 109 / 20$ , $s (29) \geq 557 / 100$ , $s (39) \geq 13 / 2$ , $s (40) \geq 13 / 2$ , $s (53) \geq 369 / 50$ , $s (55) \geq 377 / 50$ , $s (56) \geq 381 / 50$ , $s (69) \geq 841 / 100$ , $s (70) \geq 171 / 20$ and $s (72) \geq 861 / 100$ .

Four more counts follow from the same files: $s (41) \geq 13 / 2$ from the $n = 39$ file and $s (71) \geq 171 / 20$ from the $n = 70$ file by monotonicity, and $s (52) \geq 369 / 50$ and $s (68) \geq 841 / 100$ because the $n = 53$ and $n = 69$ files have exact mass below 52 and 68, and only the mass condition mentions the count.

Three more follow by monotonicity since 2 October 2026: $s (54) \geq 369 / 50$ from the $n = 53$ file, $s (57) \geq 381 / 50$ from the $n = 56$ file and $s (73) \geq 861 / 100$ from the $n = 72$ file. Nagamochi's closed form, stronger at those counts, had held their verified fields until its published proof was found incomplete (T-085).

Every file was replayed here in full on 22 September 2026 by this repository's exact verifier, sqpack.fractional.certificate, over the complete 201-direction net through the source's adapter check_with_sqpack.py.

Hosono, square-packing-bounds, building on this project: the source says "The method is not ours", credits jlevy/squares for the generator, and checks with this project's verifier.

Significance, composition and next rung

Significance: The first certificate-backed lower bounds at twelve counts from 39 to 72, above the Green values DS7 has reported without proof since 2009 and above Nagamochi's closed form by 0.02 to 0.21. A substantive family of case results at S3; the generator and verifier are this project's, run by someone else, so it is not S4's new technique. Confirmed on 11 October against T-019 and T-020 (S4): their anchor was the displacement of published values by the first certificates of a new generator, and T-020 itself set further sizes from it at S3; ten certificates at ten counts are further sizes, whoever runs the generator, and a bound family in the S4 sense is one argument at every count (T-064, T-081), not many certificates (T-045, T-046, T-056, all S3).
Composition: Ten primary certificates and seven derived counts. At $n = 41$ , 54, 57, 71 and 73 the derivation is monotonicity from the $n = 39$ , 53, 56, 70 and 72 files; at $n = 52$ and 68 it is the exact mass inequality on the $n = 53$ and $n = 69$ files, recorded as their own audited entries. Each step is one exact comparison and adds no premise, so every count keeps the replay's V3/C3. All five entries are exact-algebraic, which is one machine method.
Next rung: V4 and C4 need two adversarial AI reviews by distinct reviewers and a human oversight record, and none is retained. An interval-certified decision of the same ten files with this repository's fractional interval route, which decides weighted point certificates of this schema, would be a second machine method. No packing/tests control reads this packet; the control named is the verifier's own adversarial suite.

Hosono's replayed rectangle certificates superseded these as the verified lower bound on 2026-10-02 at $n = 40$ (T-045), at $n = 39$ , 41, 52, 53, 55 and 68 to 72 (T-070), and at $n = 56$ (T-074), the last count they held then. The three counts this entry took the same day when Karakuş's finding (T-085) moved Nagamochi's closed form out of the verified lane, $n = 54$ , 57 and 73, were superseded too, at $n = 54$ and 57 by T-074 and at $n = 73$ by T-070, as the two lines were merged on 2026-10-03; it holds none, and stays true as stated. At $n = 26$ and $n = 29$ superseded on intake, 2026-09-22, by Tokoharu's density certificates (T-047), before either held the verified field; stays true as stated.
Novelty: previously-published Present in an identified source

The cases

This result concerns 17 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
26	5.545000	5.621321	0.07632034…	open=	frontier n-026.md
29	5.810000	5.933834	0.12383346…		frontier n-029.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
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	8	0.1275		frontier n-057.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	9	0.187	open=	frontier n-073.md

Results on these cases

42 results in the register on these cases, oldest first

2005 published T-007 · 17 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-082, T-090, T-091, T-105, T-108, T-110, T-111, T-157 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
2026-08-30 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 published T-085 · 17 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 this result · 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-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 · cases 26, 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-105 and 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-045 · cases 26, 40
Rectangle-density lower bounds replayed at 15 counts in $n = 18 \dots 78$
Lower bound confirmed superseded by T-105 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 · 17 of these cases
Rectangle-density lower bounds reported for 48 counts in $n = 18 \dots 95$
Lower bound recorded superseded by T-082, T-090, T-091, T-105, T-108, 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 · 14 of these cases
Rectangle-density lower bounds replayed at 25 counts in $n = 29 \dots 95$
Lower bound confirmed superseded by 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 · 17 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 · 17 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-082, T-090, T-091, T-105, T-108, T-110, T-111, T-157 and T-205
Karakuş · Karakuş 2026 · source · register
2026-10-01 published T-068 · 12 of these cases
Rectangle-density lower bounds verified at 34 counts in $n = 19 \dots 95$
Lower bound confirmed on these cases, superseded by T-082, T-090, T-091, T-105, T-108, T-110, T-111, T-157 and T-205
Hosono after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 rectangle bounds 2026-10-01 · packet · source · review · register
2026-10-01 published T-074 · 13 of these cases
Rectangle-density lower bounds replayed at 31 counts in $n = 19 \dots 95$
Lower bound confirmed on these cases, superseded by T-082, T-090, T-091, T-105, T-108, T-110, T-111, T-157 and T-205
Hosono after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 rectangle bounds 2026-10-01 · packet · packet · source · review · register
2026-10-02 published T-077 · case 70
Rectangle-density lower bounds replayed at $n = 20$ , 42 and 70
Lower bound confirmed superseded by 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-082 · 6 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 · cases 56, 57, 69, 72
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 · 6 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-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 · 8 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-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-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 · cases 56, 57, 72, 73
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 · case 70
Complete rational construction reports at 25 counts
Upper bound confirmed superseded by T-146
Xu · ry-xu square packing 2026 · 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-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 · 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
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-180 · 12 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 and T-205
Hosono · wand125 closed form 2026-10-14 · packet · 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-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-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

Links

On this site: The frontier survey · T-044 in the results table

On GitHub, at main

Register: T-044 in results.yaml, line 3862
Evidence: E-wand125-point-bounds-report · E-wand125-point-monotone-report · E-wand125-point-source-replay · E-wand125-n052-derived-lower · E-wand125-n068-derived-lower
Proofs and certificates: certificate certificates · proof certificate.py · audit review-2026-09-22-tokoharu-density-mathematics.md · certificate cert_n53_L738.json · certificate cert_n69_L841.json
Sources: wand125 point bounds 2026 (its own site, retained copy)
Source packet: resources/web/external-square-certificates-2026-09-22/README.md
Artifacts: 7 artifacts and controls
resources/web/external-square-certificates-2026-09-22/README.md resources/web/external-square-certificates-2026-09-22/wand125-points/README.md resources/web/external-square-certificates-2026-09-22/wand125-points/src/check_with_sqpack.py resources/web/external-square-certificates-2026-09-22/receipts/wand125-points/sqpack-exact-replay.log src/sqpack/fractional/certificate.py docs/project/reviews/review-2026-09-22-external-square-certificates-integration.md tests/test_fractional_certificate.py
Case file: each case’s file is linked from its row above

T-044: Weighted point lower bounds for ten counts in n=26…72n=26…72, plus seven from the same files

The cases

Results on these cases

Links

T-044: Weighted point lower bounds for ten counts in $n = 26 \dots 72$ , plus seven from the same files