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

2005 published T-007 · 17 of these cases

s(n)≥min(⌈n​⌉,n−2⌊n​⌋+1​+1)$s(n)\ge min(\lceil \sqrt{n}\rceil ,\sqrt{n-2\lfloor \sqrt{n}\rfloor +1}+1)$ for $4\le n\le 324$$4\le n\le 324$

lower bound incomplete on these cases, superseded by T-068, T-074, T-082, T-090, T-091, T-105, T-108, T-110, T-111 and T-145

Nagamochi · Nagamochi 2005 · source · register

2026-08-29 established T-009 · case 29

$s(29)\le 5.933833\dots$$s(29)\le 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$$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$$a>3$ and $b>2$$b>2$

correction confirmed

Karakuş; chelokot · 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$$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lower bound confirmed superseded by T-068, T-074, T-082, T-090, T-091, T-105, T-108, T-110, T-111 and T-145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 · cases 26, 29

$s(11)\ge 381/100$$s(11)\ge 381/100$; $s(n)\ge 1377/250$$s(n)\ge 1377/250$ for $n=26\dots 28$$n=26\dots 28$; $s(n)\ge 571/100$$s(n)\ge 571/100$ for $n=29\dots 31$$n=29\dots 31$

lower bound confirmed superseded by T-105 and T-108

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)\le 8.82719465572975$$s(69)\le 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upper bound confirmed superseded by T-101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 · cases 26, 40

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

lower bound confirmed superseded by T-105 and T-145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 · 17 of these cases

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

lower bound recorded 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 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$$n=68$ to $307$$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-070 · 14 of these cases

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

lower bound confirmed superseded by T-068, T-074, T-082, T-090, T-091, 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 · 17 of these cases

Rectangle-certificate ceiling α·UB(n) proved for $n=1$$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 · 17 of these cases

s(n)≥1/2+n−⌊n​⌋+1/4​$s(n)\ge 1/2+\sqrt{n-\lfloor \sqrt{n}\rfloor +1/4}$ for every nonsquare $8\le n\le 324$$8\le n\le 324$

lower bound confirmed on these cases, superseded by T-068, T-074, T-082, T-090, T-091, T-105, T-108, T-110, T-111 and T-145

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$$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 · 13 of these cases

Rectangle-density lower bounds replayed at 31 counts in $n=19\dots 95$$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-077 · case 70

Rectangle-density lower bounds replayed at $n=20$$n=20$, 42 and 70

lower bound confirmed superseded by T-091

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-082 · 6 of these cases

Mixed rectangle-measure lower bounds verified at 22 counts in $n=51\dots 96$$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 · cases 56, 57, 69, 72

Mixed rectangle-measure lower bounds verified at 17 counts in $n=42\dots 96$$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 · 6 of these cases

Mixed rectangle-measure lower bounds verified at 17 counts in $n=53\dots 95$$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-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 · 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$$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-108 · case 29

Mixed rectangle-measure lower bound verified at $n=29$$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-110 · case 39

Mixed rectangle-measure lower bound verified at $n=39$$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$$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-118 · case 68

Exact rational ceiling refinements at $n=68$$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 · 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

ry-xu · ry-xu square packing 2026 · packet · register

2026-10-10 published T-132 · case 29

Mixed rectangle-measure lower bound $s(29)\ge 291/50$$s(29)\ge 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$$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-145 · case 40

$s(40)>1340000/199529=6.7158157\dots$$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