T-101: Exact certificates of 77 catalogue packings: s(n) ≤ S', 1.2e-16 to 9.8e-15 above each side

2005 published T-007 · 77 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

Nagamochi · Nagamochi 2005 · source · register

2018 published T-087 · case 37

s(37)≥53​/2+22​−1$s(37)\ge 5\sqrt{3}/2+2\sqrt{2}-1$ and s(61)≥73​/2+22​−1$s(61)\ge 7\sqrt{3}/2+2\sqrt{2}-1$, by optimal piercing

lower bound reviewed superseded by T-069

Bašić, Slivková · Basic-Slivkova 2018 · source · register

2026-09-04 published T-085 · 77 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 · 8 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lower bound confirmed superseded by T-082, T-090, T-091, T-110 and T-111

wand125 after Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 point bounds 2026 · packet · source · review · register

2026-09-22 published T-047 · case 28

$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lower bound confirmed superseded by T-107

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upper bound confirmed superseded by T-101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

2026-09-24 published T-089 · cases 83, 87

$s(83)\le 9.63475764863195$$s(83)\le 9.63475764863195$ and $s(87)\le 9.83881526994915$$s(87)\le 9.83881526994915$, Chang's packings, certified exactly

upper bound confirmed superseded by T-101

Chang, Ellsworth after Cantrell, Hajba, Stenlund, Bidwell; Chang after Ellsworth, Cantrell, Schadt, Hajba, DeVincentis · Chang n83 2026-09-24 · Chang n87 2026-09-24 · packet · packet · register

2026-09-27 published T-045 · case 28

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

lower bound confirmed superseded by T-107

wand125 after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 rectangle bounds 2026 · packet · packet · source · review · register

2026-09-27 published T-046 · 12 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-069, T-082, T-090, T-091, T-107, T-110 and T-111

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-048 · cases 50, 51

$s(50)\ge 37/5=7.4$$s(50)\ge 37/5=7.4$

lower bound confirmed

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-070 · 9 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-082, T-090, T-091, 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 · 15 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 · 77 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

Karakuş · Karakuş 2026 · source · register

2026-10-01 published T-068 · 10 of these cases

Rectangle-density lower bounds verified at 34 counts in $n=19\dots 95$$n=19\dots 95$

lower bound confirmed on these cases, superseded by T-082, T-090, T-091, T-107, T-110 and T-111

wand125 after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 rectangle bounds 2026-10-01 · packet · source · review · register

2026-10-01 published T-069 · case 37

Mixed rectangle-measure lower bounds replayed at $n=37,65,66,90,92$$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 · 9 of these cases

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

lower bound confirmed on these cases, superseded by T-082, T-090, T-091, T-107, T-110 and T-111

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 87

Mixed rectangle-measure lower bounds replayed at $n=84\dots 87$$n=84\dots 87$

lower bound confirmed superseded by T-091

wand125 after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 mixed bounds 2026-10-02 · packet · source · review · register

2026-10-02 published T-073 · cases 83, 101, 104

Linear-measure lower bounds replayed at $n=83$$n=83$ and $n=101\dots 105$$n=101\dots 105$

lower bound confirmed

wand125 after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 linear certificates 2026-10-02 · packet · source · review · register

2026-10-02 published T-075 · cases 83, 87, 88

Mixed rectangle-measure lower bounds replayed at nine counts in $n=83\dots 96$$n=83\dots 96$

lower bound confirmed

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 · 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

2026-10-02 published T-080 · cases 101, 104

Linear-measure lower bound replayed at $n=101\dots 105$$n=101\dots 105$

lower bound confirmed

wand125 after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 linear certificates 2026-10-02 · packet · source · review · register

2026-10-03 published T-082 · 7 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 51, 69, 88

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-101 this result · 77 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-107 · case 28

Mixed rectangle-measure lower bound verified at $n=28$$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-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-113 · cases 108, 129

s(n)≤Sn​$s(n)\le S_n$ at ten counts, $n=108$$n=108$ to 303, from new SQUISH rational packings

upper bound confirmed on these cases, superseded by T-125

Chaoweeraprasit after Ellsworth · SQUISH ten packings 2026-10-07 · packet · register

2026-10-07 published T-114 · case 153

$s(153)\le 7250614903299225/562949953421312$, from a SQUISH rational packing

upper bound confirmed superseded by T-127

Chaoweeraprasit after Ellsworth · SQUISH n153 2026-10-07 · packet · packet · register

2026-10-07 published T-115 · cases 129, 179, 258

s(n)≤Sn​$s(n)\le S_n$ at twelve counts, from the SQUISH update’s new and smaller rational packings

upper bound confirmed

Chaoweeraprasit after Couzo and Ellsworth · SQUISH update 2026-10-07 · packet · register

2026-10-07 published T-116 · cases 88, 108, 179

s(n)≤Sn​$s(n)\le S_n$ at nine counts, from the second SQUISH update’s new rational packings

upper bound confirmed on these cases, superseded by T-125 and T-127

Chaoweeraprasit after Ellsworth, Couzo and SQUISH · SQUISH second update 2026-10-07 · packet · register

2026-10-07 published T-119 · case 266

Three exact feasible upper bounds at n=266,270,272 from new source arrangements

upper bound confirmed

Daniel after Levy, Ellsworth, Couzo, Stead · Daniel new arrangements 2026-10-07 · packet · register

2026-10-07 published T-124 · case 28

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 · 14 of these cases

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 · 5 of these cases

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-128 · cases 108, 127

Eight complete rational refinements from Francisco Couzo

upper bound confirmed

Couzo after Xu, Chaoweeraprasit, Gupta, Ellsworth, Daniel, Levy · Couzo exact refinements 2026-10-08 · packet · register

2026-10-08 published T-130 · case 175

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-134 · case 267

Six exact rational certificates from Francisco Couzo

upper bound confirmed

Couzo after Mishapolk, Chaoweeraprasit, Gupta, Xu, Daniel, Ellsworth, Levy · Couzo exact certificates 2026-10-09 · packet · register

2026-10-09 published T-136 · cases 153, 232

Seventeen exact rational packings from SQUISH's third request

upper bound confirmed

Chaoweeraprasit after Couzo, Xu, Goebel, Schadt, Ellsworth, Hajba, Levy · SQUISH third request 2026-10-09 · packet · register

2026-10-09 published T-138 · 5 of these cases

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-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-140 · case 175

Six more exact rational certificates from Francisco Couzo

upper bound confirmed

Couzo after Chaoweeraprasit, Mishapolk, Xu, Daniel, Ellsworth, Levy · Couzo certificates 2026-10-10 · packet · register

2026-10-10 published T-142 · case 28

Mixed rectangle-measure lower bound $s(28)\ge 2297/400$$s(28)\ge 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-144 · cases 122, 124, 125

Linear-measure lower bound $s(n)\ge 563/50=11.26$$s(n)\ge 563/50=11.26$ for $n=122\dots 126$$n=122\dots 126$, reported

lower bound recorded

wand125 after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 linear n122 2026-10-10 · packet · register

2026-10-10 published T-146 · 6 of these cases

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