T-056: Smaller packings for 49 counts from $n=68$$n=68$ to $307$$307$, each certified two independent ways

2005 published T-007 · 49 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

2026-09-04 published T-085 · 49 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 · case 68

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lower bound confirmed superseded by T-068 and T-074

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

2026-09-27 published T-046 · case 68

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

lower bound recorded superseded by T-068 and T-074

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 this result · 49 of these cases

Smaller packings for 49 counts from $n=68$$n=68$ to $307$$307$, each certified two independent ways

upper bound confirmed superseded by T-098, T-113, T-115, T-116, T-117, T-118, T-119, T-125 and T-127

Couzo · franciscouzo square-packing 2026-09-27 · packet · register

2026-09-28 published T-070 · case 68

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

lower bound confirmed superseded by T-068 and T-074

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 · case 68

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 · 49 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 · case 68

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 · case 68

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-073 · cases 102, 103, 105

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-080 · cases 102, 103, 105

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-092 · 7 of these cases

Smaller packings again at seven counts from $n=208$$n=208$ to $306$$306$, each certified two independent ways

upper bound confirmed superseded by T-098, T-113, T-116, T-119 and T-127

Couzo · franciscouzo square-packing 2026-10-03 · packet · register

2026-10-05 published T-098 · 46 of these cases

Exact optima of 48 known-best packings: s(n) ≤ S', 3.5e-13 to 5.0e-11 below each printed side

upper bound confirmed

Daniel after Couzo, de Winter, Ellsworth, Levy · evand exact optima 2026-10-05 · packet · register

2026-10-07 published T-113 · 7 of these cases

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

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

2026-10-07 published T-115 · 8 of these cases

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 · 6 of these cases

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

upper bound confirmed

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

2026-10-07 published T-117 · cases 105, 292

Exact rational ceiling refinements at $n=105$$n=105$, 292

upper bound confirmed

Rehwaldt after Couzo and earlier contributors · Rehwaldt Couzo refinements 2026-10-07 · packet · 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-119 · cases 270, 272

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-120 · case 102

Reported s(102) ≤ alpha_102 with a degree-8 exact side form

upper bound recorded superseded by T-125

Daniel after Couzo · Daniel exact and local reports 2026 · packet · register

2026-10-07 published T-121 · case 106

Reported s(106) ≤ alpha_106 with a degree-32 exact side form

upper bound recorded

Daniel after Couzo · Daniel exact and local reports 2026 · packet · register

2026-10-07 published T-122 · case 152

Reported s(152) ≤ alpha_152 with a degree-40 exact side form

upper bound recorded

Daniel after Couzo · Daniel exact and local reports 2026 · packet · register

2026-10-07 published T-123 · case 177

Reported s(177) ≤ alpha_177 with a degree-32 exact side form

upper bound recorded

Daniel after Couzo · Daniel exact and local reports 2026 · packet · register

2026-10-07 published T-129 · cases 105, 130, 292

Three dated Daniel certificate reports with historical source custody

upper bound recorded superseded by T-117, T-125 and T-127

Daniel after Couzo, Levy · Daniel dated certificates 105 and 130 2026-10-07 · Daniel dated certificate 292 2026-10-07 · packet · packet · packet · register

2026-10-08 published T-125 · 8 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-127 · 12 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 · 6 of these cases

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 · cases 105, 270

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-131 · case 132

$s(132)\le 11.987099332245063\dots$$s(132)\le 11.987099332245063\dots$ from Evan Daniel's record hunt

upper bound confirmed

Daniel after Couzo, Chaoweeraprasit, Levy · Daniel record hunt 2026-10-09 · packet · register

2026-10-09 published T-134 · 5 of these cases

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 · 15 of these cases

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 · 9 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-140 · 5 of these cases

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-141 · case 132

Exact rational certificates at 132 and 308 from Evan Daniel's third record hunt

upper bound confirmed

Daniel after Fang, Arslanov, Mustafin, Shangitbayev, Couzo, Levy · Daniel record hunt 3 2026-10-10 · packet · register

2026-10-10 published T-144 · case 123

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 · 9 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