T-116: s(n)≤Sn​$s(n)\le S_n$ at nine counts, from the second SQUISH update’s new rational packings

2005 published T-007 · 9 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 · 9 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

2026-09-27 published T-046 · case 88

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

lower bound recorded superseded by T-091

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 · 6 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upper bound confirmed superseded by T-116 and T-127

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

2026-09-29 published T-058 · case 88

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 · 9 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 88

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lower bound confirmed on these cases, superseded by T-091

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 88

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lower bound confirmed on these cases, superseded by T-091

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

2026-10-02 published T-075 · case 88

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

lower bound confirmed on these cases, superseded by T-091

wand125 after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 mixed bounds afternoon 2026-10-02 · packet · source 1 · source 2 · review · register

2026-10-03 published T-082 · case 88

Mixed rectangle-measure lower bounds verified at 22 counts in $n=51\dots 96$$n=51\dots 96$

lower bound confirmed on these cases, superseded by T-091

wand125 after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 mixed bounds 2026-10-03 · packet · packet · source · review · register

2026-10-03 published T-092 · case 263

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-116

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

2026-10-04 published T-090 · case 88

Mixed rectangle-measure lower bounds verified at 17 counts in $n=42\dots 96$$n=42\dots 96$

lower bound confirmed on these cases, superseded by T-091

wand125 after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 mixed bounds 2026-10-04 · packet · source · review · register

2026-10-04 published T-091 · case 88

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 · 6 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 on these cases, superseded by T-116 and T-127

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

2026-10-05 published T-101 · cases 88, 108, 179

Exact certificates of 77 catalogue packings: s(n) ≤ S', 1.2e-16 to 9.8e-15 above each side

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

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

2026-10-07 published T-113 · cases 108, 180

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 and T-127

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

2026-10-07 published T-115 · cases 179, 263

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

upper bound confirmed on these cases, superseded by T-116 and T-127

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

2026-10-07 published T-116 this result · 9 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

2026-10-08 published T-125 · 5 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 · 7 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, 180

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

2026-10-09 published T-134 · case 263

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 207, 236, 263, 302

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 · cases 108, 180, 302

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-146 · cases 236, 263, 302

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