T-127: Fourteen rational refinements, seventeen complete source cases

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Lower bound incomplete on these cases, superseded by T-083, T-091, T-180 (reported) and T-216

Nagamochi · Nagamochi 2005 · source · 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; Vlasenko · 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Lower bound recorded superseded by T-091

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 · 12 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Upper bound confirmed superseded by T-127, T-146, T-173, T-174, T-190, T-218 and T-222

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

Hosono 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

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Lower bound confirmed on these cases, superseded by T-091

Hosono 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Lower bound confirmed on these cases, superseded by T-091

Hosono 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

Hosono 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

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

2026-10-03 published T-092 · cases 208, 209

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-174

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

Hosono 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

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 · 11 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-127, T-146, T-173, T-174, T-190, T-218 and T-222

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

2026-10-05 published T-101 · 5 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 on these cases, superseded by T-127, T-173, T-174 and T-186

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 superseded by T-174, T-186 and T-222

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-173

Chaoweeraprasit after Ellsworth · SQUISH n153 2026-10-07 · packet · 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 superseded by T-146, T-174, T-186, T-190 and T-222

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

2026-10-07 published T-116 · 7 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 superseded by T-127, T-173, T-174 and T-218

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

2026-10-07 published T-129 · case 130

Three dated Daniel certificate reports with historical source custody

Upper bound recorded superseded by T-174

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 superseded by T-127, T-146, T-173, T-174, T-186 and T-218

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

2026-10-08 published T-127 this result · 17 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 Couzo

Upper bound confirmed superseded by T-174

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

2026-10-09 published T-134 · case 237

Six exact rational certificates from Couzo

Upper bound confirmed on these cases, superseded by T-190

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

2026-10-09 published T-136 · 6 of these cases

Seventeen exact rational packings from SQUISH's third request

Upper bound confirmed on these cases, superseded by T-173, T-174, T-190, T-218 and T-222

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

2026-10-09 published T-138 · cases 108, 180

Exact witnesses at Mishapolk's printed ceilings, the smallest known at 103 and 258 when registered

Upper bound confirmed superseded by T-174

Polkovskyi after Stenlund, Friedman, Ellsworth, Chaoweeraprasit, Couzo, Xu, Levy · Mishapolk decimal poses 2026-10-09 · packet · register

2026-10-10 published T-140 · cases 209, 237

Six more exact rational certificates from Couzo

Upper bound confirmed superseded by T-174 and T-190

Couzo after Chaoweeraprasit, Polkovskyi, Xu, Daniel, Ellsworth, Levy · Couzo certificates 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

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

2026-10-10 published T-146 · cases 123, 129, 153, 236

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-147 · case 207

A packing of 207 squares from two placed patches of Xu's packing of 70

Upper bound confirmed superseded by T-173

PingYou Ltd after Xu, Daniel, Levy · PingYou n207 certificate 2026-10-11 · packet · register

2026-10-11 published T-148 · case 153

A packing of 153 squares built from Xu's packing of 70, its contacts reclosed at a new angle

Upper bound confirmed superseded by T-173

PingYou Ltd after Xu, Daniel, Levy · PingYou n153 certificate 2026-10-11 · packet · register

2026-10-11 published T-151 · cases 238, 239

Nine exact rational packings from SQUISH's fourth request

Upper bound confirmed superseded by T-174

Chaoweeraprasit after Couzo, Gupta, Ellsworth, Friedman, Levy · SQUISH fourth request 2026-10-11 · packet · register

2026-10-12 published T-160 · case 154

A smaller packing of 154 squares from Daniel's size-chaining explorer, certified exactly

Upper bound confirmed superseded by T-222

Daniel after Couzo, Levy · Daniel n154 2026-10-12 · packet · register

2026-10-12 published T-161 · case 237

Exact rational certificates at 156, 176, 237 and 271 from Couzo

Upper bound confirmed superseded by T-190

Couzo after Ostashev, Chaoweeraprasit, Fang, Daniel, Ellsworth, Levy · Couzo certificates 2026-10-12 · packet · register

2026-10-12 published T-162 · case 129

New packings of 129, 204 and 303 squares, a cut-out and two compositions, each certified exactly

Upper bound confirmed superseded by T-186

Ostashev after Nakajima, Chaoweeraprasit, Daniel, Xu, Levy · Ostashev compositions 2026-10-12 · packet · register

2026-10-12 published T-166 · case 153

A tighter rational realization of PingYou Ltd's packing of 153 squares

Upper bound confirmed superseded by T-173

PingYou Ltd after Xu, Daniel, Levy · PingYou n153 refinement 2026-10-12 · packet · register

2026-10-12 published T-167 · case 207

A tighter rational realization of PingYou Ltd's packing of 207 squares

Upper bound confirmed superseded by T-173

PingYou Ltd after Xu, Daniel, Levy · PingYou n207 refinement 2026-10-12 · packet · register

2026-10-12 published T-172 · case 154

A packing of 154 squares from Couzo, continuing Daniel's arrangement

Upper bound confirmed superseded by T-222

Couzo after Daniel, Ostashev, Fang, Ellsworth, Levy · Couzo issue-583 certificates 2026-10-13 · packet · register

2026-10-12 published T-173 · cases 153, 207

Polkovskyi's packings of 153 and 207 squares, each below the earlier refinements there

Upper bound confirmed

Polkovskyi after Xu, Chaoweeraprasit, Levy · Mishapolk exact certificates 2026-10-12 · packet · register

2026-10-13 published T-174 · 16 of these cases

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-180 · 17 of these cases

Closed-form lower bound $s(N)\ge a(N)$$s(N)\ge a(N)$ for every non-square $N\ge 49$$N\ge 49$, reported

Lower bound reviewed

Hosono · wand125 closed form 2026-10-14 · packet · packet · register

2026-10-13 published T-182 · case 154

Exact rational certificates at 154, 211 and 241 from Couzo

Upper bound confirmed on these cases, superseded by T-222

Couzo after Daniel, Ostashev, BuddenPing, Fang, Ellsworth, Levy · Couzo certificates 2026-10-13 · packet · register

2026-10-13 published T-183 · case 237

Polkovskyi's certificate for 237 squares, the smallest known side there when registered

Upper bound confirmed superseded by T-190

Polkovskyi after Friedman, Daniel, Xu, Chaoweeraprasit, Barnett, Couzo, Ostashev, Ellsworth, Fang, Levy · Mishapolk records 2026-10-14 · packet · register

2026-10-14 published T-186 · cases 129, 154

Exact certificates at twelve counts from Polkovskyi, tightening the smallest reported there

Upper bound confirmed

Polkovskyi after Ostashev, Couzo, PingYou Ltd, Levy · Mishapolk KKT refinements 2026-10-14 · packet · register

2026-10-14 published T-189 · case 237

Couzo's refinement of Polkovskyi's 237, and fifteen beyond-horizon bounds

Upper bound confirmed superseded by T-190

Couzo after Polkovskyi, Fang, Daniel, Ellsworth, Levy · Couzo certificates 2026-10-14 · packet · register

2026-10-15 published T-190 · case 237

A tighter exact certificate for 237 squares from Couzo, below his issue-607 certificate

Upper bound confirmed

Couzo after Polkovskyi, Chaoweeraprasit, Daniel, Levy · Couzo n237 2026-10-15 · packet · register

2026-10-16 published T-193 · cases 88, 123, 199

Exact algebraic sides at sixteen counts from Daniel, of packings the record credits

Upper bound confirmed

Daniel after Xu, Chaoweeraprasit, Ostashev, Couzo, PingYou Ltd, Gupta, Polkovskyi, Levy · Daniel exact current records 2026-10-16 · packet · register

2026-10-16 published T-209 · case 88

Mixed rectangle-measure lower bound $s(88)\ge 3853/400$$s(88)\ge 3853/400$ reported, on the 201-direction net

Lower bound recorded

Hosono after Tokoharu and Levy · wand125 fine-net lower bounds 2026-10-16 · packet · register

2026-10-16 published T-216 · case 108

Mixed rectangle-measure lower bound $s(108)\ge 531/50$$s(108)\ge 531/50$ 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

2026-10-16 published T-218 · case 236

Daniel's smaller n236 packing, after Chaoweeraprasit's composition, certified exactly

Upper bound confirmed

Daniel after Chaoweeraprasit, Levy · Daniel n236 2026-10-16 · packet · register

2026-10-20 published T-222 · case 154

Daniel's smaller n154 packing, continuing Couzo's #586 arrangement, certified exactly

Upper bound confirmed

Daniel after Couzo, Ostashev, Levy · Daniel n154 sweep 2026-10-20 · packet · packet · register