T-130: Five follow-up rational refinements from Couzo

2005 published T-007 · 5 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Lower bound incomplete on these cases, superseded by T-073, T-080, T-083, T-090, T-180 (reported) and T-206

Nagamochi · Nagamochi 2005 · source · register

2026-09-04 published T-085 · 5 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 86

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

Lower bound recorded superseded by T-090

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 · cases 105, 270

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

Upper bound confirmed superseded by T-174 and T-224

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

2026-09-28 published T-070 · case 86

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

Lower bound confirmed superseded by T-090

Hosono after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 rectangle bounds 2026-09-28 · packet · packet · source · review · register

2026-09-29 published T-058 · cases 84, 86

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 · 5 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 86

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Lower bound confirmed on these cases, superseded by T-090

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

2026-10-02 published T-071 · cases 84, 86

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

Lower bound confirmed superseded by T-090 and T-206

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

2026-10-02 published T-073 · case 105

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

Lower bound confirmed

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

2026-10-02 published T-075 · case 86

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-090

Hosono 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-080 · case 105

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

Lower bound confirmed

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

2026-10-03 published T-082 · case 86

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-090

Hosono 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 84, 86

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

Lower bound confirmed

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

2026-10-05 published T-094 · case 84

Mixed rectangle-measure lower bounds verified at $n=67$$n=67$ and $84$$84$

Lower bound confirmed on these cases, superseded by T-206

Hosono after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 mixed bounds 2026-10-05 · packet · source · review · register

2026-10-05 published T-098 · case 270

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-224

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

2026-10-05 published T-101 · case 175

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-186

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

2026-10-07 published T-117 · case 105

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

Upper bound confirmed superseded by T-174

Rehwaldt after Couzo and earlier contributors · Rehwaldt Couzo refinements 2026-10-07 · packet · register

2026-10-07 published T-119 · case 270

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

Upper bound confirmed on these cases, superseded by T-224

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

2026-10-07 published T-129 · case 105

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 · cases 84, 86, 105, 175

Complete rational construction reports at 25 counts

Upper bound confirmed superseded by T-174 and T-186

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

2026-10-08 published T-128 · case 105

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

2026-10-08 published T-130 this result · 5 of these cases

Five follow-up rational refinements from Couzo

Upper bound confirmed superseded by T-174, T-186 and T-224

Couzo after Xu, Daniel, Ellsworth, Levy · Couzo follow-up refinements 2026-10-08 · packet · register

2026-10-09 published T-134 · case 270

Six exact rational certificates from Couzo

Upper bound confirmed on these cases, superseded by T-224

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

2026-10-09 published T-136 · case 270

Seventeen exact rational packings from SQUISH's third request

Upper bound confirmed on these cases, superseded by T-224

Chaoweeraprasit after Couzo, Xu, Göbel, 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 superseded by T-174, T-186 and T-224

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

2026-10-10 published T-140 · cases 175, 270

Six more exact rational certificates from Couzo

Upper bound confirmed superseded by T-186 and T-224

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

2026-10-11 published T-151 · case 270

Nine exact rational packings from SQUISH's fourth request

Upper bound confirmed superseded by T-224

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

2026-10-11 published T-185 · case 175

A tighter packing of 175 squares from Couzo, superseded by Ostashev's composition

Upper bound confirmed superseded by T-186

Couzo after Xu, Daniel, Ellsworth, Levy · Couzo n175 2026-10-11 · packet · register

2026-10-12 published T-158 · case 175

New packings of 175, 232 and 264 squares built on a new 38-square rectangle, each certified exactly

Upper bound confirmed superseded by T-186

Ostashev after Nakajima, Ellsworth, Schadt, Friedman, de Winter, Daniel, Levy, Xu · Ostashev records 2026-10-12 · packet · register

2026-10-12 published T-170 · case 105

An exact contact refinement of Xu's packing of 105 squares, after Couzo's

Upper bound confirmed superseded by T-174

PingYou Ltd after Xu, Couzo, Daniel, Levy · PingYou n105 refinement 2026-10-12 · packet · register

2026-10-13 published T-174 · 5 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 · 5 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

2026-10-14 published T-186 · case 175

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

2026-10-16 published T-193 · case 175

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

Upper bound confirmed on these cases, superseded by T-186

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-206 · case 84

Mixed rectangle-measure lower bound $s(84)\ge 3769/400$$s(84)\ge 3769/400$ 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-208 · case 86

Mixed rectangle-measure lower bound $s(86)\ge 3811/400$$s(86)\ge 3811/400$ reported, on the 201-direction net

Lower bound recorded

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

2026-10-20 published T-224 · case 270

Daniel's smaller n270 packing, another arrangement found from Chaoweeraprasit's, certified exactly

Upper bound confirmed

Daniel after Chaoweeraprasit, Levy · Daniel n270 sweep 2026-10-20 · packet · register

2026-10-22 published T-229 · case 105

Polkovskyi's re-solves of the house packings at 105 and 204, below T-174 and T-225

Upper bound recorded

Polkovskyi after Couzo, Muglich, Daniel, Xu, PingYou Ltd, Levy · Mishapolk issue 719 2026-10-22 · packet · register