T-155: Exact regularized sides at 59 counts, 5.4e-20 to 9.9e-15 below the ceilings they replace

2005 published T-007 · 59 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Lower bound incomplete on these cases, superseded by T-048, T-069, T-073, T-075, T-080, T-082, T-083, T-090, T-091, T-107, T-108, T-110, T-111 and T-180 (reported)

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Lower bound reviewed superseded by T-069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

2026-08-29 established T-009 · case 29

$s(29)\le 5.93383346267692918974\dots$$s(29)\le 5.93383346267692918974\dots$, by a Krawczyk interval certificate

Upper bound confirmed superseded by T-195

Levy · register

2026-09-04 published T-085 · 59 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

2026-09-22 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Lower bound confirmed superseded by T-082, T-090, T-091, T-108, T-110 and T-111

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

2026-09-22 published T-047 · cases 28, 29

$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Lower bound confirmed superseded by T-107 and T-108

Tokoharu after Levy, Hosono, 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Upper bound confirmed superseded by T-101 and T-155

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 and T-155

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

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

2026-09-27 published T-046 · 10 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-108, T-110 and T-111

Hosono 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 · case 50

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

Lower bound confirmed

Hosono 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 · 8 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-108, T-110 and T-111

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 · 13 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

Hosono after Tokoharu, Daniel · wand125 tools 2026 · packet · register

2026-09-29 published T-083 · 59 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 · 9 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-108, T-110 and T-111

Hosono 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

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

2026-10-01 published T-074 · 8 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-108, T-110 and T-111

Hosono 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

Hosono 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

Hosono 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

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

Lower bound confirmed

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 · cases 101, 104

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 · cases 55, 69, 71, 87

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

Lower bound confirmed

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 · case 69

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

2026-10-04 published T-091 · cases 53, 54, 71, 87

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-101 · 58 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

Hosono after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 mixed bounds check2 2026-10-06 · packet · source · review · register

2026-10-06 published T-108 · case 29

Mixed rectangle-measure lower bound verified at $n=29$$n=29$, on a declared net

Lower bound confirmed

Hosono 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

Hosono 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

Hosono after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 mixed bounds check2 2026-10-06 · packet · source · review · 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

2026-10-10 published T-132 · case 29

Mixed rectangle-measure lower bound $s(29)\ge 291/50$$s(29)\ge 291/50$ reported, on a 2,073-direction declared net

Lower bound recorded

Hosono after Tokoharu and Levy · wand125 fine-net n29 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

Hosono 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

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

2026-10-12 established T-155 this result · 59 of these cases

Exact regularized sides at 59 counts, 5.4e-20 to 9.9e-15 below the ceilings they replace

Upper bound confirmed

Levy after Daniel · register

2026-10-12 published T-158 · case 264

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-162 · case 204

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

Upper bound confirmed superseded by T-225

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

2026-10-13 published T-174 · cases 204, 264

Tighter certificates at 68 counts from PingYou Ltd, of others' packings and its own

Upper bound confirmed on these cases, superseded by T-186 and T-225

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-176 · case 39

Mixed rectangle-measure lower bound $s(39)\ge 533/80$$s(39)\ge 533/80$ reported, on a 2,073-direction declared net

Lower bound reviewed

Hosono after Tokoharu and Levy · wand125 fine-net n37 n39 n44 2026-10-13 · packet · register

2026-10-13 published T-178 · case 37

Mixed rectangle-measure lower bound $s(37)\ge 2599/400$$s(37)\ge 2599/400$ reported, on a 2,073-direction declared net

Lower bound reviewed

Hosono after Tokoharu and Levy · wand125 fine-net n37 n39 n44 2026-10-13 · packet · register

2026-10-13 published T-179 · cases 145, 147, 148, 149

Linear-measure lower bound $s(n)\ge 306/25=12.24$$s(n)\ge 306/25=12.24$ for $n=145\dots 149$$n=145\dots 149$, reported

Lower bound reviewed

Hosono after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 linear n145 2026-10-13 · packet · register

2026-10-13 published T-180 · 54 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 · cases 204, 264

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

2026-10-15 published T-192 · case 28

Continuous-pose lower bound $s(28)>1149/200$$s(28)>1149/200$ reported, by two of the source's programs

Lower bound recorded

Hosono after Guzhou0806, Tokoharu, Levy · wand125 continuous n28 2026-10-15 · packet · register

2026-10-15 published T-195 · case 29

A tighter rational certificate of Schadt and Ellsworth's 29-square packing, 1.57e-36 below T-155

Upper bound confirmed

PingYou Ltd after Schadt, Ellsworth, Daniel, Levy · PingYou PR 622 certificate 2026-10-15 · packet · register

2026-10-16 published T-193 · case 204

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

Upper bound confirmed on these cases, superseded by T-225

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-202 · case 41

Mixed rectangle-measure lower bound $s(41)\ge 2721/400$$s(41)\ge 2721/400$ reported, on a 2,073-direction declared net

Lower bound recorded

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

2026-10-16 published T-215 · case 107

Mixed rectangle-measure lower bound $s(107)\ge 529/50$$s(107)\ge 529/50$ reported, on the 201-direction net

Lower bound recorded

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

2026-10-17 published T-196 · cases 227, 229, 260

de Winter reports three smaller rational upper bounds at 227, 229 and 260

Upper bound confirmed superseded by T-225

de Winter after Levy · de Winter upper bounds 2026-10-17 · packet · register

2026-10-17 published T-198 · case 229

Daniel's opening n229 dilation certificate, after Couzo and de Winter

Upper bound confirmed superseded by T-225

Daniel after Couzo, de Winter, Levy · Daniel n229 dilation 2026-10-17 · packet · register

2026-10-17 published T-226 · case 204

Couzo's 204 of #670, below Polkovskyi's T-186, since superseded by his own #675 and #678

Upper bound recorded superseded by T-225

Couzo after Polkovskyi, Daniel, Levy · Couzo issue 670 2026-10-17 · packet · packet · register

2026-10-17 published T-227 · cases 227, 229

Couzo's refinements of de Winter's 227 and 229, below T-196 at both

Upper bound recorded superseded by T-225

Couzo after de Winter, Daniel, Levy · Couzo issue 672 2026-10-17 · packet · packet · register

2026-10-18 published T-199 · case 229

Daniel's smaller n229 update, replacing the opening certificate

Upper bound confirmed superseded by T-225

Daniel after Couzo, de Winter, Levy · Daniel n229 dilation update 2026-10-18 · packet · register

2026-10-18 published T-223 · cases 204, 260

Couzo's smaller packings at 204 and 260, the two in the corpus of issue 678's 85 bounds

Upper bound confirmed superseded by T-225

Couzo after de Winter, Polkovskyi, Daniel, Fang, PingYou Ltd, Levy · Couzo issue 678 2026-10-18 · packet · register

2026-10-18 published T-228 · case 204

Couzo's 204 of #675, below his #670, since superseded by his #678

Upper bound recorded superseded by T-225

Couzo after Polkovskyi, Daniel, Levy · Couzo issue 675 2026-10-18 · packet · packet · register

2026-10-20 published T-225 · cases 204, 227, 229, 260

Daniel's solved, regularized views at 204, 227, 229 and 260, each the smallest side reported there

Upper bound confirmed

Daniel after Couzo, de Winter, Polkovskyi, Levy · Daniel regularized lists 2026-10-20 · packet · register

2026-10-22 published T-229 · case 204

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