T-058: Rectangle-certificate ceiling α·UB(n) proved for $n=1$$n=1$..100; B·UB(n) on 64 grid rows

1979 published T-011 · case 11

Trump's 1979 packing is exactly valid, so $s(11)\le 3.877083590022814\dots$$s(11)\le 3.877083590022814\dots$

Upper bound confirmed

Trump · Trump 2023 · register

2005 published T-007 · 97 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

2010-09-13 published T-004 · case 46

Bentz 2010, Theorem 8 ($s(46)\ge 7$$s(46)\ge 7$) is correct as printed, machine-audited in full

Audit confirmed

Bentz · Bentz 2010 · source · register

2010-09-13 published T-006 · case 13

$s(13)=4$$s(13)=4$

Optimality confirmed

Bentz; Daniel after Burns, Massaccesi · Bentz 2010 · evand square-packing 2026-09-28 · packet · source 1 · source 2 · review · register

2010-09-13 published T-008 · case 46

$s(46)=7$$s(46)=7$

Optimality confirmed

Bentz · Bentz 2010 · source · register

2018 published T-087 · cases 37, 61

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Lower bound reviewed superseded by T-063 and T-069

Bašić, Slivková · Basic-Slivkova 2018 · source · register

2026-08-21 published T-015 · case 17

$s(17)\ge 22529/5000=4.5058$$s(17)\ge 22529/5000=4.5058$

Lower bound confirmed superseded by T-093 and T-153 (reported)

Massaccesi after Burns · Burns–Massaccesi n17 · packet · register

2026-08-21 published T-016 · cases 18, 19

$s(n)\ge 22529/5000$$s(n)\ge 22529/5000$ for $n=18,19$$n=18,19$, by monotonicity from T-015

Lower bound confirmed superseded by T-102 and T-103

Massaccesi after Burns · Burns–Massaccesi n17 · packet · register

2026-08-24 established T-010 · case 11

s(11)≥2+4/5​$s(11)\ge 2+4/\sqrt{5}$, by a repair of Stromquist 2003's Figure 14 point set

Lower bound confirmed superseded by T-060

Levy after Stromquist · register

2026-08-25 published T-049 · case 12

$s(12)\ge 15680/3951=3.9686155\dots$$s(12)\ge 15680/3951=3.9686155\dots$

Lower bound confirmed superseded by T-095

Daniel after Burns, Massaccesi · evand square-packing 2026 · packet · source 1 · source 2 · review 1 · review 2 · 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

2026-08-30 established T-012 · case 5

Goebel's $n=5$$n=5$ packing is second-order rigid at fixed side

Rigidity confirmed

Levy · register

2026-08-30 established T-013 · case 40

Goebel's $n=40$$n=40$ packing: seven verified first-order flexes, each refused at second order

Rigidity confirmed

Levy · register

2026-08-31 established T-001 · case 17

$s(17)\ge 4426213/1000000=4.426213$$s(17)\ge 4426213/1000000=4.426213$, from a sixteen-point unavoidable set

Lower bound confirmed superseded by T-093 and T-153 (reported)

Levy after Bentz · register

2026-08-31 established T-002 · case 18

$s(18)\ge 4426213/1000000$$s(18)\ge 4426213/1000000$, by monotonicity from T-001

Lower bound confirmed superseded by T-102

Levy after Bentz · register

2026-08-31 established T-003 · cases 17, 18

The sixteen-point set's unavoidability ceiling lies in $[4426213/1000000,4427/1000)$$[4426213/1000000,4427/1000)$

Method limit confirmed

Levy after Bentz · register

2026-08-31 established T-005 · case 13

Bentz 2010, Lemma 10 is false as printed and true as corrected to $(1.74,1)$$(1.74,1)$

Correction confirmed

Levy after Bentz · source · register

2026-09-03 established T-014 · case 5

Goebel's $n=5$$n=5$ optimum is rigid at fixed side: its pose is an isolated feasible point

Rigidity confirmed

Levy · register

2026-09-04 established T-017 · case 12

$s(12)\ge 99/25=3.96$$s(12)\ge 99/25=3.96$

Lower bound confirmed superseded by T-095

Levy after Burns, Massaccesi · register

2026-09-04 established T-018 · case 11

$s(11)\ge 381/100=3.81$$s(11)\ge 381/100=3.81$

Lower bound confirmed superseded by T-060

Levy after Burns, Massaccesi · register

2026-09-04 established T-019 · cases 17, 18, 19

$s(n)\ge 459/100=4.59$$s(n)\ge 459/100=4.59$ for $n=17,18,19$$n=17,18,19$

Lower bound confirmed superseded by T-093, T-102, T-103 and T-153 (reported)

Levy after Burns, Massaccesi · register

2026-09-04 established T-020 · cases 19, 20, 21

$s(n)\ge 24/5=4.80$$s(n)\ge 24/5=4.80$ for $n=19,20,21$$n=19,20,21$

Lower bound confirmed superseded by T-052, T-103 and T-104

Levy after Burns, Massaccesi · register

2026-09-04 published T-085 · 91 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

2026-09-04 published T-086 · 8 of these cases

s(k2−2)=k$s(k^2-2)=k$ for every integer $k\ge 2$$k\ge 2$; $k=3\dots 18$$k=3\dots 18$ are the cases held here

Optimality confirmed

Vlasenko · chelokot Nagamochi counterexample 2026 · packet · source · register

2026-09-05 established T-021 · cases 20, 21

$s(n)\ge 97/20=4.85$$s(n)\ge 97/20=4.85$ for $n=20,21$$n=20,21$

Lower bound confirmed superseded by T-052 and T-104

Levy after Burns, Massaccesi · register

2026-09-06 established T-022 · case 11

s(11)≥381008100042893309449​/899996306539=3.8100257…

Lower bound confirmed superseded by T-060

Levy after Burns, Massaccesi · source · register

2026-09-08 established T-023 · case 11

Conditional exclusion: no eleven-square packing in the four-owner branch at $q=96/25$$q=96/25$

Case exclusion confirmed superseded in part by T-060

Levy · source · register

2026-09-09 established T-024 · case 11

s(11)≥3175000518400042893309449​/598960960743657=3.8166095…

Lower bound confirmed superseded by T-060

Levy after Burns, Massaccesi · source · register

2026-09-09 established T-025 · case 11

$s(11)\ge 191/50=3.82$$s(11)\ge 191/50=3.82$, by a threshold certificate

Lower bound confirmed superseded by T-060

Levy · source · register

2026-09-09 established T-026 · case 11

s(11)≥955000518400042893309449​/179696714646249=3.8264474…

Lower bound confirmed superseded by T-060

Levy · source · register

2026-09-18 established T-027 · case 18

$s(18)\ge 467/100=4.67$$s(18)\ge 467/100=4.67$

Lower bound confirmed superseded by T-102

Levy after Burns, Massaccesi · register

2026-09-19 established T-028 · case 18

$s(18)\ge 187/40=4.675$$s(18)\ge 187/40=4.675$

Lower bound confirmed superseded by T-102

Levy after Burns, Massaccesi · register

2026-09-19 established T-029 · case 18

$s(18)\ge 1871/400=4.6775$$s(18)\ge 1871/400=4.6775$

Lower bound confirmed superseded by T-102

Levy after Burns, Massaccesi · register

2026-09-19 established T-030 · case 18

$s(18)\ge 4679/1000=4.679$$s(18)\ge 4679/1000=4.679$

Lower bound confirmed superseded by T-102

Levy after Burns, Massaccesi · register

2026-09-20 established T-031 · case 11

The octagon corner class (threshold $1/2$$1/2$) holds no eleven-square packing at side $96/25$$96/25$

Case exclusion confirmed superseded by T-060

Levy · register

2026-09-20 published T-032 · case 17

$s(17)\ge 461300/99999=4.61304613\dots$$s(17)\ge 461300/99999=4.61304613\dots$, and beneath it Mira's $s(17)\ge 4613/1000$$s(17)\ge 4613/1000$

Lower bound confirmed superseded by T-093 and T-153 (reported)

Guzhou0806, Mira-acc after Levy, Burns, Massaccesi · n17 weighted certificates 2026-09-20 · packet · register

2026-09-21 published T-038 · case 17

$s(17)>461300/99853=4.6197910929\dots$$s(17)>461300/99853=4.6197910929\dots$

Lower bound confirmed superseded by T-093 and T-153 (reported)

Overbeck after Levy, Mira-acc, Guzhou0806 · Kleddamag n17 certified bound · packet · source · review · register

2026-09-21 published T-065 · case 17

$s(17)\le 4.6755300936045509\dots$$s(17)\le 4.6755300936045509\dots$, Bidwell's packing certified exactly

Upper bound confirmed

Overbeck after Levy, Mira-acc, Guzhou0806 · Kleddamag n17 certified bound · packet · source · review · register

2026-09-22 established T-033 · case 11

s(11)≥9550002073600042893309449​/359341754646249=3.8269975…

Lower bound confirmed superseded by T-060

Levy · source · register

2026-09-22 published T-037 · case 11

$s(11)>31/8=3.875$$s(11)>31/8=3.875$

Lower bound confirmed superseded by T-060

Overbeck after Levy, Guzhou0806, Mira-acc · Kleddamag n11 2026 · packet · source 1 · source 2 · review 1 · review 2 · register

2026-09-22 published T-044 · 17 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-105, T-108, T-110, T-111, T-157 and T-205

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

2026-09-22 published T-047 · 7 of these cases

$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-060, T-068, T-074, T-105, T-106, T-107, T-108 and T-109

Tokoharu after Levy, Hosono, Stromquist, Nagamochi, Burns, Massaccesi · Tokoharu density 2026 · packet · source · review · register

2026-09-23 established T-034 · case 21

$s(21)\ge 122/25=4.88$$s(21)\ge 122/25=4.88$

Lower bound confirmed superseded by T-052

Levy after Burns, Massaccesi · register

2026-09-23 published T-050 · case 21

$s(21)\ge 5000/1001=4.995004995\dots$$s(21)\ge 5000/1001=4.995004995\dots$

Lower bound confirmed superseded by T-052

Daniel after Burns, Massaccesi · evand square-packing 2026 · packet · source · review · register

2026-09-24 established T-035 · case 11

Six-plus-five packings near Trump's tilt with side ≤Uhi​$\le U_{hi}$ lie within rho of his pose

Case exclusion confirmed

Levy · register

2026-09-24 established T-036 · case 11

Trump's pose is optimal among six-plus-five packings near its tilt, unique up to symmetry

Restricted optimality confirmed superseded in part by T-060 and T-112

Levy · 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

2026-09-25 published T-039 · case 17

$s(17)>231001/50000=4.62002$$s(17)>231001/50000=4.62002$

Lower bound confirmed superseded by T-093 and T-153 (reported)

Guzhou0806 after Overbeck, Mira-acc, Levy · Guzhou0806 n17 R052 · packet · source · review · register

2026-09-26 published T-040 · case 17

$s(17)>232001/50000=4.64002$$s(17)>232001/50000=4.64002$

Lower bound confirmed superseded by T-093 and T-153 (reported)

Overbeck after Levy, Mira-acc, Guzhou0806 · Kleddamag n17 4.640020 · packet · source · review · register

2026-09-26 published T-051 · case 32

$s(32)=6$$s(32)=6$

Optimality confirmed

Daniel after Burns, Massaccesi · evand square-packing 2026 · packet · packet · packet · packet · source 1 · source 2 · review 1 · review 2 · review 3 · register

2026-09-27 published T-041 · case 17

$s(17)>466001/100000=4.66001$$s(17)>466001/100000=4.66001$

Lower bound confirmed superseded by T-093 and T-153 (reported)

Overbeck after Levy, Mira-acc, Guzhou0806 · Kleddamag n17 4.66001 · packet · source · review · register

2026-09-27 published T-045 · 15 of these cases

Rectangle-density lower bounds replayed at 15 counts in $n=18\dots 78$$n=18\dots 78$

Lower bound confirmed superseded by T-051, T-063, T-064, T-067, T-068, T-074, T-090, T-091, T-102, T-103, T-104, T-105, T-106, T-107, T-109 and T-157

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

2026-09-27 published T-046 · 48 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-062, T-063, T-064, T-066, T-067, T-068, T-069, T-074, T-082, T-090, T-091, T-094, T-097, T-102, T-103, T-104, T-105, T-106, T-107, T-108, T-109, T-110, T-111, T-157 and T-205

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-052 · case 21

$s(21)=5$$s(21)=5$, by a mixed cover of points and grid-line segments

Optimality confirmed

Daniel after Burns, Massaccesi · evand square-packing 2026-09-28 · packet · source · review · register

2026-09-27 published T-053 · case 45

$s(45)=7$$s(45)=7$, by a mixed cover of points and grid-line segments

Optimality confirmed

Daniel after Burns, Massaccesi · evand square-packing 2026-09-28 · packet · source · review · register

2026-09-27 published T-056 · case 68

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

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

2026-09-28 published T-042 · case 17

$s(17)>233009/50000=4.66018$$s(17)>233009/50000=4.66018$

Lower bound confirmed superseded by T-093 and T-153 (reported)

Guzhou0806 after Overbeck, Mira-acc, Levy · Guzhou0806 n17 R067 · packet · source · review · register

2026-09-28 published T-043 · case 17

$s(17)>116511/25000=4.66044$$s(17)>116511/25000=4.66044$

Lower bound confirmed superseded by T-093 and T-153 (reported)

Guzhou0806 after Overbeck, Mira-acc, Levy · Guzhou0806 n17 R068 · packet · source · review · register

2026-09-28 published T-048 · cases 50, 51

$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-054 · case 45

$s(45)=7$$s(45)=7$ by a second, point-only route

Simplification 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-055 · case 21

$s(21)=5$$s(21)=5$ by a point-only route

Simplification 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-062 · case 60

$s(60)=8$$s(60)=8$, by a mixed cover of points and grid-line segments

Optimality confirmed

Daniel after Burns, Massaccesi · evand square-packing 2026-10-01 · packet · source · review · register

2026-09-28 published T-063 · case 61

$s(61)=8$$s(61)=8$, by monotonicity from T-062

Optimality confirmed

Daniel after Burns, Massaccesi · evand square-packing 2026-10-01 · packet · packet · packet · source 1 · source 2 · review 1 · review 2 · register

2026-09-28 published T-070 · 25 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-062, T-066, T-068, T-074, T-082, T-090, T-091, T-094, T-108, T-110, T-111 and T-205

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 this result · 100 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-059 · case 11

Reported equality of 12028 n11 row minima reproduced by a complete bound replay

Audit confirmed

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

2026-09-29 published T-060 · case 11

Trump's eleven-square packing is globally optimal

Optimality confirmed

Ahmed after Levy, Overbeck · Ahmed n11 optimality 2026 · packet · packet · source 1 · source 2 · review 1 · review 2 · register

2026-09-29 published T-061 · case 11

$s(11)>3875000000/999999999=3.875000003875\dots$$s(11)>3875000000/999999999=3.875000003875\dots$, 3.9e-9 above $31/8$$31/8$

Lower bound confirmed superseded by T-060

Wang, Li after Overbeck, Levy · Wang Li n11 2026 · packet · source 1 · source 2 · review · register

2026-09-29 published T-083 · 85 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

2026-09-29 published T-084 · 8 of these cases

s(k2−1)=k$s(k^2-1)=k$ for every integer $k\ge 3$$k\ge 3$; $k=3\dots 18$$k=3\dots 18$ are the cases held here

Optimality confirmed

Karakuş · Karakuş 2026 · source · register

2026-09-30 published T-064 · 5 of these cases

s(k2−3)=k$s(k^2-3)=k$ for every integer $k$$k$ from 6 up; $k=6\dots 18$$k=6\dots 18$ are the cases held here

Optimality confirmed

Daniel after Burns, Massaccesi · evand square-packing 2026-10-01 · packet · packet · packet · packet · source 1 · source 2 · review 1 · review 2 · register

2026-09-30 published T-093 · case 17

$s(17)>18641771/4000000=4.66044275$$s(17)>18641771/4000000=4.66044275$

Lower bound confirmed

Guzhou0806 after Overbeck, Mira-acc, Levy · Guzhou0806 n17 R071 · packet · source · review · register

2026-10-01 published T-066 · case 59

$s(59)=8$$s(59)=8$, by a mixed cover of points and grid-line segments

Optimality confirmed

Hosono after Daniel, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 exact covers 2026-10-01 · packet · source · review · register

2026-10-01 published T-067 · cases 77, 78

$s(77)=9$$s(77)=9$, by a mixed cover extended from Daniel's $s(60)$$s(60)$ cover

Optimality confirmed

Hosono after Daniel, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 exact covers 2026-10-01 · packet · source · review · register

2026-10-01 published T-068 · 34 of these cases

Rectangle-density lower bounds verified at 34 counts in $n=19\dots 95$$n=19\dots 95$

Lower bound confirmed

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

2026-10-01 published T-069 · 5 of these cases

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 · 31 of these cases

Rectangle-density lower bounds replayed at 31 counts in $n=19\dots 95$$n=19\dots 95$

Lower bound confirmed

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 · cases 84, 85, 86, 87

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

Lower bound confirmed superseded by T-090, T-091, T-206 and T-207

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

2026-10-02 published T-072 · case 76

Mixed rectangle-measure lower bound replayed at $n=76$$n=76$

Lower bound confirmed superseded by T-091

Hosono after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 mixed bounds n76 2026-10-02 · packet · packet · source · review · register

2026-10-02 published T-073 · case 83

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

Lower bound confirmed on these cases, superseded by T-075

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

2026-10-02 published T-075 · 9 of these cases

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-076 · case 82

Linear-measure lower bound replayed at $n=82$$n=82$

Lower bound confirmed

Hosono after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 linear n82 2026-10-02 · packet · source · review · register

2026-10-02 published T-077 · cases 20, 42, 70

Rectangle-density lower bounds replayed at $n=20$$n=20$, 42 and 70

Lower bound confirmed superseded by T-090, T-091 and T-104

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

2026-10-02 published T-078 · case 12

$s(12)\ge 31360/7901=3.9691178\dots$$s(12)\ge 31360/7901=3.9691178\dots$, Daniel's certificate rescaled by $7902/7901$$7902/7901$

Lower bound confirmed superseded by T-095

Ryu after Daniel · squarepacker s12 2026 · packet · packet · source 1 · source 2 · review 1 · review 2 · register

2026-10-02 established T-079 · case 12

$s(12)\ge 15680000/3949423=3.9702002\dots$$s(12)\ge 15680000/3949423=3.9702002\dots$, Daniel's points re-weighted

Lower bound confirmed superseded by T-095

Levy after Daniel · source 1 · source 2 · review · register

2026-10-03 published T-081 · 6 of these cases

s(k2−4)=k$s(k^2-4)=k$ for every integer $k$$k$ from 5 up; $k=5\dots 18$$k=5\dots 18$ are the cases held here

Optimality reviewed

Daniel after Burns, Massaccesi · evand square-packing 2026-10-03 · packet · packet · source · review · register

2026-10-03 published T-082 · 22 of these cases

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 · 17 of these cases

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 · 17 of these cases

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-094 · cases 67, 84

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

Lower bound confirmed

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

2026-10-05 published T-095 · case 12

$s(12)\ge 7943/2000=3.9715$$s(12)\ge 7943/2000=3.9715$, Daniel's points dilated and re-weighted

Lower bound confirmed

Ryu after Daniel, Levy · squarepacker s12 2026-10-05 · packet · packet · source 1 · source 2 · review 1 · review 2 · register

2026-10-05 published T-096 · case 18

Mixed rectangle-measure lower bound replayed at $n=18$$n=18$, on a net the certificate declares

Lower bound confirmed superseded by T-102

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

2026-10-05 published T-097 · case 66

Mixed rectangle-measure lower bound verified at $n=66$$n=66$

Lower bound confirmed

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

2026-10-05 published T-098 · case 68

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-174

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

2026-10-05 published T-101 · 15 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-099 · case 18

Mixed rectangle-measure lower bound verified at $n=18$$n=18$, on a finer declared net

Lower bound confirmed superseded by T-102

Hosono after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 mixed bounds finer net 2026-10-06 · packet · source 1 · source 2 · review · register

2026-10-06 published T-100 · case 19

Mixed rectangle-measure lower bound verified at $n=19$$n=19$, past the best 18-square packing

Lower bound confirmed superseded by T-103

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

2026-10-06 published T-102 · case 18

Mixed rectangle-measure lower bound verified at $n=18$$n=18$, on the finest declared net yet

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-103 · case 19

Mixed rectangle-measure lower bound verified at $n=19$$n=19$, on the finest declared net yet

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-104 · case 20

Mixed rectangle-measure lower bound verified at $n=20$$n=20$, 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-105 · case 26

Mixed rectangle-measure lower bound verified at $n=26$$n=26$, 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-106 · case 27

Mixed rectangle-measure lower bound verified at $n=27$$n=27$, 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-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-109 · case 30

Mixed rectangle-measure lower bound verified at $n=30$$n=30$, 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

2026-10-06 established T-112 · case 11

Trump's packing is the only optimal packing of eleven squares, up to symmetry

Uniqueness confirmed

Levy after Ahmed · packet · register

2026-10-07 published T-116 · case 88

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

Chaoweeraprasit after Ellsworth, Couzo and SQUISH · SQUISH second update 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 superseded by T-174

Rehwaldt after Couzo and earlier contributors · Rehwaldt n68 refinement 2026-10-07 · packet · register

2026-10-07 published T-124 · 66 of these cases

Reported non-strict local minima for 178 source configurations

Restricted optimality recorded

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

2026-10-08 published T-125 · 5 of these cases

Complete rational construction reports at 25 counts

Upper bound confirmed superseded by T-126, T-127, T-146 and T-174

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

2026-10-08 published T-126 · case 51

Undilated n51 construction over Q(sqrt2)

Upper bound confirmed

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

2026-10-08 published T-127 · case 88

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-130 · cases 84, 86

Five follow-up rational refinements from Couzo

Upper bound confirmed superseded by T-174

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

2026-10-09 published T-138 · cases 84, 86

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-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-133 · case 40

$s(40)>335427/50000=6.70854$$s(40)>335427/50000=6.70854$, by a clipped-corner transfer on a 401-direction net

Lower bound confirmed superseded by T-157

Guzhou0806 after Hosono, Tokoharu, Levy, Stromquist, Burns, Massaccesi · Guzhou0806 n40 clipped corner 2026-10-10 · packet · packet · source 1 · source 2 · review · register

2026-10-10 published T-137 · case 70

An exact rational refinement of Xu's packing of 70 squares

Upper bound confirmed superseded by T-146

Deleeuw after Xu, Levy · Deleeuw n70 refinement 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-143 · case 30

Mixed rectangle-measure lower bound $s(30)\ge 2357/400$$s(30)\ge 2357/400$ 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-145 · case 40

$s(40)>1340000/199529=6.7158157\dots$$s(40)>1340000/199529=6.7158157\dots$, by a continuous-pose interval kernel

Lower bound confirmed superseded by T-157

Guzhou0806 after Hosono, Tokoharu, Levy, Stromquist, Burns, Massaccesi · Guzhou0806 n40 continuous pose 2026-10-10 · packet · packet · source 1 · source 2 · review · register

2026-10-10 published T-146 · case 70

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-157 · case 40

$s(40)>3359/500=6.718$$s(40)>3359/500=6.718$, by T-145's continuous-pose kernel on a density with 163 new supports

Lower bound confirmed

Guzhou0806 after Hosono, Tokoharu, Levy, Stromquist, Burns, Massaccesi · Guzhou0806 n40 new supports 2026-10-11 · packet · source 1 · source 2 · review · register

2026-10-12 published T-153 · case 17

Candidate proof that Bidwell's 1998 packing is optimal, $s(17)=4.67553009360455\dots$$s(17)=4.67553009360455\dots$

Optimality reviewed

Overbeck after Levy, Ahmed, Mira-acc, Guzhou0806, Maaz · Overbeck n17 candidate 2026 · packet · register

2026-10-12 published T-154 · case 17

Fitzharding reports a Lean proof that $s(17)=4.67553009360455\dots$$s(17)=4.67553009360455\dots$

Formalization reviewed

Fitzharding after Levy, Ahmed, MacIver · Fitzharding n17 Lean 2026 · packet · source · source build receipt · review 1 · review 2 · register

2026-10-12 established T-155 · 13 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

2026-10-13 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-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-177 · case 44

Mixed rectangle-measure lower bound $s(44)\ge 2799/400$$s(44)\ge 2799/400$ 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-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-180 · 48 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 on these cases, superseded by T-007 (reported), T-048, T-062, T-063, T-064, T-066, T-067, T-069, T-075, T-076, T-081 (reported), T-082, T-083, T-084, T-086, T-090, T-091, T-094, T-097, T-205, T-206 and T-207

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

2026-10-14 published T-184 · case 76

$s(76)=9$$s(76)=9$, by a mixed cover of points and grid-line segments

Optimality reviewed

Greff after Daniel, Hosono · bengreff s76 certificate 2026-10-14 · packet · source · review · register

2026-10-15 published T-191 · case 95

$s(95)=10$$s(95)=10$, by a mixed cover of points and grid-line segments

Optimality reviewed

Greff after Daniel, Hosono · bengreff s95 certificate 2026-10-15 · packet · source · review · 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 88

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-194 · case 19

Fitzharding reports a Lean proof that Wainwright's packing is optimal, s(19)=3+42​/3$s(19)=3+4\sqrt{2}/3$

Optimality reviewed

Fitzharding after Levy, Ahmed · Fitzharding n19 Lean 2026 · packet · source · review · register

2026-10-16 published T-200 · case 38

Mixed rectangle-measure lower bound $s(38)\ge 2631/400$$s(38)\ge 2631/400$ 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-201 · case 40

Mixed rectangle-measure lower bound $s(40)\ge 1347/200$$s(40)\ge 1347/200$ 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-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-203 · case 43

Mixed rectangle-measure lower bound $s(43)\ge 2777/400$$s(43)\ge 2777/400$ 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-204 · case 51

Mixed rectangle-measure lower bound $s(51)\ge 3011/400$$s(51)\ge 3011/400$ 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-205 · case 68

Mixed rectangle-measure lower bound $s(68)\ge 853/100$$s(68)\ge 853/100$ 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-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-207 · case 85

Mixed rectangle-measure lower bound $s(85)\ge 379/40$$s(85)\ge 379/40$ 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

2026-10-16 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-210 · case 89

Mixed rectangle-measure lower bound $s(89)\ge 1937/200$$s(89)\ge 1937/200$ 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-211 · case 91

Mixed rectangle-measure lower bound $s(91)\ge 783/80$$s(91)\ge 783/80$ 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-212 · case 90

Mixed rectangle-measure lower bound $s(90)\ge 489/50$$s(90)\ge 489/50$ reported, on the 201-direction net

Lower bound recorded

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

2026-10-16 published T-213 · case 92

Mixed rectangle-measure lower bound $s(92)\ge 493/50$$s(92)\ge 493/50$ reported, on the 201-direction net

Lower bound recorded

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