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lower bound reviewed superseded by T-063 and T-069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lower bound confirmed superseded by T-093

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lower bound confirmed superseded by T-102 and T-103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lower bound confirmed superseded by T-060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lower bound confirmed superseded by T-095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 5.933833\dots$$s(29)\le 5.933833\dots$, by a Krawczyk interval certificate

upper bound confirmed

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

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 and T-103

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; chelokot · 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

chelokot · 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

Guzhou0806, Mira 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

Kleddamag after Levy, Mira, 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

Kleddamag after Levy, Mira, 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

Kleddamag after Levy, Guzhou0806, Mira · 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-068, T-074, T-082, T-090, T-091, T-105, T-108, T-110, T-111 and T-145

wand125 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, wand125, 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

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

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

Guzhou0806 after Kleddamag, Mira, 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

Kleddamag after Levy, Mira, 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

Kleddamag after Levy, Mira, 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-145

wand125 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 and T-145

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-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-118

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

Guzhou0806 after Kleddamag, Mira, 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

Guzhou0806 after Kleddamag, Mira, 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

wand125 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

wand125 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

wand125 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 and T-111

wand125 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

wand125 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

wand125 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, Kleddamag · 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 Kleddamag, 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 Kleddamag, Mira, 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

wand125 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

wand125 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

wand125 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

wand125 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

wand125 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-075, T-090, T-091 and T-094

wand125 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

wand125 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

wand125 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

wand125 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

wand125 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

wand125 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

squarepacker 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

wand125 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

wand125 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

wand125 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

wand125 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

squarepacker 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

wand125 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

wand125 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-118

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

wand125 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

wand125 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

wand125 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

wand125 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

wand125 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

wand125 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

wand125 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

wand125 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

wand125 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

wand125 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

wand125 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

wand125 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 on these cases, 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

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

ry-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

ry-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 Francisco Couzo

upper bound confirmed

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

Mishapolk 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

wand125 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-145

Guzhou0806 after wand125, 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 Ryan Xu's packing of 70 squares

upper bound confirmed

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

wand125 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

wand125 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

Guzhou0806 after wand125, 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 Evan Daniel's regularized record lists

upper bound confirmed

Daniel after Xu, Chaoweeraprasit, Mishapolk, Levy · Daniel regularized lists 2026-10-10 · packet · register