T-261: Reported grade-C local minima for 233 source configurations of field degree at most 4

1979 published T-237 · cases 2, 3, 4, 5

$s(2),s(3),s(4)=2$$s(2),s(3),s(4)=2$ and s(5)=2+1/2​=2.70710678…$s(5)=2+1/\sqrt{2}=2.70710678\dots$, the classical first cases

Optimality confirmed

Göbel; Friedman · Göbel 1979 · Friedman DS7 · source · register

2005 published T-007 · 230 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 · case 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Lower bound reviewed superseded by T-063

Bašić, Slivková · Basic-Slivkova 2018 · source · 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Lower bound confirmed superseded by T-102 and T-103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

2026-08-25 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Lower bound confirmed superseded by T-095 and T-245 (reported)

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

2026-08-29 published T-238 · cases 2, 3, 4, 5

chelokot reports Lean proofs of $s(2),s(3),s(4)=2$$s(2),s(3),s(4)=2$, $s(5)=2.707\dots$$s(5)=2.707\dots$; Daniel, rectangle theorems

Formalization recorded

Vlasenko; Daniel · chelokot small-n Lean 2026-09-05 · Daniel small-n rectangles Lean 2026-10-22 · packet · packet · source 1 · source 2 · 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-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 · case 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 and T-245 (reported)

Levy after Burns, Massaccesi · register

2026-09-04 established T-019 · cases 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-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 · 224 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 · 16 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

2026-09-18 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

2026-09-22 published T-044 · 10 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 and T-157

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

2026-09-22 published T-047 · cases 26, 27, 30, 31

$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-105, T-106, T-109 and T-232

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

2026-09-26 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-045 · 14 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-090, T-102, T-103, T-104, T-105, T-106, T-109, T-157, T-184 and T-232

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

2026-09-27 published T-046 · 37 of these cases

Rectangle-density lower bounds verified at 48 counts in $n=18\dots 95$$n=18\dots 95$

Lower bound confirmed superseded by T-062, T-063, T-064, T-066, T-067, T-068, T-074, T-082, T-090, T-091, T-094, T-097, T-102, T-103, T-104, T-105, T-106, T-109, T-157, T-184, T-191, T-232 and T-252

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

2026-09-28 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-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 · 17 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-191 and T-252

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

2026-09-28 published T-249 · case 12

Daniel reports a Lean kernel proof that $s(12)\ge 15680/3951$$s(12)\ge 15680/3951$, the bound T-049 holds

Formalization recorded

Daniel after Burns, Massaccesi · Daniel S12H Lean 2026-09-28 · packet · source · register

2026-09-29 published T-058 · 85 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 · 210 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 · 16 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

2026-09-29 published T-234 · case 21

Hosono reports a hypothesis-free Lean proof that $s(21)=5$$s(21)=5$, by the point-only route

Formalization recorded

Hosono after Daniel, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 n21 zero-margin Lean 2026-09-29 · packet · source · register

2026-09-30 published T-064 · 13 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

2026-10-01 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 · 25 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 · cases 65, 66, 90, 92

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 · 23 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

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

Lower bound confirmed superseded by T-090, 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-184

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 · cases 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 · 6 of these cases

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-081 (reported), T-082, T-090, T-091, T-191 and T-207

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 and T-245 (reported)

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 and T-245 (reported)

Levy after Daniel · 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-081 · 14 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 · 16 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 · 14 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 · 14 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-101 · 47 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-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

2026-10-07 published T-114 · case 153

$s(153)\le 7250614903299225/562949953421312$, from a SQUISH rational packing

Upper bound confirmed superseded by T-174

Chaoweeraprasit after Ellsworth · SQUISH n153 2026-10-07 · packet · packet · register

2026-10-07 published T-115 · case 258

s(n)≤Sn​$s(n)\le S_n$ at twelve counts, from the SQUISH update’s new and smaller rational packings

Upper bound confirmed superseded by T-193

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

2026-10-07 published T-124 · 169 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 · 9 of these cases

Complete rational construction reports at 25 counts

Upper bound confirmed superseded by T-134, T-174, T-186 and T-193

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

2026-10-08 published T-127 · case 153

Fourteen rational refinements, seventeen complete source cases

Upper bound confirmed on these cases, superseded by T-174

Gupta after Chaoweeraprasit, Daniel · Gupta rational refinements 2026-10-08 · packet · register

2026-10-08 published T-128 · case 127

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 · cases 84, 86, 175

Five follow-up rational refinements from Couzo

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

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

2026-10-08 published T-233 · cases 19, 20, 26, 27

Mixed rectangle-measure lower bounds reported at $n=19,20,26,27$$n=19,20,26,27$, on a 2,073-direction net

Lower bound reviewed

Hosono after Tokoharu and Levy · wand125 fine-net lower bounds 2026-10-08 · packet · packet · register

2026-10-09 published T-134 · case 267

Six exact rational certificates from Couzo

Upper bound confirmed

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

2026-10-09 published T-136 · cases 153, 232

Seventeen exact rational packings from SQUISH's third request

Upper bound confirmed superseded by T-174 and T-256

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

2026-10-09 published T-138 · 6 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-134, T-174, T-186 and T-193

Polkovskyi after Stenlund, Friedman, Ellsworth, Chaoweeraprasit, Couzo, Xu, Gupta, Daniel, Levy · Mishapolk decimal poses 2026-10-09 · 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-174

Deleeuw after Xu, Levy · Deleeuw n70 refinement 2026-10-10 · packet · register

2026-10-10 published T-140 · case 175

Six more exact rational certificates from Couzo

Upper bound confirmed superseded by T-186

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

2026-10-10 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 · packet · source 1 · source 2 · source 3 · review 1 · review 2 · register

2026-10-10 published T-146 · cases 70, 153, 258

Exact rational certificates at fifteen counts from Daniel's regularized record lists

Upper bound confirmed on these cases, superseded by T-174 and T-193

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

2026-10-11 published T-148 · case 153

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

Upper bound confirmed superseded by T-174

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

2026-10-11 published T-150 · case 261

Exact rational refinements of Xu's packings of 146 and 261 squares

Upper bound confirmed superseded by T-174

Barnett after Xu, Daniel, Levy · Barnett refinements 2026-10-11 · packet · register

2026-10-11 published T-151 · case 176

Nine exact rational packings from SQUISH's fourth request

Upper bound confirmed superseded by T-186

Chaoweeraprasit after Couzo, Gupta, Ellsworth, Friedman, Levy · SQUISH fourth request 2026-10-11 · 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 · packet · source 1 · source 2 · source 3 · review 1 · review 2 · 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

2026-10-11 published T-232 · case 31

Mixed rectangle-measure lower bound $s(31)\ge 2391/400$$s(31)\ge 2391/400$ verified, on a 2,073-direction declared net

Lower bound confirmed

Hosono after Tokoharu and Levy · wand125 fine-net n31 2026-10-11 · packet · source · review · register

2026-10-11 published T-235 · case 59

Hosono reports a Lean kernel proof that $s(59)=8$$s(59)=8$, on Daniel's mixed-measure checker

Formalization recorded

Hosono after Daniel, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 n59 Lean kernel 2026-10-11 · packet · source · register

2026-10-12 established T-155 · 36 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 · cases 175, 232, 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 and T-256

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

2026-10-12 published T-161 · case 176

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

Upper bound confirmed superseded by T-186

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

2026-10-12 published T-165 · case 176

A tighter certificate at 176 from Couzo, published with no issue

Upper bound confirmed superseded by T-186

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

2026-10-12 published T-166 · case 153

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

Upper bound confirmed superseded by T-174

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

2026-10-12 published T-169 · case 261

A further exact refinement of Xu's packing of 261 squares, after Barnett's

Upper bound confirmed superseded by T-174

PingYou Ltd after Xu, Barnett, Daniel, Levy · PingYou n261 refinement 2026-10-12 · packet · register

2026-10-12 published T-173 · case 153

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

Upper bound confirmed superseded by T-174

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

2026-10-12 published T-261 this result · 233 of these cases

Reported grade-C local minima for 233 source configurations of field degree at most 4

Restricted optimality incomplete

Daniel after Schadt, Ellsworth, Levy · Daniel exact and local reports 2026-10-12 · packet · packet · register

2026-10-13 established T-156 · case 290

$s(290)\le 17.5355339059327376\dots$$s(290)\le 17.5355339059327376\dots$, by monotonicity from T-155, with its own exact certificate

Upper bound confirmed

Levy after Daniel · register

2026-10-13 published T-174 · 12 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-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 superseded by T-252

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 · 180 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-184 · case 76

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

Optimality confirmed

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

2026-10-14 published T-186 · cases 175, 176, 232, 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-191 · cases 95, 96

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

Optimality confirmed

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

2026-10-16 published T-193 · cases 175, 232, 258, 267

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-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-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-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

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

2026-10-16 published T-245 · case 12

Jatloe reports a Lean proof that the 4 × 3 grid is optimal, $s(12)=4$$s(12)=4$

Optimality recorded

jatloe, Gurevitch, ctjlewis, Ahmed · Jatloe n12 Lean 2026-10-16 · packet · source · review · register

2026-10-16 published T-251 · 11 of these cases

Hosono reports a Lean kernel proof of Daniel's ValidTilt9, so s(k2−4)=k$s(k^2-4)=k$ for every $k\ge 8$$k\ge 8$

Formalization recorded

Hosono after Daniel, Levy, Burns, Massaccesi · wand125 ValidTilt9 Lean 2026-10-16 · packet · source · 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 and T-257

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-227 · cases 227, 229

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

Upper bound confirmed superseded by T-225

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

2026-10-17 published T-250 · 13 of these cases

Hosono reports a Lean kernel proof of Daniel's Valid7, and so of s(k2−3)=k$s(k^2-3)=k$ for every $k\ge 6$$k\ge 6$