T-047: $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$

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 · 7 of these cases

s(n)≥min(⌈n​⌉,n−2⌊n​⌋+1​+1)$s(n)\ge min(\lceil \sqrt{n}\rceil ,\sqrt{n-2\lfloor \sqrt{n}\rfloor +1}+1)$ for $4\le n\le 324$$4\le n\le 324$

lower bound incomplete on these cases, superseded by T-060, T-068, T-074, T-105, T-106, T-107, T-108 and T-109

Nagamochi · Nagamochi 2005 · source · 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lower bound confirmed superseded by T-060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

2026-08-29 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

2026-09-04 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lower bound confirmed superseded by T-060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 published T-085 · 7 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

2026-09-06 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lower bound confirmed superseded by T-060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case exclusion confirmed superseded in part by T-060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lower bound confirmed superseded by T-060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lower bound confirmed superseded by T-060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lower bound confirmed superseded by T-060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

2026-09-20 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

2026-09-22 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 · cases 26, 29

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-105 and T-108

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

2026-09-22 published T-047 this result · 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

2026-09-24 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

2026-09-27 published T-045 · 5 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-068, T-074, T-105, T-106, T-107 and T-109

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

2026-09-27 published T-046 · 6 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-068, T-074, T-105, T-106, T-107, T-108 and T-109

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

2026-09-28 published T-070 · case 29

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

lower bound confirmed superseded by T-108

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 · 7 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 · 7 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 on these cases, superseded by T-060, T-068, T-074, T-105, T-106, T-107, T-108 and T-109

Karakuş · Karakuş 2026 · source · register

2026-10-01 published T-068 · 6 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-074 · 6 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

2026-10-05 published T-101 · case 28

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-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 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-124 · cases 11, 28, 30, 31

Reported non-strict local minima for 178 source configurations

restricted optimality recorded

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

2026-10-10 published T-132 · case 29

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

lower bound recorded

wand125 after Tokoharu and Levy · wand125 fine-net n29 2026-10-10 · packet · register

2026-10-10 published T-142 · case 28

Mixed rectangle-measure lower bound $s(28)\ge 2297/400$$s(28)\ge 2297/400$ reported, on a 2,073-direction declared net

lower bound recorded

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