T-069: Mixed rectangle-measure lower bounds replayed at $n=37,65,66,90,92$$n=37,65,66,90,92$

2005 published T-007 · 5 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lower bound incomplete on these cases, superseded by T-069, T-082, T-091 and T-097

Nagamochi · Nagamochi 2005 · source · register

2018 published T-087 · case 37

s(37)≥53​/2+22​−1$s(37)\ge 5\sqrt{3}/2+2\sqrt{2}-1$ and s(61)≥73​/2+22​−1$s(61)\ge 7\sqrt{3}/2+2\sqrt{2}-1$, by optimal piercing

lower bound reviewed superseded by T-069

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

2026-09-04 published T-085 · 5 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

2026-09-27 published T-046 · cases 37, 66, 90

Rectangle-density lower bounds reported for 48 counts in $n=18\dots 95$$n=18\dots 95$

lower bound recorded superseded by T-069, T-091 and T-097

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

2026-09-29 published T-058 · 5 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-083 · 5 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lower bound confirmed on these cases, superseded by T-069, T-082, T-091 and T-097

Karakuş · Karakuş 2026 · source · register

2026-10-01 published T-068 · cases 66, 90

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

lower bound confirmed on these cases, superseded by T-091 and T-097

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

2026-10-01 published T-069 this result · 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

2026-10-02 published T-075 · case 92

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-082

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

2026-10-03 published T-082 · cases 90, 92

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-091 · cases 90, 92

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-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-101 · case 37

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

2026-10-07 published T-124 · cases 90, 92

Reported non-strict local minima for 178 source configurations

restricted optimality recorded

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