T-077: Rectangle-density lower bounds replayed at $n=20$$n=20$, 42 and 70

The cases

Results on these cases

22 results in the register on $n=20,42,70$$n=20,42,70$, oldest first, each with what it established and how it stands now.

1. 2005 published T-007 · $n=20,42,70$$n=20,42,70$

   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$

   S3 V0 C1 lower bound incomplete on these cases, superseded by T-090, T-091 and T-104

   Nagamochi · Nagamochi 2005 · source · register

2. 2026-09-04 established T-020 · $n=20$$n=20$

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

   S4 V3 C3 lower bound confirmed superseded by T-104

   Levy after Burns, Massaccesi · register

3. 2026-09-04 published T-085 · $n=20,42,70$$n=20,42,70$

   Nagamochi 2005, Lemma 1 is false for every container with $a>3$$a>3$ and $b>2$$b>2$

   S3 V3 C3 correction confirmed

   Karakuş; chelokot · Karakuş 2026 · chelokot Nagamochi counterexample 2026 · packet · register

4. 2026-09-05 established T-021 · $n=20$$n=20$

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

   S3 V3 C3 lower bound confirmed superseded by T-104

   Levy after Burns, Massaccesi · register

5. 2026-09-22 published T-044 · $n=70$$n=70$

   Weighted point lower bounds for ten counts in $n=26\dots 72$$n=26\dots 72$, plus seven from the same files

   S3 V3 C3 lower bound confirmed superseded by T-091

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

6. 2026-09-27 published T-045 · $n=20$$n=20$

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

   S3 V3 C3 lower bound confirmed superseded by T-104

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

7. 2026-09-27 published T-046 · $n=20,42,70$$n=20,42,70$

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

   S3 V0 C0 lower bound recorded superseded by T-090, T-091 and T-104

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

8. 2026-09-28 published T-070 · $n=42,70$$n=42,70$

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

   S3 V3 C3 lower bound confirmed superseded by T-090 and T-091

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

9. 2026-09-29 published T-058 · $n=20,42,70$$n=20,42,70$

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

   S2 V3 C3 method limit confirmed

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

10. 2026-09-29 published T-083 · $n=20,42,70$$n=20,42,70$

    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$

    S3 V3 C3 lower bound confirmed on these cases, superseded by T-090, T-091 and T-104

    Karakuş · Karakuş 2026 · source · register

11. 2026-10-01 published T-068 · $n=20,42,70$$n=20,42,70$

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

    S3 V3 C3 lower bound confirmed on these cases, superseded by T-090, T-091 and T-104

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

12. 2026-10-01 published T-074 · $n=20,42,70$$n=20,42,70$

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

    S3 V3 C3 lower bound confirmed on these cases, superseded by T-090, T-091 and T-104

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

13. 2026-10-02 published T-077 this result · $n=20,42,70$$n=20,42,70$

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

    S3 V3 C3 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

14. 2026-10-03 published T-082 · $n=70$$n=70$

    Mixed rectangle-measure lower bounds verified at 22 counts in $n=51\dots 96$$n=51\dots 96$

    S3 V3 C3 lower bound confirmed on these cases, superseded by T-091

    wand125 after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 mixed bounds 2026-10-03 · packet · packet · source · review · register

15. 2026-10-04 published T-090 · $n=42$$n=42$

    Mixed rectangle-measure lower bounds verified at 17 counts in $n=42\dots 96$$n=42\dots 96$

    S3 V3 C3 lower bound confirmed

    wand125 after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 mixed bounds 2026-10-04 · packet · source · review · register

16. 2026-10-04 published T-091 · $n=70$$n=70$

    Mixed rectangle-measure lower bounds verified at 17 counts in $n=53\dots 95$$n=53\dots 95$

    S3 V3 C3 lower bound confirmed

    wand125 after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 mixed bounds evening 2026-10-04 · packet · source · review · register

17. 2026-10-05 published T-101 · $n=70$$n=70$

    Exact certificates of 77 catalogue packings: s(n) ≤ S', 1.2e-16 to 9.8e-15 above each side

    S2 V3 C3 upper bound confirmed on these cases, superseded by T-125

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

18. 2026-10-06 published T-104 · $n=20$$n=20$

    Mixed rectangle-measure lower bound verified at $n=20$$n=20$, on a declared net

    S3 V3 C3 lower bound confirmed

    wand125 after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 mixed bounds check2 2026-10-06 · packet · source · review · register

19. 2026-10-07 published T-124 · $n=20,42$$n=20,42$

    Reported non-strict local minima for 178 source configurations

    S2 V0 C0 restricted optimality recorded

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

20. 2026-10-08 published T-125 · $n=70$$n=70$

    Complete rational construction reports at 25 counts

    S3 V3 C3 upper bound confirmed

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

21. 2026-10-10 published T-137 · $n=70$$n=70$

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

    S2 V3 C3 upper bound confirmed

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

22. 2026-10-10 published T-146 · $n=70$$n=70$

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

    S2 V3 C3 upper bound confirmed

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