T-016: $s(n)\ge 22529/5000$$s(n)\ge 22529/5000$ for $n=18,19$$n=18,19$, by monotonicity from T-015

The cases

Results on these cases

22 results in the register on $n=18,19$$n=18,19$, oldest first, each with what it established and how it stands now.

1. 2005 published T-007 · $n=18,19$$n=18,19$

   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   S3 V0 C1 lower bound incomplete on these cases, superseded by T-102 and T-103

   Nagamochi · Nagamochi 2005 · source · register

2. 2026-08-21 published T-016 this result · $n=18,19$$n=18,19$

   $s(n)\ge 22529/5000$$s(n)\ge 22529/5000$ for $n=18,19$$n=18,19$, by monotonicity from T-015

   S3 V3 C3 lower bound confirmed superseded by T-102 and T-103

   Massaccesi after Burns · Burns–Massaccesi n17 · packet · register

3. 2026-08-31 established T-002 · $n=18$$n=18$

   $s(18)\ge 4426213/1000000$$s(18)\ge 4426213/1000000$, by monotonicity from T-001

   S3 V3 C3 lower bound confirmed superseded by T-102

   Levy after Bentz · register

4. 2026-08-31 established T-003 · $n=18$$n=18$

   The sixteen-point set's unavoidability ceiling lies in $[4426213/1000000,4427/1000)$$[4426213/1000000,4427/1000)$

   S2 V3 C3 method limit confirmed

   Levy after Bentz · register

5. 2026-09-04 established T-019 · $n=18,19$$n=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$

   S4 V3 C3 lower bound confirmed superseded by T-102 and T-103

   Levy after Burns, Massaccesi · register

6. 2026-09-04 established T-020 · $n=19$$n=19$

   $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   S4 V3 C3 lower bound confirmed superseded by T-103

   Levy after Burns, Massaccesi · register

7. 2026-09-04 published T-085 · $n=18,19$$n=18,19$

   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

8. 2026-09-18 established T-027 · $n=18$$n=18$

   $s(18)\ge 467/100=4.67$$s(18)\ge 467/100=4.67$

   S3 V3 C3 lower bound confirmed superseded by T-102

   Levy after Burns, Massaccesi · register

9. 2026-09-19 established T-028 · $n=18$$n=18$

   $s(18)\ge 187/40=4.675$$s(18)\ge 187/40=4.675$

   S3 V3 C3 lower bound confirmed superseded by T-102

   Levy after Burns, Massaccesi · register

10. 2026-09-19 established T-029 · $n=18$$n=18$

    $s(18)\ge 1871/400=4.6775$$s(18)\ge 1871/400=4.6775$

    S3 V3 C3 lower bound confirmed superseded by T-102

    Levy after Burns, Massaccesi · register

11. 2026-09-19 established T-030 · $n=18$$n=18$

    $s(18)\ge 4679/1000=4.679$$s(18)\ge 4679/1000=4.679$

    S3 V3 C3 lower bound confirmed superseded by T-102

    Levy after Burns, Massaccesi · register

12. 2026-09-27 published T-045 · $n=18,19$$n=18,19$

    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-102 and T-103

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

13. 2026-09-27 published T-046 · $n=18,19$$n=18,19$

    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-102 and T-103

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

14. 2026-09-29 published T-058 · $n=18,19$$n=18,19$

    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

15. 2026-09-29 published T-083 · $n=18,19$$n=18,19$

    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-102 and T-103

    Karakuş · Karakuş 2026 · source · register

16. 2026-10-01 published T-068 · $n=19$$n=19$

    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-103

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

17. 2026-10-01 published T-074 · $n=19$$n=19$

    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-103

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

18. 2026-10-05 published T-096 · $n=18$$n=18$

    Mixed rectangle-measure lower bound replayed at $n=18$$n=18$, on a net the certificate declares

    S3 V3 C3 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

19. 2026-10-06 published T-099 · $n=18$$n=18$

    Mixed rectangle-measure lower bound verified at $n=18$$n=18$, on a finer declared net

    S3 V3 C3 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

20. 2026-10-06 published T-100 · $n=19$$n=19$

    Mixed rectangle-measure lower bound verified at $n=19$$n=19$, past the best 18-square packing

    S3 V3 C3 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

21. 2026-10-06 published T-102 · $n=18$$n=18$

    Mixed rectangle-measure lower bound verified at $n=18$$n=18$, on the finest declared net yet

    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

22. 2026-10-06 published T-103 · $n=19$$n=19$

    Mixed rectangle-measure lower bound verified at $n=19$$n=19$, on the finest declared net yet

    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