T-194: Fitzharding reports a Lean proof that Wainwright's packing is optimal, s(19)=3+42​/3$s(19)=3+4\sqrt{2}/3$

The case

Results on the case

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

1. 2005 published T-007

   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   S3 V0 C1 Lower bound incomplete on this case, superseded by T-103

   Nagamochi · Nagamochi 2005 · source · register

2. 2026-08-21 published T-016

   $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   S3 V3 C3 Lower bound confirmed superseded by T-103

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

3. 2026-09-04 established T-019

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   S4 V3 C3 Lower bound confirmed superseded by T-103

   Levy after Burns, Massaccesi · register

4. 2026-09-04 established T-020

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   S4 V3 C3 Lower bound confirmed superseded by T-103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

5. 2026-09-04 published T-085

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

6. 2026-09-27 published T-045

   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   S3 V3 C3 Lower bound confirmed superseded by T-103

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

7. 2026-09-27 published T-046

   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   S3 V0 C0 Lower bound recorded superseded by T-103

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

8. 2026-09-29 published T-058

   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

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

9. 2026-09-29 published T-083

   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 this case, superseded by T-103

   Karakuş · Karakuş 2026 · source · register

10. 2026-10-01 published T-068

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

    S3 V3 C3 Lower bound confirmed on this case, superseded by T-103

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

11. 2026-10-01 published T-074

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

    S3 V3 C3 Lower bound confirmed on this case, superseded by T-103

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

12. 2026-10-06 published T-100

    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

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

13. 2026-10-06 published T-103

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

    S3 V3 C3 Lower bound confirmed

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

14. 2026-10-16 published T-194 this result

    Fitzharding reports a Lean proof that Wainwright's packing is optimal, s(19)=3+42​/3$s(19)=3+4\sqrt{2}/3$

    S5 V0 C1 Optimality reviewed

    Fitzharding after Levy, Ahmed · Fitzharding n19 Lean 2026 · packet · source · review · register