T-037: $s(11)>31/8=3.875$$s(11)>31/8=3.875$

The case

Results on the case

23 results in the register on $n=11$$n=11$, oldest first, each with what it established and how it stands now.

1. 1979 published T-011

   Trump's 1979 packing is exactly valid, so $s(11)\le 3.877083590022814\dots$$s(11)\le 3.877083590022814\dots$

   S2 V3 C3 upper bound confirmed

   Trump · Trump 2023 · register

2. 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   S3 V0 C1 lower bound incomplete on this case, superseded by T-060

   Nagamochi · Nagamochi 2005 · source · register

3. 2026-08-24 established T-010

   s(11)≥2+4/5​$s(11)\ge 2+4/\sqrt{5}$, by a repair of Stromquist 2003's Figure 14 point set

   S4 V3 C3 lower bound confirmed superseded by T-060

   Levy after Stromquist · register

4. 2026-09-04 established T-018

   $s(11)\ge 381/100=3.81$$s(11)\ge 381/100=3.81$

   S5 V3 C3 lower bound confirmed superseded by T-060

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

6. 2026-09-06 established T-022

   s(11)≥381008100042893309449​/899996306539=3.8100257…

   S5 V3 C3 lower bound confirmed superseded by T-060

   Levy after Burns, Massaccesi · source · register

7. 2026-09-08 established T-023

   Conditional exclusion: no eleven-square packing in the four-owner branch at $q=96/25$$q=96/25$

   S3 V3 C3 case exclusion confirmed superseded in part by T-060

   Levy · source · register

8. 2026-09-09 established T-024

   s(11)≥3175000518400042893309449​/598960960743657=3.8166095…

   S5 V3 C3 lower bound confirmed superseded by T-060

   Levy after Burns, Massaccesi · source · register

9. 2026-09-09 established T-025

   $s(11)\ge 191/50=3.82$$s(11)\ge 191/50=3.82$, by a threshold certificate

   S5 V3 C3 lower bound confirmed superseded by T-060

   Levy · source · register

10. 2026-09-09 established T-026

    s(11)≥955000518400042893309449​/179696714646249=3.8264474…

    S5 V3 C3 lower bound confirmed superseded by T-060

    Levy · source · register

11. 2026-09-20 established T-031

    The octagon corner class (threshold $1/2$$1/2$) holds no eleven-square packing at side $96/25$$96/25$

    S2 V3 C3 case exclusion confirmed superseded by T-060

    Levy · register

12. 2026-09-22 established T-033

    s(11)≥9550002073600042893309449​/359341754646249=3.8269975…

    S3 V3 C3 lower bound confirmed superseded by T-060

    Levy · source · register

13. 2026-09-22 published T-037 this result

    $s(11)>31/8=3.875$$s(11)>31/8=3.875$

    S5 V3 C3 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

14. 2026-09-22 published 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$

    S4 V3 C3 lower bound confirmed superseded by T-060

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

15. 2026-09-24 established T-035

    Six-plus-five packings near Trump's tilt with side ≤Uhi​$\le U_{hi}$ lie within rho of his pose

    S3 V3 C3 case exclusion confirmed

    Levy · register

16. 2026-09-24 established T-036

    Trump's pose is optimal among six-plus-five packings near its tilt, unique up to symmetry

    S3 V3 C2 restricted optimality confirmed superseded in part by T-060 and T-112

    Levy · register

17. 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

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

18. 2026-09-29 published T-059

    Reported equality of 12028 n11 row minima reproduced by a complete bound replay

    S2 V3 C3 audit confirmed

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

19. 2026-09-29 published T-060

    Trump's eleven-square packing is globally optimal

    S5 V3 C3 optimality confirmed

    Ahmed after Levy, Kleddamag · Ahmed n11 optimality 2026 · packet · packet · source 1 · source 2 · review 1 · review 2 · register

20. 2026-09-29 published T-061

    $s(11)>3875000000/999999999=3.875000003875\dots$$s(11)>3875000000/999999999=3.875000003875\dots$, 3.9e-9 above $31/8$$31/8$

    S2 V3 C3 lower bound confirmed superseded by T-060

    Wang, Li after Kleddamag, Levy · Wang Li n11 2026 · packet · source 1 · source 2 · review · register

21. 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-060

    Karakuş · Karakuş 2026 · source · register

22. 2026-10-06 established T-112

    Trump's packing is the only optimal packing of eleven squares, up to symmetry

    S3 V3 C2 uniqueness confirmed

    Levy after Ahmed · packet · register

23. 2026-10-07 published T-124

    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