T-245: Jatloe reports a Lean proof that the 4 × 3 grid is optimal, $s(12)=4$$s(12)=4$

The case

Results on the case

13 results in the register on $n=12$$n=12$, 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   S3 V0 C1 Lower bound incomplete on this case, superseded by T-095 and T-245 (reported)

   Nagamochi · Nagamochi 2005 · source · register

2. 2026-08-25 published T-049

   $s(12)\ge 15680/3951=3.9686155\dots$$s(12)\ge 15680/3951=3.9686155\dots$

   S3 V3 C3 Lower bound confirmed superseded by T-095 and T-245 (reported)

   Daniel after Burns, Massaccesi · evand square-packing 2026 · packet · source 1 · source 2 · review 1 · review 2 · register

3. 2026-09-04 established T-017

   $s(12)\ge 99/25=3.96$$s(12)\ge 99/25=3.96$

   S4 V3 C3 Lower bound confirmed superseded by T-095 and T-245 (reported)

   Levy after Burns, Massaccesi · register

4. 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

5. 2026-09-28 published T-249

   Daniel reports a Lean kernel proof that $s(12)\ge 15680/3951$$s(12)\ge 15680/3951$, the bound T-049 holds

   S2 V0 C0 Formalization recorded

   Daniel after Burns, Massaccesi · Daniel S12H Lean 2026-09-28 · packet · source · register

6. 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

7. 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   S3 V3 C3 Lower bound confirmed on this case, superseded by T-095 and T-245 (reported)

   Karakuş · Karakuş 2026 · source · register

8. 2026-10-02 published T-078

   $s(12)\ge 31360/7901=3.9691178\dots$$s(12)\ge 31360/7901=3.9691178\dots$, Daniel's certificate rescaled by $7902/7901$$7902/7901$

   S2 V3 C3 Lower bound confirmed superseded by T-095 and T-245 (reported)

   Ryu after Daniel · squarepacker s12 2026 · packet · packet · source 1 · source 2 · review 1 · review 2 · register

9. 2026-10-02 established T-079

   $s(12)\ge 15680000/3949423=3.9702002\dots$$s(12)\ge 15680000/3949423=3.9702002\dots$, Daniel's points re-weighted

   S3 V3 C3 Lower bound confirmed superseded by T-095 and T-245 (reported)

   Levy after Daniel · source 1 · source 2 · review · register

10. 2026-10-05 published T-095

    $s(12)\ge 7943/2000=3.9715$$s(12)\ge 7943/2000=3.9715$, Daniel's points dilated and re-weighted

    S3 V3 C3 Lower bound confirmed

    Ryu after Daniel, Levy · squarepacker s12 2026-10-05 · packet · packet · source 1 · source 2 · review 1 · review 2 · register

11. 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

12. 2026-10-12 published T-261

    Reported grade-C local minima for 233 source configurations of field degree at most 4

    S2 V3 C1 Restricted optimality incomplete

    Daniel after Schadt, Ellsworth, Levy · Daniel exact and local reports 2026-10-12 · packet · packet · register

13. 2026-10-16 published T-245 this result

    Jatloe reports a Lean proof that the 4 × 3 grid is optimal, $s(12)=4$$s(12)=4$

    S5 V0 C0 Optimality recorded

    jatloe, Gurevitch, ctjlewis, Ahmed · Jatloe n12 Lean 2026-10-16 · packet · source · review · register