T-049: $s(12)\ge 15680/3951=3.9686155\dots$$s(12)\ge 15680/3951=3.9686155\dots$

The case

Results on the case

10 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   S3 V0 C1 lower bound incomplete on this case, superseded by T-095

   Nagamochi · Nagamochi 2005 · source · register

2. 2026-08-25 published T-049 this result

   $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   S3 V3 C3 lower bound confirmed superseded by T-095

   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   S4 V3 C3 lower bound confirmed superseded by T-095

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

5. 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

6. 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   S3 V3 C3 lower bound confirmed on this case, superseded by T-095

   Karakuş · Karakuş 2026 · source · register

7. 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   S2 V3 C3 lower bound confirmed superseded by T-095

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

8. 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   S3 V3 C3 lower bound confirmed superseded by T-095

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

9. 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

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

10. 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