T-055: $s(21)=5$$s(21)=5$ by a point-only route

The case

Results on the case

12 results in the register on $n=21$$n=21$, 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   S3 V0 C1 lower bound incomplete on this case, superseded by T-052

   Nagamochi · Nagamochi 2005 · source · register

2. 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   S4 V3 C3 lower bound confirmed superseded by T-052

   Levy after Burns, Massaccesi · register

3. 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

4. 2026-09-05 established T-021

   $s(n)\ge 97/20=4.85$$s(n)\ge 97/20=4.85$ for $n=20,21$$n=20,21$

   S3 V3 C3 lower bound confirmed superseded by T-052

   Levy after Burns, Massaccesi · register

5. 2026-09-23 established T-034

   $s(21)\ge 122/25=4.88$$s(21)\ge 122/25=4.88$

   S3 V3 C3 lower bound confirmed superseded by T-052

   Levy after Burns, Massaccesi · register

6. 2026-09-23 published T-050

   $s(21)\ge 5000/1001=4.995004995\dots$$s(21)\ge 5000/1001=4.995004995\dots$

   S3 V3 C3 lower bound confirmed superseded by T-052

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

7. 2026-09-27 published T-052

   $s(21)=5$$s(21)=5$, by a mixed cover of points and grid-line segments

   S4 V3 C3 optimality confirmed

   Daniel after Burns, Massaccesi · evand square-packing 2026-09-28 · packet · source · review · register

8. 2026-09-28 published T-055 this result

   $s(21)=5$$s(21)=5$ by a point-only route

   S2 V3 C3 simplification confirmed

   wand125 after Daniel, Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 point and mixed bounds 2026-09-28 · packet · source · review · register

9. 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

10. 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-052

    Karakuş · Karakuş 2026 · source · register

11. 2026-10-03 published T-081

    s(k2−4)=k$s(k^2-4)=k$ for every integer $k$$k$ from 5 up; $k=5\dots 18$$k=5\dots 18$ are the cases held here

    S4 V0 C1 optimality reviewed on this case, second certificate, reported

    Daniel after Burns, Massaccesi · evand square-packing 2026-10-03 · packet · packet · source · review · register

12. 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