T-235: Hosono reports a Lean kernel proof that $s(59)=8$$s(59)=8$, on Daniel's mixed-measure checker

The case

Results on the case

11 results in the register on $n=59$$n=59$, 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   S3 V0 C1 Lower bound incomplete on this case, superseded by T-066

   Nagamochi · Nagamochi 2005 · source · register

2. 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

3. 2026-09-27 published T-046

   Rectangle-density lower bounds verified at 48 counts in $n=18\dots 95$$n=18\dots 95$

   S3 V3 C3 Lower bound confirmed superseded by T-066

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

4. 2026-09-28 published T-070

   Rectangle-density lower bounds replayed at 25 counts in $n=29\dots 95$$n=29\dots 95$

   S3 V3 C3 Lower bound confirmed superseded by T-066

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

   Hosono 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   S3 V3 C3 Lower bound confirmed on this case, superseded by T-066

   Karakuş · Karakuş 2026 · source · register

7. 2026-10-01 published T-066

   $s(59)=8$$s(59)=8$, by a mixed cover of points and grid-line segments

   S3 V3 C3 Optimality confirmed

   Hosono after Daniel, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 exact covers 2026-10-01 · packet · source · review · register

8. 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

9. 2026-10-11 published T-235 this result

   Hosono reports a Lean kernel proof that $s(59)=8$$s(59)=8$, on Daniel's mixed-measure checker

   S2 V0 C0 Formalization recorded

   Hosono after Daniel, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 n59 Lean kernel 2026-10-11 · packet · source · register

10. 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

11. 2026-10-13 published T-180

    Closed-form lower bound $s(N)\ge a(N)$$s(N)\ge a(N)$ for every non-square $N\ge 49$$N\ge 49$, reported

    S4 V0 C1 Lower bound reviewed on this case, superseded by T-066

    Hosono · wand125 closed form 2026-10-14 · packet · packet · register