T-063: $s(61)=8$$s(61)=8$, by monotonicity from T-062

The case

Results on the case

11 results in the register on $n=61$$n=61$, 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   S3 V0 C1 Lower bound incomplete on this case, superseded by T-063

   Nagamochi · Nagamochi 2005 · source · register

2. 2018 published T-087

   s(37)≥53​/2+22​−1$s(37)\ge 5\sqrt{3}/2+2\sqrt{2}-1$ and s(61)≥73​/2+22​−1$s(61)\ge 7\sqrt{3}/2+2\sqrt{2}-1$, by optimal piercing

   S3 V3 C1 Lower bound reviewed superseded by T-063

   Bašić, Slivková · Basic-Slivkova 2018 · source · 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; Vlasenko · Karakuş 2026 · chelokot Nagamochi counterexample 2026 · packet · register

4. 2026-09-27 published T-045

   Rectangle-density lower bounds replayed at 15 counts in $n=18\dots 78$$n=18\dots 78$

   S3 V3 C3 Lower bound confirmed superseded by T-063

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

5. 2026-09-27 published T-046

   Rectangle-density lower bounds reported for 48 counts in $n=18\dots 95$$n=18\dots 95$

   S3 V0 C0 Lower bound recorded superseded by T-063

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

6. 2026-09-28 published T-063 this result

   $s(61)=8$$s(61)=8$, by monotonicity from T-062

   S3 V3 C3 Optimality confirmed

   Daniel after Burns, Massaccesi · evand square-packing 2026-10-01 · packet · packet · packet · source 1 · source 2 · review 1 · review 2 · register

7. 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

8. 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   S3 V3 C3 Lower bound confirmed on this case, superseded by T-063

   Karakuş · Karakuş 2026 · source · register

9. 2026-09-30 published T-064

   s(k2−3)=k$s(k^2-3)=k$ for every integer $k$$k$ from 6 up; $k=6\dots 18$$k=6\dots 18$ are the cases held here

   S4 V3 C3 Optimality confirmed on this case, second certificate

   Daniel after Burns, Massaccesi · evand square-packing 2026-10-01 · packet · packet · 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

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-063

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