T-208: Mixed rectangle-measure lower bound $s(86)\ge 3811/400$$s(86)\ge 3811/400$ reported, on the 201-direction net

The case

Results on the case

17 results in the register on $n=86$$n=86$, 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   S3 V0 C1 Lower bound incomplete on this case, superseded by T-090

   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 reported for 48 counts in $n=18\dots 95$$n=18\dots 95$

   S3 V0 C0 Lower bound recorded superseded by T-090

   Hosono after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 rectangle bounds 2026 · wand125 rectangle bounds 2026-09-28 · packet · packet · 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   S3 V3 C3 Lower bound confirmed superseded by T-090

   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   S3 V3 C3 Lower bound confirmed on this case, superseded by T-090

   Karakuş · Karakuş 2026 · source · register

7. 2026-10-01 published T-068

   Rectangle-density lower bounds verified at 34 counts in $n=19\dots 95$$n=19\dots 95$

   S3 V3 C3 Lower bound confirmed on this case, superseded by T-090

   Hosono after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 rectangle bounds 2026-10-01 · packet · source · review · register

8. 2026-10-02 published T-071

   Mixed rectangle-measure lower bounds replayed at $n=84\dots 87$$n=84\dots 87$

   S3 V3 C3 Lower bound confirmed superseded by T-090

   Hosono after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 mixed bounds 2026-10-02 · packet · source · review · register

9. 2026-10-02 published T-075

   Mixed rectangle-measure lower bounds replayed at nine counts in $n=83\dots 96$$n=83\dots 96$

   S3 V3 C3 Lower bound confirmed on this case, superseded by T-090

   Hosono after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 mixed bounds afternoon 2026-10-02 · packet · source 1 · source 2 · review · register

10. 2026-10-03 published T-082

    Mixed rectangle-measure lower bounds verified at 22 counts in $n=51\dots 96$$n=51\dots 96$

    S3 V3 C3 Lower bound confirmed on this case, superseded by T-090

    Hosono after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 mixed bounds 2026-10-03 · packet · packet · source · review · register

11. 2026-10-04 published T-090

    Mixed rectangle-measure lower bounds verified at 17 counts in $n=42\dots 96$$n=42\dots 96$

    S3 V3 C3 Lower bound confirmed

    Hosono after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 mixed bounds 2026-10-04 · packet · source · review · register

12. 2026-10-08 published T-125

    Complete rational construction reports at 25 counts

    S3 V3 C3 Upper bound confirmed superseded by T-174

    Xu · ry-xu square packing 2026 · packet · register

13. 2026-10-08 published T-130

    Five follow-up rational refinements from Couzo

    S3 V3 C3 Upper bound confirmed superseded by T-174

    Couzo after Xu, Daniel, Ellsworth, Levy · Couzo follow-up refinements 2026-10-08 · packet · register

14. 2026-10-09 published T-138

    Exact witnesses at Mishapolk's printed ceilings, the smallest known at 103 and 258 when registered

    S3 V3 C3 Upper bound confirmed superseded by T-174

    Polkovskyi after Stenlund, Friedman, Ellsworth, Chaoweeraprasit, Couzo, Xu, Levy · Mishapolk decimal poses 2026-10-09 · packet · register

15. 2026-10-13 published T-174

    Tighter certificates at 68 counts from PingYou Ltd, of others' packings and its own

    S1 V3 C3 Upper bound confirmed

    PingYou Ltd after Friedman, Ellsworth, Daniel, Xu, Barnett, Couzo, Ostashev, Chaoweeraprasit, Gupta, Fang, Polkovskyi, Levy · PingYou PR 574 certificates 2026-10-13 · packet · register

16. 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-090

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

17. 2026-10-16 published T-208 this result

    Mixed rectangle-measure lower bound $s(86)\ge 3811/400$$s(86)\ge 3811/400$ reported, on the 201-direction net

    S3 V0 C0 Lower bound recorded

    Hosono after Tokoharu and Levy · wand125 fine-net lower bounds 2026-10-16 · packet · register