T-176: Mixed rectangle-measure lower bound $s(39)\ge 533/80$$s(39)\ge 533/80$ reported, on a 2,073-direction declared net

The case

Results on the case

13 results in the register on $n=39$$n=39$, 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   S3 V0 C1 Lower bound incomplete on this case, superseded by T-110

   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. 2026-09-22 published T-044

   Weighted point lower bounds for ten counts in $n=26\dots 72$$n=26\dots 72$, plus seven from the same files

   S3 V3 C3 Lower bound confirmed superseded by T-110

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

4. 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   S3 V0 C0 Lower bound recorded superseded by T-110

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

5. 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   S3 V3 C3 Lower bound confirmed superseded by T-110

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

6. 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

7. 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   S3 V3 C3 Lower bound confirmed on this case, superseded by T-110

   Karakuş · Karakuş 2026 · source · register

8. 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   S3 V3 C3 Lower bound confirmed on this case, superseded by T-110

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

9. 2026-10-01 published T-074

   Rectangle-density lower bounds replayed at 31 counts in $n=19\dots 95$$n=19\dots 95$

   S3 V3 C3 Lower bound confirmed on this case, superseded by T-110

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

10. 2026-10-05 published T-101

    Exact certificates of 77 catalogue packings: s(n) ≤ S', 1.2e-16 to 9.8e-15 above each side

    S2 V3 C3 Upper bound confirmed

    Daniel after Couzo, de Winter, Ellsworth, Levy · evand exact optima 2026-10-05 · packet · register

11. 2026-10-06 published T-110

    Mixed rectangle-measure lower bound verified at $n=39$$n=39$, on a declared net

    S3 V3 C3 Lower bound confirmed

    Hosono after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 mixed bounds check2 2026-10-06 · packet · source · review · register

12. 2026-10-12 established T-155

    Exact regularized sides at 59 counts, 5.4e-20 to 9.9e-15 below the ceilings they replace

    S1 V3 C3 Upper bound confirmed

    Levy after Daniel · register

13. 2026-10-13 published T-176 this result

    Mixed rectangle-measure lower bound $s(39)\ge 533/80$$s(39)\ge 533/80$ reported, on a 2,073-direction declared net

    S3 V0 C1 Lower bound reviewed

    Hosono after Tokoharu and Levy · wand125 fine-net n37 n39 n44 2026-10-13 · packet · register