T-071: Mixed rectangle-measure lower bounds replayed at $n=84\dots 87$$n=84\dots 87$

The cases

Results on these cases

24 results in the register on $n=84,85,86,87$$n=84,85,86,87$, oldest first, each with what it established and how it stands now.

1. 2005 published T-007 · $n=84,85,86,87$$n=84,85,86,87$

   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   S3 V0 C1 Lower bound incomplete on these cases, superseded by T-090, T-091, T-206 and T-207

   Nagamochi · Nagamochi 2005 · source · register

2. 2026-09-04 published T-085 · $n=84,85,86,87$$n=84,85,86,87$

   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. 2026-09-24 published T-089 · $n=87$$n=87$

   $s(83)\le 9.63475764863195$$s(83)\le 9.63475764863195$ and $s(87)\le 9.83881526994915$$s(87)\le 9.83881526994915$, Chang's packings, certified exactly

   S2 V3 C3 Upper bound confirmed superseded by T-101 and T-155

   Chang, Ellsworth after Cantrell, Hajba, Stenlund, Bidwell; Chang after Ellsworth, Cantrell, Schadt, Hajba, DeVincentis · Chang n83 2026-09-24 · Chang n87 2026-09-24 · packet · packet · register

4. 2026-09-27 published T-046 · $n=86$$n=86$

   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   S3 V0 C0 Lower bound recorded superseded by T-090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

5. 2026-09-28 published T-070 · $n=86$$n=86$

   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   S3 V3 C3 Lower bound confirmed superseded by T-090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

6. 2026-09-29 published T-058 · $n=84,85,86,87$$n=84,85,86,87$

   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 · $n=84,85,86,87$$n=84,85,86,87$

   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 these cases, superseded by T-090, T-091, T-206 and T-207

   Karakuş · Karakuş 2026 · source · register

8. 2026-10-01 published T-068 · $n=86,87$$n=86,87$

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

   S3 V3 C3 Lower bound confirmed on these cases, superseded by T-090 and T-091

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

9. 2026-10-02 published T-071 this result · $n=84,85,86,87$$n=84,85,86,87$

   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, T-091, T-206 and T-207

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

10. 2026-10-02 published T-075 · $n=85,86,87$$n=85,86,87$

    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 these cases, superseded by T-090, T-091 and T-207

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

11. 2026-10-03 published T-082 · $n=86,87$$n=86,87$

    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 these cases, superseded by T-090 and T-091

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

12. 2026-10-04 published T-090 · $n=84,86$$n=84,86$

    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

13. 2026-10-04 published T-091 · $n=87$$n=87$

    Mixed rectangle-measure lower bounds verified at 17 counts in $n=53\dots 95$$n=53\dots 95$

    S3 V3 C3 Lower bound confirmed

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

14. 2026-10-05 published T-094 · $n=84$$n=84$

    Mixed rectangle-measure lower bounds verified at $n=67$$n=67$ and $84$$84$

    S3 V3 C3 Lower bound confirmed on these cases, superseded by T-206

    Hosono after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 mixed bounds 2026-10-05 · packet · source · review · register

15. 2026-10-05 published T-101 · $n=87$$n=87$

    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

16. 2026-10-08 published T-125 · $n=84,86$$n=84,86$

    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

17. 2026-10-08 published T-130 · $n=84,86$$n=84,86$

    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

18. 2026-10-09 published T-138 · $n=84,86$$n=84,86$

    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

19. 2026-10-12 established T-155 · $n=87$$n=87$

    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

20. 2026-10-13 published T-174 · $n=84,86$$n=84,86$

    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

21. 2026-10-13 published T-180 · $n=84,85,86,87$$n=84,85,86,87$

    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 these cases, superseded by T-090, T-091, T-206 and T-207

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

22. 2026-10-16 published T-206 · $n=84$$n=84$

    Mixed rectangle-measure lower bound $s(84)\ge 3769/400$$s(84)\ge 3769/400$ verified, on the 201-direction net

    S3 V3 C3 Lower bound confirmed

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

23. 2026-10-16 published T-207 · $n=85$$n=85$

    Mixed rectangle-measure lower bound $s(85)\ge 379/40$$s(85)\ge 379/40$ verified, on the 201-direction net

    S3 V3 C3 Lower bound confirmed

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

24. 2026-10-16 published T-208 · $n=86$$n=86$

    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