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

The cases

Results on these cases

18 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   S3 V0 C1 lower bound incomplete on these cases, superseded by T-075, T-090, T-091 and T-094

   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; chelokot · 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   S2 V3 C3 upper bound confirmed superseded by T-101

   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   S3 V0 C0 lower bound recorded superseded by T-090

   wand125 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   S3 V3 C3 lower bound confirmed superseded by T-090

   wand125 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

   wand125 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-075, T-090, T-091 and T-094

   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

   wand125 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-075, T-090, T-091 and T-094

   wand125 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

    wand125 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

    wand125 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

    wand125 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

    wand125 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

    wand125 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

    ry-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 Francisco Couzo

    S3 V3 C3 upper bound confirmed

    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

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