T-209: Mixed rectangle-measure lower bound $s(88)\ge 3853/400$$s(88)\ge 3853/400$ reported, on the 201-direction net

The case

Results on the case

19 results in the register on $n=88$$n=88$, 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   S3 V0 C1 Lower bound incomplete on this case, superseded by T-091

   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   S3 V0 C0 Lower bound recorded superseded by T-091

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

4. 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

5. 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   S3 V3 C3 Lower bound confirmed on this case, superseded by T-091

   Karakuş · Karakuş 2026 · source · register

6. 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   S3 V3 C3 Lower bound confirmed on this case, superseded by T-091

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

7. 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   S3 V3 C3 Lower bound confirmed on this case, superseded by T-091

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

8. 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-091

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

9. 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-091

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

10. 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 on this case, superseded by T-091

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

11. 2026-10-04 published T-091

    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

12. 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 on this case, superseded by T-127

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

13. 2026-10-07 published T-116

    s(n)≤Sn​$s(n)\le S_n$ at nine counts, from the second SQUISH update’s new rational packings

    S3 V3 C3 Upper bound confirmed superseded by T-127

    Chaoweeraprasit after Ellsworth, Couzo and SQUISH · SQUISH second update 2026-10-07 · packet · register

14. 2026-10-08 published T-125

    Complete rational construction reports at 25 counts

    S3 V3 C3 Upper bound confirmed superseded by T-127

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

15. 2026-10-08 published T-127

    Fourteen rational refinements, seventeen complete source cases

    S3 V3 C3 Upper bound confirmed

    Gupta after Chaoweeraprasit, Daniel · Gupta rational refinements 2026-10-08 · packet · register

16. 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 on this case, superseded by T-127

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

17. 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-091

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

18. 2026-10-16 published T-193

    Exact algebraic sides at sixteen counts from Daniel, of packings the record credits

    S2 V3 C3 Upper bound confirmed

    Daniel after Xu, Chaoweeraprasit, Ostashev, Couzo, PingYou Ltd, Gupta, Polkovskyi, Levy · Daniel exact current records 2026-10-16 · packet · register

19. 2026-10-16 published T-209 this result

    Mixed rectangle-measure lower bound $s(88)\ge 3853/400$$s(88)\ge 3853/400$ 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