T-139: A more precise certificate of Xu's 126-square arrangement, from Daniel

The case

Results on the case

12 results in the register on $n=126$$n=126$, 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   S3 V0 C1 Lower bound incomplete on this case, superseded by T-083 and T-180 (reported)

   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. 2026-09-29 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

   Karakuş · Karakuş 2026 · source · register

4. 2026-10-05 published T-098

   Exact optima of 48 known-best packings: s(n) ≤ S', 3.5e-13 to 5.0e-11 below each printed side

   S2 V3 C3 Upper bound confirmed on this case, superseded by T-139

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

5. 2026-10-07 published T-113

   s(n)≤Sn​$s(n)\le S_n$ at ten counts, $n=108$$n=108$ to 303, from new SQUISH rational packings

   S3 V3 C3 Upper bound confirmed superseded by T-139

   Chaoweeraprasit after Ellsworth · SQUISH ten packings 2026-10-07 · packet · register

6. 2026-10-07 published T-115

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

   S3 V3 C3 Upper bound confirmed superseded by T-139

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

7. 2026-10-08 published T-125

   Complete rational construction reports at 25 counts

   S3 V3 C3 Upper bound confirmed superseded by T-139

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

8. 2026-10-09 published T-139 this result

   A more precise certificate of Xu's 126-square arrangement, from Daniel

   S2 V3 C3 Upper bound confirmed

   Daniel after Xu, Chaoweeraprasit · Daniel trio126 2026-10-09 · packet · register

9. 2026-10-10 published T-144

   Linear-measure lower bound $s(n)\ge 563/50=11.26$$s(n)\ge 563/50=11.26$ for $n=122\dots 126$$n=122\dots 126$, reported

   S3 V0 C0 Lower bound recorded on this case, superseded by T-180 (reported)

   Hosono after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 linear n122 2026-10-10 · packet · register

10. 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-139

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

11. 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

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

12. 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