T-120: Reported s(102) ≤ alpha_102 with a degree-8 exact side form

The case

Results on the case

14 results in the register on $n=102$$n=102$, 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   S3 V0 C1 Lower bound incomplete on this case, superseded by T-073 and T-080

   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-056

   Smaller packings for 49 counts from $n=68$$n=68$ to $307$$307$, each certified two independent ways

   S3 V3 C3 Upper bound confirmed superseded by T-146

   Couzo · franciscouzo square-packing 2026-09-27 · packet · register

4. 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   S3 V3 C3 Lower bound confirmed on this case, superseded by T-073 and T-080

   Karakuş · Karakuş 2026 · source · register

5. 2026-10-02 published T-073

   Linear-measure lower bounds replayed at $n=83$$n=83$ and $n=101\dots 105$$n=101\dots 105$

   S3 V3 C3 Lower bound confirmed

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

6. 2026-10-02 published T-080

   Linear-measure lower bound replayed at $n=101\dots 105$$n=101\dots 105$

   S3 V3 C3 Lower bound confirmed

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

7. 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   S2 V3 C3 Upper bound confirmed on this case, superseded by T-146

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

8. 2026-10-07 published T-120 this result

   Reported s(102) ≤ alpha_102 with a degree-8 exact side form

   S2 V0 C0 Upper bound recorded superseded by T-146

   Daniel after Couzo · Daniel exact and local reports 2026 · packet · register

9. 2026-10-08 published T-125

   Complete rational construction reports at 25 counts

   S3 V3 C3 Upper bound confirmed superseded by T-146

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

10. 2026-10-10 published T-146

    Exact rational certificates at fifteen counts from Daniel's regularized record lists

    S2 V3 C3 Upper bound confirmed

    Daniel after Xu, Chaoweeraprasit, Polkovskyi, Levy · Daniel regularized lists 2026-10-10 · packet · register

11. 2026-10-11 published T-159

    Another arrangement of Xu's design for 102 squares, certified exactly from its decimal layout

    S2 V3 C3 Upper bound confirmed superseded by T-146

    Calle after Xu, Levy · Calle n102 2026-10-12 · packet · register

12. 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-146

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

13. 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-073 and T-080

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

14. 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