T-197: Daniel's smaller n155 packing, started from Ostashev's packing

The case

Results on the case

12 results in the register on $n=155$$n=155$, 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   S3 V0 C1 Lower bound incomplete on this case, superseded by T-083 and T-180 (reported)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. 2026-09-27 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   S3 V3 C3 Upper bound confirmed superseded by T-197

   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

   Karakuş · Karakuş 2026 · source · register

5. 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   S2 V3 C3 Upper bound confirmed on this case, superseded by T-197

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

6. 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   S3 V3 C3 Upper bound confirmed superseded by T-197

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

7. 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   S3 V3 C3 Upper bound confirmed superseded by T-197

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

8. 2026-10-08 published T-128

   Eight complete rational refinements from Couzo

   S3 V3 C3 Upper bound confirmed superseded by T-197

   Couzo after Xu, Chaoweeraprasit, Gupta, Ellsworth, Daniel, Levy · Couzo exact refinements 2026-10-08 · packet · register

9. 2026-10-11 published T-149

   New packings of 155, 156 and 211 squares, each with an exact rational certificate

   S3 V3 C3 Upper bound confirmed superseded by T-197

   Ostashev after Nakajima, Levy, Xu · Ostashev records 2026-10-11 · 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-197

    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. 2026-10-17 published T-197 this result

    Daniel's smaller n155 packing, started from Ostashev's packing

    S3 V3 C3 Upper bound confirmed

    Daniel after Ostashev, Levy · Daniel n155 explorer 2026-10-17 · packet · register