T-220: Daniel's finite sub-grid upper bound for 183 squares, after Couzo's n182 packing

The case

Results on the case

6 results in the register on $n=183$$n=183$, 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   S3 V0 C1 Lower bound incomplete on this case, superseded by T-083 and T-180 (reported)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. 2026-10-07 published T-124

   Reported non-strict local minima for 178 source configurations

   S2 V0 C0 Restricted optimality recorded

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

5. 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

6. 2026-10-19 published T-220 this result

   Daniel's finite sub-grid upper bound for 183 squares, after Couzo's n182 packing

   S3 V3 C3 Upper bound confirmed

   Daniel after Couzo; register seed data from Levy · Daniel n183 subgrid 2026-10-19 · packet · register