T-227: Couzo's refinements of de Winter's 227 and 229, below T-196 at both

The cases

Results on these cases

11 results in the register on $n=227,229$$n=227,229$, oldest first, each with what it established and how it stands now.

1. 2005 published T-007 · $n=227,229$$n=227,229$

   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   S3 V0 C1 Lower bound incomplete on these cases, superseded by T-083

   Nagamochi · Nagamochi 2005 · source · register

2. 2026-09-04 published T-085 · $n=227,229$$n=227,229$

   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 · $n=227,229$$n=227,229$

   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-101 · $n=227,229$$n=227,229$

   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   S2 V3 C3 Upper bound confirmed on these cases, superseded by T-225

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

5. 2026-10-12 established T-155 · $n=227,229$$n=227,229$

   Exact regularized sides at 59 counts, 5.4e-20 to 9.9e-15 below the ceilings they replace

   S1 V3 C3 Upper bound confirmed on these cases, superseded by T-225

   Levy after Daniel · register

6. 2026-10-13 published T-180 · $n=227,229$$n=227,229$

   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   S4 V0 C1 Lower bound reviewed on these cases, superseded by T-083

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

7. 2026-10-17 published T-196 · $n=227,229$$n=227,229$

   de Winter reports three smaller rational upper bounds at 227, 229 and 260

   S3 V3 C3 Upper bound confirmed superseded by T-225

   de Winter after Levy · de Winter upper bounds 2026-10-17 · packet · register

8. 2026-10-17 published T-198 · $n=229$$n=229$

   Daniel's opening n229 dilation certificate, after Couzo and de Winter

   S2 V3 C3 Upper bound confirmed superseded by T-225

   Daniel after Couzo, de Winter, Levy · Daniel n229 dilation 2026-10-17 · packet · register

9. 2026-10-17 published T-227 this result · $n=227,229$$n=227,229$

   Couzo's refinements of de Winter's 227 and 229, below T-196 at both

   S2 V0 C0 Upper bound recorded superseded by T-225

   Couzo after de Winter, Daniel, Levy · Couzo issue 672 2026-10-17 · packet · packet · register

10. 2026-10-18 published T-199 · $n=229$$n=229$

    Daniel's smaller n229 update, replacing the opening certificate

    S2 V3 C3 Upper bound confirmed superseded by T-225

    Daniel after Couzo, de Winter, Levy · Daniel n229 dilation update 2026-10-18 · packet · register

11. 2026-10-20 published T-225 · $n=227,229$$n=227,229$

    Daniel's solved, regularized views at 204, 227, 229 and 260, each the smallest side reported there

    S2 V3 C3 Upper bound confirmed

    Daniel after Couzo, de Winter, Polkovskyi, Levy · Daniel regularized lists 2026-10-20 · packet · register