T-225: Daniel's solved, regularized views at 204, 227, 229 and 260, each the smallest side reported there

The cases

Results on these cases

19 results in the register on $n=204,227,229,260$$n=204,227,229,260$, oldest first, each with what it established and how it stands now.

1. 2005 published T-007 · $n=204,227,229,260$$n=204,227,229,260$

   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   S3 V0 C1 Lower bound incomplete on these cases, superseded by T-083 and T-180 (reported)

   Nagamochi · Nagamochi 2005 · source · register

2. 2026-09-04 published T-085 · $n=204,227,229,260$$n=204,227,229,260$

   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=204,227,229,260$$n=204,227,229,260$

   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=204,227,229,260$$n=204,227,229,260$

   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   S2 V3 C3 Upper bound confirmed on these cases, superseded by T-225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=204,227,229,260$$n=204,227,229,260$

   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   S1 V3 C3 Upper bound confirmed on these cases, superseded by T-225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. 2026-10-12 published T-162 · $n=204$$n=204$

   New packings of 129, 204 and 303 squares, a cut-out and two compositions, each certified exactly

   S3 V3 C3 Upper bound confirmed superseded by T-225

   Ostashev after Nakajima, Chaoweeraprasit, Daniel, Xu, Levy · Ostashev compositions 2026-10-12 · packet · register

7. 2026-10-13 published T-174 · $n=204$$n=204$

   Tighter certificates at 68 counts from PingYou Ltd, of others' packings and its own

   S1 V3 C3 Upper bound confirmed on these cases, superseded by T-225

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

8. 2026-10-13 published T-180 · $n=204,227,229,260$$n=204,227,229,260$

   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

9. 2026-10-14 published T-186 · $n=204$$n=204$

   Exact certificates at twelve counts from Polkovskyi, tightening the smallest reported there

   S1 V3 C3 Upper bound confirmed on these cases, superseded by T-225

   Polkovskyi after Ostashev, Couzo, PingYou Ltd, Levy · Mishapolk KKT refinements 2026-10-14 · packet · register

10. 2026-10-16 published T-193 · $n=204$$n=204$

    Exact algebraic sides at sixteen counts from Daniel, of packings the record credits

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

    Daniel after Xu, Chaoweeraprasit, Ostashev, Couzo, PingYou Ltd, Gupta, Polkovskyi, Levy · Daniel exact current records 2026-10-16 · packet · register

11. 2026-10-17 published T-196 · $n=227,229,260$$n=227,229,260$

    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

12. 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

13. 2026-10-17 published T-226 · $n=204$$n=204$

    Couzo's 204 of #670, below Polkovskyi's T-186, since superseded by his own #675 and #678

    S2 V0 C0 Upper bound recorded superseded by T-225

    Couzo after Polkovskyi, Daniel, Levy · Couzo issue 670 2026-10-17 · packet · packet · register

14. 2026-10-17 published T-227 · $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

15. 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

16. 2026-10-18 published T-223 · $n=204,260$$n=204,260$

    Couzo's smaller packings at 204 and 260, the two in the corpus of issue 678's 85 bounds

    S3 V3 C3 Upper bound confirmed superseded by T-225

    Couzo after de Winter, Polkovskyi, Daniel, Fang, PingYou Ltd, Levy · Couzo issue 678 2026-10-18 · packet · register

17. 2026-10-18 published T-228 · $n=204$$n=204$

    Couzo's 204 of #675, below his #670, since superseded by his #678

    S2 V0 C0 Upper bound recorded superseded by T-225

    Couzo after Polkovskyi, Daniel, Levy · Couzo issue 675 2026-10-18 · packet · packet · register

18. 2026-10-20 published T-225 this result · $n=204,227,229,260$$n=204,227,229,260$

    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

19. 2026-10-22 published T-229 · $n=204$$n=204$

    Polkovskyi's re-solves of the house packings at 105 and 204, below T-174 and T-225

    S2 V0 C0 Upper bound recorded

    Polkovskyi after Couzo, Muglich, Daniel, Xu, PingYou Ltd, Levy · Mishapolk issue 719 2026-10-22 · packet · register