T-256: Couzo's packings from Kaggle notebook packings at 232, 260 and 307, below T-186, T-225 and T-174

The cases

Results on these cases

22 results in the register on $n=232,260,307$$n=232,260,307$, oldest first, each with what it established and how it stands now.

1. 2005 published T-007 · $n=232,260,307$$n=232,260,307$

   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   S3 V0 C1 Lower bound incomplete on these cases, superseded by T-083 and T-180 (reported)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=232,260,307$$n=232,260,307$

   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 · $n=307$$n=307$

   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   S3 V3 C3 Upper bound confirmed superseded by T-256

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

4. 2026-09-29 published T-083 · $n=232,260,307$$n=232,260,307$

   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 · $n=307$$n=307$

   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   S2 V3 C3 Upper bound confirmed on these cases, superseded by T-256

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

6. 2026-10-05 published T-101 · $n=232,260$$n=232,260$

   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   S2 V3 C3 Upper bound confirmed on these cases, superseded by T-256 and T-257

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

7. 2026-10-09 published T-136 · $n=232,307$$n=232,307$

   Seventeen exact rational packings from SQUISH's third request

   S3 V3 C3 Upper bound confirmed superseded by T-256

   Chaoweeraprasit after Couzo, Xu, Göbel, Schadt, Ellsworth, Hajba, Levy · SQUISH third request 2026-10-09 · packet · register

8. 2026-10-12 established T-155 · $n=260$$n=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-257

   Levy after Daniel · register

9. 2026-10-12 published T-158 · $n=232$$n=232$

   New packings of 175, 232 and 264 squares built on a new 38-square rectangle, each certified exactly

   S3 V3 C3 Upper bound confirmed superseded by T-256

   Ostashev after Nakajima, Ellsworth, Schadt, Friedman, de Winter, Daniel, Levy, Xu · Ostashev records 2026-10-12 · packet · register

10. 2026-10-12 published T-163 · $n=307$$n=307$

    s(n² − n + 1) < n for every n ≥ 15, and n² − n + F(n) squares below side n, from Fang

    S4 V3 C1 Upper bound reviewed superseded by T-256

    Fang after Arslanov, Mustafin, Shangitbayev, Ellsworth, Stead, de Winter, Daniel, Couzo · Fang excess theorem 2026-10-13 · packet · packet · packet · source 1 · source 2 · register

11. 2026-10-12 published T-261 · $n=232,260$$n=232,260$

    Reported grade-C local minima for 233 source configurations of field degree at most 4

    S2 V3 C1 Restricted optimality incomplete

    Daniel after Schadt, Ellsworth, Levy · Daniel exact and local reports 2026-10-12 · packet · packet · register

12. 2026-10-13 published T-174 · $n=232,307$$n=232,307$

    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-256

    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 · $n=232,260,307$$n=232,260,307$

    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

14. 2026-10-14 published T-186 · $n=232$$n=232$

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

    S1 V3 C3 Upper bound confirmed on these cases, superseded by T-256

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

15. 2026-10-16 published T-193 · $n=232$$n=232$

    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-256

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

16. 2026-10-17 published T-196 · $n=260$$n=260$

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

    S3 V3 C3 Upper bound confirmed superseded by T-257

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

17. 2026-10-18 published T-223 · $n=260$$n=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-257

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

18. 2026-10-18 published T-236 · $n=307$$n=307$

    Daniel reports Lean proofs of Fang's Theorems 1 and 2, s(n2−n+1)<n$s(n^2-n+1)<n$ for $n\ge 15$$n\ge 15$

    S3 V0 C0 Formalization recorded

    Daniel after Fang, Arslanov, Mustafin, Shangitbayev, Levy · Daniel Fang and Ryu Lean 2026-10-18 · packet · source · register

19. 2026-10-20 published T-225 · $n=260$$n=260$

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

    S2 V3 C3 Upper bound confirmed on these cases, superseded by T-257

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

20. 2026-10-28 published T-255 · $n=260,307$$n=260,307$

    Couzo's certificates of ntumlnoob's Kaggle packings at 260 and 307, below T-225 and T-174

    S3 V3 C3 Upper bound confirmed superseded by T-256 and T-257

    Couzo after ntumlnoob, Daniel, Levy · Couzo a5a5d76 2026-10-28 · packet · packet · register

21. 2026-10-28 published T-256 this result · $n=232,260,307$$n=232,260,307$

    Couzo's packings from Kaggle notebook packings at 232, 260 and 307, below T-186, T-225 and T-174

    S3 V3 C3 Upper bound confirmed

    Couzo after Chilton, ntumlnoob, Lavrentev, Ellsworth, Daniel, Levy · Couzo issue 844 2026-10-28 · packet · register

22. 2026-10-29 published T-257 · $n=260$$n=260$

    Daniel's new packings of 172, 204, 260 and 293 squares, 1.64e-4 to 1.18e-2 below each house

    S3 V3 C3 Upper bound confirmed

    Daniel after Chaoweeraprasit, Gupta, Ostashev, kirilltym, Friedman · Daniel issue 845 2026-10-29 · packet · register