T-119: Three exact feasible upper bounds at n=266,270,272 from new source arrangements

The cases

Results on these cases

19 results in the register on $n=266,270,272$$n=266,270,272$, oldest first, each with what it established and how it stands now.

1. 2005 published T-007 · $n=266,270,272$$n=266,270,272$

   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   S3 V0 C1 Lower bound incomplete on these cases, superseded by T-083 and T-180 (reported)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=266,270,272$$n=266,270,272$

   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=270,272$$n=270,272$

   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   S3 V3 C3 Upper bound confirmed superseded by T-174 and T-224

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

4. 2026-09-29 published T-083 · $n=266,270,272$$n=266,270,272$

   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. 2026-10-03 published T-092 · $n=272$$n=272$

   Smaller packings again at seven counts from $n=208$$n=208$ to $306$$306$, each certified two independent ways

   S3 V3 C3 Upper bound confirmed superseded by T-174

   Couzo · franciscouzo square-packing 2026-10-03 · packet · register

6. 2026-10-05 published T-098 · $n=270,272$$n=270,272$

   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   S2 V3 C3 Upper bound confirmed on these cases, superseded by T-174 and T-224

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

7. 2026-10-05 published T-101 · $n=266$$n=266$

   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   S2 V3 C3 Upper bound confirmed on these cases, superseded by T-119

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

8. 2026-10-07 published T-119 this result · $n=266,270,272$$n=266,270,272$

   Three exact feasible upper bounds at n=266,270,272 from new source arrangements

   S3 V3 C3 Upper bound confirmed

   Daniel after Levy, Ellsworth, Couzo, Stead · Daniel new arrangements 2026-10-07 · packet · register

9. 2026-10-08 published T-130 · $n=270$$n=270$

   Five follow-up rational refinements from Couzo

   S3 V3 C3 Upper bound confirmed superseded by T-224

   Couzo after Xu, Daniel, Ellsworth, Levy · Couzo follow-up refinements 2026-10-08 · packet · register

10. 2026-10-09 published T-134 · $n=270$$n=270$

    Six exact rational certificates from Couzo

    S3 V3 C3 Upper bound confirmed on these cases, superseded by T-224

    Couzo after Polkovskyi, Chaoweeraprasit, Gupta, Xu, Daniel, Ellsworth, Levy · Couzo exact certificates 2026-10-09 · packet · register

11. 2026-10-09 published T-136 · $n=270$$n=270$

    Seventeen exact rational packings from SQUISH's third request

    S3 V3 C3 Upper bound confirmed on these cases, superseded by T-224

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

12. 2026-10-09 published T-138 · $n=270$$n=270$

    Exact witnesses at Mishapolk's printed ceilings, the smallest known at 103 and 258 when registered

    S3 V3 C3 Upper bound confirmed superseded by T-224

    Polkovskyi after Stenlund, Friedman, Ellsworth, Chaoweeraprasit, Couzo, Xu, Levy · Mishapolk decimal poses 2026-10-09 · packet · register

13. 2026-10-10 published T-140 · $n=270$$n=270$

    Six more exact rational certificates from Couzo

    S3 V3 C3 Upper bound confirmed superseded by T-224

    Couzo after Chaoweeraprasit, Polkovskyi, Xu, Daniel, Ellsworth, Levy · Couzo certificates 2026-10-10 · packet · register

14. 2026-10-11 published T-151 · $n=270,272$$n=270,272$

    Nine exact rational packings from SQUISH's fourth request

    S3 V3 C3 Upper bound confirmed superseded by T-174 and T-224

    Chaoweeraprasit after Couzo, Gupta, Ellsworth, Friedman, Levy · SQUISH fourth request 2026-10-11 · packet · register

15. 2026-10-12 published T-171 · $n=272$$n=272$

    An exact contact refinement of SQUISH's packing of 272 squares

    S1 V3 C3 Upper bound confirmed superseded by T-174

    PingYou Ltd after Chaoweeraprasit, Daniel, Levy · PingYou n272 refinement 2026-10-12 · packet · register

16. 2026-10-13 published T-174 · $n=266,270,272$$n=266,270,272$

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

    S1 V3 C3 Upper bound confirmed

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

17. 2026-10-13 published T-180 · $n=266,270,272$$n=266,270,272$

    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

18. 2026-10-16 published T-193 · $n=266$$n=266$

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

    S2 V3 C3 Upper bound confirmed

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

19. 2026-10-20 published T-224 · $n=270$$n=270$

    Daniel's smaller n270 packing, another arrangement found from Chaoweeraprasit's, certified exactly

    S3 V3 C3 Upper bound confirmed

    Daniel after Chaoweeraprasit, Levy · Daniel n270 sweep 2026-10-20 · packet · register