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

The cases

Results on these cases

13 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

   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; chelokot · 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   S3 V3 C3 upper bound confirmed superseded by T-119

   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   S3 V3 C3 upper bound confirmed superseded by T-119

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   S2 V3 C3 upper bound confirmed on these cases, superseded by T-119

   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   S2 V3 C3 upper bound confirmed on these cases, superseded by T-119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 Francisco Couzo

   S3 V3 C3 upper bound confirmed

   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 Francisco Couzo

    S3 V3 C3 upper bound confirmed

    Couzo after Mishapolk, 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

    Chaoweeraprasit after Couzo, Xu, Goebel, 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

    Mishapolk 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 Francisco Couzo

    S3 V3 C3 upper bound confirmed

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