T-117: Exact rational ceiling refinements at $n=105$$n=105$, 292

The cases

Results on these cases

14 results in the register on $n=105,292$$n=105,292$, oldest first, each with what it established and how it stands now.

1. 2005 published T-007 · $n=105,292$$n=105,292$

   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   S3 V0 C1 lower bound incomplete on these cases, superseded by T-073, T-080 and T-083

   Nagamochi · Nagamochi 2005 · source · register

2. 2026-09-04 published T-085 · $n=105,292$$n=105,292$

   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=105,292$$n=105,292$

   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   S3 V3 C3 upper bound confirmed superseded by T-117 and T-125

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

4. 2026-09-29 published T-083 · $n=105,292$$n=105,292$

   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. 2026-10-02 published T-073 · $n=105$$n=105$

   Linear-measure lower bounds replayed at $n=83$$n=83$ and $n=101\dots 105$$n=101\dots 105$

   S3 V3 C3 lower bound confirmed

   wand125 after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 linear certificates 2026-10-02 · packet · source · review · register

6. 2026-10-02 published T-080 · $n=105$$n=105$

   Linear-measure lower bound replayed at $n=101\dots 105$$n=101\dots 105$

   S3 V3 C3 lower bound confirmed

   wand125 after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 linear certificates 2026-10-02 · packet · source · review · register

7. 2026-10-07 published T-117 this result · $n=105,292$$n=105,292$

   Exact rational ceiling refinements at $n=105$$n=105$, 292

   S2 V3 C3 upper bound confirmed

   Rehwaldt after Couzo and earlier contributors · Rehwaldt Couzo refinements 2026-10-07 · packet · register

8. 2026-10-07 published T-129 · $n=105,292$$n=105,292$

   Three dated Daniel certificate reports with historical source custody

   S1 V0 C0 upper bound recorded superseded by T-117 and T-125

   Daniel after Couzo, Levy · Daniel dated certificates 105 and 130 2026-10-07 · Daniel dated certificate 292 2026-10-07 · packet · packet · packet · register

9. 2026-10-08 published T-125 · $n=105$$n=105$

   Complete rational construction reports at 25 counts

   S3 V3 C3 upper bound confirmed

   ry-xu · ry-xu square packing 2026 · packet · register

10. 2026-10-08 published T-128 · $n=105$$n=105$

    Eight complete rational refinements from Francisco Couzo

    S3 V3 C3 upper bound confirmed

    Couzo after Xu, Chaoweeraprasit, Gupta, Ellsworth, Daniel, Levy · Couzo exact refinements 2026-10-08 · packet · register

11. 2026-10-08 published T-130 · $n=105$$n=105$

    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

12. 2026-10-09 published T-136 · $n=292$$n=292$

    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

13. 2026-10-09 published T-138 · $n=105$$n=105$

    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

14. 2026-10-10 published T-146 · $n=292$$n=292$

    Exact rational certificates at fifteen counts from Evan Daniel's regularized record lists

    S2 V3 C3 upper bound confirmed

    Daniel after Xu, Chaoweeraprasit, Mishapolk, Levy · Daniel regularized lists 2026-10-10 · packet · register