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

The cases

Results on these cases

19 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   S3 V0 C1 Lower bound incomplete on these cases, superseded by T-073, T-080, T-083 and T-180 (reported)

   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; Vlasenko · 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   S3 V3 C3 Upper bound confirmed superseded by T-146 and T-174

   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

   Hosono 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

   Hosono 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   S2 V3 C3 Upper bound confirmed superseded by T-146 and T-174

   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-146 and T-174

   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 superseded by T-174

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

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

    Eight complete rational refinements from Couzo

    S3 V3 C3 Upper bound confirmed superseded by T-174

    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 Couzo

    S3 V3 C3 Upper bound confirmed superseded by T-174

    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 on these cases, superseded by T-146

    Chaoweeraprasit after Couzo, Xu, Göbel, 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 superseded by T-174

    Polkovskyi 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 Daniel's regularized record lists

    S2 V3 C3 Upper bound confirmed

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

15. 2026-10-12 published T-170 · $n=105$$n=105$

    An exact contact refinement of Xu's packing of 105 squares, after Couzo's

    S1 V3 C3 Upper bound confirmed superseded by T-174

    PingYou Ltd after Xu, Couzo, Daniel, Levy · PingYou n105 refinement 2026-10-12 · packet · register

16. 2026-10-13 published T-174 · $n=105,292$$n=105,292$

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

    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=292$$n=292$

    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. 2026-10-22 published T-229 · $n=105$$n=105$

    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