T-141: Exact rational certificates at 132 and 308 from Daniel's third record hunt

The cases

Results on these cases

15 results in the register on $n=132,308$$n=132,308$, oldest first, each with what it established and how it stands now.

1. 2005 published T-007 · $n=132,308$$n=132,308$

   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   S3 V0 C1 Lower bound incomplete on these cases, superseded by T-083 and T-180 (reported)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=132,308$$n=132,308$

   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=132$$n=132$

   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   S3 V3 C3 Upper bound confirmed superseded by T-174

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

4. 2026-09-29 published T-083 · $n=132,308$$n=132,308$

   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=132$$n=132$

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   S2 V3 C3 Upper bound confirmed on these cases, superseded by T-174

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

6. 2026-10-07 published T-124 · $n=308$$n=308$

   Reported non-strict local minima for 178 source configurations

   S2 V0 C0 Restricted optimality recorded

   Daniel after Couzo · Daniel exact and local reports 2026 · packet · register

7. 2026-10-09 published T-131 · $n=132$$n=132$

   $s(132)\le 11.987099332245063\dots$$s(132)\le 11.987099332245063\dots$ from Daniel's record hunt

   S3 V3 C3 Upper bound confirmed superseded by T-174

   Daniel after Couzo, Chaoweeraprasit, Levy · Daniel record hunt 2026-10-09 · packet · register

8. 2026-10-09 published T-134 · $n=132$$n=132$

   Six exact rational certificates from Couzo

   S3 V3 C3 Upper bound confirmed on these cases, superseded by T-174

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

9. 2026-10-09 published T-138 · $n=132$$n=132$

   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

10. 2026-10-10 published T-135 · $n=308$$n=308$

    A packing of 308 squares below the grid, from Fang

    S3 V3 C3 Upper bound confirmed superseded by T-221

    Fang after Arslanov, Mustafin, Shangitbayev, Ellsworth, Stead · Fang two-wedge certificates 2026-10-10 · packet · register

11. 2026-10-10 published T-140 · $n=132$$n=132$

    Six more exact rational certificates from Couzo

    S3 V3 C3 Upper bound confirmed superseded by T-174

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

12. 2026-10-10 published T-141 this result · $n=132,308$$n=132,308$

    Exact rational certificates at 132 and 308 from Daniel's third record hunt

    S3 V3 C3 Upper bound confirmed superseded by T-174 and T-221

    Daniel after Fang, Arslanov, Mustafin, Shangitbayev, Couzo, Levy · Daniel record hunt 3 2026-10-10 · packet · register

13. 2026-10-13 published T-174 · $n=132,308$$n=132,308$

    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

14. 2026-10-13 published T-180 · $n=132,308$$n=132,308$

    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

15. 2026-10-19 published T-221 · $n=308$$n=308$

    Daniel's smaller n308 packing, started from Chaoweeraprasit's 307 packing, certified exactly

    S3 V3 C3 Upper bound confirmed

    Daniel after Chaoweeraprasit, Levy · Daniel n308 2026-10-19 · packet · register