T-065: $s(17)\le 4.6755300936045509\dots$$s(17)\le 4.6755300936045509\dots$, Bidwell's packing certified exactly

The case

Results on the case

19 results in the register on $n=17$$n=17$, oldest first, each with what it established and how it stands now.

1. 2005 published T-007

   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   S3 V0 C1 Lower bound incomplete on this case, superseded by T-093 and T-153 (reported)

   Nagamochi · Nagamochi 2005 · source · register

2. 2026-08-21 published T-015

   $s(17)\ge 22529/5000=4.5058$$s(17)\ge 22529/5000=4.5058$

   S3 V3 C3 Lower bound confirmed superseded by T-093 and T-153 (reported)

   Massaccesi after Burns · Burns–Massaccesi n17 · packet · register

3. 2026-08-31 established T-001

   $s(17)\ge 4426213/1000000=4.426213$$s(17)\ge 4426213/1000000=4.426213$, from a sixteen-point unavoidable set

   S3 V3 C3 Lower bound confirmed superseded by T-093 and T-153 (reported)

   Levy after Bentz · register

4. 2026-08-31 established T-003

   The sixteen-point set's unavoidability ceiling lies in $[4426213/1000000,4427/1000)$$[4426213/1000000,4427/1000)$

   S2 V3 C3 Method limit confirmed

   Levy after Bentz · register

5. 2026-09-04 established T-019

   $s(n)\ge 459/100=4.59$$s(n)\ge 459/100=4.59$ for $n=17,18,19$$n=17,18,19$

   S4 V3 C3 Lower bound confirmed superseded by T-093 and T-153 (reported)

   Levy after Burns, Massaccesi · register

6. 2026-09-04 published T-085

   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

7. 2026-09-20 published T-032

   $s(17)\ge 461300/99999=4.61304613\dots$$s(17)\ge 461300/99999=4.61304613\dots$, and beneath it Mira's $s(17)\ge 4613/1000$$s(17)\ge 4613/1000$

   S3 V3 C3 Lower bound confirmed superseded by T-093 and T-153 (reported)

   Guzhou0806, Mira-acc after Levy, Burns, Massaccesi · n17 weighted certificates 2026-09-20 · packet · register

8. 2026-09-21 published T-038

   $s(17)>461300/99853=4.6197910929\dots$$s(17)>461300/99853=4.6197910929\dots$

   S3 V3 C3 Lower bound confirmed superseded by T-093 and T-153 (reported)

   Overbeck after Levy, Mira-acc, Guzhou0806 · Kleddamag n17 certified bound · packet · source · review · register

9. 2026-09-21 published T-065 this result

   $s(17)\le 4.6755300936045509\dots$$s(17)\le 4.6755300936045509\dots$, Bidwell's packing certified exactly

   S2 V3 C3 Upper bound confirmed

   Overbeck after Levy, Mira-acc, Guzhou0806 · Kleddamag n17 certified bound · packet · source · review · register

10. 2026-09-25 published T-039

    $s(17)>231001/50000=4.62002$$s(17)>231001/50000=4.62002$

    S3 V3 C3 Lower bound confirmed superseded by T-093 and T-153 (reported)

    Guzhou0806 after Overbeck, Mira-acc, Levy · Guzhou0806 n17 R052 · packet · source · review · register

11. 2026-09-26 published T-040

    $s(17)>232001/50000=4.64002$$s(17)>232001/50000=4.64002$

    S3 V3 C3 Lower bound confirmed superseded by T-093 and T-153 (reported)

    Overbeck after Levy, Mira-acc, Guzhou0806 · Kleddamag n17 4.640020 · packet · source · review · register

12. 2026-09-27 published T-041

    $s(17)>466001/100000=4.66001$$s(17)>466001/100000=4.66001$

    S3 V3 C3 Lower bound confirmed superseded by T-093 and T-153 (reported)

    Overbeck after Levy, Mira-acc, Guzhou0806 · Kleddamag n17 4.66001 · packet · source · review · register

13. 2026-09-28 published T-042

    $s(17)>233009/50000=4.66018$$s(17)>233009/50000=4.66018$

    S2 V3 C3 Lower bound confirmed superseded by T-093 and T-153 (reported)

    Guzhou0806 after Overbeck, Mira-acc, Levy · Guzhou0806 n17 R067 · packet · source · review · register

14. 2026-09-28 published T-043

    $s(17)>116511/25000=4.66044$$s(17)>116511/25000=4.66044$

    S3 V3 C3 Lower bound confirmed superseded by T-093 and T-153 (reported)

    Guzhou0806 after Overbeck, Mira-acc, Levy · Guzhou0806 n17 R068 · packet · source · review · register

15. 2026-09-29 published T-058

    Rectangle-certificate ceiling α·UB(n) proved for $n=1$$n=1$..100; B·UB(n) on 64 grid rows

    S2 V3 C3 Method limit confirmed

    Hosono after Tokoharu, Daniel · wand125 tools 2026 · packet · register

16. 2026-09-29 published T-083

    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 on this case, superseded by T-093 and T-153 (reported)

    Karakuş · Karakuş 2026 · source · register

17. 2026-09-30 published T-093

    $s(17)>18641771/4000000=4.66044275$$s(17)>18641771/4000000=4.66044275$

    S2 V3 C3 Lower bound confirmed

    Guzhou0806 after Overbeck, Mira-acc, Levy · Guzhou0806 n17 R071 · packet · source · review · register

18. 2026-10-12 published T-153

    Candidate proof that Bidwell's 1998 packing is optimal, $s(17)=4.67553009360455\dots$$s(17)=4.67553009360455\dots$

    S5 V0 C1 Optimality reviewed

    Overbeck after Levy, Ahmed, Mira-acc, Guzhou0806, Maaz · Overbeck n17 candidate 2026 · packet · register

19. 2026-10-12 published T-154

    Fitzharding reports a Lean proof that $s(17)=4.67553009360455\dots$$s(17)=4.67553009360455\dots$

    S5 V3 C1 Formalization reviewed

    Fitzharding after Levy, Ahmed, MacIver · Fitzharding n17 Lean 2026 · packet · source · source build receipt · review 1 · review 2 · register