T-042: $s (17) > 233009 / 50000 = 4.66018$

V3 C3 lower bound confirmed superseded by T-093

2026-09-28 published · Guzhou0806 after Kleddamag, Mira, Levy · $n = 17$

$s (17) > 233009 / 50000$ = 4.66018, by Guzhou0806 / N17 project's R067 release of 28 September 2026. R068 (T-043) superseded it the same day, so it never held the case's verified field.

The certificate is Kleddamag's 4.66001 charge (T-041) unchanged, budget 17000402008 units of 10^-9, at the larger parent side 32950/33287, with strict cores rebuilt over 2,808 parent-angle intervals whose endpoints contain every 4.66001 endpoint. Seventeen cores at the minimum core charge 1000026844 exceed the budget by 54340.

It was replayed here in full on 29 September 2026 by the source's paired launcher, Guzhou0806's C++ checker with Kleddamag's Node BigInt checker, and devtools.audit_guzhou_r068 found the fresh and published ledgers identical on all 2,808 rows; the two checkers are one event-cell method.

Guzhou0806 / N17 project, n17-square-packing, on Kleddamag's charge. The release names itself as made "with AI assistance".

Significance, composition and next rung

Significance: True and replayed in full, but superseded on its publication day by R068 before it held any field: a citable intermediate, +0.00017 over T-041, that changes no standing bound. Registered because it was replayed and reviewed here.
Next rung: No rung change is sought: the result never held a field, and a method-distinct decision would be spent better on R068, which runs the same charge further. Superseded on its publication day, 2026-09-28, by Guzhou0806's R068 (T-043), before it held the verified field; stays true as stated.
Novelty: previously-published Present in an identified source

The case

Case record

4.664.676

456

4.1235.123

\sqrt{n}

\sqrt{n} + 1

Proven

$4.66044$ $\leq s (17) \leq$ $4.675531$

exact

Citation record n-017

lowerGuzhou0806 after Kleddamag et al. 2026, GitHub (confirmed T-093)

upperBidwell 1998, Squares in Squares (confirmed T-065)

Open

optimality

The case record

LowerUpper

Verified

\frac{18641771}{4000000}

4.67553009…

Reported

\frac{18641771}{4000000}

4.67553009…

Gap0.01508734…

Results on the case

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

2005 published T-007
$s (n) \geq min (⌈ \sqrt{n} ⌉, \sqrt{n - 2 ⌊ \sqrt{n} ⌋ + 1} + 1)$ for $4 \leq n \leq 324$
V0 C1 lower bound incomplete on this case, superseded by T-093
Nagamochi · Nagamochi 2005 · source · register
2026-08-21 published T-015
$s (17) \geq 22529 / 5000 = 4.5058$
V3 C3 lower bound confirmed superseded by T-093
Massaccesi after Burns · Burns–Massaccesi n17 · packet · register
2026-08-31 established T-001
$s (17) \geq 4426213 / 1000000 = 4.426213$ , from a sixteen-point unavoidable set
V3 C3 lower bound confirmed superseded by T-093
Levy after Bentz · register
2026-08-31 established T-003
The sixteen-point set's unavoidability ceiling lies in $[4426213 / 1000000, 4427 / 1000)$
V3 C3 method limit confirmed
Levy after Bentz · register
2026-09-04 established T-019
$s (n) \geq 459 / 100 = 4.59$ for $n = 17, 18, 19$
V3 C3 lower bound confirmed superseded by T-093
Levy after Burns, Massaccesi · register
2026-09-04 published T-085
Nagamochi 2005, Lemma 1 is false for every container with $a > 3$ and $b > 2$
V3 C3 correction confirmed
Karakuş; chelokot · Karakuş 2026 · chelokot Nagamochi counterexample 2026 · packet · register
2026-09-20 published T-032
$s (17) \geq 461300 / 99999 = 4.61304613 \dots$ , and beneath it Mira's $s (17) \geq 4613 / 1000$
V3 C3 lower bound confirmed superseded by T-093
Guzhou0806, Mira after Levy, Burns, Massaccesi · n17 weighted certificates 2026-09-20 · packet · register
2026-09-21 published T-038
$s (17) > 461300 / 99853 = 4.6197910929 \dots$
V3 C3 lower bound confirmed superseded by T-093
Kleddamag after Levy, Mira, Guzhou0806 · Kleddamag n17 certified bound · packet · source · review · register
2026-09-21 published T-065
$s (17) \leq 4.6755300936045509 \dots$ , Bidwell's packing certified exactly
V3 C3 upper bound confirmed
Kleddamag after Levy, Mira, Guzhou0806 · Kleddamag n17 certified bound · packet · source · review · register
2026-09-25 published T-039
$s (17) > 231001 / 50000 = 4.62002$
V3 C3 lower bound confirmed superseded by T-093
Guzhou0806 after Kleddamag, Mira, Levy · Guzhou0806 n17 R052 · packet · source · review · register
2026-09-26 published T-040
$s (17) > 232001 / 50000 = 4.64002$
V3 C3 lower bound confirmed superseded by T-093
Kleddamag after Levy, Mira, Guzhou0806 · Kleddamag n17 4.640020 · packet · source · review · register
2026-09-27 published T-041
$s (17) > 466001 / 100000 = 4.66001$
V3 C3 lower bound confirmed superseded by T-093
Kleddamag after Levy, Mira, Guzhou0806 · Kleddamag n17 4.66001 · packet · source · review · register
2026-09-28 published T-042 this result
$s (17) > 233009 / 50000 = 4.66018$
V3 C3 lower bound confirmed superseded by T-093
Guzhou0806 after Kleddamag, Mira, Levy · Guzhou0806 n17 R067 · packet · source · review · register
2026-09-28 published T-043
$s (17) > 116511 / 25000 = 4.66044$
V3 C3 lower bound confirmed superseded by T-093
Guzhou0806 after Kleddamag, Mira, Levy · Guzhou0806 n17 R068 · packet · source · review · register
2026-09-29 published T-058
Rectangle-certificate ceiling α·UB(n) proved for $n = 1$ ..100; B·UB(n) on 64 grid rows
V3 C3 method limit confirmed
wand125 after Tokoharu, Daniel · wand125 tools 2026 · packet · register
2026-09-29 published T-083
$s (n) \geq 1 / 2 + \sqrt{n - ⌊ \sqrt{n} ⌋ + 1 / 4}$ for every nonsquare $8 \leq n \leq 324$
V3 C3 lower bound confirmed on this case, superseded by T-093
Karakuş · Karakuş 2026 · source · register
2026-09-30 published T-093
$s (17) > 18641771 / 4000000 = 4.66044275$
V3 C3 lower bound confirmed
Guzhou0806 after Kleddamag, Mira, Levy · Guzhou0806 n17 R071 · packet · source · review · register

T-042: s(17)>233009/50000=4.66018s(17)>233009/50000=4.66018

The case

Results on the case

Links

T-042: $s (17) > 233009 / 50000 = 4.66018$