T-043: $s (17) > 116511 / 25000 = 4.66044$

V3 C3 lower bound confirmed superseded by T-093

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

$s (17) > 116511 / 25000$ = 4.66044, by Guzhou0806 / N17 project's R068 release of 28 September 2026, continuing Kleddamag's public 4.66001 charge (T-041). It raised the verified bound by exactly 43/100000 over T-041 and leaves 0.0150901 to Bidwell's 4.67553009.

The 889 rule orbits and their weights are 4.66001's byte for byte; one zero-weight site orbit moves by 1/12500 and one weighted four-site point orbit is added, for a budget of 17000448944 units of 10^-9, and the strict cores are rebuilt over 4,991 parent-angle intervals with parents of side 115325/116511 in the container 4613/1000. Seventeen cores at the minimum core charge 1000026844 exceed the budget by 7404.

It was replayed here in full on 28 and 29 September 2026, the first local validation on record: the package says its publisher ran no local validation and did not observe its CI. Guzhou0806's own C++ checker and Kleddamag's Node BigInt checker ran over every interval, and devtools.audit_guzhou_r068 found the fresh and published ledgers identical row for row; the two checkers are one event-cell method.

Guzhou0806 / N17 project, n17-square-packing. The package credits Kleddamag for the charge, the proof and the Node BigInt checker (Kleddamag building on Squares Project (Joshua Levy), Mira and Guzhou0806), and discloses AI assistance.

Significance, composition and next rung

Significance: The case's verified lower bound, +0.00043 over T-041 from one added point orbit on an unchanged charge and a finer catalogue. A substantive case result at S3 by the T-015 and T-032 precedent; the movement is small, and this repository's replay is the first local validation of it on record.
Next rung: V4 and C4 need two adversarial AI reviews by distinct reviewers and a human oversight record, and none is retained; the mapped 2026-09-28 review is not recorded in this entry's reviews, and whether it counts is the owner's decision. The native parent-core route models only k-of-m threshold atoms and cannot read the weighted and winning-subset features, so extending it to them is the reachable second machine method. The review's code-independent exact sweep reproduced nine sampled rows; completing and retaining it would make the exact leg this repository's own but is the same method. Rung 5 needs a proof-assistant formalization reviewed by human experts. Superseded as the verified lower bound on 2026-10-05 by Guzhou0806's R071 (T-093), which keeps this charge; 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
$s (17) > 233009 / 50000 = 4.66018$
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 this result
$s (17) > 116511 / 25000 = 4.66044$
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-043: s(17)>116511/25000=4.66044s(17)>116511/25000=4.66044

The case

Results on the case

Links

T-043: $s (17) > 116511 / 25000 = 4.66044$