T-001: $s (17) \geq 4426213 / 1000000 = 4.426213$ , from a sixteen-point unavoidable set

V3 C3 lower bound confirmed superseded by T-093

2026-08-31 established · Levy after Bentz · $n = 17$

Sixteen points make [0, 4426213/1000000]^2 unavoidable for open squares of side above one, so $s (17) \geq 4426213 / 1000000$ = 4.426213.

Significance, composition and next rung

Significance: The first verified lower-bound movement in the register since Nagamochi 2005 at these n, and the strongest theorem this repository has proved itself to date; a two-case improvement in a specialist literature, 0.019 below Green's reported but sourceless value.
Next rung: V4 and C4 need two adversarial AI reviews by distinct reviewers and a human oversight record, and none is retained; the packet to review is the statement, hypotheses, proof narrative, code map and replays. Rung 5 needs a proof-assistant formalization reviewed by human experts. The stronger side at the exact ceiling is T-003's business.
Novelty: apparently-novel Not found in the recorded search, subject to its stated gaps

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V0 C1 lower bound incomplete on this case, superseded by T-093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V3 C3 lower bound confirmed superseded by T-093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 this result
$s (17) \geq 4426213 / 1000000 = 4.426213$ , 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V3 C3 lower bound confirmed superseded by T-093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
$s (17) > 116511 / 25000 = 4.66044$
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-001: s(17)≥4426213/1000000=4.426213s(17)≥4426213/1000000=4.426213, from a sixteen-point unavoidable set

The case

Results on the case

Links

T-001: $s (17) \geq 4426213 / 1000000 = 4.426213$ , from a sixteen-point unavoidable set