T-002: $s (18) \geq 4426213 / 1000000$ , by monotonicity from T-001

V3 C3 lower bound confirmed superseded by T-102

2026-08-31 established · Levy after Bentz · $n = 18$

$s (18) \geq 4426213 / 1000000$ , by monotonicity from T-001 (a packing of 18 unit squares contains a packing of 17).

Significance, composition and next rung

Significance: The same movement carried to $n = 18$ , above Nagamochi's 4.316625; the derivation adds nothing beyond T-001 but the case's frontier field moved for the first time since 2005.
Composition: Derived: the minimum over T-001 and the monotonicity step, which is one recorded line (any 18-packing contains a 17-packing) carried in the evidence limitations and the n-018 case body; both inputs sit at V3/C3, so the derived claim keeps the rungs.
Next rung: Rises with T-001; a bespoke $n = 18$ set stronger than the inherited bound would be a new result, not a rung change.
Novelty: apparently-novel Not found in the recorded search, subject to its stated gaps

The case

Case record

4.704.823

456

4.2435.243

\sqrt{n}

\sqrt{n} + 1

Proven

$4.70500$ $\leq s (18) \leq$ $4.822876$

exact

Citation record n-018

lowerwand125 after Tokoharu, Levy et al. 2026, GitHub (confirmed T-102)

upperHämäläinen 1980, Squares in Squares

Open

optimality

The case record

LowerUpper

Verified

\frac{941}{200}

\frac{7}{2} + \frac{1}{2} \sqrt{7}

Reported

\frac{941}{200}

\frac{7}{2} + \frac{1}{2} \sqrt{7}

Gap

\frac{\sqrt{7}}{2} - \frac{241}{200}

≈ 0.11787565…

Results on the case

17 results in the register on $n = 18$ , 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V0 C1 lower bound incomplete on this case, superseded by T-102
Nagamochi · Nagamochi 2005 · source · register
2026-08-21 published T-016
$s (n) \geq 22529 / 5000$ for $n = 18, 19$ , by monotonicity from T-015
V3 C3 lower bound confirmed superseded by T-102
Massaccesi after Burns · Burns–Massaccesi n17 · packet · register
2026-08-31 established T-002 this result
$s (18) \geq 4426213 / 1000000$ , by monotonicity from T-001
V3 C3 lower bound confirmed superseded by T-102
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-102
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
2026-09-18 established T-027
$s (18) \geq 467 / 100 = 4.67$
V3 C3 lower bound confirmed superseded by T-102
Levy after Burns, Massaccesi · register
2026-09-19 established T-028
$s (18) \geq 187 / 40 = 4.675$
V3 C3 lower bound confirmed superseded by T-102
Levy after Burns, Massaccesi · register
2026-09-19 established T-029
$s (18) \geq 1871 / 400 = 4.6775$
V3 C3 lower bound confirmed superseded by T-102
Levy after Burns, Massaccesi · register
2026-09-19 established T-030
$s (18) \geq 4679 / 1000 = 4.679$
V3 C3 lower bound confirmed superseded by T-102
Levy after Burns, Massaccesi · register
2026-09-27 published T-045
Rectangle-density lower bounds replayed at 15 counts in $n = 18 \dots 78$
V3 C3 lower bound confirmed superseded by T-102
wand125 after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 rectangle bounds 2026 · packet · packet · source · review · register
2026-09-27 published T-046
Rectangle-density lower bounds reported for 48 counts in $n = 18 \dots 95$
V0 C0 lower bound recorded superseded by T-102
wand125 after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 rectangle bounds 2026 · wand125 rectangle bounds 2026-09-28 · packet · packet · 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-102
Karakuş · Karakuş 2026 · source · register
2026-10-05 published T-096
Mixed rectangle-measure lower bound replayed at $n = 18$ , on a net the certificate declares
V3 C3 lower bound confirmed superseded by T-102
wand125 after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 mixed bounds finer net 2026-10-05 · packet · source · review · register
2026-10-06 published T-099
Mixed rectangle-measure lower bound verified at $n = 18$ , on a finer declared net
V3 C3 lower bound confirmed superseded by T-102
wand125 after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 mixed bounds finer net 2026-10-06 · packet · source 1 · source 2 · review · register
2026-10-06 published T-102
Mixed rectangle-measure lower bound verified at $n = 18$ , on the finest declared net yet
V3 C3 lower bound confirmed
wand125 after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 mixed bounds check2 2026-10-06 · packet · source · review · register

T-002: s(18)≥4426213/1000000s(18)≥4426213/1000000, by monotonicity from T-001

The case

Results on the case

Links

T-002: $s (18) \geq 4426213 / 1000000$ , by monotonicity from T-001